Research ArticleNatural Frequencies and Mode Shapes of MechanicallyCoupled Microbeam Resonators with an Application toMicromechanical Filters

Bashar K Hammad

Department of Mechatronics Engineering The Hashemite University PO Box 150459 Zarqa 13115 Jordan

Correspondence should be addressed to Bashar K Hammad bkhammadvtedu

Received 8 February 2013 Accepted 26 July 2013 Published 12 February 2014

Academic Editor Jeong-Hoi Koo

Copyright copy 2014 Bashar K HammadThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present a methodology to calculate analytically the mode shapes and corresponding frequencies of mechanically coupledmicrobeam resonators To demonstrate the methodology we analyze a mechanical filter composed of two beams coupled by aweak beam The boundary-value problem (BVP) for the linear vibration problem of the coupled beams depends on the numberof beams and the boundary conditions of the attachment points This implies that the system of linear homogeneous algebraicequations becomes larger as the array of resonators becomes complicated We suggest a method to reduce the large system ofequations into a smaller system We reduce the BVP composed of five equations and twenty boundary conditions to a set of threelinear homogeneous algebraic equations for three constants and the frequencies This methodology can be simply extended toaccommodate any configuration of mechanically coupled arrays To validate our methodology we compare our analytical resultsto these obtained numerically using ANSYS We found that the agreement is excellent We note that the weak coupling beam splitsthe frequency of the single resonator into two close frequencies In addition the effect of the coupling beam location on the naturalfrequencies and hence the filter behavior is investigated

1 Introduction

Mechanically coupled microbeam resonators have attractedattention recently in the microscale realm especially inRF MEMS [1ndash4] Mode shapes and frequencies of theseresonators are commonly approximated numerically usingfinite-element packages such as ANSYS COMSOL andCoventor [5 6] To the best of our knowledge there have beenfew attempts to analytically generate the natural frequenciesand mode shapes of multiresonator micromechanical struc-tures characterized by distributed-parameter systems

In this work we present an analytical methodology tofind mode shapes and the corresponding natural frequenciesthat can be applied to any system of coupled resonators Thismethodology provides closed-form expressions for modeshapes that are easier to handle more robust and accuratein further analysis of coupled-resonator systems especially indeveloping reduced-ordermodels that describe the nonlinearstatic and dynamic characteristics of microstructures Inaddition these expressions allow designers to obtain a deeper

insight into the relationship among performance metrics andthe underlying microstructure dimensions boundary condi-tions andmaterial propertiesThe advantages of closed-formexpressions of mode shapes are unattainable using finite-element packages which are constrained by convergencedrawbacks and usually are computationally expensive

Vyas and Bajaj [7] and Vyas et al [8] proposed micro-structures that depend on nonlinear modal interactionbetween microresonators They employed linear analysis toobtain analytically the natural frequencies and associatedmode shapes and defined the parameters needed to assure 1 2internal resonance inT-beam [7] and pedal-typemicrostruc-tures [8]

To discuss the proposed methodology in this paperand without loss of generality we present closed-formexpressions for the natural frequencies and mode shapesof micromechanical filters made of two clamped-clampedbeam resonators connected via a coupling beam The prob-lem formulation treats the filters as distributed-parametersystems In previous work [6 10] we used the Galerkin

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 939467 15 pageshttpdxdoiorg1011552014939467

2 Shock and Vibration

Inputresonator Output

resonatorCoupling beam

Clamped ends

Clamped ends

(a)

L

Lix

x

Lc

x Lo

L

(b)

Nx

w(x t)N

(c)

Figure 1 (a) A schematic drawing and (b) a schematic model of a filter made of two clamped-clamped microbeam resonators coupled by aweak beam and (c) a schematic diagram for one of the primary resonators

procedure to develop a reduced-ordermodel for the filter andutilized basis functions computed using the finite-elementpackage ANSYS

The remainder of this work is organized as follows wederive the governing equations of the linear vibration prob-lem in Section 21 and the associated boundary conditionsin Section 22 Then the solution of the eigenvalue problem(EVP) is discussed in Section 31 We manipulate the EVP toobtain a reduction in the system order in Section 32 Therelationship between the unknowns and the mode shapes isdiscussed in Section 33 followed by a normalization schemein Section 4 We compute the natural frequencies and modeshapes of the filter in Section 51 validate our results usingANSYS in Section 52 and compare them with these of asingle clamped-clamped beam in Section 53 We investigatethe effect of the coupling location on the natural frequency inSection 54 before we conclude this paper in Section 6

2 Problem Formulation

We compute the natural frequencies and mode shapesof a filter composed of two clamped-clamped microbeamresonators (primary beams) coupled by a microbeam asshown in Figure 1(a) Each primary resonator is dividedinto two parts at the location where the coupling beamis attached to it as shown in Figure 1(b) Consequentlythe boundary-value problem (BVP) governing the naturalfrequencies and mode shapes is composed of five equations(one equation for each part of the primary beams and onefor the coupling beam) and twenty boundary conditionsThisproblem is transformed into solving a system of twenty linearhomogeneous algebraic equations for twenty constants andthe natural frequencies Using algebraic manipulations wereduce this problem to that of solving a system of three linearhomogeneous algebraic equations for three constants and thenatural frequenciesThe determinant of the coefficientmatrix

of the reduced problem yields the characteristic equationwhich is solved for the natural frequencies Then the modeshapes are calculated

21 Governing Equations The equations of motion describ-ing the linear undamped and unforced deflection of thesegments of the primary beams and the coupling beam are

11986411986812059741199081198941

1205971199094+ 119873119894

12059721199081198941

1205971199092+ 120588119860

12059721199081198941

1205971199052= 0 where 0 lt 119909 lt 119871

119894

11986411986812059741199081198942

1205971199094+ 119873119894

12059721199081198942

1205971199092+ 120588119860

12059721199081198942

1205971199052= 0 where 119871

119894lt 119909 lt 119871

11986411986812059741199081199001

1205971199094+ 119873119900

12059721199081199001

1205971199092+ 120588119860

12059721199081199001

1205971199052= 0 where 0 lt 119909 lt 119871

119900

11986411986812059741199081199002

1205971199094+ 119873119900

12059721199081199002

1205971199092+ 120588119860

12059721199081199002

1205971199052= 0 where 119871

119900lt 119909 lt 119871

119864119868119888

1205974119908119888

1205971199094+ 119873119888

1205972119908119888

1205971199092+ 120588119860119888

1205972119908119888

1205971199052= 0 where 0 lt 119909 lt 119871

119888

(1)

where 119909 is the position along each beamrsquos axis as shown inFigures 1(b) and 1(c) 119905 is time 119908

1198941is downward transverse

deflection of the first part of input beam 1199081198942

is downwardtransverse deflection of the second part of input beam 119908

1199001

is downward transverse deflection of the first part of outputbeam 119908

1199002is downward transverse deflection of the second

part of output beam119908119888is downward transverse deflection of

the coupling beam the deflections of all parts119908 are functionsof 119909 and 119905 as shown in Figure 1(c) 119864 is Youngrsquos modulus 120588is the material density 119868 and 119868

119888are the moments of inertia

of the cross sections of the primary and coupling beamsrespectively 119860 and 119860

119888are the areas of the cross sections of

the primary and coupling beams respectively 119871119894and 119871

119900are

Shock and Vibration 3

the positions at which the coupling beam is attached to theinput and output resonators respectively 119871 and 119871

119888are the

lengths of the primary and coupling beams respectively 119873119894

is the applied compressive axial force in the input beam 119873119900

is the applied compressive axial force in the output beamand 119873

119888is the applied compressive axial force in the coupling

beam Throughout this paper the subscripts 119894 119900 and 119888 referto quantities related to the input output and coupling beamsrespectively The subscripts 1 and 2 refer to the first andsecond parts respectively of each primary beam

For convenience we introduce the nondimensional vari-ables

119908 =119908

119889 119909 =

119909

119871 =

119905

119879

ℓ119894=

119871119894

119871 ℓ

119900=

119871119900

119871 ℓ

119888=

119871119888

119871

119873119894=

1198731198941198712

119864119868 119873

119900=

1198731199001198712

119864119868 119873

119888=

1198731198881198712

119864119868119888

(2)

where the time is given by 119879 = radic1205881198601198714119864119868 and 119889 is the gapbetween primary beams and the electrode Substituting (2)into (1) and dropping the hats we obtain

12059741199081198941

1205971199094+ 119873119894

12059721199081198941

1205971199092+

12059721199081198941

1205971199052= 0 where 0 lt 119909 lt ℓ

119894

12059741199081198942

1205971199094+ 119873119894

12059721199081198942

1205971199092+

12059721199081198942

1205971199052= 0 where ℓ

119894lt 119909 lt 1

12059741199081199001

1205971199094+ 119873119900

12059721199081199001

1205971199092+

12059721199081199001

1205971199052= 0 where 0 lt 119909 lt ℓ

119900

12059741199081199002

1205971199094+ 119873119900

12059721199081199002

1205971199092+

12059721199081199002

1205971199052= 0 where ℓ

119900lt 119909 lt 1

1205974119908119888

1205971199094+ 119873119888

1205972119908119888

1205971199092+ (

ℎ

ℎ119888

)

21205972119908119888

1205971199052= 0 where 0 lt 119909 lt ℓ

119888

(3)

where ℎ and ℎ119888are the thicknesses of the primary and

coupling beams respectively

22 Boundary Conditions For the clamped (fixed) ends ofthe primary beams the bending moments and shear forcesare unrestricted but the deflections and the slopes vanishthat is

1199081198941

(0) = 01205971199081198941

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199081199001

(0) = 01205971199081199001

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199081198942

(1) = 01205971199081198942

120597119909

10038161003816100381610038161003816100381610038161003816119909=1

= 0

1199081199002

(1) = 01205971199081199002

120597119909

10038161003816100381610038161003816100381610038161003816119909=1

= 0

(4)

At the attachment point in each of the primary beams thedeflection slope and moment are continuous Hence wehave

1199081198941

(ℓ119894) = 1199081198942

(ℓ119894)

1205971199081198941

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119894

=1205971199081198942

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119894

12059721199081198941

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

=12059721199081198942

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

1199081199001

(ℓ119900) = 1199081199002

(ℓ119900)

1205971199081199001

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119900

=1205971199081199002

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119900

12059721199081199001

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

=12059721199081199002

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

(5)

The deflections of the coupling beam are the same as thedeflections of the primary beams at the attachment pointsand the slopes of the coupling beam at these attachmentpoints vanish Therefore

119908119888(0) = 119908

1198941(ℓ119894) 119908

119888(ℓ119888) = 1199081199001

(ℓ119900)

120597119908119888

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0120597119908119888

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119888

= 0

(6)

The shear forces at the ends of the coupling beam are equal tothe changes in the shear forces in the primary beams Theseconditions yield

12059731199081198941

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

minus12059731199081198942

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

=119868119888

119868

1205973119908119888

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=0

12059731199081199001

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

minus12059731199081199002

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

= minus119868119888

119868

1205973119908119888

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=ℓ119888

(7)

3 Eigenvalue Problem

31 Natural Frequencies of the Filter We assume that solu-tions of the equations of motion (3) consist of spatial andtemporal parts given as follows

1199081198941

(119909 119905) = 1206011198941

(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ119894

1199081198942

(119909 119905) = 1206011198942

(119909) exp (119895120596119905) where ℓ119894lt 119909 lt 1

1199081199001

(119909 119905) = 1206011199001

(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ119900

1199081199002

(119909 119905) = 1206011199002

(119909) exp (119895120596119905) where ℓ119900lt 119909 lt 1

119908119888(119909 119905) = 120601

119888(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ

119888

(8)

The 1206011198941is the mode shape of the first part of input beam 120601

1198942

is the mode shape of the second part of input beam 1206011199001

is themode shape of the first part of output beam 120601

1199002is the mode

shape of the second part of output beam 120601119888is themode shape

of the coupling beam and the 120596rsquos are the nondimensional

4 Shock and Vibration

natural frequencies corresponding to these mode shapesSubstituting (8) into (3) we obtain

120601119894V1198941

(119909) + 11987311989412060110158401015840

1198941(119909) minus 120596

21206011198941

(119909) = 0 where 0 lt 119909 lt ℓ119894

120601119894V1198942

(119909) + 11987311989412060110158401015840

1198942(119909) minus 120596

21206011198942

(119909) = 0 where ℓ119894lt 119909 lt 1

120601119894V1199001

(119909) + 11987311990012060110158401015840

1199001(119909) minus 120596

21206011199001

(119909) = 0 where 0 lt 119909 lt ℓ119900

120601119894V1199002

(119909) + 11987311990012060110158401015840

1199002(119909) minus 120596

21206011199002

(119909) = 0 where ℓ119900lt 119909 lt 1

120601119894V119888

(119909)+11987311988812060110158401015840

119888(119909)minus(

ℎ

ℎ119888

)

2

1205962120601119888(119909)=0 where 0 lt 119909 lt ℓ

119888

(9)

Substituting (8) into the boundary conditions as shown inSection 22 yields the following

(i) For the clamped edges

1206011198941

(0) = 0 1206011015840

1198941(0) = 0

1206011199001

(0) = 0 1206011015840

1199001(0) = 0

1206011198942

(1) = 0 1206011015840

1198942(1) = 0

1206011199002

(1) = 0 1206011015840

1199002(1) = 0

(10)

(ii) At the attachment points in the primary beams

1206011198941

(ℓ119894) = 1206011198942

(ℓ119894) 120601

1015840

1198941(ℓ119894) = 1206011015840

1198942(ℓ119894)

12060110158401015840

1198941(ℓ119894) = 12060110158401015840

1198942(ℓ119894)

(11)

1206011199001

(ℓ119900) = 1206011199002

(ℓ119900) 120601

1015840

1199001(ℓ119900) = 1206011015840

1199002(ℓ119900)

12060110158401015840

1199001(ℓ119900) = 12060110158401015840

1199002(ℓ119900)

(12)

(iii) At the attachment points in the coupling beam

120601119888(0) = 120601

1198941(ℓ119894) 120601

119888(ℓ119888) = 1206011199001

(ℓ119900) (13)

1206011015840

119888(0) = 0 120601

1015840

119888(ℓ119888) = 0 (14)

(iv) The shear force at the attachment points

120601101584010158401015840

1198941(ℓ119894) minus 120601101584010158401015840

1198942(ℓ119894) =

119868119888

119868120601101584010158401015840

119888(0) (15)

120601101584010158401015840

1199001(ℓ119900) minus 120601101584010158401015840

1199002(ℓ119900) = minus

119868119888

119868120601101584010158401015840

119888(ℓ119888) (16)

Assuming a solution to (9) of the form a exp(120590119909) yieldsthe general solution

1206011198941

(119909) = 1198621cos [120573

1198941119909] + 119862

2sin [120573

1198941119909] + 119862

1119886cosh [120573

1198942119909]

+ 1198622119886sinh [120573

1198942119909] where 0 lt 119909 lt ℓ

119894

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] + 119862

4sin [120573

1198941(1 minus 119909)]

+ 1198623119886cosh [120573

1198942(1 minus 119909)] + 119862

4119886sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 1198625cos [120573

1199001119909] + 119862

6sin [120573

1199001119909] + 119862

5119886cosh [120573

1199002119909]

+ 1198626119886sinh [120573

1199002119909] where 0 lt 119909 lt ℓ

119900

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] + 119862

8sin [120573

1199001(1 minus 119909)]

+ 1198627119886cosh [120573

1199002(1 minus 119909)] + 119862

8119886sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909] + 119862

11cosh [120573

1198882119909]

+ 11986212sinh [120573

1198882119909] where 0 lt 119909 lt ℓ

119888

(17)

The constants 120573rsquos are given by

1205731198941

= radic+119873119894

2+

1

2radic1198732

119894+ 41205962

1205731198942

= radicminus119873119894

2+

1

2radic1198732

119894+ 41205962

1205731199001

= radic+119873119900

2+

1

2radic1198732119900+ 41205962

1205731199002

= radicminus119873119900

2+

1

2radic1198732119900+ 41205962

1205731198881

= radic+119873119888

2+

1

2

radic1198732119888+ 4(

ℎ

ℎ119888

)

2

1205962

1205731198882

= radicminus119873119888

2+

1

2

radic1198732119888+ 4(

ℎ

ℎ119888

)

2

1205962

(18)

Substituting (17) into (10)ndash(16) leads to twenty linearhomogeneous algebraic equations in the twenty unknowncoefficients 119862

1 1198622 119862

12and 119862

1119886 1198622119886

1198628119886 which can

be written in matrix form as

MC = 0 (19)

where M is a 20 times 20 matrix C is a 20 times 1 vector whoseelements are the above unknown coefficients and 0 is a 20times1

zero vector The elements ofM are functions of 120573 and hence120596 and the dimensions of the filter

Shock and Vibration 5

32 Reduction of the Eigenvalue Problem Equation (19) hasnontrivial solutions C if and only if the coefficient matrixM is singular that is the determinant of the 20 times 20

matrix M is zero To reduce the cost of generating thisdeterminant and more importantly obtain a deeper insightinto the relationship among the unknowns we analyticallymanipulate the governing equation (19) To this end wesubstitute (17) into (10) solve the 119862

119895119886in terms of the 119862

119895 and

rewrite (17) as

1206011198941

(119909) = 1198621cos [120573

1198941119909] minus cosh [120573

1198942119909]

+ 1198622sin [120573

1198941119909] minus sinh [120573

1198942119909]

where 0 lt 119909 lt ℓ119894

(20)

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] minus cosh [120573

1198942(1 minus 119909)]

+ 1198624sin [120573

1198941(1 minus 119909)] minus sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

(21)

1206011199001

(119909) = 1198625cos [120573

1199001119909] minus cosh [120573

1199002119909]

+ 1198626sin [120573

1199001119909] minus sinh [120573

1199002119909]

where 0 lt 119909 lt ℓ119900

(22)

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] minus cosh [120573

1199002(1 minus 119909)]

+ 1198628sin [120573

1199001(1 minus 119909)] minus sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

(23)

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909]

+ 11986211cosh [120573

1198882119909] + 119862

12sinh [120573

1198882119909]

where 0 lt 119909 lt ℓ119888

(24)

We substitute (20)ndash(24) into (11) (12) (14) and (15) andsolve the resulting nine equations for the 119862

119895in terms of

1198621 1198625 and 119862

9 which represent the input output and

coupling beams respectivelyThen using the remaining threeboundary conditions (13) and (16) we obtain the reducedproblem

MrCr = 0 (25)

where

Mr = [

[

11989211

11989212

11989213

11989221

11989222

11989223

11989231

11989232

11989233

]

]

Cr = (

1198621

1198625

1198629

) (26)

The zero vector 0 in this case is a 3times1 vector and the elements119892119898119899

are functions of the 120573rsquos which are given in (18) Settingthe determinant of Mr equal to zero yields the characteristicequation

Det [Mr] = 0 (27)

which is solved for the natural frequencies 120596119899of the filter

33 Mode Shapes of the Filter Associated with each 120596119899

satisfying (27) is a mode shape To compute this mode shapewe substitute 120596

119899into (18) obtain numerical values for the

120573rsquos and substitute these 120573rsquos into (20)ndash(24) Then we use theboundary conditions to find numerical relations among theunknowns and end up with

11988611198621+ 11988621198625= 0 (28)

Depending on 120596119899 one of the following two cases for the

values of the constants 1198861and 1198862is obtained

Case 1 1198861

= 1198862

rArr 1198621

= minus1198625

rArr primary beamsvibrate out-of-phaseCase 2 119886

1= minus1198862

rArr 1198621

= 1198625

rArr primary beamsvibrate in-phase

By setting 1198621equal to one we find numerical values for all of

the119862rsquos in (20)ndash(24) and hence themode shape correspondingto 120596119899

4 Normalization of Mode Shapes

Because the algebraic system of equations (19) is homo-geneous it follows that if the vector C is a solution of theequation then 120572C is also a solution where 120572 is an arbitraryconstantThis implies that themode shapes are uniquewithina constant We normalize the mode shapes (ie render themunique) by setting [11]

int

1199092

1199091

(120572120601 (119909))2d119909 = 1 (29)

where 120601(119909) is the mode shape Hence the constant 120572 is givenby

120572 =1

radicint1199092

1199091

120601(119909)2d119909

(30)

Considering the topology of the filter structure investigatedin this work and the analysis of the reduced-order modelperformed in [6] we find that the constant 120572

119903associated with

120601119903and 120596

119903is given by

120572119903= (int

ℓ119894

0

1206012

1198941(119909) d119909 + int

1

ℓ119894

1206012

1198942(119909) d119909 + int

ℓ119900

0

1206012

1199001(119909) d119909

+int

1

ℓ119900

1206012

1199002(119909) d119909 + 119879

2

119888int

ℓ119888

0

1206012

119888(119909) d119909)

minus12

(31)

The parameter 1198792

119888= 119860119888119860119901is the ratio of the cross section

area of the coupling beam to that of the primary beam

5 Results and Discussion

We utilize the developed methodology to study the filterfabricated and tested by Bannon et al [9] but withoutconsidering the frequency modification factor or any adjust-ments due to fabrication processes The design parametersdimensions and material properties of the filter needed inthis analysis are obtained from [9] and listed in Table 1

6 Shock and Vibration

51 Closed-Form Expression of the Mode Shapes In this sec-tion we employ the methodology discussed in the precedingsections to present closed-form expressions for the modeshapes To avoid unnecessary repetition we show here detailsof the first mode shape for the filter specified in Table 1In this analysis because the fabrication process involves arelatively long-time (one hour) annealing of the structure [9]we assume that all of the beams are free of residual stressesthat is 119873

119894= 119873119900= 119873119888= 0

The first nondimensional natural frequency of the filter(ie the smallest solution of the characteristic equation (27))is1205961= 22363 (= 9471MHz) Substituting this frequency into

the boundary conditions we end up with 1198621= 1198625 By setting

1198621equal to one we find numerical values for all of the 119862rsquos in

(20)ndash(24) Using the normalization condition given in (31)we obtain 120572

1= 0710 and hence the following normalized

closed-form expression for the first global mode shape of thefilter

1206011198941

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119894

1206011198942

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119900

1206011199002

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = minus 0034 cos [4729119909] minus 0082 sin [4729119909]

minus 0100 cosh [4729119909] + 0082 sinh [4729119909]

where 0 lt 119909 lt ℓ119888

(32)

The normalized expression of the first mode shape of thefilter (32) is shown in Figure 2 In all mode shapes shown inthis paper we use the normalization scheme in the same waydiscussed above In addition in all mode shape figures redlines are obtained from (20) and (22) blue lines are obtainedfrom from (21) and (23) and green lines are obtained fromfrom (24)

52 Validation of the Results We use the finite-elementsoftware ANSYS and the structural elements BEAM3 andBEAM4 to validate our model andmethodologyThe degreesof freedom of elements are constrained such that only thetransverse motion is allowed Using modal analysis solution

Table 1 Filter specifications [9]

Parameter Design valuePrimary resonator length 119871 (120583m) 408Primary resonator width 119887 (120583m) 80Coupling location 119871

119894and 119871

119900(120583m) 408

Coupling beam length 119871119888(120583m) 2035

Coupling beam width 119887119888(120583m) 075

Structural thickness ℎ (120583m) 19Youngrsquos modulus 119864 (GPa) 150Polysilicon density 120588 (kgm3) 2300

we obtained natural frequencies of the filter structure andlisted them in Table 2 Inspecting the results obtained fromthe analytical approach discussed in this paper and thenumerical approach from ANSYS reveals that the agreementbetween them is excellent In addition themode shapes of thefilter structure obtained from both of the approaches (beingin-phase or out-of-phase and their order as illustrated in thenext section) are in complete agreement

53 Relation of the Mode Shapes of the Filter to Those ofSingle Clamped-Clamped Resonators In Table 2 we list thelowest 10 natural frequencies of the filter that is the lowest 10solutions of the characteristic equation (27) considering thespecifications listed in Table 1 and assuming that the structureis free from any residual stresses Due to the effect of the weakcoupling beam shown in Figure 1(a) the natural frequenciesof the single resonator are split into two close frequencies forthe filter as listed in Table 2 one frequency corresponds toan in-phase mode and the other corresponds to an out-of-phasemode In filter terminology the in-phasemode is calleda symmetric mode and the out-of-phase mode is called anantisymmetric mode Next we discuss in more detail thesemode shapes

Figures 2 and 3 show the first and second global modeshapes of the filter respectively The input and outputresonators oscillate in the first mode of a single clamped-clamped beam resonator However they oscillate in-phase inthe first mode of the filter whereas they oscillate 180∘ out-of-phase in the second mode of the filter The frequenciesof the first and second modes of the filter are shifted toslightly smaller and larger values respectively compared tothe frequency of the first mode of a single clamped-clampedresonator as shown in Table 2

The third and fourth mode shapes of the filter areshown in Figures 4 and 5 respectively The input and outputresonators oscillate in the second mode of a single clamped-clamped resonator They oscillate in-phase in the third modeand out-of-phase in the fourth mode The sixth and seventhmode shapes of the filter are shown in Figures 7 and 8respectively The input and output resonators oscillate in thethird mode of a single clamped-clamped resonator but theyoscillate out-of-phase in the sixth mode and in-phase in theseventhmode of the filter It follows from the eighth andninthmode shapes of the filter shown in Figures 9 and 10 that

Shock and Vibration 7

Table 2 Natural frequencies (in MHz) for a filter and a single resonator identical to the resonators of the filter

Filter Single resonator

Mode shape number Natural frequency Mode shape number Natural frequencyAnalytical ANSYS Difference ()

1 9471 9460 0116 1 94752 9479 9469 01053 26015 25909 0407 2 261184 26099 25991 04145 37787 37622 0437 mdash mdash6 51017 50570 0876 3 512027 51385 50930 08858 83574 82345 1470 4 846399 84453 83176 151210 104463 102740 1649 mdash mdash

0 05 10

02

04

06

08

1

12

120601i

xL

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05013

014

015

xL

120601c

(c)

0

05

1

15

1206011

0 025 05 0751

00102030405

xLxL

(d)

Figure 2 First mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

the filter oscillates in the fourth mode of a single clamped-clamped beam

Inspecting Table 2 we note that when the resonators ofthe filter oscillate in a symmetric mode of a single clamped-clamped beam the single beam frequency is split into twofrequencies one slightly higher and the other slightly lowerthan that of the single resonator In contrast when theprimary resonators vibrate in an antisymmetric mode of asingle beam both frequencies of the filter are slightly smallerthan the corresponding frequency of the single beam

For the dimensions listed in Table 1 we note that afterevery four modes there is a mode in which the vibrationsof the primary beams (input and output beams) are verysmall compared to the vibration of the coupling beam Itfollows from Figures 6 and 11 that in the fifth and tenth

modes of the filter the coupling beam oscillates in the firstand secondmodes respectively of a single beamwith slightlyflexible clamping points (the rigidity of the clamping pointsis finite) We observe that these modes appear at relativelylarger frequencies because our model considers transversevibrations of the beams that is the vibration is in thedirection perpendicular to the plane that contains the inputoutput and coupling beams However because the flexuralstiffness 119864119868 of the coupling beam in this paper in the in-plane direction is about six times smaller than its stiffness inthe transverse direction the in-plane modes have relativelysmall frequencies However due to the way these types offilters are actuated (see [6]) the in-plane modes will not beactivated thereby justifying the use of a transverse-vibrationmodel only

8 Shock and Vibration

0 05 1minus12

minus1

minus08

minus06

minus04

minus02

0

xL

120601i

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05minus016

minus008

0

008

016

xL

120601c

(c)

minus12minus06

00612

1206012

0 025 05 0751

00102030405

xLxL

(d)

Figure 3 Second mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 0503

045

06

075

xL

120601c

(c)

minus12minus06

00612

1206013

0 025 05 0751

00102030405

xLxL

(d)

Figure 4 Third mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

54 Coupling Location In this section we study the effect ofchanging the attachment location of the coupling beam to theprimary resonators on the first and second natural frequen-cies of the filter and to be more specific on their averageand difference The average and difference of the first andsecond natural frequencies are indicators of the actual center

frequency and bandwidth respectively of the filter when anelectric signal is applied to the input and output resonators Inthis section we indicate the center frequency and bandwidthof the unactuated filter by 119865

0and BW

0 respectivelyWe focus

on the first and second frequencies because it is common toexcite directly the filter with a frequency in the neighborhood

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

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2 Shock and Vibration

Inputresonator Output

resonatorCoupling beam

Clamped ends

Clamped ends

(a)

L

Lix

x

Lc

x Lo

L

(b)

Nx

w(x t)N

(c)

Figure 1 (a) A schematic drawing and (b) a schematic model of a filter made of two clamped-clamped microbeam resonators coupled by aweak beam and (c) a schematic diagram for one of the primary resonators

procedure to develop a reduced-ordermodel for the filter andutilized basis functions computed using the finite-elementpackage ANSYS

The remainder of this work is organized as follows wederive the governing equations of the linear vibration prob-lem in Section 21 and the associated boundary conditionsin Section 22 Then the solution of the eigenvalue problem(EVP) is discussed in Section 31 We manipulate the EVP toobtain a reduction in the system order in Section 32 Therelationship between the unknowns and the mode shapes isdiscussed in Section 33 followed by a normalization schemein Section 4 We compute the natural frequencies and modeshapes of the filter in Section 51 validate our results usingANSYS in Section 52 and compare them with these of asingle clamped-clamped beam in Section 53 We investigatethe effect of the coupling location on the natural frequency inSection 54 before we conclude this paper in Section 6

2 Problem Formulation

We compute the natural frequencies and mode shapesof a filter composed of two clamped-clamped microbeamresonators (primary beams) coupled by a microbeam asshown in Figure 1(a) Each primary resonator is dividedinto two parts at the location where the coupling beamis attached to it as shown in Figure 1(b) Consequentlythe boundary-value problem (BVP) governing the naturalfrequencies and mode shapes is composed of five equations(one equation for each part of the primary beams and onefor the coupling beam) and twenty boundary conditionsThisproblem is transformed into solving a system of twenty linearhomogeneous algebraic equations for twenty constants andthe natural frequencies Using algebraic manipulations wereduce this problem to that of solving a system of three linearhomogeneous algebraic equations for three constants and thenatural frequenciesThe determinant of the coefficientmatrix

of the reduced problem yields the characteristic equationwhich is solved for the natural frequencies Then the modeshapes are calculated

21 Governing Equations The equations of motion describ-ing the linear undamped and unforced deflection of thesegments of the primary beams and the coupling beam are

11986411986812059741199081198941

1205971199094+ 119873119894

12059721199081198941

1205971199092+ 120588119860

12059721199081198941

1205971199052= 0 where 0 lt 119909 lt 119871

119894

11986411986812059741199081198942

1205971199094+ 119873119894

12059721199081198942

1205971199092+ 120588119860

12059721199081198942

1205971199052= 0 where 119871

119894lt 119909 lt 119871

11986411986812059741199081199001

1205971199094+ 119873119900

12059721199081199001

1205971199092+ 120588119860

12059721199081199001

1205971199052= 0 where 0 lt 119909 lt 119871

119900

11986411986812059741199081199002

1205971199094+ 119873119900

12059721199081199002

1205971199092+ 120588119860

12059721199081199002

1205971199052= 0 where 119871

119900lt 119909 lt 119871

119864119868119888

1205974119908119888

1205971199094+ 119873119888

1205972119908119888

1205971199092+ 120588119860119888

1205972119908119888

1205971199052= 0 where 0 lt 119909 lt 119871

119888

(1)

where 119909 is the position along each beamrsquos axis as shown inFigures 1(b) and 1(c) 119905 is time 119908

1198941is downward transverse

deflection of the first part of input beam 1199081198942

is downwardtransverse deflection of the second part of input beam 119908

1199001

is downward transverse deflection of the first part of outputbeam 119908

1199002is downward transverse deflection of the second

part of output beam119908119888is downward transverse deflection of

the coupling beam the deflections of all parts119908 are functionsof 119909 and 119905 as shown in Figure 1(c) 119864 is Youngrsquos modulus 120588is the material density 119868 and 119868

119888are the moments of inertia

of the cross sections of the primary and coupling beamsrespectively 119860 and 119860

119888are the areas of the cross sections of

the primary and coupling beams respectively 119871119894and 119871

119900are

Shock and Vibration 3

the positions at which the coupling beam is attached to theinput and output resonators respectively 119871 and 119871

119888are the

lengths of the primary and coupling beams respectively 119873119894

is the applied compressive axial force in the input beam 119873119900

is the applied compressive axial force in the output beamand 119873

119888is the applied compressive axial force in the coupling

beam Throughout this paper the subscripts 119894 119900 and 119888 referto quantities related to the input output and coupling beamsrespectively The subscripts 1 and 2 refer to the first andsecond parts respectively of each primary beam

For convenience we introduce the nondimensional vari-ables

119908 =119908

119889 119909 =

119909

119871 =

119905

119879

ℓ119894=

119871119894

119871 ℓ

119900=

119871119900

119871 ℓ

119888=

119871119888

119871

119873119894=

1198731198941198712

119864119868 119873

119900=

1198731199001198712

119864119868 119873

119888=

1198731198881198712

119864119868119888

(2)

where the time is given by 119879 = radic1205881198601198714119864119868 and 119889 is the gapbetween primary beams and the electrode Substituting (2)into (1) and dropping the hats we obtain

12059741199081198941

1205971199094+ 119873119894

12059721199081198941

1205971199092+

12059721199081198941

1205971199052= 0 where 0 lt 119909 lt ℓ

119894

12059741199081198942

1205971199094+ 119873119894

12059721199081198942

1205971199092+

12059721199081198942

1205971199052= 0 where ℓ

119894lt 119909 lt 1

12059741199081199001

1205971199094+ 119873119900

12059721199081199001

1205971199092+

12059721199081199001

1205971199052= 0 where 0 lt 119909 lt ℓ

119900

12059741199081199002

1205971199094+ 119873119900

12059721199081199002

1205971199092+

12059721199081199002

1205971199052= 0 where ℓ

119900lt 119909 lt 1

1205974119908119888

1205971199094+ 119873119888

1205972119908119888

1205971199092+ (

ℎ

ℎ119888

)

21205972119908119888

1205971199052= 0 where 0 lt 119909 lt ℓ

119888

(3)

where ℎ and ℎ119888are the thicknesses of the primary and

coupling beams respectively

22 Boundary Conditions For the clamped (fixed) ends ofthe primary beams the bending moments and shear forcesare unrestricted but the deflections and the slopes vanishthat is

1199081198941

(0) = 01205971199081198941

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199081199001

(0) = 01205971199081199001

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199081198942

(1) = 01205971199081198942

120597119909

10038161003816100381610038161003816100381610038161003816119909=1

= 0

1199081199002

(1) = 01205971199081199002

120597119909

10038161003816100381610038161003816100381610038161003816119909=1

= 0

(4)

At the attachment point in each of the primary beams thedeflection slope and moment are continuous Hence wehave

1199081198941

(ℓ119894) = 1199081198942

(ℓ119894)

1205971199081198941

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119894

=1205971199081198942

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119894

12059721199081198941

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

=12059721199081198942

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

1199081199001

(ℓ119900) = 1199081199002

(ℓ119900)

1205971199081199001

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119900

=1205971199081199002

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119900

12059721199081199001

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

=12059721199081199002

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

(5)

The deflections of the coupling beam are the same as thedeflections of the primary beams at the attachment pointsand the slopes of the coupling beam at these attachmentpoints vanish Therefore

119908119888(0) = 119908

1198941(ℓ119894) 119908

119888(ℓ119888) = 1199081199001

(ℓ119900)

120597119908119888

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0120597119908119888

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119888

= 0

(6)

The shear forces at the ends of the coupling beam are equal tothe changes in the shear forces in the primary beams Theseconditions yield

12059731199081198941

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

minus12059731199081198942

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

=119868119888

119868

1205973119908119888

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=0

12059731199081199001

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

minus12059731199081199002

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

= minus119868119888

119868

1205973119908119888

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=ℓ119888

(7)

3 Eigenvalue Problem

31 Natural Frequencies of the Filter We assume that solu-tions of the equations of motion (3) consist of spatial andtemporal parts given as follows

1199081198941

(119909 119905) = 1206011198941

(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ119894

1199081198942

(119909 119905) = 1206011198942

(119909) exp (119895120596119905) where ℓ119894lt 119909 lt 1

1199081199001

(119909 119905) = 1206011199001

(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ119900

1199081199002

(119909 119905) = 1206011199002

(119909) exp (119895120596119905) where ℓ119900lt 119909 lt 1

119908119888(119909 119905) = 120601

119888(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ

119888

(8)

The 1206011198941is the mode shape of the first part of input beam 120601

1198942

is the mode shape of the second part of input beam 1206011199001

is themode shape of the first part of output beam 120601

1199002is the mode

shape of the second part of output beam 120601119888is themode shape

of the coupling beam and the 120596rsquos are the nondimensional

4 Shock and Vibration

natural frequencies corresponding to these mode shapesSubstituting (8) into (3) we obtain

120601119894V1198941

(119909) + 11987311989412060110158401015840

1198941(119909) minus 120596

21206011198941

(119909) = 0 where 0 lt 119909 lt ℓ119894

120601119894V1198942

(119909) + 11987311989412060110158401015840

1198942(119909) minus 120596

21206011198942

(119909) = 0 where ℓ119894lt 119909 lt 1

120601119894V1199001

(119909) + 11987311990012060110158401015840

1199001(119909) minus 120596

21206011199001

(119909) = 0 where 0 lt 119909 lt ℓ119900

120601119894V1199002

(119909) + 11987311990012060110158401015840

1199002(119909) minus 120596

21206011199002

(119909) = 0 where ℓ119900lt 119909 lt 1

120601119894V119888

(119909)+11987311988812060110158401015840

119888(119909)minus(

ℎ

ℎ119888

)

2

1205962120601119888(119909)=0 where 0 lt 119909 lt ℓ

119888

(9)

Substituting (8) into the boundary conditions as shown inSection 22 yields the following

(i) For the clamped edges

1206011198941

(0) = 0 1206011015840

1198941(0) = 0

1206011199001

(0) = 0 1206011015840

1199001(0) = 0

1206011198942

(1) = 0 1206011015840

1198942(1) = 0

1206011199002

(1) = 0 1206011015840

1199002(1) = 0

(10)

(ii) At the attachment points in the primary beams

1206011198941

(ℓ119894) = 1206011198942

(ℓ119894) 120601

1015840

1198941(ℓ119894) = 1206011015840

1198942(ℓ119894)

12060110158401015840

1198941(ℓ119894) = 12060110158401015840

1198942(ℓ119894)

(11)

1206011199001

(ℓ119900) = 1206011199002

(ℓ119900) 120601

1015840

1199001(ℓ119900) = 1206011015840

1199002(ℓ119900)

12060110158401015840

1199001(ℓ119900) = 12060110158401015840

1199002(ℓ119900)

(12)

(iii) At the attachment points in the coupling beam

120601119888(0) = 120601

1198941(ℓ119894) 120601

119888(ℓ119888) = 1206011199001

(ℓ119900) (13)

1206011015840

119888(0) = 0 120601

1015840

119888(ℓ119888) = 0 (14)

(iv) The shear force at the attachment points

120601101584010158401015840

1198941(ℓ119894) minus 120601101584010158401015840

1198942(ℓ119894) =

119868119888

119868120601101584010158401015840

119888(0) (15)

120601101584010158401015840

1199001(ℓ119900) minus 120601101584010158401015840

1199002(ℓ119900) = minus

119868119888

119868120601101584010158401015840

119888(ℓ119888) (16)

Assuming a solution to (9) of the form a exp(120590119909) yieldsthe general solution

1206011198941

(119909) = 1198621cos [120573

1198941119909] + 119862

2sin [120573

1198941119909] + 119862

1119886cosh [120573

1198942119909]

+ 1198622119886sinh [120573

1198942119909] where 0 lt 119909 lt ℓ

119894

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] + 119862

4sin [120573

1198941(1 minus 119909)]

+ 1198623119886cosh [120573

1198942(1 minus 119909)] + 119862

4119886sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 1198625cos [120573

1199001119909] + 119862

6sin [120573

1199001119909] + 119862

5119886cosh [120573

1199002119909]

+ 1198626119886sinh [120573

1199002119909] where 0 lt 119909 lt ℓ

119900

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] + 119862

8sin [120573

1199001(1 minus 119909)]

+ 1198627119886cosh [120573

1199002(1 minus 119909)] + 119862

8119886sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909] + 119862

11cosh [120573

1198882119909]

+ 11986212sinh [120573

1198882119909] where 0 lt 119909 lt ℓ

119888

(17)

The constants 120573rsquos are given by

1205731198941

= radic+119873119894

2+

1

2radic1198732

119894+ 41205962

1205731198942

= radicminus119873119894

2+

1

2radic1198732

119894+ 41205962

1205731199001

= radic+119873119900

2+

1

2radic1198732119900+ 41205962

1205731199002

= radicminus119873119900

2+

1

2radic1198732119900+ 41205962

1205731198881

= radic+119873119888

2+

1

2

radic1198732119888+ 4(

ℎ

ℎ119888

)

2

1205962

1205731198882

= radicminus119873119888

2+

1

2

radic1198732119888+ 4(

ℎ

ℎ119888

)

2

1205962

(18)

Substituting (17) into (10)ndash(16) leads to twenty linearhomogeneous algebraic equations in the twenty unknowncoefficients 119862

1 1198622 119862

12and 119862

1119886 1198622119886

1198628119886 which can

be written in matrix form as

MC = 0 (19)

where M is a 20 times 20 matrix C is a 20 times 1 vector whoseelements are the above unknown coefficients and 0 is a 20times1

zero vector The elements ofM are functions of 120573 and hence120596 and the dimensions of the filter

Shock and Vibration 5

32 Reduction of the Eigenvalue Problem Equation (19) hasnontrivial solutions C if and only if the coefficient matrixM is singular that is the determinant of the 20 times 20

matrix M is zero To reduce the cost of generating thisdeterminant and more importantly obtain a deeper insightinto the relationship among the unknowns we analyticallymanipulate the governing equation (19) To this end wesubstitute (17) into (10) solve the 119862

119895119886in terms of the 119862

119895 and

rewrite (17) as

1206011198941

(119909) = 1198621cos [120573

1198941119909] minus cosh [120573

1198942119909]

+ 1198622sin [120573

1198941119909] minus sinh [120573

1198942119909]

where 0 lt 119909 lt ℓ119894

(20)

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] minus cosh [120573

1198942(1 minus 119909)]

+ 1198624sin [120573

1198941(1 minus 119909)] minus sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

(21)

1206011199001

(119909) = 1198625cos [120573

1199001119909] minus cosh [120573

1199002119909]

+ 1198626sin [120573

1199001119909] minus sinh [120573

1199002119909]

where 0 lt 119909 lt ℓ119900

(22)

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] minus cosh [120573

1199002(1 minus 119909)]

+ 1198628sin [120573

1199001(1 minus 119909)] minus sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

(23)

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909]

+ 11986211cosh [120573

1198882119909] + 119862

12sinh [120573

1198882119909]

where 0 lt 119909 lt ℓ119888

(24)

We substitute (20)ndash(24) into (11) (12) (14) and (15) andsolve the resulting nine equations for the 119862

119895in terms of

1198621 1198625 and 119862

9 which represent the input output and

coupling beams respectivelyThen using the remaining threeboundary conditions (13) and (16) we obtain the reducedproblem

MrCr = 0 (25)

where

Mr = [

[

11989211

11989212

11989213

11989221

11989222

11989223

11989231

11989232

11989233

]

]

Cr = (

1198621

1198625

1198629

) (26)

The zero vector 0 in this case is a 3times1 vector and the elements119892119898119899

are functions of the 120573rsquos which are given in (18) Settingthe determinant of Mr equal to zero yields the characteristicequation

Det [Mr] = 0 (27)

which is solved for the natural frequencies 120596119899of the filter

33 Mode Shapes of the Filter Associated with each 120596119899

satisfying (27) is a mode shape To compute this mode shapewe substitute 120596

119899into (18) obtain numerical values for the

120573rsquos and substitute these 120573rsquos into (20)ndash(24) Then we use theboundary conditions to find numerical relations among theunknowns and end up with

11988611198621+ 11988621198625= 0 (28)

Depending on 120596119899 one of the following two cases for the

values of the constants 1198861and 1198862is obtained

Case 1 1198861

= 1198862

rArr 1198621

= minus1198625

rArr primary beamsvibrate out-of-phaseCase 2 119886

1= minus1198862

rArr 1198621

= 1198625

rArr primary beamsvibrate in-phase

By setting 1198621equal to one we find numerical values for all of

the119862rsquos in (20)ndash(24) and hence themode shape correspondingto 120596119899

4 Normalization of Mode Shapes

Because the algebraic system of equations (19) is homo-geneous it follows that if the vector C is a solution of theequation then 120572C is also a solution where 120572 is an arbitraryconstantThis implies that themode shapes are uniquewithina constant We normalize the mode shapes (ie render themunique) by setting [11]

int

1199092

1199091

(120572120601 (119909))2d119909 = 1 (29)

where 120601(119909) is the mode shape Hence the constant 120572 is givenby

120572 =1

radicint1199092

1199091

120601(119909)2d119909

(30)

Considering the topology of the filter structure investigatedin this work and the analysis of the reduced-order modelperformed in [6] we find that the constant 120572

119903associated with

120601119903and 120596

119903is given by

120572119903= (int

ℓ119894

0

1206012

1198941(119909) d119909 + int

1

ℓ119894

1206012

1198942(119909) d119909 + int

ℓ119900

0

1206012

1199001(119909) d119909

+int

1

ℓ119900

1206012

1199002(119909) d119909 + 119879

2

119888int

ℓ119888

0

1206012

119888(119909) d119909)

minus12

(31)

The parameter 1198792

119888= 119860119888119860119901is the ratio of the cross section

area of the coupling beam to that of the primary beam

5 Results and Discussion

We utilize the developed methodology to study the filterfabricated and tested by Bannon et al [9] but withoutconsidering the frequency modification factor or any adjust-ments due to fabrication processes The design parametersdimensions and material properties of the filter needed inthis analysis are obtained from [9] and listed in Table 1

6 Shock and Vibration

51 Closed-Form Expression of the Mode Shapes In this sec-tion we employ the methodology discussed in the precedingsections to present closed-form expressions for the modeshapes To avoid unnecessary repetition we show here detailsof the first mode shape for the filter specified in Table 1In this analysis because the fabrication process involves arelatively long-time (one hour) annealing of the structure [9]we assume that all of the beams are free of residual stressesthat is 119873

119894= 119873119900= 119873119888= 0

The first nondimensional natural frequency of the filter(ie the smallest solution of the characteristic equation (27))is1205961= 22363 (= 9471MHz) Substituting this frequency into

the boundary conditions we end up with 1198621= 1198625 By setting

1198621equal to one we find numerical values for all of the 119862rsquos in

(20)ndash(24) Using the normalization condition given in (31)we obtain 120572

1= 0710 and hence the following normalized

closed-form expression for the first global mode shape of thefilter

1206011198941

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119894

1206011198942

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119900

1206011199002

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = minus 0034 cos [4729119909] minus 0082 sin [4729119909]

minus 0100 cosh [4729119909] + 0082 sinh [4729119909]

where 0 lt 119909 lt ℓ119888

(32)

The normalized expression of the first mode shape of thefilter (32) is shown in Figure 2 In all mode shapes shown inthis paper we use the normalization scheme in the same waydiscussed above In addition in all mode shape figures redlines are obtained from (20) and (22) blue lines are obtainedfrom from (21) and (23) and green lines are obtained fromfrom (24)

52 Validation of the Results We use the finite-elementsoftware ANSYS and the structural elements BEAM3 andBEAM4 to validate our model andmethodologyThe degreesof freedom of elements are constrained such that only thetransverse motion is allowed Using modal analysis solution

Table 1 Filter specifications [9]

Parameter Design valuePrimary resonator length 119871 (120583m) 408Primary resonator width 119887 (120583m) 80Coupling location 119871

119894and 119871

119900(120583m) 408

Coupling beam length 119871119888(120583m) 2035

Coupling beam width 119887119888(120583m) 075

Structural thickness ℎ (120583m) 19Youngrsquos modulus 119864 (GPa) 150Polysilicon density 120588 (kgm3) 2300

we obtained natural frequencies of the filter structure andlisted them in Table 2 Inspecting the results obtained fromthe analytical approach discussed in this paper and thenumerical approach from ANSYS reveals that the agreementbetween them is excellent In addition themode shapes of thefilter structure obtained from both of the approaches (beingin-phase or out-of-phase and their order as illustrated in thenext section) are in complete agreement

53 Relation of the Mode Shapes of the Filter to Those ofSingle Clamped-Clamped Resonators In Table 2 we list thelowest 10 natural frequencies of the filter that is the lowest 10solutions of the characteristic equation (27) considering thespecifications listed in Table 1 and assuming that the structureis free from any residual stresses Due to the effect of the weakcoupling beam shown in Figure 1(a) the natural frequenciesof the single resonator are split into two close frequencies forthe filter as listed in Table 2 one frequency corresponds toan in-phase mode and the other corresponds to an out-of-phasemode In filter terminology the in-phasemode is calleda symmetric mode and the out-of-phase mode is called anantisymmetric mode Next we discuss in more detail thesemode shapes

Figures 2 and 3 show the first and second global modeshapes of the filter respectively The input and outputresonators oscillate in the first mode of a single clamped-clamped beam resonator However they oscillate in-phase inthe first mode of the filter whereas they oscillate 180∘ out-of-phase in the second mode of the filter The frequenciesof the first and second modes of the filter are shifted toslightly smaller and larger values respectively compared tothe frequency of the first mode of a single clamped-clampedresonator as shown in Table 2

The third and fourth mode shapes of the filter areshown in Figures 4 and 5 respectively The input and outputresonators oscillate in the second mode of a single clamped-clamped resonator They oscillate in-phase in the third modeand out-of-phase in the fourth mode The sixth and seventhmode shapes of the filter are shown in Figures 7 and 8respectively The input and output resonators oscillate in thethird mode of a single clamped-clamped resonator but theyoscillate out-of-phase in the sixth mode and in-phase in theseventhmode of the filter It follows from the eighth andninthmode shapes of the filter shown in Figures 9 and 10 that

Shock and Vibration 7

Table 2 Natural frequencies (in MHz) for a filter and a single resonator identical to the resonators of the filter

Filter Single resonator

Mode shape number Natural frequency Mode shape number Natural frequencyAnalytical ANSYS Difference ()

1 9471 9460 0116 1 94752 9479 9469 01053 26015 25909 0407 2 261184 26099 25991 04145 37787 37622 0437 mdash mdash6 51017 50570 0876 3 512027 51385 50930 08858 83574 82345 1470 4 846399 84453 83176 151210 104463 102740 1649 mdash mdash

0 05 10

02

04

06

08

1

12

120601i

xL

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05013

014

015

xL

120601c

(c)

0

05

1

15

1206011

0 025 05 0751

00102030405

xLxL

(d)

Figure 2 First mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

the filter oscillates in the fourth mode of a single clamped-clamped beam

Inspecting Table 2 we note that when the resonators ofthe filter oscillate in a symmetric mode of a single clamped-clamped beam the single beam frequency is split into twofrequencies one slightly higher and the other slightly lowerthan that of the single resonator In contrast when theprimary resonators vibrate in an antisymmetric mode of asingle beam both frequencies of the filter are slightly smallerthan the corresponding frequency of the single beam

For the dimensions listed in Table 1 we note that afterevery four modes there is a mode in which the vibrationsof the primary beams (input and output beams) are verysmall compared to the vibration of the coupling beam Itfollows from Figures 6 and 11 that in the fifth and tenth

modes of the filter the coupling beam oscillates in the firstand secondmodes respectively of a single beamwith slightlyflexible clamping points (the rigidity of the clamping pointsis finite) We observe that these modes appear at relativelylarger frequencies because our model considers transversevibrations of the beams that is the vibration is in thedirection perpendicular to the plane that contains the inputoutput and coupling beams However because the flexuralstiffness 119864119868 of the coupling beam in this paper in the in-plane direction is about six times smaller than its stiffness inthe transverse direction the in-plane modes have relativelysmall frequencies However due to the way these types offilters are actuated (see [6]) the in-plane modes will not beactivated thereby justifying the use of a transverse-vibrationmodel only

8 Shock and Vibration

0 05 1minus12

minus1

minus08

minus06

minus04

minus02

0

xL

120601i

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05minus016

minus008

0

008

016

xL

120601c

(c)

minus12minus06

00612

1206012

0 025 05 0751

00102030405

xLxL

(d)

Figure 3 Second mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 0503

045

06

075

xL

120601c

(c)

minus12minus06

00612

1206013

0 025 05 0751

00102030405

xLxL

(d)

Figure 4 Third mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

54 Coupling Location In this section we study the effect ofchanging the attachment location of the coupling beam to theprimary resonators on the first and second natural frequen-cies of the filter and to be more specific on their averageand difference The average and difference of the first andsecond natural frequencies are indicators of the actual center

frequency and bandwidth respectively of the filter when anelectric signal is applied to the input and output resonators Inthis section we indicate the center frequency and bandwidthof the unactuated filter by 119865

0and BW

0 respectivelyWe focus

on the first and second frequencies because it is common toexcite directly the filter with a frequency in the neighborhood

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

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Shock and Vibration 3

the positions at which the coupling beam is attached to theinput and output resonators respectively 119871 and 119871

119888are the

lengths of the primary and coupling beams respectively 119873119894

is the applied compressive axial force in the input beam 119873119900

is the applied compressive axial force in the output beamand 119873

119888is the applied compressive axial force in the coupling

beam Throughout this paper the subscripts 119894 119900 and 119888 referto quantities related to the input output and coupling beamsrespectively The subscripts 1 and 2 refer to the first andsecond parts respectively of each primary beam

For convenience we introduce the nondimensional vari-ables

119908 =119908

119889 119909 =

119909

119871 =

119905

119879

ℓ119894=

119871119894

119871 ℓ

119900=

119871119900

119871 ℓ

119888=

119871119888

119871

119873119894=

1198731198941198712

119864119868 119873

119900=

1198731199001198712

119864119868 119873

119888=

1198731198881198712

119864119868119888

(2)

where the time is given by 119879 = radic1205881198601198714119864119868 and 119889 is the gapbetween primary beams and the electrode Substituting (2)into (1) and dropping the hats we obtain

12059741199081198941

1205971199094+ 119873119894

12059721199081198941

1205971199092+

12059721199081198941

1205971199052= 0 where 0 lt 119909 lt ℓ

119894

12059741199081198942

1205971199094+ 119873119894

12059721199081198942

1205971199092+

12059721199081198942

1205971199052= 0 where ℓ

119894lt 119909 lt 1

12059741199081199001

1205971199094+ 119873119900

12059721199081199001

1205971199092+

12059721199081199001

1205971199052= 0 where 0 lt 119909 lt ℓ

119900

12059741199081199002

1205971199094+ 119873119900

12059721199081199002

1205971199092+

12059721199081199002

1205971199052= 0 where ℓ

119900lt 119909 lt 1

1205974119908119888

1205971199094+ 119873119888

1205972119908119888

1205971199092+ (

ℎ

ℎ119888

)

21205972119908119888

1205971199052= 0 where 0 lt 119909 lt ℓ

119888

(3)

where ℎ and ℎ119888are the thicknesses of the primary and

coupling beams respectively

22 Boundary Conditions For the clamped (fixed) ends ofthe primary beams the bending moments and shear forcesare unrestricted but the deflections and the slopes vanishthat is

1199081198941

(0) = 01205971199081198941

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199081199001

(0) = 01205971199081199001

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199081198942

(1) = 01205971199081198942

120597119909

10038161003816100381610038161003816100381610038161003816119909=1

= 0

1199081199002

(1) = 01205971199081199002

120597119909

10038161003816100381610038161003816100381610038161003816119909=1

= 0

(4)

At the attachment point in each of the primary beams thedeflection slope and moment are continuous Hence wehave

1199081198941

(ℓ119894) = 1199081198942

(ℓ119894)

1205971199081198941

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119894

=1205971199081198942

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119894

12059721199081198941

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

=12059721199081198942

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

1199081199001

(ℓ119900) = 1199081199002

(ℓ119900)

1205971199081199001

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119900

=1205971199081199002

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119900

12059721199081199001

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

=12059721199081199002

1205971199092

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

(5)

The deflections of the coupling beam are the same as thedeflections of the primary beams at the attachment pointsand the slopes of the coupling beam at these attachmentpoints vanish Therefore

119908119888(0) = 119908

1198941(ℓ119894) 119908

119888(ℓ119888) = 1199081199001

(ℓ119900)

120597119908119888

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0120597119908119888

120597119909

10038161003816100381610038161003816100381610038161003816119909=ℓ119888

= 0

(6)

The shear forces at the ends of the coupling beam are equal tothe changes in the shear forces in the primary beams Theseconditions yield

12059731199081198941

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

minus12059731199081198942

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119894

=119868119888

119868

1205973119908119888

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=0

12059731199081199001

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

minus12059731199081199002

1205971199093

1003816100381610038161003816100381610038161003816100381610038161003816119909=ℓ119900

= minus119868119888

119868

1205973119908119888

1205971199093

100381610038161003816100381610038161003816100381610038161003816119909=ℓ119888

(7)

3 Eigenvalue Problem

31 Natural Frequencies of the Filter We assume that solu-tions of the equations of motion (3) consist of spatial andtemporal parts given as follows

1199081198941

(119909 119905) = 1206011198941

(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ119894

1199081198942

(119909 119905) = 1206011198942

(119909) exp (119895120596119905) where ℓ119894lt 119909 lt 1

1199081199001

(119909 119905) = 1206011199001

(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ119900

1199081199002

(119909 119905) = 1206011199002

(119909) exp (119895120596119905) where ℓ119900lt 119909 lt 1

119908119888(119909 119905) = 120601

119888(119909) exp (119895120596119905) where 0 lt 119909 lt ℓ

119888

(8)

The 1206011198941is the mode shape of the first part of input beam 120601

1198942

is the mode shape of the second part of input beam 1206011199001

is themode shape of the first part of output beam 120601

1199002is the mode

shape of the second part of output beam 120601119888is themode shape

of the coupling beam and the 120596rsquos are the nondimensional

4 Shock and Vibration

natural frequencies corresponding to these mode shapesSubstituting (8) into (3) we obtain

120601119894V1198941

(119909) + 11987311989412060110158401015840

1198941(119909) minus 120596

21206011198941

(119909) = 0 where 0 lt 119909 lt ℓ119894

120601119894V1198942

(119909) + 11987311989412060110158401015840

1198942(119909) minus 120596

21206011198942

(119909) = 0 where ℓ119894lt 119909 lt 1

120601119894V1199001

(119909) + 11987311990012060110158401015840

1199001(119909) minus 120596

21206011199001

(119909) = 0 where 0 lt 119909 lt ℓ119900

120601119894V1199002

(119909) + 11987311990012060110158401015840

1199002(119909) minus 120596

21206011199002

(119909) = 0 where ℓ119900lt 119909 lt 1

120601119894V119888

(119909)+11987311988812060110158401015840

119888(119909)minus(

ℎ

ℎ119888

)

2

1205962120601119888(119909)=0 where 0 lt 119909 lt ℓ

119888

(9)

Substituting (8) into the boundary conditions as shown inSection 22 yields the following

(i) For the clamped edges

1206011198941

(0) = 0 1206011015840

1198941(0) = 0

1206011199001

(0) = 0 1206011015840

1199001(0) = 0

1206011198942

(1) = 0 1206011015840

1198942(1) = 0

1206011199002

(1) = 0 1206011015840

1199002(1) = 0

(10)

(ii) At the attachment points in the primary beams

1206011198941

(ℓ119894) = 1206011198942

(ℓ119894) 120601

1015840

1198941(ℓ119894) = 1206011015840

1198942(ℓ119894)

12060110158401015840

1198941(ℓ119894) = 12060110158401015840

1198942(ℓ119894)

(11)

1206011199001

(ℓ119900) = 1206011199002

(ℓ119900) 120601

1015840

1199001(ℓ119900) = 1206011015840

1199002(ℓ119900)

12060110158401015840

1199001(ℓ119900) = 12060110158401015840

1199002(ℓ119900)

(12)

(iii) At the attachment points in the coupling beam

120601119888(0) = 120601

1198941(ℓ119894) 120601

119888(ℓ119888) = 1206011199001

(ℓ119900) (13)

1206011015840

119888(0) = 0 120601

1015840

119888(ℓ119888) = 0 (14)

(iv) The shear force at the attachment points

120601101584010158401015840

1198941(ℓ119894) minus 120601101584010158401015840

1198942(ℓ119894) =

119868119888

119868120601101584010158401015840

119888(0) (15)

120601101584010158401015840

1199001(ℓ119900) minus 120601101584010158401015840

1199002(ℓ119900) = minus

119868119888

119868120601101584010158401015840

119888(ℓ119888) (16)

Assuming a solution to (9) of the form a exp(120590119909) yieldsthe general solution

1206011198941

(119909) = 1198621cos [120573

1198941119909] + 119862

2sin [120573

1198941119909] + 119862

1119886cosh [120573

1198942119909]

+ 1198622119886sinh [120573

1198942119909] where 0 lt 119909 lt ℓ

119894

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] + 119862

4sin [120573

1198941(1 minus 119909)]

+ 1198623119886cosh [120573

1198942(1 minus 119909)] + 119862

4119886sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 1198625cos [120573

1199001119909] + 119862

6sin [120573

1199001119909] + 119862

5119886cosh [120573

1199002119909]

+ 1198626119886sinh [120573

1199002119909] where 0 lt 119909 lt ℓ

119900

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] + 119862

8sin [120573

1199001(1 minus 119909)]

+ 1198627119886cosh [120573

1199002(1 minus 119909)] + 119862

8119886sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909] + 119862

11cosh [120573

1198882119909]

+ 11986212sinh [120573

1198882119909] where 0 lt 119909 lt ℓ

119888

(17)

The constants 120573rsquos are given by

1205731198941

= radic+119873119894

2+

1

2radic1198732

119894+ 41205962

1205731198942

= radicminus119873119894

2+

1

2radic1198732

119894+ 41205962

1205731199001

= radic+119873119900

2+

1

2radic1198732119900+ 41205962

1205731199002

= radicminus119873119900

2+

1

2radic1198732119900+ 41205962

1205731198881

= radic+119873119888

2+

1

2

radic1198732119888+ 4(

ℎ

ℎ119888

)

2

1205962

1205731198882

= radicminus119873119888

2+

1

2

radic1198732119888+ 4(

ℎ

ℎ119888

)

2

1205962

(18)

Substituting (17) into (10)ndash(16) leads to twenty linearhomogeneous algebraic equations in the twenty unknowncoefficients 119862

1 1198622 119862

12and 119862

1119886 1198622119886

1198628119886 which can

be written in matrix form as

MC = 0 (19)

where M is a 20 times 20 matrix C is a 20 times 1 vector whoseelements are the above unknown coefficients and 0 is a 20times1

zero vector The elements ofM are functions of 120573 and hence120596 and the dimensions of the filter

Shock and Vibration 5

32 Reduction of the Eigenvalue Problem Equation (19) hasnontrivial solutions C if and only if the coefficient matrixM is singular that is the determinant of the 20 times 20

matrix M is zero To reduce the cost of generating thisdeterminant and more importantly obtain a deeper insightinto the relationship among the unknowns we analyticallymanipulate the governing equation (19) To this end wesubstitute (17) into (10) solve the 119862

119895119886in terms of the 119862

119895 and

rewrite (17) as

1206011198941

(119909) = 1198621cos [120573

1198941119909] minus cosh [120573

1198942119909]

+ 1198622sin [120573

1198941119909] minus sinh [120573

1198942119909]

where 0 lt 119909 lt ℓ119894

(20)

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] minus cosh [120573

1198942(1 minus 119909)]

+ 1198624sin [120573

1198941(1 minus 119909)] minus sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

(21)

1206011199001

(119909) = 1198625cos [120573

1199001119909] minus cosh [120573

1199002119909]

+ 1198626sin [120573

1199001119909] minus sinh [120573

1199002119909]

where 0 lt 119909 lt ℓ119900

(22)

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] minus cosh [120573

1199002(1 minus 119909)]

+ 1198628sin [120573

1199001(1 minus 119909)] minus sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

(23)

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909]

+ 11986211cosh [120573

1198882119909] + 119862

12sinh [120573

1198882119909]

where 0 lt 119909 lt ℓ119888

(24)

We substitute (20)ndash(24) into (11) (12) (14) and (15) andsolve the resulting nine equations for the 119862

119895in terms of

1198621 1198625 and 119862

9 which represent the input output and

coupling beams respectivelyThen using the remaining threeboundary conditions (13) and (16) we obtain the reducedproblem

MrCr = 0 (25)

where

Mr = [

[

11989211

11989212

11989213

11989221

11989222

11989223

11989231

11989232

11989233

]

]

Cr = (

1198621

1198625

1198629

) (26)

The zero vector 0 in this case is a 3times1 vector and the elements119892119898119899

are functions of the 120573rsquos which are given in (18) Settingthe determinant of Mr equal to zero yields the characteristicequation

Det [Mr] = 0 (27)

which is solved for the natural frequencies 120596119899of the filter

33 Mode Shapes of the Filter Associated with each 120596119899

satisfying (27) is a mode shape To compute this mode shapewe substitute 120596

119899into (18) obtain numerical values for the

120573rsquos and substitute these 120573rsquos into (20)ndash(24) Then we use theboundary conditions to find numerical relations among theunknowns and end up with

11988611198621+ 11988621198625= 0 (28)

Depending on 120596119899 one of the following two cases for the

values of the constants 1198861and 1198862is obtained

Case 1 1198861

= 1198862

rArr 1198621

= minus1198625

rArr primary beamsvibrate out-of-phaseCase 2 119886

1= minus1198862

rArr 1198621

= 1198625

rArr primary beamsvibrate in-phase

By setting 1198621equal to one we find numerical values for all of

the119862rsquos in (20)ndash(24) and hence themode shape correspondingto 120596119899

4 Normalization of Mode Shapes

Because the algebraic system of equations (19) is homo-geneous it follows that if the vector C is a solution of theequation then 120572C is also a solution where 120572 is an arbitraryconstantThis implies that themode shapes are uniquewithina constant We normalize the mode shapes (ie render themunique) by setting [11]

int

1199092

1199091

(120572120601 (119909))2d119909 = 1 (29)

where 120601(119909) is the mode shape Hence the constant 120572 is givenby

120572 =1

radicint1199092

1199091

120601(119909)2d119909

(30)

Considering the topology of the filter structure investigatedin this work and the analysis of the reduced-order modelperformed in [6] we find that the constant 120572

119903associated with

120601119903and 120596

119903is given by

120572119903= (int

ℓ119894

0

1206012

1198941(119909) d119909 + int

1

ℓ119894

1206012

1198942(119909) d119909 + int

ℓ119900

0

1206012

1199001(119909) d119909

+int

1

ℓ119900

1206012

1199002(119909) d119909 + 119879

2

119888int

ℓ119888

0

1206012

119888(119909) d119909)

minus12

(31)

The parameter 1198792

119888= 119860119888119860119901is the ratio of the cross section

area of the coupling beam to that of the primary beam

5 Results and Discussion

We utilize the developed methodology to study the filterfabricated and tested by Bannon et al [9] but withoutconsidering the frequency modification factor or any adjust-ments due to fabrication processes The design parametersdimensions and material properties of the filter needed inthis analysis are obtained from [9] and listed in Table 1

6 Shock and Vibration

51 Closed-Form Expression of the Mode Shapes In this sec-tion we employ the methodology discussed in the precedingsections to present closed-form expressions for the modeshapes To avoid unnecessary repetition we show here detailsof the first mode shape for the filter specified in Table 1In this analysis because the fabrication process involves arelatively long-time (one hour) annealing of the structure [9]we assume that all of the beams are free of residual stressesthat is 119873

119894= 119873119900= 119873119888= 0

The first nondimensional natural frequency of the filter(ie the smallest solution of the characteristic equation (27))is1205961= 22363 (= 9471MHz) Substituting this frequency into

the boundary conditions we end up with 1198621= 1198625 By setting

1198621equal to one we find numerical values for all of the 119862rsquos in

(20)ndash(24) Using the normalization condition given in (31)we obtain 120572

1= 0710 and hence the following normalized

closed-form expression for the first global mode shape of thefilter

1206011198941

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119894

1206011198942

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119900

1206011199002

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = minus 0034 cos [4729119909] minus 0082 sin [4729119909]

minus 0100 cosh [4729119909] + 0082 sinh [4729119909]

where 0 lt 119909 lt ℓ119888

(32)

The normalized expression of the first mode shape of thefilter (32) is shown in Figure 2 In all mode shapes shown inthis paper we use the normalization scheme in the same waydiscussed above In addition in all mode shape figures redlines are obtained from (20) and (22) blue lines are obtainedfrom from (21) and (23) and green lines are obtained fromfrom (24)

52 Validation of the Results We use the finite-elementsoftware ANSYS and the structural elements BEAM3 andBEAM4 to validate our model andmethodologyThe degreesof freedom of elements are constrained such that only thetransverse motion is allowed Using modal analysis solution

Table 1 Filter specifications [9]

Parameter Design valuePrimary resonator length 119871 (120583m) 408Primary resonator width 119887 (120583m) 80Coupling location 119871

119894and 119871

119900(120583m) 408

Coupling beam length 119871119888(120583m) 2035

Coupling beam width 119887119888(120583m) 075

Structural thickness ℎ (120583m) 19Youngrsquos modulus 119864 (GPa) 150Polysilicon density 120588 (kgm3) 2300

we obtained natural frequencies of the filter structure andlisted them in Table 2 Inspecting the results obtained fromthe analytical approach discussed in this paper and thenumerical approach from ANSYS reveals that the agreementbetween them is excellent In addition themode shapes of thefilter structure obtained from both of the approaches (beingin-phase or out-of-phase and their order as illustrated in thenext section) are in complete agreement

53 Relation of the Mode Shapes of the Filter to Those ofSingle Clamped-Clamped Resonators In Table 2 we list thelowest 10 natural frequencies of the filter that is the lowest 10solutions of the characteristic equation (27) considering thespecifications listed in Table 1 and assuming that the structureis free from any residual stresses Due to the effect of the weakcoupling beam shown in Figure 1(a) the natural frequenciesof the single resonator are split into two close frequencies forthe filter as listed in Table 2 one frequency corresponds toan in-phase mode and the other corresponds to an out-of-phasemode In filter terminology the in-phasemode is calleda symmetric mode and the out-of-phase mode is called anantisymmetric mode Next we discuss in more detail thesemode shapes

Figures 2 and 3 show the first and second global modeshapes of the filter respectively The input and outputresonators oscillate in the first mode of a single clamped-clamped beam resonator However they oscillate in-phase inthe first mode of the filter whereas they oscillate 180∘ out-of-phase in the second mode of the filter The frequenciesof the first and second modes of the filter are shifted toslightly smaller and larger values respectively compared tothe frequency of the first mode of a single clamped-clampedresonator as shown in Table 2

The third and fourth mode shapes of the filter areshown in Figures 4 and 5 respectively The input and outputresonators oscillate in the second mode of a single clamped-clamped resonator They oscillate in-phase in the third modeand out-of-phase in the fourth mode The sixth and seventhmode shapes of the filter are shown in Figures 7 and 8respectively The input and output resonators oscillate in thethird mode of a single clamped-clamped resonator but theyoscillate out-of-phase in the sixth mode and in-phase in theseventhmode of the filter It follows from the eighth andninthmode shapes of the filter shown in Figures 9 and 10 that

Shock and Vibration 7

Table 2 Natural frequencies (in MHz) for a filter and a single resonator identical to the resonators of the filter

Filter Single resonator

Mode shape number Natural frequency Mode shape number Natural frequencyAnalytical ANSYS Difference ()

1 9471 9460 0116 1 94752 9479 9469 01053 26015 25909 0407 2 261184 26099 25991 04145 37787 37622 0437 mdash mdash6 51017 50570 0876 3 512027 51385 50930 08858 83574 82345 1470 4 846399 84453 83176 151210 104463 102740 1649 mdash mdash

0 05 10

02

04

06

08

1

12

120601i

xL

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05013

014

015

xL

120601c

(c)

0

05

1

15

1206011

0 025 05 0751

00102030405

xLxL

(d)

Figure 2 First mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

the filter oscillates in the fourth mode of a single clamped-clamped beam

Inspecting Table 2 we note that when the resonators ofthe filter oscillate in a symmetric mode of a single clamped-clamped beam the single beam frequency is split into twofrequencies one slightly higher and the other slightly lowerthan that of the single resonator In contrast when theprimary resonators vibrate in an antisymmetric mode of asingle beam both frequencies of the filter are slightly smallerthan the corresponding frequency of the single beam

For the dimensions listed in Table 1 we note that afterevery four modes there is a mode in which the vibrationsof the primary beams (input and output beams) are verysmall compared to the vibration of the coupling beam Itfollows from Figures 6 and 11 that in the fifth and tenth

modes of the filter the coupling beam oscillates in the firstand secondmodes respectively of a single beamwith slightlyflexible clamping points (the rigidity of the clamping pointsis finite) We observe that these modes appear at relativelylarger frequencies because our model considers transversevibrations of the beams that is the vibration is in thedirection perpendicular to the plane that contains the inputoutput and coupling beams However because the flexuralstiffness 119864119868 of the coupling beam in this paper in the in-plane direction is about six times smaller than its stiffness inthe transverse direction the in-plane modes have relativelysmall frequencies However due to the way these types offilters are actuated (see [6]) the in-plane modes will not beactivated thereby justifying the use of a transverse-vibrationmodel only

8 Shock and Vibration

0 05 1minus12

minus1

minus08

minus06

minus04

minus02

0

xL

120601i

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05minus016

minus008

0

008

016

xL

120601c

(c)

minus12minus06

00612

1206012

0 025 05 0751

00102030405

xLxL

(d)

Figure 3 Second mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 0503

045

06

075

xL

120601c

(c)

minus12minus06

00612

1206013

0 025 05 0751

00102030405

xLxL

(d)

Figure 4 Third mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

54 Coupling Location In this section we study the effect ofchanging the attachment location of the coupling beam to theprimary resonators on the first and second natural frequen-cies of the filter and to be more specific on their averageand difference The average and difference of the first andsecond natural frequencies are indicators of the actual center

frequency and bandwidth respectively of the filter when anelectric signal is applied to the input and output resonators Inthis section we indicate the center frequency and bandwidthof the unactuated filter by 119865

0and BW

0 respectivelyWe focus

on the first and second frequencies because it is common toexcite directly the filter with a frequency in the neighborhood

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

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4 Shock and Vibration

natural frequencies corresponding to these mode shapesSubstituting (8) into (3) we obtain

120601119894V1198941

(119909) + 11987311989412060110158401015840

1198941(119909) minus 120596

21206011198941

(119909) = 0 where 0 lt 119909 lt ℓ119894

120601119894V1198942

(119909) + 11987311989412060110158401015840

1198942(119909) minus 120596

21206011198942

(119909) = 0 where ℓ119894lt 119909 lt 1

120601119894V1199001

(119909) + 11987311990012060110158401015840

1199001(119909) minus 120596

21206011199001

(119909) = 0 where 0 lt 119909 lt ℓ119900

120601119894V1199002

(119909) + 11987311990012060110158401015840

1199002(119909) minus 120596

21206011199002

(119909) = 0 where ℓ119900lt 119909 lt 1

120601119894V119888

(119909)+11987311988812060110158401015840

119888(119909)minus(

ℎ

ℎ119888

)

2

1205962120601119888(119909)=0 where 0 lt 119909 lt ℓ

119888

(9)

Substituting (8) into the boundary conditions as shown inSection 22 yields the following

(i) For the clamped edges

1206011198941

(0) = 0 1206011015840

1198941(0) = 0

1206011199001

(0) = 0 1206011015840

1199001(0) = 0

1206011198942

(1) = 0 1206011015840

1198942(1) = 0

1206011199002

(1) = 0 1206011015840

1199002(1) = 0

(10)

(ii) At the attachment points in the primary beams

1206011198941

(ℓ119894) = 1206011198942

(ℓ119894) 120601

1015840

1198941(ℓ119894) = 1206011015840

1198942(ℓ119894)

12060110158401015840

1198941(ℓ119894) = 12060110158401015840

1198942(ℓ119894)

(11)

1206011199001

(ℓ119900) = 1206011199002

(ℓ119900) 120601

1015840

1199001(ℓ119900) = 1206011015840

1199002(ℓ119900)

12060110158401015840

1199001(ℓ119900) = 12060110158401015840

1199002(ℓ119900)

(12)

(iii) At the attachment points in the coupling beam

120601119888(0) = 120601

1198941(ℓ119894) 120601

119888(ℓ119888) = 1206011199001

(ℓ119900) (13)

1206011015840

119888(0) = 0 120601

1015840

119888(ℓ119888) = 0 (14)

(iv) The shear force at the attachment points

120601101584010158401015840

1198941(ℓ119894) minus 120601101584010158401015840

1198942(ℓ119894) =

119868119888

119868120601101584010158401015840

119888(0) (15)

120601101584010158401015840

1199001(ℓ119900) minus 120601101584010158401015840

1199002(ℓ119900) = minus

119868119888

119868120601101584010158401015840

119888(ℓ119888) (16)

Assuming a solution to (9) of the form a exp(120590119909) yieldsthe general solution

1206011198941

(119909) = 1198621cos [120573

1198941119909] + 119862

2sin [120573

1198941119909] + 119862

1119886cosh [120573

1198942119909]

+ 1198622119886sinh [120573

1198942119909] where 0 lt 119909 lt ℓ

119894

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] + 119862

4sin [120573

1198941(1 minus 119909)]

+ 1198623119886cosh [120573

1198942(1 minus 119909)] + 119862

4119886sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 1198625cos [120573

1199001119909] + 119862

6sin [120573

1199001119909] + 119862

5119886cosh [120573

1199002119909]

+ 1198626119886sinh [120573

1199002119909] where 0 lt 119909 lt ℓ

119900

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] + 119862

8sin [120573

1199001(1 minus 119909)]

+ 1198627119886cosh [120573

1199002(1 minus 119909)] + 119862

8119886sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909] + 119862

11cosh [120573

1198882119909]

+ 11986212sinh [120573

1198882119909] where 0 lt 119909 lt ℓ

119888

(17)

The constants 120573rsquos are given by

1205731198941

= radic+119873119894

2+

1

2radic1198732

119894+ 41205962

1205731198942

= radicminus119873119894

2+

1

2radic1198732

119894+ 41205962

1205731199001

= radic+119873119900

2+

1

2radic1198732119900+ 41205962

1205731199002

= radicminus119873119900

2+

1

2radic1198732119900+ 41205962

1205731198881

= radic+119873119888

2+

1

2

radic1198732119888+ 4(

ℎ

ℎ119888

)

2

1205962

1205731198882

= radicminus119873119888

2+

1

2

radic1198732119888+ 4(

ℎ

ℎ119888

)

2

1205962

(18)

Substituting (17) into (10)ndash(16) leads to twenty linearhomogeneous algebraic equations in the twenty unknowncoefficients 119862

1 1198622 119862

12and 119862

1119886 1198622119886

1198628119886 which can

be written in matrix form as

MC = 0 (19)

where M is a 20 times 20 matrix C is a 20 times 1 vector whoseelements are the above unknown coefficients and 0 is a 20times1

zero vector The elements ofM are functions of 120573 and hence120596 and the dimensions of the filter

Shock and Vibration 5

32 Reduction of the Eigenvalue Problem Equation (19) hasnontrivial solutions C if and only if the coefficient matrixM is singular that is the determinant of the 20 times 20

matrix M is zero To reduce the cost of generating thisdeterminant and more importantly obtain a deeper insightinto the relationship among the unknowns we analyticallymanipulate the governing equation (19) To this end wesubstitute (17) into (10) solve the 119862

119895119886in terms of the 119862

119895 and

rewrite (17) as

1206011198941

(119909) = 1198621cos [120573

1198941119909] minus cosh [120573

1198942119909]

+ 1198622sin [120573

1198941119909] minus sinh [120573

1198942119909]

where 0 lt 119909 lt ℓ119894

(20)

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] minus cosh [120573

1198942(1 minus 119909)]

+ 1198624sin [120573

1198941(1 minus 119909)] minus sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

(21)

1206011199001

(119909) = 1198625cos [120573

1199001119909] minus cosh [120573

1199002119909]

+ 1198626sin [120573

1199001119909] minus sinh [120573

1199002119909]

where 0 lt 119909 lt ℓ119900

(22)

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] minus cosh [120573

1199002(1 minus 119909)]

+ 1198628sin [120573

1199001(1 minus 119909)] minus sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

(23)

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909]

+ 11986211cosh [120573

1198882119909] + 119862

12sinh [120573

1198882119909]

where 0 lt 119909 lt ℓ119888

(24)

We substitute (20)ndash(24) into (11) (12) (14) and (15) andsolve the resulting nine equations for the 119862

119895in terms of

1198621 1198625 and 119862

9 which represent the input output and

coupling beams respectivelyThen using the remaining threeboundary conditions (13) and (16) we obtain the reducedproblem

MrCr = 0 (25)

where

Mr = [

[

11989211

11989212

11989213

11989221

11989222

11989223

11989231

11989232

11989233

]

]

Cr = (

1198621

1198625

1198629

) (26)

The zero vector 0 in this case is a 3times1 vector and the elements119892119898119899

are functions of the 120573rsquos which are given in (18) Settingthe determinant of Mr equal to zero yields the characteristicequation

Det [Mr] = 0 (27)

which is solved for the natural frequencies 120596119899of the filter

33 Mode Shapes of the Filter Associated with each 120596119899

satisfying (27) is a mode shape To compute this mode shapewe substitute 120596

119899into (18) obtain numerical values for the

120573rsquos and substitute these 120573rsquos into (20)ndash(24) Then we use theboundary conditions to find numerical relations among theunknowns and end up with

11988611198621+ 11988621198625= 0 (28)

Depending on 120596119899 one of the following two cases for the

values of the constants 1198861and 1198862is obtained

Case 1 1198861

= 1198862

rArr 1198621

= minus1198625

rArr primary beamsvibrate out-of-phaseCase 2 119886

1= minus1198862

rArr 1198621

= 1198625

rArr primary beamsvibrate in-phase

By setting 1198621equal to one we find numerical values for all of

the119862rsquos in (20)ndash(24) and hence themode shape correspondingto 120596119899

4 Normalization of Mode Shapes

Because the algebraic system of equations (19) is homo-geneous it follows that if the vector C is a solution of theequation then 120572C is also a solution where 120572 is an arbitraryconstantThis implies that themode shapes are uniquewithina constant We normalize the mode shapes (ie render themunique) by setting [11]

int

1199092

1199091

(120572120601 (119909))2d119909 = 1 (29)

where 120601(119909) is the mode shape Hence the constant 120572 is givenby

120572 =1

radicint1199092

1199091

120601(119909)2d119909

(30)

Considering the topology of the filter structure investigatedin this work and the analysis of the reduced-order modelperformed in [6] we find that the constant 120572

119903associated with

120601119903and 120596

119903is given by

120572119903= (int

ℓ119894

0

1206012

1198941(119909) d119909 + int

1

ℓ119894

1206012

1198942(119909) d119909 + int

ℓ119900

0

1206012

1199001(119909) d119909

+int

1

ℓ119900

1206012

1199002(119909) d119909 + 119879

2

119888int

ℓ119888

0

1206012

119888(119909) d119909)

minus12

(31)

The parameter 1198792

119888= 119860119888119860119901is the ratio of the cross section

area of the coupling beam to that of the primary beam

5 Results and Discussion

We utilize the developed methodology to study the filterfabricated and tested by Bannon et al [9] but withoutconsidering the frequency modification factor or any adjust-ments due to fabrication processes The design parametersdimensions and material properties of the filter needed inthis analysis are obtained from [9] and listed in Table 1

6 Shock and Vibration

51 Closed-Form Expression of the Mode Shapes In this sec-tion we employ the methodology discussed in the precedingsections to present closed-form expressions for the modeshapes To avoid unnecessary repetition we show here detailsof the first mode shape for the filter specified in Table 1In this analysis because the fabrication process involves arelatively long-time (one hour) annealing of the structure [9]we assume that all of the beams are free of residual stressesthat is 119873

119894= 119873119900= 119873119888= 0

The first nondimensional natural frequency of the filter(ie the smallest solution of the characteristic equation (27))is1205961= 22363 (= 9471MHz) Substituting this frequency into

the boundary conditions we end up with 1198621= 1198625 By setting

1198621equal to one we find numerical values for all of the 119862rsquos in

(20)ndash(24) Using the normalization condition given in (31)we obtain 120572

1= 0710 and hence the following normalized

closed-form expression for the first global mode shape of thefilter

1206011198941

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119894

1206011198942

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119900

1206011199002

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = minus 0034 cos [4729119909] minus 0082 sin [4729119909]

minus 0100 cosh [4729119909] + 0082 sinh [4729119909]

where 0 lt 119909 lt ℓ119888

(32)

The normalized expression of the first mode shape of thefilter (32) is shown in Figure 2 In all mode shapes shown inthis paper we use the normalization scheme in the same waydiscussed above In addition in all mode shape figures redlines are obtained from (20) and (22) blue lines are obtainedfrom from (21) and (23) and green lines are obtained fromfrom (24)

52 Validation of the Results We use the finite-elementsoftware ANSYS and the structural elements BEAM3 andBEAM4 to validate our model andmethodologyThe degreesof freedom of elements are constrained such that only thetransverse motion is allowed Using modal analysis solution

Table 1 Filter specifications [9]

Parameter Design valuePrimary resonator length 119871 (120583m) 408Primary resonator width 119887 (120583m) 80Coupling location 119871

119894and 119871

119900(120583m) 408

Coupling beam length 119871119888(120583m) 2035

Coupling beam width 119887119888(120583m) 075

Structural thickness ℎ (120583m) 19Youngrsquos modulus 119864 (GPa) 150Polysilicon density 120588 (kgm3) 2300

we obtained natural frequencies of the filter structure andlisted them in Table 2 Inspecting the results obtained fromthe analytical approach discussed in this paper and thenumerical approach from ANSYS reveals that the agreementbetween them is excellent In addition themode shapes of thefilter structure obtained from both of the approaches (beingin-phase or out-of-phase and their order as illustrated in thenext section) are in complete agreement

53 Relation of the Mode Shapes of the Filter to Those ofSingle Clamped-Clamped Resonators In Table 2 we list thelowest 10 natural frequencies of the filter that is the lowest 10solutions of the characteristic equation (27) considering thespecifications listed in Table 1 and assuming that the structureis free from any residual stresses Due to the effect of the weakcoupling beam shown in Figure 1(a) the natural frequenciesof the single resonator are split into two close frequencies forthe filter as listed in Table 2 one frequency corresponds toan in-phase mode and the other corresponds to an out-of-phasemode In filter terminology the in-phasemode is calleda symmetric mode and the out-of-phase mode is called anantisymmetric mode Next we discuss in more detail thesemode shapes

Figures 2 and 3 show the first and second global modeshapes of the filter respectively The input and outputresonators oscillate in the first mode of a single clamped-clamped beam resonator However they oscillate in-phase inthe first mode of the filter whereas they oscillate 180∘ out-of-phase in the second mode of the filter The frequenciesof the first and second modes of the filter are shifted toslightly smaller and larger values respectively compared tothe frequency of the first mode of a single clamped-clampedresonator as shown in Table 2

The third and fourth mode shapes of the filter areshown in Figures 4 and 5 respectively The input and outputresonators oscillate in the second mode of a single clamped-clamped resonator They oscillate in-phase in the third modeand out-of-phase in the fourth mode The sixth and seventhmode shapes of the filter are shown in Figures 7 and 8respectively The input and output resonators oscillate in thethird mode of a single clamped-clamped resonator but theyoscillate out-of-phase in the sixth mode and in-phase in theseventhmode of the filter It follows from the eighth andninthmode shapes of the filter shown in Figures 9 and 10 that

Shock and Vibration 7

Table 2 Natural frequencies (in MHz) for a filter and a single resonator identical to the resonators of the filter

Filter Single resonator

Mode shape number Natural frequency Mode shape number Natural frequencyAnalytical ANSYS Difference ()

1 9471 9460 0116 1 94752 9479 9469 01053 26015 25909 0407 2 261184 26099 25991 04145 37787 37622 0437 mdash mdash6 51017 50570 0876 3 512027 51385 50930 08858 83574 82345 1470 4 846399 84453 83176 151210 104463 102740 1649 mdash mdash

0 05 10

02

04

06

08

1

12

120601i

xL

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05013

014

015

xL

120601c

(c)

0

05

1

15

1206011

0 025 05 0751

00102030405

xLxL

(d)

Figure 2 First mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

the filter oscillates in the fourth mode of a single clamped-clamped beam

Inspecting Table 2 we note that when the resonators ofthe filter oscillate in a symmetric mode of a single clamped-clamped beam the single beam frequency is split into twofrequencies one slightly higher and the other slightly lowerthan that of the single resonator In contrast when theprimary resonators vibrate in an antisymmetric mode of asingle beam both frequencies of the filter are slightly smallerthan the corresponding frequency of the single beam

For the dimensions listed in Table 1 we note that afterevery four modes there is a mode in which the vibrationsof the primary beams (input and output beams) are verysmall compared to the vibration of the coupling beam Itfollows from Figures 6 and 11 that in the fifth and tenth

modes of the filter the coupling beam oscillates in the firstand secondmodes respectively of a single beamwith slightlyflexible clamping points (the rigidity of the clamping pointsis finite) We observe that these modes appear at relativelylarger frequencies because our model considers transversevibrations of the beams that is the vibration is in thedirection perpendicular to the plane that contains the inputoutput and coupling beams However because the flexuralstiffness 119864119868 of the coupling beam in this paper in the in-plane direction is about six times smaller than its stiffness inthe transverse direction the in-plane modes have relativelysmall frequencies However due to the way these types offilters are actuated (see [6]) the in-plane modes will not beactivated thereby justifying the use of a transverse-vibrationmodel only

8 Shock and Vibration

0 05 1minus12

minus1

minus08

minus06

minus04

minus02

0

xL

120601i

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05minus016

minus008

0

008

016

xL

120601c

(c)

minus12minus06

00612

1206012

0 025 05 0751

00102030405

xLxL

(d)

Figure 3 Second mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 0503

045

06

075

xL

120601c

(c)

minus12minus06

00612

1206013

0 025 05 0751

00102030405

xLxL

(d)

Figure 4 Third mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

54 Coupling Location In this section we study the effect ofchanging the attachment location of the coupling beam to theprimary resonators on the first and second natural frequen-cies of the filter and to be more specific on their averageand difference The average and difference of the first andsecond natural frequencies are indicators of the actual center

frequency and bandwidth respectively of the filter when anelectric signal is applied to the input and output resonators Inthis section we indicate the center frequency and bandwidthof the unactuated filter by 119865

0and BW

0 respectivelyWe focus

on the first and second frequencies because it is common toexcite directly the filter with a frequency in the neighborhood

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

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Shock and Vibration 5

32 Reduction of the Eigenvalue Problem Equation (19) hasnontrivial solutions C if and only if the coefficient matrixM is singular that is the determinant of the 20 times 20

matrix M is zero To reduce the cost of generating thisdeterminant and more importantly obtain a deeper insightinto the relationship among the unknowns we analyticallymanipulate the governing equation (19) To this end wesubstitute (17) into (10) solve the 119862

119895119886in terms of the 119862

119895 and

rewrite (17) as

1206011198941

(119909) = 1198621cos [120573

1198941119909] minus cosh [120573

1198942119909]

+ 1198622sin [120573

1198941119909] minus sinh [120573

1198942119909]

where 0 lt 119909 lt ℓ119894

(20)

1206011198942

(119909) = 1198623cos [120573

1198941(1 minus 119909)] minus cosh [120573

1198942(1 minus 119909)]

+ 1198624sin [120573

1198941(1 minus 119909)] minus sinh [120573

1198942(1 minus 119909)]

where ℓ119894lt 119909 lt 1

(21)

1206011199001

(119909) = 1198625cos [120573

1199001119909] minus cosh [120573

1199002119909]

+ 1198626sin [120573

1199001119909] minus sinh [120573

1199002119909]

where 0 lt 119909 lt ℓ119900

(22)

1206011199002

(119909) = 1198627cos [120573

1199001(1 minus 119909)] minus cosh [120573

1199002(1 minus 119909)]

+ 1198628sin [120573

1199001(1 minus 119909)] minus sinh [120573

1199002(1 minus 119909)]

where ℓ119900lt 119909 lt 1

(23)

120601119888(119909) = 119862

9cos [120573

1198881119909] + 119862

10sin [120573

1198881119909]

+ 11986211cosh [120573

1198882119909] + 119862

12sinh [120573

1198882119909]

where 0 lt 119909 lt ℓ119888

(24)

We substitute (20)ndash(24) into (11) (12) (14) and (15) andsolve the resulting nine equations for the 119862

119895in terms of

1198621 1198625 and 119862

9 which represent the input output and

coupling beams respectivelyThen using the remaining threeboundary conditions (13) and (16) we obtain the reducedproblem

MrCr = 0 (25)

where

Mr = [

[

11989211

11989212

11989213

11989221

11989222

11989223

11989231

11989232

11989233

]

]

Cr = (

1198621

1198625

1198629

) (26)

The zero vector 0 in this case is a 3times1 vector and the elements119892119898119899

are functions of the 120573rsquos which are given in (18) Settingthe determinant of Mr equal to zero yields the characteristicequation

Det [Mr] = 0 (27)

which is solved for the natural frequencies 120596119899of the filter

33 Mode Shapes of the Filter Associated with each 120596119899

satisfying (27) is a mode shape To compute this mode shapewe substitute 120596

119899into (18) obtain numerical values for the

120573rsquos and substitute these 120573rsquos into (20)ndash(24) Then we use theboundary conditions to find numerical relations among theunknowns and end up with

11988611198621+ 11988621198625= 0 (28)

Depending on 120596119899 one of the following two cases for the

values of the constants 1198861and 1198862is obtained

Case 1 1198861

= 1198862

rArr 1198621

= minus1198625

rArr primary beamsvibrate out-of-phaseCase 2 119886

1= minus1198862

rArr 1198621

= 1198625

rArr primary beamsvibrate in-phase

By setting 1198621equal to one we find numerical values for all of

the119862rsquos in (20)ndash(24) and hence themode shape correspondingto 120596119899

4 Normalization of Mode Shapes

Because the algebraic system of equations (19) is homo-geneous it follows that if the vector C is a solution of theequation then 120572C is also a solution where 120572 is an arbitraryconstantThis implies that themode shapes are uniquewithina constant We normalize the mode shapes (ie render themunique) by setting [11]

int

1199092

1199091

(120572120601 (119909))2d119909 = 1 (29)

where 120601(119909) is the mode shape Hence the constant 120572 is givenby

120572 =1

radicint1199092

1199091

120601(119909)2d119909

(30)

Considering the topology of the filter structure investigatedin this work and the analysis of the reduced-order modelperformed in [6] we find that the constant 120572

119903associated with

120601119903and 120596

119903is given by

120572119903= (int

ℓ119894

0

1206012

1198941(119909) d119909 + int

1

ℓ119894

1206012

1198942(119909) d119909 + int

ℓ119900

0

1206012

1199001(119909) d119909

+int

1

ℓ119900

1206012

1199002(119909) d119909 + 119879

2

119888int

ℓ119888

0

1206012

119888(119909) d119909)

minus12

(31)

The parameter 1198792

119888= 119860119888119860119901is the ratio of the cross section

area of the coupling beam to that of the primary beam

5 Results and Discussion

We utilize the developed methodology to study the filterfabricated and tested by Bannon et al [9] but withoutconsidering the frequency modification factor or any adjust-ments due to fabrication processes The design parametersdimensions and material properties of the filter needed inthis analysis are obtained from [9] and listed in Table 1

6 Shock and Vibration

51 Closed-Form Expression of the Mode Shapes In this sec-tion we employ the methodology discussed in the precedingsections to present closed-form expressions for the modeshapes To avoid unnecessary repetition we show here detailsof the first mode shape for the filter specified in Table 1In this analysis because the fabrication process involves arelatively long-time (one hour) annealing of the structure [9]we assume that all of the beams are free of residual stressesthat is 119873

119894= 119873119900= 119873119888= 0

The first nondimensional natural frequency of the filter(ie the smallest solution of the characteristic equation (27))is1205961= 22363 (= 9471MHz) Substituting this frequency into

the boundary conditions we end up with 1198621= 1198625 By setting

1198621equal to one we find numerical values for all of the 119862rsquos in

(20)ndash(24) Using the normalization condition given in (31)we obtain 120572

1= 0710 and hence the following normalized

closed-form expression for the first global mode shape of thefilter

1206011198941

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119894

1206011198942

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119900

1206011199002

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = minus 0034 cos [4729119909] minus 0082 sin [4729119909]

minus 0100 cosh [4729119909] + 0082 sinh [4729119909]

where 0 lt 119909 lt ℓ119888

(32)

The normalized expression of the first mode shape of thefilter (32) is shown in Figure 2 In all mode shapes shown inthis paper we use the normalization scheme in the same waydiscussed above In addition in all mode shape figures redlines are obtained from (20) and (22) blue lines are obtainedfrom from (21) and (23) and green lines are obtained fromfrom (24)

52 Validation of the Results We use the finite-elementsoftware ANSYS and the structural elements BEAM3 andBEAM4 to validate our model andmethodologyThe degreesof freedom of elements are constrained such that only thetransverse motion is allowed Using modal analysis solution

Table 1 Filter specifications [9]

Parameter Design valuePrimary resonator length 119871 (120583m) 408Primary resonator width 119887 (120583m) 80Coupling location 119871

119894and 119871

119900(120583m) 408

Coupling beam length 119871119888(120583m) 2035

Coupling beam width 119887119888(120583m) 075

Structural thickness ℎ (120583m) 19Youngrsquos modulus 119864 (GPa) 150Polysilicon density 120588 (kgm3) 2300

we obtained natural frequencies of the filter structure andlisted them in Table 2 Inspecting the results obtained fromthe analytical approach discussed in this paper and thenumerical approach from ANSYS reveals that the agreementbetween them is excellent In addition themode shapes of thefilter structure obtained from both of the approaches (beingin-phase or out-of-phase and their order as illustrated in thenext section) are in complete agreement

53 Relation of the Mode Shapes of the Filter to Those ofSingle Clamped-Clamped Resonators In Table 2 we list thelowest 10 natural frequencies of the filter that is the lowest 10solutions of the characteristic equation (27) considering thespecifications listed in Table 1 and assuming that the structureis free from any residual stresses Due to the effect of the weakcoupling beam shown in Figure 1(a) the natural frequenciesof the single resonator are split into two close frequencies forthe filter as listed in Table 2 one frequency corresponds toan in-phase mode and the other corresponds to an out-of-phasemode In filter terminology the in-phasemode is calleda symmetric mode and the out-of-phase mode is called anantisymmetric mode Next we discuss in more detail thesemode shapes

Figures 2 and 3 show the first and second global modeshapes of the filter respectively The input and outputresonators oscillate in the first mode of a single clamped-clamped beam resonator However they oscillate in-phase inthe first mode of the filter whereas they oscillate 180∘ out-of-phase in the second mode of the filter The frequenciesof the first and second modes of the filter are shifted toslightly smaller and larger values respectively compared tothe frequency of the first mode of a single clamped-clampedresonator as shown in Table 2

The third and fourth mode shapes of the filter areshown in Figures 4 and 5 respectively The input and outputresonators oscillate in the second mode of a single clamped-clamped resonator They oscillate in-phase in the third modeand out-of-phase in the fourth mode The sixth and seventhmode shapes of the filter are shown in Figures 7 and 8respectively The input and output resonators oscillate in thethird mode of a single clamped-clamped resonator but theyoscillate out-of-phase in the sixth mode and in-phase in theseventhmode of the filter It follows from the eighth andninthmode shapes of the filter shown in Figures 9 and 10 that

Shock and Vibration 7

Table 2 Natural frequencies (in MHz) for a filter and a single resonator identical to the resonators of the filter

Filter Single resonator

Mode shape number Natural frequency Mode shape number Natural frequencyAnalytical ANSYS Difference ()

1 9471 9460 0116 1 94752 9479 9469 01053 26015 25909 0407 2 261184 26099 25991 04145 37787 37622 0437 mdash mdash6 51017 50570 0876 3 512027 51385 50930 08858 83574 82345 1470 4 846399 84453 83176 151210 104463 102740 1649 mdash mdash

0 05 10

02

04

06

08

1

12

120601i

xL

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05013

014

015

xL

120601c

(c)

0

05

1

15

1206011

0 025 05 0751

00102030405

xLxL

(d)

Figure 2 First mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

the filter oscillates in the fourth mode of a single clamped-clamped beam

Inspecting Table 2 we note that when the resonators ofthe filter oscillate in a symmetric mode of a single clamped-clamped beam the single beam frequency is split into twofrequencies one slightly higher and the other slightly lowerthan that of the single resonator In contrast when theprimary resonators vibrate in an antisymmetric mode of asingle beam both frequencies of the filter are slightly smallerthan the corresponding frequency of the single beam

For the dimensions listed in Table 1 we note that afterevery four modes there is a mode in which the vibrationsof the primary beams (input and output beams) are verysmall compared to the vibration of the coupling beam Itfollows from Figures 6 and 11 that in the fifth and tenth

modes of the filter the coupling beam oscillates in the firstand secondmodes respectively of a single beamwith slightlyflexible clamping points (the rigidity of the clamping pointsis finite) We observe that these modes appear at relativelylarger frequencies because our model considers transversevibrations of the beams that is the vibration is in thedirection perpendicular to the plane that contains the inputoutput and coupling beams However because the flexuralstiffness 119864119868 of the coupling beam in this paper in the in-plane direction is about six times smaller than its stiffness inthe transverse direction the in-plane modes have relativelysmall frequencies However due to the way these types offilters are actuated (see [6]) the in-plane modes will not beactivated thereby justifying the use of a transverse-vibrationmodel only

8 Shock and Vibration

0 05 1minus12

minus1

minus08

minus06

minus04

minus02

0

xL

120601i

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05minus016

minus008

0

008

016

xL

120601c

(c)

minus12minus06

00612

1206012

0 025 05 0751

00102030405

xLxL

(d)

Figure 3 Second mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 0503

045

06

075

xL

120601c

(c)

minus12minus06

00612

1206013

0 025 05 0751

00102030405

xLxL

(d)

Figure 4 Third mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

54 Coupling Location In this section we study the effect ofchanging the attachment location of the coupling beam to theprimary resonators on the first and second natural frequen-cies of the filter and to be more specific on their averageand difference The average and difference of the first andsecond natural frequencies are indicators of the actual center

frequency and bandwidth respectively of the filter when anelectric signal is applied to the input and output resonators Inthis section we indicate the center frequency and bandwidthof the unactuated filter by 119865

0and BW

0 respectivelyWe focus

on the first and second frequencies because it is common toexcite directly the filter with a frequency in the neighborhood

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

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6 Shock and Vibration

51 Closed-Form Expression of the Mode Shapes In this sec-tion we employ the methodology discussed in the precedingsections to present closed-form expressions for the modeshapes To avoid unnecessary repetition we show here detailsof the first mode shape for the filter specified in Table 1In this analysis because the fabrication process involves arelatively long-time (one hour) annealing of the structure [9]we assume that all of the beams are free of residual stressesthat is 119873

119894= 119873119900= 119873119888= 0

The first nondimensional natural frequency of the filter(ie the smallest solution of the characteristic equation (27))is1205961= 22363 (= 9471MHz) Substituting this frequency into

the boundary conditions we end up with 1198621= 1198625 By setting

1198621equal to one we find numerical values for all of the 119862rsquos in

(20)ndash(24) Using the normalization condition given in (31)we obtain 120572

1= 0710 and hence the following normalized

closed-form expression for the first global mode shape of thefilter

1206011198941

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119894

1206011198942

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119894lt 119909 lt 1

1206011199001

(119909) = 0710 cos [4729119909] minus cosh [4729119909]

minus 0702 sin [4729119909] minus sinh [4729119909]

where 0 lt 119909 lt ℓ119900

1206011199002

(119909) = 0707 cos [4729 (1 minus 119909)] minus cosh [4729 (1 minus 119909)]

minus0695 sin [4729 (1minus119909)]minussinh [4729 (1minus119909)]

where ℓ119900lt 119909 lt 1

120601119888(119909) = minus 0034 cos [4729119909] minus 0082 sin [4729119909]

minus 0100 cosh [4729119909] + 0082 sinh [4729119909]

where 0 lt 119909 lt ℓ119888

(32)

The normalized expression of the first mode shape of thefilter (32) is shown in Figure 2 In all mode shapes shown inthis paper we use the normalization scheme in the same waydiscussed above In addition in all mode shape figures redlines are obtained from (20) and (22) blue lines are obtainedfrom from (21) and (23) and green lines are obtained fromfrom (24)

52 Validation of the Results We use the finite-elementsoftware ANSYS and the structural elements BEAM3 andBEAM4 to validate our model andmethodologyThe degreesof freedom of elements are constrained such that only thetransverse motion is allowed Using modal analysis solution

Table 1 Filter specifications [9]

Parameter Design valuePrimary resonator length 119871 (120583m) 408Primary resonator width 119887 (120583m) 80Coupling location 119871

119894and 119871

119900(120583m) 408

Coupling beam length 119871119888(120583m) 2035

Coupling beam width 119887119888(120583m) 075

Structural thickness ℎ (120583m) 19Youngrsquos modulus 119864 (GPa) 150Polysilicon density 120588 (kgm3) 2300

we obtained natural frequencies of the filter structure andlisted them in Table 2 Inspecting the results obtained fromthe analytical approach discussed in this paper and thenumerical approach from ANSYS reveals that the agreementbetween them is excellent In addition themode shapes of thefilter structure obtained from both of the approaches (beingin-phase or out-of-phase and their order as illustrated in thenext section) are in complete agreement

53 Relation of the Mode Shapes of the Filter to Those ofSingle Clamped-Clamped Resonators In Table 2 we list thelowest 10 natural frequencies of the filter that is the lowest 10solutions of the characteristic equation (27) considering thespecifications listed in Table 1 and assuming that the structureis free from any residual stresses Due to the effect of the weakcoupling beam shown in Figure 1(a) the natural frequenciesof the single resonator are split into two close frequencies forthe filter as listed in Table 2 one frequency corresponds toan in-phase mode and the other corresponds to an out-of-phasemode In filter terminology the in-phasemode is calleda symmetric mode and the out-of-phase mode is called anantisymmetric mode Next we discuss in more detail thesemode shapes

Figures 2 and 3 show the first and second global modeshapes of the filter respectively The input and outputresonators oscillate in the first mode of a single clamped-clamped beam resonator However they oscillate in-phase inthe first mode of the filter whereas they oscillate 180∘ out-of-phase in the second mode of the filter The frequenciesof the first and second modes of the filter are shifted toslightly smaller and larger values respectively compared tothe frequency of the first mode of a single clamped-clampedresonator as shown in Table 2

The third and fourth mode shapes of the filter areshown in Figures 4 and 5 respectively The input and outputresonators oscillate in the second mode of a single clamped-clamped resonator They oscillate in-phase in the third modeand out-of-phase in the fourth mode The sixth and seventhmode shapes of the filter are shown in Figures 7 and 8respectively The input and output resonators oscillate in thethird mode of a single clamped-clamped resonator but theyoscillate out-of-phase in the sixth mode and in-phase in theseventhmode of the filter It follows from the eighth andninthmode shapes of the filter shown in Figures 9 and 10 that

Shock and Vibration 7

Table 2 Natural frequencies (in MHz) for a filter and a single resonator identical to the resonators of the filter

Filter Single resonator

Mode shape number Natural frequency Mode shape number Natural frequencyAnalytical ANSYS Difference ()

1 9471 9460 0116 1 94752 9479 9469 01053 26015 25909 0407 2 261184 26099 25991 04145 37787 37622 0437 mdash mdash6 51017 50570 0876 3 512027 51385 50930 08858 83574 82345 1470 4 846399 84453 83176 151210 104463 102740 1649 mdash mdash

0 05 10

02

04

06

08

1

12

120601i

xL

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05013

014

015

xL

120601c

(c)

0

05

1

15

1206011

0 025 05 0751

00102030405

xLxL

(d)

Figure 2 First mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

the filter oscillates in the fourth mode of a single clamped-clamped beam

Inspecting Table 2 we note that when the resonators ofthe filter oscillate in a symmetric mode of a single clamped-clamped beam the single beam frequency is split into twofrequencies one slightly higher and the other slightly lowerthan that of the single resonator In contrast when theprimary resonators vibrate in an antisymmetric mode of asingle beam both frequencies of the filter are slightly smallerthan the corresponding frequency of the single beam

For the dimensions listed in Table 1 we note that afterevery four modes there is a mode in which the vibrationsof the primary beams (input and output beams) are verysmall compared to the vibration of the coupling beam Itfollows from Figures 6 and 11 that in the fifth and tenth

modes of the filter the coupling beam oscillates in the firstand secondmodes respectively of a single beamwith slightlyflexible clamping points (the rigidity of the clamping pointsis finite) We observe that these modes appear at relativelylarger frequencies because our model considers transversevibrations of the beams that is the vibration is in thedirection perpendicular to the plane that contains the inputoutput and coupling beams However because the flexuralstiffness 119864119868 of the coupling beam in this paper in the in-plane direction is about six times smaller than its stiffness inthe transverse direction the in-plane modes have relativelysmall frequencies However due to the way these types offilters are actuated (see [6]) the in-plane modes will not beactivated thereby justifying the use of a transverse-vibrationmodel only

8 Shock and Vibration

0 05 1minus12

minus1

minus08

minus06

minus04

minus02

0

xL

120601i

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05minus016

minus008

0

008

016

xL

120601c

(c)

minus12minus06

00612

1206012

0 025 05 0751

00102030405

xLxL

(d)

Figure 3 Second mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 0503

045

06

075

xL

120601c

(c)

minus12minus06

00612

1206013

0 025 05 0751

00102030405

xLxL

(d)

Figure 4 Third mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

54 Coupling Location In this section we study the effect ofchanging the attachment location of the coupling beam to theprimary resonators on the first and second natural frequen-cies of the filter and to be more specific on their averageand difference The average and difference of the first andsecond natural frequencies are indicators of the actual center

frequency and bandwidth respectively of the filter when anelectric signal is applied to the input and output resonators Inthis section we indicate the center frequency and bandwidthof the unactuated filter by 119865

0and BW

0 respectivelyWe focus

on the first and second frequencies because it is common toexcite directly the filter with a frequency in the neighborhood

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

Table 2 Natural frequencies (in MHz) for a filter and a single resonator identical to the resonators of the filter

Filter Single resonator

Mode shape number Natural frequency Mode shape number Natural frequencyAnalytical ANSYS Difference ()

1 9471 9460 0116 1 94752 9479 9469 01053 26015 25909 0407 2 261184 26099 25991 04145 37787 37622 0437 mdash mdash6 51017 50570 0876 3 512027 51385 50930 08858 83574 82345 1470 4 846399 84453 83176 151210 104463 102740 1649 mdash mdash

0 05 10

02

04

06

08

1

12

120601i

xL

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05013

014

015

xL

120601c

(c)

0

05

1

15

1206011

0 025 05 0751

00102030405

xLxL

(d)

Figure 2 First mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

the filter oscillates in the fourth mode of a single clamped-clamped beam

Inspecting Table 2 we note that when the resonators ofthe filter oscillate in a symmetric mode of a single clamped-clamped beam the single beam frequency is split into twofrequencies one slightly higher and the other slightly lowerthan that of the single resonator In contrast when theprimary resonators vibrate in an antisymmetric mode of asingle beam both frequencies of the filter are slightly smallerthan the corresponding frequency of the single beam

For the dimensions listed in Table 1 we note that afterevery four modes there is a mode in which the vibrationsof the primary beams (input and output beams) are verysmall compared to the vibration of the coupling beam Itfollows from Figures 6 and 11 that in the fifth and tenth

modes of the filter the coupling beam oscillates in the firstand secondmodes respectively of a single beamwith slightlyflexible clamping points (the rigidity of the clamping pointsis finite) We observe that these modes appear at relativelylarger frequencies because our model considers transversevibrations of the beams that is the vibration is in thedirection perpendicular to the plane that contains the inputoutput and coupling beams However because the flexuralstiffness 119864119868 of the coupling beam in this paper in the in-plane direction is about six times smaller than its stiffness inthe transverse direction the in-plane modes have relativelysmall frequencies However due to the way these types offilters are actuated (see [6]) the in-plane modes will not beactivated thereby justifying the use of a transverse-vibrationmodel only

8 Shock and Vibration

0 05 1minus12

minus1

minus08

minus06

minus04

minus02

0

xL

120601i

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05minus016

minus008

0

008

016

xL

120601c

(c)

minus12minus06

00612

1206012

0 025 05 0751

00102030405

xLxL

(d)

Figure 3 Second mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 0503

045

06

075

xL

120601c

(c)

minus12minus06

00612

1206013

0 025 05 0751

00102030405

xLxL

(d)

Figure 4 Third mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

54 Coupling Location In this section we study the effect ofchanging the attachment location of the coupling beam to theprimary resonators on the first and second natural frequen-cies of the filter and to be more specific on their averageand difference The average and difference of the first andsecond natural frequencies are indicators of the actual center

frequency and bandwidth respectively of the filter when anelectric signal is applied to the input and output resonators Inthis section we indicate the center frequency and bandwidthof the unactuated filter by 119865

0and BW

0 respectivelyWe focus

on the first and second frequencies because it is common toexcite directly the filter with a frequency in the neighborhood

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

0 05 1minus12

minus1

minus08

minus06

minus04

minus02

0

xL

120601i

(a)

0 05 10

02

04

06

08

1

12

xL

120601o

(b)

0 025 05minus016

minus008

0

008

016

xL

120601c

(c)

minus12minus06

00612

1206012

0 025 05 0751

00102030405

xLxL

(d)

Figure 3 Second mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 0503

045

06

075

xL

120601c

(c)

minus12minus06

00612

1206013

0 025 05 0751

00102030405

xLxL

(d)

Figure 4 Third mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

54 Coupling Location In this section we study the effect ofchanging the attachment location of the coupling beam to theprimary resonators on the first and second natural frequen-cies of the filter and to be more specific on their averageand difference The average and difference of the first andsecond natural frequencies are indicators of the actual center

frequency and bandwidth respectively of the filter when anelectric signal is applied to the input and output resonators Inthis section we indicate the center frequency and bandwidthof the unactuated filter by 119865

0and BW

0 respectivelyWe focus

on the first and second frequencies because it is common toexcite directly the filter with a frequency in the neighborhood

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

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VLSI Design

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Shock and Vibration

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DistributedSensor Networks

International Journal of

Shock and Vibration 9

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus04

minus02

0

02

04

xL

120601c

(c)

minus12minus06

00612

1206014

0 025 05 0751

00102030405

xLxL

(d)

Figure 5 Fourth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus02

minus01

0

01

02

xL

120601i

(a)

0 05 1minus02

minus01

0

01

02

xL

120601o

(b)

0 025 050

2

4

6

8

xL

120601c

(c)

02468

1206015

0 025 05 0751

00102030405

xLxL

(d)

Figure 6 Fifth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

of these frequencies In case highermodes are excited directlythe center frequency and bandwidth of the filter will changeFigure 12 shows variations of the first and second naturalfrequencies of the filter with the attachment location Whenthe coupling beam is attached to the clamping points of theprimary beams the frequencies of the filter are the same

as the frequencies of a single resonator This is expectedbecause in this case the two primary resonators are notactually coupled As the attachment location moves awayfrom the clamping point towards the middle of the primarybeams the first natural frequency decreases and the secondnatural frequency increases until they reach their extrema

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus06

minus03

0

03

06

xL

120601c

(c)

minus12minus06

00612

1206016

0 025 05 0751

00102030405

xLxL

(d)

Figure 7 Sixth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus1

minus05

0

05

xL

120601c

(c)

minus12minus06

00612

1206017

0 025 05 0751

00102030405

xLxL

(d)

Figure 8 Seventh mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

at the middle of the primary beams As the coupling beammoves away from the middle to the other clamping pointsof the primary resonators the first and second frequenciesmove closer to each other until they become equal to thefundamental natural frequency of a single resonator at theclamping points

Variation of the center frequency 1198650of the filter is shown

in Figure 13 We note that this variation is insensitive tothe attachment location it varies only 23 kHz through thewhole range On the other hand the bandwidth BW

0of

the filter shown in Figure 14 changes significantly as thecoupling location sweeps the whole length of the primary

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05minus16

minus08

0

08

16

xL

120601c

(c)

minus16minus08

00816

1206018

0 025 05 0751

00102030405

xLxL

(d)

Figure 9 Eighth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601i

(a)

0 05 1minus12

minus08

minus04

0

04

08

12

xL

120601o

(b)

0 025 05

minus04

0

04

08

xL

120601c

(c)

minus12minus06

00612

0 025 05 0751

00102030405

xLxL

1206019

(d)

Figure 10 Ninth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

beams The widest bandwidth is realized when the couplingbeam is attached to the midpoints of the primary resonatorsand it becomes narrower as the attachment location movestowards the clamping points Because wireless systems trans-mit and receive signals within a narrow bandwidth allocated

to the user in a communications environment crowded withinterferers [12] attaching the coupling beam close to theclamping points is desirable because it yields a narrow band-width even though the center frequency is not a maximumthere

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

0 05 1minus04

minus02

0

02

04

xL

120601i

(a)

0 05 1minus04

minus02

0

02

04

xL

120601o

(b)

0 025 05minus8

minus4

0

4

8

xL

120601c

(c)

minus8minus4

048

12060110

0 025 05 0751

00102030405

xLxL

(d)

Figure 11 Tenth mode shape of the filter (a) input (b) output (c) coupling beams and (d) combination of (a) (b) and (c)

0 10 20 30 40

92

94

96

98

Nat

ural

freq

uenc

y120596i

(MH

z)

Coupling location (120583m)

Figure 12 Variation of the first (dashed line) and second (solid line)natural frequencies of the filter with the attachment location

The behavior of the bandwidth in Figure 14 is explainedusing the following equation [2 9]

BW0= 1198650

119896cs119896cc119896cl

(33)

where 119896cs is the coupling spring constant 119896cc is the nor-malized coupling coefficient between the resonator tanks for

0 10 20 30 40

9477

9476

9475

F0

(MH

z)

Coupling location (120583m)

Figure 13 Variation of the center frequency of the unactuated filterwith the attachment location

a given filter type and 119896cl is the resonator stiffness at thecoupling location For a given type of filter and specifiedlength and width of the coupling beam the only importantfactor that determines the bandwidth is 119896cl So that forlocations close to the clamping ends the stiffness of theprimary beam is high due to the shortness of the resonatorresulting in a narrow bandwidth But when the attachment

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

600

450

300

150

00 10 20 30 40

Coupling location (120583m)

BW0

(kH

z)

Figure 14 Variation of the bandwidth of the unactuated filter withthe attachment location

locations are at the midpoints of the primary resonators thestiffness is the smallest (because the length has the largestvalue) leading to the widest bandwidth

Based on Figures 13 and 14 we plot the quality factor1198760filter of the filter using the following equation

1198760filter =

1198650

BW0

(34)

and show the result in Figure 15 It is clear that as theattachment location moves closer to the middle of theprimary beams 119876

0filter becomes smaller and smaller and itis the highest close to the clamping points Figures 14 and 15are in qualitative agreement with the simulation results of thequality factor and bandwidth of the actual filter reported in[9]

It is important to mention that the actual filter centerfrequency bandwidth and quality factor are obtained whenthe filter is excited by proper electric signals applied tothe primary resonators However the designer can obtainqualitative ideas about the effect of a variety of parameterson the filter performance from Figures 12ndash15 for unactuatedfilters A quantitative description needs a high-level modelin which the interaction between the electric force and thedeflection of the beams is taken into consideration

Next we discuss the effect of the axial load on the firstand second natural frequencies of the filter and to be morespecific on the center frequency 119865

0and bandwidth BW

0

of the unactuated filter For the filter structure under studybecause the primary beams are in direct contact with thesubstrate the effect of temperature variation on these beamsis much more than its effect on the coupling beam So thatfor the purpose of analysis and discussion here we assumethat the coupling beam is free of any stress and vary theaxial stress in the primary beams Increasing the tensile load

(negative values of the axial load) shifts the center frequencyof the unactuated filter 119865

0to higher values as shown in

Figure 16 But in the case of increasing the compressiveload (positive values of the axial load) the center frequencyshifts to smaller values and eventually it reaches zero whenthe primary beams buckle Moreover increasing the tensileload produces a wider filter bandwidth BW

0 as shown in

Figure 17 But more interestingly as the compressive loadincreases the bandwidth BW

0decreases slightly before it

expands for large compressive loadsWe have assumed ideal clamped-clamped boundary con-

ditions in this paper In reality these ends have finite rigidityand some structures have step-up type [13] or special design[9] supports Consequently the actual natural frequencies arelower than those of amicrobeamwith ideal clamped-clampedends Bannon et al [9] and Wong [3] used a frequency mod-ification factor obtained using the finite-element package toaccount for the difference between the natural frequencies ofideal clamped-clamped beams and those of the actual beamsthey have fabricated

6 Conclusions

We solved the linear undamped and unforced vibrationproblem of micromechanical filters and obtained closed-form expressions for their natural frequencies and modeshapes The model described in this work treats the filter asa distributed-parameter system For a micromechanical filtermade of two clamped-clamped beam resonators connectedvia a coupling beam we solved a boundary-value problem(BVP) composed of five equations and twenty boundaryconditions for its natural frequencies and mode shapesInstead of dealing with such a large system of equations wesuggested a method to reduce the problem to a set of threelinear homogeneous algebraic equations for three constantsand the frequencies In addition to time and cost savings dueto dealing with smaller system the relation between the filterstructure parameters became clearer Setting the determinantof the reduced coefficient matrix equal to zero we obtainedthe characteristic equation which we solved for the naturalfrequencies Then we computed the constants and hence themode shapes

We compared our analytical procedure investigated inthis paper with a numerical approach by using the finite-element package ANSYS The agreement was excellentvalidating our methodology The closed-form expressionsobtained in this work are easier to handle more robustand accurate They are especially valuable in developingreduced-order models for the nonlinear static and dynamiccharacteristics of filters using the Galerkin procedure

Due to the fact that the coupling beam is weak thenatural frequencies of the single resonator are split into twoclose frequencies for the filter one frequency correspondsto an in-phase mode and the other corresponds to an out-of-phase mode Moreover we discussed the effect of theattachment position on the bandwidth and quality factorof the unactuated filter and found that our results are inqualitative agreement with published results We noted that

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

010

100

1000

10000

100000

1000000

10 20 30 40

Q0

filte

r

Coupling location (120583m)

Figure 15 Variation of the quality factor of the unactuated filterwiththe attachment location

12

10

8

6

4

F0

(MH

z)

Axial load (N)

Tension Compression

minus0015 minus001 minus0005 0 0005 001 0015

Figure 16 Variation of the center frequency of the unactuated filterwith axial load in the primary beams

the bandwidth is very sensitive to the coupling locationwhereas the center frequency is not Although the results inthis paper are developed for an unactuated filter they givethe designer a qualitative idea about the effect of a varietyof parameters on its performance Actual filter specifica-tions (center frequency bandwidth and quality factor) areobtained when electric signals are applied to the input andoutput resonators The latter is the subject of other works[6 14] In this work we focused on the first two frequenciesof the filter and discussed its center frequency and bandwidthbecause it is usually excited directly with a frequency in

16

14

12

10

8

Tension Compression

Axial load (N)minus0015 minus001 minus0005 0 0005 001 0015

BW0

(MH

z)Figure 17 Variation of the bandwidth of the unactuated filter withaxial load in the primary beams

the neighborhood of these frequencies In the case wherehigher modes are directly excited the definition of the centerfrequency and bandwidth of the filter should be modified

In addition we studied the effect of axial loads appliedto the beam resonators on the natural frequencies andbandwidth of the filter More interestingly we found that asthe compressive load increases from no axial load case thebandwidth of the unactuated filter becomes narrower beforeit becomes larger for large compressive loads

Finally we emphasize that although the model presentedin this paper for the vibration problem is restricted tomicromechanical filters made of two clamped-clamped beamresonators connected via a coupling beam they can be easilymodified and adjusted to model any mechanically coupledmicrobeam arrays such as but not limited to free-freemicrobeamarray higher-order filters and arrayswith bridgesbetween nonadjacent resonators [15 16]

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] V B Chivukula and J F Rhoads ldquoMicroelectromechanicalbandpass filters based on cyclic coupling architecturesrdquo Journalof Sound and Vibration vol 329 no 20 pp 4313ndash4332 2010

[2] K Wang and C T-C Nguyen ldquoHigh-order medium frequencymicromechanical electronic filtersrdquo Journal of Microelectrome-chanical Systems vol 8 no 4 pp 534ndash557 1999

[3] A-C Wong VHF microelectromechanical mixer-filters [PhDthesis] University of Michigan 2001

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15

[4] M U DemirciMicromechanical composite array resonators andfilters for communications [PhD thesis] University ofMichigan2005

[5] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada August 2004

[6] B K Hammad E M Abdel-Rahman and A H NayfehldquoModeling and analysis of electrostaticMEMSfiltersrdquoNonlinearDynamics vol 60 no 3 pp 385ndash401 2010

[7] A Vyas and A K Bajaj ldquoMicroresonators based on 12 internalresonancerdquo in Proceedings of the ASME International Mechani-cal Engineering Congress and Exposition Paper IMECE2005-81701 Orlando Fla USA 2005

[8] A Vyas D Peroulis and A K Bajaj ldquoDynamics of a nonlin-ear microresonator based on resonantly interacting flexural-torsionalmodesrdquoNonlinearDynamics vol 54 no 1-2 pp 31ndash522008

[9] F D Bannon J R Clark and C T-C Nguyen ldquoNguyen high-QHF microelectromechanical filtersrdquo IEEE Journal of Solid-StateCircuits vol 35 no 4 pp 512ndash526 2000

[10] B K Hammad E M Abdel-Rahman and A H NayfehldquoCharacterization of a tunable MEMS RF filterrdquo in Proceedingsof the ASME International Mechanical Engineering Congress andExposition Paper IMECE2006-14136 Chicago Ill USA 2006

[11] L Meirovitch Fundamentals of Vibrations McGraw-Hill NewYork NY USA 1st edition 2001

[12] H J de los Santos RF MEMS Circuit Design for Wireless Com-munications Artech House Boston Mass USA 1st edition2002

[13] Q Meng M Mehregany and R L Mullen ldquoTheoreticalmodeling of microfabricated beams with elastically restrainedsupportsrdquo Journal of Microelectromechanical Systems vol 2 no3 pp 128ndash137 1993

[14] B K Hammad A H Nayfeh and E M Abdel-RahmanldquoA study of subharmonic excitation of mechanically coupledmicrobeams for filtrationrdquo in Proceedings of the ASME Interna-tional Mechanical Engineering Congress and Exposition (IMECErsquo08) Paper IMECE2008-66728 pp 371ndash384 Boston MassUSA November 2008

[15] S-S Li M U Demirci Y-W Lin Z Ren and C T-CNguyen ldquoBridged micromechanical filtersrdquo in Proceedings ofthe IEEE International Ultrasonics Ferroelectrics and FrequencyControl Joint 50Anniversary Conference pp 280ndash286MontrealCanada 2004

[16] S S Li Y -W Lin Z Ren and C T-C Nguyen ldquoSelf-switchingvibrating micromechanical filter bankrdquo in Proceedings of theJoint IEEE International Frequency ControlPrecision Time andTime Interval Symposium pp 135ndash141 Vancouver Canada2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of