Vibration Analysis in Aerospace

ANALYSIS OF THE EFFECTS OF LAMINATE DEPTH AND MATERIALPROPERTIES ON THE DAMPING ASSOCIATED WITH LAYERED STRUCTURES

IN A PRESSURIZED ENVIRONMENT

Vincent O. S. Olunloyo1, Olatunde Damisa2, Charles A. Osheku2, Ayo A. Oyediran3

1Department of Systems Engineering, Faculty of Engineering, University of Lagos

2Department of Mechanical Engineering, Faculty of Engineering, University of Lagos

3AYT Research Corp., McLean VA. 22102, USA

E-mail: [email protected]

Received September 2008, Accepted February 14, 2009

No. 06-CSME-10, E.I.C. Accession 2929

ABSTRACT

In aerodynamic and machine structures, one of the effective ways of dissipating unwantedvibration or noise is to exploit the occurrence of slip at the interface of structural laminates wheresuch members are held together in a pressurised environment. The analysis and experimentalinvestigation of such laminates have established that when subjected to either static or dynamicloading, non-uniformity in interface pressure can have significant effect on both the energydissipation and the logarithmic damping decrement associated with the mechanism of slip damping.Such behaviour can in fact be effectively exploited to increase the level of damping available in sucha mechanism. What has however not been examined is to what extent is the energy dissipationaffected by the relative sizes or the material properties of the upper and lower laminates? In thispaper the analysis is extended to incorporate such effects. In particular, by invoking operationalmethods, it is shown that variation in laminate thickness may provide less efficacious means ofmaximizing energy dissipation than varying the choice of laminate materials but that either of theseeffects can in fact dwarf those associated with non-uniformity in interface pressure. To achieve this,a special configuration is required for the relative sizes and layering of the laminates. In particular, itis shown that for the case of two laminates, the upper laminate is required to be thinner and harderthan the lower one. These results provide a basis for the design of such structures.

ANALYSE DES EFFETS DES PROFONDEURS LAMINEES ET DES PROPRIETESMATERIELLES SUR LE DECREMENT ASSOCIES A DES COUCHES

STRUCTURELLES DANS UN ENVIRONNEMENT PRESSURISE

RESUME

Dans l’aerodynamique et la structure des machines, une des facons efficaces de dissiper lesvibrations ou les bruits non desires est d’exploiter la presence du glissement au niveau de l’interfacelaminee ou de tels elements sont tenus ensemble dans un environnement pressurise. L’analyse et larecherche experimentale de cette lamination a etabli que, assujettie a un chargement statiqueou dynamique, la non uniformite dans l’interface pressurisee peut avoir un effet significatif, a la fois,sur la dissipation de l’energie et le decrement d’amortissement logarithmique associe avec lemecanisme de glissement du decrement. Un tel evenement peut etre utilise efficacement pouraugmenter le niveau de decrement disponible dans un tel mecanisme. Ce qui n’a pas ete etudie estl’ampleur de la dissipation de l’energie utilisee par les parties superieures ou inferieures laminees.Dans cet article, l’analyse s’est appliquee a introduire de tels effets. En particulier, en utilisant cesmethodes, on demontre que les variations dans les epaisseurs laminees pourraient donner desmoyens moins efficaces dans la maximisation de la dissipation de l’energie. Une configurationspeciale est requise pour des tailles specifiques et les couches laminees. En particulier, il est demontreque dans le cas de deux couches laminees, la couche superieure doit etre plus mince et dure que lacouche inferieure. Ces resultats nous procurent une base pour le design de telles structures.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 2, 2010 165

1. INTRODUCTION

The mechanism of damping as a means of controlling undesirable effects of vibration hasreceived considerable attention in the literature over the years. Within this context, slip dampingis a mechanism exploited for dissipating noise and vibration energy in aerodynamic andmachine structures. There are in fact, several ways of effecting such damping; including theintroduction of either constrained, unconstrained and even viscoelastic layers. One of suchtechniques is layered construction made possible by externally applied pressure that holds themembers together at the interface. Such layers could also either be jointed or held together byappropriately spaced bolts. Under such circumstances, the profile of the interface pressureassumes a significant role, especially in the presence of slip, to dissipate the vibration energy. Afull account of the nature of the pressure profile can be found in [1–3] where the work of Gouldand Mikic [4] and Ziada and Abd [5] as well as Nanda and Behera [6] are discussed in furtherdetails.

Right from the time of Goodman and Klumpp [7] who were credited with one of the earliestworks on slip damping, the emphasis has usually revolved around the maximum amount ofenergy dissipation that could be arranged and for the case of dynamic loading, what level oflogarithmic damping decrement could be achieved. However all the early workers includingMasuko et al [8], Nishiwaki et al [9, 10] and Motosh [11] limited their investigations to the caseof uniform or constant intensity of pressure distribution at the interface.

More recently, there have been attempts to relax the restriction of uniform interface pressureto allow for more realistic pressure profiles that are encountered in practice. Such attemptsinclude both experimental and numerical treatments such as the work of Shin et al [12], Songet al [13], Nanda [14, 15] as well as Nanda and Behera [16, 17].

The analytical analysis of the effect of non-uniform interface pressure distribution on themechanism of slip damping for layered beams was also recently examined for both static anddynamic loads. In particular, whereas the investigation in Damisa et al [1, 3] was limited to the

Nomenclature

b width of laminated beamsdifferential operator

E1 modulus of rigidity of lower laminateE2 modulus of rigidity of upper laminateF applied end force amplitudeh1 depth of lower laminate beamh2 depth of upper laminate beamI1 moment of inertia for lower laminateI2 moment of inertia for upper laminateL length of Laminated beamsP clamping pressure at the interface of

the laminated beamst time coordinateu1 displacement of the lower laminateu2 displacement of the upper laminateW dynamic responseWF transverse response in Fourier plane

x space coordinate along the beam interfacey space coordinate perpendicular to the

beam interface

Greek lettersratio of Young’s moduli of laminates

m coefficient of friction at the interfaceof the laminated beams

j dummy variabler1 density of lower laminate materialr2 density of upper laminate material(sx)1 bending stress at the lower half of the

laminates(sx)2 bending stress at the upper half of thelaminatestxz shear stress at the interface of thelaminated beamsY ratio of the laminate thicknesses


case of linear pressure profile, the static analysis in Olunloyo et al [2] included other forms ofinterfacial pressure distributions such as polynomial or hyperbolic representations and theresults obtained demonstrated that the effects of such distributions in comparison with thelinear profile were largely incremental in nature and no fundamental differences were found.This provides additional justification for the linear pressure profile selected for the cantileverarchitecture used in our present investigation.

The results of the analysis of the cantibeam in [1–3] revealed that when the beam laminatesare of the same material and thickness, non-uniformity in interface pressure can for examplehave significant effect on the mechanism of slip damping for static load while the energydissipation and the logarithmic damping decrement associated with dynamic loads aresignificantly influenced by the nature of the interfacial pressure profile between the laminates.What has not been studied is the effect of asymmetry either in the dimensions of the laminatesor in the choice of materials for the upper and lower laminates.

The aim of the present work is to extend earlier dynamic analysis to cover the case where theupper and lower laminates need not be of the same dimensions, neither do they need to be of thesame material so as to accommodate the use of composites and study the additional effects thatmight arise in the context of energy dissipation and logarithmic damping decrement.

2.1. Governing Differential EquationFor the case of a layered beam of two dissimilar materials and laminate thicknesses, Osheku

[18] has shown that the governing equations of motion corresponding to microscopic slip at theinterface can be derived as:

L4W

Lx4z

b1zb2

2

� �L2W

Lt2~

a1za2

2

� � dp

dxð1Þ

where the following have been defined

a1~6m

E1h12

, b1~r1bh1

E1I1

a2~6m

E2h22

, b2~r2bh2

E2I2

2.2. Problem Definition and Method of AnalysisOne of the principal reasons for this investigation is to determine to what extent the structure

illustrated in Fig. 1 below can serve as an energy dissipating mechanism. In particular, the

Fig. 1. Coordinate axes and geometry for layered beam of dissimilar laminates under dynamic load.


objective here, is to examine analytically the effect of the nature of load, laminate depth ratioand material properties variation on:

(i) the dynamic response of a clamped layered beam made from dissimilar materials and heldtogether by some externally applied non-uniform force;

(ii) the profile of interfacial slip;

(iii) the slip energy dissipation under dynamic conditions;

(iv) logarithmic damping decrement associated with mechanism of slip damping in suchlayered structures.

A general theory of the energy dissipation properties of press-fit-joints in the presence ofcoulomb friction as originally developed by Goodman and Klumpp provides the basis for thephysics of the problem. The contact conditions between the two layers are:

(i) there is continuity of stress distributions at the interface to sufficiently hold the two layerstogether both in the pre- and post- slip conditions.

(ii) a stick elastic slip with presence of coulomb friction occurs at the interface of the sandwichelastic beams to dissipate energy and does not remain constant as a function of some othervariable such as spatial distance, time or velocity.

3. ANALYSIS OF DYNAMIC RESPONSE FOR LINEAR INTERFACE PRESSUREPROFILE

When we take the Laplace transform of the governing differential Eq. (1), we obtain

d4 *W x,sð Þdx4

zb1zb2

2

� �s2 *WW x,sð Þ{sW 0ð Þ{ W

.0ð Þ

� �~

a1za2

2s

� � dP

dxð2Þ

where the Laplace transform viz:

~..ð Þ ~

ð?0

.ð Þe{stdt , .ð Þ ~1

2pi

ðgzi?

g{i?

~..ð Þestds ð2aÞ

has been invoked.

If the corresponding analysis is also limited to the case of linear pressure variation at theinterface viz:

P xð Þ ~ P0 1ze

Lx

� �ð2bÞ

then substitution for the pressure in Eq. (2) gives

d4*W x,sð Þdx4

zb1zb2

2

� �s2*W x,sð Þ{sW 0ð Þ{ W

.0ð Þ

� �~

a1za2

2s

� �P0e

Lð3Þ

the next step is to introduce the Fourier finite sine transform namely:


.½ �F~

ðL0

.½ �sinnpx

Ldx ; .½ � ~

2

L

X?n~1

.½ �F sinnpx

Lð4Þ

from which the following relation can be inferred viz:

*W xxxx x,sð Þ� �F

~n4p4

L4

*W x,sð Þ� �F

{n3p3

L3

*W 0,sð Þz {1ð Þnz1*

W L,sð Þ� �

ð5Þ

The operating boundary conditions at the ends of the beam in the Laplace transform plane are:

*W 0,sð Þ ~

d

dx

*W 0,sð Þ ~

d2

dx2

*W L,sð Þ ~ 0 ð6Þ

and use of the first and third conditions in the preceding equation reduce Eq. (5) to

*W xxxx x,sð Þ� �F

~n4p4

L4

*W x,sð Þ� �F

{n3p3

L3{1ð Þnz1*

W L,sð Þz np

L

*W xx 0,sð Þ

ð7Þ

so that on assuming zero initial conditions for W, the Fourier sine transform of Eq. (3) gives theresult

n4p4

L4

*W F ln,sð Þz b1zb2

2

� �s2*W F ln,sð Þ ~

a1za2

2s

� �P0e

np1z {1ð Þnz1� �

zn3p3

L3{1ð Þnz1*

W L,sð Þ{ np

L

*W xx 0,sð Þ

ð8Þ

To further simplify Eq. (8), one can proceed to evaluate the term*Wxx 0,sð Þ by applying the

Goodman and Klumpp end condition in the spatial-state form as

ð0{h1

t xzð Þ1 x,tð Þzðh2

0

t xzð Þ2 x,tð Þ

0B@

1CAdz ~

F tð Þb

at x~L ð9Þ

so that using the out of plane shear stress relations namely

t xzð Þ1 x,tð Þ ~1

2E1 z2zh1z� �

W1xxx x,tð Þz mp

h1zzh1ð Þ ð10aÞ

and

t xzð Þ2 x,tð Þ ~1

2E2 z2{h2z� �

W2xxx x,tð Þ{ mp

h2z{h2ð Þ ð10bÞ


makes it possible to rewrite Eq. (9) in the Laplace transform plane as

ðL0

ð0{h1

1

2E1 z2zh1z� �*

W xxx x,sð Þdzdxz

ðL0

ð0{h1

mp

sh1zzh1ð Þdzdx

ð11Þ

Integration of this equation then reveals that the bending moment of the Euler-Bernoulli’sclamped laminated beams admits the form

*Wxx 0,sð Þ ~

12*F sð Þ

b E1h13zE2h2

3� �{

6mP

s

h1zh2

E1h13zE2h2

3

� �1zeð Þ

!L ð12Þ

These results clearly indicate that the value for the expression (12) cannot be fully determineduntil the forcing function F tð Þ is fully specified. Consequently, further analysis is limited to thefollowing cases namely:

(a) F tð Þ~F0H t{t0ð Þ where, H(t), is the Heaviside function and

(b) F tð Þ~F0eivt

3.1. Case of Heaviside Loading FunctionFor case (a) above, the forcing function is F tð Þ~F0H t{t0ð Þ which gives the Laplace

transform as*F sð Þ~ F0

se{t0s:F0

*H sð Þ.

By recalling the only unutilized boundary condition in Eq. (6), vizd

dx

*W 0,sð Þ~0 and guided

by the Laplace transform of the forcing function, the expression for the bending moment can berewritten as

*Wxx 0,sð Þ ~

12F0*H sð Þ

b E1h13zE2h2

3� �{

6mP0

s

h1zh2

E1h13zE2h2

3

� �1zeð Þ

!L ð13Þ

Hence, the corresponding response of the Euler-Bernoulli’s laminated beam in the Fourier-Laplace transform plane as presented in Eq. (8) admits the form

*W

Fln,sð Þ ~

n3p3

L3{1ð Þnz1

s*W 1 L,sð Þ{ np

L

12F0*H sð Þ

b E1h13zE2h2

3� �L

znp

L

6mP0

s

h1zh2

E1h13zE2h2

3

� �1zeð ÞL

za1za2

2

� �P0e

1z {1ð Þnz1

np

!

0BBBBBBBBB@

1CCCCCCCCCA

b1zb2

2

� �s s{ivnð Þ szivnð Þ

ð14Þ

where v2n~

2n4p4

b1zb2ð ÞL4is the natural frequency of vibration of the clamped dissimilar layered

beam as can be derived by setting the right hand side of Eq. (8) to zero.


The next step is to evaluate the Fourier inversion of Eq. (14) as

*W x,sð Þ~

2s*W L,sð Þ

P?n~1

{1ð Þnz1sinnp�xx

np

{2L3 12F0*H sð Þ

b E1h13zE2h2

3� �{

6mP0

s

h1zh2

E1h13zE2h2

3

� � ! P?n~1

sinnp�xx

n3p3

z2L3 6mP0e

s

h1zh2

E1h13zE2h2

3

� �X?n~1

sinnp�xx

n3p3

zL3

32a1za2ð ÞP0e

X?n~1

sinnp�xx

n5p5

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

b1zb2

2

� �s s{ivnð Þ szivnð Þ L4

n4p4

ð15Þ

where

�xx~x

L

To further simplify the series in Eq. (15), one can invoke the well known closed form Fourierseries representations namely:

�xx~1

p

X?n~1

{1ð Þnz1

nsinnp�xx , V 0v�xxv1 ð16aÞ

X?n~1

sinn�xx

n3~

p2�xx

6{

p�xx2

4z

�xx3

12, V 0v�xxv2 ð16bÞ

and X?n~1

sinn�xx

n5~

p4�xx

90{

p�xx3

36z

p�xx4

48{

�xx5

240, V 0v�xxv2 ð16cÞ

Consequently, Eq. (15) can be expressed in the form

*W �xx,sð Þ~

2s*W 1 L,sð Þ�xxz

L3

32a1za2ð ÞP0e

�xx

45{

2�xx3

9z

�xx4

3{

2�xx5

15

� �

{2L3 12F0*H sð Þ

b E1h13zE2h2

3� �{

6mP0

s

h1zh2

E1h13zE2h2

3

� � !�xx

6{

�xx2

4z

�xx3

12

� �

z2L3 6mP0e

s

h1zh2

E1h13zE2h2

3

� ��xx

6{

�xx2

4z

�xx3

12

� ��

0BBBBBBBBB@

1CCCCCCCCCA

b1zb2

2


n4p4

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

ð17Þ


By imposing the boundary conditiond*W 0,sð Þ

d�xx~0 in Eq. (17) the deflection at the end of the

laminated Euler-Bernoulli’s cantilever beam can be evaluated in the Laplace domain as

*W L,sð Þ~ L3

s

2F0

b E1h13zE2h2

3� �{mP0

h1zh2

E1h13zE2h2

3

� �1zeð Þ

{3

32ð Þ 45ð ÞmP0e1

E1h12z

1

E2h22

� �� 0BBB@

1CCCA

8>>><>>>:

9>>>=>>>;

ð18Þ

Thus, one can now express Eq. (17) in the form

*W �xx,sð Þ~

2L3 12F0

b E1h13zE2h2

3� �{

6mP0 h1zh2ð ÞE1h1

3zE2h23

� � 1zeð Þ !

�xx

6{

�xx2

4z

�xx3

12

� �

zL3 3

16ð ÞmP0e1

E1h12z

1

E2h22

� ��xx

45{

2�xx3

9z

�xx4

3{

2�xx5

15

� �0BBBB@

1CCCCA

b1zb2

2


n4p4

ð19Þ

which may be rearranged as

*W �xx,sð Þ ~

L3

2F0

b E1h13zE2h2

3� �{

mP0 h1zh2ð ÞE1h1

3zE2h23

� � !

3�xx2{�xx3� �

zmP0e h1zh2ð ÞE1h1

3zE2h23

� � 3�xx2{�xx3� �

zmP0e1

E1h12z

1

E2h22

� �{

�xx3

24z

�xx4

16{

�xx5

40

� �

0BBBBBBBBB@

1CCCCCCCCCA

b1zb2

2


n4p4

ð20Þ

On the other hand by invoking Laplace inversion

W �xx,ttð Þ ~1

2pi

ðgzi?

g{i?

*W �xx,sð Þesttds ð21Þ

where

tt~t{t0

the dynamic response in state-space domain can be evaluated as


W �xx,ttð Þ ~ L3 1{cosvnttð Þ

2F0

b E1h13zE2h2

3� �{

mP0 h1zh2ð ÞE1h1

3zE2h23

� � !

3�xx2{�xx3� �

zmP0e h1zh2ð ÞE1h1

3zE2h23

� � {3�xx2z�xx3� �

zmP0e1

E1h12z

1

E2h22

� �{

�xx3

24z

�xx4

16{

�xx5

40

� �

0BBBBBBBBB@

1CCCCCCCCCAð22Þ

or as

W �xx,tð Þ ~ F1 tð Þ�DD1{m�PP0

�DD2

� �3�xx2{�xx3� �

zm�PP0e {3�DD2�xx2{ �DD2{�DD3

12

� ��xx3

� �z�DD3

�xx4

8{

�xx5

20

� �0B@

1CA ð23Þ

where the following have been used

�DD1~2

1zcY3� � , �DD2~

1zY

1zcY3, �DD3~

1zc{1Y{2� �

2, F1 tð Þ~ 1{cos2ptð Þ ð23aÞ

in conjunction with the non-dimensionalized parameters viz:

W �xx,tð Þ~ W �xx,tð ÞE1bh13

L3F0; �PP0~

P0

F=bh1

; tt~2p

vn

t; c~E2

E1; Y~

h2

h1ð24Þ

3.2. Case of Harmonic Loading FunctionFor this case, the forcing function is F(t) 5 F0eivt. This gives its Laplace transform as

*F sð Þ~ F0

s{ivð25Þ

where, v is the excitation frequency. On the other hand,

*W x,sð Þ~

2s*W L,sð Þ�xx

{2L3 12sF0*H sð Þ

s{ivð Þb E1h13zE2h2

3� �{ 1zeð Þ 6mP0 h1zh2ð Þ

E1h13zE2h2

3

� � !L1

z3

16mP0eL3 1

E1h12z

1

E2h22

� �L2

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

b1zb2

2


n4p4

ð26Þ

where


L1~�xx

6{

�xx2

4z

�xx3

12

� �; L2~

�xx

45{

2�xx3

9z

�xx4

3{

2�xx5

15

� �

By imposing the boundary conditiond*W 0,sð Þ

d�xx~0 in Eq. (17) one can evaluate the deflection

at the end of the laminated Euler-Bernoulli’s cantilever beam in the Laplace domain as:

*W L,sð Þ~

L3

2F0

s{ivð Þb E1h13zE2h2

3� �{ 1zeð Þ 6mP0 h1zh2ð Þ

s E1h13zE2h2

3� �

!

zmP0e

480

1

E1h12z

1

E2h22

� �0BBBB@

1CCCCA

b1zb2

2


n4p4

ð27Þ

Subsequent substitution into Eq. (26) and carrying out the Laplace inversion for the non-dimensionalized variable gives the result

W �xx,tð Þ ~

2�xx{12L1ð Þ

2F2 tð Þ1zcY3� �{ 1zeð Þ m�PP0 1zYð Þ

1zcY3

� �F1 tð Þ

26664

37775

{�xx

240{

3

16L2

� �m�PP0e 1zc{1Y{2

� �F1 tð Þ

0BBBBBBBB@

1CCCCCCCCA

ð28Þ

which on introducing some of the non-dimensionalized parameters earlier used, can be re-arranged as

W �xx,tð Þ ~

�DD1F2 tð Þ{m�PP0�DD2F1 tð Þ

� �3�xx2{�xx3� �

zm�PP0eF1 tð Þ {3�DD2�xx2{ �DD2{�DD3

12

� ��xx3

� �z�DD3

�xx4

8{

�xx5

20

� �0B@

1CA ð29Þ

where �DD1,�DD2,�DD3 are as previously defined.

It is also convenient to introduce the additional normalizations

v

vn

~g;2pt

vn

~t ð30Þ

where g can be regarded as the associated frequency ratio of the driving load.

This facilitates the rearrangement of the earlier expressions for F1(t) and F2(t) as

F1 tð Þ~F1 tð Þ~ 1{cos2ptð Þ ð31Þ

and


F2 tð Þ~F2 tð Þ~ 1

1{g2ð Þcos 2pgtð Þ{cos2ptð Þ

zi sin 2pgtð Þzgsin2ptð Þ

� ð32Þ

4. COMPUTATION OF DYNAMIC SLIP

In this section, we can proceed to compute the slip associated with the motion under study forthe two cases being considered.

4.1. Case of Heaviside Loading FunctionThe relative dynamic slip at the interface of the laminated beams is given by

Du x,ttð Þ ~ u2 x,tt,0{ð Þ{u1 x,tt,0zð Þ ð33Þ

Following Goodman and Klumpp, this can also be written as

Du x,ttð Þ ~ E1{1

ðx0

sxð Þ1 j,tt,0{ð Þ �

dj{E2{1

ðx0

sxð Þ2 j,tt,0zð Þ �

dj ð34Þ

so that on substituting the relevant bending stress relations in state-space domain namely:

sxð Þ1 x,z,ttð Þ~{E1 2zzh1ð Þ

2

L2W1 x,ttð ÞLx2

{mPav x{Lð Þ

h1ð35aÞ

and

sxð Þ2 x,z,ttð Þ~{E2 2z{h1ð Þ

2

L2W2 x,ttð ÞLx2

zmPav x{Lð Þ

h2ð35bÞ

as listed in Eqs. (35a) and (35b) above, Eq. (34) can now be expressed in the form

Du x,ttð Þ ~

ð�xx0

h2

2

L2W2 j,ttð ÞLj2

zh1

2

L2W1 j,ttð ÞLj2

z

zm�PPav

E2h2j{Lð Þz m�PPav

E1h1j{Lð Þ

8>>><>>>:

9>>>=>>>;

dj ð36Þ

so that Eq. (36) is then integrated to give

Du x,ttð Þ ~

h2

2

L2W2 x,ttð ÞLx2

zh1

2

L2W1 x,ttð ÞLx2

zmP0

E2h2

x2

2{Lx

� �z

mP0

E1h1

x2

2{Lx

� �

zmP0e

E2h2L

x3

3{

x2

2L

� �z

mP0e

E1h1L

x3

3{

x2

2L

� �

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

ð37Þ


which that on introducing the usual non-dimensionalised parameters, gives

D�uu �xx,tð Þ ~ �DD4L �WW

Lxzm�PP0

�DD5 �xx2{2�xx� �

ze2

3�xx3{�xx2

� �� ð38Þ

where the following parameters have been introduced

�DD4~1zY{1

2

� �, �DD5~

1zc{1Y{1

2

� �

Thus by using the results for W from Eq. (23), we can rewrite Eq. (38) as

D�uu �xx,tð Þ~F1 tð Þm�PP0

m�PP0 3�kk1F1 tð Þz�DD5

� �{3�kk3F1 tð Þ

� ��xx2{2�xx� �

zm�PP0e

{6�kk1F1 tð Þ�xxz 3�kk1{�kk4

4

� �F1 tð Þ{�DD5

� ��xx2

z�kk4

2F1 tð Þz 2

3�DD5

� ��xx3{

�kk4

2F1 tð Þ�xx4

26664

37775

2666664

3777775 ð39Þ

Here the following have also been defined

�kk1~1zYð Þ 1zY{1

� �2

4 1zcY3� � ,

�kk3~1zY{1� �2

2 1zcY3� � , �kk4~

1zc{2Y{2� �

1zY{1� �2

8

4.2. Case of Harmonic Forcing FunctionFollowing the procedure introduced in [3], it is now possible to compute the slip under

harmonic load by recalling Eq. (38) and substituting for—W from Eqs. (29), (31) and (32) to

obtain

D�uu~

m�PP0 3�LL2F1 tð Þz�LL4

� �{3�LL1F2 tð Þ

� �xx2{2�xx� �

zm�PP0e {6�LL2F1 tð Þ�xxz 3�LL2{�LL3

4

� �F1 tð Þ{�DD5

� �3�xx2

� �

z �DD3F1 tð Þ

4{

2

3

� ��xx3{�LL3

F1 tð Þ2

�xx4

8>>>>><>>>>>:

9>>>>>=>>>>>;

ð40Þ

where the following have been defined

�LL1~1zY{1

1zcY3

� �, �LL2~

1zYð Þ 1zY{1� �

2 1zcY3� �

!, �LL3~

1zc{1Y{2� �

1zY{1� �

4

!ð40aÞ


5. ENERGY DISSIPATION

5.1. Case of Heaviside Forcing FunctionIt can be recalled from earlier work [3] that the energy dissipated per cycle, following

Goodman and Klumpp is given by the relation

D ~ 4mb

ðp=2vn

0

ðL0

P xð ÞDu x,ttð Þdxdt ð41Þ

which can also be expressed as

D ~ 4m

ð1=40

ð10

P �xxð ÞD�uu �xx,tð Þd�xxdt ð42Þ

whereby on recalling the linear pressure profile and substituting for D�uu from Eq. (39) we canevaluate the right hand side of relation (42) to give.

D~D�1zD

�2 ð43Þ

with

D�1~ 1z

e

2

� � 8

11�kk3m�PP0{m2 �PP2

0

8

11�kk1z

2

3�DD5

� �� ð43aÞ

and

D�2~

m2P20e {

12

11�kk1z

�DD5

6z

3

110�kk4

� �

ze4

11�kk3m�PP0{m2 �PP2

0

4

11�kk1z

1

3�DD5

� �� 8>>><>>>:

9>>>=>>>;

ð43bÞ

5.2. Case of Harmonic Forcing FunctionFollowing the procedure outlined in the preceding section and substituting the corresponding

values for D�uu for the harmonic excitation from Eq. (40) into Eq. (42), we obtain the energydissipated through slip for this case as

D~D1zD2zD3zD4 ð44Þ

where,

D1~ 8D gð ÞmP0L1{m2P2

0

8

11L2z

2

3L3

� �� ð45aÞ


�DD2~e 4D gð Þm�PP0�LL1{m2 �PP2

0

4

11�LL2z

1

3�LL3

� �� ð45bÞ

�DD3~em2 �PP20 {

12

11�LL2z

3

110�LL3z

1

6�DD5

� �ð45cÞ

�DD4~e2m2 �PP20 {

6

11�LL2z

3

220�LL3z

1

12�DD5

� �ð45dÞ

and where

D gð Þ ~1

1{g2ð Þsin

p

2g

� �2pg

{1

2p

0@

1A{i

cosp

2g

� �{1

2pgz

g

2p

0@

1A

24

35 ð45eÞ

5.3. Analysis of Optimum Energy Dissipation for The Case of Harmonic Load ExcitationFrom Eqs. (44) and (45), it can be deduced that

mPopt~�LL1 2zeð ÞD gð Þ

4

11�LL2z

1

3�LL3

� �ze

8

11�LL2z

101

660�LL3{

1

12�DD5

� �ze2 3

11�LL2{

3

440�LL3{

1

24�DD5

� ��

~JdD gð Þ

where

Jd~Jd e,Y,cð Þ

~�LL1 2zeð Þ

4

11�LL2z

1

3�LL3

� �ze

8

11�LL2z

101

660�LL3{

1

12�DD5

� �ze2 3

11�LL2{

3

440�LL3{

1

24�DD5

� ��

Thus, the optimal slip energy is computed as

�DDopt~X4

n~1

�DDnopt ð46Þ

where,

�DD1opt~JdD2 8�LL1{Jd

8

11�LL2z

2

3�LL3

� �� ð47aÞ


�DD2opt~eJdD2 4�LL1{

Jd

3312�LL2z11�LL3

� �� ð47bÞ

�DD3opt~eJ2

dD2

330{360�LL2z9�LL3z55�DD5

� ð47cÞ

�DD4opt~e2J2

dD2

660{360�LL2z9�LL3z55�DD5

� ~

e

2�DD3opt ð47dÞ

5.4. Analysis of Optimum Thicknesses Ratio of Laminate for Energy Dissipation for The Caseof Harmonic load Excitation

The optimum thicknesses ratio of laminate for maximum dissipation of energy can becomputed by employing term wise differentiation via Eq. (46), viz;

LLY

X4

n~1

Dnopt Y,c,eð Þ !

~ U1 Y,c,eð ÞzU2 Y,c,eð Þð Þ~0, V c~E2

E1ð48aÞ

where;

U1 Y,c,eð Þ~ D2

2Jd

LJd

LY12

11ez

6

11e2

� ��LL2{

12

110ez

3

220e2

� ��LL3{

11

65ez

11

130e2

� ��LL4

zJd2 12

11ez

6

11e2

� �1{Y{2{2cY3{4cY{6cY2

2 1zcY3� �2

!

z3

110ez

3

220e2

� �Y{2zc{1Y{3z3c{1Y{4

4

� �z

11

65ez

11

130e2

� �c{1Y{2

26666666664

37777777775

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;ð48bÞ

and

U2 Y,c,eð Þ~ D2

8�LL1{Jd

8

11�LL2z

2

3�LL3

� �ze 4�LL1{Jd

4

11�LL1z

�LL3

3

� �� LJd

LY

{Jd 8z4eð Þ 3cY2z4cYzY{2

1zcY3� �2

!{Jd

8

11�LL2z

2�LL3

3z

4�LL2e

11z

2�LL3e

3

� �LJd

LY

{Jd

8

112zeð Þ 1{Y{2{2cY3{4cY{6cY2

2 1zcY3� �2

!{

2ze

3

� �Y{2zc{1Y{3z3c{1Y{4

4

� �" #

266666666664

377777777775

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;ð48cÞ

Here;

LJd

LY~

{H Y,c,eð Þ 3cY2z4cYzY{2

1zcY3� �2

!{�LL1

LH Y,c,eð ÞLY

H2 Y,c,eð Þ ð48dÞ

while;

(48b)

(48c)


H Y,c,eð Þ

~4

11�LL2z

1

3�LL3

� �ze

8

11�LL2z

101

660�LL3{

1

12�DD5

� �ze2 3

11�LL2{

3

440�LL3{

1

24�DD5

� ��

and

LH Y,c,eð ÞLY

~

4

111zeð Þz 3

11e2

� �1{Y{2{2cY3{4cY{6cY2

2 1zcY3� �2

!z

3

11z

101

660e{

3

440e2

� �Y{2zc{1Y{3z3c{1Y{4

4

� � !

z1

24ez

1

48e2

� �c{1Y{2

� �8>>>><>>>>:

9>>>>=>>>>;

Consequently, the optimum thicknesses ratio Ymin is a solution to the transcendentalequation namely;

U1 Ymin,c,eð ÞzU2 Ymim,c,eð Þ~0 ; V c~E2

E1ð49Þ

5.5. Analysis of Optimum Moduli Ratio at Optimum Thicknesses Ratio of Laminate forEnergy Dissipation for The Case of Harmonic load Excitation

The optimum moduli ratio at optimum thicknesses ratio of laminate for maximumdissipation of energy can be computed by employing term wise differentiation via Eq. (48), viz;

Fig. 2. Energy dissipation as a function of interfacial pressure gradient.


LLc

LLY

X4

n~1

Dnopt Y,c,eð Þ !��

Ymin

0@

1A~

LLc

U1 Ymin,c,eð ÞzU2 Ymin,c,eð Þð Þ� �

~0 ð50aÞ

where;

LLc

U1 Ymin,c,eð Þð Þ~ D2

24

11ez

12

11e2

� � �LL2 Ymin,cð Þ LJd Ymin,cð ÞLc

:LJd

LY

��Y~Ymin

zJd Ymin,cð Þ LLc

LJd

LY

��Y~Ymin

!0BBBB@

1CCCCA

z12

110ez

3

220e2

� �c{2 1zYminð Þ

4z

11

65ez

11

130e2

� �cYmin

{1

2

12

11ez

6

11e2

� � zJd2 Ymin,cð ÞH1 Ymin,cð Þ

z2Jd Ymin,cð Þ: LJd Ymin,cð ÞLc

1{Ymin{2{2cYmin

3{4cYmin{6cYmin2

2 1zcY3� �2

!0BB@

1CCA

z3

110ez

3

220e2

� �{c{2Ymin

{3{3c{2Ymin{4

4

� �{

11

65ez

11

130e2

� �c{2Ymin

{2

266666666666666666666664

377777777777777777777775

8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>;

ð50bÞ

Here,

H1 Ymin,cð Þ~

1zcYmin3

� �2� �

{Ymin3{2Ymin{3Ymin

2� �

{ 1{Y{2{2cY3{4cY{6cY2� �

Y3minzcY6

min

� �1zcYmin

3� �4

0@

1A


(50b)


while

LU2 Ymin,c,eð ÞLc

~

D2

{8Ymin

3zYmin2

� �1zcYmin

3� �2

!{

LJd Ymin,cð ÞLc

8

11�LL2 Ymin,cð Þz 2

3�LL3 Ymin,cð Þ

� �{Jd Ymin,cð Þ 8

11

LLc

�LL2 Ymin,cð Þ� �

z2

3

LLc

�LL3 Ymin,cð Þ� ��

ze

{4Ymin

3zYmin2

� �1zcYmin

3� �2

!{ Jd Ymin,cð Þ:LJd Y,cð Þ

LY

��Y~Ymin

!{

4

11

Ymin3zYmin

2� �

1zcYmin3

� �2

!{

c{2 Y{2minzY{3

min

� �12

!

{4

11�LL1 Ymin,cð Þz�LL3 Ymin,cð Þ� � LJd Ymin,cð Þ

Lcz 4�LL1 Ymin,cð Þ{Jd Ymin,cð Þ 4

11�LL1 Ymin,cð Þz 1

3�LL1 Ymin,cð Þ

� �� LLc

LJd Y,cð ÞLY

��Y~Ymin

!0BBBBB@

1CCCCCA

{ 8z4eð Þ 3cYmin2z4cYminzYmin

{2

1zcYmin3

� �2

!LJd Ymin,cð Þ

LczJd Ymin,cð Þ

1zcYmin3

� �23Y2

minz4Ymin

� �{2 3cY2

minz4cYminzY{2min

� �Y3

minzcYmin6

� �1zcY3

minð Þ4

! !

{8z4eð Þ

11�LL2z

2zeð Þ�LL3

3

� �LJd Ymin,,cð Þ

Lc:LJd Y,cð Þ

LY

��Y~Ymin

!zJd Ymin,,cð Þ: L

Lc

LJd Y,cð ÞLY

��Y~Ymin

! !

{ Jd Ymin,,cð Þ:LJd Y,cð ÞLY

��Y~Ymin

!2ze

11

� �2Y3

minzY2minzY4

min

1zcY3minð Þ2

!{

2ze

3

� �c{2 Y{2

minzY{3min

� �4

!

{8

112zeð Þ 1{Y2

min{2cY3min{4cYmin{6cYmin

2

2 1zcY3minð Þ2

!{

2ze

3

� �Y{2

minzc{1Y{3minz3c{1Y{4

min

4

� �" #LJd Ymin,cð Þ

Lc

zJd Ymin,cð Þ1zcY3

min

� �2Y3

minz2Yminz3Y2min

� �z Y3

minzcY6min

� �1{Y2

min{2cY3min{4cYmin{6cYmin

2� �

1zcY3minð Þ4

!" # !

266666666666666666666666666666666666666664

377777777777777777777777777777777777777775

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

On the other hand;

LJd

LY~

{H Y,c,eð Þ 3cY2z4cYzY{2

1zcY3ð Þ2� �

{�LL1LH Y,c,eð Þ

LY

H2 Y,c,eð Þ ð50dÞ


(50c)


where;

H Y,c,eð Þ

~4

11�LL2z

1

3�LL3

� �ze

8

11�LL2z

101

660�LL3{

1

12�DD5

� �ze2 3

11�LL2{

3

440�LL3{

1

24�DD5

� �� and

Fig. 5. Energy dissipation as a function of interfacial pressure.



LH Y,c,eð ÞLY

~

4

111zeð Þz 3

11e2

� �1{Y{2{2cY3{4cY{6cY2

2 1zcY3� �2

!z

3

11z

101

660e{

3

440e2

� �Y{2zc{1Y{3z3c{1Y{4

4

� � !

z1

24ez

1

48e2

� �c{1Y{2

� �8>>>><>>>>:

9>>>>=>>>>;


Fig. 8. Energy dissipation as a function of c for g 5 0.1; and Y 5 1.


Consequently, the optimum moduli ratio cmin is a solution of the transcendental equationnamely;

Y1 Ymin,c,eð ÞzY2 Ymim,c,eð Þ~0; V c~E2

E1~1

� �ð51Þ

6. LOGARITHMIC DAMPING DECREMENT FOR THE CASE OF HARMONICLOAD EXCITATION

Following similar procedural analysis introduced in an earlier paper [3], the logarithmicdamping for this generalised problem can be defined as

d~1

2Ln 1z

D

U1zU2

� �ð52Þ

where for this case, U1 is the energy introduced by the bending moment and can in this case becalculated from the theorem of Castigliano as

U1~27 H1ei4pg{2H2mP0 1zeð Þei2pgzH3m2P2

0 1z2eze2� �n o

ð53aÞ

On the other hand U2 is the energy stored from the deflection at the free end which can alsobe computed in this case from the theory of strength of materials as

�UU2~

H1ei4pg{2mP0 H2ze

240240H2zH4

� �� ei2pg

zm2P2

0 H3z240H3zH5

120

� �ez

2402H3z480H5zH6

2402

� �e2

� �8>><>>:

9>>=>>; ð53bÞ

Fig. 9. Energy dissipation as a function of c for g 5 0.1; and Y 5 0.45.


In writing the results in Eqs. (52), (53a) and (53b) we have made use of the non-dimensionalized variables viz:

Un~UnEbh3

F20 L3

; n ~ 1, 2

H1~4

1zcY3� �2

, H2~2 1zYð Þ1zcY3� �2

, H3~1zYð Þ2

1zcY3� �2

Fig. 10(ab) Energy dissipation as a function of Y for g 5 0.1; and c 5 1. Energy dissipation as afunction of Y for g 5 0.1; and c 5 2.0.


H4~1zc{1Y{2� �

1zcY3� � , H5~

1zc{1Y{2� �

1zYð Þ2 1zcY3� � , H6~

1zc{1Y{2� �2

4

7. ANALYSIS OF RESULTS

In general, the results indicate that both the displacement and slip behave, by and large, aswas reported earlier in [3]. What is of interest here is to ascertain to what extent can we influencethe amount of energy dissipated either by our choice of different materials for the upper andower laminates or by varying the thicknesses of the upper and lower laminates? These effects arebest simulated by varying the values of c and Y these being the relative ratios of the Young’s moduli

Fig. 10(cd) Energy dissipation as a function of Y for g 5 0.1; and c 5 4. Energy dissipation as afunction of Y for g 5 0.1; and c 5 6.


and laminate thicknesses respectively (c~E2

E1; Y~

h2

h1). In such circumstances, it stands to reason

that for the special case Y 5 c 5 1, we expect to recover the earlier results reported in [3].

In particular, as a check, the results reported in [3] for the slip energy dissipation profiles arecorrectly recovered from the present work as the special case c 5 1, Y 5 1. In this respect, it canbe observed that Figs. 2–4 confirm the convergence of both solutions for the sample cases g 5

0.1, 0.5, and 0.85 respectively.

Figures. 5–7 on the other hand, amply demonstrate that given some fixed values of c and Y,the amount of energy dissipated through slip progressively decreases as g increases as wasreported in [3], whenever g is restricted within the pre-resonance zone.

In this case, it is also confirmed that as the value of g increases the value of energy dissipationdrops for a prescribed value of the normalised pressure mP. More importantly, it is observed

Fig. 11. Energy dissipation as a function of c for g 5 0.1; and Y50.35.

Fig. 12. Energy dissipation as a function of Y for g 5 0.5; and c 5 1.


that the relative ordering of the energy curves as a function of the pressure gradient e appearsreversed from the usual pattern where negative pressure gradient at the laminate interfaceinduces higher energy dissipation.

However, part of the exercise here is to know how the ratio of laminate thicknesses, Y andmodulus of rigidity of laminate materials c influence the level of dissipation of the vibrationenergy. The effect of the ratio of modulus of rigidity of laminate materials c is shown in Figs. 8,9, 11, 13 and 14 for different values of g and Y whilst the effect of laminate thicknesses Y isshown in Figs. 10, 12, 15, and 16. We find that, more energy is dissipated in almost all casesthan the results earlier reported in [3] for which we had Y 5 c 5 1. The results also indicate thatin general, the modulating role of pressure gradient becomes relatively insignificant ascompared with some of the other parameters under consideration.

Fig. 13. Energy dissipation as a function of c for g 5 0.5; and Y51.

Fig. 14. Energy dissipation as a function of Y for g 5 0.5; and c 5 0.35.


In fact, by studying Figs. 8–13, it is clear that for energy dissipation, it is more efficacious toplay with either Y or c than to settle for the symmetric case Y 5 c 5 1 as this gives less energyfor a given value of g than either of the two other cases.

It is also clear that in comparison more energy can be dissipated by varying c as against Ysince for any given g, the maximum dissipated energy from the c curve is larger than thatobtained from the Y curve.

Moreover, by carefully selecting both Y or c much larger energy dissipation can in fact bearranged than in any of the cases discussed above.

For example by choosing a value of c . 1, one can arrange a much higher value for energydissipation and vice-versa.

Fig. 15. Energy dissipation as a function of Y for g 5 0.5; and c 5 2.5.

Fig. 16. Energy dissipation as a function of (Y,c) for g 5 0.1; and e 5 0.


From the engineering and economic points of view, it is much easier to have a hold on Y thanc since the former requires the preparation of laminates of certain thickness ratios which is astraightforward matter whereas it is sometimes a formidable problem to find appropriatelaminate materials that will give a prescribed c ratio knowing that there might be otherengineering, economic or environmental requirements to be met in the use of such materials.

The pattern of results for energy dissipation also suggests that for effective energy dissipationthere is a preferred arrangement of the laminates. For example, it is observed from Figs. 10, 12,

Table (a). Optimum energy, derived from plot of �DDopt versus Y as modulated by g , c for the casee~0.

g c~0:075 c~0:25 c~0:35 c~1 c~2 c~4 c~6

0.1 �DDopt50.0958 �DDopt50.1451 �DDopt5 0.1601 �DDopt50.2160 �DDopt50.2346 �DDopt50.2464 �DDopt50.21210.5 �DDopt50.0936 �DDopt50.1418 �DDopt50.1565 �DDopt50.2111 �DDopt50.2293 �DDopt50.2408 �DDopt50.20720.85 �DDopt50.0895 �DDopt50.1356 �DDopt50.1495 �DDopt50.2018 �DDopt50.2191 �DDopt50.2301 �DDopt50.19812.5 �DDopt50.0528 �DDopt50.0800 �DDopt50.0883 �DDopt50.1191 �DDopt50.1294 �DDopt50.1359 �DDopt50.1169

Table (b). Optimum energy, derived from plot of �DDoptversus Y as modulated by g , c for the casee~{0:2.

g c~0:075 c~0:25 c~0:35 c~1 c~2 c~4 c~6

0.1 �DDopt50.0854 �DDopt50.1362 �DDopt50.1533 �DDopt50.2134 �DDopt50.2311 �DDopt50.2614 �DDopt50.23330.5 �DDopt50.0834 �DDopt50.1331 �DDopt50.1498 �DDopt50.2085 �DDopt50.2259 �DDopt50.2554 �DDopt50.22800.85 �DDopt50.0797 �DDopt50.1272 �DDopt50.1432 �DDopt50.1993 �DDopt50.2159 �DDopt50.2441 �DDopt50.21792.5 �DDopt50.0471 �DDopt50.0751 �DDopt50.0846 �DDopt50.1177 �DDopt50.1275 �DDopt50.1441 �DDopt50.1286

Table (d). Optimum energy, derived from plot of �DDopt versus c as modulated by g , Y for the casee~0.

g Y~0:075 Y~0:25 Y~0:35 Y~0:45 Y~1 Y~4 Y~6


Table (c). Optimum energy, derived from plot of �DDopt versus Y as modulated by g , c for the casee~0:2.

g c~0:075 c~0:25 c~0:35 c~1 c~2 c~4 c~6



15 and 16 that the optimum value of Y for which you have maximum dissipation usually occurssomewhere between 0.35 and 0.55 and certainly below the value Y 5 1. Since Y is the ratio h2/h1, this means that the thickness of the upper laminate h2 should always be less than that of thelower laminate as illustrated in Tables (a–f). Similarly, if the ratio of Young’s Moduli of thelaminates c 5 E2/E1 has to be greater than 1, the harder laminate is to be placed on top of thesofter one. Thus when dealing with composite laminates, for optimal energy dissipation, it isimperative that the softer material (i.e. with Youngs Modulus E1) be placed under the harderlaminate.

Table (e). Optimum energy, derived from plot of �DDopt versus c as modulated by g , Y for the casee~{0:2.

g Y~0:075 Y~0:25 Y~0:35 Y~0:45 Y~1 Y~4 Y~6


Table (f). Optimum energy, derived from plot of �DDopt versus c as modulated by g , Y for the casee~0:2.

g Y~0:075 Y~0:25 Y~0:35 Y~0:45 Y~1 Y~4 Y~6


Fig. 17. Energy dissipation as a function of (Y,c) for g 5 0.1; and e 5 20.2.


However there is the separate but related issue of the global ordering of these effects viz: ofthe two additional options available for the dissipation of unwanted energy which is moreefficacious; is it by varying the laminate materials, or by varying the thicknesses of thelaminated slabs?

This question is best answered by studying the 3-D plots in Figs. 17–19 which show that whereasin any given case, there is always an optimum laminate thickness ratio Y below 1 for energydissipation the same cannot be said for c as the amount of energy dissipated seems to increasemonotonically well beyond 1 before the dissipated energy peaks. Thus if one is restricted to the use

Fig. 18. Energy dissipation as a function of (Y,c) for g 5 0.5; and e 5 20.2.

Fig. 19. Plot of Y1zY2~0 as a function of Y for g 5 5.5; and c 50.85.


of a single material for the laminates one can maximise energy dissipation by keeping the ratio of thelaminate thicknesses within the neighbourhood of the optimal value as illustrated in Figs. (19–20)

via the relation U1 Ymin,c,eð ÞzU2 Ymim,c,eð Þ~0 ; V c~E2

E1~1

� �for any given g. In the

alternative, for a desired level of energy dissipation achievable from uniform laminate sizes onecan conceivably obtain the same level of energy dissipation by using a cheaper material but nowvarying the thickness ratio of the upper to lower laminate to advantage.

8. SUMMARY AND CONCLUSION

In this paper we have revisited the problem of using a layered structural member as amechanism for dissipating unwanted vibration or noise, be it in an aerodynamic or machinestructure.

Earlier work had established that some of the factors influencing the level of energydissipation include the nature of the pressure distribution profile at the interface of thelaminates as well as the nature of the external force to which the structure is subjected.

In fact, prior to this paper it was well known that a negative pressure gradient in a cantibeamtends to increase the level of energy dissipation whereas an enhanced frequency ratio of thedriving load tends to reduce the amount of energy dissipation that can be arranged via slip atthe laminate interface. These observations however presume that both upper and lowerlaminates are of the same thickness and are made from the same material.

When such restrictions are removed, two new effects arise and are the subject of this paper.Our findings in fact confirm that each of these factors can independently be exploited toenhance the level of energy dissipation that can be arranged. In other words such increases canbe arranged either by using different materials for the upper and lower laminates in a prescribedfashion or by retaining the same material for both laminates but varying the individual ratios ofthe laminate thicknesses in a defined manner.

Another conclusion from the present work is that for effective energy dissipation, it is better tosimultaneously play with choice of the laminate materials and their thickness ratios rather than tinkerwith any one of them by itself. In fact there are many instances when choice of material alone eclipses

Fig. 20. Plot of Y1zY2~0 as a function of Y for g 5 0.5; and c 52.


whatever gains can be made from playing with interfacial pressure gradient. This underscores thewisdom in the search for composites in the construction of such laminates.

The strategy here is to exploit the advantage of composite structures to dissipate vibrationenergy via slip damping especially in aerodynamic structures where the effect of weight ofstructural member becomes significant.

The conclusion, then, is that for maximum energy dissipation, you need laminates of differentmaterials and of different thicknesses. This makes the use of composites inevitable.

These results can be positively exploited in the design of aerodynamic and machinestructures.

9. REFERENCES

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Vibration Analysis in Aerospace

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