inextensional beam vibration

7/27/2019 inextensional beam vibration

1/9

This docum ent is dow nloaded at: 2013-03-10T18:20:44Z

Title N onlinear V ibrations of Inextensional B eam s

A uthor(s) Takahashi, K azuo; N oguchi, Y utaka

C itation , (9), pp.35-42; 1977

Issue D ate 1977-07

U R L http://hdl.handle.net/10069/23887

R ight

N A O S ITE : N agasaki U niversity's A cadem ic O utput S ITE

http://naosite.lb.nagasaki-u.ac.jp


2/9

REPORTS OF THE FACULTY OF ENGINEERING, NAGASAKI UNIVERSITY No. 9 (1977) 35

Nonlinear Vibrations of Inextensional Beams

byKazuo TAKAHASHI* and Yutaka NOGUCHI**

In this paper, the nonlinear vibrations of inextensional beams are reportedbased 'upon the simplified equation of motion derived by Yoshimura andUemura. This problem is analyzed by the application of Galerkin's method,in which the deflection of the beam is assumed from a series of the productof the normal mode of the corresponding linear beam, and an unspecifiedfunction' of the time resulting in nonlinear ordinary differential equation issolved by the harmonic balance method. Numerical results are presentedfor the nonlinear free vibrations of the clamped-free beam and the free-freebeam.

1. INTRODUCTIONIn the previous researches on the non -

linear vibrations of beams, the nonlinearities were characterized either by nonlinear geometry (i. e., the usual linear.. 8 2 y ft'approxlmatlOn K = -8x 2- ' or curva ure ISreplaced by. K = ~ ~ , the nonlinear exactform, because of the presence of largedeformations), or by an axial tension whichdepends on lateral displacement. In thecase of beams having immovable endssuch as a hinged -hinged beam, clamped -hinged beam or clamped -clamped beam,the nonlinearity is caused by an axialtension and the small slope assumptionwhich reduces the exact expression for thebeam curvature to linear one is adopted .Such a beam is considered by WoinowskyKrieger' etc. 1- a) Th e governing nonlinear*Department of Civil Engineering**Departmen t of Structural Engineering

equation of motion is given by the wellknown one, the nonlinearity shows a hardening spring.On the contrary to those, beams havingfree-free ends, clamped-free ends or hinged -roller ends are assumed to be free tomove in the axial direction. The effectof large curvature is the nonlinear termin this problem. Under the assumptionof the inextensional motion, taking intothe rotatory inertia into account, freevibrations of abeam with hinged-rollerends were treated by means of a Rayleighmethod by Takahashi. 4) Yoshimura andUemura5 ) derived nonlinear equations ofmotion considering both the axial forceand the nonlinear curvature by usingvector analysis of strain, and solvednonlinear free vibrations of h i n g e ~ - r o l l e rbeam and clamped -free beam under theassumption of inextensional motion.


3/9

36 Nonlinear Vibrations of Inextensional Beams

Wooda1l6 ) considered a similar problem byusing a Galerkin method and a perturbationmethod and a finite difference method,and AtlurF) .investigated the nonlinearelasticity term arising from moderatelylarge curvatures, and nonlinear inertia.In hese analyses, the nonlinearity showsa weakly softening spring. On the otherhand, Wagner8 ) formulated a similar inextensional problem considering both longitudinal and transverse inertias by usinga Hamilton's principle and solved nonlinearfree vibrations of a clamped-free beam anda free-free beam. Kat09 ) solved nonlinearfree vibrations of a clamped-free beamwith Wagner's equation of motion. Aravamudan and MurthylO) used the samemethod and considered both contributionsof the nonlinear curvature and the axialtension. According to the Wagner typeapproach, the influence of the large amplitude on the frequency is similar to thatof a hardening spring and this result is notcoincide with the results of Takahashi etc.Wagner type approach is assumed to beerroneous because the subsidiary conditionis not taken into account correctly in theformulation of the basic equation.

Th e nonlinearity due to the nonlinearcurvature is smaller than that due to anaxial tension. The small parameter methodsuch as a perturbation method is employedas to a method of solution assuming that thesystem is single degree-of-freedom. The

ef fects of change of the vibration mode,the strain distribution and the higher orderterm of the deflection, are still not investigated.

In this paper, by using the simplifiedequation of motion derived by Yoshimuraand Uemura, nonlinear vibrations of aclamped-free beam and a free-free beam

are reported. A multiple degree-of-freedomapproach is applied to this problem by usinga Galerkin method and the harmonicbalance method.2. METHOD OF SOLUTION

The nonlinear equations for the undamped slender beam, which consider nonlinear curvature and axial tension androtatorY inertia derived by Yoshimura andUemura5 ), will be given as follows:

aT _ ( l + c) NK + (X-PA a2u)ax at 2(1)

y' ( a y) l +u '1 + c + Z - PA aP 1 + c = 0 , (2)aM a2 cpax + N +G-pI aP = 0 , (3)

where N, T and M are the axial force,the shear force and the bending momentat a point x, P is is the mass density perunit length, A the cross-sectional area ofthe beam, I the cross -sectional moment ofinertia, c normal strain, K curvature, uand y the deflections of any point x inthe axial and transverse directions, cp theangle between the tangent to the medianline and the x axis, and t the time, andx is space co-ordinate along the axis of abeam after deformation, and X, Y and Gare external forces.

By the aid of the vector analysis ofstrain, the exact curvature and strain ofthe neutral axis are given byK= { ~ " - ( 1 + ~ ) - ~ ~ } /ax2 ax ax2 ax

{ 1 + ~ ~ f + ( : ~ r} 3/2,C = {( 1 + ) + ( r} 1/2 - 1. (4)The assumptions used to further develop

the equations of motion are


4/9

T AKAHASH AND NOGUCHI 37

(1) The beam material is homogeneous,isotropic and obeys Hooke's law.

(2) The cross -section of the beam is uniform.

(3) A cross-section, originally plane, remains plane and normal to the deformedaxis.

(4) The deformation in the cross-sectionalplane is negligible.

(5) External and internal damping areneglected.

(6) The beam is initially straight.(7) Rotatory inertia is negligible.(8) Elongation of the middle plane is

negligible.(9) No external tangential force or moment

acts on the beam.The axial force T and the bending moment M are given by

T=EAc, M=EIK. (5)Using the above mentioned assumptions,Eq. 4 can be rewritten as follows:

K= //1-(aYlax)2, C=O. (6)Combining Eqs. (1), (2), (3), (5) and (6), theresulting equation is differentiated withrespect to the x co-ordinate, we obtain

Adopting the nonlinear terms up to anorder of five, the following equation forthe present problem is given by

P A ~ y - + EI { ~ ~ + ( ~ y _ ) 3 +2 ayat 2 ax4 ax2 axa2 y a3y (ay )3 a2y a3yax 2 ax3- +2 ax ax2--axS- +3

(8)A normal mode solution is assumed asfo11ows:

coy = e 2: Xi(x)Ti(t),

i =1(9)

where e is the length of the beam, Ti(t) an unknown function of the time, andXi (x) a space variable satifying the geometric boundary conditions of the beam,which denotes the associated linear problem. Therefore, it follows that

(10)where W i is the natural linear frequencyof i-th mode. From Eq. (8), the followingequation can be obtained with the help of Eq.(10) ,as

co coTn+an2Tn+ ~ ~ ~ ,enklmTkTITm

k= l l = lm= lco co

+ ~ ~ ~ 2: ~ r y n k l m q s T k T I T mk=11= lm=lq=18=1

T q Ts =POnCOSWTwhere

(11)

( I )2,en f ( dX k d 2X2a n = A. n A.l , klm= 0 2 ~ ~dX 2 m + d 2 X, d 2 X 1 d 2 Xm) X d ::1 ( 4 )d;2 d;2 (if2-- d;2 n s A.I Cnn _ f 1 ( dX k dX 1 dX md2Xqd 3 Xsry klmqs - 0 2 c r r - c r r d f ~ d ~ 3 -

3 d 2 X k d 2 X 1 d 2 Xm dX q dX s )+ d;2 -d;2 -df2CIr d ~ = - - Xnd;1(A. 14 Cn ),On= f:Xnd;/(A.14Cn),Cn=S:Xn2d;,T=Wlt,co= nlw 1 ,p=po e aiEl,A.l = ei/pAw l 2lEI, ;=xl e

Since there is no known exact solution ofEq. (11), the harmonic balance method isused here. For Duffing type of Eq. (11), asolution of the form is assumed as

coTn = ~ b S nCOSSWT,

8=1,3(12)

where b s n is an amplitude component andthe w requency ratio.


5/9


Substituting Eq . (12) into Eq. (11) and apllying the harmonic balance ~ m e t h o d , a se tof nonlinear algebraic equations will beobtained and solved by the Newton Raphson method.3. NONLINEAR FREE VIBRATIONS

The equation for the first second modescan be obtained from Eq. (11) in th e following form in th e cases of the clamped-freebeam and the free-free beam as shown inFig. 1

Fig. 1 Coordinate system an d definitionof amplitude2 2 2

Tn+ a n2T n+ ~ L: L: jjnklmT . TITmk=1 l=18=1

2 2 2 2 2+ ~ L: ~ ~ ~ 'YnklmqsTkTITm

k= l l= 1m=lq= l s= l(13)

Table 1 Coefficients of an , [ink 1m an d On of Eq. (13)n a.. 1l si 12 S'f22 S ~ 2 2 0.,.1 1.00000 -0.40891 ~ 4 . 8 0 4 3 0 -4.3887510 1 -0.40619 10 1 0.12667(a ) 1 12 6.26689 0.66595 -6.35774 -4.7436210 -1.3 5675 10 -0.070201 1.00000 -1.98812 -1. 0139110 2 -4.3429910 3 - 0.88648 10 4 0.0(b) -0.1622210 2 3 42 5.40392 0.61022 -0.32177 10 -0.7 3117 10 0.0

(a)clamped-free beam (b)free-free beam

Table 2 Coefficients of rnk Irrq s of Eq. (13)'Y\ '1"\ Jt\ Y"; 1 2 2 2 Y"j 2 2 2 2 Y ~ 2 2 2 2Y11111 Y11112 Yll122

1 0.66595 - 7 ~ 0 2 4 4 3 -0.8014 810 2 -2.916 88 10 2 -0.64398 10 3 -0.4463810 3(a ) 22 0.18923 -9.79669 -1. 2955510 2 -7.27552 10 -1.4279310 3 -1.29 853 10 3':""r1 -8.01996 -6.5085810 2 -4.2336610 4 -2.7573510 5 -0.39535 10 6 -7.3828410 6(b) 2 42 1.22059 -1.0575210 -0.5137 210 -1.4151810 5 -Q. 752061 06 -4.1156 510 6

(a)clamped-free beam (b) free-free beam

Coefficients an, jjn klm , 'YnklmqS and On aresummarized in Tables 1 and 2. In thesecases, the value c in Fig. 1 is taken tounity for each mode. Employing the firstsecond harmonics as to time variable, Eq .(12) ca n be e x p r e s s e d ~ a s i.ollows:

(14)Substituting Eq. (14) into Eq. (13) and usingthe harmonic balance method, the followin g four nonlinear coupled equations areobtained as(a 2 n - 00 2 )a n+ jjn 1 1 1f 1 1 1 + jjn 11 2 11 2 + jjn 122

f 122 + f3n 2 2 2 222 + 'Yn111 1 1 1 1 1 1 1 + 'Yn111 12f 1 1 112 + Y n1 1122 f 11 122 + 'Yn11 2 2 f 11222 +'Yn12222fI2222+'Yn22222f222,22=OnP, (a 2 n- 900 2) b n + jjn 1 1 1gil l + jjn 112g il 2 + jjn 122g 122 + jjn 2 2 2g 222 + 'Yn11 1 1 1gIl l 1 1 + 'Yn1 1 112g il 112 + 'Yn1 1122gil l 2 2 + 'Yn11222 gil 2 2 2 +'Y n 12 2 22g 12 2 2 2 + 'Yn 2 2 2 ,2 g 2 2 2 2 = 0 , (15)where

1f 1 Jk = -4-{3ala jak+2(a lbjb k+ b laj b k+bibjak)+aia jbk+albjak+bia jak} ,


6/9

TAKAHASHI AND NOGUCHI 39

1giJ k = -4-{a ia ja k+3b I b jb k +2(a iaj b k+aibjak+biajak)} ,

1f i j k Im=16 {lOa 1aja ka lam+6b ib j bkb Ibm+5(a i aja ka lbm+aiaj akblam+aiajbka lam+ a I b j a ka I am+ b a j a ka Iam) + 6 (a a jakblbm+aiaj bka lbm+a lbjaka lbm+ b iajaka lbm+alaj bkb lam+aibjakblam+ b Ia j a kb I am+ a i b j b ka Iam+ b i a j bka Iam + b Ibjakalam+3(b ib jbkalam+ b lbjakblam+ biaj bkblam+albj bkb1am+ bibjaka lbm+ b 1aj bka lbm+a ib jbka lbm+a lb jb ka lbm+ b iajakb Ibm+aibjakb bm+aiajbkblbm) +6(b ib jbkb lam+ bib jbka 1bm+ b ib ,akb lbm+ biaj bkblbm+aibjbkblbm )} ,

1gl j k Im=16 {5a iaja ka lam+ 10b ib jbkb lbm+6(a 1aja ka lbm+aiaj akb lam+a1aj b kalam+a ibjakalam+ b ia'j akalam ) +3(a iajakb I bm+aia.j bka lbm+a ibjaka lbm+b iajakalbm+aiaj bkblam+a ibjakb lam+b la j akb lam) +6(b b jbkalam+ bib jakb

am+biaJ bkb lam+a i b jbkb lam+b ibja k'a lbm+ b ia j bka lbm+aib Jbka lbm+ b iajakb I bm+a ib jakb lbm+aiaj bkb Ibm)} .

Th e non dimensional amplitude is definedby the following equation when the timeWT is equal to n7t(n=O, 1,2, .... ;.)

(16)

(1 ) Clamped -Free BeamFrequency' ratio co versus various ampli

tude ratio c/ t of the first natural frequency when the nonlinear term of the fiveorder is neglected is shown in Table 3(a). As the nonlinear restoring force/3 1 111 of the first natural frequency isnegative as shown in Table 1 (a ) , i t isevident from Table 3 (a) that the influenceof the large. amplitude is similar to thatof a softening spring. The frequency ratiodecreases with increase of the amplituderatio. Th e frequency ratio in Table 3 (b )shows the frequency ratio of the corresponding solution of the single degree-of-

Table 3 Frequency ratios of th e clamped-free beam

cIt Ca) (b ) (c ) (d ) (e) (f) (g )0.0 1.0000 1. 000'0 1.0000 1.0000 1.0000 1. 0000 1.00000.1 0.9985 0.9985 0.9985 0.9938 0.9980 0.9977 1.0-0T50.2 0.9939 0.9939 0.9937 0.9752 0.9918 0.9908 1.00610.3 0.9862 0.9861 0.9854 0.9432 0.9815 0.9793 1. 01380.4 0.9756 0.9752 0.9729 0.8965 0.9669 0.9632 1. 02450.5 0.9620 0.9610 0.9554 0.8327 0.9478 0.9425 1.0383

(a) present solution(2 degree of freedom,2 term solution)(b) present solution(l degree of freedom) ,perturbation method(c) present solution(2 degree of freedom,Z term solution),higher order solution(d) energy method by T.Takahashi(e ) Galerkin method by Yoshimura and Uemura(f) Ga1erkin method by Bau.er(h) perturbation method by Kato


7/9


freedom obtained by a perturbation method.As the magnitude of the nonlinear term inthis problem is small, it will be seen fromTable 3 (a) and :(b) that the differencebetween the multiple degree-of -freedomsolution and the single degree-of -freedomis very small. This means that the effectof the change of the normal mode due toamplitude on frequency is small.

The frequency ratio in Table 3 (c) showsthe frequency ratio which considers thenonlinear term up to an order of five.The solution of Table 3 (c) becomes slightly smaller than that of Table 3 (a). Thatis , the effect of the order of five is greaterthat of the change of the mode.

Table 3 (d), (e), (0 and (g ) show results obtained by various basic equationsand methods of solution, i. e., (d ) showsthe Takahashi's solution obtained by theenergy method which neglects the axialdisplacement and uses an approximatecurvature JC = {I - +( r} ,(e)shows Yoshimura and Uemura's resultwhose basic equation is the same one upto an order of three of Eq. (9) and solvedby a Galerkin method. However, the spacefunction is not coincide with the beamfunction. (0 shows the result obtained byBauer' basic equation which neglects theaxial displacement and inertia force. The

Table 4 Normal mode of th eclamped-free beam

~ 0.0 0.1 0.30.0 0.0000 0.0000 0.00000.2 0.0639 0.0639 0.06450.4 0.2299 0.2301 0.23140.6 0.4611 0.4613 0.46290.8 0.7255 0.7256 0.72661.0 1.0000 1. 0000 1. 0000

0.50.00000.06570.23440.46620.72861. 0000

~ 0.0 0.1 0.3 0.50.0 1.0000 1.0000 1.0000 1.00000.2 0.7254 0.7260 0.7304 0.7394

0.4 0.4611 0.4625 0.4738 0.50810.6 0.2299 0.2307 0.2385 0.27400.8 0.0639 0.0640 0.0656 0.07841.0 0.0000 0.0000 0.0000 0.0000

Table 5 Strain distribution of th eclamped-free beam

last column (g) shows the result obtainedby Wagner's basic equation. The nonlineareffect shows a hardening spring type inthis case only. This discrepancy arises froma incorrect variational problem.

The normal mode and strain distributionof the clamped -free beam are shown inTables 4 and 5. I t will be seen that thesevary slightly from those of the linear one.

(2) Free-Free BeamFrequency ratios of the free-free beam

are shown in Table 6. Table 6 (a ) showsthe frequency ratio of the present solutionwhich considers the nonlinear term up toan order of three, and (b ) shows the ratioconsidering the contribution of an orderof five , while (c) shows' the ratio of thecorresponding solution obtained by thesingle degree-of -freedom approach. The

Table 6 F r ~ q u e n c y ratios of thefree-free beam

c/R- (a ) (b )0.0 1.0000 1.00000.1 0.9926 0.99230.2 0.9706 0.96700.3 0.9360 0.91790.4 0.8907 0.84010.5 0.8383 0.7200

(c)1. 00000.99250.97020.93290.88070.8136


8/9

TAKAHASHI AND NOGUCHI 41

nonlinear term ( 3 1 1 1 1 of the free-free beamis_ greater than that of the clamped-freebeam as shown in Table 1 (a). The frequency ratios show that the free-free beamexhibits much change in frequency withamplitude than does the clamped-free beam.The difference between the multiple degreeof -freedom solution and the single degree-

of-freedom solution comes out with increase of amplitude. This means that themode of vibration which changes withamplitude modifies the frequency. Thecontribution the order of five can not benegligible as shown in Table 6 (b).The normal mode of the free-free beamis shown in Table 7.

Table 7 Normal mode of th e free-free beam

. ~ 0.0 0.1 0.2 0.3 0.4 0.50.0 1.0000 1.0000' 1.0000 1.0000 1.0000 1.0000

0.1 0.5372 0.5373 0.5391 0.5433 0.5475 0.5568

0.2 0.0977 0.0987 0.10D5 0.1063 0.1129 0.1228

0.3 -0.2720 -0.2718 -0.2712 -0.2695 -0.2670 -0.26720.4 -0.5202 -0.5213 -0.5233 -0.5307 -0.5372 -0.5489

0.5 -0.6078 -0.6096 -0.6130 -0.6240 -0.6368 -0.6530

4. CONCLUSIONSIn this paper, the nonlinear vibrations

of the clamped-free beam and the freefree beam are analyzed. A summary ofthe results is as follows:(1) Clamped -Free Beam(a) The influence of the large amplitudeon the frequency of the clamped -free beamis similar to- that of a weakly softeningspring.(b ) The influence of the change of thenormal mode with the amplitude on thenonlinear frequency is small. The difference of the solutions between the mUltiplemode approach and the single mode approach is small. The solution of the multipIe mode approach gives slightly smallerfrequency ratio than that of the singlemode approach.

(c) The effect of the order of five ofthe deflection modifies the frequency ratioobtained by the solution of the order ofthree.(d) The previous solutions give differentresults. This reason is due to the occasionwhen the higher term is neglected or aincorrect variational problem.(e ) The normal mode and strain distribution vary with amplitude. These changes are cosiderably small.(2) Free -Free Beam(a ) Frequency ratios of the free-free beamare much 'affected by the amplitude thanthat of the clamped-free beam.(b) The frequency ratio must be calculated by the multiple mode approach whichcosiders the higher order of the deflection.


9/9


REFERENCES(1) Woinowsky-Krieger; S., The Effect of

an Axial Force on th e Vibration of HingedBars, J. App1. Mech., Vol. 17 (1950), pp.35-36.

(2) Sato, K., Nonlinear Vibration of Beamswith Clamped Ends and with One EndClamped, Other End Simply Supported End;,Trans. Japan Soc. Mech. Engrs., Vol. 34,No. 259 (1968), pp. 418-426

(3) Takahashi, K., Nonlinear Free Vibrationsof Beams, Theoretical an d Applied Mechan

. ics, Vol. 24 (1976), pp. 109-120(4) Takahashi, T ., On th e EffeCt of Elastic

Curvature on Large Amplitude Frequencyof Elastic Bars, J. Japan Soc. Aeronautics,Vol. 10, No. 95 (1943), pp. 189-193.

(5) Yoshimura, Y. and Uemura, M., TheEffect of Amplitudes on th e Lateral Vibrationof Rods, Rep. Inst. Sci. Technol. Tokyo,Vol. 2 (1948), pp. 57-61.

(6) Woodall, K., On th e Large AmplitudeOscillations of a .Thin Elastic Beam, Int.J. Nonline,ar Mech. Vol. 1 (1966), pp.217-238.

(7) Atluri, S., Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects, J. Appl. Mech., Vol. 40 (1973), pp.122-126.

(8) Wagner, H., Large Amplitude Free Vibrations of a Beam, J. Appl. Mech., Vol. 32(1965),PP. 887-892.

(9) Kato, M., Large Amplitude Vibration ofa Clamped-Free Beam, Trans. Japan Soc .Mech. Engrs., Vol. 34, No. 257 (1968), pp.67-72.

(10) Aravamudam, K. S. and Murthy, P. N.,Nonlinear Vibration of Beams with TimeDependent Boundary Conditions, Int. J.Nonlinear Mech., Vol. 8 (1973), pp ... 9 S ~212.

inextensional beam vibration

Documents

Transcript of inextensional beam vibration