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ApplicationoftheRitzmethodtotheanalysisofnon-linearfreevibrationsofbeams

ARTICLEinJOURNALOFSOUNDANDVIBRATION·DECEMBER1987

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Journal of Sound and Vibration (1987) 114(1), 91-101

APPLICATION OF THE RITZ METHOD TO THE

ANALYSIS OF NON-LINEAR FREE VIBRATIONS OF BEAMS

R. LEWANDOWSKI

Technical University of Poznan, Poznan, Poland

(Received 23 May 1985, and in revised form 1 May 1986)

This paper presents an analytical solution for geometrically non-linear free vibrations of beams with elastically supported ends in the horizontal direction. The equation of motion is obtained by employing Hamilton's principle and assuming that horizontal inertia forces can be neglected. The Ritz method, with a continuum solution and an iterative procedu'e, are used for determining the frequencies and non-linear modes of vibrations. The orthogonality conditions for these modes are also discussed. Numerical results for various beam boundary conditions are presented and compared with available results.

1. INTRODUCTION The problem of non-linear free vibrations of beams with immovable ends has attracted many researchers in the past. Both a continuum approach [1-6] and finite element methods [7-12] have been used.

The solution can be obtained in three ways. The assumption that the non-linear mode shape of the vibration is the same as the linear one is the basis of the first method (seereferences [1,3-5]). This enables one to write a modal equation of Duffing's type (intime) whose exact solution is a cosine elliptic function, and a period or frequency of the oscillation is given by an elliptic integral.

In the second class of solutions [6,7,10,11,13], the time factor of the problem isassumed to be a cosine harmonic function. The frequency and mode of vibrations areobtained by solving approximately the differential equation in the space variable.

Recently Bhashyam and Prathap [8], and Sarma and Varadan [9] reduced the problemto a so-called non-linear eigenvalue problem by the assumption that w(x, t) = -a^w^x, /)at the point of a maximum amplitude of a transverse displacement w, and interpreted <unot as a frequency but as a coefficient of proportionality between w and w in this point.The solution of the non-linear eigenvalue problem defines <D2 and a non-linear mode ofvibration.

The idea of a non-linear normal mode was introduced by Rosenberg [14]. Szemplinska-Stupnicka [15-18] used this mode to analyze several types of non-linear vibrations, andincluded an analysis for beams.

This paper presents an analytical solution for geometrically non-linear free vibrationsof beams with their ends elastically supported in the horizontal direction. The problemis formulated as a variational one, Hamilton's principle being used, and the solution ofthe problem is found by using the second of the three classes of methods described above.The Ritz method, with a continuum solution and an iterative procedure, is used fordetermining any frequencies and non-linear modes. The orthogonality conditions for

910022-460X/87/010091+11 $03.00/0 @ 1987 Academic Press Inc. (London) Limited

92 R- LEWANDOWSKI

these modes are discussed too. At the end, numerical results for various beams includingelastically supported ones are presented and compared with available results.

2. EQUATION OF MOTION The non-linear strain displacement relations of a beam are (see reference [8])

e^=^9w/Sxf+Su/8x, ^=8tw/^xl, (1)

where e^ is the axial strain, i/^ is the curvature, and u and w are the axial and transversedisplacements, respectively. Using Hooke's law one can write the following relation for

the elastic strain energy U:

U= \ ^EAel+Wl+Ss^dx

Jo

. ['^[^i^yi^^^y^^y^nidx (2) h[2\.Sx 2\Qx} j 2 \Qx2} \.9x 2\Qx)\}

Here E, A, I, S, and ; are Young's modulus, the area of the cross-section, the moment of inertia of the cross-section, the initial axial force, and the length of the beam,

respectively. The kinetic energy of the beam is given by

T^f'O^+^df, (3)

2 Jo

where m is the mass per unit length of the beam. Applying Hamilton's principle, 6\',\ L d( = 0, to the Lagrangian L=T-Uofthe system

gives @/^4w/ax4-5^2w/^x2-(^/^x){@A[au/^x+i(^w/flx)2](^w/^x)}+mfl2w/<5^2=0,

m Q^u/Qt2 - W9x){EA[Qu/9x+^9w/9x)2]} = 0. (4)

With the horizontal inertia forces omitted in the second equation of the system (4), one

has

N=EA[9u/^x+^9w/Qx)2]= const. (5)

Integrating expression (5) between the limits 0 and / and taking into consideration that u(0, t) = KoN(.t) and u(l, t) = -K,N(t) one can write

EA C'/awV ,f,\ N=@- (@) dx, (6)

UK Jo \9x/

where K =I+(EA/I)(KO+K,), and KO, and K, are the flexibilities of the left and right ends,.respectively. It is evident that the axial force depends only on time and is independent

of x. Inserting expressions (5) and (6) into the first of equations (4) yields the equation of

motion for a large amplitude free oscillation of the beam, in the following form:

^w ^w r ^['(^YA '\Qlw n m ^,^^EI^-[S^\^) dxj^=0. (7)

NON-LINEAR FREE VIBRATIONS OF BEAMS 93

3. SOLUTION

The aim here is to find the periodic solutions of the problem. The approximate solution

thus is assumed to have the form

w(x, t) = av(x) cos at, (8)

where v(x) is a so-called "non-linear normal mode" (see references [14,18]) and v(x) = 1

for freely chosen x = x, a denotes the amplitudes of vibrations at the point x = x, and w

is the radian frequency of the oscillations. The solution (8) satisfies the initial and boundary

conditions but it does not satisfy exactly equation (7). Applying the Ritz method yields

the condition

Joe(x, () cos <u? d(<t)() = 0,(9)

where s(x, t) denotes the residual of equation (7) after substituting into it the approximatesolution (8). From equation (9) one obtains

"@^-^[^J^HS-",or, in a dimensionless form,

d\/d^-y2d\/d^-t3^=0, (11)

where

S=x/l, ^=^ml'/EI, y^l^C+S^EI, ^=|,

C = (/MEAa2/!!2^ [' (dv/dfi2 df, (12)

Jo

which constitutes, with boundary conditions, a so-called "non-linear eigenvalue problem". A similar equation with p. = 1 may be written if the following properties of the time

function at the point of the reversal of motion (the point of maximum amplitude) are used:

w^(x, ;i) = -w^^x, tt), w(x, ti) = 0. (13)

This method was first suggested by Mei [7], The solution of equation (11), or (10), has the form

v(fi = Di cosh 8^+ DZ sinh 8^+ D-,, cos v^+ D^sin v@, (14)

because C in equation (12) can be treated as an unknown constant and equation (11)can thus be regarded as a linear one. The above D, denotes constants dependent onboundary conditions, and

8=^/^/(74/4)+/34-^-(y2/2), ^yT^^^CTTz). (15)

The parameters 8 and c are functions of C, and will be determined, for a given value of

C, from the characteristic equation, which can be generally written in the form

/(W,r03))=0. (16)

This equation can be obtained in the same way as in linear dynamic analysis and it is

possible to solve it numerically.

94 R. LEWANDOWSKI

The proposed method for determining C is an iterative one. Assume, that a certain

approximation for C is known, denoted by C, (f is the index of the iteration steps).

Initially C = 0 or is equal to C for the previous value of a, if one is calculating C for

the set of values of a. Equations (11), (12), and (14)-(16) are used to compute OyV,, o,(^),

Cf+i corresponding to the chosen C,-. Inserting C = C,+i one repeats the iteration until

[C,+i-C,|s@eC;.

By using expressions (6) and (8) the dimensionless amplitude of the axial force

l^N^JEI can be written in the form I2 N^/ El@@ (a/I)2 ('(dv/d^dS.

2.K Jo

One can see that it does not depend explicitly on the parameter ^. The following three end conditions cases have been studied.

3.1. SIMPLY SUPPORTED BEAM

For the end conditions u(0)=u(l)=0 and El d^v/d^^o^EI d^/d^^O, one

has @>i = @>3 = 0, and Di sinh 8 + D^ sin v = 0 and S2D@^ sinh 8 - v^D^ sin v = 0, which leads

to the following results;

v=n-n-, v(S)=D4sin-irnf, C =(EAa2/4l2K)t^,n2'JT2D'i,

w2=a)2+(EAa2fMn4v4/4l2Km)Dl, n=l,2,..., (17)

where co2=(n2@rr2/l2ml(n2@^@2/l2)EI+S~\ denotes the linear frequency of the beam and

D4 = 1/sin -n-nf, i=x/l, has been obtained from the condition u(^) = 1.

In this case the iterative process is superfluous and the mode of vibration is independent

of the vibration amplitude.

3.2. SIMPLY SUPPORTED-CLAMPED BHAM

For this case the end conditions u(0)=u(l)=0, El d2v/df\^o=0, and du/d^iO

are satisfied if Di = Dy = 0 and

Dz sinh 6 + 04 sin v = 0, oD^ cosh 5 + D^ v cos v = 0. (18)

The characteristic equation and the mode of vibration have the forms

tgh8-(8/v)tgv=0, (19)

v(^) = Dy. sinh 8^+ D^ sin v^, (20)

where one obtains D^ and Ds, by solving the equations D; sinh 8 + Ds, sin v = 0 and

Dzsinh 8i"+D4sin v^= 1. Inserting equation (20) into the fourth of equation (12) gives

EAa2/^/ , , 8 + cosh 5 sinh 8 _, 8 sinh 5 cos v+ v cosh 8 sin v C=^[D28@@@@28@@@@+2WSV@@@@@@82^2@@@@@@

^^+sin^cos.\ ^

NON-LINEAR FREE VIBRATIONS OF BEAMS 95

3.3. CLAMPED-CLAMPED BEAM

Moving the origin of co-ordinates to the middle of the beam and applying the boundary conditions v(-0-5)= v(0-5)=0, and du/d^=i/2=dy/d^i/:>=0, one can write

D, cosh (5/2) + D3 cos (v/2) = 0, SD, sinh (8/2) - Dj v sin (v/2) == 0,

Dz sinh (5/2)+D4 sin (y/2)=0, 5D;,cosh (5/2)+D4i/cos (v/2)=0, (22)

and the characteristic equation takes the form

[tg(@'/2)+(5/r)tgh(5/2)][(5/y)tg(^/2)-tgh(5/2)]=0. (23)

Symmetric and antisymmetric modes of vibrations are expressed as

v(^) = D, cosh 8^+Dy cos v^ v(f) = Dy sinh 8^+ D^ sin v^ (24a, b)

where D, satisfies the equations

D, cosh (5/2) + Ds cos (v/2) = 0, Di cosh 5^+ Dy cos yj= 1, (25a)

DZ sinh (5/2)+D4 sin (2//2)=0, Dzsinh 5^+D4sin vH= 1. (25b)

The constant C is equal to

EAa2^ f 2 / . 5 ,55\ 4(/5 _ _ 5 f/ v S\ c=^/2^LD@Tmh2cosh2-2)-,^52DlD3cosh2cos2(5tg2-ytgh2}

n /y c v\~\ @^Dsvl^-sm^cos-^} \, (26a)

gAa^f -/ . 5 ,55\4^__ 5 vl 8 v\ C = @^- [^D^^smh ^ cosh ^+^ +-^r^ 0^4 cosh ^ cos ^ ( 5 tgh ^+ v tg ^1

-.-, I v sin y y\"l +D^^2+^-COS2)J' (26b)

for the symmetric and antisymmetric modes, respectively. It can be noticed that the differences between the linear and "non-linear" modes of

vibrations depend only on the parameters 5 and v. It is well-known that the modes of vibrations thus obtained for a simply supported

beam are orthogonal (see reference [4]) but for the other cases the situation is not clear.The problem can be briefly discussed as follows.

Assume that w,(x, t) and w,(x, t) are two different known solutions with correspondingaxial forces N, and N,, amplitudes a, and a,, frequencies a, and &@@ and modes ofvibration v,(x) and v,(x). The following relationship then is to be satisfied:

| w,(x,t)im^(t)+EI^-[S+N^,t)]^}dx=0. (27)

@'0 I OX r)x )Substituting equation (8) into equation (27) shows that condition (27) is not satisfied,because equation (8) is only an approximate solution of equation (7). It will be satisfied,however, if the procedure of the Ritz method is applied to equation (27). One then obtains

J:-^^'-^-^^!:^)'-]^}-0- ^Proceeding similarly, one can write

^,W[E,^-^M-\S^ fWJ^L.-o. (29) @'o I dx L 2/K Jo \dx/ J dx- J

96 R. LEWANDOWSKI

Adding expressions (28) and (29), integrating by parts and using the boundary conditions

gives

^-^j;^.^^!:^)2^^:^)2.]!;^,.((n2-^2) wzw Jo ' ' -Jo 2lK L Jo \dx/ Jo \<"/ J Jo aj.

This condition will be satisfied, and then the "non-linear modes" of vibration will be

orthogonal, if d^vjdx^-^av^m, (30)

or if @A^f'/d^V^@A^r'/d^\2^ ^

"11.. \ \ Av I '>lie - V (\v i

@A^f'/d^\2 @A^r'/d^\2^

2lK Jo \dx/ 2/K Jo \dx/

Condition (30) is satisfied for a simply supported beam and for freely chosen a, and a,.For a simply supported-clamped and clamped-clamped beam, the "non-linear modestof vibration will be orthogonal if for some given values of a, and a, the axial forces N,and Nr (averaged over time) are equal.

4. NUMERICAL RESULTS

First, in this section, the results for a simply supported beam obtained by the Ritz and

Mei methods are compared with an exact solution. For the fundamental frequency, the

relation (17) between the ratios W/WL and a/i can be rewritten in the form

((./^l+^MXa/O2, (32)

where i=v7/A, ^=0-5, K=I and S=0. The exact backbone curve equation is given by (see references [1,13])

(^W^/W+i^/O2]/?2^2, Tr/2), (33)

where k2=3i(a/i)2/2[l+3t(a/i)2] and F( ) is the elliptic integral of the first kind. Figure

1 shows the ratio W/WE for /A = 0-75 (Ritz method) and ^ = 1 (Mei method). Both methods

give too large values of w, but for the Ritz method the error is considerably smaller. The

Figure 1. Increase of the fundamental frequency error for the Ritz (@@) and Mei (---) methods vs.

amplitude of the vibration ratio a/i.

NON-LINEAR FREE VIBRATIONS OF BEAMS 97

Mei method is, in fact, a collocation-type method with freely chosen collocation points

?=0, n/u, 2-ir/w. The approximate solution parameters determined on the basis of this

method do not necessarily assign a minimum value to the functional J;2 Ldt. In respect

to the solution parameters, the minimum of the functional is obtained in the Ritz method

(see reference [18]), and for that reason@when the same form of approximate solution

is used in the two methods@the error of the non-linear frequency of vibrations obtained

by using the Ritz method is smaller.

Recently Bhashyam and Prathap [8], and Sarma and Varadan [9] interpreted w2 not

as a frequency but only as a proportionality coefficient between w and w at the point of

maximum amplitude of the transverse displacement. In this context a method for calculat-

ing the frequency, which of course is a very important characteristic feature of the motion,

is not generally available. For a simply supported beam the relation between the period

of vibrations and this coefficient depends on the ratio (a/i) (see Figure 1). For the reasons

mentioned above, the author prefers the Ritz method.

Table 1 gives the non-linear frequency ratio (w/w,)2 for the fundamental mode of a

simply supported beam as obtained by several methods. It can be observed that the Ritz

method is in good agreement with the exact results and those obtained by Mei [7], who

made use of a finite element formulation. The differences between the results obtained

for /j.=l from equations (31) and (32) are presented in Figure 1. As was discussed

recently [19], in the finite element formulation presented in reference [7] there are a few

inaccuracies which have probably reduced the errors shown in Figure 1 to small values.

TABLE 1

Comparison with published results for a simply supported beam

(w/wi)2

Present analysis Rao et at. Mei Srinivasan Cheung Exact

a+/( /x=0-75 [10] [7] [2] [18] [1]

1-0 1-1875 1-1855 1-1857 1-1874 1-1864 1-1864 2-0 1-7500 1-7211 1-7379 1-7477 1-7366 1-7366 3-0 2-6875 2-5670 2-6439 2-6798 2-6422 2-6429

+ a is the amplitude in the middle of the beam.

Tables 2 and 3 give the variation of the non-linear frequency ratio (<u/<u,)2 and the

amplitude of the non-linear axial force ratio (Nl2/ El) with an increase of amplitude of

the vibration ratio (a/i) for simply supported-clamped and clamped-clamped beams,

respectively. In both tables the new results for the fundamental mode are compared with

others presented in references [5,9]. Good agreement is achieved for the axial force, but

there are significant differences between the frequency values reported in reference [9]

and those presented here. As can be observed, the normal force amplitude does not

depend appreciably on the value of the parameter /z. Tables 4 and 5 present (w/w,)2 for four dominant modes for simply supported-clamped,

and clamped-clamped beams, respectively. It can be seen, that the second and fourth

frequencies for a clamped-clamped beam change with the amplitude ratio in the same

way as the first and second frequencies fora simply supported-clamped beam, respectively.

Shown in brackets are results obtained by Rao et al. [10]. The normal force amplitudes

presented in Tables 4 and 5 are correct only when one can neglect the interactions among

particular forms of vibrations.

98 R. LEWANDOWSKI

TABLE 2

Frequency and axial force ratio for a simply supported-clamped beam (5==0)

(w/u,)2 . N^/EI

'@@@@@@@@@@@@@@@p^^' present

Sarma et at. Evensen analysis Sarma et al. analysis

at/i [9] [5]_______^=0-75_______[9]_______^=0-75

0.2 1-0053 1-0040 1-0040 0-1103 0-1103 0.4 1-0213 1-0160 1-0610 0-4407 0-4408

0.6 1-0479 1-0360 1-0360 0-9902 0-9908 0-8 1-0850 1-0641 1-0638 1-7569 1-7588 1.0 1.1323 1-1001 1-0995 2-7387 2-7433 1.5 1-2947 1-2253 1-2220 6-1167 6-1365

2.0 1.5175 1-4005 1-3908 10-7771 10.8303 2.5 1-7978 1-6258 1-6038 16-6787 16.7863 3.0 2-1331 1-9011 1-8592 23-7907 23.9716 3.5 2.5217 2-2265 2-1556 32-0941 32-3620 4.0 2-9625 2-6020 2-4919 41-5779 41-9428

4.5 3-4550 3-0275 2-8675 52-2454 52-7063 5.o 3.9989 3-5031 3-2819 64-0989 64-6515

+ a is the amplitude in the middle of the beam.

TABLE 3Frequency and axial force ratio for a clamped-clamped beam (5=0)

(^Nl2EI~

/;PresentPresent Sarma et al.EvensenanalysisSarma et al. analysis

a\li [9][5]^=0-75[9]_______^=0-75

0.2 1-0024 1-0018 1-0018 0-0955 0.09760.4 1-0096 1-0072 1-0072 0-3902 0-39020.6 1-0216 1-0162 1-0162 0-8779 0-87790.8 1.0383 1-0288 1-0287 1-5606 1-56061.0 1-0598 1-0450 1-0449 2.4381 2-4383

(1-0439)1.5 1.1343 1-1012 1-1009 5.4839 5.48472.0 1-2382 1-1800 1-1789 9-7452 9-7474

(1-1731)2.5 1.3708 1-2812 1-2788 15-2203 15-22473.0 1-5320 1-4050 1-4002 21.9087 21-9156

(1-3820)3.5 1-7211 1-5512 1-5430 29-8106 29-82004.0 1.9377 1-7200 1-7066 38-9281 38-93724.5 2-1816 1-9112 1-8910 49-2639 49-27045.0 2-4522 2-1250 2-0957 60-8218 60-8208

+ a is the amplitude in the middle of the beam.

NON-LINEAR FREE VIBRATIONS OF BEAMS 99

TABLE 4

Frequency ratio for a simply supported-clamped beam (5=0)

________________(wf^f__________

First mode Second mode Third mode Fourth modea/i x=0-5l x= 0.251 jc=0-81 jc=0-61

0-2 1-0040 1.0058 1-0056 1-00660-4 1-0160 1-0231 1-0225 1-02650-6 1.0360 1.0517 1.0506 1-05920.8 1.0638 1.0915 1-0901 1-10491-0 1-0995 1-1421 1-1411 1-1630

(1-1080)1-5 1-2220 1.3142 1-3190 1-36162-0 1-3908 1-5487 1.5694 1-6345

(1.4021)2-5 1-6038 1.8434 1-8919 1.98103-0 1.8592 2-1970 2-2855 2.4018

(1-8387)3-5 2-1556 2-6096 2-7497 2.8981

TABLE 5

Frequency ratio for a clamped-clamped beam (5=0)

_______________(^)2______

First mode Second mode Third mode Fourth modea/i x=0.51 x=0x251 x=0.51 x= 0-6251

0-2 1-00181.00401-00511-00580-4 1-00721.01601-02031-02310-6 1-01621-03601-04541-05170-8 1-02871-06381-08021.09151-0 1.04491-09951-12461-1421

(1-0439)(1-1080)(1-1319)1-5 1-10091-22201-27541.31422-0 1-17891.39091.48081.5487

(1-1731)(1-4021)(1.4864)2-5 1-27881.60381-73871.84343-0 1.40021-85922-04842-1970

(1-3820)(1-8387)(2.0150)/3-5 1.54302-15562-40972-6096

The effects of a variable axial restraint and slenderness ratio (//;) for a simply supported-clamped beam have been discussed recently by Prathap [20]. He has found that a softeningtype of beam behaviour takes place at low values of spring constant and slendernessratio, while at high values of these parameters a hardening type of behaviour occurs. Thebackbone curves for a few values of the parameter 5,= (KO+K,)(EA/I), for the funda-mental mode, are shown in Figures 2-4. It is evident that there are also significantnon-linear effects in cases of large support flexibility.

All the above results were obtained for e = 0-0001, and five or less iterations are neededto achieve this accuracy.

R. LEWANDOWSKI

1.0 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 1-9

<>>/0>[

Figure 2. Variation of tii/ai, for a simply supported beam for various a/i and S,.

1-0 1-1 1-2 1-3 1-4 1-5 1-6 1-7 tU/fa^

Figure 3. Variation of w/a, for a simply supported-clamped beam for various a/i and S,.

Figure 4. Variation of M/&), for a clamped beam for various a/i and 5,

NON-LINEAR FREE VIBRATIONS OF BEAMS 101

5. CONCLUSION

The Ritz method has been used for studying non-linear free vibrations of beams. The

analytical solution for a non-linear eigenvalue problem is given, and the orthogonality

conditions for non-linear modes of vibration are presented. The effects of both large

amplitude and partially movable ends have been considered.

The Frequency increases considerably with the amplitude of vibration when the support

has a large flexibility in the horizontal direction. It is also of interest that for the

fundamental mode shape parameter 5 ss 0-05, the beam considered behaves like one with

immovable ends.

REFERENCES

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14. R. M. ROSENBERG 1966 Advances in Applied Mechanics 9, 154-242. On non-linear vibrations of systems with many degrees of freedom.

15. W. SZEMPLINSKA-STUPNICKA 1975 International Journal ofNon-linear Mechanics 10,289-304. A study of main and secondary resonances in non-linear multi-degree-of-freedom vibrating systems.

16. W. SZEMPLINSKA-STUPNICKA 1978 Journal of Sound and Vibration 58, 347-361. The general- ized harmonic balance method for determining the combination resonance in the parametric dynamic systems.

17. W. SZEMPLINSKA-STUPNICKA 1980 International Journal of Non-linear Mechanics IS, 407-415. The resonant vibration of homogeneous non-linear system.

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