On nonlinear free vibrations of simply supported uniform beams

Journal of Sound and Vibration (1992) W(3),

527-531

ON NONLINEAR FREE VIBRATIONS OF SIMPLY SUPPORTED UNIFORM BEAMS S. R. R. PILLAI AND B. NAGESWARA RAO Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum-695 022, India (Received 31 January 1991, and in$nalform

12 August 1991)

The problem of large amplitude free vibrations of simply supported uniform beams with immovable ends is examined. The equation of motion for this problem is found to be of Dufling type. The solution of the equation of motion; that is, the frequency us. amplitude relation, is examined here by several methods including the perturbation method, the elliptic function method, the harmonic balance method, and the method in which one assumes simple harmonic oscillations. It is found that the second approximation of the harmonic

balance method yields good results which are in excellent agreement with the exact solution.

1. INTRODUCTION

In recent years Gajbir Singh et al. [l] have discussed various existing analytical formulations and presented results for the problem of large amplitude free vibrations of simply supported beams with immovable ends. The restoring force function in the equation of motion is found to be a cubic polynomial. The equation of motion for the present problem is known as Duffing’s equation which is a non-linear second order ordinary differential equation. In this paper, we present alternative solutions (such as those obtained by the time doman method, the Galerkin method and the harmonic balance method) of the equation of motion, to obtain similar results and to compare the relative advantages of various methods, such as the perturbation method, the elliptic function method, the finite element method, and the method in which one assumes simple harmonic oscillations.

2. EQUATION OF MOTION The equation of motion for the titled problem is [l]

d2rv/dr2 + &f(

IV) = 0,

(1)

f(W) = w+ (/3/a) w3.

(2)

restoring force function, f( IV), is

The linear frequency is Q)~= ,I$%, a = lr4EI/L4,

and

p = x4EA/4L4,

/3/a = 1/4(Z/A) = 1/4p2.

A is the area of the cross-section, E is the Young’s modulus, Z is the moment of inertia, L

is the length and m is the mass per unit length of the beam, p is the radius of gyration of the section and r represents time. 527 0022-460X/92/240527 + 05 $08.00/0

0 1992 Academic Press Limited

528

S. R.

R. PILLAl

AND

B. NAGESWARA

RAO

The initial conditions for equation (1) are w= w,,,,

d W/T=O.

(3)

Multiplying equation (1) by 2 d W/dz and integrating yields the energy balance equation in the form (d W/dz)*=wt{

1 +&J/a)(

W;?,,,+ W*)}( W;,-

W’).

(4)

Since, the energy balance equation (4) is derived from the equation of motion (I), the solution of the problem can be obtained by solving any one of these equations. The solutions of the problem obtained here by assuming simple harmonic oscillations, by Galerkin’s method, and by the harmonic balance method, is described in what follows.

3. SOLUTION

VIA THE SIMPLE HARMONIC

OSCILLATIONS

ASSUMPTION

If the beam is assumed to be executing simple harmonic oscillations, then w= W,“, cos (wr).

(5)

The velocity is d W/dz = - Wmaxw sin (or). In obtaining the velocity, the frequency w in equation (5) is treated as constant (independent of time 5). At r = 0 one has W= W,,, and d W/dz =O, the initial conditions given in equation (3). The function W in equation (5) satisfies the initial condition (3), but it does not exactly satisfy the non-linear differential equations (1) and (4). Substitution of equation (5) in equation (1) and (4) yields the frequency ratio (w/w=) in terms of time (r) and maximum amplitude ( W,,,,) as, respectively, (w/eQ2 = I+ (P/a) cl,, = 1+ a ( w,,Jp)*

(o/Q*=

1 +@/a>

Wl,{l

= 1 + $( K”ax/p)‘{

cos2 (wt)

cos2 (wr) = 1+ $ ( w/p)2,

(6a)

+cos* (wr)} 1 + aIs2 cmz>> = I+ $ { ( K,,,/P~*

+ ( W/P)“}.

(6b)

The upper and lower bounds to the actual solution of the problem presented in reference [I] can be obtained by substituting r = 0 in equation (6a) and r = a/2@ in equation (6b) as @/wL= { 1 +:t wm?,~/p)‘} “2,

w/at=

{ 1 +A< w”,.~/p)2}

I’*.

(79 8)

The restoring forcef( W) in equation (2) is an odd function (i.e., f(- W) = -f(W)). The behaviour of the oscillations is the same for both negative and positive amplitudes. The magnitude of the maximum amplitude, W,,, can be found from equation (5) at r = 0, K/O, 2w/o. Substituting these values of r in equations (6a) and (6b) gives the same frequency ratio as that obtained from equation (7). Since the assumed solution (5) does not exactly satisfy the equation of motion (1) and the energy balance equation (4), one can obtain an infinite number of frequency ratio relations from equations (6) by specifying the values of re[O, z/20]. However, equations (6a) and (6b) both show that the frequency o is a function of time r. From a theoretical point of view this is not logical. In such a situation, without loss of generality, one can obtain the frequency ratio as a function of maximum amplitude by substituting equations with respect to time over a quarter period (i.e., from r = 0 to r = a/20~), obtaining, respectively, (@/@L)z = I+ $3 w,“&929

w%)*

= I+ a

~~xlP)2.

Pa, b)

ON NON-LINEAR

FREE VIBRATION

OF BEAMS

529

If the solution assumed in equation (5) is closer to the exact solution of the problem, one should obtain almost identical results from equations (1) and (4). The approximate solutions (9) obtained from equations (1) and (4) by this time domain method are seen to be in good agreement to each other. In general, the solution of the problem should be obtained at any instant of time, not at a single instant of time as in reference [l]. Since the energy balance equation (4) is obtained from the equation of motion (l), the solution of the problem can be obtained by solving either of these equations. With the simple harmonic oscillations assumption, the solution of the problem can be improved by following Galerkin’s procedure, as follows.

4. GALERKIN’S METHOD The approximate function m is chosen as the function Win equation (5) which satisfies the initial conditions (3) and for which the period is 2x/w, implying that, starting from r=O, one period is completed at r =2a/o. From equation (l), the residual ~=d~~/dr~+w;f(W),

(10)

with respect to the trial function, C#J=cos(wr), is orthogonalized

(11)

as (E, 4) = 0, T = 2n/o : i.e.,

Using equations (5) and (11) in (12), one obtains (@/@32= 1 +f(Plo)WZox=

1 +&(W,,,lp)2.

(13)

It can also be verified that by multiplying the equation of motion (1) by W, substituting into it the simple harmonic solution (5) and then integrating the resulting equation with respect to time from r = 0 to r =2x/w, one can obtain the approximate solution given in equation (13), which is the same as the perturbation solution [2]. The same result can be obtained from the energy balance equation (4) by dividing it by ( W;“,, - W’), substituting into it the function W defined in equation (5) and integrating it over a period. Since the coefficients of the non-linear terms in the differential equation in general do not involve small parameter, the usual perturbation techniques are inapparopriate [3]. Mickens [4] has indicated that the only generally applicable technique in such situations is the method of harmonic balance, which is described next.

5. HARMONIC

The first approximation

BALANCE

METHOD

of the function Win equation (1) is written in the form W(r) =a1 cos (Of),

(14)

where uI (= W,,,,,) is the amplitude of W. Implementation of the method of harmonic balance [4], which involves equating the coefficients of cos (or) in equation (1) and neglecting the term in cos (3rur), results in expression (13) for CU/CU~ again.

530

S. R. R. PILLAI

AND

B. NAGESWARA

RAO

A second approximation to the frequency o, and a check on the reasonableness of the method of harmonic balance in this case, is now achieved by writing W(r)=az

cos (wr)+b,

cos (3wr).

(15)

After extensive use of trigonometric identities and application of the method of harmonic balance, with only terms involving cos (wr) and cos (3wr) retained, two simultaneous equations for w in terms of the amplitudes a2 and bZ are obtained. The assumption (to be checked subsequently) is now made that the ratio bJa2 is small, so that coefficient terms involving higher powers b:, bi, . . . , etc., can be neglected. The results (w/w,)~=

1+ &C 1+ W4Wd2

= I+ ~WP)(

Km/p),

(16)

with b&z~=(a~/p)2(128+21(az/p)2}-‘.

(17)

0
(18)

This actually satisfies

for all values of a2/p, attaining zero as az/p -+ 0 and tending to & as al/p + co. By equation (18), the ratio bz/a2 for the solution (15) is thus indeed small, whatever a*/~ is. Here K,,,/P

=

(a2

+

b2)lp

=

( I+

For a specified value of the maximum amplitude, from the cubic equation 22(~2/p)~ - 21( K,,,/p)Wp)

(19)

b2la2Na2lp).

W,,,/p,

+ 128Wp)

the value of a2/p is obtained

- 128( K,,/P)

= 0.

(20)

Equation (20) is obtained from equations (17) and (19) by eliminating b2/a2. Otherwise, one can specify a value of u2/p in equation (17) to find b2/a2 and substitute it in equation (19) to obtain w,,,,/p. Then one can use these values in equation (16) to obtain the frequency ratio w/w L. From equations (16) and (18), one can thus find that the second approximate solution of the harmonic balance method, lies in the interval presented by { 1 +a

%,x/p>“>

< (m/%)2

< { 1+ i% wmlP>“>.

(21)

6. RESULTS AND DISCUSSION

The frequency ratios (o/w~) for the specified values of the maximum amplitude ratio have been calculated for the problem of non-linear free vibrations of a simply supported uniform beam with immovable ends and are presented in Table 1. The solution of the equation of motion by Galerkin’s approach with the simple harmonic oscillations assumption (13) is in reasonably good agreement with the elliptic function result [5]. The converged mode shape obtained from the dynamic finite element matrix equations by an iterative process with the harmonic oscillations assumption [6] and the trigonometric functions used for the mode shape as in reference [5] yield the same results, which indicates that the converged mode shape of reference [6] is almost identical to that of the trigonometrc functions used in reference [5]. The second approximation (15) of the harmonic balance method is found to give good results which are in excellent agreement with the exact solution of the modal equation [5]. The various methods discussed in the paper are useful for obtaining the solution of equation of motion, which is of Duffing type. It has been shown that a unique approximate solution for the problem with the simple harmonic oscilations assumption can be obtained ( W,,,/p)

531

ON NON-LINEARFREE VIBRATIONOF BEAMS TABLE 1

Frequency ratios (W/WL) and the maximum amplitude ( W,,,Jp) for a simply supported uniform beam with immovable ends; (a) comparison of solutions obtained by assuming simple harmonic oscillations; (b) comparison of solutions obtained by other metho& (4

Present study Equation Energy balance of motion [ 11, equation [ 11, Wwx/p equation (7) equation (8) 1 3 5

1.1180 (1*1183)i 1.0828 2.6926

I Equation of motion, equation (9a)

Energy balance equation (9b)

Galerkin’s method, equation ( 13)

1.0607

1.0801

1.0753

1.0897

1.4577 2.0310

1.5811 2.2730

1.5512 2.2150

1.6394 2.3848

(h)

q/p,

W-/p

1 3 5

equation (20) o-99341 2.9190 4.8183

Perturbation solution [2]

Elliptic function result [5]

Finite element solution [6]

Present study, harmonic balance method, equation ( 16)

1.0897 1.6394 2.3848

1.0892 1.6257 2.3501

1.0892 1.6257 2.3502

1.0892 1.6254 2.3489

t Result from reference [6] obtained by using the dynamic finite element matrix equations. by using either of the two equations (1) and (4) : i.e., the equation of motion and the energy balance equation. It can he concluded from the present study that the upper and lower bound solutions for the problem with the simple harmonic oscillations assumption from equations (1) and (4) found in reference [l] and the upper bound value presented in reference [6], found by using the finite element method, are not logical from a theoretical point of view. Representation of the frequency ratio explicitly in terms of amplitude as suggested in this paper, and using the method of harmonic balance, can provide not only a check of a computer model but also a means by which the effect of a parameter change on a system can he readily gauged, which is useful in design process. REFERENCES 1. GAJBIRSINGH, A. K. SHARMAand G. VENKATESWARA RAO 1990 Journal of Sound and Vibration 142, 77-85. Large amplitude free vibrations of a beam-a discussion on various formulations

and assumptions. P. C. DUMIR and A. BHASKAR1988 Journalof Soundand Vibration 123,517-527. Some erroneous finite element formulations of nonlinear vibrations of beams and thin plates. R. E. MICKENS1981 An Introduction to Nonliner Oscillations. Cambridge University Press. R. E. MICKENS1984 Journal of Sound and Vibration 94,456-460. Comments on the method of

harmonic balance. 1950 Transactions of the American Society of Me&mica1 Engineers S. WOINOWSKY-KRIEGER 72, 35-36. The effect of an axial force on the vibration of hinged bars. GAJBIR SINGH, G. VENKATESWARARAO and N. G. R. IYENGAR 1990Journal of Sound and

Vibration 143, 351-355. Re-investigation of large amplitude free vibrations of beams using finite elements.