random vibrations of a nonlinear elastic beam

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 36, NUMBER 11 NOVEMBER 1964

Random Vibrations of a Nonlinear Elastic Beam

RIC•AKD E. HERBERT*

University of Florida, Gainesville, Florida

(Received 30 January 1964)

The theory of the Markoff process and the associated Fokker-Planck equation is used to investigate the large vibrations of a hinged, axially restrained, elastic beam driven by white noise. An expression for the joint probability-density function of the first N modal amplitudes is obtained and used to derive an approximate expression for the mean-squared displacement. Numerical integration of the exact expression for the mean-squared displacement indicates a reduction of this quantity as compared to the linear theory. Calcula- tions also show a range of applicability of the derived approximate expression. Furthermore, computations show that the first node still gives a good estimate of the mean-squared displacement but the coupling effects of the nodes are so strong, for sufficiently large deflections, that the effect of the higher modes must be taken into account when computing the mean-squared value of the first mode.

A

½, ½t E

h

I

L

q(x,t) q,• N

N(x,y) No

P8

T

V, Vs, V•., Vs

w(t)

LIST OF SYMBOLS

area of the cross section

see Eq. (11) moments appearing in the Fokker- Planck equation normalizing constants elastic modulus

thickness of a rectangular beam moment of inertia of the cross section

length of beam load per unit length of beam see Eq. (6) number of terms retained in all series

expansions space correlation function spectral density of the load transitional probability density function

stationary probability density function radius of gyration of the cross section see Eq. (14) kinetic energy Potential energy-total, of bending, of the load and of stretching, respectively deflection of beam

modal amplitudes

*Present address: The College of Aeronautics, Cranfield, England.

w

0-0 2

O-c 2

<>

vector with components (w•, ß ß .w•v, •, .- transverse damping coefficient see Eq. (30) see Eq. (23) density of beam mean-squared value of the first mode of the linear problem mean-squared deflection at the center of the beam

natural frequency of the first mode of the linear problem denotes ensemble average

INTRODUCTION

HE large free vibrations of an axially restrained, pin-ended elastic beam have been studied by

several authors? 4 More recently, Wah 5 has studied the response of such a beam to a transient deterministic loading. Until now, little attention has been given to the

x S. Woinowsky-Kreiger, "The Effect of an Axial Force on the Vibration of Hinged Bars," J. Appl. Mech. 17, 35-36 (1950).

•' D. Burgreen, "Free Vibrations of a Pin-Ended Column with Constant Distance between Pin-Ends," J. Appl. Mech. 18, 135-139 (1951).

a A. C. Eringen, "On the Nonlinear Vibrations of Elastic Bars," Quart. J. Appl. Math. 10, 361-369 (1952).

4 p. H. McDonald, "Nonlinear Dynamic Coupling in a Beam Vibration," J. Appl. Mech. 22, 573-578 (1956).

5 T. Wah, "Dynamic Response of Beams with Large Ampli- tudes," J. Aerospace Sci. 27, 877-878 (1960).

2090

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RANDOM VIBRATIONS OF A NONLINEAR BEAM 2091

vibrations of such a beam subjected to a randomly distributed load. The difficulty of the problem is, of course, due to the membrane stress, which causes a nonlinear coupling of the modes.

Until recently, only approximate methods have been used for investigating the response of nonlinear multi- degree-of-freedom systems driven by random forcing functions. Chuang and Kazda 6 have shown how, under suitable restrictions, the Fokker-Planck diffusion equation can be used to find exact expressions for the probability-density functions governing the responses of some nonlinear servo systems. Ariaratnam 7,8 has since used this equation to find the probability-density functions of the responses of some nonlinear spring systems and of the responses of a set of masses attached to a weightless nonlinear string.

It is the purpose of this paper to use the Fokker- Planck equation to investigate the large vibrations of a hinged, axially restrained elastic beam driven by a randomly distributed load. The joint N dimensional probability-density function of the first N modal amplitudes is obtained. Using this, an approximate expression for the mean-squared displacement of the beam is derived. Numerical computations are then given to compare the linear, approximate nonlinear, and the exact nonlinear expressions for the mean- squared response over a range of parameters.

I. FORMULATION OF THE PROBLEM

Consider an elastic beam with pin-ended supports that are restrained from motion. Let the deflection be

represented by N n7roC

w(x,t)= Y'. w,•(t) sin--. (1)

We are considering that only the first N modes of the beam are excited. However, subsequently we will as- sume that the power spectral density of the load is that of white noise. This implies that all modes are excited and contribute to the response of the beam. If, for white-noise excitation, the infinite series representing quantities such as mean-squared displacement, mean- squared stress, etc., converge, then the results can be made as accurate as desired by taking the number of terms N sufficiently large. Furthermore, Crandall and Yildiz ø have shown that for the linear problem the various mean-squared quantities can be made finite (i.e., the series representing these quantities can be made to converge) by selecting the appropriate beam model and the appropriate damping mechanism. For

6 K. Chuang and L. F. Kazda, "A Study of Nonlinear Systems with Random Inputs," Trans. AIEE 78, 100-105 (1959).

7 S. T. Ariaratnam, "Random Vibrations of Nonlinear Sus- pensions," J. Mech. Eng. Sci. 2, 195-201 (1960).

8 S. T. Ariaratnam, "Response of a Loaded Nonlinear String to Random Excitation," J. Appl. Mech. 29, 483-485 (1962).

0 S. H. Crandall and A. Yildiz, "Random Vibrations of Beams," J. Appl. Mech. 29, 267-275 (1962).

example, a Bernoulli-Euler beam with transverse damping subject to spatially uncorrelated, white-noise excitation will have finite mean-squared displacements but infinite mean-squared stresses. On the other hand, a Rayleigh beam with rotatory damping subject to the same excitation will have finite mean-squared displacements and stresses. 1ø

It is reasonable to expect similar results for the nonlinear problem. Therefore, as a first solution to the problem, we consider a Bernoulli-Euler beam with transverse damping. The various energies of such a beam are the following:

Kinetic energy'

pA;z'( &V•2dx pAL :• T= •j0 \•t/ : 4 Z(tb•) s, (2) where dot indicates differentiation with respect to time. Potential energy of bending'

4L • •=l

Potential energy of stretching'

VS=•L•ak•xx/ J 32L.• L ,•, n2w'•2 ' (4)

Potential energy of the load'

fo L v,,--

We write

N n7roc

q(x,t)= Y'. q,•(t) sin. , (6)

where we have terminated the series at the same N as for the deflection. Then,

L•

Vu- Y'. q,•w,•. (7) /z-•l

The Lagrangian L= T-- V, where V = VB+ Vs-+- V•, may now be written as

pAL L•-- Z (•)n) 2--• Z n47ddn 2

4 ,•=1 4L • ,•=1

Z + Z q.w (8) 32LaL •--• -•- •.

The Lagrangian equations of motion are

(d/dr) (aLlOrb,,)-- (aLlOw,,) = 0, (9)

where the w• are considered as generalized coordinates. Substitution of Eq. (8) into (9) yields the following N

•0 See Ref. 9, Table III.


2092 R. E. HERBERT

equations governing the modal amplitudes'

/2•+•,+w02n 4 1+ - Y'. m2w• w,•= a,•. pA L 4R2n 2 m=•

(•o)

We have introduced a linear viscous-damping term that corresponds to transverse damping. Also,

wo2=•-4EI/pAL 4, a,•=q,,/pA. (11)

Equation (10) is a nonlinear stochastic differential equation. We seek the probability-density function P s(w•,w2," ',w2v) governing the modal amplitudes. To this end, we employ the Fokker-Planck equation.

II. PROBABILITY-DENSITY FUNCTION OF THE MODAL AMPLITUDES

To employ the Fokker-Plank equation, we make the following assumptions ß (1) the input to the system is Gaussian with zero mean; (2) the random impulses acting on the system at any two consecutive time intervals are statistically independent. Assumption (2) makes it necessary for us to take

(am(t)a,•(t-kr))= 2Rm,•(r)/Lp2A 2 (12) or

{q(x,t)q(y, t+r))= N(x,y)•(r), (13)

where Rm, is some function of m and n that is related to the space correlation function N(x,y) by the equation

2foLfoL mrx nry Rm,•=-- N(x,y) sin-- sin--dxdy. (14) L L L

With these assumptions, it can be shown u that the transitional probability-density function P(w,t[ w0,t0) must satisfy the Chapman-Kolmogoroff equation in the form

P(w, tlWo, to) = f P(w, tlw',r)P(w',rlwo, r)aw'. (lS) From this equation, the following Fokker-Planck diffusion equation can be derived 7,n'

OP 22v 0 •.2v 22v 02

Ot ,,•-10wm •=1 m=l OwmOw,• •(bm,,•P), (16)

where

WN+m =tbm, 1

bin= lim •xt•o At

1

b m, ,, = lim --(/Xw m/XW,,). at--,O At

(17)

n D. Middleton, Statistical Communication Theory (McGraw- Hill Book Co., Inc., New York, 1961), p. 46.

n T. K. Caughey, "Derivation and Application of the Fokker- Planck Equation to Discrete Nonlinear Dynamic Systems Subjected to White Random Excitation," J. Acoust. Soc. Am. 35, 1683-1692 (1963).

With the aid of Eq. (10), the various moments bm and bm.• can be found to be

bin-

bN+m-- -- --WN+m pA

-wo2m 4 1+ Y'. n2w,? Wm, (18) 4R2m 2 ,--1

b m , ,• = b •v + m , ,• = b m , 2V + ,• = O ,

b2v+m.2V+,•= 2Rm,•/LpM2, all m, n_< N,

so that the Fokker-Planck equation for our system becomes

OP

Ot - I2 -

m=• Owm m=• OW2v+m[- pA

-½wo2m 4 1-• 5-. n2w,? Wm 4R2m 2 ,•=1

1 •v •v 02 (Rm,•P) -+-• Y'. •, (19) ß

Lo2A 2 ,•=1 m=l OW•V+mOW•v+,•

Ariaratnam s obtained a similar equation governing the probability-density function of the responses of a set of masses on a weightless nonlinear string. •y direct substitution, the stationary solution of Eq. (19) with

Rmn = 2No•m• (20)

can be verified to be

1

P(w)=7 exp n=l

•. n4w,?-½-- •. •. m2n2w•w,? 2•02 ,=1 8R 2 ,=1

(21)

where

•ro 2= 2NoL•/fiEI•r 4 (22) and

u=•LoA/4No. (23)

Equation (20) implies that

(q(x,t)q(y, tq-r)}= 2Nob(x-y)b(r), (24)

so that the load is completely uncorrelated in time and space. It is evident that the velocity variables tb• have a Gaussian distribution. Integration of Eq. (21) over all these variables gives the expression

P•(w•, . . . ,w•v)

i 2_•0•[ • 1 • • • =- exp -- • n%•+ • • • m•n•w•2w• • . 6 n=l 8R 2 n=l m=l

(25)


RANDOM VIBRATIONS OF A NONLINEAR BEAM 2093

It is evident from this result that the effect of the

membrane stress, characterized by the radius of gyration R, is to cause the probability-density functions of the modal amplitudes to become non-Gaussian and statistically dependent. This causes a "squashing in" of the density functions and thereby reduces the variances of the modal amplitudes.

III. MEAN-SQUARED RESPONSE

An important function characterizing random motion is the mean-squared response. For our beam, we

have m11-x flAl'X

sin sin•. (26) L L

Because Eq. (25) is an even function, the wm will be uncorrelated so that, while the membrane stress causes the modal amplitudes to become statistically dependent, it still leaves them linearly independent. Equation (26) then reduces to

N m11-x

<w2(x,t)) = Y'. <win 2) sin s--, (27) where

(win • ) = - dw • . . . w m 2 exp -- • n4w,• 2d-• Y'. Y'. n2r2w,•2r • dw•v. ,,=• 8R 2 •=• n=•

(28)

Since this expression is not readily integrable, we must employ either numerical integration or some other approximate technique. If we expand the function

1 .¾ :v } exp -- • • rn2nSwmSw•2 16•02R 2 n=• •-•

in a MacClaurin series and retain only the first two terms, then the integration indicated in Eq. (28) can be performed. After a little juggling, we are led to

f o-o2 I(1) 2 1 4 ld__8 q 1---- Z +2Y'.--d---Y'.-- --

(7jjm2 } = 0-02 16R 2 • •4 •12 •2 •//,4..j •/•4 0_0 2 • (•_,.

(29)

Substitution of Eq. (29) into (27) then yields an approximate expression for the mean-squared response.

IV. NUMERICAL COMPUTATIONS

Numerical computations have been performed for the mean-squared deflection at midspan over a range of the two parameters •02 and R. The results are presented in graphical form in Figs. 1 and 2.

.12

.O8

//

½• -- R i/• / ff•/' ............. • _-

' R--I/8 APPROXIMATE FORMULA NUMERICAL INTEGRATION

LINEAR THEORY

.10

.06

.04

.02.

0 .02 .04 .06 .08 .10 .12 .14 .16

FIG. 1. Mean-squared deflection at midspan for small non- linearities. R, •0, and •c carry the dimensions of length.

In Fig. 1, the mean-squared deflection--as determined by the linear theory, the approximate formula (29), and numerical integration of Eq. (28)--is plotted against •02 with R= «. For the range of •02 considered, it was found, as in the linear case, that sufficient accuracy is obtained with N= 1. It was also found that the approximate formula is valid over a small range of

1.2

0.8

-"•:3 ...•. :--'

..- ..-' ..._ •.•.-5

? ,?

FIRST MODE

' FIRST AND THIRD MODES • LINEAR THEORY I

1.0

0.6

0.4

0.2

0 1.0 2.0 3.0 4.0

Fro. 2. Mean-squared deflection at midspan for large non- linearities and with R = «. R, ½0, and •c carry the dimensions of length.


2094 R. E. HERBERT

•02. As long as the difference between linear and nonlinear theories is not greater than 10%, this formula gives an excellent estimate of the true mean-squared deflection. This is to be expected in view of the approxi- mation made in deriving formula (29).

Also plotted in Fig. ! is the mean-squared deflection determined by numerical integration with N= 1 and R= • and •. These curves indicate that for lower values of the radius of gyration the curve of the nonlinear theory begins to deviate sooner from the straight line of the lineary theory. This is to be expected since with diminishing R the role of bending diminishes. Further- more, if the beam were rectangular, then h 2= 12R 2, so that

•= (•c2/h2)•= 1/2R(o-•/3) •. (30)

Now for R= «, the nonlinear theory begins to appreci- ably deviate (i.e., greater than 10%) from the linear theory at about •02=0.3 or at •=0.1. For R=•, we have appreciable deviation at about •02=0.009 or •=0.11, and for R=• at •02=0.003 or •=0.13. Since, as is well-known, the linear theory of beams is valid only for w/h<<l, these results are reasonable.

In Fig. 2, the mean-squared deflection as determined from the linear theory and from numerical integration of Eq. (28) is plotted against larger values of •02 with R= «. Of the three curves, the uppermost represents numerical integration with N-1. The lowest curve represents {wt 2) as determined by numerical integration with N-3. The middle curve represents the total deflection; i.e., {w•2)+ {w3 •) with N= 3. Two obser- variations are evident from these curves. Firstly, taking /V= 1 is no longer valid for such large values of •02. Secondly, as in the linear theory, {w• 2) gives a fairly close estimate (a few percent) of the total mean- squared deflection. However, in computing {wt 2) it is now necessary to consider the effects of the second and third modes. That is to say, the nonlinear coupling is so strong, for the range of parameters considered in Fig. 2, that the second and third modal coefficients have a

significant effect on the mean-squared value of the first modal coefficient. Thus, the beam could still be treated

as a one-degree-of-freedom system if the appropriate nonlinear spring constant could be found.

It should be pointed out that for a rectangular beam with R= « we have h •= 3;so that with •,•= 1.1, the highest value in the graph of Fig. 3, we have

•= (•,•/h•)• 0.6. (31)

This value is not in excess of the applicability of the nonlinear theory considered nor of the values of practical interest.

V. CONCLUSIONS

In this paper, the effects of the membrane stress on the response of a simply supported beam subject to white-noise excitation have been studied by means of the Fokker-Planck equation. The joint probability- density function of the first N modal amplitudes of vibration has been obtained. From this, an expression for the mean-squared displacement has been expressed in integral form. Assuming small nonlinearity, an approximate expression for the mean-squared displacement has been obtained.

Numerical computations have indicated a reduction of the mean-squared displacement due to the nonlinearity. Also, the approximate formula developed has been shown to be valid over a small range of the parameters. Finally, the numerical computations have shown that, when the deflections are sufficiently large, but still within the applicability of the nonlinear theory considered and the realm of practical interest, the first mode still represents a good estimate of the total mean-squared deflection, but the effect of the higher modes must be considered in calculating the response of the first mode.

ACKNOWLEDGMENTS

This study was supported by a U.S. Air Force Office of Scientific Research grant and conducted under the general supervision of Dr. William A. Nash of the University of Florida.


random vibrations of a nonlinear elastic beam

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