On a perturbation problem in structural dynamics∗

'nt. J. Non-Linear Mechanics, Vol. 3, pp. 215-225. Pergamon Press 1968. Printed in Great Britain

ON

A PERTURBATION PROBLEM STRUCTURAL DYNAMICS*

IN

CLIVE L. DYM? and MAURICEL. RASMUSSEN Stanford University, Stanford, California ~bstraet--Perturbation solutions for the differential equation governing dynamic buckling of an elastic column are discussed. A region of validity is suggested for a solution of Hoff's, and two alternate solutions are presented. Results of both of these solutions are compared with results obtained by numerical integration of the full equation. INTRODUCTION

EXACTmathematical solutions for physical problems are m general very difficult to obtain. This secn~ to be especially true now, when the most important (and most interesting) problems are non-linear in nature. The number of tools available for handling these complex problems is small. In structural mechanics and elasticity, popular and powerful methods have developed through the use of the variational principles of mechanics. Another powerful tool which has received much attention in physics in general [1] and in fluid mechanics in particular [2], is the perturbation expansion. Generally, one expands about a basic solution which is well known or easily obtainable. The expansion is often in the form of a power series in some small parameter that expresses the physical differences between the basic problem and the problem being examined. In many instances a straightforward expansion is not uniformly valid. Here one speaks of a "singular perturbation problem". Two fairly general methods have been developed for this type of problem, although a complete mathematical basis is not yet available for either method. For details the reader should refer to the text by Van Dyke [2]. TIIE PROBLEM

In the present paper, a p r o b l e m originally analyzed in some detail by Hoff [3, 4] will be considered in the light of the above discussion. We consider a perfectly elastic column, slightly bent from the straight, as it is compressed in a testing machine whose head moves d o w n w a r d at a constant velodty. It is assumed in the derivation that the time required for a pressure wave to traverse the length of the c o l u m n is small enough, in c o m p a r i s o n to the time ne~',.ded for the lateral buckling displacements, that the longitudinal inertia effects can be entirely disregarded. The differential equation governing the transverse m o t i o n of the c o l u m n is

e:

+

+ ''se

= o

(,>

* This work was done at Stanford University, supported by a contract of the Air Force Office of Scientific Research with Dr. Nicholas J. Hoff of the Department of Aeronautics and Astronautics. ? Now at: Department of Engineering, State University of New York at Buffalo, Buffalo, New York. 215

216

CLIVI~ L. DYM and M^URICE L. RASMUSSEN

where E1 is the bending stiffness of the column, A the cross-sectional area, # the mass per unit volume of the column material, x is the axial coordinate, t is time, y the lateral displacemerit, and Yo the initial deviation from straightness. The compressive force in the column can be calculated from the compressive force due to the loading head moving downward at constant velocity c, minus the relief in the axial stress due to the lateral deflection : L

o

where L is the length of the column. The initial deviation of the column center line is represented as 7~X

Yo=pesin~

at

t=0

(3)

where p is the radius of gyration of the column cross-section, and e is the dimensionless amplitude of the initial imperfection. A second initial condition is chosen as ~Y=0 dt

at

t=0

(4)

The column in the test machine is assumed to be pinned at both ends. Thus we have as boundary conditions d2y y = ~-~x2= 0 at x = O , L (5) A deflection function appropriate to these boundary conditions can be assumed in the form x

y = p f sin ~

(6)

where f is the dimensionless total deflection. With the assumption (6), and the use of (2), the governing differential equation derived by Hoff [3] reads f"(~) + 12 [1 f 3 + (1 - ¢)f - ~e2- f - el = 0

(7)

where ¢ is the dimensionless time coordinate defined as ct =

- -

(8)

L£~

and Es is the Euler strain, i.e. the strain corresponding to the Euler stress: EE = ~2/(L/p)2

(9)

12 is a dimensionless dynamic similarity number representative of the speed of the testing machine head: 12 = ~2E~E/~c2)

where E is Young's modulus.

(10)

On a perturbation problem in structural dynamics

217

The initial conditions appropriate to equation (7) become, from equations (3) and (4): f =e, f'=0 at ~ = 0 . (11) Finally from equation (10) let us note that f2 ~ 0 represents extremely rapid loading. For normal testing machine loading, ~ is on the order of 10 l°. (For the purposes of this short note, it is sufficient to examine the behavior of the solution to the differential equation (7) for dimensionless time ~ from zero to approximately unity, for varying values of f2 and e. To calculate the load in the column, equation (2), in dimensionless form, gives: P I P E = ~ -- ~ f 2

_ e2).

Some results displaying the variation of load are presented in Figs 7, 8, and 9, for the various solutions that are discussed in this note.) PREVIOUS SOLUTIONS

The first solution to the set of equations (7) and (11) was obtained by Hoff 13] for extremely rapid loading, i.e. I2 very small (Fig. 1). It was found that for these high loading speeds, the value of the maximum load could exceed the value of the Euler critical load. This has also been borne out by experiment [5]. The case of slow loading was studied by Hoff in two different contexts. In the first 16], the W.K.B. method was used to demonstrate that rapid oscillations of very small amplitude, about the static solution, constituted a good approximation to the solution of the problem; the number of oscillations found by this method agreed very well with the numbers derived from the linearized solution valid at the onset of the loading process. 50O

4,OO

f

IO0

0

FIG. 1. Non-dimensional lateral displacement amplitude f as function of non-dimensional time in very rapid loading.

218

CLIVE L. D'CMand MAUmCEL. RASMUSSEN

A second case of slower loading, which we shall treat here in more detail, was solved by asymptotic expansion in powers of l/f2 E4]. Since slow loading implies large fL a new parameter ca = 1/~ was introduced. Thus equation (7) becomes: 1 3 caf"(~) + -~ f + ( 1 -

e2 ~ ) f - ~- f - e -- O

(12)

The initial conditions (11) are of course unchanged.

/

to

[ t

2

I~o. Z Static non.-dim~sional lam'al displacemem amplitude fo and end load ratio P/Pe for axially compressed

perfect column.

A solution was attempted in the form of a power series in ca: f(~) = fo(~) + cafl(~) + ca2f2(~) + ...

(13)

Quite obviously, then, fo(O represents the static solution. It is found as the solution to the cubic equation ~ f o3 + (1 - ~ -

e4) f o - e =

0

(14)

We can further find that f°(~) =

dfo 2f2o d~ = 2e + f-----~

(15)

The solution to the cubic equation (14) is shown in Fig. 3 for a perfect column, i.e. for e = 0. In reality e is never zero since it is not possible to manufacture a "perfect" column. This being the case, we note from equation (15) that the second initial condition, that of zero initial lateral velocity, can never be satisfied. However, at ~ = 0, it is evident from equation (14) that fo = e will satisfy this equation. Thus, one initial condition is satisfied, the other violated--and even this does not seem serious, for at small e, f ~ -~ e, at ¢ = 0. Having obtained fo, and being able to find its derivatives with respect to ~, the first three terms of (11) have been constructed. It is well to note here that the construction of the

On a perturbation problem m structural dynamics

219

perturbation expansion allows the calculation of succeeding approximations from preceding terms from algebraic, and not differential, equations. Then fl = 8f~ F f 3 _ 4 e t [

L(fo + 2e)d

(16a) '

and f2 =

I(

2f°7 1 foa + 2e)gj (640ea - 2352e2fo3 +

l164ef 6 -

7 3 f 9)

(16b)

Then for given values of the loading parameter 09 and the imperfection amplitude e, plots of the displacement f and the load ratio P / P r . (where PE is the Euler load) against the dimensionless time ~ were obtainecL Examples of these are shown in Figs. 3 and 7. These results have been obtained by Hoff [4].* From these curves and from further calculations, it was seen that as co becomes smaller, for a fixed e, the curve became smoother and the non-uniformities vanish. This would seem reasonable, for as co becomes very small, the loading is slow, and the inertia term must be unimportant. However, it was also noted that for a fixed o9, non-uniformities became evident again as e was allowed to become smaller and smaller. Various sets of calculations indicated the following: CO

e

10 -~

>I 10 - 3 10 -'~, 10 -5

convergence no convergence

10 -6, 1.5 x 10 -6

10 -2 l0 - a

convergence n o convergence

10- 3

0"25, 10-1 I0- 2

convergence no convergence

In short, convergence was achieved whenever og
2.

In addition, it seems possible to pass to the limit o9 = 0 and fred satisfying results. However, a different tack must he tried for investigation of the cases of vanishingly small imperfections. P R E S E N T S O L U T I O N S (1)

As was seen above, and as is evident from Fig, 4. the expansion (13) becomes invalid as o9 increases, or as e becomes very small for a fLXedvalue ofog. In both cases this may be traced to the fact that in the region just before ~ -- 1, the inertia term becomes important. This is evidently true for increasing o9, as the loading speed increases, and not so obviously true for very small imperfections. In the latter case the inertia term is important because the deflections of the column, * Figure 3 is a slightly modified version of Fig. 18 of Ref. [4]. When making up Fig. 18 of Ref. [4"1 for Dr. Hoff, the present senior author inadvertently altered the scale. The shapes of the curves are the same, and the pertinent points of Dr. HotVs discussion are unaltered.

220

CLIVE L. DYI~ a n d MAURICE L. RASMUSSEN

though small, increase very rapidly at buckling. Evidence of this is given in Table 1, where the results of a numerical integration of the full equation are presented. It is seen that in the region ¢ = 1 + , the increase in deflection is much more dramatic for smaller imperfections. Since the interest here is in small imperfections, it seemed logical to attempt an expansion in powers of the imperfection amplitude e : f(~) = yt(~)e + y2(~)e2 + ... + y,(~)e" (17) Substituting the expansion into equation (12), we find the following system of differential equations: O(e) coy'i(¢) + (1 -- ¢)Yl = 1 O(e 2) coy~(.~)+ (1 - ~)Y2 = 0 O ( e 3) o ) y ~ ( ~ ) + (1 - - ~)Y3 = ¼Yt(1 - y~)

(18)

O(e*) ogy~(~)+ (i - ¢)y, = ¼Y2(1 - 3y~) TAm~ 1.

C O M P A R I S O N OF I~I'I~CT OF INITIAL DEVIATION AMPLITUDE; P R E S E N T SOLUTIONS

co---~ 10-7;

ALSOC O M P A R I S O N

W I T H FIRST OF

f(~) Coefficients ~ e e = 10 -3 0.1 0'3 0-5 0.7 0-9 0-92 0-94 0.96 0.97 0-98 0-99 14)0 14)1 14)2 14)3 1-04

1"111 1"428 2.000 3.333 9.996 1'249 1-663 2"483 3-282 4.779 8.158 1"593 2'369 3"038 3'627 4'124

x x x x x x x x x x x

101 l0 t 101 101 l0 t 101 102 102 102 102 102

Exact e - ~ 10 -7 1"Ill 1•429 2-001 3"332 9-999 1"250 1'665 2'492 3"310 4"895 9.002 2.775 2'612 1.043 4"456 3"814

x x x x x x x x x x x

101 l0 t 101 101 101 101 102 10sl0 s 106 106

e = 10 - 9 1"112 1'431 2.009 3-321 1.001 1'251 1.665 2'495 3'312 4'895 9.001 2'774 2"612 1.043 1"379 2-689

x x x x x x x x x x x x

101 101 101 10 ~ 101 101 101 102 l0 s l0 s 107 l0 s

Approx. ['18] l'111 I '428 24)00 3,333 9,998 1,250 1,665 2-492 3,310 4.896 9"002 2,775 2"612 1-043 1'379 4'726

x x x x x x x x x × ×

101 10 l 10 t 101 10 l 10 l 102 l0 s 105 107 109

The appropriate initial conditions for this set of equations are : y~(0) = 1,

y'l(0) = 0,

y,(0) = y',(0) = 0,

n/> 2

{19)

Immediately it will be seen that Y2(¢) is identically zero, for it is the solution of a secondorder homogeneous equation with homogeneous initial conditions. Then it also follows that Y,(~) vanishes, for with Y2(~) identically zero, the equation for Y,d¢) is now homogeneous. It appears that we shall then have only odd powers of e in our asymptotic expansion. The first of equations (18) is of the Bessel type, and its solution can be written in terms of familiar functions. Since co ,~ 1, it would appear that asymptotic expansions might be used to facilitate numerical calculation of the Bessel functions. This is unfortunately not the case in the region of greatest interest, in the neighborhood of ~. = 1. For this reason, the

221

On a perturbation problem in structural dynamics

first of differential equations (18 was solved numerically on the B-5500 Computer at Stanford University. The solution of this equation, multiplied by e, represents a first approximation to the exact solution, which has been obtained numerically to check the validity of the expansion (I 7). The numerical results obtained with this expansion indicate that convergence is relatively good for0~<~. ~< 1 + i f ~ / 9 > e 2.

Some of these results are shown in Figs. 4, 5, 8 and 9. It is to be noted that the present expansion is not good for ~ > 1 +, for then the exponential behavior of the Bessel Functions leads to rapid increases in deflection which are not indicated by the exact solution. It might also be noted that taking only the first term of (17), results in accepting a direct linearization of the differential equation (12) with respect to the dependent variable f(~). PRESENT SOLUTIONS (2)

It is seen in Fig. 4 that the solution presented above fails in the region around ~ = 1 because of the exponential increase in the deflection in" this region. The rapid increase is due to the exponential behavior of the Bessel Functions in the solution of equation (18) at the turning point (~ = 1). It would appear, then, that a multiplicative correction factor that would decay as time increases, might level out the above solution and make it uniformly valid. Thus a solution was attempted in the form f(~) = u(¢)v(¢)

(20)

Since we have introduced two functions in the place of one, an arbitrary condition is n o w at our disposal for stipulating conditions on one or the other of the two functions. Substituting (20) into the differential equation (12) yields co,~" +

4

v

+ co(2u'v' + uv") +

/./3/33= e

(21)

In view of the arbitrary condition mentioned earlier, we shall take cou" +

(1

e2

- "~ -

~)

(22)

u = e

and enforce the initial conditions u(0) = e,

u'(0) = 0

(22a)

Comparison with equations (17, 18, 19) shows that u(~ is virtually identical with the first approximation solution presented above. The only difference is a slight shift in the turning point from ~ = 1 to ~ = 1 - e2/4. For the range of e of interest here, this shift is aegligible. With these conditions enforced, what remains is a differential equation for v : (D/MY + 2(Du'v' + ~ u 3 v 3 + e(v -

1) = 0

(23)

solution to this equation was sought in the form of a series for small co : v =/30 + (Dr1 + co2v2 + ...

(24)

0,30

0.30

0"25

0.25

0,20

_

f v~ (e=~ J)

Presentsolution(ll

/

/

0.20

=,=10~

--

--

0.15 0,15 O.lO 0.10 @{3'5

o:

0-00

~=i.5

O05

E

O'(3

)6

-0.05

I

I

O97

@98

0"~0 =

0.20

0.201--

0-15

o.J51--

O-C

,<

>

c

o.zsl--

f vs.~ le=~ ~) Exocl solul/o~

0.05 D

I

1"(30

FI6.4

0.30

0-10

I

0-99

~e

FIG. 3

0.25

xlO-6

f vs. ((e=lO-3) Prese~ sobt~n (2)

~g c z

o.1ol--

~

=1.5xI0-6

~

"--.~,l.Sx10-6

L

I

0.97

I

O~J6

FiG. 5

I

0~9

[

1.00

0.97

I

oz~

I

0.99

I

i.o(3

FIG. 6

FIGs. 3--6. Non-dimensi0nal lateral displacement a m p l i t u d e f vs. non-dimensional time ~ for slowly compressed imperfect column.

On o perturbation problem in structural dynamics 1.00

0.99

o.ge

0-97 )7

I

r

~

04~9

bOO

I'01

FIG. 7 i.oo

/ u = l ' 5 x I0-6 P/Pc v& { ! e = 10-3)

~

0-99

D-

oxJe

0"97

0.~

009

t.O0

1,01

FIG. 8 1.00 •

~~

E r e c t solutlon

~

/

,,

~

.~

,.~

FIG. 9 FIGs. 7-9. Load ratio

P/Pe vs. non-dimensions] time ~ for slowly compressed imperfect column.

223

224

CLIVEL. DYMand MAUmCEL. RASMUSSEN

The first approximation is then readily found as the solution to the algebraic equation u3 _

3 + 4e Vo

Vo

1

=

(25)

Some interesting features are obtainable from this equation. In view of the initial conditions (22a), it is seen that at ~ = 0 e2 -2- v3 + Vo = 1

or vo --- 1. Further, differentiating (26) with respect to the time variable yields 332,

332,

,

~eV°U u + 4eU VoVo+ vo=O so that at ~ = 0, v3 = 0.

Thus the total deflection function very closely satisfies the initial conditions. In addition, as co ~ 0 it is seen from (22) that in the neighbourhood of the turning point u ---. 3c. Then from (25) (4e~ ~

vo ~- ~,U3] or, at the turning point

f(~=l-~)=uv

o=(4e) ~

(26)

But from the full differential equation (12), with 09 = 0, we find at the turning point, t f 3 _ e = 0, which leads to the same results given in (26). Some of the results obtained using this solution are given in Fig. 6. It can be seen by comparison with Figs. 3 and 4 that this solution tends to correct the first approximation towards the exact solution in the region ~ ~ 1, but yields a value that is too low. Naturally, the results are better as ~o --. 0. For ~ > 1, the present solution (2) breaks down and is not valid. CONCLUSIONS For completeness, although this investigation has been concerned primarily with the time interval around ~ = 1, it is of interest to point out features of the behavior of the column before and after this time. It appears (see Table 1, for example) that for very small imperfections the column deflections increase monotonically. Then during the critical interval in the region of the point ~ = 1, the increase in deflections is monotonic and rapid, followed by slow oscillatory behavior about an increasing mean afterwards. The load in the column increases monotonically until this critical time, and then it oscillates in a similar manner as the deflection.

On a perturbation problem in structural dynamics

225

For the particular time interval of interest, bounds have been indicated for the use of .-ither of the expansions discussed. It must be remembered that these bounds were obtained "empirically" by comparison of the various numerical results obtained with results of aumerical integration of the full equation. In addition it can be stated that the first expansion suggested here will be valid in the :egion 0 ~< ~ <~ 1 for the bounds indicated, but not beyond this. It appears that the ex9ansion in powers of co will be valid beyond the critical region, if e, co are within the bounds ;uggested earlier. The second solution appears to be valid only for ~ < 1. Perhaps the most important conclusion is an implicit warning that great care must be :aken in the use of perturbation or asymptotic expansions. It is quite possible that non~atisfaction of boundary or initial conditions, or deletion of an important term in the eading approximation can lead to a solution that is not uniformly valid. This is especially :rue when two small parameters appear in the problen~ lcknowledgements--The authors are very grateful to Dr. Nicholas J. Hoff and Dr. I-Dee Chang for many Lelpful and informative discussions.

REFERENCT.S I] IL BELL~N, Perturbation Techniques in Mathematics, Physics, and Engineering. Holt, Rinehart & Winston 0964). !] M. V^tq DYic~ Perturbation Methods in Fluid Mechanics. Academic Press (1964). t] N. J. Hovr, The dynamics of the buckling of elastic columns. J. appL Mech. lg (1951). gl N. J. HOFF, Dynamic stability of structures: keynote address at the Int. Conf. on Dynamic Stability of Structures, Northwestern University, October 18, 1965. (Also, Stanford University Department of Aeronautics and Astronautics Report SUDAER No. 251, October 1965). l N. L. Horr, S. V. NAROOand B. EmcrdoN, The maximum load supported by an elastic column in a rapid compression test. Proc. 1st U.S. Natn. Cong. appL Mech., p. 419. AS/VIE (1952). ;] N. J. HOFF, Buckling and stability: the forty-first Wilbur Wright memoria~ lecture. JI R. aeronaut. Soc. 58 3 (1954). (Received 5 May 1967) /~am6----On discute des solutions de perturbation pour r6quation diff6rentielle r6gissant la dynamique du ambement d'une colonne 6lastique. On sugg6re tree r~gion de validit6 pour lane solution de l'&luation de Hoff : on montre deux solutions diff~rentes. On compare les r&ultats de ces deux solutions avec ceux obtenus par kt6gration num6rique de l'&luation compl6te. mmmmmfamng--Fiir die Differentialgleiehung, die dynamisehes Knieken einer elastisehen Siule beschreibt, ,-rden St6rungsl6gungen er6nert. Ein Gfiitigkeitslgebiet fiir eine L6sung nach Hoff wird vorgeschlagen~ und ¢ei andere L6sungen werden angegeben. Ergebnisse von beiden diesen L6sungen werden mit Ergebnissen rglichen, die durch numerisehe Integration der vollstlindigen Gleichung erhalten wurden. 5 e T p a ~ - O S c y m ~ a m T c ~ nepTyp6au~tOHH~e pemeHng ~d~dpepeHmtaJ~br~oro ypaBHerli4rl, rtoTOpOMy )~/HH~eTca ~rIHal~l,tqecrmlI npo~oabnHg ~srH6 y n p y r o r o cToaSa. IIpe~aaraercn oSaacrb cnpa~/XannOCTn pemearm Xodp~be a npano~;aTca 3Ba ~pyrHx petueHaa. PevyabTaTra o6onx pemeaatt ,aBHHBaIOTCfl C pe3y~bTaTaM~ IIOJIyqeHHhlMH HyTeM q~lcaeHHoro HHTerpHpOBaHHH no~Horo ypaBHeH~.