Applied Acoustics 17 (1984) 345-356

Forced Vibrations of a Cmtttmumm B e n with Ends Elastically R e s t r a ~ Against Rotation

P. A. A. Laura, P. L. Verniere de Irassar and G. M. Ficcadenti

Institute of Applied Mechanics, 8111-Puerto Belgrano Naval Base (Argentina)

(Received: 24 August, 1983)

S U M M A R Y

A survey of the literature shows that the titleproblem has not been studied to any great extent. In the present paper an approximate solution is obtained in the case of a beam with ends elastically restrained against rotation and an intermediate support. A sinusoidally varying excitation is assumed.

INTRODUCTION

Excellent studies dealing with free vibrations of continuous beams are available. 1 - s

The case of ends elastically restrained against rotation has been treated recently.6 The present study deals with a forced vibrations condition and it has been considered convenient to analyze two different situations (see Fig. I.

The situations shown in Fig. I are dealt with in a unified manner by approximating the deflection amplitude using simple polynomial co- ordinate functions which satisfy the essential boundary conditions. The Ritz method is then used to determine the coefficients of expansion.

345

Applied Acoustics 0003-682X/84/$03.00 © Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain

346 P. A. A. Laura, P. L. Verniere de Irassar, G. M. Ficcadenti

~" = ; L

II)

L. II --I I - - ~ ! I L_ L ~!

( I l l J F~. l° Continuous beam subject to forced vibrations.

APPROXIMATE SOLUTION

Neglecting damping, one expresses the response of the structural element by means of the approximate expression: 6

Wa1(l ) =0; dWa~ l d2Waxl

w(x, t) ~ wa(x, t) = [Ao(~isx s + ~14x 4 + ~13x 3 + ~2 x2 + ~x)

-t- AI(~26 x6 + (x2sx 5 Jr ¢x24x 4 + x 2 -I- ot21x)]

x cosoJt--(Wal + Wa2) coscot (I)

The ui~'s satisfy the boundary conditions (Fig. I):

I d2Wall Wa,(O ) =0; dWaldx x=o = ~bl ~[x=o

Wax(n) = 0 (2)

Forced vibrations of a continuous beam with ends restrained 347

I41.2(0) = 0;

wo2(,t) = o

dWa2 d2Wa2 dx x= 0 ---- ~ d'x2 x = o

(3)

d W.21 d2 W"2 Wo2(1) =0; dx I~=1=-¢~2 dx2 ~=i

where ~I = a /L; dp' 1 = dpEl/L; dp' z = dp2El/L. Expressions for the %'s are given in the Appendix.

CASE OF A CONCENTRATED LOAD (Fig. l(I))

The maximum potential and kinetic energies are given by the well known expressions:

l'+. = - ; -(d2w~ 2 d,~ + - - l (dW~2l I :o

, (de,l= I +~-22 \ d i_ / I~--L - P o ~ , (4(a))

1 f] W2 Tm,,x =~ PAo co2 d~ (4(b))

Accordingly, one must now minimize the functional:

or[W.] = f ] / ' d2W. \ 2 1 /'dWo~21 1 /'dWo'~2l ) - 2Po WoD< = x, - r i 2 fox W. 2 dx (5)

to the A{s. In eqn(5): x = $ / L ; P o = P o L 3 / E I ; with respect f~2__ (pAo/EI )co2L +. From the minimization requirement:

0J 0A, [I4/.] =0 ( /=0, l) (6)

one obtains a system of two linear equations in A o and A l- The algorithmic procedure is quite straightforward but rather lengthy

to be included here. Figures 2, 3 and 4 show results for the deflection and bending moment

348 P. A. A. Laura, P. L. Vern£ere de lrassar, G. M. Ficcadenti

2~i0 ~I

4x10 -m

6x10 "~'

Iwl/3/

/ \ i \ / \

I \

I I k

I I k

/ k I I

I # I I

I I

I I

I I

I I

/ \ \ /

/ I ~ ~

Exact

+ + + --~m:O.

. . . . . ~-~l: 0.5

~i : o g

x

Sotution (s t l t ic case)

~ / iE~ i o~,L z = ~,79

0.I

0.2

0.3

F~.Z.

t~. "~" / %

/ ," \ \ / t \

~ ~ . ~ . ~ . . \~

/ I \ / /

\ \ /

,~--x

Two span, simply supported beam subject to conccntratad load (Po cosmt).

Forced vibrations of a continuous beam with ends restrained 349

0.10

020

1 ~ . 3 .

\ ' \"~- ,7 . / / I " ' ~ " ~ " / I

I ,. I

I I

\, / I

//~%% \ /

I \ / \

/ \

I t I \

I

(. g

Exact So.ion (static ease)

+ +~-~- =0.

. . . . . ~-- = 0,5

=09

4'

/ / / \ \ / \

',(%_.y"._. / X I I \

\ i I

\ 7 "

\ \

Two span, clamped beam subject to concentrated load (Po cos cot).

350 P. A. A. Laura, P. L. Verniere de lrassar, G. M. Ficcmi~ti

lx10 -a

2 x ~ a

%

0 , 0

0.11

IM.J/ 'P.L

Fig. 4.

p-% / \

/ \ I

/ \\ I \

/ \ / \

/

! I / I

\ %, . . . ~ f //

I

I \ i \ i \ / \ /

~ x

+ + 4"-'~" =0.

. . . . . tO =0.5

. . . . . . ~ = 0 . 9

l eo cos

. / / ~ ~Ow Lz = 47,05 %/ EZ

Two span, clamped, simply supported case subject to concentrated load (Po cosmt).

Forced vibrations of a continuous beam with end~ restrained 351

amplitudes for three different conditions at the beam ends: (a)simply supported; (b) clamped and (c) clamped--simply supported.

Results have also been presented for the static case in order to have an idea of the accuracy of the approach. In the case of the bending moment the difference is appreciable but, as shown in other studies, the situation improves considerably as co increases. 7'8

LOAD fl cos o~t ACTING OVER A PORTION OF THE BEAM (Fig. 1(II))

Now, the functional is given by:

i 1 {d2W \2 + 1 (dWo 2[ ,[w.]= j o - - dx J Ix=o

+ ,;\ dx j [x= -2Po W2dx-"2 y2 Wyd (7)

where/~o = poL4/El . The procedure is similar to that previously explained. Figures 5, 6 and 7 show variations of deflection and bending moment

amplitudes for the same combinations of end supports as previously considered and for co/co~ =0, 0.5 and 0.9 (co I is the fundamental frequency).

In the static case one observes that the agreement between exact and approximate results is not as good as in the previous problem. Hopefully, the accuracy improves as the forcing frequency increases.

CONCLUSIONS

The results presented in this paper relate to displacement and bending moment response. The graph of the bending moment distribution has discontinuities in gradient and certainly the continuous functions assumed in this paper cannot possibly represent these. In the circumstances, the degree of agreement is reasonable in the static case.

Admittedly, the results corresponding to bending moment amplitude must be considered as first order approximations, useful from a structural designer's viewpoint. On the other hand, the displacement amplitudes

352 P. A, A. Laura, P, L. Verniere de Irassar, G. M. Ficcadenti

16x~ 4

/ \ / \

/ \ // \

/ ~ /

\ / I' Exact Solution (static case)

,~ t " + + =0. \ /"

/ ~d =0.5

. _ / . . . . . . ~ = 0.9

L ~ , ~ e I

2 xlO-" 4xe -~

l~.S.

/ / / \

/ \ / \

i / \ \ I %

/ \

, ~ ~.

i I

\ /"

\ / • i /

~ . . r X

Two span, s ~ p l y supported case subject to Po cos o~t over the left span.

Forced vibrations of a continuous beam with ends restrained 353

4x~'

/ \ / / \ I \ /

/ ~ \

/ / \

/ , / . / ~ . ~ \

/ /

/ \ i

/

X

Exact Solution (static case)

+ + + ~,, =0.

= 0.5 (0,

=0,9

_4110 -z

_ 2~10 -2~

k i l

r,,

• r I

[ / S/ "\\

\ % / I

% /

\ 1.

~ ' - X

Fig. 6. Two span, clamped beam subject to Po cosoJt over the left span.

354 P. A. A. Laura, P. L. Verniere de lrassar, G. M. Ficcadenti

2x10-4~ 4x10 -4

_8x10 "z

- 2 -6x10

_4x10 -~

-Z -2x10

IN I~oLZ I

t \ / \

/ \ I \

/ \ / \

I

i \

I ...-TLZ='...

I \ / \ /

/ \ / N ,,

,,.~-x

E n d ' Solution ($'l"~m ease,)

÷ + + ~ ' l =0.

. . . . . ~---- = 09 60 I

~ . _ ~ . _ ~ Pe c°~ ~¢

I ! I I I I I

/ \ \

I

/ \ \ , /b

r v X

Two span clamped, simply supponmt cme subject to Po cos cot over ttg left span.

Forced vibrations of a continuous beam with ends restrained 355

obta ined following the present approach are o f interest in acoustics applications, e.g. sound radiation calculations.

R E F E R E N C E S

1. D.J. Gorman, Free lateral vibration analysis of double-span uniform beams, Int. J. Mech. Sci., 16 (1974), pp. 345-51.

2. D.J. Gorman, Free vibration analysis of beams and shafts, John Wiley, New York, 1975.

3. S. S. Chen and M. W. Wambsganss, Design guide for calculating natural frequencies of straight and curved beams on multiple supports, Components Technology Division, Argonne National Laboratory, Argonne, Illinois, USA. Report ANL-CT-74-06, 1974.

4. H. Chung, Analysis method for calculating vibration characteristics of beams with intermediate supports, Nuclear Engineering and Design, 63 (1981), pp. 55-80.

5. R. D. Blevins, Formulas for natural frequency and mode shape, Van Nostrand Reinhold Co., New York, 1975.

6. P.A.A. Laura, P. Verniere de Irassar and G. Ficcadenti, A note on transverse vibrations of continuous beams subject to an axial force and carrying concentrated masses, Journal of Sound and Vibration, 86(2) (1983), pp. 279-84.

7. P. A. A. Laura and R. Duran, A note on forced vibrations on a clamped rectangular plate, Journal of Sound and Vibration, 42 (1975), pp. 129-35.

8. G. B. Warburton, Comment on 'A note on forced vibrations of a clamped rectangular plate', Journal of Sound and Vibration, 43(3) (1976), pp. 461-6.

A P P E N D I X ~ D E T E R M I N A T I O N OF THE ~,~ s

0111 = 2 ~ 0 t 1 2

Ct12 = 1

0(13 = - - 2 4 ' 1 - - 1 - Oil5 - Oil4

ns(-4~'~ - 12~'x 4) ~ - 4 ~ - 1) + e / 3 ( 8 ~ + 3 + 4 0 ~ b ~ + 1 8 ~ )

+ ~/2( - 2 - 1 4 ~ ) + ~ - 4 ~ - 2 8 ~ )

cq,, = -- ~/s(6~)~ + I) + q'*(14~ + 2) + qs( _ 8~b~ - I)

356

0C24

P. A. A. Laura, P. L. Verniere de lrassar, G. M. Ficcadenti

-,76(3 + 84,~ + 18,/6 + 40,/,~¢~) +,15(4 + 10¢~ + 28~6 + 6 0 ~ ¢ D - q 2 ( I + lObby)- ~(2¢p' + 20¢~b~)

t/4(1 + lObby) -- r/5(2 + 18~b~) + t/6(1 + 8~b~)

0~25 -~--

24,~, ~, x _ , , rf(6$'~ + 1 0 ~ + 2 + vt ,e2, r f (4 + lO~b' t + 28~b~ + 605 t$2) + t/2(2 + 18~b~) + r/(4~b'~ + 36¢',¢~)

t/4(1 + 10#~) - t;5(2 + 18~b~) + t/6(l + 8~b~)

- ~s(2 + 6 ~ + lO~b~ + 2 4 ~ b ~ ) + t/4(8#'~ + 3 + 18#~ + 40~b't#~) - ~/2(1 + 8~b~) - r/(2~b~ + 16~b~b~)

a2~ = , ' 0 + 10¢~) - ~s(2 + ]8¢~) + ~6(] + 8¢~)