LETTER TO THE EDITOR

Comments on “Calculation of the natural frequencies and steady state re- sponse of thin plates in bending by an improved rectangular element.” [H. M. Negm and E. A. Armanios. Comprrt. Sttwc’tures 17(l), 139-147, 19831

P. A. A. LAURA, L. EKCOLI and R. 0. GROW

Institute of Applied Mechanics, Puerto Belgtano Naval Base 81 i I, Argentina

(Received 5 A&l 1984)

INTRODUCTION

The purpose of the present Letter is twofold, first to congratulate the authors far their interesting paper and useful improved rectangular element 1 I] and second to express their vigorous support of the opinion of well known experts [2, 3J regarding the basic importance of analytical developments in me- chanics, and as a matter of fact in all fields of sci- ences.

As stated by Dym in an excellent paper “.-. while great strides have been made in the development of general purpose computer codes, mostly based on finite element approaches, there is developing an increasing sense that such codes are not the an- swer for all problems, that there are still unresolved questions that are best answered by cooperation and collaboration between the analytical mechan- ician and the numerical analyst.”

It may be instructive for the educator as well as for the researcher and the professional engineer to analyze some of Negm and Armanios’ results at the light of the philosophy expressed in 12, 31.

As shown in [4, 51 a very simple and efficient approximate variational approach allows for the de- termination of the fundamentl frequency of vibra- tion of rectangular plates for very general combi- nations of boundary conditions. As shown in [61 the entire algorithmic procedure can be carried out on a pocket programmable calculator.

It will be seen here that the same approach can be used to determine vibration amplitudes when the plate is subjected to a uniformly applied sinusoidal

excitation of the type p. cos w t. The results are in very good agreement with those obtained in [I).

Ap~r5ximate determi~arioff of the pfatc response In the case where w is smailer than the second

natural frequency of the plate one takes the ap-

proximation:

+ a3 (f)’ •t a4 ($1 [P, (5) + ($ + p3 (f)’ + P4 (;)‘l (I)

where the QS and ps are determined in such a man- ner as to satisfy the governing boundary conditions (Fig. I):

W(0, y) = W(ri, _v) = W(.u, 0) = W(.u, h) = 0

(2)

(3a)

(3b)

The &s are flexibity coeff%ents defined by means of equations (3) which express the propo~ionaIity between edge rotation and acting normal bending moment.

Substituting (I) in (2) and (3) one obtains:

2

a’ = G’

013 = -2 3kz + Sk, + k,kz + 12

k,(kz + 6) ’

4/i, + 4kI + k,k2 + 12 ci4 =

k,(kz + 6)

1379

1380 P. A. A. LAURA et (11.

Table I, Values of Wlp,,d/D at the center of a square plate: Comparison of results for several value\ of wiol I

Bomdary Conditions e Exact Equation (1)

I 0.125 0.004128 1 0.004138 0.004195 ssss

where

The expressions for the Bis are similar to the pre- vious one. They are obtained substituting /ir and k2

by LJ = $D and X.4 = $D, respectively. 3 4

Making use of Galerkin’s variational formulation one obtains:

A ,X,

= [24&$P2 ($)‘,e’+“i ($$

P24Q24 + 24pnQ2Pla-j [I - (z)'],

(4)

where w, , : fundamental circular frequency

P14 = P2 +

PM = alp3

Q24 = Qz +

QM = PIQ,

P+h

P1= y +

P4 = y +

Ps = y +

Ph = 7 +

3WP1 + 6a4P4,

f P4 + a.?Pc + LY4Ph.

3&Q? + 6P4Q4.

+ Q4 + PJQS + P4Qh.

I

i+:++ 6

_k+!3+!$

f+T+!?.

The parameters Q2 thru Qh are obtained from the

expressions for Pz thru P6 changing the (Y;S by the

PiS.

This algorithm has been implemented in a

TEXAS TI-59 pocket programmable calculator in a similar form as that described in [6]. A comparison of dimensionless desplacement amplitudes at the center of a square plate is depicted in Table I for different combinations of edge conditions and for several values of the parameter forcing frequency/ fundamental frequency.

The agreement is very good, from an engineering viewpoint for all cases considered. On the other hand the accuracy of the polynomial (I) improves as w/w,, increases.

CONCLUSIONS

As stated in [6] “. . . the authors are well aware

of the fact that digital computers have made pos- sible the solution of problems which were impos- sible to solve thirty years ago. But proper, rational balance must exist between digital computer ap- proaches and approximate analytical techniques.”

The authors of [I] certainly deserve credit for their computer oriented solution but educators, professional engineers and engineering students should be also aware of the possibility of approx- imate analytical solutions such as the present one which is classical, very simple, efIicient and rather elegant. As shown in [7] stress resultants can be determined also using the same approach and with sufficient engineering accuracy.

Fig. I. Mechanical system under study.

Letter to the Editor 1381

REFERENCES

I. H. M. Negm and E. A. Armanios, Calculation of the natural frequencies and steady state response of thin plates in bending by an improved rectangular element. Compcct. Strrrc~ttr~s 17t I ), 139-147 (1983).

2. C. L. Dym. Analysis and modeling in mechanics: An informal view. Ct)rnrr~~. ~rrff~,r~fr~~.s 16 (l-4). IOI- 107 t 1983).

3. R. L. Eshleman, The decline of approximate meth- ods-its implications for computation, The Shock and Vibration Digest 13(6). 2 (I981 1.

4. P. A. A. Laura, L. E. Luisoni and C. Filipich. A note on the determination of the fundamental frequency of

vibration of thin, rectangular plates with edges pos- sessing different rotational flexibility coefficients. J. Sound Vihrrction 59(2), 355-368 (1977).

5. P. A. A. Laura and R. 0. Grossi, Transverse vibrations of rectangular plates with edges elastically restrained against translation and rotation. J. .!+rmd Vihrtrtion 75(l), 101-108 (1981).

6. P. A. A. Laura and R. 0. Grossi. Solving some struc- tural dynamics problems using a pocket programmable calculator. Znt. J. Me&. En~ng ~~~~~~f~i~~~ lR3). l79- 187 t 1983).

7. P. A. A. Laura and R. Duran, A note on forced vibra- tions of a clamped rectangular plate. J. Sorrnd VihrtrticJn 42(l). 129-135 (1975).