Comments on “lowest natural extensional frequency of clamped elliptic arcs”

Journal of Sound and Vibration (1974) 37(1), 146-150

COMMENTS ON "LOWEST NATURAL EXTENSIONAL FREQUENCY OF CLAMPED ELLIPTIC ARCS" The writers wish to congratulate the authors of reference [1] for their interesting investigation. It appears that equation (9) [I] implies an approximation which has not been treated by most authors but has been thoroughly discussed by the writers [2, 3]. Since in references [2] and [3] the effect of normal forces has been neglected, this factor will be taken into account in the present discussion.

-•W§

(dW/as}ds

O+ (aOlas)ds

u+(au/Ss)ds

\

/:?

\

I \

I \1

x4

Figure 1. Analysis o f an infinitesimal are segment.

The following relations can be derived directly from Figure 1 by using the well-known Bernouilli beam theory and appropriate geometric considerations: a0 as Ow

M El' u

N

~'s= X +E-A' ~u

.... 3s

(la)

w

R

+ O.

(Ib) (lc)

From equation (lb) one obtains (2) 146

LETTERSTO THE EDITOR

147

Multiplying equation (lc) by R results in the expression Oil

--

ar

= - w + OR.

(3)

Accordingly 0 z u -Ow O0 OR + 0~-- 7 . ar 2 = ar + Rg-;; oq~ oq~

(4)

O0 O0 R ~ = R ~s = - ' - ~ M

(5)

From equation (la) one obtains

and from equation (lc)

~ a. w l{a,+w). r

(6)

From equations (4) and (6) the following expression results for M: (7) The strain energy for extensional vibrations is accordingly I I'M 2

1

N 2

( 1 [0 2u

Ow

laR{au + w

f?w ,,]

+EA jkar

)]2

de

~-.

If R is a constant (circular center line) equation (8) yields

f 1 (aZu a,v~2 U=EIj-~t-~+~) ds+EAf /tOw ~-u} ~2dr--~,

(9)

which agrees with equation (9) of Wang and Moore's paper [1]. It would be interesting to investigate the effect of the variation of R in the numerical results obtained by the authors. Solid Mechanics Laboratory, Universidad Nacional del Sur, Bahia Blanca, Argenthta (Received 13 February 1974)

E. ROMANELLI P . A . A . LAURA

REFERENCES

i

1. T. M. WANG and J. A. MOORE 1973 Journal of Sound attd Vibration 30, 1-7. Lowest natural extensional frequency of clamped elliptic arcs. 2. E. ROMANELLI1972 Journal of the Acoustical Society of America 51, 1486--1489. Comments on "Lowest natural frequencies of elastic hinged arcs". 3. E. ROMANELUand P. A. LAURA 1972 Journal of Sound and Vibration 24, 17-22. Fundamental frequencies of non-circular elastic, hinged arcs.