Applied Aeoustics 25 (1988) 225-234

Vibrations of Rings of Variable Cross Section

P. A. A. Laura

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

C. P. Fil ipich

Mechanical Systems Analysis Group, Universidad Tecnol6gica Nacional, 8000~Bahia Blanca, Argentina

R. E. Ross i & J. A. Reyes

Department of Engineering, Universidad Nacional del Sur, 8000-Bahia Blanca, Argentina

(Received 14 September 1987; revised version accepted 19 April 1988)

ABSTRACT

The present paper deals with the determhlation of the lower natural frequencies of vibration of rings of variable cross-sectional area using three approximate schemes:

--using polynomial coordinate functions in the angular coordinate in order to approximate the fundamental mode shape

--expanding the ring response in terms of a sinusoidal truncated series --by means of afinite element algorithm.

When using the first two procedures the Ritz method is applied in order to obtain the frequency equation. In general veo' good agreement is obtained between the eigenvalues predicted by the three approaches.

I N T R O D U C T I O N

The dynamic behavior of ring-like structures is of interest in several technological applications: electric motors and transformers, transducers, automotive and aircraft tires, etc.

225 Applied Acoustics 0003-682X/88/$03.50 (~) 1988 Elsevier Science Publishers Ltd, England. Printed in Great Britain

226 1'. A. ,t. Laura el al.

(a)

h~ c h e

/ ( + \ \

' "" / / ( r hl I= ,: ~ - - l/~j

(b)

.i / \ "/

Fig. 1,

/'~,R

N , Fundamental Mode

Shape

Ring- type e lements on non-un i fo rm cross section execut ing in-plane ax isymmetr ic vibrations.

Free vibrations of circular rings of uniform cross section have been extensively studied in the literature ~ -2 but apparently very few studies have been performed on vibrating rings of non-uniform cross section.

The present study deals with the determination of the lower frequencies of vibration of rings of non-uniform cross section (see Fig. 1) using bending theory. 2 Accordingly they must be considered as first order approximations but nevertheless quite useful for design purposes. A very complete ring theory has been used in an excellent paper by Soedel and co-workers) This study deals with a uniform ring which deviates from axisymmetry due to a local mass or stiffness non-uniformity.

APPROXIMATE ANALYTICAL SOLUTION

As previously stated, rings of non-uniform cross section will be studied--see Fig. l(a) and (b). Only axisymmetric modes will be considered in the present investigation.

Since the Rayleigh-Ritz method will be employed it is convenient to write

Vibrations of rings of variable cross section 227

down, as a first step, the expressions for the maximum strain energy, Ureas, and for the maximum kinetic energy Km,~.

where

Accordingly R fn/2 M 2

gmax= 42- J o EI(~) 4 R (~,2 Me(a)

- - d ~ = ~ j 0 - i(.:~)--d~ (1)

E10'- EI(~) [ a')] M = R- R2 (w" + w), (')' - c%-j

w,u: tangential and radial displacement amplitudes respectively (w' = u)

It is assumed that no normal force is acting on any cross section of the ring.'*

0: rotation amplitude of a given cross section

0=-~(w+u')=- (w+w") (see Ref. 4)

M: bending moment amplitude

I(~),A(~): variable moment of inertia and respectively.

Accordingly E f~,.,2

U . . . . = 4 ~ 3 I(~)(w" + w') 2 d~

On the other hand, the maximum kinetic energy is given by

4 Rp e )2 £i '2 Xmax= 2 A(~)(v"2 + w'2) do~

Solution by means of Rayleigh's Optimization Technique

Requiring that

Umax= Km.x

one obtains Rayleigh's quotient which for the present conveniently expressed in the form

f~ /2

i(~)w'" + w') 2 d:~ {12 = pA 1 (_o284 _

f ~,/2 EI1 a(cO(w 2 + w '2) d:~

where A(:0 = a(~)A1 and I(:0 =./(~)11-

cross sectional area,

(2)

(3)

(4)

case may be

t5)

228 P . A . A . Laura et al,

The fundamenta l mode shape will be approximated , in the interval 0 _< ~ <_ rt/2 by means o f the funct ional relat ion 4

w ~ w,(~) = :~>~ + A~ d + B~ (6i

where, in order to satisfy the geometric b o u n d a r y condit ions, Fig. l(c):

w(0) = w = 0(0) = = 0

one must have

i7~

Subst i tut ing eqn (6) in eqn (5) one obtains

and since (8) const i tutes an upper bound, by requiring

one is able to determine the op t imum fundamenta l f requency coefficient corresponding to the funct ional relation {6). s

l f j (~) = a(c0 = 1 (case of cons tan t cross section) the previously explained procedure yields fl~ = 2-71 which differs in the order of 1% from the exact eigenvalue (f~ ~ac, = 2"683}. The min imu m value of ~2~ is a t ta ined for ~' = 4-08. J

Solution by means of the Ritz Method using a sinusoidal expansion

It seems convenient to make use of the exact solut ion of vibrat ing rings in the case o f un i form cross section and axisymmetr ic modes. 2

Accordingly the coord ina te funct ions chosen in the present s tudy are

w,, = A~ sin nee (n = 2,4,6 . . . . ) (10a)

Accordingly

and

t u,, = hA. cos n~[ = w.} (10b)

RO,, = - A.(1 - n2) sin n0~ = [ --(w. + w',i)] (10c)

w , = ~--~ w,,= / ~7' A,,sinn~ ( l l i L_a

n = 2 , 4 . ~ n 2 , 4 A ~

Vibrations qf rings o[ ~ar}'able cross section 229

In the case of the ring shown in Fig. l(a) it is convenient to define the parameter r = h2/h 1. Accordingly

A 2 I. r3 - r: - - = ( 1 2 )

A t 11

The expressions for the maximum strain energy (Um~×) and the maximum kinetic energy (Km~,x) now become

f2 ] (W',, + W~')2 do~ + r 3 (it,, + w~')2 d:x 13) , 2R 4 ]_ )

K m a x = 2 ~ - - - (w~ + w~2)d'~ + r (w2 + wf)dz~ 14 )

Ritz' classical method requires that the functional

J[w,,] = Um.x[W.] -- Km,,×[w,~ ] (1 5)

be a minimum with respect to lhe A.'s. From the condition

~J[w.] 8Um.~[wa] 8Km.x[W.] - 1 6 )

8A. 8A. 8A.

one obtains a linear system of equations in the A'.s. From the non-triviality condition a determinantal equation in the frequency coefficients ~

corresponding, in the present case, to axisymmetric modes is obtained. It is a simple task to show that the partial derivative of U ...... with respect to

an arbitrary coefficient A. is given by

4Ell ?A,, n = 2,4-,6

+;7{[ 2 n = 2,4,~,

An,,,

( 1 7 )

230 P.A.A. Laura et al.

S i m i l a r l y

4P (nzA 1 OA. n = 2 , z t , 6

n = 2 . 4 . 6

+ ( " / 2 f ] / , ~ A. sinne] sinme d ~0 k_k

t t = 2 . 4 . 6

+i ~ ~ n A " c ° s n e l m c ° s m ° ~ } d e

n : 2 , 4 , 6

(18)

In the p r e s e n t s t u d y it w a s dec ided , fo r s impl ic i ty , to t a k e e o = ~/4.

A c c o r d i n g l y the i n t eg ra l s a p p e a r i n g in the e n e r g y e x p r e s s i o n s a re

f] /4 s m (n + m)n/4 . sin (n - m)~/4 cos ne c o s m e d e = . . . . . . . ~ .... 4

2(n + m) 2(n - m)

ff /4 sin (n + re)n~4 sin (n - m)~/4 sin ne sin m e d e . . . . . . . . . . . . . . . . . + -- - . . . . . . . . .

2(n + m) 2(n - m)

f[ /2 [ s i n ( , , +m)Tr /4 s i n ( n - n 0 r r / 4 ] .,4 cos ne cos m e d e = - [ - - - S i ~ + - ; ] i - - - + 2(n - mt J

f[ ,.2 [ s i n ( n + re)K/4 s i n ( n - - m ) r c / 4 ] /4 sin ne sin mc~ d e = - [ _~n-+ ~ + 2~--m-i J

fo r n ¢ m a n d

f ~/4 f rr, 2 7[ cos 2 na d e = ~8; =,,4 cos 2 na d e = -8

fO/4 7r, f ,'2 ~ ..... sin 2 ne d e = 8-; sin 2 ne d e rr / 4 8

w h e n n = m.

Vibrations o f rings of variable cross section 231

After some s t ra igh t forward a l though lengthy algebraic manipu la t ion one can show that

where

2R 4 ~ U r e a x =

4 E I 1 (3A,. n = 2 , 4 , 6

2 63Krnax= 4P ~OzA 1 8A, .

n = 2 , 4 , 6

for n ¢ m and

A . U , , . (19)

A , K , , . (20)

Unto = (fin,. + Yn,.)( n -- n3)( rn -- m3)( 1 -- r3)

Kn,. = [(7n,. -- [~n,.) + nm(Tn,. + ]~n,.)]( 1 - r)

sin (n + m)~z/4, sin (n - m)g /4

/~n,. = 2(n + m) ' 7n,. = 2(n - m)

7t U",, = ~ ( m - m3)Z(1 + r 3)

Kin, . = -~(m 2 + 1)(1 + r)

when n = m. If one wishes a f irst-order approx ima t ion for the fundamenta l f requency

coefficient one takes the first term o f eqn (11) and obta ins

/ ~ 1 /1 + r 3 f~l = , x / E I I ~°, R2 = 2"683X/ - 1 ~ - r (21)

which, in the case o f uni form cross section, yields the exact eigenvalue.

F I N I T E E L E M E N T S O L U T I O N

The finite element solut ion was ob ta ined considering rectilinear beam elements (Fig. 2). The c o m p o n e n t s o f the displacement vector used are:

(a) Transverse displacement u - A x 3 + BX 2 + C x + D (22a)

(b) Axial d isplacement w = E x + F (22b)

Calcula t ions were per formed using 8 elements for the quad ran t o f the mechanical system since the accuracy o f the results was qui te good f rom the point o f view o f the goal o f the present study.

232 P. A. ,4. Laura et al.

/ /

/ /

\ \

Fig. 2.

J

\ \ \

k.

j'

/ //

/ / j "

s

Finite element modelling of the system shown in Fig, l(a).

NUMERICAL RESULTS

Table 1 shows results of~21 for the ring-type structures shown in Fig. l(a) and l(b) using (I) Rayleigh's optimization procedure and the modal shape defined in relation (6); (II) the Ritz method and sinusoidal coordinate expansion (for the configuration shown in Fig. l(a)); and (lII) the finite element technique.

In the case of Fig. l(b} the parameter r is defined as the ratio h2/hi . . . .

The agreement between the three sets of numerical values is very good from a practical viewpoint, especially if one takes into account the simplicity of the functional relation (6) employed when making use of Rayleigh's optimization technique and the fact that six sinusoids were used when applying the Ritz method.

Table 2 gives a clear indication of the rate of convergence of the Ritz method when using sinusoidal coordinate functions. It is observed that the difference is very small between the results obtained using four- and six-term

Vibrations o f rings o f variable cross section 233

TABLE 1 Comparison of Fundamental Frequency Coefficients

E ~ I {DI j~2

for the Ring-Type Structural Elements Shown in Fig. 1

Configuration Method Values o f r

1"00 1"10 120 1.30 1"40 1.50

Fig. I(a)

Fig. l(b)

(I) a 271 2.86 3-01 3-17 334 352 (4.08)* (3.95)* (3-85)* (376)* (369)* (363)*

(I1) h 2683 2.8:2 2.97 312 328 3.44 (III) c 269 2-83 297 3 12 328 3.43

(1) a 2-71 2'85 3.00 3-14 3"29 343 (4-08)* (3.94)* (3"83)* (375)* (3-67)* (3.62)*

(III) c 2"69 2"82 2"96 3"09 3-22 335

a (I): Rayleigh's optimization technique; ( )* values of ~,. h (II): Ritz method (six terms). c (Ill): Finite elements results.

TABLE 2 Analysis of the Rate of Convergence of the First Two Natural Frequency Coefficients Corresponding to Axisymmetric Modes in the Case of the Configuration Shown in Fig. l(a)

Using Two, Four and Six Sinusoidal Coordinate Functions

r = 110 r = 1"20 r = 130

(2) (4) (6) (2) (4) (6) (2) (4) (6)

~1 2.822 2.822 2.822 2.970 2.969 2.969 3.127 3.123 3-122 ~ l 15"332 15"250 15"237 16"208 15"893 15'845 17"165 16"487 16"389

r = 1"40 r = 1"50

(2) (4) (6) (2) (4) (6)

~)1 3-292 3 " 2 8 2 3.279 3.464 3.445 3-439 ~1 18"190 17-034 16.880 19-275 17.369 17"239

234 P.A.A. Laura et al.

solutions, respectively in the case of the first two eigenvalues corresponding to axisymmetric modes.

It is also concluded that Rayleigh's optimization procedure yields excellent accuracy, from an engineering viewpoint, for the fundamental eigenvalue especially if one takes into account the simplicity of the functional relation employed. Obviously the accuracy can be improved if additional polynomial coordinate functions are employed.

A C K N O W L E D G E M E N T

The present investigation has been sponsored by C O N IC E T Research and Development Program PID 3009400.

R E F E R E N C E S

1. Prescott, J., Applied Elasticity. London, Longmans, Green and Company, t 924. 2. Soedel, W., Vibrations Of Shells and Plates. Marcel Dekker, Inc., New York, NY,

1981. 3. Allaei, D., Soedel, W. & Yang, T. Y., Natural frequencies and modes of rings that

deviate from perfect axisymmetry. J. Sound and Vibration, 3 (1986) 9 28. 4. Laura, P. A. A., Filipich, C. P. & Cortinez, V. H., In-plane vibrations of an

elastically cantilevered circular arc with a tip mass. J. Sound and Vibration, 115(3) (1987) 437 46.

5. Bert, C. W., Use symmetry in applying the Rayleigb Schmidt method to static and free vibrations problems, lndustr. Mathematics, 34 (1984) 65 7.