Ingenieur-Archiv 46 (1977) 245-- 251 Ingenieur- Archiv

�9 by Springer-Verlag 1977

Dynamic Behaviour of the Ring Subjected to Distributed Impulsive Load

S. Suzuki, Nagoya

S u l c n m a r y :

Dynamic l)ehaviour of the ring subjected to distributed impulsive load is investigated. The lii]g is represented by the centroidal line and the problem is solved one-dimensionally. Impulsive load is assumed to be the step function with respect to time. The effects of shearing torce and rotatory inertia to dynamic load factors with respect to axial force and bending moment are studied.

From the results of theoretical analysis, it became evident that shearing force and rotatory inertia can not be negIected in such a problem as this, and that shear coefficient varies remarkably by the dimensions of the ring and the loading conditions.

U b e r s i c h t :

ls wird das dynamische Vcrh;_dten eines Rmges m]tersucht, der einer verteilten Stol31ast untcrworIen ist. Der Ring wird (lurch seine Mittellinie ersetzt, so dab das Problem als eindimensionai behandelt werden kann. Die StoBlast wird als Sprungfunktion beziiglich der Zeit angenommen. Die Einwirkungen von Schubkr[iften und Drehtrgtgheit auf die dynamischen Lastfaktoren werden far Achsialkraft und Biegemoment untersucht. Die Ergebnisse zeigen, dab SchubkrSAte nnd DrehtrS.gheit nicht vernachlS~ssigt werden dfirfen. Ferner zeigt sich, dab der Schubbeiwert stark yon den IRingabmessungen und den Belastungsbedingungen abhS~ngt.

1. In troduct ion

The problem of vibrations of free rings is of basic importance in gears, electrical machines and stiffened shells. Therefore, vibrational studies of rings have been given much attention.

Many papers have already been published on the dynamics of rings. Previously, the author E]~ investigated the dynamic behaviour of the ring subjected to distributed impulsive loads along inner and outer edges of the ring, and Chou [~J analysed tt~e layered rings.

In these papers mentioned above, rings are assumed to be constant in thickness and are analysed as two-dimensional problem. But, since it is very difficult to solve exactly the cases such as toroids [3], problems are analysed one dimensionally, representing the rings by the centroidal lines. In such a way, Seidal [4J, Kirkhope E53 and Rao [6] discussed the effects of shearing force and rotatory inertia to frequency. To simplify calculus, all of them treated the case where mid-surface extension is neglected.

In this paper, analysis is carried out for the rings subjected to distributed impulsive loads, taking into account of shearing force, rotatory inertia and extensional force and the relation- ships between dimensions and maximum stresses induced in the ring are obtained. Impulsive loads are assumed to be the step function with respect to time. Shear coefficient k, on which Cowper [7J discussed in detail previously, is also studied for the case of arc beam.

2. F u n d a m e n t a l Equat ions and The ir So lu t ions

Dimension and co-ordinate of the ring are illustrated in Fig. 1. The cross section of the ring is assumed to be symmetrical about X - - X ' axis. The distributed impulsive load is assumed

t8

246 Ingenieur-Archiv 46 (t977)

to be the step Iunction with respect to time and to be Z G cos ~*0, where G is constant and is positive integer except unity.

The equation of motion can be derived by considering the element of the ring in Fig. 2 a. The differential equations for rotational motion and for translatory motion in radial and tangential directions of this element are

OM O~rp OF 02u c~ P 32v o~- + tVR = o R I ~ , O) - ~ + P -- qR = e A R - ~ , (2) o-o--- F = ~ A R ~ , (3)

where R, A, I, Q, M, F, P, u, v and ~0 are radius of centroidaI line, sectional area, moment of inertia, density, bending moment, shearing force, axial force, radial and tangential deflec- tions and angle of rotation, respectively.

q(o)

Fig. 1. Co-ordinate and dimension of the ring

qtO~

b o t

Fig. 3. (a) Element of the ring showing displacements and forces, (b) Relationship between q and t ime

At first, the case will be considered where shearing force and rota tory inertia are neglected. From the elementary theory of bending of circular rings, axial force, bending moment and shearing force can be expressed in terms of the displacement components as

P = ~ - ~ 6 - u , M--mloo~ +5o ' F- - N\~6~+ oo21 (4)

where E is Young's modulus. Substitution of (4) into (2) and (3) gives

- a \00, + 008 ] + 5o - u - qZ = T ~ ' 00~ 00 + a ~ 0 ~ + 00~ ] = ~ ' (5)

t where T = ~ - ~ / ~ , a = I / A R 2 and A = R2/EA. Now, we put

CO

q - - Z G cos ~ 0 , n q= 1 (6)

In this case, the deformations will take the form

CO CO

u : Z % cos nO, v : Z v,~sin nO, (7)

where u, and v, are tile functions with respect to T 0nly. Substitution of (7) into (5) gives

u~ + (an~ + 1) u . - n ( ~ + 1) ~. = - r v~ - ~(~n~ + 1) u . + n~(~ + 1) v~ = o , (8)

where the primes denote the differentiation with respect to T. Therefore, the fundamental equation with respect to u~ becomes

u~," + (n 2 + 1) (an 2 + 1)u;'~ + a~,~(n ~ - - 1) 2 % = -- n~(a + 1) G2 . (9)

From (9), a + l

% = C 1 cos ~T + C 2 sin a T + Ca cos f iT + C4 sin f iT a(n ~ - 1)2G ~ ' (lo)

S. Suzuki: Dynamic Behaviour of the Ring Subjected to Distributed Impulsive Load

where 2fl~J = (n~ + 1) (t/fr 2 -}- 1) 2~ ~(7r -~- 1)~ @t~Z2 2t- 1)2 __ 4an~(n ~ _ 1)g

stants. With the aid of (8), initial conditions wii1 be given as

. ~ = s 1 6 3 / at r o. ,g; - - - q , / t !

247

and Ci are con-

-1], / (11)

Using (11), Ci are determined and u,, and v. will be expressed in the following form

a + 1 [{(a + ~)f12 _ a 0 ~ _ 1)2} cost:C+ {a(n 2 - 1) 2 - (~+ 1)~2}cos~r %/q.)~ - - . ( ~ 2 -~)~ (~ + ~) (fi~ _ ~)

v,&,,Z - ~'~ + ~ j ~ oo~ ~r - ~ cos ~r } ~(~-171 ~ ~ - 1 �9

From (4) and. (12), axial force and bending moment will be obtained as follows:

{ (/~2 @ ~2 __ 1) COS o ~ r - - (~x 2 @ n 2 - - 1) c o s f l y ] c o s ~ 0 ;,

(12)

(13)

] cos ~0 + l j ~ . (14)

3. Effects of Shearing Force and Rotatory Inertia

Consider the case where shearing force and rota tory inertia are taken into consideration. Axial force, shearing force and bending moment can be expressed as

P = - K \ ~ d - - u , F = ~ - ~ - + v - - R ~ , M - - R 00" (15)

Substitution of (15)into (1), (2) and (3) gives I

b \002 + 00

• O02

02R cp b [Ou ) O~R 9

O02 + a ~ - - ' ~ O0 T V - - R ~ OT 2 ,

~-g / + o 6 - u - q 2 - o r ~ ,

oo - - b + v - - R o y - - OT~ ,

where b k G / E and G and k are shear modulus and shear coefficient, respectively. As in the previous section, u, v and q) will take the form

(16)

(17)

(lS)

From (20),

r = -- (n2b @ 1) r @ ~r -}- 1) v n - - 7?,b~n - - qn,~,,

v,, = n ( b + l ) u . - - ( n 2 + b ) v , , + b % .

n(2abn ~ + 2an 2 - - ab - - a + b) u . = ave" + (abn ~ + 3an 2 + ab + b) v;i + ! [

@ n2(abn 2 + 2an 2 - - a + b) v / + na(b + 1) q~2. ]

(2o)

(21)

18"

where u., v. and ~v~ are functions with respect to T only. Substitution of (19) into (16), (x7) and (18) gives

u = 2J u,, cos n0 , v = ~ v,, sin nO, Rqv = ~ ~. sin nO , (19)

248 ingenleur-Archiv 46 (1977)

Therefore, the fundamental equation with respect to v. becomes

av~ .... + (abn 2 + 2 a n 3 + ab + a + b) v'~ 't' + {n~(2ab753 + a n 2 - ab + a + b) + ab + b} v~ +

-h u2(n 3 - - 1) 3 ab% = - - n(abu 3 + an 3 + b) q,,), . (22)

From (22), % = C l c o s ~ 7 " + C 3 s i n c 2 T + C a c o s , S T + C 4 s i n / 3 T +

an2(b + 1) + b + C5 cos 71" + C 6 sin 71" abn(,z: -- 1) 2 q ' ) ' ' (23)

where C/a re constants and ~ , ~3 and y3 are the roots of the following equation, where ~,/3 and Y are positive:

- - - - _ _ - - X a b/q, 3 -~- 2/'/, 3 -@ b -]- 1 ~- ~ - x 3 -]- 2 2bn2 _]_ n2 3 Ju 1 -f- ~f- b ~- a

- - u2(n 2 - - 1) 3 b = o . (24)

With the aid of (21), initial conditions will be given as tP i t r t ~ s m

C ' = - ~ ( 5 + ~) qnz Using (25), Ci will be determined in tile following form.

a t r = o . (25)

q = - ( G + =~7~ ,~ ) / (7 - ~<~)(~,~-~<~),

G = (B, + B~<~r~)/(/~ i - ~<~) (~,~ - 7 ) ,

G = - - (B, + B=~<~7)/(;r ~ - ~ ) ( # - - 7 ) ,

where B 1 = n(b + 1) q,X and B 2 = - - {au2(b + 1) + b} G~/nab(u 2 - - 1) 2,

C 2 = o j

C, -- o ,

C ~ : o ,

(26)

4. Num e r i ca l Analysis

Numerical calculations have been carried out for several cases. Attention is paid to axial force and bending moment acted in the ring and the relationships between these maximum values and dimensions will be studied.

At first, the value of k must be examined. Cowper [7] analysed the straight beam three- dimensionally and obtained the values of k for various sections. But, for arc beam, the value of k varies remarkably by the loading condition and dimensions. These will be discussed in detail in Appendix (section 6). In this section, calculations will be carried out on the assumption

of b = 1/3. The case will be t reated where the load is assumed to be q(1 + cos 20). The relationships

between T and P at 0 = o are illustrated in Fig. 3 for the case of a = o.1. Solid line and dotted

r r

-I

F i~ . 3.

8=0 a=o.i

] i i

i (- ' , :, , ,

T

i

I I '

o I !

v

Rela t ionsh ips be tween t i m e and axia l force a t 0 = o

S. Suzuki: Dynamic Behaviour of the Ring Subjected to Distr ibuted Impuls ive Load 249

line indicate tile cases where shearing force and ro ta tory inertia are taken into consideration (case A) and are neglected (case ]3), respectively. Periods are nearly equal for both cases, but the extreme value for case B is a little larger than tha t for case A. The relationships between T and M at 0 = o a r e i l lustrated in Fig. 4 for the case of a = o.1. The period for case A is longer than tha t for case B.

The relationships between a and dynamic load factor (D.L.F.) with respect to P and M at 0 = o are i l lustrated in Fig. 5. The values of D.L.F. for P are, in general, much larger than those for M.

The distributions of the values of D.L.F. for a = o.1 along the circumferential direction are i l lustrated in Fig. 6. The values for M are independent of 0 for both case A and t3. On the contrary, the values for P va ry remarkably.

0 = o

a=o . l 0.8

I / % / '~ / '~ / i/' J \ 0.4 / , , ,,, /' f l i ' , ' \ i

',,,, X ,' I 0 20 30 40

T

Fig . 4. Relat ionships between t ime and bending m o m e n t at 0 = o

I..C ..,.J ,d

3 . 6

3.4

2.2

2.0

0=0

0.1 0.2 0.3 a

Fig . 5. Rela t ionships be tween a and D.L.F. wi th respect to P and M at 0 = o

a=O.l

J

3

z .......... ~ . . . . . . . . .

l i

o 'f,~ '~2 e

Fig . 6. Dis t r ibu t ion of D.L.F. wi th respect to ? and M for a = o. 1 along the circumferential direction

Although the simple case is t reated in this section, the case where arbi t rary distributed impulsive load is applied will easily be analysed in the same way.

Since the ring is t rea ted one dimensionally, the results in this paper will be useful as design data. In order to obtain more accurate results, we must analyse the ring as two or three- dimensional problem.

5. Conclus ions

The following conclusions will be drawn from the present results. (1) The value of D.L.F. with respect to axial force becomes very large. Axial force can not be neglected in such a problem as this.

250 Ingenieur-Archiv 46 (1977)

(2) Shearing force and ro ta tory inertia have remarkable influence on the value of bending moment . (3) The value of k for arc beam is much different from tha t for straight beam obtained by Cowper.

6. A p p e n d i x : Shear Coeff icient k for Arc B e a m

Cowper [7] analysed straight beam three-dimensionally and discussed the value of k in detail. Following his method, the value of k for arc beam will be investigated. Since the values of displacement and stress must be known for obtaining the value of k, the case of the ring with constant rectangular cross section will be treated.

The center of the ring is taken as origin. Stress-displacement relation is

0v v ~ 0u z 0f V ~- r 00 G (A1)

where u, v, r and G are radial displacement (positive in radially outward direction), tangential displacement, shearing stress and shear modulus, respectively. Now, we put u and v as

u = U + % , v = V + ( r - - R) q~ + Vo, (A2)

where R, U, V and ~ are mean radius, mean displacements in the radial and tangential direc- tions and mean angle 0I rotation of a cross section, respectively, and they are expressed as

I t follows from (A2) and (A3) tha t

f f Uo dA = o , f f v o dA = o . (A4)

Substi tution of (A2) into (A1) and integration over a cross section gives

r dA -- V A + Rq~A + oO - - d A . (A5)

Therefore, the following equation will be introduced:

R O0 R + ~ = ~ r ~ - - o r / d A = ~ - ~ r - - Or/ d A , (A6)

where Q = f f z dA. From (A6), k will be obtained in the following form:

/ff ( k = I ~ r - - G o r / d A " (A7)

Next, consider the case where the distr ibuted static load q cos nO is applied along the outer edge of the ring. Since previously the author [11 analysed the ring subjected to distributed impulsive load ~ q, cos n0, this results will be used. Then, for n = 2, z and v become

n

TIN = {(1 -{- 3 e2) ~2 _j_ (3 -~ 82) e4~ -4 - - (1 @ 82 -~- 2~ 4) - - (2 -]- ~2 _}_ el) e2~-2}/(1 __ e2)8, (AS)

tqr2 ~ 3 + V (1 -~- 3 e2) ~a _ e4 V / ~ = [3(1 + v) 3 (3 @ ~2) ~}-a __ (1 + 82 + 2e4) T] - -

~-~(2~ + ~, + ~ +~)~,-1}/(~_ ~)~ (A9)

where q, r 2 and v are inner and outer radius and Poisson's ratio, respectively, and where = rl/r 2 and ~ = r/r 2. Since v o and Q are obtained from (A2), (A3), (A8) and (A9), the value

of k will easily be determined. The relationships between k and e are il lustrated in Fig. 7 for n = 2, 3 and 4, assuming

v = 0. 3. The values of k are about 0.86 in the range of e ~ 0.6, which are nearly equal to the

S. Suzuki: Dynamic Behaviour of the Ring Subjected to Distributed Impulsive Load 251

value ob ta ined by Cowper. But , as the va lue of e decreases and n increases, the va lue of k

varies r emarkab ly . T h a t is, the va lue of k for arc b e a m is not cons tant .

The i l lus t ra ted resul ts are those for the case of the r ing wi th r ec t angu la r section. But , i t

will be considered t h a t s imilar results will be ob ta ined for the o ther kinds of section.

i.O - n=2

0.8

0.6

0.2 0.! 0.05 0.01 (3 r i i

0.2 0.6 LO E

Fil~. 7. Relationships between k and

R e f e r e n c e s

1~ Suzuki, S.: Dynamic Elastic Response of a Ring to Transient Pressure Loading. Trans. ASME, J. Appl. Mech., Ser. E, 33 (1966) p. 261

2. Chou, S.; Greif, R.: Numerical Solution of Stress Waves in Layered Media. AIAA J. 6 (1968) p. lO67 3. Lincoln, J.; Volterra, E. : Experimental and TheoreticM Determination of Frequencies of Elastic Toroids.

Experim. Mech. 24 (1967) p. 211 4. Seide], B. ; Erdelyi, E.: On the Vibration of a Thick Ring in its Own Plane. Trans. ASME, J. Appl. Mech.,

Ser. I3, 86 (1964) p. 24 ~ 5. Kirkhope, J. :'Simple Frequency Expression for the In-Plane Vibration of Thick Circular Rings. J. Acoust.

Soci. Am. 59 (1976) P. 86 6. Rao, S.; Sundararajan, V.: In-Plane Flexural Vibrations of Circular Rings. Trans. ASME J. Appl. Mech.,

Ser. E, 36 (1969) p. 620 7. Cowper, G.: The Shear Coefficient ill Timoshenko's Beam Theory. Trans. ASME, J. Appl. Mech., Ser. E,

33 (1966) p. 335

Received September 14, 1976

Prof. Shin-ichi Suzuki Dept. of Aeronautics Nagoya University Furo-cho, Chikusa-ku, Nagoya, Japan