Impact of elastic cylinders subject to lateral constraint

Pergamon

Mechanics Research Communications, Vol. 27, No. 6, pp. 669--678, 2000 Copyright © 2000 Elsevier Science Ltd Printed m the USA. All rights reserved 0093-6413/00IS--see front matter

PII: 80093-6413(00)00147-6

IMPACT OF ELASTIC CYLINDERS S U B J E C T T O LATERAL C O N S T R A I N T

M. C. Harrison, R. P. Menday and W. A. Green Department of Mathematical Sciences, Loughborough University, Loughborough, Leics., L E l l 3TU, U.K.

/Received 31 May 2000: accepted for print 20 September 2000)

Introduction

The study of the impact of two elastic bodies has a history going back for over a hundred and thirty years to when St. Venant (see Todhunter and Pearson [1]) solved the problem of longitudinal impact of two elastic rods. St. Venant's solution involves a system of waves propagating in the rods following the iinpact and it predicts that when the colliding rods part company some of the energy is in the form of internal vibration of the rods. This results in an effective coefficient of restitution which depends on the density and the Young's modulus of each of the two rods but which is independent of the impact velocities. The predicted duration of the impact is also independent of the impact velocities and is equal to the time taken for the elastic wave to travel down one or other of tile rods and back to the contact surface. In the case of impact of a moving slug of finite length on a semi-infinite rod at rest, the slug will rebound only if its acoustic impedance is less than that of the semi-infinite rod. Otherwise, the slug and the rod will remain in contact for all time with the compressive stress at the interface decaying asymptotically to zero. In the former case, the rebound occurs when the stress wave set up in the slug returns to the interface and the effective coefficient of restitution is a function of the ratio of the two acoustic impedances. This paper deals with the problem of the longitudinal impact of a moving slug on a semi-infinite rod when, at the instant of impact, the slug is attached to a static body by a continuous distribution of shear springs over its lateral surface. The problem arises in the investigation of the impact of a vibrating b e a m against a static beam or against a stop when the moving b e a m is modelled as a system of short slugs joined to each other by shear springs (see [2]). In the case when the acoustic impedance of the slug is less than that of the rod it is shown that the presence of the springs has no influence on the duration of the impact but leads to an increase in the effective coefficient of restitution as compared with the unrestrained case. When the acoustic impedance of the slug is greater than that of the rod, the springs cause the interface stress to drop to zero in a finite time and the slug and rod move apart but the concept of an effective coefficient of restitution no longer has meaning. 669

670

M.C. HARRISON, R. P. MENDAY and W. A. GREEN

Theory

A rod or slug of length 1 of an elastic material with Y0ung's modulus E and density p moving with speed V impacts a semi-infinite rod of the same cross section composed of elastic material with Young's modulus /) and density p at time { = 0. The moving rod is attached to an outer sheath through a continuous distribution of shear springs and the sheath is assumed to be brought to rest instantaneously at the moment of impact. The coordinate system O2 is chosen to lie along the common axis of the two rods with the origin O at the interface and so that at the instant of impact the slug occupies the region - l _< ~ < 0. Then, assuming that plane sections of the rods remain plane in the subsequent motion, the position Y:({) at time { of the cross section of the slug which was at position J( at time t = 0 is given by the expression

~({) = 2 + v{ + ~(2, {),

(1)

where ~()(, {) satisfies the equation c 2 02~

pZ(V{+~)

= 02~

022

0{2.

(2)

Here, c = v / ( ( E / p ) is the speed of longitudinal waves in the slug and the parameter p is given in terms of the stiffness k of the shear springs per unit area of the slug surface, the circumference of the slug b and the area of cross section of the slug A by the expression p2 = k b / A p . The position }({) of the cross section of the impacted rod which was at position X before the impact, is given by

~({) = 2 + ~(2, {),

(3)

c~2 02~ 0~,~ ~_~ - 0 { 2 ,

(4)

where ~t(2, {) satisfies the equation

OX

and e = V~/)/fi). Introducing non-dimensional quantities x, 2, X, ){, t, u, ~, by the relations x = 2/1, 2 = ~ / l , X = f ( / 1 , ) ( = ) ( / l ,

t = ct/1, u = c(t/V1, ~ = c ~ / V l ,

equation (2) takes the non-dimensional form 02u OX 2

q2(t + u) -

02u Ot 2 ,

(5)

where q = p l / c . The corresponding non-dimensional equation for the semi-infinite rod is 02~

OX ~

02fi -- c~ 2 -

Ot 2 ,

(6)

IMPACT ON CONSTRAINED CYLINDER where a =

671

c/~. Equations (5) and (6) are to be solved subject to the intial conditions Ou 5 i - (x, o ) = 0~ ( 2 , 0 ) = 0.

~(x,o) = ~ ( 2 , 0 ) = o ,

(7)

Writing z = pc/p~, the boundary conditions of continuity of displacement and stress at the interface are

t + u(O,t) = fL(O,t),

Ou t) = -g-~(O,t), O~ c~z-:d~(O,

(8)

the condition of zero stress at the free end of the slug is

Ou O X ( - 1 , t ) = 0,

(9)

and the solution in the semi-infinite rod must satisfy the condition

oa 2

o2(

,

t) + a 0~ "X t) ~-~(

,

=

o,

(lO)

which ensures that the waves propagate away from the interface. Taking Laplace transforms with respect to time, equations (5) and (6) become d2fi dX 2

q2 (s 2 + q2)~ = --s2,

(11)

d2u dX----2 - s2c~2~ = 0,

(12)

where the carat denotes the Laplace transform and s is the transform paranmter. The transformed boundary conditions (8), (9) and (10) become 1

d/t

s~ + ~(o, s) - ~(o, s), ~dt~, -

du

~ - y ( o , s) = ~ ( o , s), 1

, s) = 0,

13) (14)

d~. (2, s) + sa~ = 0. dX

(15)

The solutions to equations (11) and (12) which satisfy conditions (13), (1.4) and (15) are

s cosh [,X(1 + X)]

¢~(X,s) = - A2(scosh A + Az sinh A) z sinh A e x p ( - s a X ) u(X, s) = A(s cosh A + Az sinh A)'

q2 s2.~2'

(16)

(17)

where A = x/rfi + q2. The solutions (16) and (17) are derived on the assumption that the slug and the rod remain in contact for all time. In fact, the solutions obtained by taking the inverse Laplace

672

M.C. HARRISON, R. P. MENDAY and W. A. GREEN

transforms of (16) and (17) remain valid only as long as the stress at the interface is nonpositive. When the solution predicts a positive (tensile) contact stress between the slug and the rod they part company and it is necessary to modify the boundary conditions so as to satisfy the traction free condition on both the slug and the rod at the faces which were in contact. The Laplace transform of the nondimensional stress at the interface may be derived from equation (16) or (17) and is given by

dXJ x=o (~z \dX/x=o

A(scoshA + Azsinh)Q'

(is)

In order to determine the parting time it. is necessary to invert this transform solution.

Unrestrained impact

The case of unrestrained impact is obtained from the general solution by taking q = 0 so that k = .s. Equation (16) then becomes

a ( x , ~) =

cosh [s(1 + X)] .2(cosh.~ + z sinh.~)

(19)

Expanding the right hand side of equation (19) as art infinite series of negative exponentials and inverting term by term leads to the solution

,t(X,t)-

(z+l)

\z+l]

[(t-2r+X)H(t-2r'+X) + (t - 2 ( , . + 1) - X ) H ( t

(20)

- 2(~- + 1) - X)],

where H(.) is the Heaviside unit flmction. It follows from equation (1) that the nondimensignal velocity in the slug is given by

z(X,t) = 1

1 fifz l'~[H(t_2r+X)+H(t_2(r+l)_X)], (z+l)~=o\Z+l /

(21)

and the nondimensional stress or(X, t) in the slug is given by

cr(X,t)- (z+1 1)fi

( ~~ 1- )1

[H(t-2r+X)-H(t

2(r+l)-X)].

(22)

F~0

At the interface X = 0, equation (22) reduces to ~(0, t) -

(1 + ~)

H(t) - 2 ~

~;~-7)~

t~(t - 2,.)

.

(23)

"c=l

For any finite value of t > 2, the summation in equation (22) may be evaluated to give a(0, t) =

( z + l ) ( ~+1)'

for 2n < t < 2 ( n + 1 ) ,

(24)

IMPACT ON CONSTRAINED CYLINDER

673

For z > 1 the stress is always negative (compression) and the slug and rod remain in contact for all time. W h e n the slug and rod are of the s a m e material so that z = 1 the stress drops to zero at time t = 2 and the slug is at rest in contact with the rod for all subsequent time. E q u a t i o n (24) predicts a positive (tensile) stress at the interface for t > 2 when z < 1 and it is then necessary to examine the solutions (21) and (22)) for t < 2. These become 1

(z + 1 ) [ H ( t + X) + H(t - 2 - X)],

~(X, t) -- 1 and

a(X,t) -

(25)

1

(z + l)[H(t + X ) - H ( t - 2 - X)]'

(26)

From equation (26) it is clear t h a t the stress in the slug is zero for all X <_ (t - 2) and in particular the stress t h r o u g h o u t the slug is zero at time t = 2. E q u a t i o n (25) shows t h a t at t = 2 the velocity has the uniform value - ( 1 - z)/(1 + z) t h r o u g h o u t the slug and the slug rebounds from the impact with an effective coefficient of restitution of (1 - z)/(1 + z).

m m

o ........

i

. . . . . . . ¢. . . . -0.2

~-0.5

-1

0

,

,

,

2

4

6

-0.4[

,

,

,

0

2

4

6

8

8

o 0

...............

t- . . . . . . . . .

-0,2

N-0s

-0.4 0

0

2

.........

4

i. . . . . . . . .

6

8

0

-- --.--:

2

4

0 . . . . . . . . .

6

8

"! ........

~ -0.2

"~-0.

-t3.4 0

2

4

6

8

0

2

..... m

=

,,:::::

-0.4 -i

0

,

,

2

4

6

8

6

8

"~:---::

o

~.-o2

-0.5

4

.... 1 .

0

2

4

.

.

.

6

8

FIG 1. Interface stress as a function of non-dimensional t i m e for (a) z = 0.633, (b) z = 1.9 and q -- 0.1, 0.233, 0.367, 0.5. T h e solid lines denote compressive stress and the dashed lines denote a non-physical tensile stress.

674

M.C. HARRISON, R. P. MENDAY and W. A. GREEN 0.1

O'J-0.1

-02

-0.3

~-0.4

>-

~ ,-0.5 E -0.6

-07

1

1.5

2

__

.---t

25

3

-0.8

-09

-'I 0

015

115 z

2.5

2oi +=oo,o 18

_.+

16

.....

/I

14 :=

q=0.168 q = 0.200

S /S

8

- --

i......... : /Y

,,~-~,-/,,"~

6 4 2 05

1

1.5 Z

2

2.5

3

FIG 2. Mean parting velocity (upper diagram) and parting times (lower diagram) as a function of z for different values of q.

IMPACT ON CONSTRAINED CYLINDER

675

Transform inversion

The nondimensional contact stress at the interface is obtained by inverting the transform solution (18) and it is formally given by the expression ~(0, t)

1 /~+~oo s sinh A 2~ri J~-ioc A(s cosh A + Az sinh A) e S t d s

(27)

The parameter A has branch points at s = ± i q but since the integrand in equation (27) is even in/k it has no singularities at the branch points and the integral can be expressed in terms of the poles of the integrand in the negative half plane. The integrand has simple poles at the roots of the equation tanhA-

3 zA"

(28)

This equation has an infinite number of roots and in the case where q = 0 these are given by -13+iklr z> 1 (k=0,1,2, .), (29) s = - 1 3 + i ( 2 k + 1)r/2, z < 1 o

where/3 = (1/2)In[(1 + z)/(1 - z)]. In the general case of q ¢ 0 the roots of equation (28) have to be determined numerically. The stress at the interface has been calculated numerically [3] using 41 roots of equation (28) for a range of values of q and representative results for z < 1 and for z > 1 are shown in Figure 1. The time at which this stress falls to zero (the parting time) is plotted against z in Figure 2. Also shown in Figure 2 are plots of the mean parting velocity of the slug as functions of z from which it is possible to calculate an effective coefficient of restitution e. The results shown in Figure 2 relate to values of q < < 1, which is appropriate to the study of the impact of a beam modelled as a collection of slugs attached to their neighbours and next nearest neighbours through a continuous distribution of elastic springs. For values of q > 1 and z < 1 it is possible to derive an approximate solution based on the first pair of conjugate roots only. Thus, Figure 3 shows the real root and first few conjugate pairs of roots of equation (28) for the value of z = 0.5 and for q increasing from 0 to 5. The roots for q = 0 are indicated by small circles and the arrows show the direction of movement of the roots as q increases. This figure is typical of results for other values of z < 1 and it is apparent that as q increases the negative real part of the first pair of conjugate roots rapidly approaches zero and this pair of roots accordingly gives the dominant contribution to the response for large values of q. For z < 1 and q large, an approximation to the first pair of conjugate roots is given by the expression s=pexp

i~

1±

,

(30)

where ~ = v/q 2 + 7r2/4. Figure 4 shows the mean parting velocity of the slug as a function of q for three values of z. These solutions have been derived using the first pair of conjugate roots only. They clearly show that the effective coefficient of restitution approaches the value 1 at large q for all values of z < 1.

676

M . C . H A R R I S O N , R. P. M E N D A Y and W. A. G R E E N

10

......................................................

:

. . . . . . . . . . i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-10

-2.5

-3

-2

-1.5

...........

:. . . . . . . . . . i

-1

-0.5

0

real

FIG 3 M o t i o n of s o m e of t h e p o l e s of t h e i n t e g r a n d in e q u a t i o n (27) for z -- 0.5 as q = 0 --+ 5.

-0.97

-,4 O

,-4 ¢v - 0 . 9 8

-0.99

-1

,

0

.

.

.

.

I

50

,

,

,

,

J

i00

J

,

,

,

J

150

.

.

.

.

•

,

200

.

.

.

.

.

i

250

.

.

.

.

_ l _ _

300

q

FIG 4 A p p r o x i m a t e v a l u e of t h e m e a n s l u g p a r t i n g v e l o c i t y as a f u n c t i o n o f q for z = 0.5 ( b o t t o m curve); z = 0.7 ( m i d d l e c u r v e ) a n d z = 0.9 ( t o p curve).

IMPACT ON CONSTRAINED CYLINDER

677

Discussion

From Figures 1 and 2, it may be seen that for z <__1, the slug and rod part company at the non-dimensional time t = 2, when the wave set up in the rod at the instant of impact returns to the interface. The slug then rebounds with a non-zero effective coefficient of restitution e which varies slightly with q for the range of values shown here. For values of z > 1 and any non-zero value of q, the interface stress always drops to zero but the parting time increases with increasing z, corresponding to an increasing number of passages of the stress wave to and fro in the slug. Moreover, in this circumstance, the parting time is no longer associated with an integral number of traverses of the wave in the slug, the effect of the restraining springs being to induce parting at non-integer values of the nondimensional time t. The plots of the mean velocity in Figure 2 show an apparent positive value for some ranges of q when z > 1. Since the rod is of semi-infinite length it will have zero mean velocity and a positive mean velocity of the slug then implies a negative (nonsensical) value of e. This circumstance arises because the local velocity of the rod at the contacting interface is greater than that of the slug so that the bodies part company whilst the mean velocity of the slug is still positive. The concept of effective coefficient of restitution then becomes meaningless. In the case of the free impact of two rods, Rayleigh [4] reports that the experimental investigations of Voigt indicate that the St.Venant predictions do not hold when the two impacting rods are of significantly different lengths. It then appears that the rods move apart with an effective coefficient of restitution unity and Rayleigh postulates that the impact is governed by the Hertz theory of contact. The Hertzian theory (see Love [5]) is concerned with impact times which are long in comparison with the travel time of elastic waves through the impacting bodies. It assumes that the impact has the form of a quasistatic deformation of the two bodies in the region of contact and it gives rise to a perfectly elastic collision in which the effective coefficient of restitution is unity. The predicted time of collision is inversely proportional to the fifth root of the impact velocity with the constant of proportionality depending on the elastic constants and on the geometry of the colliding bodies. Wagstaff [6] has carried out an experimental investigation of the impact of rods with rounded ends in an a t t e m p t to determine the range of validity of the Hertz theory. He has established experimentally a relation between the duration of contact and the impact velocity. Wagstaff shows that the duration of contact is proportional to a linear function of the length of the longer rod and that it depends on the velocity according to an inverse power which itself is a linear function of the length to diameter ratio. For sufficiently large values of the length to diameter ratio the duration of contact predicted by Wagstaff's results is independent of the impact velocity and, as the ratio of the diameter of the rods to the radius of curvature of the impacting ends tends to zero, it is directly proportional to the length of the longest rod. The St. Venant theory which is employed here is based on a one dimensional model of wave propagation in the rods. This theory holds for flat ended rods in the case when the length to diameter ratio tends to infinity and it is in agreement with the asymptotic behaviour predicted by Wagstaff.

678

M.C. HARRISON, R. P. MENDAY and W. A. GREEN

References

1. I. Todhunter and K. Pearson, A history of the theory of elasticity and of the strength of materials, Dover, New York (1960) 2. M. C. Harrison, R. P. Menday and W. A. Green, Discretised models of impacting beams, to appear in Proceedings of Dynamic Systems and Applications, 3, (2000) 3. R. P. Menday, The forced vibration of a partially delaminated beam, Ph.D. Thesis, Loughborough University (1999) 4. Lord Rayleigh (J. W. Strutt), Phil. Mag. 11,283 (1906) 5. A. E. H. Love, A treatise on the theory of elasticity (4th ed.), C.U.P., Cambridge (1927) 6. J. E. P. Wagstaff, Proc. Roy. Soc. A105, 544 (1924)