- Email: [email protected]
Dynamic behaviour of thin cylindrical shells collided with dampers
NUCLEAR ENGINEERING AND DESIGN 24 (1973) 188-194. NORTH-HOLLAND PUBLISHING COMPANY
DYNAMIC BEHAVIOUR OF THIN CYLINDRICAL SHELLS COLLIDED WITH DAMPERS Shin-ichl SUZUKI Department of A eronauttc Engineering, Nagoya Umverszty, Nagoya, Japan Received 24 November 1972
Dynamic behaviour of thin cylindrical shells of finite length is investigated for the case where they collide with a spring with constant velocity. The relationships between dimensions of a cylinder, spring constant and maximum values of dynamic displacements and stresses are obtained. The fundamental equations of motion are derived, taking into account of radial drsplacement, and are solved by Laplace transformation method. From the results of theoretical analysis, it became evident that impulsive stresses are damped considerably by a spring.
1. Introduction Control rods have to be inserted when nuclear reactors are overheated and collide with a grid plate. From the structural point of view, the author [ 1] previously investigated the behaviour o f dynamic stresses acting in the grid plate and found that extraordinary large dynamic stresses were induced. It will easily be considered that the control rod itself will be subjected to large impulsive stresses. In order to make dynamic stresses small, it is desirable to use springs or dash-pots as dampers. However, few papers have been published in this area. It is important from the engineering point o f view to investigate the effects of sizes of cyhnder and characteristics of damper on dynamic stresses. Several papers have been published on the longitudinal impact of thin cylindrical shells. Berkowitz [2] treated problems two-dimensionally and discussed the propagation of stress waves in a semi-infinite cylinder collided with a rigid wall, and Testa and Bleich [3] studied the same case, taking into account of solid viscosities. Conway and Jakubowskl [4] analysed the case where the same two rods of finite length collide and compared the theoretical results with experimental ones. In the present paper, the cases are treated where thin cylindrial shells of finite length collide with a spring with constant velocity. The cylindrical shell is assumed to be elastic and displacement in the radial direction is taken into consideration. The relationships are obtained between dimensions of cylinder, spring constant and maximum stresses acting in the cylinder. Attention is paid to stresses in longitudinal and circumferential directions. Solutions are found by the Laplace transformation method.
2. Formulation of the problem Consider the case where a thin cylindrical shell collides with a spring with constant velocity V0. Coordinates and displacements of a cylinder are illustrated in fig. 1. The equations of motion of a cylinder under axially symmetric conditions are Oo (/Ox = p ~)2Ux/Ot2 ,
- Oo/R = p 02Ur/Ot 2 .
(1)
S. Suzuki, Thin cylindrical shells colhded with dampers
ux 0
189
i
=X
I
Vo Fig. 1. Dimensions and coordinates of thin-walled cylindrical shell. Stresses o~ and a o m the longitudinal and circumferential directions are, respectively
o~ -
1
-_-p2 I(-~-X'-
R u
r
,
o o-
1 --v 2
+ p
'~-]
(2)
.
Substituting eq. (2) into eq. (1) yields a2ux P
"at 2
E _
--~1
[a2Ux v aur_] __+_ tax 2 R --~-]'
a2Ur P---
at 2
--E
[Ur
aUx.~ u l - 122 \R 2 + ~ -~-x] "
Introducing dimensionless variables ~ = x/l and T = ( t f l ) x / E / p ( l a 2 u x l a T 2 = a2uxla~ 2 + (v/v) aurla~ ,
(3)
- p 2) gives
a2urlaT 2 = - UriC/2 - (v/T) aUx/a~ ,
(4)
where 7 = R / I . The initial conditions for the cylinder moving axially with velocity V0 prior to impact are u x = u r = aUr/aT = 0 ,
aUx/aT = - V ,
(5)
where V= Vol/x/E/p(1 - u 2 ) . Laplace transformation with respect to T of eq. (4)gwes, after replacing p by ~o, r - p2 U-x = u- "x + (v/T) u-;,
p2 fir = ffrl~/2 + (121")')U - 'x '
(6, 7)
where primes indicate differentiation with respect to ~. From eq. (7), fir is u-r = {u')'/(72p2 - 1) }ux .
(8)
The fundamental equation with respect to U-x will easily be obtained by substituting eq. (8) into eq. (6). u-x +{p2(,),Zp2 _ 1)/(,),2p2 _ 1 + u2)} U-x = {(,),2p2 _ 1)/(,),2p2 _ 1 + u2)} V.
3. Solution
Since a cyhnder is free at ~ = 1, o~ becomes zero. That is t
Ux/! + ( v / R ) u r = O .
(9)
190
S. Suzuki, Thin cylindrical shells collided with dampers
Using eq. (8), yields u-'x = 0 , and since a cylinder continues to contact with spring at ~ = 0 after impact, we obtain 2 rrRho~ = k u x.
With the aid~of eqs. (2) and (8), this relation becomes E 2rrRh ~/2p 2 _ 1 + v 2 - , 1 - v2 k l ,y2p2 _ 1 Ux + f i x = 0 . -
Therefore, the transformed boundary conditions are -, ~,2p2 _ 1 + !)2 Ux=O ate=l; fix,,/2p2_ 1 b'yU'x = 0 ,
(10)
where b = { E / ( I - v 2) }21rh/k. The following two cases will be considered. (1)3,2p 2 < 1 - v
2
(or~'2p2>l),
(2) 1 - v 2 <3,2p 2 < 1. Analysis will be carried out for each case. 3.1. The case ,y2p 2 < 1 - v 2 or 3,2p 2 > 1
Eq. (9) will be put as U-x + °t2ffx = {(72p2 -- 1)/(3'2p2 -- 1 + rE)) V,
(I 1)
where a 2 = p2(72p2 - 1)/(3,2p 2 - 1 + v2). From eq. (11), Ux = C1 cos oa~ + C 2 sin odl + V/p 2 ,
(12)
where C 1 and C 2 are constants. With the aid ofeq. (10), C 1 and C 2 will be determined and ~x becomes fix/V = - {a cos a( 1 - ~j)}/p2(_ b,yp2 sin ot + ot cos or) + 1/p2.
(13)
From eq. (18), ffr / V = - (v~'~ 2 sm a ( l -/j)}/p2(,y2p2 _ 1) ( - b 3~p2 sin ot + ot cos a).
(14)
The inversion theorem gives Ux = ~ V
2or cos a(1 -- ~) s i n p T p3[-2b'ysinct+~l
+v2(1 - v 2 ) / ( ' / 2 p 2 -
,
(15)
1 +v2)2) (a -1 c o s t x - 1/sin or)]
where p are the roots of eq. (16): - b ' y p 2 sina + a c o s a = 0 .
In the same way, u r is obtained from eq. (14) in the following form.
(16)
S. Suzuki, Thin cylindrical shells collided with dampers
191
2or2 sin a(1 -~j) s i n p T
u_z = v T ~ V
(17)
p3(~2p2 _ 1) [ - 2b7 sin a + {1 + v2(1 - v2)/(212p 2 - 1 + v2) 2) (or -1 cos ot - 1/sin or)]
Substituting eqs. (16) and (17) into eq. (2), o~ and o 0 are expressed as
o~
sin a( 1 - ~) sin p T
= 2 ~
V0 x/EP/( 1
-
,
(18)
p [ - 2 / ~ sin a + (1 + v2(1 - v2)/('r2p 2 - 1 + v2) 2} (a -1 cos a - 1/sin a)]
v 2)
I,'3'2t~2 Sift a(1 --. 0 sin p T
o0
Vox&p/(1 ,,2)
p(,),2p2 _ 1) [ - 263' sin a + (1 + v2(1 - v2)/('y2p 2 - 1 + v2) 2} (or -1 cosot - 1 / sin or)]
_
(19)
3.2. The ease 1 - v2 < 72p 2 < 1 Eq. (9) will be put as (20)
Ux - ct2Ux = {(72p2 - 1)/(3'2p2 - 1 + v2)) V , where - a2 = p2(72p2 _ 1)/(72p2 _ 1 + v 2) . F r o m eq. (20),
(21)
fix = CI cosh t ~ + C 2 sinh cd~ + V/p 2 ,
where C 1 and C 2 are constants. With the aid o f e q . (10), C 1 and C 2 can be determined and fix and fir become
Ux/V =
- {o~ cosh ot(l - ~)}/p2 (o~ cosh a -
bvp 2 sinh
(22)
o~) + I//o2,
Ur/V = (v~l a 2 sinh (x(1 - ~))/p2(3'2p2 - 1) (or cosh a - bTp 2 sinh a).
(23)
The inversion theorem gives - 2 a cosh a(1 - Osin p T
Ux= ~
V
p B [ 2 b T s i n h a + (1 + v 2 ( 1 - p 2 ) / ( 3 ' 2 p 2 -
ur = ~ V
1 +v2)2 } ( a - I cosh ot - 1/sinh a)]
,
2vTa 2 s m h a ( 1 - ~ ) sin p T p3(,y2p2 _ 1) [267 smh a + (1 + v2(1 - v2)/(~/2p 2 - 1 + v2) 2) (ct -1 cosh ot - 1/sinh or)]
(24)
(25)
where p are the roots of eq. (26): a cosh t~ -
bTp 2 sinh
(26)
a = 0.
Substituting eqs. (24) and (25) into eq. (2), o t and % are expressed as = - 2 ~
VoX/E o/(1 -- 1)2)
sinh ot(l - ~)
sinpT
p [2b'y sinh ot + (1 + v2(1 - v2)/(72p 2 - 1 + v2) 2} (or-1 cosh ot - 1/sinh or)]
, (27)
S. Suzuki, Thin cylindrical shells collided with dampers
192
°0 v~ =2~ t;, =~/-~S~, _ VoV~V/t~_
p(72p 2
sinh _ 1) [2b7
+ oe
{1
v72ot 2 +v2( 1 sinh a(lv2)/(72p 2-~) sin p T - 1 + v2) 2} (or-1 cosh oe - 1 [ sinh oe)] (28)
Adding the results of these cases (1) and (2) mentioned above, the final forms of displacements and stresses will be obtained.
4. Numerical analysis Numerical calculations have been carried out for several cases. As is seen in eqs. (15), (18) and (19), u x, o~ and o 0 are expressed in the series form. Therefore, convergence of these series must be investigated. It becomes evident that twenty terms should be summed to obtain the solutions which give satisfactory results for the practical purposes for the cases b :~ 0. On the other hand, convergence becomes very slow for the case b = 0 and five hundred terms must be taken. The relationship between T a n d o~ at ~ = 0 for the cases of'), = 0.1 and b = 0, 10 and 30 are illustrated in fig. 2. For the case b = 0, the values of o~ increases remarkably immediately after the cylinder collides and decreases suddenly m the neighbourhood of T = 2. But, as the value of b increases, o~ change gradually and their maximum values decrease. The dotted line indicates the result for the case where radius displacement is neglected on the assumption o f b = 0. The relationships between T and o~ at ~ = 0.5 for the cases o f y = 0.1 and b = 0, 10 and 30 are illustrated in fig. 3. T = 1 ~s the time necessary for the stress wave to propagate the distance 1. Therefore, the stresses rise in
~
Y=O.I
~
0.8
~
I 0.4
T
0.8
w
0
i
I
2
o
3'
\;
Fig. 2. The relationships between time a n d o~ at ~ = 0 (for the case ~, = 0.1)
.0.4I-b=5 b=o. ~ o
Y=O.I
b=lO
0.4
o
I
3
Fig. 3. The relationships between time a n d o~ at ~ = 0.5 (for the case ~/= 0 1 ).
~=0,I
Fig. 4. The relationships between time and % (for the case ~ = 0.1).
S. Suzuki, Thin cylindrical shells collided with dampers
193
the neighbourhood of T = 0.5*. Stress changes remarkably and its maximum value is large for b = 0. The relationships between T and o 0 for the case of 3' = 0.1 are illustrated in fig. 4. The solid lines indicate the cases for b = 0 and ~ = 0 and 0.5. Their maximum values become very small in comparison with those of of. The dotted line indicates the case for b = 0 and ~ = 0. It is found that the damping effect is very large. At ~ = 0, displacement u x is equal to 2 ~ R h o ~ / k . Therefore, the values o f u x will easily be obtained by the results illustrated in fig. 2. Next, the distribution o f oo to the axial direction will be investigated. The relationships between o r and ~ at T = 0.5 are iUustrated in fig. 5~. Solid lines indicate the cases f o r 7 = 0.1 and b = 0,5 and 10. For the case b = 0, the value of of decreases gradually as the value of ~ increases and afterwards increases and reaches a considerably largo value at ~ = 0.5. On the other hand, for the cases o f b = 5 and 10, o f decreases and their values become very small at ~ = 0.5. Tho dotted line and chain line indicate the cases for 7 = 0.2, 0.025 and b = 0, respectively. For the case o f b = 0, o~ change remarkably as the value o f 7 decreases. The maximum value of o~ at ~ = 0 can be obtained in fig. 2. The relationships between their maximum values, 7 and b are illustrated in fig. 6. The values of 7 are taken as 0.025, 0.1 and 0.2. For all cases these values decrease remarkably as the value o f b incroases, and decrease gradually when b exceed some values. That is, in order to make impulsive stress small, it is not necessary to use a very soft spring. The rate o f decrement becomes large as the value of 7 increases.
1.2 T = o.5
.....
-A--
-I:" - -
~ ' = 0.1
....
~'= 0.2
--'--
)'=0.025
b-o b=:o
I
0.4
0
0.2
06
0.4
Fig. 5. D i s t r ~ u t i o n s of of in the axial direction at T = 0.5.
1.2
1.2
~
Y~0.025
E
~--0.8
b~ f
7 - O.O25
b~"
0.4
o.4
I
..,
0
0.8
i
i
i
50
I00
150
b
Fig. 6. The relationships b e t w e e n 7, b and m a x i m u m values o f of at f = 0.
i
i
50
I00
I
b
150
Fig. 7. The relationships b e t w e e n 7 , b a n d m a x i m u m values of a f at f = 0.5.
* In the present case, radial displacement is taken into consideration. Therefore, stress rises at a little earlier time than T = 0.5. It is a well-known fact that stress rises at T = 0.5 b y the classical theory, which is indicated b y the d o t t e d line.
194
s. Suzukt, Thin cylindrical shells collgted with dampers
The relationships between maximum values of o~ at ~ = 0.5, ~ and b are illustrated in fig. 7. Although the tendency is much similar to that in fig. 6, their values are, in general, smaller than those at ~ = 0. Let us consider the value o f b . Since b is equal to 27rhE/(1 - v2)k, the value o f k is nearly equal to that o f E when b is taken as 14 on the assumption o f h = 2 ram. The maximum value ofo~ for b = 14 becomes much smaller than that for b = 0. That is, it will easily be understood that, in order to make the value of impulsive stress small, it is sufficiently effective even to use such a very stiff spring as this one.
5. Conclusions The following conclusions can be drawn from the theoretical results o f the present work. (1) The analytical results in the present paper are much different from those obtained by classical theory. (2) The maximum values of o¢ are very large for the case where a cylindrical shell collides with a rigid wall, but these values decrease remarkably as the value o f k decreases and the rate o f decrement becomes large as the value o f R / l increases. (3) The effect of damping for impulsive stresses Is sufficiently large even for the case where the value o f k is large. (4) The maximum values of o 0 are, in general, much smaller than those of o¢.
Acknowledgement The author wishes to thank Dr. Kajita in Nagoya University for his help with numerical calculations.
Notation The following symbols are used in this paper. b
= (El1 - v 2) 21rh/k
E h k l R
= Young's modulus = thickness o f a cyhndrical shell = spring constant = length of a cylindrical shell = mean radius o f a cylindrical shell (tfl) x/E/p(1 - v2) = time = displacement m the longitudinal direction = displacement in the radial direction
T
t ux
ur Ux
= ~00 Ux e - P T dT
V0
= initial velocity o f a cylindrical shell = V 0 l / x / E / p ( l -/.,2)
V
Greek letters ~/
= R fl
# of o0 v
= density = stress in the longitudinal direction = stress in the circumferential direction = Polsson's ratio = x/l
References [1 ] S. Suzuki, Dynamic Response of Circular Plates Subjected to Transverse Impulsive Loads, Ingenieur-Archiv, 40, 2, (1971) 131-144. [2] H. Berkowitz, Longitudinal Impact ofa Semi-Inf'mlte Elastic Cylindrical Shell. Trans. of ASME, E30 (1963) 347-354. [3] R. Testa and H. Bleich, Longitudinal Impact of a SemiInfinite, Cylindrical, Viscoelastic Shell. Trans. of ASME, E32 (1965) 813-820. [4] H. Conway and M. Jakubowski, Axial Impact of Short Cylindrical Bars. Trans. of ASME, E36 (1969) 809-813.
DYNAMIC BEHAVIOUR OF THIN CYLINDRICAL SHELLS COLLIDED WITH DAMPERS Shin-ichl SUZUKI Department of A eronauttc Engineering, Nagoya Umverszty, Nagoya, Japan Received 24 November 1972
Dynamic behaviour of thin cylindrical shells of finite length is investigated for the case where they collide with a spring with constant velocity. The relationships between dimensions of a cylinder, spring constant and maximum values of dynamic displacements and stresses are obtained. The fundamental equations of motion are derived, taking into account of radial drsplacement, and are solved by Laplace transformation method. From the results of theoretical analysis, it became evident that impulsive stresses are damped considerably by a spring.
1. Introduction Control rods have to be inserted when nuclear reactors are overheated and collide with a grid plate. From the structural point of view, the author [ 1] previously investigated the behaviour o f dynamic stresses acting in the grid plate and found that extraordinary large dynamic stresses were induced. It will easily be considered that the control rod itself will be subjected to large impulsive stresses. In order to make dynamic stresses small, it is desirable to use springs or dash-pots as dampers. However, few papers have been published in this area. It is important from the engineering point o f view to investigate the effects of sizes of cyhnder and characteristics of damper on dynamic stresses. Several papers have been published on the longitudinal impact of thin cylindrical shells. Berkowitz [2] treated problems two-dimensionally and discussed the propagation of stress waves in a semi-infinite cylinder collided with a rigid wall, and Testa and Bleich [3] studied the same case, taking into account of solid viscosities. Conway and Jakubowskl [4] analysed the case where the same two rods of finite length collide and compared the theoretical results with experimental ones. In the present paper, the cases are treated where thin cylindrial shells of finite length collide with a spring with constant velocity. The cylindrical shell is assumed to be elastic and displacement in the radial direction is taken into consideration. The relationships are obtained between dimensions of cylinder, spring constant and maximum stresses acting in the cylinder. Attention is paid to stresses in longitudinal and circumferential directions. Solutions are found by the Laplace transformation method.
2. Formulation of the problem Consider the case where a thin cylindrical shell collides with a spring with constant velocity V0. Coordinates and displacements of a cylinder are illustrated in fig. 1. The equations of motion of a cylinder under axially symmetric conditions are Oo (/Ox = p ~)2Ux/Ot2 ,
- Oo/R = p 02Ur/Ot 2 .
(1)
S. Suzuki, Thin cylindrical shells colhded with dampers
ux 0
189
i
=X
I
Vo Fig. 1. Dimensions and coordinates of thin-walled cylindrical shell. Stresses o~ and a o m the longitudinal and circumferential directions are, respectively
o~ -
1
-_-p2 I(-~-X'-
R u
r
,
o o-
1 --v 2
+ p
'~-]
(2)
.
Substituting eq. (2) into eq. (1) yields a2ux P
"at 2
E _
--~1
[a2Ux v aur_] __+_ tax 2 R --~-]'
a2Ur P---
at 2
--E
[Ur
aUx.~ u l - 122 \R 2 + ~ -~-x] "
Introducing dimensionless variables ~ = x/l and T = ( t f l ) x / E / p ( l a 2 u x l a T 2 = a2uxla~ 2 + (v/v) aurla~ ,
(3)
- p 2) gives
a2urlaT 2 = - UriC/2 - (v/T) aUx/a~ ,
(4)
where 7 = R / I . The initial conditions for the cylinder moving axially with velocity V0 prior to impact are u x = u r = aUr/aT = 0 ,
aUx/aT = - V ,
(5)
where V= Vol/x/E/p(1 - u 2 ) . Laplace transformation with respect to T of eq. (4)gwes, after replacing p by ~o, r - p2 U-x = u- "x + (v/T) u-;,
p2 fir = ffrl~/2 + (121")')U - 'x '
(6, 7)
where primes indicate differentiation with respect to ~. From eq. (7), fir is u-r = {u')'/(72p2 - 1) }ux .
(8)
The fundamental equation with respect to U-x will easily be obtained by substituting eq. (8) into eq. (6). u-x +{p2(,),Zp2 _ 1)/(,),2p2 _ 1 + u2)} U-x = {(,),2p2 _ 1)/(,),2p2 _ 1 + u2)} V.
3. Solution
Since a cyhnder is free at ~ = 1, o~ becomes zero. That is t
Ux/! + ( v / R ) u r = O .
(9)
190
S. Suzuki, Thin cylindrical shells collided with dampers
Using eq. (8), yields u-'x = 0 , and since a cylinder continues to contact with spring at ~ = 0 after impact, we obtain 2 rrRho~ = k u x.
With the aid~of eqs. (2) and (8), this relation becomes E 2rrRh ~/2p 2 _ 1 + v 2 - , 1 - v2 k l ,y2p2 _ 1 Ux + f i x = 0 . -
Therefore, the transformed boundary conditions are -, ~,2p2 _ 1 + !)2 Ux=O ate=l; fix,,/2p2_ 1 b'yU'x = 0 ,
(10)
where b = { E / ( I - v 2) }21rh/k. The following two cases will be considered. (1)3,2p 2 < 1 - v
2
(or~'2p2>l),
(2) 1 - v 2 <3,2p 2 < 1. Analysis will be carried out for each case. 3.1. The case ,y2p 2 < 1 - v 2 or 3,2p 2 > 1
Eq. (9) will be put as U-x + °t2ffx = {(72p2 -- 1)/(3'2p2 -- 1 + rE)) V,
(I 1)
where a 2 = p2(72p2 - 1)/(3,2p 2 - 1 + v2). From eq. (11), Ux = C1 cos oa~ + C 2 sin odl + V/p 2 ,
(12)
where C 1 and C 2 are constants. With the aid ofeq. (10), C 1 and C 2 will be determined and ~x becomes fix/V = - {a cos a( 1 - ~j)}/p2(_ b,yp2 sin ot + ot cos or) + 1/p2.
(13)
From eq. (18), ffr / V = - (v~'~ 2 sm a ( l -/j)}/p2(,y2p2 _ 1) ( - b 3~p2 sin ot + ot cos a).
(14)
The inversion theorem gives Ux = ~ V
2or cos a(1 -- ~) s i n p T p3[-2b'ysinct+~l
+v2(1 - v 2 ) / ( ' / 2 p 2 -
,
(15)
1 +v2)2) (a -1 c o s t x - 1/sin or)]
where p are the roots of eq. (16): - b ' y p 2 sina + a c o s a = 0 .
In the same way, u r is obtained from eq. (14) in the following form.
(16)
S. Suzuki, Thin cylindrical shells collided with dampers
191
2or2 sin a(1 -~j) s i n p T
u_z = v T ~ V
(17)
p3(~2p2 _ 1) [ - 2b7 sin a + {1 + v2(1 - v2)/(212p 2 - 1 + v2) 2) (or -1 cos ot - 1/sin or)]
Substituting eqs. (16) and (17) into eq. (2), o~ and o 0 are expressed as
o~
sin a( 1 - ~) sin p T
= 2 ~
V0 x/EP/( 1
-
,
(18)
p [ - 2 / ~ sin a + (1 + v2(1 - v2)/('r2p 2 - 1 + v2) 2} (a -1 cos a - 1/sin a)]
v 2)
I,'3'2t~2 Sift a(1 --. 0 sin p T
o0
Vox&p/(1 ,,2)
p(,),2p2 _ 1) [ - 263' sin a + (1 + v2(1 - v2)/('y2p 2 - 1 + v2) 2} (or -1 cosot - 1 / sin or)]
_
(19)
3.2. The ease 1 - v2 < 72p 2 < 1 Eq. (9) will be put as (20)
Ux - ct2Ux = {(72p2 - 1)/(3'2p2 - 1 + v2)) V , where - a2 = p2(72p2 _ 1)/(72p2 _ 1 + v 2) . F r o m eq. (20),
(21)
fix = CI cosh t ~ + C 2 sinh cd~ + V/p 2 ,
where C 1 and C 2 are constants. With the aid o f e q . (10), C 1 and C 2 can be determined and fix and fir become
Ux/V =
- {o~ cosh ot(l - ~)}/p2 (o~ cosh a -
bvp 2 sinh
(22)
o~) + I//o2,
Ur/V = (v~l a 2 sinh (x(1 - ~))/p2(3'2p2 - 1) (or cosh a - bTp 2 sinh a).
(23)
The inversion theorem gives - 2 a cosh a(1 - Osin p T
Ux= ~
V
p B [ 2 b T s i n h a + (1 + v 2 ( 1 - p 2 ) / ( 3 ' 2 p 2 -
ur = ~ V
1 +v2)2 } ( a - I cosh ot - 1/sinh a)]
,
2vTa 2 s m h a ( 1 - ~ ) sin p T p3(,y2p2 _ 1) [267 smh a + (1 + v2(1 - v2)/(~/2p 2 - 1 + v2) 2) (ct -1 cosh ot - 1/sinh or)]
(24)
(25)
where p are the roots of eq. (26): a cosh t~ -
bTp 2 sinh
(26)
a = 0.
Substituting eqs. (24) and (25) into eq. (2), o t and % are expressed as = - 2 ~
VoX/E o/(1 -- 1)2)
sinh ot(l - ~)
sinpT
p [2b'y sinh ot + (1 + v2(1 - v2)/(72p 2 - 1 + v2) 2} (or-1 cosh ot - 1/sinh or)]
, (27)
S. Suzuki, Thin cylindrical shells collided with dampers
192
°0 v~ =2~ t;, =~/-~S~, _ VoV~V/t~_
p(72p 2
sinh _ 1) [2b7
+ oe
{1
v72ot 2 +v2( 1 sinh a(lv2)/(72p 2-~) sin p T - 1 + v2) 2} (or-1 cosh oe - 1 [ sinh oe)] (28)
Adding the results of these cases (1) and (2) mentioned above, the final forms of displacements and stresses will be obtained.
4. Numerical analysis Numerical calculations have been carried out for several cases. As is seen in eqs. (15), (18) and (19), u x, o~ and o 0 are expressed in the series form. Therefore, convergence of these series must be investigated. It becomes evident that twenty terms should be summed to obtain the solutions which give satisfactory results for the practical purposes for the cases b :~ 0. On the other hand, convergence becomes very slow for the case b = 0 and five hundred terms must be taken. The relationship between T a n d o~ at ~ = 0 for the cases of'), = 0.1 and b = 0, 10 and 30 are illustrated in fig. 2. For the case b = 0, the values of o~ increases remarkably immediately after the cylinder collides and decreases suddenly m the neighbourhood of T = 2. But, as the value of b increases, o~ change gradually and their maximum values decrease. The dotted line indicates the result for the case where radius displacement is neglected on the assumption o f b = 0. The relationships between T and o~ at ~ = 0.5 for the cases o f y = 0.1 and b = 0, 10 and 30 are illustrated in fig. 3. T = 1 ~s the time necessary for the stress wave to propagate the distance 1. Therefore, the stresses rise in
~
Y=O.I
~
0.8
~
I 0.4
T
0.8
w
0
i
I
2
o
3'
\;
Fig. 2. The relationships between time a n d o~ at ~ = 0 (for the case ~, = 0.1)
.0.4I-b=5 b=o. ~ o
Y=O.I
b=lO
0.4
o
I
3
Fig. 3. The relationships between time a n d o~ at ~ = 0.5 (for the case ~/= 0 1 ).
~=0,I
Fig. 4. The relationships between time and % (for the case ~ = 0.1).
S. Suzuki, Thin cylindrical shells collided with dampers
193
the neighbourhood of T = 0.5*. Stress changes remarkably and its maximum value is large for b = 0. The relationships between T and o 0 for the case of 3' = 0.1 are illustrated in fig. 4. The solid lines indicate the cases for b = 0 and ~ = 0 and 0.5. Their maximum values become very small in comparison with those of of. The dotted line indicates the case for b = 0 and ~ = 0. It is found that the damping effect is very large. At ~ = 0, displacement u x is equal to 2 ~ R h o ~ / k . Therefore, the values o f u x will easily be obtained by the results illustrated in fig. 2. Next, the distribution o f oo to the axial direction will be investigated. The relationships between o r and ~ at T = 0.5 are iUustrated in fig. 5~. Solid lines indicate the cases f o r 7 = 0.1 and b = 0,5 and 10. For the case b = 0, the value of of decreases gradually as the value of ~ increases and afterwards increases and reaches a considerably largo value at ~ = 0.5. On the other hand, for the cases o f b = 5 and 10, o f decreases and their values become very small at ~ = 0.5. Tho dotted line and chain line indicate the cases for 7 = 0.2, 0.025 and b = 0, respectively. For the case o f b = 0, o~ change remarkably as the value o f 7 decreases. The maximum value of o~ at ~ = 0 can be obtained in fig. 2. The relationships between their maximum values, 7 and b are illustrated in fig. 6. The values of 7 are taken as 0.025, 0.1 and 0.2. For all cases these values decrease remarkably as the value o f b incroases, and decrease gradually when b exceed some values. That is, in order to make impulsive stress small, it is not necessary to use a very soft spring. The rate o f decrement becomes large as the value of 7 increases.
1.2 T = o.5
.....
-A--
-I:" - -
~ ' = 0.1
....
~'= 0.2
--'--
)'=0.025
b-o b=:o
I
0.4
0
0.2
06
0.4
Fig. 5. D i s t r ~ u t i o n s of of in the axial direction at T = 0.5.
1.2
1.2
~
Y~0.025
E
~--0.8
b~ f
7 - O.O25
b~"
0.4
o.4
I
..,
0
0.8
i
i
i
50
I00
150
b
Fig. 6. The relationships b e t w e e n 7, b and m a x i m u m values o f of at f = 0.
i
i
50
I00
I
b
150
Fig. 7. The relationships b e t w e e n 7 , b a n d m a x i m u m values of a f at f = 0.5.
* In the present case, radial displacement is taken into consideration. Therefore, stress rises at a little earlier time than T = 0.5. It is a well-known fact that stress rises at T = 0.5 b y the classical theory, which is indicated b y the d o t t e d line.
194
s. Suzukt, Thin cylindrical shells collgted with dampers
The relationships between maximum values of o~ at ~ = 0.5, ~ and b are illustrated in fig. 7. Although the tendency is much similar to that in fig. 6, their values are, in general, smaller than those at ~ = 0. Let us consider the value o f b . Since b is equal to 27rhE/(1 - v2)k, the value o f k is nearly equal to that o f E when b is taken as 14 on the assumption o f h = 2 ram. The maximum value ofo~ for b = 14 becomes much smaller than that for b = 0. That is, it will easily be understood that, in order to make the value of impulsive stress small, it is sufficiently effective even to use such a very stiff spring as this one.
5. Conclusions The following conclusions can be drawn from the theoretical results o f the present work. (1) The analytical results in the present paper are much different from those obtained by classical theory. (2) The maximum values of o¢ are very large for the case where a cylindrical shell collides with a rigid wall, but these values decrease remarkably as the value o f k decreases and the rate o f decrement becomes large as the value o f R / l increases. (3) The effect of damping for impulsive stresses Is sufficiently large even for the case where the value o f k is large. (4) The maximum values of o 0 are, in general, much smaller than those of o¢.
Acknowledgement The author wishes to thank Dr. Kajita in Nagoya University for his help with numerical calculations.
Notation The following symbols are used in this paper. b
= (El1 - v 2) 21rh/k
E h k l R
= Young's modulus = thickness o f a cyhndrical shell = spring constant = length of a cylindrical shell = mean radius o f a cylindrical shell (tfl) x/E/p(1 - v2) = time = displacement m the longitudinal direction = displacement in the radial direction
T
t ux
ur Ux
= ~00 Ux e - P T dT
V0
= initial velocity o f a cylindrical shell = V 0 l / x / E / p ( l -/.,2)
V
Greek letters ~/
= R fl
# of o0 v
= density = stress in the longitudinal direction = stress in the circumferential direction = Polsson's ratio = x/l
References [1 ] S. Suzuki, Dynamic Response of Circular Plates Subjected to Transverse Impulsive Loads, Ingenieur-Archiv, 40, 2, (1971) 131-144. [2] H. Berkowitz, Longitudinal Impact ofa Semi-Inf'mlte Elastic Cylindrical Shell. Trans. of ASME, E30 (1963) 347-354. [3] R. Testa and H. Bleich, Longitudinal Impact of a SemiInfinite, Cylindrical, Viscoelastic Shell. Trans. of ASME, E32 (1965) 813-820. [4] H. Conway and M. Jakubowski, Axial Impact of Short Cylindrical Bars. Trans. of ASME, E36 (1969) 809-813.