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Axisymmetric vibration of a long cylinder
Journal of Sound and Vibration (1995) 186(5), 711–721
AXISYMMETRIC VIBRATION OF A LONG CYLINDER A. K. G Reactor Safety Division, Bhabha Atomic Research Centre, Bombay 400 085, India (Received 12 April 1994, and in final form 21 September 1994) The axisymmetric vibration of a thick elastic cylinder has been studied under plane-strain conditions. The solution for forced vibration is obtained by using the Laplace transform. Results are presented for natural frequency and dynamic stresses for various types of loading. 7 1995 Academic Press Limited
1. INTRODUCTION
Cylindrical structures are widely used in pressure vessels, heat exchangers and nuclear reactor’s containments. Transient loading of these structures could occur during off-normal or accident conditions. In several instances the response could be axisymmetric: e.g., the overpressurization of a reactor containment following a loss of coolant accident. Furthermore, in many cases the cylinder may be long enough to permit the use of plane-strain conditions. Besides the stresses due to various other loads, the dynamic stresses induced by the accident are required to be withstood by the structure. The problem of axisymmetric radial vibration of a thick cylinder using the equations of the theory of elasticity has been addressed by Cinelli [1, 2], Sneddon [3] and Buorgois [4]. Nagaya [5] has determined the natural frequencies only by using a Fourier expansion collocation method. The free vibration problem has also been solved by Gazis [6]. Cinelli [1] used a Hankel transform and presented a closed form solution. The radial deflection, u, is given as u(r, t)=s F(ji )
C(r, ji )
G(ji , t),
i
where C(r, ji ) is a natural mode of free vibration with natural frequency directly proportional to ji and the summation is taken over all the modes. F and G depend on the inner and outer radii of the cylinder. This equation yields zero radial stresses at both the boundaries irrespective of the applied pressure—accounted for by G(ji , t). This is expected, since C(r, ji ) is a natural mode shape derived from the condition of a stress-free boundary. Numerical results support this observation. Whereas the Hankel transform in this study has been verified to be true, the same could not be said about its inverse. The basis of the latter analysis could not be found in reference [2], which was cited in reference [1]. Sneddon [3] analyzed the response of the cylinder for an internal pressure, pi varying with time t as pi=A0 (1−cos vt). Buorgois [4] analyzed the response of the cylinder for an internal pressure varying as a unit step function. The solutions in both references [3] and [4] are based on the Laplace transform. The transform of the responses given in references 711 0022–460X/95/400711+11 $12.00/0
7 1995 Academic Press Limited
. .
712
T 1 Non-dimensional natural frequencies
n
Mode ZXXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXXV k 1 2 3 4 5
0·30
1·02 1·05 1·10 1·25 1·50 2.00
0·8945E+00 0·8823E+00 0·8607E+00 0·8074E+00 0·7364E+00 0·6335E+00
0·1571E+03 0·6284E+02 0·3143E+02 0·1259E+02 0·6332E+01 0·3218E+01
0·3142E+03 0·1257E+03 0·6284E+02 0·2515E+02 0·1259E+02 0·6319E+01
0·4712E+03 0·1885E+03 0·9425E+02 0·3771E+02 0·1887E+02 0·9449E+01
0·6283E+03 0·2513E+03 0·1257E+03 0·5027E+02 0·2514E+02 0·1258E+02
0·25
1·02 1·05 1·10 1·25 1·50 2·00
0·9328E+00 0·9195E+00 0·8990E+00 0·8424E+00 0·7655E+00 0·6554E+00
0·1571E+03 0·6284E+02 0·3143E+02 0·1260E+02 0·6343E+01 0·3233E+01
0·3142E+03 0·1257E+03 0·6284E+02 0·2515E+02 0·1260E+02 0·6327E+01
0·4712E+03 0·1885E+03 0·9425E+02 0·3771E+02 0·1887E+02 0·9454E+01
0·6283E+03 0·2513E+03 0·1257E+03 0·5027E+02 0·2515E+02 0·1259E+02
0·20
1·02 1·05 1·10 1·25 1·50 2·00
0·9588E+00 0·9448E+00 0·9232E+00 0·8636E+00 0·7853E+00 0·6687E+00
0·1571E+03 0·6284E+02 0·3143E+02 0·1216E+02 0·6351E+01 0·3246E+01
0·3142E+03 0·1257E+03 0·6284E+02 0·2515E+02 0·1260E+02 0·6334E+01
0·4712E+03 0·1885E+03 0·9425E+03 0·3771E+02 0·1887E+02 0·1259E+01
0·6283E+03 0·2513E+03 0·1257E+03 0·5027E+02 0·2515E+02 0·9459E+02
0·15
1·02 1.05 1·10 1.25 1·50 2.00
0·9741E+00 0·9612E+00 0·9374E+00 0·8774E+00 0·7968E+00 0·6757E+00
0·1571E+03 0·6284E+02 0·3144E+02 0·1261E+02 0·6359E+01 0·3257E+01
0·3142E+03 0·1257E+03 0·6284E+02 0·2515E+02 0·1260E+02 0·6340E+01
0·4712E+03 0·1885E+03 0·9425E+02 0·3771E+02 0·1887E+02 0·9462E+01
0·6283E+03 0·2513E+03 0·1257E+03 0·5028E+02 0·2515E+02 0·1259E+02
ZXXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXXV 6 7 8 9 10 0·30 1.02 0·7854E+03 0·9425E+03 0.1100E+04 0.1257E+04 0·1414E+04 1.05 0·3142E+03 0·3770E+03 0·4398E+03 0·5027E+03 0·5655E+03 1.10 0.1571E+03 0·1885E+03 0·2199E+03 0·2513E+03 0·2827E+03 1.25 0·6284E+02 0·7540E+02 0·8797E+02 0·1005E+03 0·1131E+03 1.50 0.3143E+02 0·3771E+02 0·4399E+02 0·5027E+02 0·5655E+02 2.00 0·1572E+02 0·1886E+02 0·2200E+02 0·2514E+02 0·2828E+02 0.25
1.02 1.05 1.10 1.25 1.50 2.00
0·7854E+03 0·3142E+03 0.1571E+03 0.6284E+02 0·3143E+02 0·1573E+02
0·9425E+03 0·3770E+03 0·1885E+03 0·7540E+02 0·3771E+02 0·1886E+02
0·1100E+04 0·4398E+03 0.2199E+03 0·8797E+02 0·4399E+02 0·2200E+02
0·1257E+04 0·5027E+03 0·2513E+03 0·1005E+03 0·5027E+02 0·2514E+02
0·1414E+04 0·5655E+04 0·2827E+03 0·1131E+03 0·5656E+02 0·2828E+02
0·20
1.02 1.05 1.10 1.25 1.50 2.00
0·7854E+03 0·3142E+03 0·1571E+03 0·6284E+02 0·3143E+02 0·1573E+02
0.9425E+03 0·3770E+03 0·1885E+03 0·7540E+02 0·3771E+02 0·1887E+02
0·1100E+04 0·4398E+03 0·2199E+03 0·8797E+02 0·4399E+02 0·2201E+02
0·1257E+04 0·5027E+03 0·2513E+03 0·1005E+03 0·5027E+02 0·2515E+02
0·1414E+04 0·5655E+03 0·2827E+03 0·1131E+03 0·5656E+02 0·2829E+02
0·15
1.02 1.05 1.10 1.25 1.50 2.00
0·7854E+03 0·3142E+03 0·1571E+03 0·6284E+02 0·3143E+02 0·1573E+02
0·9425E+03 0·3770E+03 0·1885E+03 0·7541E+02 0.3771E+02 0·1887E+02
0·1100E+04 0·4398E+03 0·2199E+03 0·8797E+02 0·4399E+02 0·2201E+02
0·1257E+04 0·5027E+03 0·2513E+03 0·1005E+03 0·5027E+02 0·2515E+02
0·1414E+04 0·5655E+03 0·2827E+03 0·1131E+03 0·5656E+02 0·2829E+02
713
Figure 1. Variation of non-dimensional hoop stress on the inner boundary for internal pressure varying as a step function.
[3] and [4] tallies with the results presented in this paper. However, the time-domain solution presented in these works [3, 4] could not be derived. No numerical result has been presented in any of the works referred to [1–4]. As mentioned before, the works mentioned in references [5, 6] pertain to free vibration alone. This paper presents a solution of the axisymmetric free and forced vibration of a cylinder under the plane-strain condition. The solution for forced vibration is obtained by using the Laplace transform. Results are presented for the natural frequency and dynamic stresses for various types of loading.
T 2 Peak amplification factors with step function loading
n
k ZXXXXXXXXXXXXCXXXXXXXXXXXXV 1·02 1·05 1·10
0·30 0·25 0·20 0·15
0·202E+01 0·202E+01 0·201E+01 0·201E+01
0·204E+01 0·202E+01 0·202E+01 0·201E+01
0·208E+01 0·206E+01 0·204E+01 0·204E+01
n
1·25
1·50
2·0
0·30 0·25 0·20 0·15
0·218E+01 0·213E+01 0·210E+01 0·207E+01
0·228E+01 0·222E+01 0·215E+01 0·207E+01
0·242E+01 0·229E+01 0·217E+01 0·207E+01
. .
714
T 3 Convergence with step function loading; n=0·300E+00
k 1·05 1·50 2·00
NR ZXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXV 1 2 4 6 8 0·203E+01 0·222E+01 0·219E+01
0·204E+01 0·227E+01 0·238E+01
0·204E+01 0·227E+01 0·240E+01
0·204E+01 0·228E+01 0·241E+01
k
10
20
30
50
1·05 1·50 2.00
0·204E+01 0·228E+01 0·241E+01
0·204E+01 0·228E+01 0·242E+01
0·204E+01 0·228E+01 0·242E+01
0·204E+01 0·228E+01 0·242E+01
0·204E+01 0·228E+01 0·241E+01
2. THEORY
The equation for axisymmetric motion of a cylinder under the plane-strain condition can be written as [7]
0 1
1 2u 2 1 1u u =c + , 1t 2 1r 1r r
(1)
where the dilational wave speed, c, is given by c 2=(l+2m)/7. l and m are the Lame´ constants and 7 is the density. r and t denote the radius and time respectively. The variables
Figure 2. Variation of non-dimensional hoop stress on the inner boundary for internal pressure varying as a sinusoid; f(t)=sin(au); a=0·6.
715
can be normalized as x=r/b and u=ct/b, where b is the outer radius of the cylinder. Then equation (1) can be rewritten as
0 1
1 2u 1 1u u = + , 1u 2 1x 1x x
(2)
The initial conditions are taken as zero: i.e., u=0
and
1u/1u=0
u=0.
(3)
sr (1, u)=0.
(4, 5)
at
The boundary conditions are taken as sr (a/b, u)=p0 f(u),
Here a is the inner radius of the cylinder and sr denotes the radial stress: sr (x, u)=(1/b)[(l+2m) 1u/1x+lu/x].
(6)
2.1. A solution of equation (2) is sought in the form u=U(x) exp(jvt)=U(x) exp(jvbu/c).
(7)
Substituting equation (7) in equation (2) yields
0
1
1 2U 1 1U b 2v 2 1 + + − 2 U=0. 1x 2 x 1x c2 x
(8)
The solution of equation (8) is U=AJ1 (bvx/c)+BY1 (bvx/c).
(9)
By substituting this into conditions (4) and (5), the natural frequency equation is obtained:
$ 0 1
0 1%$
av av av +(m−1) J1 J c 0 c c
$ 0 1
=
0 1
0 1%
bv bv bv +(m−1)Y1 Y c 0 c c
0 1%$
bv bv bv J +(m−1)J1 c 0 c c
0 1
0 1%
av av av Y +(m−1)Y1 c 0 c c
.
(10)
2.2. With the Laplace transform of u(x, u) defined as u¯=u(r, s)=
g
a
e−suu(x, u) du,
(11)
0
the transform of equation (2) can be written as
0 1
d2u¯ 1 du¯ 1 + − s 2+ 2 u¯=0. dx 2 x dx x
(12)
The solution of equation (12) is u¯=AI1 (sx)+BK1 (sx).
(13)
. .
716
Figure 3. Variation of maximum non-dimensional hoop stress on the inner boundary for internal pressure varying as a sinusoid with non-dimensional excitation frequency; k=2, ——, n=0·3; – – –, n=0·2, n=0.15.
The transforms of the boundary conditions (4) and (5) are written as s¯ r (a/b, s)=p0(s), f
s¯ r (1, s)=0.
(14, 15)
A bar over a variable denotes its Laplace transform. By making use of the transformed boundary conditions, i.e., equations (14) and (15), the constants A and B are evaluated: A=
(s)a f G2 (z2 ) 1b , s G1 (z1 )G2 (z2 )−G1 (z2 )G2 (z1 )
B=
−f(s)a1 b G1 (z2 ) . s G1 (z1 )G2 (z2 )−G1 (z2 )G2 (z1 )
(16, 17)
Here, z=sx, z1=sa/b and z2=s, G1 (z)=I'1 (z)+(m/z)I1 (z),
G2 (z)=K'1 (z)+(m/z)K1 (z).
m=l/(l+2m)=n/(1−n),
(18, 19)
a1=p0 /(l+2m).
I1 (z) and K1 (z) are the modified Bessel functions of order one. A prime denotes differentiation with respect to z. The transformed stresses can be written as
6
1u¯ lu¯ s s + = (l+2m)[AG1 (z)+BG2 (z)], 1z z b b
6
1u¯ u¯ s s +(l+2m) = (l+2m)[AG3 (z)+BG4 (z)], 1z z b b
s¯ p= (l+2m)
s¯ u= l
7
(20)
7
(21)
717
where G3 (z)=mI'1 (z)+(1/z)I1 (z),
G4 (z)=mK'1 (z)+(1/z)K1 (z).
(22, 23)
Simplifying, one has s¯ u=p0(s) f
G2 (z2 )G3 (z)−G1 (z2 )G4 (z) , G1 (z1 )G2 (z2 )−G1 (z2 )G2 (z1 )
s¯ r=p0(s) f
G2 (z2 )G1 (z)−G1 (z2 )G2 (z) . G1 (z1 )G2 (z2 )−G1 (z2 )G2 (z1 )
(24, 25)
Explicit evaluation of su (x, u) is given in what follows. The same procedure applies to the evaluation of sr and u as well. From equation (24) it is seen that su is of the form s¯ u=p0(s)N(s)/D(s). f The inverse of the Laplace transform is obtained through residues as su =
g
u
f(u−t) s i
0
exp(si t)N(si ) dt , (dD/ds)(si )
(26)
where the summation is taken over all the poles of su : i.e., the zeros of D(s). By making use of the relations among the Bessel functions Jn (jz), Yn (jz) and the modified Bessel functions In (z) and Kn (z) (j=z−1) [8], the final result can be written as su=−2p0
a bx
g
u
0
f(u−t) s
vi N1 (vi ) sin (vi t) dt. D1 (vi )
(27)
Figure 4. Variation of maximum non-dimensional hoop stress on the inner boundary for internal pressure varying as a sinusoid with non-dimensional excitation frequency; k=1·5. Key as figure 3.
. .
718
Figure 5. Variation of maximum non-dimensional hoop stress on the inner boundary for internal pressure varying as a sinusoid with non-dimensional excitation frequency; k=1·05. Key as figure 3.
The roots (si ) of D(s) and the natural frequencies (vi ) of the cylinder are related by si=jvi and N1=[y2 Y0 (y2 )+(m−1)Y1 (y2 )][myJ0 (y)+(1−m)J1 (y)] −[y2 J0 (y2 )+(m−1)J1 (y2 )][myY0 (y)+(1−m)Y1 (y)],
(28)
D1=[(m+1)y1 J0 (y1 )−y12 J1 (y)][y2 Y0 (y2 )+(m−1)Y1 (y2 )] +[y1 J0 (y1 )+(m−1)J1 (y1 )][(m+1)y2 Y0 (y)−y22 Y1 (y2 )] −[(m+1)y2 J0 (y2 )−y22 J1 (y1 )][y1 Y0 (y1 )+(m−1)Y1 (y1 )] −[y2 J0 (y2 )+(m−1)J1 (y2 )][(m+1)y1 Y0 (y1 )−y1 Y1 (y)].
(29)
In these equations (28) and (29) y=vx, y1=va/b and y2=v. Explicit results for the integral in equation (27) are given for certain types of loading, as follows: with I=
g
u
f(u−t) sin (vi−t) dt,
0
one has (i) for sinusoidal input, f(t)=sin (Vt)=sin (au) I=
g
u
0
(i.e., a=Vb/c),
sin (a(u−t)) sin (vi t) dt
(30)
=[1/(a 2−vi 2 )][a sin (vi u)−vi sin (au)]
when a$vi
when a=vi
(1/2a)[sin (au)−au(cos (au)]
719
;
(31, 32)
(ii) for exponentially decaying input, f(t)=1−exp(−t/t)=1−exp(−au)
(i.e., a=b/ct),
(33)
I=[(1−cos vi u)/vi ]+[1/(a +v )][vi cos (vi u)−a sin (vi u)−vi exp(−au)]; 2
2 i
(34)
(iii) for step input, f(t)=1
I=(1−cos vi u)/vi .
(i.e., f(u)=1),
(35, 36)
3. NUMERICAL ANALYSIS
The non-dimensional natural frequency b=(a/b)v'=(bv/c)(1/k),
k=b/a,
(37)
has been evaluated. The first ten values of b for various values of k and n are given in Table 1. These values depend only on k and n. The stress results are normalized by dividing the dynamic stress by the corresponding static stress for the peak value of the pressure. This ratio, called the dynamic load amplification factor, is again found to depend only on k, n and the type of loading. The positions are also presented in normalized co-ordinates. For static internal pressure p0 the hoop stress at x(=r/b) is su,static=p0
01 a b
2
1 x 2+1 , x 2 (1−a 2/b 2 )
(38)
Figure 6. Variation of non-dimensional hoop stress on the inner boundary for internal pressure varying as f(t)=1−exp(−au), a=1.
. .
720
T 4 Peak amplification factors for exponentially decaying input (equation (33))
k 20·0 10·0 5·0 2·0 1·0 0·5 0·25 0·1 0·05 0·01 0·005 0·001
n=0·3 ZXXXXCXXXXV 2·0 1·5 1·05 2·46 2·43 2·38 2·21 1·95 1·67 1·48 1·35 1·31 1·27 1·27 1·26
2·30 2·29 2·26 2·14 1·91 1·63 1·42 1·27 1·22 1·18 1·18 1·17
2·03 2·03 2·02 1·94 1·77 1·51 1·29 1·14 1·08 1·04 1·03 1·03
n=0·2 ZXXXXCXXXXV 2·0 1·5 1·05 2·23 2·22 2·16 1·99 1·75 1·51 1·34 1·23 1·19 1·16 1·16 1·15
2·16 2·15 2·12 1·99 1·74 1·50 1·32 1·19 1·15 1·11 1·11 1·10
2·02 2·02 2·00 1·92 1·73 1·47 1·27 1·12 1·07 1·03 1·03 1·02
n=0·15 ZXXXXCXXXXV 2·0 1·5 1·05 2·15 2·13 2·07 1·93 1·72 1·46 1·28 1·18 1·15 1·12 1·11 1·11
2·11 2·09 2·06 1·92 1·72 1·46 1·28 1·16 1·12 1·08 1·08 1·07
2·01 2·01 1·99 1·90 1·67 1·44 1·26 1·11 1·07 1·03 1·02 1·01
In Figure 1 is shown the non-dimensional hoop stress on the inner boundary as a function of the non-dimensional time for the cylinder subjected to an internal pressure applied as a step input (i.e., f(u)=1). It is seen that the dynamic stress oscillates about the corresponding static stress (i.e., a non-dimensional stress of 1). The amplitude of the steady state oscillation remains unaltered since there is no damping. The peak amplification factor for various k and n is presented in Table 2. The amplification factor increases asymptotically with k and n. The convergence of the solution with number of modes is presented in Table 3. In Figure 2 is shown the non-dimensional hoop stress on the inner boundary as a function of the non-dimensional time for the cylinder subjected to an internal pressure applied as a sinusoidal input with (Vb/c=a=0·6). In Figures 3–5 is shown the variation of the maximum value of non-dimensional hoop stress (on the inner boundary) with the normalized non-dimensional excitation frequency (a/kb1 ). Due to the absence of damping the peak value increases without bound around the resonant frequencies. The results are presented for three values of k=2, 1·5 and 1·05. The values considered are expected to cover the parameters encountered in common usage. In Figure 6 is shown the variation of the non-dimensional hoop stress on the inner boundary as a function of the non-dimensional time for the cylinder subjected to an internal pressure varying as f(t)=1−exp(−au) for a=1. The dynamic response gradually builds up to the steady state value. The peak amplification factor for various values of k, n and a are presented in Table 4.
4. CONCLUSIONS
The non-dimensional hoop stress, i.e., the dynamic hoop stress normalized with respect to the corresponding static value for the peak value of the applied pressure, is a measure of the dynamic amplification. The equivalent static loads used in the design process can now be easily obtained by multiplying the static stress by the peak dynamic amplification factors.
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REFERENCES 1. G. C 1966 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics, 33(4), 825–830. Dynamic vibrations and stresses in elastic cylinders and spheres. 2. G. C 1965 International Journal of Engineering Science 3, 539–559. An extension of the finite Hankel transform and applications. 3. I. N. S 1951 Fourier Transforms, New York; McGraw-Hill. 4. R. A. B 1973 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics, 40(4), 1140–1141. Application of the generalized airy stress function to problems of elastic vibrations of hollow cylinders. 5. K. N 1983 Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 50, 757–764. Vibrations of a thick-walled pipe or a ring of arbitrary shape in its plane. 6. D. G. G 1958 Journal of the Acoustical Society of America 30, 786–794. Exact analysis of the plane strain vibrations of thick walled hollow cylinders. 7. A. E. H. L 1944 A Treatise on the Mathematical Theory of Elasticity. Fourth edition. New York: Dover. 8. G. N. Watson 1952 A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press.
AXISYMMETRIC VIBRATION OF A LONG CYLINDER A. K. G Reactor Safety Division, Bhabha Atomic Research Centre, Bombay 400 085, India (Received 12 April 1994, and in final form 21 September 1994) The axisymmetric vibration of a thick elastic cylinder has been studied under plane-strain conditions. The solution for forced vibration is obtained by using the Laplace transform. Results are presented for natural frequency and dynamic stresses for various types of loading. 7 1995 Academic Press Limited
1. INTRODUCTION
Cylindrical structures are widely used in pressure vessels, heat exchangers and nuclear reactor’s containments. Transient loading of these structures could occur during off-normal or accident conditions. In several instances the response could be axisymmetric: e.g., the overpressurization of a reactor containment following a loss of coolant accident. Furthermore, in many cases the cylinder may be long enough to permit the use of plane-strain conditions. Besides the stresses due to various other loads, the dynamic stresses induced by the accident are required to be withstood by the structure. The problem of axisymmetric radial vibration of a thick cylinder using the equations of the theory of elasticity has been addressed by Cinelli [1, 2], Sneddon [3] and Buorgois [4]. Nagaya [5] has determined the natural frequencies only by using a Fourier expansion collocation method. The free vibration problem has also been solved by Gazis [6]. Cinelli [1] used a Hankel transform and presented a closed form solution. The radial deflection, u, is given as u(r, t)=s F(ji )
C(r, ji )
G(ji , t),
i
where C(r, ji ) is a natural mode of free vibration with natural frequency directly proportional to ji and the summation is taken over all the modes. F and G depend on the inner and outer radii of the cylinder. This equation yields zero radial stresses at both the boundaries irrespective of the applied pressure—accounted for by G(ji , t). This is expected, since C(r, ji ) is a natural mode shape derived from the condition of a stress-free boundary. Numerical results support this observation. Whereas the Hankel transform in this study has been verified to be true, the same could not be said about its inverse. The basis of the latter analysis could not be found in reference [2], which was cited in reference [1]. Sneddon [3] analyzed the response of the cylinder for an internal pressure, pi varying with time t as pi=A0 (1−cos vt). Buorgois [4] analyzed the response of the cylinder for an internal pressure varying as a unit step function. The solutions in both references [3] and [4] are based on the Laplace transform. The transform of the responses given in references 711 0022–460X/95/400711+11 $12.00/0
7 1995 Academic Press Limited
. .
712
T 1 Non-dimensional natural frequencies
n
Mode ZXXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXXV k 1 2 3 4 5
0·30
1·02 1·05 1·10 1·25 1·50 2.00
0·8945E+00 0·8823E+00 0·8607E+00 0·8074E+00 0·7364E+00 0·6335E+00
0·1571E+03 0·6284E+02 0·3143E+02 0·1259E+02 0·6332E+01 0·3218E+01
0·3142E+03 0·1257E+03 0·6284E+02 0·2515E+02 0·1259E+02 0·6319E+01
0·4712E+03 0·1885E+03 0·9425E+02 0·3771E+02 0·1887E+02 0·9449E+01
0·6283E+03 0·2513E+03 0·1257E+03 0·5027E+02 0·2514E+02 0·1258E+02
0·25
1·02 1·05 1·10 1·25 1·50 2·00
0·9328E+00 0·9195E+00 0·8990E+00 0·8424E+00 0·7655E+00 0·6554E+00
0·1571E+03 0·6284E+02 0·3143E+02 0·1260E+02 0·6343E+01 0·3233E+01
0·3142E+03 0·1257E+03 0·6284E+02 0·2515E+02 0·1260E+02 0·6327E+01
0·4712E+03 0·1885E+03 0·9425E+02 0·3771E+02 0·1887E+02 0·9454E+01
0·6283E+03 0·2513E+03 0·1257E+03 0·5027E+02 0·2515E+02 0·1259E+02
0·20
1·02 1·05 1·10 1·25 1·50 2·00
0·9588E+00 0·9448E+00 0·9232E+00 0·8636E+00 0·7853E+00 0·6687E+00
0·1571E+03 0·6284E+02 0·3143E+02 0·1216E+02 0·6351E+01 0·3246E+01
0·3142E+03 0·1257E+03 0·6284E+02 0·2515E+02 0·1260E+02 0·6334E+01
0·4712E+03 0·1885E+03 0·9425E+03 0·3771E+02 0·1887E+02 0·1259E+01
0·6283E+03 0·2513E+03 0·1257E+03 0·5027E+02 0·2515E+02 0·9459E+02
0·15
1·02 1.05 1·10 1.25 1·50 2.00
0·9741E+00 0·9612E+00 0·9374E+00 0·8774E+00 0·7968E+00 0·6757E+00
0·1571E+03 0·6284E+02 0·3144E+02 0·1261E+02 0·6359E+01 0·3257E+01
0·3142E+03 0·1257E+03 0·6284E+02 0·2515E+02 0·1260E+02 0·6340E+01
0·4712E+03 0·1885E+03 0·9425E+02 0·3771E+02 0·1887E+02 0·9462E+01
0·6283E+03 0·2513E+03 0·1257E+03 0·5028E+02 0·2515E+02 0·1259E+02
ZXXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXXV 6 7 8 9 10 0·30 1.02 0·7854E+03 0·9425E+03 0.1100E+04 0.1257E+04 0·1414E+04 1.05 0·3142E+03 0·3770E+03 0·4398E+03 0·5027E+03 0·5655E+03 1.10 0.1571E+03 0·1885E+03 0·2199E+03 0·2513E+03 0·2827E+03 1.25 0·6284E+02 0·7540E+02 0·8797E+02 0·1005E+03 0·1131E+03 1.50 0.3143E+02 0·3771E+02 0·4399E+02 0·5027E+02 0·5655E+02 2.00 0·1572E+02 0·1886E+02 0·2200E+02 0·2514E+02 0·2828E+02 0.25
1.02 1.05 1.10 1.25 1.50 2.00
0·7854E+03 0·3142E+03 0.1571E+03 0.6284E+02 0·3143E+02 0·1573E+02
0·9425E+03 0·3770E+03 0·1885E+03 0·7540E+02 0·3771E+02 0·1886E+02
0·1100E+04 0·4398E+03 0.2199E+03 0·8797E+02 0·4399E+02 0·2200E+02
0·1257E+04 0·5027E+03 0·2513E+03 0·1005E+03 0·5027E+02 0·2514E+02
0·1414E+04 0·5655E+04 0·2827E+03 0·1131E+03 0·5656E+02 0·2828E+02
0·20
1.02 1.05 1.10 1.25 1.50 2.00
0·7854E+03 0·3142E+03 0·1571E+03 0·6284E+02 0·3143E+02 0·1573E+02
0.9425E+03 0·3770E+03 0·1885E+03 0·7540E+02 0·3771E+02 0·1887E+02
0·1100E+04 0·4398E+03 0·2199E+03 0·8797E+02 0·4399E+02 0·2201E+02
0·1257E+04 0·5027E+03 0·2513E+03 0·1005E+03 0·5027E+02 0·2515E+02
0·1414E+04 0·5655E+03 0·2827E+03 0·1131E+03 0·5656E+02 0·2829E+02
0·15
1.02 1.05 1.10 1.25 1.50 2.00
0·7854E+03 0·3142E+03 0·1571E+03 0·6284E+02 0·3143E+02 0·1573E+02
0·9425E+03 0·3770E+03 0·1885E+03 0·7541E+02 0.3771E+02 0·1887E+02
0·1100E+04 0·4398E+03 0·2199E+03 0·8797E+02 0·4399E+02 0·2201E+02
0·1257E+04 0·5027E+03 0·2513E+03 0·1005E+03 0·5027E+02 0·2515E+02
0·1414E+04 0·5655E+03 0·2827E+03 0·1131E+03 0·5656E+02 0·2829E+02
713
Figure 1. Variation of non-dimensional hoop stress on the inner boundary for internal pressure varying as a step function.
[3] and [4] tallies with the results presented in this paper. However, the time-domain solution presented in these works [3, 4] could not be derived. No numerical result has been presented in any of the works referred to [1–4]. As mentioned before, the works mentioned in references [5, 6] pertain to free vibration alone. This paper presents a solution of the axisymmetric free and forced vibration of a cylinder under the plane-strain condition. The solution for forced vibration is obtained by using the Laplace transform. Results are presented for the natural frequency and dynamic stresses for various types of loading.
T 2 Peak amplification factors with step function loading
n
k ZXXXXXXXXXXXXCXXXXXXXXXXXXV 1·02 1·05 1·10
0·30 0·25 0·20 0·15
0·202E+01 0·202E+01 0·201E+01 0·201E+01
0·204E+01 0·202E+01 0·202E+01 0·201E+01
0·208E+01 0·206E+01 0·204E+01 0·204E+01
n
1·25
1·50
2·0
0·30 0·25 0·20 0·15
0·218E+01 0·213E+01 0·210E+01 0·207E+01
0·228E+01 0·222E+01 0·215E+01 0·207E+01
0·242E+01 0·229E+01 0·217E+01 0·207E+01
. .
714
T 3 Convergence with step function loading; n=0·300E+00
k 1·05 1·50 2·00
NR ZXXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXXV 1 2 4 6 8 0·203E+01 0·222E+01 0·219E+01
0·204E+01 0·227E+01 0·238E+01
0·204E+01 0·227E+01 0·240E+01
0·204E+01 0·228E+01 0·241E+01
k
10
20
30
50
1·05 1·50 2.00
0·204E+01 0·228E+01 0·241E+01
0·204E+01 0·228E+01 0·242E+01
0·204E+01 0·228E+01 0·242E+01
0·204E+01 0·228E+01 0·242E+01
0·204E+01 0·228E+01 0·241E+01
2. THEORY
The equation for axisymmetric motion of a cylinder under the plane-strain condition can be written as [7]
0 1
1 2u 2 1 1u u =c + , 1t 2 1r 1r r
(1)
where the dilational wave speed, c, is given by c 2=(l+2m)/7. l and m are the Lame´ constants and 7 is the density. r and t denote the radius and time respectively. The variables
Figure 2. Variation of non-dimensional hoop stress on the inner boundary for internal pressure varying as a sinusoid; f(t)=sin(au); a=0·6.
715
can be normalized as x=r/b and u=ct/b, where b is the outer radius of the cylinder. Then equation (1) can be rewritten as
0 1
1 2u 1 1u u = + , 1u 2 1x 1x x
(2)
The initial conditions are taken as zero: i.e., u=0
and
1u/1u=0
u=0.
(3)
sr (1, u)=0.
(4, 5)
at
The boundary conditions are taken as sr (a/b, u)=p0 f(u),
Here a is the inner radius of the cylinder and sr denotes the radial stress: sr (x, u)=(1/b)[(l+2m) 1u/1x+lu/x].
(6)
2.1. A solution of equation (2) is sought in the form u=U(x) exp(jvt)=U(x) exp(jvbu/c).
(7)
Substituting equation (7) in equation (2) yields
0
1
1 2U 1 1U b 2v 2 1 + + − 2 U=0. 1x 2 x 1x c2 x
(8)
The solution of equation (8) is U=AJ1 (bvx/c)+BY1 (bvx/c).
(9)
By substituting this into conditions (4) and (5), the natural frequency equation is obtained:
$ 0 1
0 1%$
av av av +(m−1) J1 J c 0 c c
$ 0 1
=
0 1
0 1%
bv bv bv +(m−1)Y1 Y c 0 c c
0 1%$
bv bv bv J +(m−1)J1 c 0 c c
0 1
0 1%
av av av Y +(m−1)Y1 c 0 c c
.
(10)
2.2. With the Laplace transform of u(x, u) defined as u¯=u(r, s)=
g
a
e−suu(x, u) du,
(11)
0
the transform of equation (2) can be written as
0 1
d2u¯ 1 du¯ 1 + − s 2+ 2 u¯=0. dx 2 x dx x
(12)
The solution of equation (12) is u¯=AI1 (sx)+BK1 (sx).
(13)
. .
716
Figure 3. Variation of maximum non-dimensional hoop stress on the inner boundary for internal pressure varying as a sinusoid with non-dimensional excitation frequency; k=2, ——, n=0·3; – – –, n=0·2, n=0.15.
The transforms of the boundary conditions (4) and (5) are written as s¯ r (a/b, s)=p0(s), f
s¯ r (1, s)=0.
(14, 15)
A bar over a variable denotes its Laplace transform. By making use of the transformed boundary conditions, i.e., equations (14) and (15), the constants A and B are evaluated: A=
(s)a f G2 (z2 ) 1b , s G1 (z1 )G2 (z2 )−G1 (z2 )G2 (z1 )
B=
−f(s)a1 b G1 (z2 ) . s G1 (z1 )G2 (z2 )−G1 (z2 )G2 (z1 )
(16, 17)
Here, z=sx, z1=sa/b and z2=s, G1 (z)=I'1 (z)+(m/z)I1 (z),
G2 (z)=K'1 (z)+(m/z)K1 (z).
m=l/(l+2m)=n/(1−n),
(18, 19)
a1=p0 /(l+2m).
I1 (z) and K1 (z) are the modified Bessel functions of order one. A prime denotes differentiation with respect to z. The transformed stresses can be written as
6
1u¯ lu¯ s s + = (l+2m)[AG1 (z)+BG2 (z)], 1z z b b
6
1u¯ u¯ s s +(l+2m) = (l+2m)[AG3 (z)+BG4 (z)], 1z z b b
s¯ p= (l+2m)
s¯ u= l
7
(20)
7
(21)
717
where G3 (z)=mI'1 (z)+(1/z)I1 (z),
G4 (z)=mK'1 (z)+(1/z)K1 (z).
(22, 23)
Simplifying, one has s¯ u=p0(s) f
G2 (z2 )G3 (z)−G1 (z2 )G4 (z) , G1 (z1 )G2 (z2 )−G1 (z2 )G2 (z1 )
s¯ r=p0(s) f
G2 (z2 )G1 (z)−G1 (z2 )G2 (z) . G1 (z1 )G2 (z2 )−G1 (z2 )G2 (z1 )
(24, 25)
Explicit evaluation of su (x, u) is given in what follows. The same procedure applies to the evaluation of sr and u as well. From equation (24) it is seen that su is of the form s¯ u=p0(s)N(s)/D(s). f The inverse of the Laplace transform is obtained through residues as su =
g
u
f(u−t) s i
0
exp(si t)N(si ) dt , (dD/ds)(si )
(26)
where the summation is taken over all the poles of su : i.e., the zeros of D(s). By making use of the relations among the Bessel functions Jn (jz), Yn (jz) and the modified Bessel functions In (z) and Kn (z) (j=z−1) [8], the final result can be written as su=−2p0
a bx
g
u
0
f(u−t) s
vi N1 (vi ) sin (vi t) dt. D1 (vi )
(27)
Figure 4. Variation of maximum non-dimensional hoop stress on the inner boundary for internal pressure varying as a sinusoid with non-dimensional excitation frequency; k=1·5. Key as figure 3.
. .
718
Figure 5. Variation of maximum non-dimensional hoop stress on the inner boundary for internal pressure varying as a sinusoid with non-dimensional excitation frequency; k=1·05. Key as figure 3.
The roots (si ) of D(s) and the natural frequencies (vi ) of the cylinder are related by si=jvi and N1=[y2 Y0 (y2 )+(m−1)Y1 (y2 )][myJ0 (y)+(1−m)J1 (y)] −[y2 J0 (y2 )+(m−1)J1 (y2 )][myY0 (y)+(1−m)Y1 (y)],
(28)
D1=[(m+1)y1 J0 (y1 )−y12 J1 (y)][y2 Y0 (y2 )+(m−1)Y1 (y2 )] +[y1 J0 (y1 )+(m−1)J1 (y1 )][(m+1)y2 Y0 (y)−y22 Y1 (y2 )] −[(m+1)y2 J0 (y2 )−y22 J1 (y1 )][y1 Y0 (y1 )+(m−1)Y1 (y1 )] −[y2 J0 (y2 )+(m−1)J1 (y2 )][(m+1)y1 Y0 (y1 )−y1 Y1 (y)].
(29)
In these equations (28) and (29) y=vx, y1=va/b and y2=v. Explicit results for the integral in equation (27) are given for certain types of loading, as follows: with I=
g
u
f(u−t) sin (vi−t) dt,
0
one has (i) for sinusoidal input, f(t)=sin (Vt)=sin (au) I=
g
u
0
(i.e., a=Vb/c),
sin (a(u−t)) sin (vi t) dt
(30)
=[1/(a 2−vi 2 )][a sin (vi u)−vi sin (au)]
when a$vi
when a=vi
(1/2a)[sin (au)−au(cos (au)]
719
;
(31, 32)
(ii) for exponentially decaying input, f(t)=1−exp(−t/t)=1−exp(−au)
(i.e., a=b/ct),
(33)
I=[(1−cos vi u)/vi ]+[1/(a +v )][vi cos (vi u)−a sin (vi u)−vi exp(−au)]; 2
2 i
(34)
(iii) for step input, f(t)=1
I=(1−cos vi u)/vi .
(i.e., f(u)=1),
(35, 36)
3. NUMERICAL ANALYSIS
The non-dimensional natural frequency b=(a/b)v'=(bv/c)(1/k),
k=b/a,
(37)
has been evaluated. The first ten values of b for various values of k and n are given in Table 1. These values depend only on k and n. The stress results are normalized by dividing the dynamic stress by the corresponding static stress for the peak value of the pressure. This ratio, called the dynamic load amplification factor, is again found to depend only on k, n and the type of loading. The positions are also presented in normalized co-ordinates. For static internal pressure p0 the hoop stress at x(=r/b) is su,static=p0
01 a b
2
1 x 2+1 , x 2 (1−a 2/b 2 )
(38)
Figure 6. Variation of non-dimensional hoop stress on the inner boundary for internal pressure varying as f(t)=1−exp(−au), a=1.
. .
720
T 4 Peak amplification factors for exponentially decaying input (equation (33))
k 20·0 10·0 5·0 2·0 1·0 0·5 0·25 0·1 0·05 0·01 0·005 0·001
n=0·3 ZXXXXCXXXXV 2·0 1·5 1·05 2·46 2·43 2·38 2·21 1·95 1·67 1·48 1·35 1·31 1·27 1·27 1·26
2·30 2·29 2·26 2·14 1·91 1·63 1·42 1·27 1·22 1·18 1·18 1·17
2·03 2·03 2·02 1·94 1·77 1·51 1·29 1·14 1·08 1·04 1·03 1·03
n=0·2 ZXXXXCXXXXV 2·0 1·5 1·05 2·23 2·22 2·16 1·99 1·75 1·51 1·34 1·23 1·19 1·16 1·16 1·15
2·16 2·15 2·12 1·99 1·74 1·50 1·32 1·19 1·15 1·11 1·11 1·10
2·02 2·02 2·00 1·92 1·73 1·47 1·27 1·12 1·07 1·03 1·03 1·02
n=0·15 ZXXXXCXXXXV 2·0 1·5 1·05 2·15 2·13 2·07 1·93 1·72 1·46 1·28 1·18 1·15 1·12 1·11 1·11
2·11 2·09 2·06 1·92 1·72 1·46 1·28 1·16 1·12 1·08 1·08 1·07
2·01 2·01 1·99 1·90 1·67 1·44 1·26 1·11 1·07 1·03 1·02 1·01
In Figure 1 is shown the non-dimensional hoop stress on the inner boundary as a function of the non-dimensional time for the cylinder subjected to an internal pressure applied as a step input (i.e., f(u)=1). It is seen that the dynamic stress oscillates about the corresponding static stress (i.e., a non-dimensional stress of 1). The amplitude of the steady state oscillation remains unaltered since there is no damping. The peak amplification factor for various k and n is presented in Table 2. The amplification factor increases asymptotically with k and n. The convergence of the solution with number of modes is presented in Table 3. In Figure 2 is shown the non-dimensional hoop stress on the inner boundary as a function of the non-dimensional time for the cylinder subjected to an internal pressure applied as a sinusoidal input with (Vb/c=a=0·6). In Figures 3–5 is shown the variation of the maximum value of non-dimensional hoop stress (on the inner boundary) with the normalized non-dimensional excitation frequency (a/kb1 ). Due to the absence of damping the peak value increases without bound around the resonant frequencies. The results are presented for three values of k=2, 1·5 and 1·05. The values considered are expected to cover the parameters encountered in common usage. In Figure 6 is shown the variation of the non-dimensional hoop stress on the inner boundary as a function of the non-dimensional time for the cylinder subjected to an internal pressure varying as f(t)=1−exp(−au) for a=1. The dynamic response gradually builds up to the steady state value. The peak amplification factor for various values of k, n and a are presented in Table 4.
4. CONCLUSIONS
The non-dimensional hoop stress, i.e., the dynamic hoop stress normalized with respect to the corresponding static value for the peak value of the applied pressure, is a measure of the dynamic amplification. The equivalent static loads used in the design process can now be easily obtained by multiplying the static stress by the peak dynamic amplification factors.
721
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