- Email: [email protected]
Moving loads on elastic cylindrical shells
Journal of Sound and Vibration (1976) 49(2), 215-220
MOVING
LOADS
ON ELASTIC
CYLINDRICAL
SHELLS
C.-C. HUANG Department of Mechanical Engineering, University of Western Australia, Nedlands, Western Australia 6009, Australia (Received 25 May 1976, and in revised form 24 July 1976)
This paper presents a theoretical analysis of the axially-symmetric, steady-state response of a linearly-elastic, homogeneous, infinitely-long, cylindrical shell, subjected to a ring load traveling at a constant velocity. The Fourier transform method in conjunction with the contour integral has been applied to obtain the steady-state response. A numerical illustration is given. 1. INTRODUCTION This paper presents a theoretical analysis of the axially-symmetric steady-state response of a linearly-elastic, homogeneous, infinitely-long, cylindrical shell subjected to a ring load traveling at a constant velocity. The shell is modelled by the refined theory derived in [l]. Further, the Fourier transform method in conjunction with the contour integral has been , applied to obtain the steady-state response. Loadings of this nature are of interest in several practical applications: for example, in conjunction with the forming operation of shell casing, interference effects in mechanism containing shells, shell turbine design and localised pressure discontinuity. The existing works on the dynamic response of shells based on the classical shell theory have been reported by Mann-Nachbar [2], Bhuta [3], Jones [4], Tang [5], Reismann [6], and Liao and Kessel [7]. Later, Reismann and Padlog [8] applied the refined shell theory (which includes both the shell and rotatory inertia effects) developed in reference [l] and worked out an example for a finite shell subjected to a traveling ring load. It is noted that reference [8] deals with the shell of finite length by use of the method of eigenfunction expansion, whereas herein a shell of infinite length is treated and the results presented are in greater detail. The same mathematical technique has previously been used by Adler and Reismann [9] for a different type of problem-the dynamic response of moving line loads on an elastic strip. In their paper, the intimate, organic relationship which exists between the motion of forced flexural waves and corresponding free flexural wave propagation has been emphasised. Also, it has been shown that the effects of small viscous damping on the resulting forced vibration can be obtained by a simple modification of the undamped case. The present paper is an extension of reference [2] to investigate theoretically the dynamic response of the shell of infinite length acted upon by a traveling load. 2. ANALYSIS
2.1. NON-DIMENSIONALISED GOVERNINGEQUATIONS One obtains from reference [I] the following equations above (a list of notations is given in the Appendix) :
w13,3m,1=
based on the assumptions
mx3~&x1 + v%x1> 215
stated
(1)
216
C.-C. HUANG
where
912= (v)(a/ao,g13 = w/12) wv), 2- (s2)(1 + h2/12), _tzz2 = a2laf -92, =--(S24(a/at>, Y,,= a/at,-spsl = (w/12) a2lag2,T,, = -a/x, q1 = ayap,
933= (w/12)
a2/ar2I,
(2)
9,,= 1,
LJ12 = 0,
$21 = 0,
42, = s2, 4,, = s2,
= 12/r,
932= 0,
9,, = 12s2/r,
Fl=O,
4,,
333=12s2,
F,=s'p,c!@ - VT), r;;=O,
where 6 is the Dirac delta function. For the infinitely long structure, it is assumed
(3)
that
lim {d} = 0. e-rm 2.2.
(4)
CO-ORDINATE TRANSFORMATION
To obtain the steady-state response of the shell subjected to a ring load, one may refer the deformation to a translating co-ordinate system attached to the moving load by means of the transformation q = 5 - 2rr. Upon change of the independent variables, equation (1) becomes
bl3X3M~N3Xl = W13X3~~(?)~3Xl + v33Xl~
(5)
where a 11 = a2/ar12,
al2 = (v)
a/arl,aI3= (P/12) a2ja12,
azl=-(swa/a~), cl22 = az/atfz - (s~)(I + h2/12), a31 = (h2s2/12) pll =
ayaq,
c(32 = -a/av,
v2a2jatf2, I312 = 0,
a33 = (h2~2/12)
a23 =
a/atl,
a/arl - 1,
(6)
pi3= (212~2) a2/ar2,
pz2= ~va2/a+, B23= O, = 12SW a2/a12. pJ1= (12.72~2) a2/ar2,832=O, fi33
821= 09
(7)
2.3. FOURIER TRANSFORM Equation (5) is solved by the Fourier transform method. The Fourier transform for the displacement vector are defined as
{d*} = (27~)-~/~ J’ {d(q)) eJaVdr], --m
{d(q)} = (27~-“~
i {d*(a)} e-‘“‘da. al
transform
and inverse
(8)
(9)
217
MOVING LOADS ON SHELLS
The transformed equations become u* = (p/a)
(aj){(h2/12 - Z20”) (1 + vs2) a2 + v}/p,
W*= (plfi)a2{(h2/12 4x*=(p/d%)(aj) where j = &l
- Z2v2) [(--I -I-u’) + (P/12 - Z2v2)] s2a2 - (1 - n2)}/fi,
[-(v)(h2/12 - Z2u2).s2a2 - (1 - u2)a2]/p,
(10)
and p=(a,a4+a2a2+a,)a2,
(11)
in which a,=(1
-u’)(l
a2=(s2/r2)(1
-s2~2)(z2S2U2-h2s2/12)+(h2/12-z2v2)(1 +Zz2/12)(1 -v2)(Z2s2~2-Zz2~2/12)-(1
-?zY)?, -u*)(l
-s2v2)-
- (Z2s2v2 - h2?/12)(sv)2 + (P/12 - Z2u2)2(1+ h2/12)s4 + (1 - 0”) + + 2(h2/ 12 - 12 0’) (v?), a, = -(s2)(I + h2/12)(1 - u’) + (sv)2.
(12)
2.4. INVERSE TRANSFORM The improper integral in equation (8) can be evaluated by the contour integral in conjunction with Jordan’s Lemma as follows [9] : 2zj 2 res+, m{d*(a)>da = 2nj c res_ 9 -m
I
?
(13)
?>O I
where 2 res+ and 2 res- are the residues evaluated on the positive and negative imaginary axes, respectively. It is noted that the integral does not exist for the real zones of p = 0. This difficulty can be overcome by introducing an energy dissipation mechanism into the governing differential equation. By doing so, the zeroes will move away from the real axes and the integral can then be evaluated. The directions in which small viscous damping causes the real zeroes to move can be determined by imposing a viscous damping term in the governing equations and actually carrying out the calculations, which results in considerably more computational effort. The problem can also be resolved by simple modification of the undamped system in two different ways, as follows. (4 Substituting ai = (zi + 6z,) into the polynomial for the slightly damped system p. = 0 (which corresponds to equation (11) for the damped system) and neglecting terms of order Szf, one obtains the expression of 6z,. By studying the sign of the imaginary part of 6Zi, the directions of the movement of the real root due to small damping can be determined. for (b) u > 21,,,one can determine whether the roots af contribute to the response in r~> 0 or q < 0 by comparing the group and phase velocities. For instance, if the phase velocity is greater than the group velocity (velocity of energy transport) for a particular mode, the root will contribute to the response in the region YZ < 0. 3. NUMERICAL RESULTS From the following numerical data for the shell parameters, h = 0.01666, one has Z = 0.0048 and S = 1.8459.
lc2= 0.86,
v = o-3,
G/E = 318,
218
C.-C. HUANG 5-
4-
i
I
I
I :
I 3": 0 x
-
G
2-
I-
Figure 1. Roots of .4 = 0. C, Classical theory; T, refined theory; ----, roots.
pure imaginary roots; -,
real
Figure 1 shows the real and pure imaginary roots of /J = 0 versus the non-dimensional velocity o for both the classical and refined shell theories. As v increases indefinitely the higher values of z1 increase monotonically for the classical theory, whereas the upper branch of the curve for the refined theory behaves entirely differently from that of the classical theory. This clearly indicates the inadequacy of the classical theory in simulating the behavior of the higher modes. Figure 2 shows typical response curves for both theories for v less than the critical velocity W. For v < G, the roots z1 consist of four pairs of complex conjugate numbers whose locations in Fourier transform space are symmetric about both the real and imaginary axes. Because of this symmetry, the response curves are symmetric about the axis of the ring load. As the velocity u reaches v,,, the deflection surface becomes unbounded. For the case v > v,, the computations carried out for u = O-6 and u = O-8 show that the inclusion of small damping makes the two real roots deviate slightly in the positive imaginary
-15
-10
-5
Figure 2. Deflection profile, v = 0.1. ----,
0
5
Classical theory; -,
IO
15
Timoshenko theory.
219
MOVING LOADS ON SHELLS
Figure 3. Reflection profile. u = 06. ----,
Classical theory; -,
refined theory.
Figure 4. Deflection profile. u = 0.8. ----,
Classical theory; -,
refined theory.
direction in Fourier transform space. The calculated values of the group velocity also show that they are less than their corresponding phase velocities. Thus, the two real zeroes will contribute to the region q < 0. The response curves for v = 0.6 and u = O-8 are presented respectively in Figure 3 and Figure 4. The discrepancies between the two theories are quite obvious in q > 0 because the waves propagating upstream are of short wave lengths (in which discrepancies occur). On the other hand, high correlations exist in q < 0, since waves propagating downstream are of long wave lengths. Mathematically, in q -C0 the response is dominated by the lower modes (in which high correlations exist between the two theories), whereas in q > 0 the response is solely determined by the upper branch of the curves in Figure 1 (which are quite different for the two theories). In the present investigation, attention is restricted to the deflection characteristic of a shell. If one is concerned with the stress characteristics, it is expected that the shear force and bending moment responses obtained from both theories will be quite different as illustrated in reference [8] for shells of finite length. REFERENCES 1. G. HERRMANN and I. MIRSKY1956 Journal of&plied Mechanics 23,563-568. Three-dimensional and shell theory analysis of axially symmetric motions of cylinders.
220
C.-C. HUANG
2. P. MANN-NACHBAR1962 Journal of Aerospace Sciences 29,648-657.
3. 4. 5. 6.
7. 8. 9.
On the role of bending in the dynamic response of thin shells to moving discontinuous loads. P. G. BHUTA 1963 Journal of the Acoustical Society of America 35, 25-30. Transient response of a thin elastic cylindrical shell to a moving shock wave. J. P. JoNEs~~~P. G. BHLJTA1964JournaZofAppZiedMechanics31,105-111. Responseofcylindrical shells to moving loads. S. C. TANG 1965 Journalof the Engineering Mechanics Division, American Society of Civil Engineers, 91,97-122. Dynamic response of a tube under moving pressure. H. REISMANN1965 Development in mechanics. Proceedings of the Eighth Midwestern Mechanics Conference. Pergamon Press, pp. 349-363. Response of a prestressed cylindrical shell to moving pressure load. E. N. K. LIAO and P. G. KESSEL1972 Journal of Applied Mechanics 39, 227-234. Response of pressurised cylindrical shells subjected to moving loads. H. REISMANNand J. PADLOG 1967 Journal of the Franklin Institute 284, 308-319. Forced axisymmetric motions of cylindrical shells. A. A. ADLERand H. R~ISMANN1974 Journal of Applied Mechanics 41,713-718. Moving loads on an elastic plate strip. APPENDIX D E h h
EJt3/[12(1 - v”)] Young’s modulus
: NOTATION ICY shear coefficient v Poisson’s ratio
h/R
LZ xlR
thickness of shell
p
Z2 h2/12 jt/=I ?&, R/d M,, i;l2 ii&x 5 a,,(1 + z/R)sds -r/z s/2 m.%, 1 gz& + z/R) dz
z2 $
specific mass Et 2/[p(l - v2) R2]
angle of rotation of cross-section Fourier transform of &
42 NXX
NxxU
-
v2ME~)
Nee &SO - v2ME6 512 i% j coeedz -ii/z
m j,
Qx fz R
s2 t u
A Rk2
GA)
radial surface traction k/k2 Gh) ii12 s gxz(l + z/R) dz -ii12 mean radius of shell E/[K~ G(l - v2)] time
u/R u Vp[(l - v’)/E]~” B velocity of ring load
a, ti axial and radial displacement components u*, w* Fourier transforms of u and w, respectively w i+,lR n co-ordinate in axial direction .z distance in radial direction in the middle surface z1 roots of p = 0 {d] {u, w,d.%)
MOVING
LOADS
ON ELASTIC
CYLINDRICAL
SHELLS
C.-C. HUANG Department of Mechanical Engineering, University of Western Australia, Nedlands, Western Australia 6009, Australia (Received 25 May 1976, and in revised form 24 July 1976)
This paper presents a theoretical analysis of the axially-symmetric, steady-state response of a linearly-elastic, homogeneous, infinitely-long, cylindrical shell, subjected to a ring load traveling at a constant velocity. The Fourier transform method in conjunction with the contour integral has been applied to obtain the steady-state response. A numerical illustration is given. 1. INTRODUCTION This paper presents a theoretical analysis of the axially-symmetric steady-state response of a linearly-elastic, homogeneous, infinitely-long, cylindrical shell subjected to a ring load traveling at a constant velocity. The shell is modelled by the refined theory derived in [l]. Further, the Fourier transform method in conjunction with the contour integral has been , applied to obtain the steady-state response. Loadings of this nature are of interest in several practical applications: for example, in conjunction with the forming operation of shell casing, interference effects in mechanism containing shells, shell turbine design and localised pressure discontinuity. The existing works on the dynamic response of shells based on the classical shell theory have been reported by Mann-Nachbar [2], Bhuta [3], Jones [4], Tang [5], Reismann [6], and Liao and Kessel [7]. Later, Reismann and Padlog [8] applied the refined shell theory (which includes both the shell and rotatory inertia effects) developed in reference [l] and worked out an example for a finite shell subjected to a traveling ring load. It is noted that reference [8] deals with the shell of finite length by use of the method of eigenfunction expansion, whereas herein a shell of infinite length is treated and the results presented are in greater detail. The same mathematical technique has previously been used by Adler and Reismann [9] for a different type of problem-the dynamic response of moving line loads on an elastic strip. In their paper, the intimate, organic relationship which exists between the motion of forced flexural waves and corresponding free flexural wave propagation has been emphasised. Also, it has been shown that the effects of small viscous damping on the resulting forced vibration can be obtained by a simple modification of the undamped case. The present paper is an extension of reference [2] to investigate theoretically the dynamic response of the shell of infinite length acted upon by a traveling load. 2. ANALYSIS
2.1. NON-DIMENSIONALISED GOVERNINGEQUATIONS One obtains from reference [I] the following equations above (a list of notations is given in the Appendix) :
w13,3m,1=
based on the assumptions
mx3~&x1 + v%x1> 215
stated
(1)
216
C.-C. HUANG
where
912= (v)(a/ao,g13 = w/12) wv), 2- (s2)(1 + h2/12), _tzz2 = a2laf -92, =--(S24(a/at>, Y,,= a/at,-spsl = (w/12) a2lag2,T,, = -a/x, q1 = ayap,
933= (w/12)
a2/ar2I,
(2)
9,,= 1,
LJ12 = 0,
$21 = 0,
42, = s2, 4,, = s2,
= 12/r,
932= 0,
9,, = 12s2/r,
Fl=O,
4,,
333=12s2,
F,=s'p,c!@ - VT), r;;=O,
where 6 is the Dirac delta function. For the infinitely long structure, it is assumed
(3)
that
lim {d} = 0. e-rm 2.2.
(4)
CO-ORDINATE TRANSFORMATION
To obtain the steady-state response of the shell subjected to a ring load, one may refer the deformation to a translating co-ordinate system attached to the moving load by means of the transformation q = 5 - 2rr. Upon change of the independent variables, equation (1) becomes
bl3X3M~N3Xl = W13X3~~(?)~3Xl + v33Xl~
(5)
where a 11 = a2/ar12,
al2 = (v)
a/arl,aI3= (P/12) a2ja12,
azl=-(swa/a~), cl22 = az/atfz - (s~)(I + h2/12), a31 = (h2s2/12) pll =
ayaq,
c(32 = -a/av,
v2a2jatf2, I312 = 0,
a33 = (h2~2/12)
a23 =
a/atl,
a/arl - 1,
(6)
pi3= (212~2) a2/ar2,
pz2= ~va2/a+, B23= O, = 12SW a2/a12. pJ1= (12.72~2) a2/ar2,832=O, fi33
821= 09
(7)
2.3. FOURIER TRANSFORM Equation (5) is solved by the Fourier transform method. The Fourier transform for the displacement vector are defined as
{d*} = (27~)-~/~ J’ {d(q)) eJaVdr], --m
{d(q)} = (27~-“~
i {d*(a)} e-‘“‘da. al
transform
and inverse
(8)
(9)
217
MOVING LOADS ON SHELLS
The transformed equations become u* = (p/a)
(aj){(h2/12 - Z20”) (1 + vs2) a2 + v}/p,
W*= (plfi)a2{(h2/12 4x*=(p/d%)(aj) where j = &l
- Z2v2) [(--I -I-u’) + (P/12 - Z2v2)] s2a2 - (1 - n2)}/fi,
[-(v)(h2/12 - Z2u2).s2a2 - (1 - u2)a2]/p,
(10)
and p=(a,a4+a2a2+a,)a2,
(11)
in which a,=(1
-u’)(l
a2=(s2/r2)(1
-s2~2)(z2S2U2-h2s2/12)+(h2/12-z2v2)(1 +Zz2/12)(1 -v2)(Z2s2~2-Zz2~2/12)-(1
-?zY)?, -u*)(l
-s2v2)-
- (Z2s2v2 - h2?/12)(sv)2 + (P/12 - Z2u2)2(1+ h2/12)s4 + (1 - 0”) + + 2(h2/ 12 - 12 0’) (v?), a, = -(s2)(I + h2/12)(1 - u’) + (sv)2.
(12)
2.4. INVERSE TRANSFORM The improper integral in equation (8) can be evaluated by the contour integral in conjunction with Jordan’s Lemma as follows [9] : 2zj 2 res+, m{d*(a)>da = 2nj c res_ 9 -m
I
?
(13)
?>O I
where 2 res+ and 2 res- are the residues evaluated on the positive and negative imaginary axes, respectively. It is noted that the integral does not exist for the real zones of p = 0. This difficulty can be overcome by introducing an energy dissipation mechanism into the governing differential equation. By doing so, the zeroes will move away from the real axes and the integral can then be evaluated. The directions in which small viscous damping causes the real zeroes to move can be determined by imposing a viscous damping term in the governing equations and actually carrying out the calculations, which results in considerably more computational effort. The problem can also be resolved by simple modification of the undamped system in two different ways, as follows. (4 Substituting ai = (zi + 6z,) into the polynomial for the slightly damped system p. = 0 (which corresponds to equation (11) for the damped system) and neglecting terms of order Szf, one obtains the expression of 6z,. By studying the sign of the imaginary part of 6Zi, the directions of the movement of the real root due to small damping can be determined. for (b) u > 21,,,one can determine whether the roots af contribute to the response in r~> 0 or q < 0 by comparing the group and phase velocities. For instance, if the phase velocity is greater than the group velocity (velocity of energy transport) for a particular mode, the root will contribute to the response in the region YZ < 0. 3. NUMERICAL RESULTS From the following numerical data for the shell parameters, h = 0.01666, one has Z = 0.0048 and S = 1.8459.
lc2= 0.86,
v = o-3,
G/E = 318,
218
C.-C. HUANG 5-
4-
i
I
I
I :
I 3": 0 x
-
G
2-
I-
Figure 1. Roots of .4 = 0. C, Classical theory; T, refined theory; ----, roots.
pure imaginary roots; -,
real
Figure 1 shows the real and pure imaginary roots of /J = 0 versus the non-dimensional velocity o for both the classical and refined shell theories. As v increases indefinitely the higher values of z1 increase monotonically for the classical theory, whereas the upper branch of the curve for the refined theory behaves entirely differently from that of the classical theory. This clearly indicates the inadequacy of the classical theory in simulating the behavior of the higher modes. Figure 2 shows typical response curves for both theories for v less than the critical velocity W. For v < G, the roots z1 consist of four pairs of complex conjugate numbers whose locations in Fourier transform space are symmetric about both the real and imaginary axes. Because of this symmetry, the response curves are symmetric about the axis of the ring load. As the velocity u reaches v,,, the deflection surface becomes unbounded. For the case v > v,, the computations carried out for u = O-6 and u = O-8 show that the inclusion of small damping makes the two real roots deviate slightly in the positive imaginary
-15
-10
-5
Figure 2. Deflection profile, v = 0.1. ----,
0
5
Classical theory; -,
IO
15
Timoshenko theory.
219
MOVING LOADS ON SHELLS
Figure 3. Reflection profile. u = 06. ----,
Classical theory; -,
refined theory.
Figure 4. Deflection profile. u = 0.8. ----,
Classical theory; -,
refined theory.
direction in Fourier transform space. The calculated values of the group velocity also show that they are less than their corresponding phase velocities. Thus, the two real zeroes will contribute to the region q < 0. The response curves for v = 0.6 and u = O-8 are presented respectively in Figure 3 and Figure 4. The discrepancies between the two theories are quite obvious in q > 0 because the waves propagating upstream are of short wave lengths (in which discrepancies occur). On the other hand, high correlations exist in q < 0, since waves propagating downstream are of long wave lengths. Mathematically, in q -C0 the response is dominated by the lower modes (in which high correlations exist between the two theories), whereas in q > 0 the response is solely determined by the upper branch of the curves in Figure 1 (which are quite different for the two theories). In the present investigation, attention is restricted to the deflection characteristic of a shell. If one is concerned with the stress characteristics, it is expected that the shear force and bending moment responses obtained from both theories will be quite different as illustrated in reference [8] for shells of finite length. REFERENCES 1. G. HERRMANN and I. MIRSKY1956 Journal of&plied Mechanics 23,563-568. Three-dimensional and shell theory analysis of axially symmetric motions of cylinders.
220
C.-C. HUANG
2. P. MANN-NACHBAR1962 Journal of Aerospace Sciences 29,648-657.
3. 4. 5. 6.
7. 8. 9.
On the role of bending in the dynamic response of thin shells to moving discontinuous loads. P. G. BHUTA 1963 Journal of the Acoustical Society of America 35, 25-30. Transient response of a thin elastic cylindrical shell to a moving shock wave. J. P. JoNEs~~~P. G. BHLJTA1964JournaZofAppZiedMechanics31,105-111. Responseofcylindrical shells to moving loads. S. C. TANG 1965 Journalof the Engineering Mechanics Division, American Society of Civil Engineers, 91,97-122. Dynamic response of a tube under moving pressure. H. REISMANN1965 Development in mechanics. Proceedings of the Eighth Midwestern Mechanics Conference. Pergamon Press, pp. 349-363. Response of a prestressed cylindrical shell to moving pressure load. E. N. K. LIAO and P. G. KESSEL1972 Journal of Applied Mechanics 39, 227-234. Response of pressurised cylindrical shells subjected to moving loads. H. REISMANNand J. PADLOG 1967 Journal of the Franklin Institute 284, 308-319. Forced axisymmetric motions of cylindrical shells. A. A. ADLERand H. R~ISMANN1974 Journal of Applied Mechanics 41,713-718. Moving loads on an elastic plate strip. APPENDIX D E h h
EJt3/[12(1 - v”)] Young’s modulus
: NOTATION ICY shear coefficient v Poisson’s ratio
h/R
LZ xlR
thickness of shell
p
Z2 h2/12 jt/=I ?&, R/d M,, i;l2 ii&x 5 a,,(1 + z/R)sds -r/z s/2 m.%, 1 gz& + z/R) dz
z2 $
specific mass Et 2/[p(l - v2) R2]
angle of rotation of cross-section Fourier transform of &
42 NXX
NxxU
-
v2ME~)
Nee &SO - v2ME6 512 i% j coeedz -ii/z
m j,
Qx fz R
s2 t u
A Rk2
GA)
radial surface traction k/k2 Gh) ii12 s gxz(l + z/R) dz -ii12 mean radius of shell E/[K~ G(l - v2)] time
u/R u Vp[(l - v’)/E]~” B velocity of ring load
a, ti axial and radial displacement components u*, w* Fourier transforms of u and w, respectively w i+,lR n co-ordinate in axial direction .z distance in radial direction in the middle surface z1 roots of p = 0 {d] {u, w,d.%)