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Torsional wave propagation in an orthotropic magneto-elastic hollow circular cylinder
Torsional Wave Propagation in an Orthotropic Magneto-elastic Hollow Circular Cylinder Abo-el-nour
N. Abd-alla
Department of Mathematics Faculty of Science Sohag 82516, Egypt
Transmitted by John Casti
ABSTRACT The aim of this work is to present numerical data on the velocities of torsional vibrations of a perfectly conducting orthotropic hollow circular cylinder in the presence of an axail magnetic field. The extremes of the geometry are investigated. The values of the characteristic numbers governing the velocities of the first five modes for various geometries of the circular cylinder are numerically computed. The phase velocities of the disturbances for the different wave numbers corresponding to the first five modes are calculated. It is found that torsional wave phase velocities increase with decreasing wall thickness of the hollow circular cylinder for all modes and increase with the higher modes of motion.
1.
INTRODUCTION The effect of electromagnetic
fields on the motion of a deformable solid
has been the topic for a number of investigations in recent years [I-S]. These investigations have many theoretical and practical applications in several fields like geophysics, seismology, acoustics, optics, and plasma physics. Among many important considered studies, the problems of torsional vibration in a pure elastic, isotropic or anisotropic and homogeneous circular cylinder was considered in [J-8]. The corresponding studies when the body is nonhomogeneous isotropic or anisotropic, in the presence of thf magnetic field have been considered analytically in [9, lo]. Moreover, in [II] an attempt has been made to solve the problem of torsional vibrations of a nonhomogeneous magnetostrictive circular cylinder. APPLZED MATHEMATICS AND COMPUTATION 0 Elsevier Science Inc., 1994 655 Avenue of the Americas, New York, NY 10010
63:281-293
(1994)
281 0096-3003/94/$7.00
A. N. ABD-ALLA
282
In this paper, the numericaldata on the velocitiesof torsionalvibrationsof a perfectly conducting orthotropic hollow circular cylinder are given in the presence of an axial magnetic field. The extremes of the geometry are investigated.This means the case when the radius of the hole in the cylinder approacheszero and the case when the wall thicknessapproaches zero. The values of the characteristicnumbers governing the velocitiesof the first five modes for various geometries of the circular cylinder are numericallycorn-‘ puted. The phase velocities of the disturbances for the different wave numbers for the first five modes are also calculated.It is found that torsional wave phase v&&ties increase with decreasing wall thicknessof the hollow circularcylinderfor all modes and increasewith the higher modes of motion. 2.
FUNDAMENTAL
EQUATIONS
We take the Maxwell equationsgoverning the electromagneticfield in the absence of the displacementcurrent, with the notationsof Lorentz-Heaviside units,in the form I dB
1
=-J;
VXE
VxE=--cat
c
(1)
PxB=O;
B=pJ.
We also have Ohm’s law:
J =u
1 au E+---xB
(2)
9 I
where a, B, and E, respectively?represent magnetic field intensity,magnetic induction, and electric field intensityvectors, and J is the current density vector; CG,, C, and c, resp4ztively,denote magnetic permeability, electricalconductivityof the body, and the velocityof light, and u represents the displacementvector in the strainedsolid. The characteristicmatrixof orthotropicbody, provided that the principal elasticaxes coincidewith the axes of coordinates,has the form
[c,B3=
Cl1
Cl2
Cl3
0
0
0
Cl2
%2
%3
0
0
0
Cl3
c23
c33
0
0
0
0
0
0
CM
0
0
0
0
0
0
Q.3
0
0
0
0
0
0
cfgj
Torsional Wave Propagation
283
where cy and p take the values 1,2,3,4.5,6. They may be replaced by 11,22,33,23 or 32,31 or 13,12, or 21, respectively. We have, thus, nine independent constants. The stress-strain relations for an orthotropic body in cylindrical coordinates are given by
Trf = cllerr + cleeee + c13ezz, TM= q2err + C&&.j+ c23ezzr Tzz = q3err + C&&j + c33e=z,
(3)
Tr, = adder,,
The seain-mecha&al
e IT =-
T82
=
C@ez,
Tre
=
C66ere*
displacement relations are dU
eee
>
=--
ldo
au;
t1 - 9
9
(4)
..-,it~+~~,+e~=~~~~,,
e re =-
Eprr 1
do 1 P_
ll,;z.
1 au
1
The stress equations of motion for the cylinder are 1 aT,e
8%
-_t_--
ar
r aT-e -+-&-
CL -+-_ dr
do
+-
aT,=
ax
1 aTee r
t30
1 aTo, r
d0
+-
+-
+
T,,
-
Tee
r
dTrz dz
dt2 ’
d2v 2 + -Tre + Fe = pat2
r
dT== + 1 T dz
d2t1
+Fr=p-
1_
rz
+ F = z
(5)
’
d2~ p
at2
l
A. N. ABD-ALLA
284
Tij, eij denote Cauchy stress and strain components, u = In o-(5) (24,u, tc), p is mass density and F = ( F,, Q, F,) is the body force per unit vohme. If them are no body forces apart from the Lorentz force, we take
F=JxB.
(6)
We readily see tram (l)_(6) that the electromagneticfield interactswith the mechanicalfield The electromagneticfield equationsin vacuum are
(PZ - g)(E*,h*,=0, 1 ah VxE*=-;--'
V x h* =--
1 im*
c
where h*
is the
at *
perturbation in the magnetic field and E* is the electric
field in vacuum. 3. TORSIONAL
DISTURBA
CES IN A PERFECT
CONDUCTOR
Suppose that the z-axis is taken along the axis of the body, which in the present analysisis a hollow circular cylinder.The displacementvector u is also assumed to have its only nonvanishingcomponent o, which is independent of 8. Let the induced magnetic field h be small as compared with the applied primaq~fie!d 9,; we can put
H=a,+h,
(8)
where Ho is acting parallel to z-axis (second and higher powers of h as well as the products of h and u may be neglected). If the body is a perfect conductor for electricity (i.e., a * 0~) (2) gives dU
E=
-atXB=
dV
[
-g,o,o -peH
where we have used (l>, and (2.9) with H = 1f&l.
1
9
(9)
Torsional Wave Propagation
Eliminating
E from (11, and (9) and using cl>,, we get
1
do
O,H--,o
h =
.
[
We can easily see thal (3) is satisfied. Equations (11,) ( 114 together with (10) give
=
JxB=JxPcIPo
[o, -H2$0].
(11)
Eliminating eij from (3) and (4) and substituting the resulting components of stress in the equations of motion (5) and using (6) and (111, we get
azii = P at”
(12)
9
which can be rewritten in the form 1 do + z - -r dr dr2 r2
d”u
1
1 a”v Z---P
d2V
+ (a? - a;)--_
p2 at2’
az2
where
+-,
c55 5x
2 =a2
H2 , C66
p2
=
E!?!_
P
In view of (9) and (lo), we take
E* = (E*,O,O)
and
h* = (O,h*,O).
(13)
A. N. ABD-ALLA
286 Equations(7) then mduce to
(QL - ;$)(E*.h*)
= 0,
and c aE*
ah”
-=--
at
az
r
(15) l
For harmonic torsional vibrations,(13) and (14) may be satisfiedby taking E* inthefom
v,h*,and
Substituting(16) into (13) and (14) yield dV1 -++-
drg
pw” + (H’
1 dbr,
r dr
- ~49~
1 -0
c66
r- )
#I=0
(171
and
I
d2
--I-
dr”
Id
---+s2
c#&=o,
k = 2,3,
(18)
1
where 9 denotes 27r/h (A is the wave length) and o denotes 29r times frequencyof the wave and .$ = “2
4.
-
$_
(19)
SOLUTION OF THE PROBLEM
Suppose that the given body is a hollow circular cylinder with r = a and r= E as inner and outer boundaries. The solution of (17) may be taken as
41 = A],( nr) + BY,(nr),
(20)
Torsional Wave Propagation
287
where we have set
n: =
n2 =nf+n2,,
lo2
--
P2
ni = aiq2,
atq2,
(21)
A and B are arbitrary constants and Jr and Y, are the Bessel functions of the first and second kind, respectively. The corresponding displacement is
where
o = [ A],( nr) + BY,( nr)] exp[i(qz
+ ot)].
P)
If the boundaries r = a and r = b of the cylinder which separate the body from the free-space are stress free, then the required boundary conditions are
T,, + Al,, = M;
and
on r =a
r=b,
(9
where M,, and M$ are Maxwell stresses in the body and in vacuum, respectively. It is easily seen that M,, = M,*, = 0 [l]. Then (23) together with (3) yields e,, = 0, which, with the help of (4) and (14), gives
= 0
and
for r = a
r=b.
Substituting (20) into (24) and using the recurrence functions, we get
(24)
relations for Bessel
AD,, + BD,, = 0
(=I
AD,, + BD,, = 0, where i-~-i:have set Dir = 5JCW
- VW,
42
=
W5)
-
D,
=
h%,(W
-
2y,w (26)
D2,
withh
=
hSJo(W
= a/b, 5 = nb.
2]dhS),
-
2Y,(h5)
A. N. ABD-ALLA
288
The condition for a nontrivial solution of (25) is that the determinant of the coefficients of these integration constants must vanish, which leads to the following freq-lency equation:
(27) The roots of this equation give the vahresof the characteristicnumbers e for the torsionaloscillationsof an orthotropichollow circular cylinder. Knowkdge of the c vahressatistjping (2’0 allows computationof the wave v&cities for Werent modes of motion since
k = %ru/A is the wave number. Also, the nondimensionalphase veloci~ c* = q//3 is given by
where
(2 9) For isotropicbodies, es5 =
=j~a.ndar
= 1. Therefore, (28) reduces
to
where p is the modulusof rigidity. In the absence of magnetic field H = 0. So, a, = 0 and we get an expressionfor the phase velocitywhich agrees with the result given in 15,pp. 7801. For a solid cylinder(i.e., the inner diameteris zero, so h = O), the solution of (15) which is consistent with the necessity that 4I is finite on the axis of the cylinder is
TorsionalWave Propagation
289 TABLE
VALUES OF THE CHARACTERISTIC
1
NUMBER
5 FOR DIFFERENT
VALUES OF
h
Mode 1
h 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2
5.142 5.221 5.470 5.966 6.814 8.227 10.720 15.855 31.482
8.457 8.804 9.600 10.894 12.856 15.904 21.071 31.490 62.865
3
4
11.739 12.494 13.906 15.999 19.046 23.694 31.501 47.174 94.270
15.044 16.268 18.290 21.165 25.281 31.515 41.952 62.869 125.680
5 18.383 20.094 22.713 26.357 31.545 49.349 52.411 78.570 157.093
The corresponding displacement is u =A](nr)exp[i(qz
+ ot)].
If the boundaries are stress free, we get, as in the previous case,
n 4,(r) 1 z [Yq=O
for r = b.
Using (31) and (32), we tibiain
Id 5) = m 5) - 21,w = 0,
w
which agrees with what is well known for solid cylinder in torsion [S]. Values of the roots of this case are given in Table 1 when h = 0. 5.
THE
NUMERICAL
RESULTS
The roots of (27) give the values of the characteristic numbers 5 governing the velocity of the torsional vibrations of a perfectly conducting orthotropic elastic hollow circular cylinder in the presence of an axial magnetic field. These roots, which correspond to various modes, are numerically calculated by a “method of halving the interval, ” also known as the bisection method or Bolzano method [l2]. The first five modes of 5 are tabulated in Table 1 and
A. N. ABD-ALLA
0
Ql
a2
a3
a4
a7
OS
0.8
0.9
a=aP FIG. L
1
2
Wues
of the chwucteristic numbers & for different values of h.
3
4
6 5 Wave number
7
FIG. 2. Phase velocity c * for different values of wavenumber numbers 5 take the values corresponding to h = 0.1.
8
9
10
k when the characteristic
291
Torsional Wave Propagation TABLE 2 l’HASE VELOCITY C* FOR DIFFERENT CHARACTERISTIC
VALUES OF WAVENUMBER
k
NUMBERS 6 TAKE THE VALUES CORRESPONDING
WHEN THE TO
h = 0.1
Mode k 1 2 3 4 5 6 i 8 9 10
1
2
3
2.452 1.855
3.054 2.258 1.920 1.iZi
3.551 2.596
3.989 2.897
2.18’7 1.950 1.i93 1.861 1.595 1.528 1.474 1.429
2.426 2.152 1.9969 1I337 1.i3i 1.658 1.1549 1.540
1.69i i .468 1.377 1.314 1.266 1.229 1.200 l.li6
1.609 1.509 1.441 1.388 1.345 1.309
4
5
4.388 3.172 2.645 2.338 2.132 1.983 1.969 1.X9 l.iO6 1.645
TABLE 3 PHASE VELOCITY C* FOR DIFFERENT CHARACTERISTIC
VALUES OF \VAVENUMBER
k
NUMBERS 5 TAKE THE VALUES CORRESPONDING
WHEN THE TO 12 = 0.9
Mode
1
k 1
2 3 4 5 6 i 8 9 10
2
5.688
7.983
4Oi5
5.683 4.672
3.3’71 2.956 2.6ii 2.473 2.316 2.192 2.090 2.004
4.0'72 3.666 3.368 3.138 2.954 2.802
2.6i5
3
4
9.i54 6.928 5.683 4.943 4.441 4.072 3.786 3.557 3.368 3.209
11.249 ‘7.982 6.539 5.682 5.099 4.671 4.339 4.072 3.851 3.666
5 12.568 8.912 7.296 6.336 5.682 5.201 4.828 4.528 4.281 4.072
shown in Figure 1 for different values of h. Moreover, the numerical results of the phase velocity c* for different wave numbers corresponding to the first five modes, taking the characteristic numbers e at h = 0.1 and h = 0.9 are given in Tables 2 and 3 and are shown in Figures 2 and 3. Computational work is carried out for the following data [3]: p = 4.7 g/cm”, cS = 0.66 X 1012 dyn/cm2, ce6 = 0.67 X 1012 dyn/cm2, and H = lo6 oerested. The tabular values presented are calculated to the nearest 0.00001 units, to which they are believed accurate.
A. N. ABD-ALLA
1
2
3
4
5
6
7
8
9
10
WalRimamlYer FIG. 3. Phase wkscity c* for different wahws of wawnumber k when the characteristic the ldues correspon~g to h = 0.9.
This shows that torsionalwave phase velocitiesincrease with decreasing w-all&chess of the hollow cylinder for all modes and increase with the higher modes of motion.
1 A. K. Abd-alla and G. A. Maugin, Nonlinear magnetoacousticequations, J. Acoustic !bc. America 82:1746-1752(1987). 2 A. C. Eringen and G. A. Maugin, Electrodynamics of Continua I, II, Sptinger-Verlag, New York, 199G. 3 V. Z. Parton and B. A. Kudryavtscv,Electrrnnagnetoehsticity, Piezzelectrics and Electricity Conducting Solids, Gordon and Breach, New York, 198% 4 A. C. Eringen and E. S, Sahubi, Ehstodynamks I, II, Academic Press, New York, l974-19’7.5. 5 A. I. Beltzer, &x&ics of Solids, Springer-Verlag, New York, 1988. 6 E. S. Suhubi, Small torsionaloscillationsof a circularcylinder with finite e!etirc conductivityin a constantaxial magnetic field, i@t. J. Engrg. Sti. 2:44l-455 (19651.
Torsional Wave Propagation 7 8
293
S. Kaliski, On a conception of basic solutions for orthotropic elastic and anelastic bodies, Arch. Mcch. 1(11):45-60 (1959). S. Kaliski, Solution of the equations of motion of an anisotropic body in a magnetic field assuming finite electric conductivity, Arch. Mech. 3(12):%X3
( 1960). 9
D. S. Chandrasekharaiak, On magneto-elastic torsional vibrations of nonhomogeneous circular cylinder and cylindrical shell, Math. Student 42(2):144-148 (1974). 10 D. S. Chandrasekh,araiak, Magneto-elastic torsional vibrations of an orthotropic cylinder and cylindrical shell, Indian J. Phys. 46:197-202 (1972). 11 A. N. Abd-alla and L. Debnath, The torsional vibrations of nonhomogeneous magnetostrictive circular cylinder, submitted for publication. 12 C. F. Ceralg and P. 0. Wheatiey, Applied Numerical Anahpis. 4th ed., AddisonWesley, New York, 1989.
N. Abd-alla
Department of Mathematics Faculty of Science Sohag 82516, Egypt
Transmitted by John Casti
ABSTRACT The aim of this work is to present numerical data on the velocities of torsional vibrations of a perfectly conducting orthotropic hollow circular cylinder in the presence of an axail magnetic field. The extremes of the geometry are investigated. The values of the characteristic numbers governing the velocities of the first five modes for various geometries of the circular cylinder are numerically computed. The phase velocities of the disturbances for the different wave numbers corresponding to the first five modes are calculated. It is found that torsional wave phase velocities increase with decreasing wall thickness of the hollow circular cylinder for all modes and increase with the higher modes of motion.
1.
INTRODUCTION The effect of electromagnetic
fields on the motion of a deformable solid
has been the topic for a number of investigations in recent years [I-S]. These investigations have many theoretical and practical applications in several fields like geophysics, seismology, acoustics, optics, and plasma physics. Among many important considered studies, the problems of torsional vibration in a pure elastic, isotropic or anisotropic and homogeneous circular cylinder was considered in [J-8]. The corresponding studies when the body is nonhomogeneous isotropic or anisotropic, in the presence of thf magnetic field have been considered analytically in [9, lo]. Moreover, in [II] an attempt has been made to solve the problem of torsional vibrations of a nonhomogeneous magnetostrictive circular cylinder. APPLZED MATHEMATICS AND COMPUTATION 0 Elsevier Science Inc., 1994 655 Avenue of the Americas, New York, NY 10010
63:281-293
(1994)
281 0096-3003/94/$7.00
A. N. ABD-ALLA
282
In this paper, the numericaldata on the velocitiesof torsionalvibrationsof a perfectly conducting orthotropic hollow circular cylinder are given in the presence of an axial magnetic field. The extremes of the geometry are investigated.This means the case when the radius of the hole in the cylinder approacheszero and the case when the wall thicknessapproaches zero. The values of the characteristicnumbers governing the velocitiesof the first five modes for various geometries of the circular cylinder are numericallycorn-‘ puted. The phase velocities of the disturbances for the different wave numbers for the first five modes are also calculated.It is found that torsional wave phase v&&ties increase with decreasing wall thicknessof the hollow circularcylinderfor all modes and increasewith the higher modes of motion. 2.
FUNDAMENTAL
EQUATIONS
We take the Maxwell equationsgoverning the electromagneticfield in the absence of the displacementcurrent, with the notationsof Lorentz-Heaviside units,in the form I dB
1
=-J;
VXE
VxE=--cat
c
(1)
PxB=O;
B=pJ.
We also have Ohm’s law:
J =u
1 au E+---xB
(2)
9 I
where a, B, and E, respectively?represent magnetic field intensity,magnetic induction, and electric field intensityvectors, and J is the current density vector; CG,, C, and c, resp4ztively,denote magnetic permeability, electricalconductivityof the body, and the velocityof light, and u represents the displacementvector in the strainedsolid. The characteristicmatrixof orthotropicbody, provided that the principal elasticaxes coincidewith the axes of coordinates,has the form
[c,B3=
Cl1
Cl2
Cl3
0
0
0
Cl2
%2
%3
0
0
0
Cl3
c23
c33
0
0
0
0
0
0
CM
0
0
0
0
0
0
Q.3
0
0
0
0
0
0
cfgj
Torsional Wave Propagation
283
where cy and p take the values 1,2,3,4.5,6. They may be replaced by 11,22,33,23 or 32,31 or 13,12, or 21, respectively. We have, thus, nine independent constants. The stress-strain relations for an orthotropic body in cylindrical coordinates are given by
Trf = cllerr + cleeee + c13ezz, TM= q2err + C&&.j+ c23ezzr Tzz = q3err + C&&j + c33e=z,
(3)
Tr, = adder,,
The seain-mecha&al
e IT =-
T82
=
C@ez,
Tre
=
C66ere*
displacement relations are dU
eee
>
=--
ldo
au;
t1 - 9
9
(4)
..-,it~+~~,+e~=~~~~,,
e re =-
Eprr 1
do 1 P_
ll,;z.
1 au
1
The stress equations of motion for the cylinder are 1 aT,e
8%
-_t_--
ar
r aT-e -+-&-
CL -+-_ dr
do
+-
aT,=
ax
1 aTee r
t30
1 aTo, r
d0
+-
+-
+
T,,
-
Tee
r
dTrz dz
dt2 ’
d2v 2 + -Tre + Fe = pat2
r
dT== + 1 T dz
d2t1
+Fr=p-
1_
rz
+ F = z
(5)
’
d2~ p
at2
l
A. N. ABD-ALLA
284
Tij, eij denote Cauchy stress and strain components, u = In o-(5) (24,u, tc), p is mass density and F = ( F,, Q, F,) is the body force per unit vohme. If them are no body forces apart from the Lorentz force, we take
F=JxB.
(6)
We readily see tram (l)_(6) that the electromagneticfield interactswith the mechanicalfield The electromagneticfield equationsin vacuum are
(PZ - g)(E*,h*,=0, 1 ah VxE*=-;--'
V x h* =--
1 im*
c
where h*
is the
at *
perturbation in the magnetic field and E* is the electric
field in vacuum. 3. TORSIONAL
DISTURBA
CES IN A PERFECT
CONDUCTOR
Suppose that the z-axis is taken along the axis of the body, which in the present analysisis a hollow circular cylinder.The displacementvector u is also assumed to have its only nonvanishingcomponent o, which is independent of 8. Let the induced magnetic field h be small as compared with the applied primaq~fie!d 9,; we can put
H=a,+h,
(8)
where Ho is acting parallel to z-axis (second and higher powers of h as well as the products of h and u may be neglected). If the body is a perfect conductor for electricity (i.e., a * 0~) (2) gives dU
E=
-atXB=
dV
[
-g,o,o -peH
where we have used (l>, and (2.9) with H = 1f&l.
1
9
(9)
Torsional Wave Propagation
Eliminating
E from (11, and (9) and using cl>,, we get
1
do
O,H--,o
h =
.
[
We can easily see thal (3) is satisfied. Equations (11,) ( 114 together with (10) give
=
JxB=JxPcIPo
[o, -H2$0].
(11)
Eliminating eij from (3) and (4) and substituting the resulting components of stress in the equations of motion (5) and using (6) and (111, we get
azii = P at”
(12)
9
which can be rewritten in the form 1 do + z - -r dr dr2 r2
d”u
1
1 a”v Z---P
d2V
+ (a? - a;)--_
p2 at2’
az2
where
+-,
c55 5x
2 =a2
H2 , C66
p2
=
E!?!_
P
In view of (9) and (lo), we take
E* = (E*,O,O)
and
h* = (O,h*,O).
(13)
A. N. ABD-ALLA
286 Equations(7) then mduce to
(QL - ;$)(E*.h*)
= 0,
and c aE*
ah”
-=--
at
az
r
(15) l
For harmonic torsional vibrations,(13) and (14) may be satisfiedby taking E* inthefom
v,h*,and
Substituting(16) into (13) and (14) yield dV1 -++-
drg
pw” + (H’
1 dbr,
r dr
- ~49~
1 -0
c66
r- )
#I=0
(171
and
I
d2
--I-
dr”
Id
---+s2
c#&=o,
k = 2,3,
(18)
1
where 9 denotes 27r/h (A is the wave length) and o denotes 29r times frequencyof the wave and .$ = “2
4.
-
$_
(19)
SOLUTION OF THE PROBLEM
Suppose that the given body is a hollow circular cylinder with r = a and r= E as inner and outer boundaries. The solution of (17) may be taken as
41 = A],( nr) + BY,(nr),
(20)
Torsional Wave Propagation
287
where we have set
n: =
n2 =nf+n2,,
lo2
--
P2
ni = aiq2,
atq2,
(21)
A and B are arbitrary constants and Jr and Y, are the Bessel functions of the first and second kind, respectively. The corresponding displacement is
where
o = [ A],( nr) + BY,( nr)] exp[i(qz
+ ot)].
P)
If the boundaries r = a and r = b of the cylinder which separate the body from the free-space are stress free, then the required boundary conditions are
T,, + Al,, = M;
and
on r =a
r=b,
(9
where M,, and M$ are Maxwell stresses in the body and in vacuum, respectively. It is easily seen that M,, = M,*, = 0 [l]. Then (23) together with (3) yields e,, = 0, which, with the help of (4) and (14), gives
= 0
and
for r = a
r=b.
Substituting (20) into (24) and using the recurrence functions, we get
(24)
relations for Bessel
AD,, + BD,, = 0
(=I
AD,, + BD,, = 0, where i-~-i:have set Dir = 5JCW
- VW,
42
=
W5)
-
D,
=
h%,(W
-
2y,w (26)
D2,
withh
=
hSJo(W
= a/b, 5 = nb.
2]dhS),
-
2Y,(h5)
A. N. ABD-ALLA
288
The condition for a nontrivial solution of (25) is that the determinant of the coefficients of these integration constants must vanish, which leads to the following freq-lency equation:
(27) The roots of this equation give the vahresof the characteristicnumbers e for the torsionaloscillationsof an orthotropichollow circular cylinder. Knowkdge of the c vahressatistjping (2’0 allows computationof the wave v&cities for Werent modes of motion since
k = %ru/A is the wave number. Also, the nondimensionalphase veloci~ c* = q//3 is given by
where
(2 9) For isotropicbodies, es5 =
=j~a.ndar
= 1. Therefore, (28) reduces
to
where p is the modulusof rigidity. In the absence of magnetic field H = 0. So, a, = 0 and we get an expressionfor the phase velocitywhich agrees with the result given in 15,pp. 7801. For a solid cylinder(i.e., the inner diameteris zero, so h = O), the solution of (15) which is consistent with the necessity that 4I is finite on the axis of the cylinder is
TorsionalWave Propagation
289 TABLE
VALUES OF THE CHARACTERISTIC
1
NUMBER
5 FOR DIFFERENT
VALUES OF
h
Mode 1
h 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2
5.142 5.221 5.470 5.966 6.814 8.227 10.720 15.855 31.482
8.457 8.804 9.600 10.894 12.856 15.904 21.071 31.490 62.865
3
4
11.739 12.494 13.906 15.999 19.046 23.694 31.501 47.174 94.270
15.044 16.268 18.290 21.165 25.281 31.515 41.952 62.869 125.680
5 18.383 20.094 22.713 26.357 31.545 49.349 52.411 78.570 157.093
The corresponding displacement is u =A](nr)exp[i(qz
+ ot)].
If the boundaries are stress free, we get, as in the previous case,
n 4,(r) 1 z [Yq=O
for r = b.
Using (31) and (32), we tibiain
Id 5) = m 5) - 21,w = 0,
w
which agrees with what is well known for solid cylinder in torsion [S]. Values of the roots of this case are given in Table 1 when h = 0. 5.
THE
NUMERICAL
RESULTS
The roots of (27) give the values of the characteristic numbers 5 governing the velocity of the torsional vibrations of a perfectly conducting orthotropic elastic hollow circular cylinder in the presence of an axial magnetic field. These roots, which correspond to various modes, are numerically calculated by a “method of halving the interval, ” also known as the bisection method or Bolzano method [l2]. The first five modes of 5 are tabulated in Table 1 and
A. N. ABD-ALLA
0
Ql
a2
a3
a4
a7
OS
0.8
0.9
a=aP FIG. L
1
2
Wues
of the chwucteristic numbers & for different values of h.
3
4
6 5 Wave number
7
FIG. 2. Phase velocity c * for different values of wavenumber numbers 5 take the values corresponding to h = 0.1.
8
9
10
k when the characteristic
291
Torsional Wave Propagation TABLE 2 l’HASE VELOCITY C* FOR DIFFERENT CHARACTERISTIC
VALUES OF WAVENUMBER
k
NUMBERS 6 TAKE THE VALUES CORRESPONDING
WHEN THE TO
h = 0.1
Mode k 1 2 3 4 5 6 i 8 9 10
1
2
3
2.452 1.855
3.054 2.258 1.920 1.iZi
3.551 2.596
3.989 2.897
2.18’7 1.950 1.i93 1.861 1.595 1.528 1.474 1.429
2.426 2.152 1.9969 1I337 1.i3i 1.658 1.1549 1.540
1.69i i .468 1.377 1.314 1.266 1.229 1.200 l.li6
1.609 1.509 1.441 1.388 1.345 1.309
4
5
4.388 3.172 2.645 2.338 2.132 1.983 1.969 1.X9 l.iO6 1.645
TABLE 3 PHASE VELOCITY C* FOR DIFFERENT CHARACTERISTIC
VALUES OF \VAVENUMBER
k
NUMBERS 5 TAKE THE VALUES CORRESPONDING
WHEN THE TO 12 = 0.9
Mode
1
k 1
2 3 4 5 6 i 8 9 10
2
5.688
7.983
4Oi5
5.683 4.672
3.3’71 2.956 2.6ii 2.473 2.316 2.192 2.090 2.004
4.0'72 3.666 3.368 3.138 2.954 2.802
2.6i5
3
4
9.i54 6.928 5.683 4.943 4.441 4.072 3.786 3.557 3.368 3.209
11.249 ‘7.982 6.539 5.682 5.099 4.671 4.339 4.072 3.851 3.666
5 12.568 8.912 7.296 6.336 5.682 5.201 4.828 4.528 4.281 4.072
shown in Figure 1 for different values of h. Moreover, the numerical results of the phase velocity c* for different wave numbers corresponding to the first five modes, taking the characteristic numbers e at h = 0.1 and h = 0.9 are given in Tables 2 and 3 and are shown in Figures 2 and 3. Computational work is carried out for the following data [3]: p = 4.7 g/cm”, cS = 0.66 X 1012 dyn/cm2, ce6 = 0.67 X 1012 dyn/cm2, and H = lo6 oerested. The tabular values presented are calculated to the nearest 0.00001 units, to which they are believed accurate.
A. N. ABD-ALLA
1
2
3
4
5
6
7
8
9
10
WalRimamlYer FIG. 3. Phase wkscity c* for different wahws of wawnumber k when the characteristic the ldues correspon~g to h = 0.9.
This shows that torsionalwave phase velocitiesincrease with decreasing w-all&chess of the hollow cylinder for all modes and increase with the higher modes of motion.
1 A. K. Abd-alla and G. A. Maugin, Nonlinear magnetoacousticequations, J. Acoustic !bc. America 82:1746-1752(1987). 2 A. C. Eringen and G. A. Maugin, Electrodynamics of Continua I, II, Sptinger-Verlag, New York, 199G. 3 V. Z. Parton and B. A. Kudryavtscv,Electrrnnagnetoehsticity, Piezzelectrics and Electricity Conducting Solids, Gordon and Breach, New York, 198% 4 A. C. Eringen and E. S, Sahubi, Ehstodynamks I, II, Academic Press, New York, l974-19’7.5. 5 A. I. Beltzer, &x&ics of Solids, Springer-Verlag, New York, 1988. 6 E. S. Suhubi, Small torsionaloscillationsof a circularcylinder with finite e!etirc conductivityin a constantaxial magnetic field, i@t. J. Engrg. Sti. 2:44l-455 (19651.
Torsional Wave Propagation 7 8
293
S. Kaliski, On a conception of basic solutions for orthotropic elastic and anelastic bodies, Arch. Mcch. 1(11):45-60 (1959). S. Kaliski, Solution of the equations of motion of an anisotropic body in a magnetic field assuming finite electric conductivity, Arch. Mech. 3(12):%X3
( 1960). 9
D. S. Chandrasekharaiak, On magneto-elastic torsional vibrations of nonhomogeneous circular cylinder and cylindrical shell, Math. Student 42(2):144-148 (1974). 10 D. S. Chandrasekh,araiak, Magneto-elastic torsional vibrations of an orthotropic cylinder and cylindrical shell, Indian J. Phys. 46:197-202 (1972). 11 A. N. Abd-alla and L. Debnath, The torsional vibrations of nonhomogeneous magnetostrictive circular cylinder, submitted for publication. 12 C. F. Ceralg and P. 0. Wheatiey, Applied Numerical Anahpis. 4th ed., AddisonWesley, New York, 1989.