- Email: [email protected]
Transient response of an elastic solid to a moving torsional load in a cylindrical bore an approximate solution
OOZO-7225184 $3.W+.W 0 1984 Pergamon Press Ltd.
In!. 1.Eqng Sci. Vol. 22,No. 3,pp.211-284. 1984 Printed inGreat Brilain.
TRANSIENT RESPONSE OF AN ELASTIC SOLID TO A MOVING TORSIONAL LOAD IN A CYLINDRICAL BORE AN APPROXIMATE
Department
of Mechanical
SOLUTION
KAZUMI WATANABE Engineering, Technical College, Yamagata 992, Japan
(Communicated
Yamagata
University,
Yonezawa,
by I. N. SNEDDON)
Abstract-This paper is an attempt to apply Cagniard’s technique to a transient problem of a dispersive medium. The transient response of an elastic solid to a moving torsional ring load in a cylindrical bore is considered. By making an approximation to the modified Bessel function, Cagniard’s technique is applied. An approximate solution, which has explicit expressions for the wave front geometry and has a more suitable form for numerical computations, is obtained. The applicability of the solution is also discussed.
INTRODUCTION IT
IS WELL-KNOWN
that the most powerful inversion method is Cagniard’s[ I] for the transient problem for a non-dispersive elastic medium. However, there seem to have been nd attempts to apply the technique to that of the dispersive elastic medium (including the geometrical dispersion). This paper presents an attempt to do so. A transient problem of a bored elastic solid, in which the dispersion of waves takes place, is considered and an approximate solution is obtained. The problem of the response of an elastic solid to a moving load in a cylindrical bore is one of the most important problems in engineering. The steady-state response of the bored elastic solid to normal, tangental and torsional ring loads was discussed by Parnes[2,3]. But, no transient problems of the elastic solid, in which the dispersion takes place due to the presence of the cylindrical bore, were considered, so far. The present paper considers the transient response of the solid to a moving torsional ring load. The load is suddenly applied to the cylindrical surface and then moves along the bore with uniform velocity. The Laplace and Fourier integral technique is used and an approximation in the transformed space is made for the modified Bessel function K,. Then, Cagniard’s technique is applied and the Laplace inversion is performed with the aid of simple inversion formulas, but by inspection. The applicability of the approximate solution obtained is discussed in some details. FORMULATION
Let us consider a cylindrical bore of radius a in an elastic solid. A cylindrical coordinate system (r, 8, z) is taken so that z-axis lies on the axis of the cylindrical bore. The cylindrical surface of the bore is I = a (see Fig. 1). A torsional ring load of magnitude q. is suddenly applied on a circle at z = 0 and then moves along the bore with uniform velocity c. Boundary conditions at the cylindrical surface are c,fJ
=
4”tq.z - ct)H(t) .
urr = urz =
’
0
r=a
(1)
where 6( ) and H( ) are Dirac’s delta function and Heaviside’s unit step function, respectively. The application of the load causes an axially symmetric torsional deformation. The nonvanishing component of the displacement is U,, only. Equation of motion in terms of the 271
K. WATANABE
278
Flp I Geometry
displacement
of the problem.
is
and constitutive
equations
are
(3) where C, is the velocity of M-wave and p is the torsional quiescent condition is employed. That is uH
2Lo. at
t=o
’
Consequently, the title problem is reduced to determine and (4). Applying Laplace transform,
f*(s) =
foxf(t) exp (-
f(t) = (2ni)- ’
rigidity. As an initial condition,
.
a
(4)
U, under the conditions of eqns (l), (3)
st) dt
1Rr f*(s) exp (st) ds
(5.a)
(5.b)
and Fourier transform,
(6.4 (6.b) to eqns (l)-(3), we get fi*= ”
_$. CL
=* I=_-.__;40 U r” s - ic[
1
.-
Km-)
P(s-ic5) Kdpa) K-(P) K:(@I)
(7)
(8)
Transient response of an elastic
solid
219
(9) where K,( ), n = 1,2 is the modified Bessel function p = ([‘+
of the second kind and
s~/c:)“~; Re (/3) IO
(10)
where Re denotes the real part in the bracket. AN APPROXIMATE
INVERSION
The direct application of Cagniard’s technique is impossible because of the dispersive nature of the problem. It is very troublesome to obtain the exact inversions of eqns (7)-(9). The standard inversion integrals of the Laplace and Fourier transforms gives a form of the double singular integrals and then its solution has no explicit expressions for wave front. So, we shall present here an approximate inversion procedure to make Cagniard’s technique applicable. As an example, its procedure is developed only for the displacement of eqn (7). Now, let us introduce an approximate formula[4] of the Bessel function,
4)
(n,
E e-z i
K,,(i)-
(22)”
n, -0
The last fraction in eqn (7) is approximated
-\I(~)e~z{l+$!$+~+O(z~4))
(11)
as
(12) where D(x) = 128x2 + 240x + 105. Substituting
eqn (12) into eqn (7), we can get an approximate
=* U,, _ _f
5 d
. )
1 PCS
Its formal Fourier inversion
-
k3
(13) form of fiz,
emp(r-uJ 1 _ 240(13a) + 15 I 48(~a) a Wa) r I mm)
15 Wa)
E
2
.
(14)
01 r
yields
(15)
where
(16) (17)
17 =;
I I
I;=!
exp
- y (Pa + ini)} ( -I(l-i~n)(u+p)(y+q)dn x
exp
1
(18)
- F (Pa + ini))
s I -xcr(l-iv7))(y+p)(ytq)
dv
(19)
280
K. WATANABE
and where y = asculc,Y,(Y=(n’+l)“‘;
~=$a,
p=p-I,
Re(cu)rO
<==z/u
(20)
(21)
Y= c/c,
(22)
A factorization of eqn (13), D(x) = 128(x + p)(x + 4)
(23)
where (24) is introduced in eqns (16)-(19). As a next step, we convert the real axis integrals of eqns (16)-(19) to the integrals along the Cagniard’s contour with the variable transform, n --f u,
where R =
(p= + p)“*
(27)
The integrand in each of these equations has branch points at n = 2 i, and a simple pole at n = - i/v. Two branch cuts are introduced along the imaginary axis in the complex q-plane (see Fig. 2). In this complex plane, the denominator D(y) has no zeros because of We(a) 2 0. The pole lies always on the lower cut as the torsional load moves subsonically, v < 1. The Cagniard’s contour does not cross the cut and then eqns (16)-(19) are converted to exp (- UK/C,) dv+ cr+(l - ivq+) dud’
xexp ( a+(l-
fR
asuic,) by+)
(28) 16y++ 7
’ (Y++ p){y++ 4)
x
I exp( - mu/c,) (- nsu/c,J iV~+)(y++ p)(y++ I a+(1- exp R (I-
h+)(Y+
.!i!&,
+ p)(y+ + 9)
z-a
=fRe
R
.gdir du
du
d.‘l’du q) du
(2%
(30)
(31)
where
a+= {(v+)‘+ l}“‘,
y+ = am+/c,.
(32)
Then, the formal Laplace inversions of these equations are given by Z
I, = 28e
I2 = 2Re
1 Z-’ lexp(-asu/c,) R cu’(livYl+) I s I a+(l T*ivn+)Z-I{ s(s +lii;ch
adze 1 exp (- asuic,)} x $$- du
(33)
(34)
Transient response of an elastic solid
281
ReVi’) .
I
I
\
\ \ \
-
I
\
\
\
\
Fig. 2. Capniard’s
contour
13= 2Re
(35)
I4 = 2Re
where
7’ = (aa+/c,)-‘. The simple inversion
(37)
formulas
2-l lexp(-as)
=H(t-a)
(38)
LYE-’ exp(-us)}=H(t-n)~~+~e-b(f~~“‘--~e-~(’-ai) t c) t s(s + b)(s
(39)
Is
1
1
1
-~e-“(f-“‘t~e~‘(f-U’) Lf-’ I(s t h)(st c) eXp(--as)}=H(t-a){
(40)
are used. Thus, we get
I, = ~H(T - R) Re
1
(~‘(1- ivv+)
, !!!!!du du
(41)
I2 = ~H(T - R) Re (42) j,=-
2 -H(T-R)Re. P-4
T
1 e-P+trflo+ _ e -qf+-uYrr$$du R a+(1 - ilq”) I >
I
(43)
282
I
1, = ~H(T - R) We
K.WATA~ABE
_‘+_-
1
7
R cu+(l-ivy’)
pq
1
I’(’ UVn+
PC=,’
1 4(P - 4)
e--y(7--eIla+ I
where 7 =
c,tta.
(45)
Some manipulations give more concise expressions for Ii, j = 1,2,3,4. Then, we get an approximate solution as follows: (46)
13=
&
{G(p,5,w,q)-
Ghd,cv)J
(49)
(51)
G(P&~;w) =
I
R
T
1 (1 -
_ put7- u) x
u2)pZ+ (vu -
[yexp I
2 u-5
2
I
((7 - u)u’(uZ - R3
X-
ut_cz
i
(52)
DISCUSSIONS
An approximation of eqn (11) enables us to apply the Cagniard’s technique to a transient probIem for a dispersive medium. The approximate solution, which has an explicit expression for wave front and a more profitable form for numerical computations, is obtained in eqns (46)-(52). A comparison of this solution with an exact one is not available because no exact solutions of this problem were known, so far. But, the applicable region and time of this solution should be discussed as we have neglected the higher terms of O(Y4) in eqn (II). As a check of the solution, we introduce a parameter, 15 128p’l4 E = 100 15 3 15 z,-~l,+-~I,-,r, 8p 128~
(53)
which gives the percentage of the last term contribution to the total displacement of eqn (46). Figure 3 shows the computations of E with time at three observing points, R = 2.5, 5 and 10. The contribution of the last term is very small near the wave front but increases rapidly with time. The variation of distance R has dominant characters for the accuracy of the solution, however, that of the moving velocity is not so effective. Figure 4 shows the variation of E with time at an arc on R = 5. The contribution of the last term increases with angle \el. The
Transient
response
of an elastic
283
solid
N ,i
K. WATANABE
284
--iI-- R Fig 5. Response of the displacement.
Fig. 6. Applicable region of the approximate solution
approximate solution is most accurate on r-axis but is most incorrect on the cylindrical surface. From these figures, it may be concluded that the approximate solution is one of the wave front expansion and its applicable region and time within 1% accuracy are 5 < R, \$I~45” and T - R I 1.5. CONCLUSIONS
Transient response of a bored elastic solid to a moving torsional load is considered. An approximate solution is obtained to show the solution procedure, which is an attempt to apply Cagniard’s technique to the transient problem of the dispersive medium. An approximation is introduced in the transformed space and then very severe restrictions are imposed on the application of the approximate solution. However, the restrictions may be relaxed by taking account of the higher order terms in the approximation. REFERENCES [I] [2] [3] [4]
A. R. R. G.
T. DeHOOP, Appl. SC;. Res. 8-B, 349-356 (1960). PARNES. J. Appl. Meek.. Trans. ASME, E. 36-1, 21-58 (1969). PARNES, Int. J. Solids Structures 16. 653-670 (1980). N. WATSON, Theory of Bessel Functions. 2nd Edn. p. 202. Cambridge (1966). (Received 21 Januarv 1982)
In!. 1.Eqng Sci. Vol. 22,No. 3,pp.211-284. 1984 Printed inGreat Brilain.
TRANSIENT RESPONSE OF AN ELASTIC SOLID TO A MOVING TORSIONAL LOAD IN A CYLINDRICAL BORE AN APPROXIMATE
Department
of Mechanical
SOLUTION
KAZUMI WATANABE Engineering, Technical College, Yamagata 992, Japan
(Communicated
Yamagata
University,
Yonezawa,
by I. N. SNEDDON)
Abstract-This paper is an attempt to apply Cagniard’s technique to a transient problem of a dispersive medium. The transient response of an elastic solid to a moving torsional ring load in a cylindrical bore is considered. By making an approximation to the modified Bessel function, Cagniard’s technique is applied. An approximate solution, which has explicit expressions for the wave front geometry and has a more suitable form for numerical computations, is obtained. The applicability of the solution is also discussed.
INTRODUCTION IT
IS WELL-KNOWN
that the most powerful inversion method is Cagniard’s[ I] for the transient problem for a non-dispersive elastic medium. However, there seem to have been nd attempts to apply the technique to that of the dispersive elastic medium (including the geometrical dispersion). This paper presents an attempt to do so. A transient problem of a bored elastic solid, in which the dispersion of waves takes place, is considered and an approximate solution is obtained. The problem of the response of an elastic solid to a moving load in a cylindrical bore is one of the most important problems in engineering. The steady-state response of the bored elastic solid to normal, tangental and torsional ring loads was discussed by Parnes[2,3]. But, no transient problems of the elastic solid, in which the dispersion takes place due to the presence of the cylindrical bore, were considered, so far. The present paper considers the transient response of the solid to a moving torsional ring load. The load is suddenly applied to the cylindrical surface and then moves along the bore with uniform velocity. The Laplace and Fourier integral technique is used and an approximation in the transformed space is made for the modified Bessel function K,. Then, Cagniard’s technique is applied and the Laplace inversion is performed with the aid of simple inversion formulas, but by inspection. The applicability of the approximate solution obtained is discussed in some details. FORMULATION
Let us consider a cylindrical bore of radius a in an elastic solid. A cylindrical coordinate system (r, 8, z) is taken so that z-axis lies on the axis of the cylindrical bore. The cylindrical surface of the bore is I = a (see Fig. 1). A torsional ring load of magnitude q. is suddenly applied on a circle at z = 0 and then moves along the bore with uniform velocity c. Boundary conditions at the cylindrical surface are c,fJ
=
4”tq.z - ct)H(t) .
urr = urz =
’
0
r=a
(1)
where 6( ) and H( ) are Dirac’s delta function and Heaviside’s unit step function, respectively. The application of the load causes an axially symmetric torsional deformation. The nonvanishing component of the displacement is U,, only. Equation of motion in terms of the 271
K. WATANABE
278
Flp I Geometry
displacement
of the problem.
is
and constitutive
equations
are
(3) where C, is the velocity of M-wave and p is the torsional quiescent condition is employed. That is uH
2Lo. at
t=o
’
Consequently, the title problem is reduced to determine and (4). Applying Laplace transform,
f*(s) =
foxf(t) exp (-
f(t) = (2ni)- ’
rigidity. As an initial condition,
.
a
(4)
U, under the conditions of eqns (l), (3)
st) dt
1Rr f*(s) exp (st) ds
(5.a)
(5.b)
and Fourier transform,
(6.4 (6.b) to eqns (l)-(3), we get fi*= ”
_$. CL
=* I=_-.__;40 U r” s - ic[
1
.-
Km-)
P(s-ic5) Kdpa) K-(P) K:(@I)
(7)
(8)
Transient response of an elastic
solid
219
(9) where K,( ), n = 1,2 is the modified Bessel function p = ([‘+
of the second kind and
s~/c:)“~; Re (/3) IO
(10)
where Re denotes the real part in the bracket. AN APPROXIMATE
INVERSION
The direct application of Cagniard’s technique is impossible because of the dispersive nature of the problem. It is very troublesome to obtain the exact inversions of eqns (7)-(9). The standard inversion integrals of the Laplace and Fourier transforms gives a form of the double singular integrals and then its solution has no explicit expressions for wave front. So, we shall present here an approximate inversion procedure to make Cagniard’s technique applicable. As an example, its procedure is developed only for the displacement of eqn (7). Now, let us introduce an approximate formula[4] of the Bessel function,
4)
(n,
E e-z i
K,,(i)-
(22)”
n, -0
The last fraction in eqn (7) is approximated
-\I(~)e~z{l+$!$+~+O(z~4))
(11)
as
(12) where D(x) = 128x2 + 240x + 105. Substituting
eqn (12) into eqn (7), we can get an approximate
=* U,, _ _f
5 d
. )
1 PCS
Its formal Fourier inversion
-
k3
(13) form of fiz,
emp(r-uJ 1 _ 240(13a) + 15 I 48(~a) a Wa) r I mm)
15 Wa)
E
2
.
(14)
01 r
yields
(15)
where
(16) (17)
17 =;
I I
I;=!
exp
- y (Pa + ini)} ( -I(l-i~n)(u+p)(y+q)dn x
exp
1
(18)
- F (Pa + ini))
s I -xcr(l-iv7))(y+p)(ytq)
dv
(19)
280
K. WATANABE
and where y = asculc,Y,(Y=(n’+l)“‘;
~=$a,
p=p-I,
Re(cu)rO
<==z/u
(20)
(21)
Y= c/c,
(22)
A factorization of eqn (13), D(x) = 128(x + p)(x + 4)
(23)
where (24) is introduced in eqns (16)-(19). As a next step, we convert the real axis integrals of eqns (16)-(19) to the integrals along the Cagniard’s contour with the variable transform, n --f u,
where R =
(p= + p)“*
(27)
The integrand in each of these equations has branch points at n = 2 i, and a simple pole at n = - i/v. Two branch cuts are introduced along the imaginary axis in the complex q-plane (see Fig. 2). In this complex plane, the denominator D(y) has no zeros because of We(a) 2 0. The pole lies always on the lower cut as the torsional load moves subsonically, v < 1. The Cagniard’s contour does not cross the cut and then eqns (16)-(19) are converted to exp (- UK/C,) dv+ cr+(l - ivq+) dud’
xexp ( a+(l-
fR
asuic,) by+)
(28) 16y++ 7
’ (Y++ p){y++ 4)
x
I exp( - mu/c,) (- nsu/c,J iV~+)(y++ p)(y++ I a+(1- exp R (I-
h+)(Y+
.!i!&,
+ p)(y+ + 9)
z-a
=fRe
R
.gdir du
du
d.‘l’du q) du
(2%
(30)
(31)
where
a+= {(v+)‘+ l}“‘,
y+ = am+/c,.
(32)
Then, the formal Laplace inversions of these equations are given by Z
I, = 28e
I2 = 2Re
1 Z-’ lexp(-asu/c,) R cu’(livYl+) I s I a+(l T*ivn+)Z-I{ s(s +lii;ch
adze 1 exp (- asuic,)} x $$- du
(33)
(34)
Transient response of an elastic solid
281
ReVi’) .
I
I
\
\ \ \
-
I
\
\
\
\
Fig. 2. Capniard’s
contour
13= 2Re
(35)
I4 = 2Re
where
7’ = (aa+/c,)-‘. The simple inversion
(37)
formulas
2-l lexp(-as)
=H(t-a)
(38)
LYE-’ exp(-us)}=H(t-n)~~+~e-b(f~~“‘--~e-~(’-ai) t c) t s(s + b)(s
(39)
Is
1
1
1
-~e-“(f-“‘t~e~‘(f-U’) Lf-’ I(s t h)(st c) eXp(--as)}=H(t-a){
(40)
are used. Thus, we get
I, = ~H(T - R) Re
1
(~‘(1- ivv+)
, !!!!!du du
(41)
I2 = ~H(T - R) Re (42) j,=-
2 -H(T-R)Re. P-4
T
1 e-P+trflo+ _ e -qf+-uYrr$$du R a+(1 - ilq”) I >
I
(43)
282
I
1, = ~H(T - R) We
K.WATA~ABE
_‘+_-
1
7
R cu+(l-ivy’)
pq
1
I’(’ UVn+
PC=,’
1 4(P - 4)
e--y(7--eIla+ I
where 7 =
c,tta.
(45)
Some manipulations give more concise expressions for Ii, j = 1,2,3,4. Then, we get an approximate solution as follows: (46)
13=
&
{G(p,5,w,q)-
Ghd,cv)J
(49)
(51)
G(P&~;w) =
I
R
T
1 (1 -
_ put7- u) x
u2)pZ+ (vu -
[yexp I
2 u-5
2
I
((7 - u)u’(uZ - R3
X-
ut_cz
i
(52)
DISCUSSIONS
An approximation of eqn (11) enables us to apply the Cagniard’s technique to a transient probIem for a dispersive medium. The approximate solution, which has an explicit expression for wave front and a more profitable form for numerical computations, is obtained in eqns (46)-(52). A comparison of this solution with an exact one is not available because no exact solutions of this problem were known, so far. But, the applicable region and time of this solution should be discussed as we have neglected the higher terms of O(Y4) in eqn (II). As a check of the solution, we introduce a parameter, 15 128p’l4 E = 100 15 3 15 z,-~l,+-~I,-,r, 8p 128~
(53)
which gives the percentage of the last term contribution to the total displacement of eqn (46). Figure 3 shows the computations of E with time at three observing points, R = 2.5, 5 and 10. The contribution of the last term is very small near the wave front but increases rapidly with time. The variation of distance R has dominant characters for the accuracy of the solution, however, that of the moving velocity is not so effective. Figure 4 shows the variation of E with time at an arc on R = 5. The contribution of the last term increases with angle \el. The
Transient
response
of an elastic
283
solid
N ,i
K. WATANABE
284
--iI-- R Fig 5. Response of the displacement.
Fig. 6. Applicable region of the approximate solution
approximate solution is most accurate on r-axis but is most incorrect on the cylindrical surface. From these figures, it may be concluded that the approximate solution is one of the wave front expansion and its applicable region and time within 1% accuracy are 5 < R, \$I~45” and T - R I 1.5. CONCLUSIONS
Transient response of a bored elastic solid to a moving torsional load is considered. An approximate solution is obtained to show the solution procedure, which is an attempt to apply Cagniard’s technique to the transient problem of the dispersive medium. An approximation is introduced in the transformed space and then very severe restrictions are imposed on the application of the approximate solution. However, the restrictions may be relaxed by taking account of the higher order terms in the approximation. REFERENCES [I] [2] [3] [4]
A. R. R. G.
T. DeHOOP, Appl. SC;. Res. 8-B, 349-356 (1960). PARNES. J. Appl. Meek.. Trans. ASME, E. 36-1, 21-58 (1969). PARNES, Int. J. Solids Structures 16. 653-670 (1980). N. WATSON, Theory of Bessel Functions. 2nd Edn. p. 202. Cambridge (1966). (Received 21 Januarv 1982)