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Torsional waves in a circular cylinder of piezoelectric (622) crystal class
Int. J. Engng Sci. Vol.7,
pp. 737-745.
Pergamon Press 1969.
Printed in Great Britain
TORSIONAL WAVES IN A CIRCULAR CYLINDER PIEZOELECTRIC (622) CRYSTAL CLASS
OF
H. S. PAUL and B. SRlNlVASA RAO Department of Mathematics, Indian Institute of Technology, Madras-36, India Abstract-Torsional wave motion when displacements over the cross section of a circular cylinder involve nodal circles has been studied in the case of piezoelectric (622) crystal class. Solutions are obtained in the following cases: (a) when initial motion is prescribed; and (b) when external stresses are prescribed. I. INTRODUCTION STUDY of wave propagation in piezoelectric crystals is needed owing to its wide applications in resonators and transducers. The torsional crystal method of measuring shear viscosity and elasticity of liquids is well known[l]. Torsional waves can be produced in a piezoelectric crystal of class (622). p-quartz and ice belong to this class. The torsional vibration of a cricular cylinder of piezoelectric P-quartz has been discussed by Paul [2]. Kolsky [3] has indicated how Pochhammer-Chree theory of torsional wave propagation can be obtained as a particular case of complex torsional wave propagation involving nodal cylinders.. Jones [4] has discussed a general method of obtaining exact solutions to problems involving complex torsional vibrations in elastic bodies when initial motion or external stresses are prescribed. This paper deals with complex modes of torsional waves in a circular cylinder of piezoelectric (622) crystal class, treating the vibration discussed in [2] as the fundamental mode. Solutions are obtained for an infinite bar when initial displacement and velocity are prescribed and for a semi-infinite bar when external stresses and electric field are prescribed.
THE
2. BASIC
EQUATIONS
We use the cylindrical polar coordinate system (r, 0, z) with the axis of the cylinder as the z-axis. The radius of the cylinder is taken to be a. In the case of torsion problem the only non-vanishing component of displacement is the cross-radial component which can be taken as independent of 8. We may also take the electric potential as independent of 8. The non-vanishing piezoelectric relations can be expressed in terms of displacement v and electric potential V as
To2=
c44~,r +
e14V,,r
Tro= css(v,, - v/r) Q. =
e14v.,-
a
-33
=
l11V,r
(2.1)
v,,
where c, , e,j, lil are elastic, piezoelectric and dielectric constants respectively and comma followed by subscripts denotes partial derivatives with respect to those subscripts. 737
738
H. S. PAUL
and B. SRINIVASA
RAO
The equations of motion reduce to lieu.,, + cG,V2a+ e14VP,, = s v,tt
(2.2)
where V2 = d2/8fl + r-l alar - rp2 and s denotes the density of the material. Gauss’s equation Div D = 0 becomes e14(a,~+ v/r).,-
ll1 (v,,, + r-‘~.,)
- kv,rz
= 0.
(2.3)
Eliminating I/ between (2.2) and (2.3) we obtain [a,,, -
~~/C44)~,,,1,,,-
(d~44ku72~)dt+
(Cl+E+k:4)
(V22r),Z,+~l~V4v = 0
(2.4)
where kf, = ef,I (c44d.
cl = c~~/c~~;E = E~JE~~ and
(2.5)
Considering the curved surface of the cylinder free from tractions, the mechanical boundary condition will be T,.@=O
at
r=a.
If the curved surface is coated with electrodes boundary condition will be V=O
at
(2.6) which are shorted, the electrical
r=a.
(2.7)
The solution of the problem[2] suggests that we can seek solution of equation (2.4) satisfying boundary conditions (2.6) and (2.7) in the following form: u = 5 {/z,[Z,(h~a)]-‘Z,(har)
+h~[J,(h,a)]-?l,
(h,r)}
Y,(z,
t)
(2.8)
n=1 V
is given by v= i ,i,
{Q,hn[h~l,(h:a)l-‘l,(h~r)
-P,h:,[h,J2(h,a)l-‘J,(h,r))
Y,(z,
f)
(2.9)
where Jj, Ij are Bessel and modified Bessel functions of order j, P, and Qn are constants and h2,and hL2are given by the relation h2+h” 11 n = 0 ’
(2.10)
It can be shown by expanding J,,(h,a) _J,(h,a) Z,(hAa) and Z,(hAa) in power series and simplifying the expressions, that the left hand side of the phase velocity equation P,hL2J,,(h,a)Z,(h&)
-QnhiJJz(hna)Zo(hha)
=0
(2.11)
has a factor (Jr;+ Q) for all values of h, and h;. When equation (2.10) holds we find that P, = Qn and hence we can rewrite (2.9) as
Torsional waves in a circular cylinder
I/=
i i
P,(h”[~Z,~&d J-“Z,(hhr)-hl,[h,J,(h,u)]_‘Jo(h,r)} Y,(z, t>.
739
(2.12)
n=1
Using relations (2.8) and (2.12) in equation (2.4) we find -I-(s/c,,)!z~ Y,,t,-I-c,&4,Y, = 0. (2.13) [Yn,zz- (~/C44)Yn,ttl,zr-~2n(c,+~+~T4) yn,z, The general solutions (2.8) and (2.12) may be considered as an extension of problem 121. 3. INFINITE
BAR WHEN INITIAL MOTION IS PRESCRIBED
We proceed to obtain solution of equation (2.13) in specific cases. First we consider the case when Yn(z, 0) and (a/at) Y,,(z, 0) can be expressed in terms of given functions. Let the initial displacement and velocity be given for a bar of infitiite length as
v=g,(r,z);
u,t= g,(r, z).
(3.1)
Assuming term by term differentiation in equation (2.8) we have g,(r, 2) = B$IH,(r) Y,(G 0)
(3.2)
gz(r, z) = 5I H,(r) Y?E(Z. 0) i
(3.3)
where (3.4) and differentiation with respect to t is denoted by the dot. The functions H,(r) can be shown to have the following orthogonal properties [5]. (3.5) (3.6) when (2.10) holds (uide Appendix). These relations enable us to obtain Y, (z, 0) and I’,( z, 0) from equations (3.2) and (3.3) as
fn(z,0) =
(~,a)-’
(
g,(r, z)Zf,(r)rdr
(3.8)
where R, = IQa[N,(r)]*rdr.
(3.9)
We now proceed to the solution of equation (2.13) when initial value functions Y,(z, 0) and Y,(z, 0) are known. For convenience let
H. S. PAUL
740
and B. SRINIVASA
RAO
Y,(z, r) = U(z, t) and h, = h. Changing the independent (2.13) becomes
variable z to the dimensionless
[U&E - (~/C~,) U,ttl,s*- h2(c,+E+kf4)U,zs+
(3.10) variable x( = z/a) equation
(s/c44)h2EU,tt+c,Eh4U=
0. (3.11)
Let the given initial functions be (3.12)
Y,(z, 0) = U,(x); Y?I(z, 0) = U,(x). Considering the Laplace transform
u given by
I 0me-ht U(x, t) dt = u(x, A) we
(3.13)
have the subsidiary equation
[(D*-&)(P-&)]u=
hc-2[D2UO+~(ha)2UO]
+c-2[D2CI,+E(hu)2U,]
(3.14)
where D = (d/h);
m:+m;
= c-~A~+ (hu)2(c,+e+
k:4); (3.15)
rnfrni = r(ha)2[c-2A2+cl(ha)2];
c2 = c,~/(su’).
In the present analysis we consider the approximate m: =
C-*(A2+b2)
and
values of rn: and rni as
rni = E(hu)2
(3.16)
where (3.17)
h2 = c’(hu)2(c,+k;4).
If the quantity k:&(hu)4 is added in the third expression of (3.15), the roots in (3.16) will be exact. These approximate roots are justifiable when the radius of the cross section of the cylinder is taken to be so small that (ha) and (h’u) are both less than unity for all values of II and terms containing fourth powers of these quantities are negligible. In addition to that we have k,,2 = O-0029 and E = OS% (we consider the dielectric constants of a-quartz since those of p-quartz are not available [6]). Hence the term k:c ( hu)4 becomes very small indeed. We now derive the Green’s function G (x, 6) using (i) the boundary condition (3.18)
0 is finite as x + +m, (ii) the continuity conditions [G]$Z$ = 0; [G,,]$-+; = 0; [G,&-‘; and (iii) the jump condition
= 0
(3.19)
741
Torsional waves in a circular cytinder
(3.20)
[G,,,I -$3= 1 and obtain when ml and mz are positive
G(x,U = 2(m:!-mf)-‘(m:l~exp[-mzfx-~)~-m;‘exp~-m,(x-~)J),x = 2(m!-m$)-‘{m;‘exp
[m2(x-t)]
-m;‘exp
> 5 (3.21)
[q(x--t)]},x
< r
(3.24) +q
(c-*e(ha)2jtlo(x--crt)+U*(x+crt)] 0
+c-“(d2/dr12)[U,(x-crl)+U,(x+rrl)l}N(X)&I? where N(X) = (mS-- t#~)-‘[BZ~lexp f--~~Cq) - my’ eXp (-@Crj) 1.
(3.25)
Evaluation of U(x, t) involves finding the inverse Laplace transform of the function N(A) which is given in the Appendix. The Laplace transform tables given in [7) are used. We then have
+ Ul(x-tcq)]}
X
(b2-K4)-1&@sin
[t(b’-P)1’2]
exp (-mzmj) d+~
0
t-v
X[U~(x-qry))+ U,~x+cr))l) X[ I sin [u(b2-Ka)1~2]J,{b[(t-u)2-~2]~~2}d~]dT) 0
742
H. S. PAUL
and B. SRINIVASA
{m:[Uo(x-q)
+ Uo(x+q,l
RAO
+c-‘(d*ldrl”)
0 X
[U,(x-q)
+ Uo(x+q)]}
xexp (-m*c-q)
X (b2-K2)-1/2~;2sin
d+*-K’)-q
{mf[r/,(x-q)
[,(b2-K2)1/2]
+ Uo(x+q)l
0 t--l)
+c-*(d2/~*)[U0(~-Cr))+U0(~+~)]~[
j sin[U(b2-K2)112] 0
X J,{b[(t
-
u)*
-
du] dn)
~*I”*}
(3.26)
where K = cm,. 4. STRESS
WAVES
(3.27)
IN A SEMI-INFINITE
BAR
We next proceed to the case where Y,(O, t) and (a/dz) Y,,(O, t) are expressed in terms of given functions. We consider the problem of generating stress waves in a semiinfinite bar (z > 0) by applying a prescribed shear stress and electric field at the end z = 0. Thus Toz= iga(r,t)
at
z= 0
(4.1)
V,, = ig4(r, t)
at
z = 0.
(4.2)
and
If term by term differentiation
g,(r,
t)
=
c44
g4(f-9t) = f,
of equations (2.8) and (2.12) is permissible, we get
jl
H,(r)Y;(O,
I)
+e14
,;,
H?l(r)PJ?l(O,
(4.3)
f)
PilH,(r)Yn(O,f)
(4.4)
where prime denotes differentiation with respect to z. Equations (4.3) and (4.4) together with the orthogonal relations (3.5) and (3.6) give Y,(O,
t) = (R,, a)-’ i g4(rr t)&(r)rdr
(4.5)
0
YA(O,
t)
*
(c44
KJ-’
p
gdr,
t)ff,(r)rdr-
(c44
Rd-1e14RnluYn(0,
t)
(4.6)
0
where R,, = f P, [Hn(r)]*rdr. 0
(4.7)
743
Torsional waves in a circular cylinder
If Yn(z, 0) and Y=(z,0) are zero for all Z, we can obtain solutions of equation (2.13), when Y,(O, t) and Y’(0, t) are prescribed, in the following manner. We employ the same notation as in section 3. Using the Laplace transform with respect to t as before, we have from equation (3.11). [(D’-- m:) (P-
m”z)]u = 0.
(4.8)
The boundary condition Gisfiniteasx-,
03
(4.9)
gives (4.10)
i;T= A exp (- m,x) + B exp (- RQX) where m, and m, are given by equation (3.16), and A, B are arbitrary constants. Let V(0, t) =fi(t)
and
(4.11)
V’(0, t) =f2(t)
and let their Laplace transforms bef;l(A)andJ;,(h) respectively. We then have
-
~=.mh
n~,)-~[rn, exp (--m2x) - llzzexp (-mlx)]
+.&(A) (m1-m2)-l[exp
(-m,x) -exp (-m,x) 1 (4.12)
= O,(x,X)+Gz(x,X),say. Taking the inverse transforms [7] (vide Appendix), we obtain Utx, t> = U(0, t) exp (-vm,x) +c exp (-m2x) / [F?~~U(O, u) + U’(0, u)] {J(Jb(t-- u)] 0 t-a +KZ(b2-F)-if2~
sin [w(h2-K2)1’2] dw
JO[b(t--u-w)] 0
+K(b2-K2)-112sin
[(t--~)(b2--K~)‘~~]) du
t-x/c --c
$
[m,U(O, u) + U’(0, ~)]J~(b[(t--)~-X21C2]“~)
du
0
- CK2(b2-
W)-“2
1
[m,U(O,
u)
+
U’(0,
u)]
0 t-r--x/c
X
Ef
sin [w(b2- ~2)1~2]JO{b[(1-~--_)2-xz/dL]1iZ}dw]d~
0
-cK(b2-K2)-1’2
j [m,U(O, u) -i- U’(0, 0
a)]
H. S. PAUL
744
Xsin
and B. SRINIVASA
RAO
du+bKx(b2-K2)-1’2
[(t-u-~/~)(6*-K~)~~*]
i [m2U(0, u) 0
I-U-S/C +u’(o.u)]x[J
sin [w(~~-K~)~~~][(~-u~w)~-x~/c~]-~~~ 0 dw]du.
+J,~b[~t-u-w)~-~~/~~]~‘~~ Acknowledgmenr-The
(4.13)
authors wish to thank the reviewer for his comments on the manuscript. REFERENCES
[II W. P. MASON,
Piezoelectric Crystals and their Application lo Ulrrasonics, p. 339. Van Nostrand
(1956). PI H. S. PAUL, Arch. Mech. Slos. 14. 127 (1962). H. KOLSKY, Stress Waues in Solids, p. 65, Dover (1963). R. P. N. JONES, Q. Jl. Mech. appf. Math. 12,325 (1959). N. W. McLACHLAN, Bessel Functions,for Engineers, OxJ Engng Sci. Ser. pp. 195,203 (1955). C. S. BROWN, R. C. KELL, R. TAYLOR and L. A. THOMAS, Proc. Insrn elect. Engrs B109, 99 (1962). Mathematics, __ PP. 353-355. [71 H. S. CARSLAW and J. C. JAEGER. Ooerational Methods in Aooked .. Oxford University Press (1953).
r31 [41 IS1 [61
(Received 26 September 1968)
APPENDIX
= ~h,l~(h~)/[l~(h~)(h~+h~z)(h~-h~*)]+h~J,(h.a)/[J,(h.a)(h~+h2)(h~-hZ,)]} X ah,hL (h2, + h2)
(A.11
+{h~J,(h,a)/[J,(h,a)(hf+hr)(hK-hZ,)]-h,l,(haa)ll,(h6a)(h~-hC)(h~+h9.)]] X
ah.hA (h2.+ hT)
= 0, since hj + h;* = 0 for allj. 2.
Eoaluation ofa-‘{N(h)) a-‘{(mt-mf)-‘}
= cZ(bP-KP)-1f2sin[~(bs-~*)“*]
(A.21
where H(r) is Heaviside’s unit step function. Hence a-‘[m,(m:-ml)] = fl(bS-Ks)--/t =0, &{N(A)}
+exp
(-m,q)
(-‘I j sin[u(b2--K*)‘/*] II
XJ,{b[(t-u)~-~Z]l~Z}
du,
fort > 7
(A.3)
for1<7)
is obtained from (A.2) and (A.3).
Alsoa-‘{AN(X)}
=
$[cI-~{N(A)]
since
N(A)
is independentof
1.
3. Eoafuarion ofa-‘{ I!?~(&A) + 02(x, A)} Since[(A2+bz)*l*-K]--l= (A*+b*)-L~*+~*(A*+bZ)-*~*(A*+bZ-K*)-’+K(AZ+b*-K*)-’
(‘4.4)
74.5
Torsional waves in a circular cylinder we have (i) a-*{[(~~+b~)~‘*-K]-~}
=J0(bt)+Kf(b*-K*)-1’2
jJ,[b(t-u)]
sin[u(bP-P)ll*]
du
i +K(b2-P)-1’2sin
[t(b*-K2)1’2]
(A.5)
and (ii) a-‘{ [ (A*+ b*)1f2-K]--l +K’(bZ-K1)-“2
exp [-(x/c)
(AP+b2)11*]} = J,[b(tP-X1/CP)“*]H(r-x/c)
t--1/C I
sin[u(~-P)1~e]J,{b[(f-u)2-XP/C2]1~2}
du
0 I_-l,C
+K(bS-K*)-“Psin[(r-x/c)(b~-K2)~‘P]-Kb(be-K2)-L12(x/c)
j 0
[(t-~)*-X2/~?]-1’2
Xsin[~(b*--K*)~~~]J~{b[(t-~)*-_~+/~]~~*}du.
(A.6)
Application of Duhamel’s theorem using inverse transforms given in (A.5) and (A.6) enables evaluation of o-l{ u,(x, A)} anda-‘{ ol,(x, A)} and hencea-*{ r/,(x, A) + f12(x, A)}. RCume-On a etudie le mouvement ondulatoire de torsion lorsque les deplacements dans la section droite dun cylindre circulaire comportent des cercles nodaux darts le cas de cristaux pi&o-tlectriques de la classe (622). On obtient des solutions dam les cas suivants: (a) lorsque le mouvement initial est impose; et (b) lorsque les contraintes extemes sont impos&s. Zusammenfassuug-Torsionswellenbewegung unter Einbeziehung von Knotenkreisen durch Verriickungen iiber den Querschnitt eines kreisfiirmigen Zylinders wurde fiir den Fall einer piezoelektrischen (622) Kristallklasse untersucht. Losungen fur die folgenden F%lle werden erhalten: (a) dass die Anfangsbewegung gegeben ist; und (b) dass die Aussenspannungen gegeben sind. Sunmario-Si di un cilindro soluzioni per sollecitazioni
t studiato il movimento d’onda torsionale allorche si verificano spostamenti trasversali su circolare the interessano cerchi nodali nel case di cristalli piezoelettrici (622). Si sono ottenute i seguenti casi: (a) quando 5 prescritto il movimento iniziale; (b) quando si prescrivono esteme.
A6crparcr-_Ann Kpncranna nbe303neKTpnYecKoro Knacca (622) HccneayeTcn nmueme KpyTxueti BombI, Korna nepeMemenHa no noneperHoMy ceKenHm Kpyrnoro unnminpa can3aHbt c y3noabtMn Kpyrahni. lIony9arorcr peureHHn nna cnenymusx nsyx cnyraee: (a) Korna 3aKnanbmaeTcr Haqanbkme nmxeme, (6) Korna 3aKnanbIEtaK)TCIIHapyXHble HanprxeHun.
pp. 737-745.
Pergamon Press 1969.
Printed in Great Britain
TORSIONAL WAVES IN A CIRCULAR CYLINDER PIEZOELECTRIC (622) CRYSTAL CLASS
OF
H. S. PAUL and B. SRlNlVASA RAO Department of Mathematics, Indian Institute of Technology, Madras-36, India Abstract-Torsional wave motion when displacements over the cross section of a circular cylinder involve nodal circles has been studied in the case of piezoelectric (622) crystal class. Solutions are obtained in the following cases: (a) when initial motion is prescribed; and (b) when external stresses are prescribed. I. INTRODUCTION STUDY of wave propagation in piezoelectric crystals is needed owing to its wide applications in resonators and transducers. The torsional crystal method of measuring shear viscosity and elasticity of liquids is well known[l]. Torsional waves can be produced in a piezoelectric crystal of class (622). p-quartz and ice belong to this class. The torsional vibration of a cricular cylinder of piezoelectric P-quartz has been discussed by Paul [2]. Kolsky [3] has indicated how Pochhammer-Chree theory of torsional wave propagation can be obtained as a particular case of complex torsional wave propagation involving nodal cylinders.. Jones [4] has discussed a general method of obtaining exact solutions to problems involving complex torsional vibrations in elastic bodies when initial motion or external stresses are prescribed. This paper deals with complex modes of torsional waves in a circular cylinder of piezoelectric (622) crystal class, treating the vibration discussed in [2] as the fundamental mode. Solutions are obtained for an infinite bar when initial displacement and velocity are prescribed and for a semi-infinite bar when external stresses and electric field are prescribed.
THE
2. BASIC
EQUATIONS
We use the cylindrical polar coordinate system (r, 0, z) with the axis of the cylinder as the z-axis. The radius of the cylinder is taken to be a. In the case of torsion problem the only non-vanishing component of displacement is the cross-radial component which can be taken as independent of 8. We may also take the electric potential as independent of 8. The non-vanishing piezoelectric relations can be expressed in terms of displacement v and electric potential V as
To2=
c44~,r +
e14V,,r
Tro= css(v,, - v/r) Q. =
e14v.,-
a
-33
=
l11V,r
(2.1)
v,,
where c, , e,j, lil are elastic, piezoelectric and dielectric constants respectively and comma followed by subscripts denotes partial derivatives with respect to those subscripts. 737
738
H. S. PAUL
and B. SRINIVASA
RAO
The equations of motion reduce to lieu.,, + cG,V2a+ e14VP,, = s v,tt
(2.2)
where V2 = d2/8fl + r-l alar - rp2 and s denotes the density of the material. Gauss’s equation Div D = 0 becomes e14(a,~+ v/r).,-
ll1 (v,,, + r-‘~.,)
- kv,rz
= 0.
(2.3)
Eliminating I/ between (2.2) and (2.3) we obtain [a,,, -
~~/C44)~,,,1,,,-
(d~44ku72~)dt+
(Cl+E+k:4)
(V22r),Z,+~l~V4v = 0
(2.4)
where kf, = ef,I (c44d.
cl = c~~/c~~;E = E~JE~~ and
(2.5)
Considering the curved surface of the cylinder free from tractions, the mechanical boundary condition will be T,.@=O
at
r=a.
If the curved surface is coated with electrodes boundary condition will be V=O
at
(2.6) which are shorted, the electrical
r=a.
(2.7)
The solution of the problem[2] suggests that we can seek solution of equation (2.4) satisfying boundary conditions (2.6) and (2.7) in the following form: u = 5 {/z,[Z,(h~a)]-‘Z,(har)
+h~[J,(h,a)]-?l,
(h,r)}
Y,(z,
t)
(2.8)
n=1 V
is given by v= i ,i,
{Q,hn[h~l,(h:a)l-‘l,(h~r)
-P,h:,[h,J2(h,a)l-‘J,(h,r))
Y,(z,
f)
(2.9)
where Jj, Ij are Bessel and modified Bessel functions of order j, P, and Qn are constants and h2,and hL2are given by the relation h2+h” 11 n = 0 ’
(2.10)
It can be shown by expanding J,,(h,a) _J,(h,a) Z,(hAa) and Z,(hAa) in power series and simplifying the expressions, that the left hand side of the phase velocity equation P,hL2J,,(h,a)Z,(h&)
-QnhiJJz(hna)Zo(hha)
=0
(2.11)
has a factor (Jr;+ Q) for all values of h, and h;. When equation (2.10) holds we find that P, = Qn and hence we can rewrite (2.9) as
Torsional waves in a circular cylinder
I/=
i i
P,(h”[~Z,~&d J-“Z,(hhr)-hl,[h,J,(h,u)]_‘Jo(h,r)} Y,(z, t>.
739
(2.12)
n=1
Using relations (2.8) and (2.12) in equation (2.4) we find -I-(s/c,,)!z~ Y,,t,-I-c,&4,Y, = 0. (2.13) [Yn,zz- (~/C44)Yn,ttl,zr-~2n(c,+~+~T4) yn,z, The general solutions (2.8) and (2.12) may be considered as an extension of problem 121. 3. INFINITE
BAR WHEN INITIAL MOTION IS PRESCRIBED
We proceed to obtain solution of equation (2.13) in specific cases. First we consider the case when Yn(z, 0) and (a/at) Y,,(z, 0) can be expressed in terms of given functions. Let the initial displacement and velocity be given for a bar of infitiite length as
v=g,(r,z);
u,t= g,(r, z).
(3.1)
Assuming term by term differentiation in equation (2.8) we have g,(r, 2) = B$IH,(r) Y,(G 0)
(3.2)
gz(r, z) = 5I H,(r) Y?E(Z. 0) i
(3.3)
where (3.4) and differentiation with respect to t is denoted by the dot. The functions H,(r) can be shown to have the following orthogonal properties [5]. (3.5) (3.6) when (2.10) holds (uide Appendix). These relations enable us to obtain Y, (z, 0) and I’,( z, 0) from equations (3.2) and (3.3) as
fn(z,0) =
(~,a)-’
(
g,(r, z)Zf,(r)rdr
(3.8)
where R, = IQa[N,(r)]*rdr.
(3.9)
We now proceed to the solution of equation (2.13) when initial value functions Y,(z, 0) and Y,(z, 0) are known. For convenience let
H. S. PAUL
740
and B. SRINIVASA
RAO
Y,(z, r) = U(z, t) and h, = h. Changing the independent (2.13) becomes
variable z to the dimensionless
[U&E - (~/C~,) U,ttl,s*- h2(c,+E+kf4)U,zs+
(3.10) variable x( = z/a) equation
(s/c44)h2EU,tt+c,Eh4U=
0. (3.11)
Let the given initial functions be (3.12)
Y,(z, 0) = U,(x); Y?I(z, 0) = U,(x). Considering the Laplace transform
u given by
I 0me-ht U(x, t) dt = u(x, A) we
(3.13)
have the subsidiary equation
[(D*-&)(P-&)]u=
hc-2[D2UO+~(ha)2UO]
+c-2[D2CI,+E(hu)2U,]
(3.14)
where D = (d/h);
m:+m;
= c-~A~+ (hu)2(c,+e+
k:4); (3.15)
rnfrni = r(ha)2[c-2A2+cl(ha)2];
c2 = c,~/(su’).
In the present analysis we consider the approximate m: =
C-*(A2+b2)
and
values of rn: and rni as
rni = E(hu)2
(3.16)
where (3.17)
h2 = c’(hu)2(c,+k;4).
If the quantity k:&(hu)4 is added in the third expression of (3.15), the roots in (3.16) will be exact. These approximate roots are justifiable when the radius of the cross section of the cylinder is taken to be so small that (ha) and (h’u) are both less than unity for all values of II and terms containing fourth powers of these quantities are negligible. In addition to that we have k,,2 = O-0029 and E = OS% (we consider the dielectric constants of a-quartz since those of p-quartz are not available [6]). Hence the term k:c ( hu)4 becomes very small indeed. We now derive the Green’s function G (x, 6) using (i) the boundary condition (3.18)
0 is finite as x + +m, (ii) the continuity conditions [G]$Z$ = 0; [G,,]$-+; = 0; [G,&-‘; and (iii) the jump condition
= 0
(3.19)
741
Torsional waves in a circular cytinder
(3.20)
[G,,,I -$3= 1 and obtain when ml and mz are positive
G(x,U = 2(m:!-mf)-‘(m:l~exp[-mzfx-~)~-m;‘exp~-m,(x-~)J),x = 2(m!-m$)-‘{m;‘exp
[m2(x-t)]
-m;‘exp
> 5 (3.21)
[q(x--t)]},x
< r
(3.24) +q
(c-*e(ha)2jtlo(x--crt)+U*(x+crt)] 0
+c-“(d2/dr12)[U,(x-crl)+U,(x+rrl)l}N(X)&I? where N(X) = (mS-- t#~)-‘[BZ~lexp f--~~Cq) - my’ eXp (-@Crj) 1.
(3.25)
Evaluation of U(x, t) involves finding the inverse Laplace transform of the function N(A) which is given in the Appendix. The Laplace transform tables given in [7) are used. We then have
+ Ul(x-tcq)]}
X
(b2-K4)-1&@sin
[t(b’-P)1’2]
exp (-mzmj) d+~
0
t-v
X[U~(x-qry))+ U,~x+cr))l) X[ I sin [u(b2-Ka)1~2]J,{b[(t-u)2-~2]~~2}d~]dT) 0
742
H. S. PAUL
and B. SRINIVASA
{m:[Uo(x-q)
+ Uo(x+q,l
RAO
+c-‘(d*ldrl”)
0 X
[U,(x-q)
+ Uo(x+q)]}
xexp (-m*c-q)
X (b2-K2)-1/2~;2sin
d+*-K’)-q
{mf[r/,(x-q)
[,(b2-K2)1/2]
+ Uo(x+q)l
0 t--l)
+c-*(d2/~*)[U0(~-Cr))+U0(~+~)]~[
j sin[U(b2-K2)112] 0
X J,{b[(t
-
u)*
-
du] dn)
~*I”*}
(3.26)
where K = cm,. 4. STRESS
WAVES
(3.27)
IN A SEMI-INFINITE
BAR
We next proceed to the case where Y,(O, t) and (a/dz) Y,,(O, t) are expressed in terms of given functions. We consider the problem of generating stress waves in a semiinfinite bar (z > 0) by applying a prescribed shear stress and electric field at the end z = 0. Thus Toz= iga(r,t)
at
z= 0
(4.1)
V,, = ig4(r, t)
at
z = 0.
(4.2)
and
If term by term differentiation
g,(r,
t)
=
c44
g4(f-9t) = f,
of equations (2.8) and (2.12) is permissible, we get
jl
H,(r)Y;(O,
I)
+e14
,;,
H?l(r)PJ?l(O,
(4.3)
f)
PilH,(r)Yn(O,f)
(4.4)
where prime denotes differentiation with respect to z. Equations (4.3) and (4.4) together with the orthogonal relations (3.5) and (3.6) give Y,(O,
t) = (R,, a)-’ i g4(rr t)&(r)rdr
(4.5)
0
YA(O,
t)
*
(c44
KJ-’
p
gdr,
t)ff,(r)rdr-
(c44
Rd-1e14RnluYn(0,
t)
(4.6)
0
where R,, = f P, [Hn(r)]*rdr. 0
(4.7)
743
Torsional waves in a circular cylinder
If Yn(z, 0) and Y=(z,0) are zero for all Z, we can obtain solutions of equation (2.13), when Y,(O, t) and Y’(0, t) are prescribed, in the following manner. We employ the same notation as in section 3. Using the Laplace transform with respect to t as before, we have from equation (3.11). [(D’-- m:) (P-
m”z)]u = 0.
(4.8)
The boundary condition Gisfiniteasx-,
03
(4.9)
gives (4.10)
i;T= A exp (- m,x) + B exp (- RQX) where m, and m, are given by equation (3.16), and A, B are arbitrary constants. Let V(0, t) =fi(t)
and
(4.11)
V’(0, t) =f2(t)
and let their Laplace transforms bef;l(A)andJ;,(h) respectively. We then have
-
~=.mh
n~,)-~[rn, exp (--m2x) - llzzexp (-mlx)]
+.&(A) (m1-m2)-l[exp
(-m,x) -exp (-m,x) 1 (4.12)
= O,(x,X)+Gz(x,X),say. Taking the inverse transforms [7] (vide Appendix), we obtain Utx, t> = U(0, t) exp (-vm,x) +c exp (-m2x) / [F?~~U(O, u) + U’(0, u)] {J(Jb(t-- u)] 0 t-a +KZ(b2-F)-if2~
sin [w(h2-K2)1’2] dw
JO[b(t--u-w)] 0
+K(b2-K2)-112sin
[(t--~)(b2--K~)‘~~]) du
t-x/c --c
$
[m,U(O, u) + U’(0, ~)]J~(b[(t--)~-X21C2]“~)
du
0
- CK2(b2-
W)-“2
1
[m,U(O,
u)
+
U’(0,
u)]
0 t-r--x/c
X
Ef
sin [w(b2- ~2)1~2]JO{b[(1-~--_)2-xz/dL]1iZ}dw]d~
0
-cK(b2-K2)-1’2
j [m,U(O, u) -i- U’(0, 0
a)]
H. S. PAUL
744
Xsin
and B. SRINIVASA
RAO
du+bKx(b2-K2)-1’2
[(t-u-~/~)(6*-K~)~~*]
i [m2U(0, u) 0
I-U-S/C +u’(o.u)]x[J
sin [w(~~-K~)~~~][(~-u~w)~-x~/c~]-~~~ 0 dw]du.
+J,~b[~t-u-w)~-~~/~~]~‘~~ Acknowledgmenr-The
(4.13)
authors wish to thank the reviewer for his comments on the manuscript. REFERENCES
[II W. P. MASON,
Piezoelectric Crystals and their Application lo Ulrrasonics, p. 339. Van Nostrand
(1956). PI H. S. PAUL, Arch. Mech. Slos. 14. 127 (1962). H. KOLSKY, Stress Waues in Solids, p. 65, Dover (1963). R. P. N. JONES, Q. Jl. Mech. appf. Math. 12,325 (1959). N. W. McLACHLAN, Bessel Functions,for Engineers, OxJ Engng Sci. Ser. pp. 195,203 (1955). C. S. BROWN, R. C. KELL, R. TAYLOR and L. A. THOMAS, Proc. Insrn elect. Engrs B109, 99 (1962). Mathematics, __ PP. 353-355. [71 H. S. CARSLAW and J. C. JAEGER. Ooerational Methods in Aooked .. Oxford University Press (1953).
r31 [41 IS1 [61
(Received 26 September 1968)
APPENDIX
= ~h,l~(h~)/[l~(h~)(h~+h~z)(h~-h~*)]+h~J,(h.a)/[J,(h.a)(h~+h2)(h~-hZ,)]} X ah,hL (h2, + h2)
(A.11
+{h~J,(h,a)/[J,(h,a)(hf+hr)(hK-hZ,)]-h,l,(haa)ll,(h6a)(h~-hC)(h~+h9.)]] X
ah.hA (h2.+ hT)
= 0, since hj + h;* = 0 for allj. 2.
Eoaluation ofa-‘{N(h)) a-‘{(mt-mf)-‘}
= cZ(bP-KP)-1f2sin[~(bs-~*)“*]
(A.21
where H(r) is Heaviside’s unit step function. Hence a-‘[m,(m:-ml)] = fl(bS-Ks)--/t =0, &{N(A)}
+exp
(-m,q)
(-‘I j sin[u(b2--K*)‘/*] II
XJ,{b[(t-u)~-~Z]l~Z}
du,
fort > 7
(A.3)
for1<7)
is obtained from (A.2) and (A.3).
Alsoa-‘{AN(X)}
=
$[cI-~{N(A)]
since
N(A)
is independentof
1.
3. Eoafuarion ofa-‘{ I!?~(&A) + 02(x, A)} Since[(A2+bz)*l*-K]--l= (A*+b*)-L~*+~*(A*+bZ)-*~*(A*+bZ-K*)-’+K(AZ+b*-K*)-’
(‘4.4)
74.5
Torsional waves in a circular cylinder we have (i) a-*{[(~~+b~)~‘*-K]-~}
=J0(bt)+Kf(b*-K*)-1’2
jJ,[b(t-u)]
sin[u(bP-P)ll*]
du
i +K(b2-P)-1’2sin
[t(b*-K2)1’2]
(A.5)
and (ii) a-‘{ [ (A*+ b*)1f2-K]--l +K’(bZ-K1)-“2
exp [-(x/c)
(AP+b2)11*]} = J,[b(tP-X1/CP)“*]H(r-x/c)
t--1/C I
sin[u(~-P)1~e]J,{b[(f-u)2-XP/C2]1~2}
du
0 I_-l,C
+K(bS-K*)-“Psin[(r-x/c)(b~-K2)~‘P]-Kb(be-K2)-L12(x/c)
j 0
[(t-~)*-X2/~?]-1’2
Xsin[~(b*--K*)~~~]J~{b[(t-~)*-_~+/~]~~*}du.
(A.6)
Application of Duhamel’s theorem using inverse transforms given in (A.5) and (A.6) enables evaluation of o-l{ u,(x, A)} anda-‘{ ol,(x, A)} and hencea-*{ r/,(x, A) + f12(x, A)}. RCume-On a etudie le mouvement ondulatoire de torsion lorsque les deplacements dans la section droite dun cylindre circulaire comportent des cercles nodaux darts le cas de cristaux pi&o-tlectriques de la classe (622). On obtient des solutions dam les cas suivants: (a) lorsque le mouvement initial est impose; et (b) lorsque les contraintes extemes sont impos&s. Zusammenfassuug-Torsionswellenbewegung unter Einbeziehung von Knotenkreisen durch Verriickungen iiber den Querschnitt eines kreisfiirmigen Zylinders wurde fiir den Fall einer piezoelektrischen (622) Kristallklasse untersucht. Losungen fur die folgenden F%lle werden erhalten: (a) dass die Anfangsbewegung gegeben ist; und (b) dass die Aussenspannungen gegeben sind. Sunmario-Si di un cilindro soluzioni per sollecitazioni
t studiato il movimento d’onda torsionale allorche si verificano spostamenti trasversali su circolare the interessano cerchi nodali nel case di cristalli piezoelettrici (622). Si sono ottenute i seguenti casi: (a) quando 5 prescritto il movimento iniziale; (b) quando si prescrivono esteme.
A6crparcr-_Ann Kpncranna nbe303neKTpnYecKoro Knacca (622) HccneayeTcn nmueme KpyTxueti BombI, Korna nepeMemenHa no noneperHoMy ceKenHm Kpyrnoro unnminpa can3aHbt c y3noabtMn Kpyrahni. lIony9arorcr peureHHn nna cnenymusx nsyx cnyraee: (a) Korna 3aKnanbmaeTcr Haqanbkme nmxeme, (6) Korna 3aKnanbIEtaK)TCIIHapyXHble HanprxeHun.