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Thermally induced vibrations of a viscoelastic plate
JournnlofSound and Vibration(1980) 73(1),31-39
THERMALLY
INDUCED
VISCOELASTIC J. MAZUMDAR, Department
of Applied
VIBRATIONS
OF A
PLATE
D. HII.L AND D. L. CLEMENTS
Mathematics, The University of Adelaide, Adelaide, South Australia 5001. Australia (Receitied 12 March 1980)
A method for the study of thermally induced vibrations of a viscoelastic plate of arbitrary shape is proposed. The method is based upon the concept of iso-amplitude contour lines on the surface of the plate. It is shown that the time behaviour can be found by assuming a normal mode expansion in terms of the eigenfunctions for the associated elastic plate problem, and the deflection is obtained by using the elastic-viscoelastic analogy. As an illustration of the technique, the vibration of a viscoelastic circular plate under a thermal shock at its centre is discussed, all details of which are illustrated by graphs.
1. INTRODUCTION
The application of plates in modern nuclear technology and welding engineering makes the classical perfectly elastic theory often inadequate for rational design. Under high ternperatures the material exhibits the phenomenon of creep and thus mechanical properties of plates must be time dependent as well as temperature dependent. Therefore, one of the main concerns of the analytical and experimental development in recent years has been the study of thermally induced dynamic behaviour of viscoelastic plates. A review of the literature seems to indicate that although a few investigations of stress analysis of viscoelastic plates at elevated temperatures have been made in the past [l--4], very little has been done on the dynamic analysis of thermally induced motion of viscoelastic plates. Some work has recently been done on vibration analysis of viscoelastic plates without taking thermal effects [5,6] into consideration. The purpose of the present investigation is to develop a simple and yet sufficiently accurate method for the analysis of thermally induced vibrations of viscoelastic plates. The usual assumptions of small deflections and linear, homogeneous, isotropic viscoelastic properties are made. It is further assumed that within the range of temperature changes the viscoelastic material parameters remain constant. The pertinent equations for the vibration analysis of viscoelastic plates with effects of temperature are first presented in section 2. This is then followed in section 3 by an illustration of the method. 2. DERIVATION
OF THE BASIC EQUATIONS
Consider a viscoelastic plate of constant thickness h and uniform density p situated in a non-stationary temperature field. The plate is referred to a system of rectangular coordinates x, y, z such that the horizontal plane OXY coincides with the middle surface of the plate in the undeformed configuration. The deflection w (x, y, r) of the plate, which is a function of the spatial coordinates, x and y, and the temporal variable, 7, is taken as 31
0022-460X/80/210037 +09 $02.00/O
0 I980 Academic
Press Inc. (London
1Limited
32
J. MAZUMDAR.
D. HII.1.
AND
D. I.
CLEMENTS
positive downward in the z-direction. An incremental increase in temperature, T’(x, y, z, r), in the plate will produce a state in which thermal stresses are evident. The stress-strain equations are given by [2] P’Sij = Q’F~,,
P”ukk = Q”(ekk - 3a,T’),
(I, 3)
where (or is the coefficient of linear thermal expansion of the material, S,, and pi, are the components of the deviatoric stress and strain tensors, respectively, (Tkkand ei;k are the traces of the stress and strain tensors, and P’, Q’, P” and Q” are linear differential operators in time variables. For small thickness, the temperature variation across the thickness of the plate can be considered to be linear: i.e., T’(x, y, z, 7) = zT(x, y, 7).
(3)
The temperature fields are assumed to be such as to produce a deflection which is small. At any instant of time, ro, intersections of the middle surface of the plate with the parallels t = constant lead to the formation of lines of equal deflection u (x, y) = constant, which are iso-amplitude contours. A consideration of forces in the z-direction inside and on a contour u(x, y) = constant, including the effect of inertia forces which are especially important for suddenly induced temperature fields, gives rise to the dynamical equation [5] f
W+jj
2
ph$dR=O.
(4)
a.
cu
The temperature term is introduced through the transverse forces V,,, the expressions for moments and shearing forces being as follows: M* = -~(p)[~2Wl~X2)+CL(p)(~2Wl~y2~1-~(p)[1
+/J(p)IwT,
My = -~(P)[a2wlaY2)+CL(p)(~2WlaX2)1-~(p)[1
+PL(p)lwT
Ky = -My, = D(pNl-/4pm2wl~x~Y, +/.~(p)lw@T/dxL
0, = -o(p)(a/ax)[(a’~lax~)+(a’w/ay’)l-D(p)[l
Q,= -D~p~~~l~y~~~~2~l~x2~+~~2~l~~‘~l-~~~~~~+~~(p)lwW/~yh
(5)
where p = d/h, and D(p) and p (p) are viscoelastic operators corresponding, respectively, to the elastic constants D and CL.By using the procedure outlined in reference [5], the dynamical equation (4) can finally be reduced to the integro-differential form
C”
fL
where t = uf + U; and the expressions for RI, Fl and G; are listed in reference [5]. The solution of the above equation can be found in the same way as for the free vibration problem as follows. Firstly, the eigenfunctions M’i and the eigenfrequencies Ai of the corresponding elastic problem under homogeneous boundary conditions are determined. It is well-known that this eigenvalue problem is self-adjoint and hence the associated eigenfunctions form a complete set and are mutually orthogonal. The viscoelastic solution is then obtained by assuming that the deflection w and the temperature T appearing in the
THERMALLY
INDUCED
PLATE
33
VIBRATIONS
above equation can be expressed as a linear sum of the eigenfunctions
Wi in the form
T= z Ti(T)W,(II).
W = f gi(7)Wi(u)t i=l
t 7)
1=l
Since Wi is a solution of the corresponding elastic equation, differentiating with respect to u and using expressions (7) yields
equation (6)
Here use has been made of Green’s Theorem in establishing the result -a au
f(
VT&. Jr
cc.
(9)
Lu
Using the heat conduction equation for a plate, one can write equation (8) in the form (10) where u is the specific heat and k is the thermal conductivity. Equation (10) will uncouple into an infinite number of linear differential equations for the gi(7) if one uses the expression (7) for T and then the orthogonality properties of the eigenfunctions W, : viz., Ll) In the next section, a procedure is outlined to indicate how the above equation can be solved for the gj(T) for a particular viscoelastic material.
3. ILLUSTRATION 3. I. EXAMPLE OF A CIRCULAR PLATE Some information concerning the engineering aspects of viscoelastic plates can be obtained by studying the behaviour of simple models. Therefore, as an example of the application of this technique, the vibration of a viscoelastic circular plate, of radius a, under a thermal shock at its centre, and with radiation at the boundary into a medium at zero temperature, can now be discussed. The boundary of the plate is assumed to be completely fixed. Problems of this type very frequently occur in both welding engineering and in nuclear reactors. Since only symmetric modes of vibration are involved in this example, the eigensolutions for the corresponding circular elastic problem can be easily obtained from those of an elliptical elastic plate [5], W = I,(ki)Jo(kif)-
J,,(ki)In(kf),
112)
where f2=l-U
=(X2+~*)/a2,
k; = pha%;
113)
and the ki’S satisfy the determinant Jo(k)
h,(k) =
J&(k)
IL(k)
I
o,
(14)
34
J. MAZUMDAR.
The orthogonality
D. HILL
AND
D. I,. (‘LEMENTS
relation in this case is
I
f Wi Wj df
=
i
6;jB;,
15,
where Bi =$Ig(kj)[Ji(kj)+JT(kj)]+tJi(kj)[Ig(kj)-I:(kj)] well-known Bessel functions. 3.2.
TEMPERATURE
and Jo, J1, IO and I1 are the
EFFECTS
With the eigenfunctions and eigenvalues thus having been determined, the next task is to analyze the effects of temperature in the region of the plate. The distribution of temperature can be obtained by using any of the standard techniques in heat conduction problems [7,8]. Laura et al. [9, lo] used an interesting conformal mapping technique for the study of two dimensional heat conduction problems whereas Mazumdar [ 1 l] proposed a new method which is based upon the concept of isothermal contour lines on the surface of the plate. In the present analysis this latter method is used for evaluation of the temperature T appearing in equation (lo), which, in accordance with the results given in reference [ll], is of the form
(16) where 9 is the strength of the point heat source, the w, are the positive roots of wJ,(wa) = hrJo(wa), and hr = H/k, H being the exterior conductivity. T = f
(17)
Equation (16) can also be written as
c,,Jo(o,nf) e -cz”fT,
(18)
n=l
where the notation c,, =qw;/~a~(h: has been used. Upon using the orthogonality relationship
+w:)J;(w,a)
(19)
relation (15) and expression (7) for T one obtains the
I
f TWj df = BiTi(
(20)
0 which, in conjunction
with equations (18) and (12), becomes (21)
where the quantities 4, are given by 1
1
An= b(kj) \ fJo(kif)Jo(wnaf) df -Jo(kj) [ fIo(kif)Jo(wnaf) df. 0
0
(22)
THERMALLY
INDUCED
PLATE
3s
VIBRATIONS
By using the following formulae for Bessel integrals [12], I
~yJo(p)Jb((y)-PJo(~y)Jb(P)
J
xJo(cyx)J,(/3x) dx =
p2_cu2
’
0
1
aJoW)Ibb)
J
the expressions obtained. 3.3.
TIME
xIo((~x)Jo@x) dx = --
-PJI,Wob)
(23)
p2ca’
for the Iin’s can be evaluated
exactly and hence Tj(T) can be easily
FUNCTIONS
In order to solve equation (11) for the time functions gi(T), it is assumed that the viscoelastic properties of the plate are of the “Kelvin” type, for which one has the relationship D(p)lD
(24)
= 1 +(n/G)p,
where n and G are material constants. Here it has been assumed that Poisson’s ratio is a constant quantity. The differential equation (11) thus takes the form of a second order ordinary differential equation, {p2+(77/G)A:p+A:}gi(7)=-_(
(~~41
+~L~/khl{p
+(~/GJp21~ji(~).
125)
where Ti(r) is as given by equation (21). In order to solve this differential equation with the aid of the Laplace transform method, one needs to know the corresponding initial conditions. By using the method developed by DeLeeuw [13] for determining initial conditions for the time equation of a viscoelastic plate, which is based on an earlier method of Boley and Weiner [7] for a general viscoelastic stress-strain law in linear operator form, one obtains the initial conditions gj(O) = -A(nIG)Ti(O),
g~(0)=A[{(~*/G2)A~~1}T~(O)-(~/G)T~(O)]~
where the prime denotes differentiation
(26)
with respect to T and A is given by
A = ad1
+k)D/kh.
(27)
Equation (25) then has the following Laplace transform, incorporating tions (26): Pj(s)gj(s)=-A(s+ps*)~(s),
the initial condi(28)
where Pj(S)=S*+PA:S+A;
=(S-Sl)(S-Sl).
P=dG
(29) and the overscores denote Laplace transforms, s being the Laplace transform parameter. The solution to equation (28) can be obtained by inverse Laplace transformation:
S+pS’ (s - Sl)(S - s&s +
I
c’co’,,
(30)
J. MAZUMDAR.
36
D. HILL
AND
D.
L. CLEMENTS
642-
Radius,
Figure (D/a3q)w.
r
1. Deflection profiles for a circular viscoelastic plate under thermal (a) Initial deflection profile, 7 = Ofs; (b) deflection profile at 0.06 s.
-4
2. Amplitude
at the centre;
0
Figure
shock
1
1
I
I
I
2
3
4
of vibrations
1 5
1 6
with time. /3 = 0.0025;
1 7
T* = (l/a’dD/phT,
w*
THERMALLY
INDUCED
PLATE
37
VIBRATIONS 24F
'
'
'
'
'
'
'
0
0.5
r*x IO5 Figure
3. (a) Variation /3 = 0.1.
of gl(r)
'-1
(b)
20 -
I.0
I.5
2.0
r*x IO 7
corresponding
to the first mode;
lb) variation
of gZ(7) corresponding
tcl the
second mode.
16 14 12 Q
?? : I0
I
6 6 4 2 0 -2 -4 -6
Figure
4. Variation
of (a) g,(s)
and (b) g2(T) for selected
values of the damping
parameter
p.
yields (31)
where Al,, A2”, A3,, are a set of constants depending on the roots of the denominator of equation (30) as well as on the damping parameter /3. Clearly, when the two roots sl and s2 are two complex conjugates with negative real parts the result corresponds to a damped oscillatory motion.
J.
38
MAZUMDAR,
D. HILL
AND
D. I,. C‘LEMENTS
Substituting expression (31) for gi(7) in equation (71, then yields the deflection
ampli-
tude at any point in the plate. 3.4.
RESULTS
Figure l(a) is a graph of the deflection of the plate just after the initial shock has been applied. As time proceeds the general shape of the deflection remains similar in form but with the amplitude reducing, as is evident from Figure l(b). In Figure 2, the amplitude of vibration is plotted against time for various points in the plate. Figures 3(a) and 3(b) show the variations of the time functions gr(r) and g2(r) corresponding to the first and second modes of vibration, respectively. The third modal function g3(r) was found to have values less than 5 x 1O-9 with successive modes giving successively less contribution to the final answers, so that a good approximation can be obtained by using only the first two modes of vibration. Figures 4(a) and (b) show the effects of variation of the damping parameter p. It can be observed that for values of /3 less than 2/hi( = 0*017), the plate undergoes damped oscillation.
4. CONCLUSION
A method has been presented for the study of thermally induced vibrating motion of viscoelastic plates. It has been shown that the method is useful for approximation of the response, as the problem can be separated into two parts: (i) finding spatial functions which approximate the mode shapes in vibration and satisfy the boundary conditions and an orthogonality relation, and (ii) solving a time response equation which depends on the viscoelastic model chosen, The mode shapes are chosen to be the vibrational modes of the associated elastic problem, so that if these and the corresponding frequencies can be found for a plate of a particular geometry, then the associated viscoelastic problem can be easily solved.
ACKNOWLEDGMENTS The authors wish to acknowledge the support of the Australian Research Grants Committee who sponsored this investigation under Grant No. F78/15085. The authors also wish to thankfully acknowledge the valuable comments of J. Hewitt during the preparation of the manuscript.
REFERENCES and E. H. LEE 1960 Transactions of the Society of Rheology IV, 233-263. Stress analysis for linear viscoelastic materials with temperature variation. R. MUKI and E. STERNBERG 1961Journal of Applied Mechanics 28, 193-207. On transient thermal stresses in viscoelastic materials with temperature-dependent properties. W. NOWACKI 1962Advances in Aeronautical Sciences 4,947-969. Transient thermal stresses in viscoelastic plates and shells. T. R. TAUCHERT 1968 Acta Mechanica 6, 239-252. Transient temperature distributions in viscoelastic solids subject to cyclic deformations. J. S. HEWrnand J.MAZUMDAR 1974JournalofSoundand Vibration 33,319-333. Vibration of viscoelastic plates under transverse load by the method of constant deflection contours. K. NAGAYA 1977Journal of Engineering for Industry, Transactions of the American Society of Mechanical Engineers, Series B 99,404-409. Vibrations and dynamic response of viscoelastic plates on non-periodic elastic supports. B. A. BOLEY and J.H. WEINER 1960 Theory of Thermal Stresses. New York: John Wiley.
1. L. W. MORLAND 2. 3. 4. 5. 6.
7.
THERMALLY
INDUCED
PLATE
VIBRATIONS
8. P. BISWAS 1978 Journal of Sound and Vibration 59,304-306. 9. 10. 11. 12. 13.
39
Thermally-induced vibrations of a triangular slab on elastic foundation. P. A. LAURA and A. J. FAULSTICH, JR. 1968 International Journal Heat Mass Transfer '11, 297-303. Unsteady heat conduction in plates of polygonal shape. P. A. LAURA and R. ERCOLI 1972 NuclearEngineering and Design 23, 1-9. A solution of the unsteady diffusion equation in an arbitrary, doubly connected region. J. MAZUMDAR 1974 NuclearEngineeringand Design 31,383-390. A method for the study of transient heat conduction in plates of arbitrary cross section. G. N. WATSON 1944 A Treatise on the Theory of BesselFunctions. Cambridge University Press, second edition. S. E. DELEEUW 1961 Ph.D. Thesis, Michigan State Unicersity. Behaviour of viscoelastic plates under the action of in-plane forces.
THERMALLY
INDUCED
VISCOELASTIC J. MAZUMDAR, Department
of Applied
VIBRATIONS
OF A
PLATE
D. HII.L AND D. L. CLEMENTS
Mathematics, The University of Adelaide, Adelaide, South Australia 5001. Australia (Receitied 12 March 1980)
A method for the study of thermally induced vibrations of a viscoelastic plate of arbitrary shape is proposed. The method is based upon the concept of iso-amplitude contour lines on the surface of the plate. It is shown that the time behaviour can be found by assuming a normal mode expansion in terms of the eigenfunctions for the associated elastic plate problem, and the deflection is obtained by using the elastic-viscoelastic analogy. As an illustration of the technique, the vibration of a viscoelastic circular plate under a thermal shock at its centre is discussed, all details of which are illustrated by graphs.
1. INTRODUCTION
The application of plates in modern nuclear technology and welding engineering makes the classical perfectly elastic theory often inadequate for rational design. Under high ternperatures the material exhibits the phenomenon of creep and thus mechanical properties of plates must be time dependent as well as temperature dependent. Therefore, one of the main concerns of the analytical and experimental development in recent years has been the study of thermally induced dynamic behaviour of viscoelastic plates. A review of the literature seems to indicate that although a few investigations of stress analysis of viscoelastic plates at elevated temperatures have been made in the past [l--4], very little has been done on the dynamic analysis of thermally induced motion of viscoelastic plates. Some work has recently been done on vibration analysis of viscoelastic plates without taking thermal effects [5,6] into consideration. The purpose of the present investigation is to develop a simple and yet sufficiently accurate method for the analysis of thermally induced vibrations of viscoelastic plates. The usual assumptions of small deflections and linear, homogeneous, isotropic viscoelastic properties are made. It is further assumed that within the range of temperature changes the viscoelastic material parameters remain constant. The pertinent equations for the vibration analysis of viscoelastic plates with effects of temperature are first presented in section 2. This is then followed in section 3 by an illustration of the method. 2. DERIVATION
OF THE BASIC EQUATIONS
Consider a viscoelastic plate of constant thickness h and uniform density p situated in a non-stationary temperature field. The plate is referred to a system of rectangular coordinates x, y, z such that the horizontal plane OXY coincides with the middle surface of the plate in the undeformed configuration. The deflection w (x, y, r) of the plate, which is a function of the spatial coordinates, x and y, and the temporal variable, 7, is taken as 31
0022-460X/80/210037 +09 $02.00/O
0 I980 Academic
Press Inc. (London
1Limited
32
J. MAZUMDAR.
D. HII.1.
AND
D. I.
CLEMENTS
positive downward in the z-direction. An incremental increase in temperature, T’(x, y, z, r), in the plate will produce a state in which thermal stresses are evident. The stress-strain equations are given by [2] P’Sij = Q’F~,,
P”ukk = Q”(ekk - 3a,T’),
(I, 3)
where (or is the coefficient of linear thermal expansion of the material, S,, and pi, are the components of the deviatoric stress and strain tensors, respectively, (Tkkand ei;k are the traces of the stress and strain tensors, and P’, Q’, P” and Q” are linear differential operators in time variables. For small thickness, the temperature variation across the thickness of the plate can be considered to be linear: i.e., T’(x, y, z, 7) = zT(x, y, 7).
(3)
The temperature fields are assumed to be such as to produce a deflection which is small. At any instant of time, ro, intersections of the middle surface of the plate with the parallels t = constant lead to the formation of lines of equal deflection u (x, y) = constant, which are iso-amplitude contours. A consideration of forces in the z-direction inside and on a contour u(x, y) = constant, including the effect of inertia forces which are especially important for suddenly induced temperature fields, gives rise to the dynamical equation [5] f
W+jj
2
ph$dR=O.
(4)
a.
cu
The temperature term is introduced through the transverse forces V,,, the expressions for moments and shearing forces being as follows: M* = -~(p)[~2Wl~X2)+CL(p)(~2Wl~y2~1-~(p)[1
+/J(p)IwT,
My = -~(P)[a2wlaY2)+CL(p)(~2WlaX2)1-~(p)[1
+PL(p)lwT
Ky = -My, = D(pNl-/4pm2wl~x~Y, +/.~(p)lw@T/dxL
0, = -o(p)(a/ax)[(a’~lax~)+(a’w/ay’)l-D(p)[l
Q,= -D~p~~~l~y~~~~2~l~x2~+~~2~l~~‘~l-~~~~~~+~~(p)lwW/~yh
(5)
where p = d/h, and D(p) and p (p) are viscoelastic operators corresponding, respectively, to the elastic constants D and CL.By using the procedure outlined in reference [5], the dynamical equation (4) can finally be reduced to the integro-differential form
C”
fL
where t = uf + U; and the expressions for RI, Fl and G; are listed in reference [5]. The solution of the above equation can be found in the same way as for the free vibration problem as follows. Firstly, the eigenfunctions M’i and the eigenfrequencies Ai of the corresponding elastic problem under homogeneous boundary conditions are determined. It is well-known that this eigenvalue problem is self-adjoint and hence the associated eigenfunctions form a complete set and are mutually orthogonal. The viscoelastic solution is then obtained by assuming that the deflection w and the temperature T appearing in the
THERMALLY
INDUCED
PLATE
33
VIBRATIONS
above equation can be expressed as a linear sum of the eigenfunctions
Wi in the form
T= z Ti(T)W,(II).
W = f gi(7)Wi(u)t i=l
t 7)
1=l
Since Wi is a solution of the corresponding elastic equation, differentiating with respect to u and using expressions (7) yields
equation (6)
Here use has been made of Green’s Theorem in establishing the result -a au
f(
VT&. Jr
cc.
(9)
Lu
Using the heat conduction equation for a plate, one can write equation (8) in the form (10) where u is the specific heat and k is the thermal conductivity. Equation (10) will uncouple into an infinite number of linear differential equations for the gi(7) if one uses the expression (7) for T and then the orthogonality properties of the eigenfunctions W, : viz., Ll) In the next section, a procedure is outlined to indicate how the above equation can be solved for the gj(T) for a particular viscoelastic material.
3. ILLUSTRATION 3. I. EXAMPLE OF A CIRCULAR PLATE Some information concerning the engineering aspects of viscoelastic plates can be obtained by studying the behaviour of simple models. Therefore, as an example of the application of this technique, the vibration of a viscoelastic circular plate, of radius a, under a thermal shock at its centre, and with radiation at the boundary into a medium at zero temperature, can now be discussed. The boundary of the plate is assumed to be completely fixed. Problems of this type very frequently occur in both welding engineering and in nuclear reactors. Since only symmetric modes of vibration are involved in this example, the eigensolutions for the corresponding circular elastic problem can be easily obtained from those of an elliptical elastic plate [5], W = I,(ki)Jo(kif)-
J,,(ki)In(kf),
112)
where f2=l-U
=(X2+~*)/a2,
k; = pha%;
113)
and the ki’S satisfy the determinant Jo(k)
h,(k) =
J&(k)
IL(k)
I
o,
(14)
34
J. MAZUMDAR.
The orthogonality
D. HILL
AND
D. I,. (‘LEMENTS
relation in this case is
I
f Wi Wj df
=
i
6;jB;,
15,
where Bi =$Ig(kj)[Ji(kj)+JT(kj)]+tJi(kj)[Ig(kj)-I:(kj)] well-known Bessel functions. 3.2.
TEMPERATURE
and Jo, J1, IO and I1 are the
EFFECTS
With the eigenfunctions and eigenvalues thus having been determined, the next task is to analyze the effects of temperature in the region of the plate. The distribution of temperature can be obtained by using any of the standard techniques in heat conduction problems [7,8]. Laura et al. [9, lo] used an interesting conformal mapping technique for the study of two dimensional heat conduction problems whereas Mazumdar [ 1 l] proposed a new method which is based upon the concept of isothermal contour lines on the surface of the plate. In the present analysis this latter method is used for evaluation of the temperature T appearing in equation (lo), which, in accordance with the results given in reference [ll], is of the form
(16) where 9 is the strength of the point heat source, the w, are the positive roots of wJ,(wa) = hrJo(wa), and hr = H/k, H being the exterior conductivity. T = f
(17)
Equation (16) can also be written as
c,,Jo(o,nf) e -cz”fT,
(18)
n=l
where the notation c,, =qw;/~a~(h: has been used. Upon using the orthogonality relationship
+w:)J;(w,a)
(19)
relation (15) and expression (7) for T one obtains the
I
f TWj df = BiTi(
(20)
0 which, in conjunction
with equations (18) and (12), becomes (21)
where the quantities 4, are given by 1
1
An= b(kj) \ fJo(kif)Jo(wnaf) df -Jo(kj) [ fIo(kif)Jo(wnaf) df. 0
0
(22)
THERMALLY
INDUCED
PLATE
3s
VIBRATIONS
By using the following formulae for Bessel integrals [12], I
~yJo(p)Jb((y)-PJo(~y)Jb(P)
J
xJo(cyx)J,(/3x) dx =
p2_cu2
’
0
1
aJoW)Ibb)
J
the expressions obtained. 3.3.
TIME
xIo((~x)Jo@x) dx = --
-PJI,Wob)
(23)
p2ca’
for the Iin’s can be evaluated
exactly and hence Tj(T) can be easily
FUNCTIONS
In order to solve equation (11) for the time functions gi(T), it is assumed that the viscoelastic properties of the plate are of the “Kelvin” type, for which one has the relationship D(p)lD
(24)
= 1 +(n/G)p,
where n and G are material constants. Here it has been assumed that Poisson’s ratio is a constant quantity. The differential equation (11) thus takes the form of a second order ordinary differential equation, {p2+(77/G)A:p+A:}gi(7)=-_(
(~~41
+~L~/khl{p
+(~/GJp21~ji(~).
125)
where Ti(r) is as given by equation (21). In order to solve this differential equation with the aid of the Laplace transform method, one needs to know the corresponding initial conditions. By using the method developed by DeLeeuw [13] for determining initial conditions for the time equation of a viscoelastic plate, which is based on an earlier method of Boley and Weiner [7] for a general viscoelastic stress-strain law in linear operator form, one obtains the initial conditions gj(O) = -A(nIG)Ti(O),
g~(0)=A[{(~*/G2)A~~1}T~(O)-(~/G)T~(O)]~
where the prime denotes differentiation
(26)
with respect to T and A is given by
A = ad1
+k)D/kh.
(27)
Equation (25) then has the following Laplace transform, incorporating tions (26): Pj(s)gj(s)=-A(s+ps*)~(s),
the initial condi(28)
where Pj(S)=S*+PA:S+A;
=(S-Sl)(S-Sl).
P=dG
(29) and the overscores denote Laplace transforms, s being the Laplace transform parameter. The solution to equation (28) can be obtained by inverse Laplace transformation:
S+pS’ (s - Sl)(S - s&s +
I
c’co’,,
(30)
J. MAZUMDAR.
36
D. HILL
AND
D.
L. CLEMENTS
642-
Radius,
Figure (D/a3q)w.
r
1. Deflection profiles for a circular viscoelastic plate under thermal (a) Initial deflection profile, 7 = Ofs; (b) deflection profile at 0.06 s.
-4
2. Amplitude
at the centre;
0
Figure
shock
1
1
I
I
I
2
3
4
of vibrations
1 5
1 6
with time. /3 = 0.0025;
1 7
T* = (l/a’dD/phT,
w*
THERMALLY
INDUCED
PLATE
37
VIBRATIONS 24F
'
'
'
'
'
'
'
0
0.5
r*x IO5 Figure
3. (a) Variation /3 = 0.1.
of gl(r)
'-1
(b)
20 -
I.0
I.5
2.0
r*x IO 7
corresponding
to the first mode;
lb) variation
of gZ(7) corresponding
tcl the
second mode.
16 14 12 Q
?? : I0
I
6 6 4 2 0 -2 -4 -6
Figure
4. Variation
of (a) g,(s)
and (b) g2(T) for selected
values of the damping
parameter
p.
yields (31)
where Al,, A2”, A3,, are a set of constants depending on the roots of the denominator of equation (30) as well as on the damping parameter /3. Clearly, when the two roots sl and s2 are two complex conjugates with negative real parts the result corresponds to a damped oscillatory motion.
J.
38
MAZUMDAR,
D. HILL
AND
D. I,. C‘LEMENTS
Substituting expression (31) for gi(7) in equation (71, then yields the deflection
ampli-
tude at any point in the plate. 3.4.
RESULTS
Figure l(a) is a graph of the deflection of the plate just after the initial shock has been applied. As time proceeds the general shape of the deflection remains similar in form but with the amplitude reducing, as is evident from Figure l(b). In Figure 2, the amplitude of vibration is plotted against time for various points in the plate. Figures 3(a) and 3(b) show the variations of the time functions gr(r) and g2(r) corresponding to the first and second modes of vibration, respectively. The third modal function g3(r) was found to have values less than 5 x 1O-9 with successive modes giving successively less contribution to the final answers, so that a good approximation can be obtained by using only the first two modes of vibration. Figures 4(a) and (b) show the effects of variation of the damping parameter p. It can be observed that for values of /3 less than 2/hi( = 0*017), the plate undergoes damped oscillation.
4. CONCLUSION
A method has been presented for the study of thermally induced vibrating motion of viscoelastic plates. It has been shown that the method is useful for approximation of the response, as the problem can be separated into two parts: (i) finding spatial functions which approximate the mode shapes in vibration and satisfy the boundary conditions and an orthogonality relation, and (ii) solving a time response equation which depends on the viscoelastic model chosen, The mode shapes are chosen to be the vibrational modes of the associated elastic problem, so that if these and the corresponding frequencies can be found for a plate of a particular geometry, then the associated viscoelastic problem can be easily solved.
ACKNOWLEDGMENTS The authors wish to acknowledge the support of the Australian Research Grants Committee who sponsored this investigation under Grant No. F78/15085. The authors also wish to thankfully acknowledge the valuable comments of J. Hewitt during the preparation of the manuscript.
REFERENCES and E. H. LEE 1960 Transactions of the Society of Rheology IV, 233-263. Stress analysis for linear viscoelastic materials with temperature variation. R. MUKI and E. STERNBERG 1961Journal of Applied Mechanics 28, 193-207. On transient thermal stresses in viscoelastic materials with temperature-dependent properties. W. NOWACKI 1962Advances in Aeronautical Sciences 4,947-969. Transient thermal stresses in viscoelastic plates and shells. T. R. TAUCHERT 1968 Acta Mechanica 6, 239-252. Transient temperature distributions in viscoelastic solids subject to cyclic deformations. J. S. HEWrnand J.MAZUMDAR 1974JournalofSoundand Vibration 33,319-333. Vibration of viscoelastic plates under transverse load by the method of constant deflection contours. K. NAGAYA 1977Journal of Engineering for Industry, Transactions of the American Society of Mechanical Engineers, Series B 99,404-409. Vibrations and dynamic response of viscoelastic plates on non-periodic elastic supports. B. A. BOLEY and J.H. WEINER 1960 Theory of Thermal Stresses. New York: John Wiley.
1. L. W. MORLAND 2. 3. 4. 5. 6.
7.
THERMALLY
INDUCED
PLATE
VIBRATIONS
8. P. BISWAS 1978 Journal of Sound and Vibration 59,304-306. 9. 10. 11. 12. 13.
39
Thermally-induced vibrations of a triangular slab on elastic foundation. P. A. LAURA and A. J. FAULSTICH, JR. 1968 International Journal Heat Mass Transfer '11, 297-303. Unsteady heat conduction in plates of polygonal shape. P. A. LAURA and R. ERCOLI 1972 NuclearEngineering and Design 23, 1-9. A solution of the unsteady diffusion equation in an arbitrary, doubly connected region. J. MAZUMDAR 1974 NuclearEngineeringand Design 31,383-390. A method for the study of transient heat conduction in plates of arbitrary cross section. G. N. WATSON 1944 A Treatise on the Theory of BesselFunctions. Cambridge University Press, second edition. S. E. DELEEUW 1961 Ph.D. Thesis, Michigan State Unicersity. Behaviour of viscoelastic plates under the action of in-plane forces.