Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 1, pp. 255-266, Warsaw 2010 VIBRATION OF NON-HOMOGENEOUS VISCO-ELASTIC CIRCULAR PLATE OF LINEARLY VARYING THICKNESS IN STEADY STATE TEMPERATURE FIELD Arun K. Gupta M. S. College, Mathematics Department, Saharanpur, India e-mail: gupta arunnitin@yahoo.co.in Lalit Kumar Kisan (PG) College, Mathematics Department, Simbhaoli, Ghaziabad (U.P.) India e-mail: abbutyagi@yahoo.com An analysis is presented for free vibration of a non-homogeneous visco- elastic circular plate with linearly varying thickness in the radial direc- tion subjected to a linear temperature distribution in thatdirection.The governingdifferential equationofmotion for free vibration is obtainedby the method of separation of variables. Rayleigh-Ritz’s method has been applied. Deflection, time period and logarithmic decrement correspon- ding to the first twomodes of vibrations of a clamped non-homogeneous visco-elastic circular plate for various values of non-homogeneity para- meter, taper constant and thermal gradients are obtained and shown graphically for the Voigt-Kelvinmodel. Keywords:non-homogeneous,visco-elastic, circularplate, variable thick- ness, steady state temperature field 1. Introduction In recent years, an interest towards the effect of temperature on vibration of plates of variable thickness are often encountered in engineering applications. Their use in machine design, nuclear reactor technology, naval structures and acoustical components is quite common. The reason for these is that during heating up periods, structures are exposed to high intensity heat fluxes and material propertiesundergosignificant changes; inparticular the thermal effect can not be taken as negligible. 256 A.K. Gupta, L. Kumar Many analyses show that plate vibrations are based on non-homogeneity of materials. Non-homogeneity can be natural or artificial. Non-homogeneous materials such as plywood, delta wood, fiber-reinforced plastic, etc. are used in engineering design and technology to strengthen the construction. There are some artificial non-homogeneous materials such as glass epoxy and boron epoxy in steel alloys for making rods in nuclear reactors. Consideration of visco-elastic behaviour of the plate material, together with its variation in thickness, of structural components not only ensures re- duction in the rate and size but also meets desirability for high strength in various technological situations in the aerospace industry, ocean engineering and electronic and optical equipment. In a survey of the recent literature, the authors have found that no work deals with vibration of non-homogeneous visco-elastic circular plates of va- riable thickness subject to thermal gradient. Several authors (Li and Zhou, 2001, Tomar and Gupta, 1983, 1985; Tomar and Tewari, 1981) studied the effect of thermal gradient on vibration of a homogeneous plate of variable thickness. Singh and Saxena (1995) discussed the transverse vibration of qu- arter of a circular plate with variable thickness. It is well known (Hoff, 1958) that in the presence of thermal gradient, the elastic coefficient of homogeneous materials becomes a function of space variables. Lal (2003) studied transver- se vibrations of orthotropic non-uniform rectangular plates with continuously varying density. Warade and Deshmukh (2004) discussed thermal deflection of a thin clamped circular plate due to partially distributive heat supply. So- botka (1971) discussed rheology of orthotropic visco-elastic plates. Gupta and Khanna (2007) studied the effect of linearly varying thickness on vibration of visco-elastic rectangular plates of variable thickness. Recently, Gupta andKu- mar (2008) analysed vibration of non-homogeneous visco-elastic rectangular plates with linearly varying thickness. The presentwork dealswith vibration of clamped non-homogeneous visco- elastic circular plates with linearly varying thickness in the radial direction subjected to a linear temperature distribution in this direction for the Voigt- Kelvinmodel.Thenon-homogeneity is assumed to arise due to linear variation in density of the platematerial in the radial direction. Rayleigh-Ritz’s method hasbeenapplied toderive the frequencyequationof theplate.Thetimeperiod, deflection and logarithmic decrement for the first twomodes of vibrations are calculated for various values of thermal constants, non-homogeneity parameter and taper constant at different points of a clamped non-homogeneous visco- elastic circular plate with linearly varying thickness. Vibration of non-homogeneous visco-elastic circular plate... 257 2. Equation of transverse motion The axisymmetric motion of a circular plate of the radius a is governed by the equation (Leissa, 1969) r ∂ ∂r [1 r ( ∂ ∂r (rMr)−Mθ )] = ρh ∂2w ∂t2 (2.1) The resultant moments Mr and Mθ for a polar visco-elastic material of the plate are Mr =−D̃D (∂2w ∂r2 + ν r ∂w ∂r ) Mθ =−D̃D (1 r ∂w ∂r +ν ∂2w ∂r2 ) (2.2) where D= Eh3 12(1−ν2) (2.3) and D̃ is the visco-elastic operator. The deflection w can be sought in the form of product of two functions as follows w(r,θ,t)=W(r,θ)T(t) (2.4) where W(r,θ) is the deflection function and T(t) is the time function. Using equations (2.2) and (2.4) in (2.1), one gets rD ∂4W ∂r4 + ( D+2r ∂D ∂r )∂3W ∂r3 + ( −2 D r +(1+ν) ∂D ∂r +r ∂2D ∂r2 )∂2W ∂r2 + + ( 2 D r2 − 1+ν r ∂D ∂r +ν ∂2D ∂r2 )∂W ∂r −ρhp2W =0 (2.5) d2T dt2 +p2D̃T =0 where p2 is a constant. These equations are expressions for transverse motion of a non- homogeneous circular plate with variable thickness and a differential equation of the time function for free vibration of the visco-elastic plate, respectively. 3. Analysis of equation of motion Assuming a steady temperature field in the radial direction for a circular plate as τ = τ0 ( 1− r a ) (3.1) 258 A.K. Gupta, L. Kumar where τ denotes the temperature excess above the reference temperature at any point at the distance r/a from the centre of the circular plate of the ra- dius a and τ0 denotes the temperature excess above the reference temperature at r=0. The temperature dependence of the modulus of elasticity for most struc- tural materials is given as (Nowacki, 1962) E(τ) =E0(1−γτ) (3.2) where E0 is the value of Young’s modulus at the reference temperature, i.e. τ = 0 and γ is the slope of variation of E with τ. The module variation, in view of expressions (3.1) and (3.2), becomes E(r)=E0 [ 1−α ( 1− r a )] (3.3) where α= γτ0 (0¬α< 1) is a parameter known as thermal gradient. The expression for themaximumstrain energy Vmax andmaximumkinetic energy Tmax in the plate, when it vibrates with themode shape W(r,θ), are given as (Leissa, 1969) Vmax = 1 2 2π∫ 0 a∫ 0 D {(∂2W ∂r2 + 1 r ∂W ∂r + 1 r2 ∂2W ∂θ2 )2 + −2(1−ν) [∂2W ∂r2 (1 r ∂W ∂r + 1 r2 ∂2W ∂θ2 ) − ( ∂ ∂r (1 r ∂W ∂θ ))2]} r dθdr (3.4) Tmax = 1 2 p2 2π∫ 0 a∫ 0 ρhW2r dθdr It is assumed that the thickness and non-homogeneity varies in the r- direction only, consequently the thickness h, non-homogeneity ρ and flexural rigidity D of the plate become a function of r only. Assume themode shape as (Ramaiah and Kumar, 1973) W(r,θ)=W1(r)cosθ (3.5) taking W1(r)= rW1(r) as the integration contains a negative power of r and introduce non-dimensional quantities R= r a h= h a W = W1 a D= D a3 ρ= ρ a (3.6) Vibration of non-homogeneous visco-elastic circular plate... 259 Now let us assume the thickness and non-homogeneity of the plate to be h(R)=h0(1−βR) ρ(R)= ρ0(1−α3R) (3.7) where h0 =h|R=0 and ρ0 = ρ|R=0. Using equations (3.5), (3.6) and (3.7) in equations (3.4), one gets Vmax = πa5E0h 3 0 24(1−ν2) 1∫ 0 (1−α+αR)(1−βR)3 {( 3 dW dR +R d2W dR2 )2 + −2(1−ν) [dW dR (dW dR +R d2W dR2 )]} RdR (3.8) Tmax = πa8p2ρ0h0 2 1∫ 0 (1−α3R)(1−βR)R 3W 2 dR 4. Solutions and frequency equation Rayleigh-Ritz technique requires that the maximum strain energy must be equal to themaximumkinetic energy. It is, therefore, necessary for theproblem under consideration that δ(Vmax −Tmax)= 0 (4.1) forarbitraryvariation of W satisfying relevantgeometric boundaryconditions. For a circular plate clamped at the edges r= a, i.e. R=1, the boundary conditions are W = dW dR =0 at R=1 (4.2) and the corresponding two terms of deflection function is taken as W(R)=C1(1−R) 2+C2(1−R) 3 (4.3) where C1 and C2 are undetermined coefficients. Now using equations (3.8) in equation (4.1), one has δ(V1−p 2ℓT1)= 0 (4.4) 260 A.K. Gupta, L. Kumar where V1 = 1∫ 0 (1−α+αR)(1−βR)3 {( 3 dW dR +R d2W dR2 )2 + −2(1−ν) [dW dR (dW dR +R d2W dR2 )]} RdR (4.5) T1 = 1∫ 0 (1−α3R)(1−βR)R 3W 2 dR Here ℓ= 12(1−ν2)ρ0a 3 E0h 2 0 (4.6) Equation (4.4) involves the unknowns C1 and C2 arising due to substitution of W(R) from (4.3). These unknowns are to be determined from equation (4.4), for which ∂ ∂Cn (V1−p 2ℓT1)=0 n=1,2 (4.7) Equation (4.7) simplifies to the form bn1C1+ bn2C2 =0 n=1,2 (4.8) where bn1, bn2 (n=1,2) involve the parametric constant and frequency para- meter. For a non-trivial solution, the determinant of the coefficient of equation (4.8) must be zero. Thus, one gets the frequency equation as ∣∣∣∣∣ b11 b12 b21 b22 ∣∣∣∣∣=0 (4.9) where b11 =2(F1+B1p 2) b12 = b21 =F2+B2p 2 b22 =2(F3+B3p 2) Here F1, F2, F3 are functions of α, β and B1,B2,B3 are functions of α3. Frequency equation (4.9) is a quadratic one with respect to p2 fromwhich two values of p2 can be found. Choosing C1 = 1, one obtains C2 = −F4/F5 where F4 = 2(F1 + p 2B1), F5 =F2+p 2B2, therefore W(R)= (1−R)2− F4 F5 (1−R)3 (4.10) Vibration of non-homogeneous visco-elastic circular plate... 261 5. Time function of vibration of non-homogeneous visco-elastic plate The time function of free vibration of the visco-elastic plate is defined by general ordinary differential equation (2.5)2. Its form depends on the visco- elastic operator D̃. ForKelvin’smodel, one hasGupta andKumar (2008) and Sobotka (1978) D̃≡ 1+ η G d dt (5.1) Taking the temperature dependence of shear modulus G and visco-elastic coefficient η in the same form as that of Young’s modulus, one has G(R)=G0[1−α1(1−R)] η(R)= η0[1−α2(1−R)] (5.2) where G0 is the shear modulus and η0 is the visco-elastic constant at some reference temperature, i.e. at τ =0. Using equations (5.1) and (5.2) in equation (2.5)2, one obtains d2T dt2 +p2q dT dt +p2T =0 (5.3) where q= η0[1−α2(1−R)] G0[1−α1(1−R)] Equation (5.3) is a differential equation of the second order for the time func- tion T . Solving equation (5.3), one gets T(t)= e− p 2 qt 2 (e1 cosst+e2 sinst) (5.4) where s2 = p2− 1 4 p4q2 and e1, e2 are integration constants. Assuming that the initial conditions are T =1 and dT dt =0 at t=0 (5.5) and using condition (5.5) in equation (5.4), one gets T(t)= e− p 2 qt 2 ( cosst+ p2q 2s sinst ) (5.6) 262 A.K. Gupta, L. Kumar Thus, the deflection w(r,θ,t) may be expressed as w(r,θ,t) =W(R)e− p 2 qt 2 ( cosst+ p2q 2s sinst ) cosθ (5.7) The time period of vibration of the plate is given by K = 2π p (5.8) where p is the frequency given by equation (4.9). The logarithmic decrement of vibration is given by Λ= ln w2 w1 (5.9) where w1 is thedeflection at anypoint of theplate at the timeperiod K =K1, and w2 is the deflection at the same point at the time period succeeding K1. 6. Results and discussion The deflection w, time period K and logarithmic decrement Λ are computed for a non-homogeneous clamped visco-elastic circular plate with linearly va- rying thickness for different values of taper constant β, thermal constants α, α1,α2 and non-homogeneity constant α3 and different points for the first two modes of vibrations. The results are shown in Figs. 1-6. For numerical computation, the following material parameters are used (Nagaya, 1977): E0 =7.08 ·10 10n/m2, G0 =2.682 ·10 10n/m2, η0 =1.4612 · 106n.s/m2, ρ0 =2.8 ·10 3kg/m3, ν =0.345. The thickness of the plate at the center is taken as h0 =0.01m. Figure 1 shows that the time period K of the first twomodes of vibration decreases with an increase in the non-homogeneity parameter α3, and whe- never the taper constant β and thermal constant α increase then the time period increases for the first two modes of vibrations. Figures 2 and 3 show that the deflection w starts from themaximumvalue to decrease to zero for the firstmode of vibration, but for the secondmode of vibration the deflection starts from the minimum value to grow and decrease again to finally become zero for a fixed value of θ and increasing R for the initial time 0K and 5K and uniform thickness. Figures 4 and 5 show thatwhen the taper constant β increases, the deflec- tion for the firstmodeof vibrationfirstly increases tomaximumthendecreases Vibration of non-homogeneous visco-elastic circular plate... 263 Fig. 1. Variation of time period with non- homogeneity constant of visco-elastic non- homogeneous circular plate with linearly varying thickness Fig. 2. Transverse deflection w vs. R of visco-elastic non-homogeneous circular plate with linearly varying thickness at initial time 0K Fig. 3. Transverse deflection w vs. R of visco-elastic non-homogeneous circular plate with linearly varying thickness at initial time 5K 264 A.K. Gupta, L. Kumar and finally becomes zero, but for the second mode of vibration the deflection starts from the minimum value to increase and then decrease again finally reaching zero for a fixed value of θ and increasing R for the initial time 0K and 5K. Fig. 4. Transverse deflection w vs. R of visco-elastic non-homogeneous circular plate with linearly varying thickness at initial time 0K Fig. 5. Transverse deflection w vs. R of visco-elastic non-homogeneous circular plate with linearly varying thickness at initial time 5K Figure 6 shows that the logarithmic decrement Λ decreases with an incre- ase in Rbut it remains the same for afixedvalueof Randdifferentvalues of θ. It can be seen in Fig.6 that as the non-homogeneity parameter α3 increases, the logarithmic decrement Λ decreases for the first twomodes of vibration. Vibration of non-homogeneous visco-elastic circular plate... 265 Fig. 6. Logarithmic decrement Λ vs. R of visco-elastic non-homogeneous circular plate with linearly varying thickness References 1. Gupta A.K., Khanna A., 2007, Vibration of visco-elastic rectangular plate with linearly thickness variations in both directions, J. Sound and Vibration, 301, 3/5, 450-457 2. GuptaA.K.,KumarL., 2008,Thermaleffectonvibrationofnon-homogenous visco-elastic rectangular plate of linear varying thickness,Meccanica, 43, 47-54 3. Hoff N.J., 1958, High Temperature Effect in Aircraft Structures, Pergamon Press, NewYork 4. Lal R., 2003, Transverse vibrations of orthotropic non-uniform rectangular plates with continuously varying density, Indian J. Pure Appl. Math., 34, 4, 587-606 5. Leissa A.W., 1969, Vibration of Plates, NASA SP-160, U.S. Govt. Printing Office 6. Li S.R., Zhou Y.H., 2001, Shooting method for non linear vibration and thermal buckling of heated orthotropic circular plate, J. Sound and Vibration, 248, 2, 379-386 7. Nagaya K., 1977, Vibration and dynamic response of viscoelastic plates on non-periodic elastic supports, Journal of Engineering for Industry, Trans. of the ASME, Series B99, 404-409 8. Nowacki W., 1962,Thermoelasticity, PergamonPress, NewYork 9. Ramaiah G.R., Kumar V.K., 1973, Natural frequencies of polar orthotropic plates, J. Sound and Vibration, 26, 4, 517-531 266 A.K. Gupta, L. Kumar 10. Singh B., Saxena V., 1995, Transverse vibration of quarter of circular plate with variable thickness, J. Sound and Vibration, 183, 1, 49-67 11. Sobotka Z., 1971, Rheology of orthotropic visco-elastic plates, Proc. of 5th International Congress on Rheology, Univ. of Tokyo Press, Tokyo Univ., Park Press, Baltimore, 175-184 12. Sobotka Z., 1978, Free vibration of visco-elastic orthotropic rectangular pla- tes,Acta Technica, CSAV, 6, 678-705 13. Tomar J.S., Gupta A.K., 1983, Thermal effect on frequencies of an ortho- tropic rectangular plate of linearly varying thickness, J. Sound and Vibration, 90, 3, 325-331 14. Tomar J.S., Gupta A.K., 1985, Thermal effect on axisymmetric vibrations of an orthotropic circular plate of parabolically varying thickness, Indian J. Pure Appl. Math., 16, 5, 537-545 15. Tomar J.S., Tewari V.S., 1981, Effect of thermal gradient on frequencies of a circular plate of linearly varying thickness, J. Non-Equilibrium. Thermody- namics, 6, 115-122 16. Warade R.W., Deshmukh K.C., Thermal deflection of a thin clamped cir- cular plate due to a partially distributive heat supply,Ganita, 55, 2, 179-186 Drgania niejednorodnej lepko-sprężystej płyty kołowej o liniowo zmiennej grubości i ustalonym polu temperatury Streszczenie W pracy przedstawiono analizę drgań swobodnych niejednorodnej lepko- sprężystej płyty kołowejo liniowo zmiennej grubościwkierunkupromieniowymi pod- danej polu temperatury o liniowym rozkładzie w tym kierunku. Konstytutywne rów- nanie różniczkowe ruchu dla drgań swobodnych otrzymano poprzez separację zmien- nych. Zastosowano metodę Rayleigha-Ritza. W wyniku analizy wyznaczono ugięcie płyty, okres drgań i logarytmiczny dekrement tłumienia dwóch pierwszych postaci drgań dla warunkówbrzegowych odpowiadających zamocowaniu niejednorodnej pły- ty na brzegu.Wyniki przedstawiono graficzniew funkcji parametru niejednorodności, stałej zawężania grubości oraz zmiennego gradientu temperatury przy wykorzystaniu modelu reologicznegoKelvina-Voigta opisującego właściwości materiału płyty. Manuscript received December 23, 2008; accepted for print November 2, 2009