EFFECT OF DAMPING AND THERMAL GRADIENT ON VIBRATIONS OF ORTHOTROPIC RECTANGULAR PLATE OF VARIABLE THICKNESS

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1 Available at Appl. Appl. Math. ISSN: Vol. 1, Issue 1 (June 17), pp Applications and Applied Mathematics: An International Journal (AAM) EFFECT OF DAMPING AND THERMAL GRADIENT ON VIBRATIONS OF ORTHOTROPIC RECTANGULAR PLATE OF VARIABLE THICKNESS 1 U.S. Rana and Robin Abstract 1 Department of Mathematics D.A.V. College, Dehradun drusrana@ahoo.co.in & Department of Mathematics DIT Universit, Dehradun robin1986sr@gmail.com Received June, 16; Accepted Januar 1, 17 In this present paper, damped vibrations of an orthotropic rectangular plate resting on elastic foundation with thermal gradient is modeled, considering variable thickness of plate. Following Le`v approach, the governed equation of motion is solved numericall using quintic spline technique with clamped and simpl supported edges. The effect of damping parameter and thermal gradient together with taper constant, densit parameter and elastic foundation parameter on the natural frequencies of vibration for the first three modes of vibration are depicted through Tables and Figures, and mode shapes have been computed for fied value of plate parameter. It has been observed that the rate of decrease of frequenc parameter with damping parameter D k for C-SS plate is higher than that for C-C plate keeping all other parameter fied. Kewords: Damping; Thermal gradient; Taperness; Elastic foundation; Orthotrop MSC 1 No.: 41A15, 81V99 1. Introduction Structural damping is essential for design engineering as damping materials enhance the performance of sstem b increasing structural stiffness and thermal stabilit and the effect of temperature on mechanics of solid bodies has acquired great interest because of rapid development in space technolog, high speed atmospheric flights and nuclear energ applications. Numerous studies on free vibration for isotropic/orthotropic plates of uniform/nonuniform thickness with or without temperature effect have been reported. Among these, Laura 1

2 Rana and Robin and Gutierrez (198) discussed vibration analsis of a rectangular plate subjected to a thermal gradient. Tomar and Gupta (1983, 1985) analzed the effect of thermal gradient on frequencies of an orthotropic rectangular plate with variable thickness. In this series, Gupta et al. (7) observed the thermal effect on vibration of non-homogeneous orthotropic rectangular plate having bi-directional parabolicall varing thickness. Effect of non-homogeneit on thermall induced vibration of orthotropic visco-elastic rectangular plate of linearl varing thickness studied b Gupta and Singhal (1). Gupta et al. (13) had evaluated the effect of thermal gradient on the vibration of parallelogram plate with linearl varing thickness in both directions and thermal effect in linear form onl. Effect of thermal gradient on vibration of nonhomogeneous orthotropic trapezoidal plate of varing thickness has been studied b Gupta and Sharma (13). Khanna and Singhal (14) analzed the thermal effect on vibration of tapered rectangular plate. Gupta and Jain (14) analzed eponential temperature effect on frequencies of a rectangular plate of non-linear varing thickness using quintic spline technique. Plates resting on elastic foundation have applications in pressure vessels technolog such as petrochemical, marine and aerospace industr, building activities in cold regions and aircraft landing in arctic operations discussed b Mcfadden and Bennett (1991), and Civalek and Acar (7). In a series of papers, Lal et al., (1) have studied the transverse vibrations of a rectangular plate of eponentiall varing thickness resting on an elastic foundation. Transverse vibration of non-homogeneous orthotropic rectangular plate with variable thickness was discussed b Lal and Dhanpati (7). In man applications of vibration and wave theor the magnitudes of the damping forces are small in comparison with the elastic and inertia forces but these small forces ma have ver great influence under some special situations. Recentl, Robin and Rana (13) analzed the damped vibrations of isotropic/orthotropic rectangular plate with varing thickness resting on elastic foundation. The stud of damped vibration of infinite plate with variable thickness resting on elastic foundation was carried out b Rana and Robin (15). In realit all the vibrations are damped vibration and foundations of underling vibrational problems are elastic in nature, and thermal effects on the damped vibrations of orthotropic rectangular plates of variable thickness with elastic foundations is not considered b researchers but it plas a ke role in vibrational problems. Keeping this in view thermall induced damped vibrations of orthotropic rectangular plates with eponentiall thickness variation resting on Winkler foundation is presented here using quintic spline method on the basis of classical plate theor. Effect of damping parameter and thermal gradient together with thickness variation and foundation parameter on the frequencies has been illustrated for the first three modes of vibration for two different combinations of clamped, and simpl supported correct to four decimal places.. Mathematical Formulation Consider a non- homogeneous orthotropic rectangular plate of length a, breath b, thickness h(,) and densit ρ, with resting on a winkler- tpe elastic foundation k f occuping the domain a, b in the plane. The -and aes are taken along the principal directions and z aes is perpendicular to the plane. The middle surface being z = and the origin is at one of the corners of the plate. The differential equation which governs the damped transverse vibration of such plates is given b

3 AAM: Intern. J., Vol 1, Issue 1 (June 17) w w w H w H w D w D w D D 4 H D D w w D1 w D1 w D w w w 4 h K K, f w t t where (1) D E h / (1 ), D E h / (1 ), D E h /1(1 ), H D D, D G h 3 /1, E E, w(,, t) is the transverse deflection, t is the time, and E, E,, and G are material constants in proper directions defined b an orthotropic stress-strain law. Let the two opposite edges = and = b of the plate be simpl supported and thickness h h(, ) varies eponentiall along the length i.e., in the direction of -ais. Thus, h is independent of i.e., h h( ). Figure 1. Boundar conditions and vertical cross-section of the plate For a harmonic solution, the deflection function w satisfing the condition at = and = b, is assumed m t w(,, t) W ( )sin e cos pt, () b where p is the circular frequenc of vibration and m is a positive integer. Furthermore, for elasticall non-homogeneous material, it is assumed that Young s moduli E, E and densit are functions of space variable onl, and shear modulus is G E E / (1 ).

4 4 Rana and Robin Thus, Equation (1) becomes 3 6 Ehh ' 3 Eh h '' 6 h h ' E ' h E '' 3 ( iv) 3 m 3 Eh W 6 Eh h ' h E ' W ''' ( '' Eh W b cos 3 pt G(1 ) h ) m Eh h ' h E ' (1 )(3 h Gh ' h G ') W ' b m b m b 3 E h h '' 6 E hh ' 6 h h ' E ' h E '' W cos pt Eh 4 (3) ( p )cos pt 1(1 ) h 1(1 ) cos p sin pt K fw pt W. 1(1 ) k psin pt cos pt Introducing the non-dimensional variables a ; ; ; ; ; m, h E W H X E W a a a a a b and equating the coefficient of sin(pt) and cos(pt) independentl to zero, Equation (3) reduces to 3 3 ( iv) 3 3 E H H '' 6 E HH ' 6 H H ' E H E '' E H W 3 E H H ' H E ' W ''' W '' 3 3 ( E H G(1 ) H ) E H H ' H E ' (1 )(3 H GH ' H G ') W ' E H 3 E H H '' 6 E HH ' 6 H H ' E ' H E '' k W 1(1 ) a p H ak f 4H. (4) The eponential temperature variation for the plate is assumed in the following form

5 AAM: Intern. J., Vol 1, Issue 1 (June 17) 5 T e e e 1 T, (5) where T is the temperature ecess above the reference temperature at an point and T is the temperature ecess above the reference temperature at the end =. Furthermore, most of the engineering materials are found to have a linear relationship between the modulus of elasticit and temperature as discussed b Nowacki (196). Therefore, we get E E(1 T), (6) where E is the modulus of elasticit of the material at the reference temperature and is a constant. If the temperature at = 1 is assumed as the reference temperature, the modulus variable becomes E ( ) E 1 e e e 1, (7) where,( 1), is thermal gradient. T Substituting X e e e e X H He ; E E1 1 ; E E 1 ; e, e1 e1 where 1 H H, E E, E E,, X X X and is the taper constant due to eponentiall varing thickness of plate, and equating the coefficients, the following equation is formed d 4 W 3 d W 4 1 d W 3 dw 3 4, A A A A A W dx dx dx dx (8) where

6 6 Rana and Robin X e e A 1, e 1 X X e e e A1 31, e1 e1 e e e e e A e 1 e 1 e 1 X X X E (1 ) 1, E 1 X E X e e e A 3 (1 ) 3 E 1, 1 e 1 e 1 X X X 4 E e e e e e A (6 1) E 1 e 1 e 1 e 1 ( 4 ) X ( ) X 3 X 3 DKI * e I * e 1 E fe C *, (1 ) K a(1 ) K 1(1 ) a p DK, I, C, E,, E * 1 * 1 f 3 f 1 H H E1 E1 and, D K, E are frequenc parameter, damping parameter and elastic foundation parameter, f respectivel. is the densit of the plate and E1, Ethe Young s moduli in proper directions at =.The solution of Equation (8) together with boundar conditions at the edge = and = 1 constitutes a two-point boundar value problem. As the PDE has several plate parameters, it becomes quite difficult to find its eact solution. Keeping this in mind which is comple for the purpose of computation, the quintic spline interpolation technique is used. According to the spline technique, suppose W() be a function with continuous derivatives in [, 1] and interval [, 1] be divided into n subintervals b means of points X i such that... 1, X X1 X X n where X 1, Xi ix ( i,1,,..., n). n Let the approimating function W( X ) for the W() be a quintic spline with the following properties: (i) W( X ) is a quintic polnomial in each interval ( Xk, Xk 1). (ii) W( X ) W( X ), k,1,,..., n. k

7 AAM: Intern. J., Vol 1, Issue 1 (June 17) dw d W d W d W (iii),, and 3 4 dx dx dx dx are continuous. In view of the above aioms, the quintic spline takes the form 4 n1 i 5 i j j, (9) i j W ( X ) a a ( X X ) b ( X X ) where ( X X ) j, if X X ( X X j), if X X j j, and a s, b s are constants.,, i j Thus, for the satisfaction at the n th knot, Equation (8) reduces to A a [ A ( X X ) A ] a [ A ( X X ) A ( X X ) A ] a 4 4 m m 3 m [ A ( X X ) 3 A ( X X ) 6 A ( X X ) 6 A ] a n1 j 3 4 m 3 m r 1 3 [ A ( X X ) 4 A ( X X ) 1 A ( X X ) 4 A ( X X ) 4 A ] a m 3 m m 1 m 4 b [ A ( X X ) 5 A ( X X ) A ( X X ) 6 A ( X X ) j 4 m j * 3 m j * m j * 1 A ( X X ) ]. m j * 1 m j * (1) For the sstem (1) contains (n + 1) homogeneous equation with (n + 5) unknowns, a i, and b j, j =, 1,,, (n - 1), can be represented in matri form as [A][B]= [], (11) where [A] is a matri of order (n+1) (n+5), while [B] and [] are column matrices of order (n5). 3. Boundar Conditions and Frequenc Equation The following two cases of boundar conditions have been considered: (i) (C-C): clamped at both the edge X = and X = 1. (ii) (C-SS): clamped at X = and simpl supported at X = 1. The relations that should be satisfied at clamped and simpl supported, respectivel, are

8 8 Rana and Robin W dw, W dx dw. (1) dx Appling the boundar conditions C-C to the displacement function b Equation (9) one obtains a set of four homogeneous equations in terms of (n+5) unknown constants which can be written as [B cc ][B]= [], (13) where B cc is a matri of order 4(n+5). Therefore, Equation (11) together with the Equation (13) gives a complete set of (n+5) homogeneous equations having (n+5) unknowns which can be written as A [ B] []. cc B (14) For a non-trivial solution of Equation (11), the characteristic determinant must vanish, i.e., A cc B. (15) Similarl, for (C-SS) plate the frequenc determinant can be written as A ss B, (16) where B ss is a matri of order 4(n+5). 4. Numerical Results and Discussions The frequenc Equations (15) and (16) have been solved to get the values of the frequenc parameter Ω for various values of plate parameters. B putting m = 1 in frequenc equations, the numerical results have been computed for first three modes of vibrations. The elastic constants for the plate material ( ORTHOI ) are taken as E 11 Mpa, E 5 1 Mpa,., At the edge X =, the thickness h have been considered as.1. To choose the appropriate interval X, a computer program was developed for the evaluation of frequenc parameter Ω, and run for and for different sets of the values of parameters for both the boundar conditions. Figure represents the percentage error in the numerical values of frequenc parameter Ω up to fourth decimal places with the increase in the number of nodes. In all the computation n = 14 has been fied to achieve the decimal accurac.

9 % Error in frequenc parameter % Error in frequenc parameter AAM: Intern. J., Vol 1, Issue 1 (June 17) % Error in frequenc parameter for c-ss-c-ss plate. 4 % Error in frequenc parameter for c-ss-ss-ss plate n (number of nodes) (a) n(numbers of nodes) (b) Figure. Percentage error in frequenc parameter Ω: (a) for c-c plate (b) c-ss plate for a/b =.5, α =.4, β =.4, E f =., D k =.1;, First mode;, Second mode;,third mode ; Percentage error = [(Ω n - Ω 14 )/ Ω 14 ]1; The results are presented in Tables 1-3 and Figures 3-5, for different values of damping parameter D =.,.1,.,.3,.4, and.5, thermal gradient ɳ =.,.,.4,.6, k foundation parameter E f =.,., densit parameter =.5 and taper paremeter =.5 for both the boundar conditions C-C and C-SS, b taking the fied values of aspect ratio a/ b=1.. It is found that the values of frequenc parameter Ω for a C-C plate are greater than that for C-SS plate for the same set of values of plate parameters. Table 1 shows the values of frequenc parameter Ω with the increasing value of damping parameter D k for the fied value of densit parameter β =.5 and foundation parameter E f =., for first three modes of vibration of C-C and C-SS plates. Figure 3a shows the behavior of frequenc parameter Ω. It decreases with the increasing values of damping parameter D k for two different values of taper parameter α = ±.5, thermal gradient ɳ =.,.5 and densit parameter β =.5 for both the plates. The rate of decrease of Ω with damping parameter D k for C-SS is higher than that for C-C plate keeping all other plate parameters fied. This rate increases with the increase in the value of non taper parameter α and thermal gradient ɳ. A similar inference can be drawn from Figures 3b and 3c, when the plate is vibrating in the second mode as well as in the third mode of vibration ecept that the rate of decrease of Ω with D k is lesser as compared to the first mode. Tables and 3, provide the inference of thermal gradient ɳ on frequenc parameter Ω for two values of damping parameter D k =., and.1, taper parameter α = ±.5 densit parameter β = ±.5 and foundation parameter E f =.,. for the fied value of aspect ratio a/b = 1.. It is noticed that the frequenc parameter Ω decreases continuousl with the increasing value of thermal gradient ɳ for C-C and C-SS plates, whatever be the value of other plate parameters. It is

10 1 Rana and Robin found that the rate of decrease of frequenc parameter Ω for C-C plate is higher than C-SS plate for three modes. Figure 4a gives the inference of thermal gradient ɳ on frequenc parameter Ω for the first mode of vibration for fied values of β =.5 and α =.5. This rate decreases with the increase in the value of foundation parameter E f, and also increases with the increase in the value of damping parameter D k and the number of modes, as is clear from Tables 3b and 3c when the plate is vibrating in the second and third mode of vibration. Table 1. Values of frequenc parameter Ω for C-C and C-SS plate for a/b= 1. E f =., β=.5 α=-.5,ɳ=. α=-.5,ɳ=.5 α=.5,ɳ=. α=.5,ɳ=.5 MODE C-C C-SS C-C C-SS C-C C-SS C-C C-SS I Dk=. II III I Dk=.1 II III I Dk=. II III I Dk=.3 II III I Dk=.4 II III I Dk=.5 II III

11 AAM: Intern. J., Vol 1, Issue 1 (June 17) 11 Table. Values of frequenc parameter Ω for C-C and C-SS plate for a/b=1., α=.5, β=.5 Dk=., E f=. Dk=.1, Ef=. Dk=., Ef=. Dk=.1, Ef=. MODE C-C C-SS C-C C-SS C-C C-SS C-C C-SS I η=. II III I η=. II III I η=.4 II III I η=.6 II III Table 3. Values of frequenc parameter Ω for C-C and C-SS plate for a/b=1., E f =., D k =.1 α=-.5,β=-.5 α=-.5,β=.5 α=.5,β=-.5 α=.5,β=.5 MODE C-C C-SS C-C C-SS C-C C-SS C-C C-SS I η=. II III I η=. II III I η=.4 II III I η=.6 II III

12 frequenc parameter 1 Rana and Robin From Figure 4b, the effect of thermal gradient ɳ is found to decrease the frequenc parameter Ω; however, the rate of decrease gets increased to more than twice of the first mode for both the boundar conditions. In case of third mode, this rate of decrease further increases and becomes nearl twice of the second mode as is evident from Figure 4c. Figure 5a provides graphs of frequenc parameter Ω verses thermal gradient ɳ for the first mode of vibration. There is a continuous decrease in the value of frequenc parameter Ω for fied values of E f =. and damping parameter D k =.1 for both the boundar conditions. This rate decreases with the increase in the value of densit parameter β; it increases with the increase in the value of taper parameter α and in increases the number of modes, as is clear from Tables 3b and 3c, when the plate is vibrating in the second and third mode of vibration. From Figure 5b, the effect of thermal gradient ɳ is found to decrease the frequenc parameter Ω; however, the rate of decrease gets increased to more than twice of the first mode for both the boundar conditions. In case of third mode, this rate of decrease further increases and becomes nearl twice of the second mode as is evident from Figure 5c.The normalized displacements for the two boundar conditions C-C and C-SS are plotted in Figures 6 and 7, respectivel. The plate thickness varies parabolicall in X-direction and the plate is considered resting on elastic foundations E f =. with damping parameter D k =.1. Mode shapes for a rectangular plate i.e, a/b =.5 have been computed and observed that the nodal lines are seen to shift towards the edge, i.e., X = 1 as the edge X = increases in thickness for both the plates. No special change was seen in the pattern of nodal lines b taking different values of β and E f. The normalized displacements were differing onl at the third or fourth place after decimal for both the boundar conditions. (a) (b) (c) Damping parameter Figure 3. Natural frequencies for c-c and c-s plates: (a) First mode (b) Second mode (c) Third mode,for a/b=1.,β=.5, E f =.,,C-C; --,C-SS; Δ, α=-.5,ɳ=.,, α=-.5,ɳ=.5;, α=.5,ɳ=.;, α=.5,ɳ=.5

13 frequenc parameter frequenc parameter AAM: Intern. J., Vol 1, Issue 1 (June 17) 13 (a) (b) (c) thermal gradient Figure 4. Natural frequencies for c-c and c-s plates: (a) First mode (b) Second mode (c) Third mode,for a/b=1.,β=.5, α=.5,,c-c; --,C-SS; Δ,D k =., E f =.,,,D k =.1, E f =.;,D k =., E f =.;,D k =.1, E f =. (a) (b) (c) thermal gradient Figure 5. Natural frequencies for c-c and c-s plates: (a) First mode (b) Second mode (c) Third mode,for a/b=1., Ef=., D k =.1,,C-C; --,C-SS; Δ, α=-.5,β=-.5;, α=-.5,β=.5;, α=.5,β=-.5;, α=.5,β=.5

14 W norm W norm 14 Rana and Robin C-C plate X Figure 6. Normalized displacements for c-ss-c-ss plate, for a/b=1; h=.1, β=.5,d k =.5, E f =.;,First mode;, Second mode ; Third mode,, α=-.5,ɳ=.5;, α=.5,ɳ=.5; o, α=.5,ɳ=. C-S plate X Figure 7. Normalized displacements for c-ss-ss-ss plate, for a/b=1; h=.1, β=.5,d k =.5, E f =.;,First mode;, Second mode ; Third mode,, α=-.5,ɳ=.5;, α=.5,ɳ=.5; o, α=.5,ɳ=.

15 AAM: Intern. J., Vol 1, Issue 1 (June 17) Conclusion In the present stud results are computed using MATLAB within the permissible range of parameters up to the desired accurac (1-8 ), which validates the actual phenomenon of vibrational problem. The rate of decrease of frequenc parameter Ω with thermal gradient ɳ, decreases with the increase in the value of foundation parameter and non-homogeneit parameter and this rate increases with the increase in the value of taper parameter and damping parameter. Temperature effect, variation in thickness, elastic foundation, damping parameter and nonhomogeneit parameter are of great interest because engineers often tr to know natural frequenc and modes of vibration before finalizing the design of a structure or machine. Our endeavor is to provide a mathematical model for analzing the vibrational behavior of orthotropic rectangular plate resting on elastic foundation for different values of thermal gradient, taperness and damping parameter. Thus, the present stud ma be helpful in the determination of natural frequencies and mode shapes b proper choice of plate parameters so that robustness of structure can be determined for improved structural design. REFERENCES Civalek, O. and Acar, M.H. (7). Discrete singular method for the analsis of mindlin plates on elastic foundation, International Journal of Press Vessels Piping, Vol. 84, pp Gupta, A.K., Johri, T. and Vats, R.P. (7). Thermal effect on vibration of non-homogeneous orthotropic rectangular plate having bi-directional parabolicall varing thickness, proceeding of international conference in world congress on engineering and computer science 7, San-Francisco, USA, 4-6 Oct, 7, pp Gupta, A.K. and Singhal, P. (1). Effect of non-homogeneit on thermall induced vibration of orthotropic visco-elastic rectangular plate of linearl varing thickness, Applied Mathematics, Vol. 1, pp Gupta, A.K., Rana, K. and Gupta, D.V. (13). Effect of thermal gradient on vibration of non homogeneous parallelogram plate of linearl varing thickness in both directions, Journal of Mathematical Sciences and Applications, Vol. 1, No. 3, pp Gupta, A.K. and Sharma, S. (13). Effect of thermal gradient on vibration of non-homogeneous orthotropic trapezoidal plate of linearl varing thickness, Ain Shams Engineering Journal, Vol. 4, No. 3, pp Gupta, A.K and Jain, M. (14). Eponential temperature effect on frequencies of a rectangular plate of non-linear varing thickness: a quintic spline technique, Journal of Theoretical and Applied Mechanics, Vol. 5, No. 1, pp Khanna, A. and Singhal, A. (14). Thermal effect on vibration of tapered rectangular plate, Journal of Structure, Hinwawi Publishing Corporation, id Lal, R., Gupta, U.S. and Goel, C. (1). Chebshev polnomials in the stud of transverse vibration of non-uniform rectangular orthotropic plates, the shock and vibration digest, Vol. 33, No., pp Lal, R. and Dhanpati (7). Transverse vibrations of non homogeneous orthotropic rectangular plates of variable thickness: a spline technique, journal of sound and vibration, Vol. 36, pp Laura, P.A.A. and Gutierrez, R.H. (198). Vibration analsis on a rectangular plate subjected to a thermal gradient, Journal of Sound and Vibration, Vol. 7, pp. 63.

16 16 Rana and Robin Mcfadden, T.T and Bennett, F.L. (1991). Construction in cold regions, Wile, New York. Nowacki, W. (196). Thermo elasticit, Pergamon Press, New York. Robin and Rana, U.S. (13). Damped vibrations of rectangular plate of variable thickness resting on elastic foundation: a spline technique, journal of applied and computational mathematics, Vol., No. 3, Robin and Rana, U.S. (13). Numerical stud of damped vibration of orthotropic rectangular plates of variable thickness, Journal of Orissa Mathematical Societ, Vol. 3, No., pp Rana, U.S. and Robin (15). Damped vibrations of parabolic tapered non-homogeneous infinite rectangular plate resting on elastic foundation, International Journal of engineering, Vol. 8, No. 7, pp Tomar, J.S. and Gupta, A.K. (1983). Thermal effect of frequencies of an orthotropic rectangular plate of linearl varing thickness, Journal of Sound and Vibration, Vol. 9, No. 3, pp Tomar, J.S. and Gupta, A.K. (1985). Thermal gradient on frequencies of orthotropic rectangular plate whose thickness varies in two directions, Journal of Sound and Vibration, Vol. 98, No., pp