Acta Polytechnica https://doi.org/10.14311/AP.2024.64.0297 Acta Polytechnica 64(4):297–313, 2024 © 2024 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague NON-HOMOGENEITY EFFECT ON THE VIBRATION OF THE RECTANGULAR VISCO-ELASTIC PLATE SUBJECTED TO THE LINEAR TEMPERATURE EFFECT WITH QUADRATIC THICKNESS VARIATION IN BOTH DIRECTIONS Sudhanshu Aggarwal National Post Graduate College Barhalganj, Department of Mathematics, Gorakhpur-273402, Uttar Pradesh, India correspondence: sudhanshu30187@gmail.com Abstract. The present work investigates the effect of non-homogeneity on the vibration of a rectan- gular visco-elastic plate. The plate is subjected to linear temperature variation in the x-direction with quadratic thickness variation in both directions. The quadratic variation has been considered in the material density of the plate only along the x-axis, and it is assumed that non-homogeneity transpires because of this variation. The governing differential equation is solved using the Rayleigh-Ritz technique. All four edges of the plate should be clamped to drive the frequency equation. Deflection and time period have been evaluated for several combinations of values of the thermal constant, constant of nonhomogeneity, taper constants, and length-to-width ratio (aspect ratio) for the first two vibration modes of the clamped plate. The results presented are compared with those found in the literature. Keywords: Rayleigh-Ritz technique, taper constants, clamped plate, differential equation, thermal constant. 1. Introduction Due to their practical and scholarly interests, the vibration problems of elasticity have been the subject of effort or research for a considerable amount of time. Machines often produce heat as a result of vibration, which reduces machine efficiency. As a result, it is critical to investigate variations in temperature on vibration-affected plates. There are a number of materials, found in nature that are nonhomogeneous (e.g. delta wood, plywood, timber, fibre reinforced plastics, etc.). The use of these natural nonhomogeneous materials in the indus- try is generally due to their nature to strengthen the construction. Glass epoxy and boron epoxy in the steel alloys are examples of the artificial nonhomoge- neous materials used in the manufacture of nuclear reactor rods. The non-homogeneity in the materials arises due to imperfections in the material, them being composite materials, or work under elevated tempera- tures [1]. Sharma and Sarkar [2, 3] suggested another common way to introduce the non-homogeneity in the plate by attaching a mass, spring, or a spring- mass system to it. Today, scientists and engineers are developing new materials to meet the demands of high temperatures and high strength services. Dural- ium (Alloy of Aluminium, Copper, Magnesium and Manganese) is the material that is widely used in the defence industry (aircraft fittings, space booster tank age, forgings, pistons of aircraft engine, compressor rings, and impellers of the jet engine) due to its very attractive idiosyncrasy like the high strength, and the light weight. Duralium material parameters are used for the numerical computation in this study. In concrete, most of the materials used in industrial applications become visco-elastic due to the very high temperature rise and these materials deviate from the Hooke’s law of elasticity, which states that within the elastic limits, the stress is directly proportional to the strain. The behaviour (elastic or viscous) of the material depends on the two parameters, the tem- perature and the frequency (the rate of the loading). Ceramics (rocks, concrete, inorganic glassy materials, piezoelectric ceramics), biological composite materials (wood, plant seeds, tissue), and many polymers (plas- tic, rubber, vinyl, acrylics, silicones, adhesive, etc.) are the generally used visco-elastic materials. Although the uniform thickness plates are com- monly used, the use of variable thickness plates is also common, such as rocket fins, the wing panels in the aerospace engineering, disc wheels and car body panels in mechanical engineering, building walls, and floors in civil engineering, off-shore platforms, and ship decks in marine engineering, and printed circuit boards in electronics, in example. Plates of the vari- able thickness have the wide practice in many fields of the engineering due to the some very attractive idiosyncrasy of their like the high strength, and the stiffness, the reduction of the cost, and the minimizing the size, and the weight. Gupta and Kumar [4] used the Galerkin’s technique and analysed the vibration of a non-homogeneous visco-elastic plate (rectangular) in the thermal envi- ronment with the linear variation of the thickness. Gupta and Khanna [5, 6] studied the vibration of the variable thickness rectangular visco-elastic plate using 297 https://doi.org/10.14311/AP.2024.64.0297 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Sudhanshu Aggarwal Acta Polytechnica the Rayleigh-Ritz technique. Gupta et al. [7] deter- mined the fundamental frequencies and the deflection by analysing the vibration of the non-homogeneous orthotropic plate (rectangular) having linear thickness variations in both the directions with the thermal gra- dient effect. Gupta and Kaur [8] used the Rayleigh- Ritz method to determine the thermal gradient effect on the vibration of the variable thickness visco-elastic rectangular plate with clamped boundary conditions. The monograph of Leissa [1] has an excellent litera- ture on the vibration of the plates for the scholars and it is very convenient to curb the duplication of the research handiwork in the plate vibrations. Singh and Saxena [9] have determined the vibration (transverse) of the rectangular plate and they have considered the bidirectional thickness variation in this study. Laura and Gutierrez [10] analysed the vibra- tion of the rectangular plate with the thermal gradient effect. Gupta et al. [11] considered the Rayleigh-Ritz technique for analysing the vibrations of the variable thickness rectangular visco-elastic plate. Gupta and Aggarwal [12–14] discussed the linear temperature effect on the vibration of a non-homogeneous visco- elastic plate (rectangular) with variable thickness in two directions. Gupta and Kaur [15] have considered the linear temperature variation and studied its ef- fect on the free transverse vibration of the clamped visco-elastic rectangular plate exponential thickness variation in both directions. Nagaya [16] found so- lutions to the problems of vibration and transient response of the non-periodic elastically supported con- tinuous visco-elastic plates. Gupta et al. [17] analysed the thermal effect on the vibration problem of the clamped visco-elastic rectangular plate. Bhat [18] de- termined the natural frequencies of the rectangular plates using the Rayleigh-Ritz method. Tomar and Gupta [19] studied the vibration prob- lem of an orthotropic rectangular plate with the lin- early varying thickness and determined the thermal effect on the frequencies of the plate considered. Saini and Lal [20] used the general differential quadrature (GDQ) method to analyse the transverse vibration of the variable thickness non-homogeneous rectangu- lar plates. Gupta et al. [21] considered the linearly varying thickness non-homogeneous rectangular plate and analysed its vibration. Sobamowo et al. [22] ap- plied the three-dimensional differential transformation method to the problem of thermally induced vibration of the non-homogeneous rectangular plate with the varying thickness. Shabnam et al. [23] studied the problem of free vibration of thin plates with the vari- able thickness and determined its solution using the two-dimensional differential transformation method. Singhal and Ruby [24] discussed the thermally induced vibrations of the tapered non-homogeneous rectangu- lar plate. Yeh et al. [25] used the finite difference, and the differential transformation methods to determine the solution to the free vibration problem of the plate. Khanna and Singhal [26] studied the vibration prob- lem of a clamped visco-elastic thin plate (rectangu- lar). Jafari and Azhari [27] discussed the problem of free vibration of visco-elastic plate (Mindlin) and determined the natural frequency along with viscous damping. Amabili et al. [28] obtained the solutions of the problems of nonlinear vibration of fractional visco- elastic rectangular plates with damping. Bhardwaz et al. [29] determined the time period for the vibration problem of a non-homogeneous parallelogram plate (skew) with two-dimensional parabolic temperature variations. Abdikarimov et al. [30] discussed the prob- lem of dynamic stability of a orthotropic visco-elastic plate (rectangular) with arbitrary thickness variation. This work aims to analyse the non-homogeneity ef- fect of vibrations on the rectangular visco-elastic plate subjected to a linear temperature effect with quadratic thickness variation in both directions. The deflec- tion (w) and time period (tp) have been evaluated for several combinations of the values of the thermal constant (α), constant of the non-homogeneity (α1), taper constants (β1, β2) and the length-to-width ratio (a b ) for the first two vibration modes of a clamped plate. Graphs and tables are used for presenting the corresponding results. The rest of this paper is organ- ised as follows. Assumptions of the study are provided in Section 2. Equation of the motion and the method of the analysis are presented in Section 3. We present differential equation of the time function with its so- lution in Section 4. All the results and discussions are presented in Section 5. The comparison of the results of the present study with the previous study is provided in Section 6. The last section (Section 7) presents a brief conclusion. 2. Assumptions of the study Author has considered the following assumptions for this study: • The quadratic variation has been considered in the material density of the plate only along the x-axis. In addition, it is supposed that the non-homogeneity occurs due to this variation. • It is supposed that all the four edges of the plate are clamped for deriving the frequency equation with the use of the Rayleigh-Ritz technique. According to this technique, the maximum strain energy must be equal to the maximum kinetic energy. • The linear variation in the temperature of the plate has been considered only along the x-axis. • The quadratic variation has been considered for the thickness of the plate along the x-axis and the y-axis. • The linear visco-elastic behaviour is of the Kelvin type. • The assumption of the small deflection is assumed. • Duralium material (Alloy of Aluminium, Copper, Magnesium, and Manganese) parameters are used for the numerical computation. 298 vol. 64 no. 4/2024 Non-Homogeneity Effect on the Vibration of . . . 3. Equation of the motion and the method of the analysis The author discusses the equation of the motion and the method of the analysis in this section. For the free vibration of the plate, the transverse motion and the time function are given by the follow- ing equations, respectively, as [15]:   D ( ∂4W ∂x4 + 2 ∂4W ∂x2∂y2 + ∂4W ∂y4 ) +2 ∂D ∂x ( ∂3W ∂x3 + ∂3W ∂x∂y2 ) +2 ∂D ∂y ( ∂3W ∂y3 + ∂3W ∂x2∂y ) + ∂2D ∂x2 ( ∂2W ∂x2 + ν ∂2W ∂y2 ) + ∂2D ∂y2 ( ∂2W ∂y2 + ν ∂2W ∂x2 ) +2 (1 − ν) ∂2D ∂x∂y ∂2W ∂x∂y  − ρhk2W = 0  , (1) and: T̈ + k2D̃T = 0, (2) where: D = Eh3 12 (1 − ν2) , (3) is the flexural rigidity, ν is the Poisson’s ratio, E is the Young modulus of elasticity, ρ is the density, h is the thickness of the plate, D̃ is the unique Rheological operator, k is the frequency, T is the time function, W is the deflection function, T̈ is the second derivative of the time function with respect to time t. Let’s consider a symmetrical plate (rectangular) with all four edges clamped at length a and width b, see Figure 1. Assuming that the plate under consideration has the linear temperature variation only along the x-axis: τ = τ0 ( 1 − x a ) , (4) where τ0 and τ denote the temperature increment above the reference temperature at any point at the end x = a and at the distance ( x a ) , respectively. The modulus of the elasticity (temperature depen- dent) can be expressed as [16]: E = E0 (1 − γτ) , (5) where E0 is the modulus of elasticity at τ = 0 (reference temperature), γ is the slope of the variation of E with τ . Figure 1. Rectangular plate with length a and width b. Substituting the Equation (4) into the Equation (5) provides: E = E0 [ 1 − α ( 1 − x a )] , (6) where the parameter α = γτ0 is called the thermal gradient and has a value in the interval [0, 1]. Suppose that the plate under consideration has a quadratic thickness variation in both directions: h = h0 ( 1 + β1 x2 a2 ) ( 1 + β2 y2 b2 ) , (7) where β1 is the taper constant along the x-axis, β2 is the taper constant along the y-axis, h0 is the thickness of the plate at x = y = 0. Assuming that the density ρ of the plate under consideration has the parabolic variation along the x-axis, given by: ρ = ρ0 ( 1 + α1 x2 a2 ) , (8) where α1 is the non-homogeneity constant, ρ0 is the value of the density at x = 0. The deflection function (w) can be considered in the following form for the free vibrations (transverse) of the plate [12]: w(x, y, t) = T (t) W (x, y). (9) Leissa [1] defined the strain energy S and the kinetic energy P as: S = 1 2 ∫ a 0 ∫ b 0 D [ ( ∂2W ∂y2 )2 + ( ∂2W ∂x2 )2 + 2 (1 − ν) ( ∂2W ∂x∂y )2 + 2ν ∂2W ∂x2 ∂2W ∂y2 ] dxdy, (10) and P = 1 2k2 ∫ a 0 ∫ b 0 ρhW 2dxdy. (11) 299 Sudhanshu Aggarwal Acta Polytechnica The expression of the flexural rigidity using the Equations (6) and (7) (assuming ν is the constant) in the Equation (3) can be written as: D = 1 12 (1 − ν2) [ E0h0 3 { 1 − α ( 1 − x a )} ( 1 + β1 x2 a2 )3 ( 1 + β2 y2 b2 )3 ] . (12) Substituting Equations (7) and (8) into the Equa- tion (11) gives: P = 1 2ρ0h0k2 ∫ a 0 ∫ b 0 ( 1 + α1 x2 a2 ) ( 1 + β1 x2 a2 ) ( 1 + β2 y2 b2 ) W 2dxdy. (13) The principle of the Rayleigh-Ritz technique [14] provides: δ(S − P ) = 0, (14) The assumption that the four edges of the rectan- gular plate are clamped gives: (W )x=0 = (W )x=a = ( ∂W ∂x ) x=0 = ( ∂W ∂x ) x=a = 0, (W )y=0 = (W )y=b = ( ∂W ∂y ) y=0 = ( ∂W ∂y ) y=b = 0. (15) Taking the deflection function W (x, y) as [13]: W (x, y) = { [(y b ) (x a ) ( 1 − y b ) ( 1 − x a )]2 [ L1 + L2 (y b ) (x a ) ( 1 − y b ) ( 1 − x a )] } , (16) where both constants L1 and L2 are satisfied by the Equation (15). Now, taking the non-dimensional quantities in the capital letters as [13]: h̄ = ( h a ) , W̄ = ( W a ) , X = (x a ) , Y = (y a ) . (17) Using Equation (17) in Equation (16) gives: W = [ Y (a b ) X ( 1 − Y (a b )) (1 − Y ) ]2 [ L1 + L2Y (a b ) X ( 1 − Y (a b )) (1 − X) ] . (18) Using Equations (12), and (17) in Equations (10) and (13) gives: S =C ∫ 1 0 ∫ b a 0 [ [ {1 − α (1 − X)} ( 1 + β1X2)3 ( 1 + β2Y 2 a2 b2 )3] [ ( ∂2W̄ ∂Y 2 )2 + ( ∂2W̄ ∂X2 )2 + 2 (1 − ν) ( ∂2W̄ ∂X∂Y )2 + 2ν ∂2W̄ ∂X2 ∂2W̄ ∂Y 2 ] ] dXdY, (19) where C = E0h̄0 3 a3 24(1−ν2) and: P =1 2ρ0h̄0k2a5 ∫ 1 0 ∫ b a 0 ( 1 + α1X2) ( 1 + β1X2) ( 1 + β2Y 2 a2 b2 ) W̄ 2dXdY. (20) Using Equations (19) and (3) in Equation (14) gives: δ ( S1 − n2k2P1 ) = 0, (21) where: S1 = ∫ 1 0 ∫ b/a 0 [ [ {1 − α (1 − X)} ( 1 + β1X2)3 ( 1 + β2Y 2 a2 b2 )3] [ ( ∂2W̄ ∂Y 2 )2 + ( ∂2W̄ ∂X2 )2 + 2 (1 − ν) ( ∂2W̄ ∂X∂Y )2 + 2ν ∂2W̄ ∂X2 ∂2W̄ ∂Y 2 ] ] dXdY, (22) and: P1 = ∫ 1 0 ∫ b/a 0 ( 1 + α1X2) ( 1 + β1X2) ( 1 + β2Y 2 a2 b2 ) W̄ 2dXdY. (23) Here: n2 = { 12ρ0 ( 1 − ν2) a2 E0h0 2 } . (24) The two unknown parameters L1 and L2 that are appearing by substituting W from Equation (16) into equation (21) can be computed as: ∂ ∂B1 (S1 − n2k2P 1) = 0, and ∂ ∂B2 (S1 − n2k2P 1) = 0. (25) Solution of Equation (25) provides:{ d11L1 + d12L2 = 0, d21L1 + d22L2 = 0, (26) 300 vol. 64 no. 4/2024 Non-Homogeneity Effect on the Vibration of . . . S.N. Parameters Values 1. Young modulus of the elasticity (N m−2), E 7.08 × 1010 2. Shear modulus (N m−2), G 2.682 × 1010 3. Visco-elastic constant (Ns m−2), η 1.4612 × 106 4. Density (kg m−3), ρ 2.80 × 103 5. Poisson’s ratio, ν 0.345 Table 1. Duralium material (Alloy of Aluminum, Copper, Magnesium, and Manganese) parameters that are used for the numerical computation [16]. where d11, d12, d21, d22 involve the frequency param- eter and the parametric constants. The non-trivial solution of the Equation (26) is obtained by: ∣∣∣∣d11 d12 d21 d22 ∣∣∣∣ = 0. (27) Equation (27) is a quadratic equation in k2 and it is known as the frequency equation. Solving Equa- tion (27) gives the two values of k2. Now, if we choose L1 = 1 and substitute it into Equation (26), we have L2 = − ( d11 d12 ) . Using the above values for L1, and L2 in Equa- tion (18) suggests that: W = [ Y (a b ) X ( 1 − Y (a b )) (1 − X) ]2 [ 1 + ( −d11 d12 ) Y (a b ) X ( 1 − Y (a b )) (1 − X) ] . (28) 4. Differential equation of the time function with its solution The author has derived the differential equation of the time function and then he has given its solution in this section. The unique rheological operator D̃ for the consid- eration of linear visco-elastic model (model of the Kelvin) is given as [13]: D̃ ≡ [ 1 + ( η G ) ( d dt )] , (29) where η is visco-elastic constant, G is shear modulus. Using Equation (29) in Equation (2) gives: T̈ + k2 ( η G ) Ṫ + k2T = 0. (30) The solution of the Equation (30) will be of the form: T (t) = ea1t(A cos b1t + B sin b1t), (31) where:  a1 = −k2 2 ( η G ) , b1 = k √ 1 − ( ρη 2G )2 , A, B = constants. (32) Consider the following initial conditions for deter- mining the values of A and B: At t = 0, T = 1 and Ṫ = 0. (33) Using Equation (33) in Equation (31) provides:A = 1, B = −a1 b1 . (34) After substituting the values of A, B into Equa- tion (31), one obtains: T (t) = ea1t ( cos b1t − a1 b1 sin b1t ) . (35) Thus, w(x, y, t) can be expressed using Equa- tions (28) and (35) in Equation (9) as: w = { Y (a b ) X ( 1 − Y (a b )) (1 − X) }2 { 1 − ( b11 b12 ) Y (a b ) X ( 1 − Y (a b )) (1 − X) } { ea1t ( cosb1t − ( a1 b1 ) sinb1t )} . (36) For the vibration of the plate, the relation of the time period (tp) with the frequency (k) is expressed by [13]: tp = 2π k , (37) where the frequency k is easily determined using Equa- tion (27). 5. Results and discussions In the present work, the time period (tp) and the deflection (w) are determined for various combinations of the values of (α), ( a b ) , (β1, β2), and (α1) for the first two modes of vibration and are given in Tables 2–12. The quadratic variation has been considered in the material density of the plate only along the x-axis. In the calculation, the thickness at the centre of the rectangular plate was taken as 0.01 m. The material (Duralium) parameters used for the computational work are presented in Table 1. Table 2 deals with the variation of tp (*10−6 in seconds) with different values of α (0.0, 0.2, 0.4, 0.6, 0.8, 1.0), and constant a b = 1.5 for all X and Y . The continuous increments are observed in the values of 301 Sudhanshu Aggarwal Acta Polytechnica α1 = 0.0 α1 = 0.4 β1 = 0.0 β2 = 0.0 β1 = 0.4 β2 = 0.2 β1 = 0.0 β2 = 0.0 β1 = 0.4 β2 = 0.2 Mode → I II I II I II I II α ↓ 0.0 6 462 1 623 5 324 1 332 6 897 1 728 5 543 1 433 0.2 6 823 1 731 5 513 1 438 7 246 1 856 5 849 1 496 0.4 7 236 1 863 5 874 1 497 7 733 2 002 6 194 1 617 0.6 7 737 2 002 6 213 1 603 8 497 2 176 6 626 1 759 0.8 8 443 2 168 6 584 1 737 9 220 2 285 7 082 1 846 1.0 9 231 2 304 7 117 1 863 9 668 2 439 7 530 1 988 Table 2. Variation of tp (*10−6 in seconds) with different α and constant a b = 1.5 for all X and Y see Figure 2. 0 2000 4000 6000 8000 10000 12000 0 0.2 0.4 0.6 0.8 1 First Mode α1=0,β1=0,β2=0 Second Mode α1=0,β1=0,β2=0 First Mode α1=0,β1=0.4,β2=0.2 Second Mode α1=0,β1=0.4,β2=0.2 First Mode α1=0.4,β1=0,β2=0 Second Mode α1=0.4,β1=0,β2=0 First Mode α1=0.4,β1=0.4,β2=0.2 Second Mode α1=0.4,β1=0.4,β2=0.2 Figure 2. Variation of tp (*10−6 in seconds) with different α and constant a b = 1.5 for all X and Y . α1 = 0.0 α1 = 0.4 β1 = 0.0 β2 = 0.0 α = 0.0 β1 = 0.4 β2 = 0.2 α = 0.3 β1 = 0.0 β2 = 0.0 α = 0.0 β1 = 0.4 β2 = 0.2 α = 0.3 Mode → I II I II I II I II a b ↓ 0.5 15 559 3 878 13 873 3 417 16 023 4 008 14 296 3 557 1.0 10 411 2 638 9 657 2 471 10 860 2 747 9 971 2 588 1.5 6 462 1 623 5 697 1 465 6 897 1 728 6 021 1 515 2.0 3 724 905 3 513 873 4 082 961 3 864 957 2.5 2 451 576 2 391 560 2 771 658 2 739 651 Table 3. Variation of tp (*10−6 in seconds) with different a b for all X and Y , see Figure 3. tp as α increases from 0.0 to 1.0 for both modes of vibration. The graph shown in Figure 2 supports the results of Table 2. Table 3 is concerned with the variation of tp (*10−6 in seconds) with different a b (0.5, 1.0, 1.5, 2.0, 2.5) for all X and Y . The continuous decrements are observed in the values of tp as a b increases from 0.5 to 2.5 for modes of vibration. The results of Table 3 are supported by the graph shown in Figure 3. Table 4 shows the variation of tp (*10−6 in seconds) with different β1(0.0, 0.2, 0.4, 0.6, 0.8, 1.0) and con- stant a b = 1.5 for all X and Y . It is observed that as the value of β1 increases from 0.0 to 1.0, the value of tp decreases for both modes of vibration. The graph in Figure 4 supports the results of Table 4. Table 5 shows the variation of tp (*10−6 in seconds) with different β2(0.0, 0.2, 0.4, 0.6, 0.8, 1.0) and con- stant a b = 1.5 for all X and Y . From this table, it can clearly be seen that the value of tp decreases as β2 increases from 0.0 to 1.0 for both the vibration 302 vol. 64 no. 4/2024 Non-Homogeneity Effect on the Vibration of . . . 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0.5 1 1.5 2 2.5 First Mode α1=0,β1=0,β2=0,α=0 Second Mode α1=0,β1=0,β2=0, α=0 First Mode α1=0,β1=0.4,β2=0.2,α=0.3 Second Mode α1=0,β1=0.4,β2=0.2,α=0.3 First Mode α1=0.4,β1=0,β2=0,α=0 Second Mode α1=0.4,β1=0,β2=0,α=0 First Mode α1=0.4,β1=0.4,β2=0.2,α=0.3 Second Mode α1=0.4,β1=0.4,β2=0.2,α=0.3 Figure 3. Variation of tp (*10−6 in seconds) with different a b for all X and Y . α1 = 0.0 α1 = 0.4 β2 = 0.0 α = 0.0 β2 = 0.2 α = 0.3 β2 = 0.0 α = 0.0 β2 = 0.2 α = 0.3 Mode → I II I II I II I II β1 ↓ 0.0 6 462 1 623 6 360 1 564 6 897 1 728 6 702 1 701 0.2 6 123 1 566 6 011 1 503 6 543 1 618 6 336 1 600 0.4 5 784 1 489 5 697 1 465 6 216 1 568 6 021 1 515 0.6 5 402 1 373 5 278 1 327 5 930 1 453 5 591 1 437 0.8 5 087 1 285 4 923 1 213 5 433 1 365 5 242 1 343 1.0 4 736 1 207 4 642 1 128 5 141 1 297 4 971 1 260 Table 4. Variation of tp (*10−6 in seconds) with different β1 and constant a b = 1.5 for all X and Y , see Figure 4. 0 1000 2000 3000 4000 5000 6000 7000 8000 0 0.2 0.4 0.6 0.8 1 First Mode α1=0,β2=0,α=0 Second Mode α1=0,β2=0, α=0 First Mode α1=0,β2=0.2,α=0.3 Second Mode α1=0,β2=0.2,α=0.3 First Mode α1=0.4,β2=0,α=0 Second Mode α1=0.4,β2=0,α=0 First Mode α1=0.4,β2=0.2,α=0.3 Second Mode α1=0.4,β2=0.2,α=0.3 Figure 4. Variation of tp (*10−6 in seconds) with different β1 and constant a b = 1.5 for all X and Y . 303 Sudhanshu Aggarwal Acta Polytechnica α1 = 0.0 α1 = 0.4 β1 = 0.0 α = 0.0 β1 = 0.4 α = 0.3 β1 = 0.0 α = 0.0 β1 = 0.4 α = 0.3 Mode → I II I II I II I II β2 ↓ 0.0 6 462 1 623 6 033 1 512 6 897 1 728 6 504 1 681 0.2 6 102 1 543 5 697 1 465 6 537 1 609 6 021 1 515 0.4 5 761 1 462 5 318 1 324 6 193 1 537 5 576 1 446 0.6 5 384 1 356 4 866 1 243 5 903 1 429 5 167 1 333 0.8 5 049 1 253 4 613 1 117 5 413 1 316 4 791 1 211 1.0 4 703 1 184 4 264 1 058 4 998 1 251 4 448 1 135 Table 5. Variation of tp (*10−6 in seconds) with different β2 and constant a b = 1.5 for all X and Y , see Figure 5. 0 1000 2000 3000 4000 5000 6000 7000 8000 0 0.2 0.4 0.6 0.8 1 First Mode α1=0,β1=0,α=0 Second Mode α1=0,β1=0, α=0 First Mode α1=0,β1=0.4,α=0.3 Second Mode α1=0,β1=0.4,α=0.3 First Mode α1=0.4,β1=0,α=0 Second Mode α1=0.4,β1=0,α=0 First Mode α1=0.4,β1=0.4,α=0.3 Second Mode α1=0.4,β1=0.4,α=0.3 Figure 5. Variation of tp (*10−6 in seconds) with different β2 and constant a b = 1.5 for all X and Y . α = 0.0 α = 0.3 β1 = 0.0 β2 = 0.0 β1 = 0.4 β2 = 0.2 β1 = 0.0 β2 = 0.0 β1 = 0.4 β2 = 0.2 Mode → I II I II I II I II α1 ↓ 0.0 6 462 1 623 5 324 1 332 7 083 1 833 5 697 1 465 0.2 6 695 1 687 5 511 1 426 7 291 1 890 5 889 1 504 0.4 6 897 1 728 5 543 1 433 7 489 1 930 6 021 1 515 0.6 7 096 1 786 5 747 1 522 7 714 1 996 6 172 1 607 0.8 7 321 1 834 5 864 1 597 7 973 2 066 6 290 1 673 1.0 7 509 1 888 5 933 1 654 8 211 2 127 6 443 1 741 Table 6. Variation of tp (*10−6 in seconds) with different α1 and constant a b = 1.5 for all X and Y , see Figure 6. modes. The graph shown in Figure 5 shows that the period (tp) decreases as β2 increases for both modes of vibration. Table 6 deals with the variation of tp (*10−6 in seconds) with different α1(0.0, 0.2, 0.4, 0.6, 0.8, 1.0) and constant a b = 1.5 for all X and Y . It is observed that the value of tp increases as α1 increases from 0.0 to 1.0 for both modes of vibration. The results shown in Table 6 are supported by the graph shown in Figure 6. Tables 7–9 show the value of w (*10−5) for the constant a b = 1.5 and for different X and Y with three different variations of the values of α, β1, β2 and α1 given by (α = β1 = β2 = α1 = 0.0), (α = β1 = β2 = 0.0 , α1 = 0.4), (α = 0.2 , β1 = 0.3, β2 = 0.4, α1 = 0.4), respectively. From these tables, it can clearly be seen that as X̄ increases for different values of Ȳ , the value of w first increases and then decreases to zero for the first mode of vibration. For the second mode of vibration, it can be seen that the 304 vol. 64 no. 4/2024 Non-Homogeneity Effect on the Vibration of . . . 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 0.2 0.4 0.6 0.8 1 First Mode α=0,β1=0,β2=0 Second Mode α=0,β1=0,β2=0 First Mode α=0,β1=0.4,β2=0.2 Second Mode α=0,β1=0.4,β2=0.2 First Mode α=0.3,β1=0,β2=0 Second Mode α=0.3,β1=0,β2=0 First Mode α=0.3,β1=0.4,β2=0.2 Second Mode α=0.3,β1=0.4,β2=0.2 Figure 6. Variation of tp (*10−6 in seconds) with different α1 and constant a b = 1.5 for all X and Y . Y = 0.20 Y = 0.60 Mode → I II I II X̄ ↓ T im e = 0. 0 × t p 0.0 0.0 0.0 0.0 0.0 0.2 126.32 38.87 427.32 14.52 0.4 303.71 6.12 1420.12 26.73 0.6 303.71 6.12 1420.12 26.73 0.8 126.32 38.87 427.32 14.52 1.0 0.0 0.0 0.0 0.0 T im e = 5. 0 × t p 0.0 0.0 0.0 0.0 0.0 0.2 55.12 1.38 187.31 0.53 0.4 132.32 0.21 623.16 1.01 0.6 132.32 0.21 623.16 1.01 0.8 55.12 1.38 187.31 0.53 1.0 0.0 0.0 0.0 0.0 Table 7. w(*10−5) for constant a b = 1.5 and α = β1 = β2 = α1 = 0.0 and for different X and Y , see Figure 7. value of w for Y = 0.2, the value changes frequently as it first increases and then decreases, then repeats this trend by increasing again and finally becoming zero as X̄ increases but for Y = 0.6, with the increase in X, the value of w varies as it first increases and then it decreases until it reaches zero. The graphs in Figures 7, 8 and 9 support the results of Tables 7–9. Tables 10–12 represent the deflection w (*10−5) for X = Y = 0.2 and for three different cases, namely (for different a b and α = β1 = β2 = α1 = 0.0), (for different a b and α = β1 = β2 = 0.0, α1 = 0.4) and (for different a b and α = 0.3, β1 = 0.3, β2 = 0.4, α1 = 0.4) respectively. From these tables, it can be seen that the value of w increases continuously at time 0.0× tp, it is also observed that with the increase there is slight decrease at the time 5.0 × tp for the first mode of vibration but it is also noticed that the decrease follows the increase in the value of a b from 0.5 to 2.5, at time 0.0 × tp and at time 5.0 × tp for the second mode of vibration. These results are also supported by the graphs shown in Figures 10–12. 6. Comparison of the results of the present study with the previous study The time function and the deflection results obtained in the present study are compared with those of the reference [13]. The comparison of the results of these 305 Sudhanshu Aggarwal Acta Polytechnica 0 200 400 600 800 1000 1200 1400 1600 1800 0 0.2 0.4 0.6 0.8 1 First Mode Time=0.K Y=0.2 First Mode Time=0.K Y=0.6 First Mode Time=5.K Y=0.2 First Mode Time=5.K Y=0.6 (a). First mode. 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 Second Mode Time=0.K Y=0.2 Second Mode Time=0.K Y=0.6 Second Mode Time=5.K Y=0.2 Second Mode Time=5.K Y=0.6 (b). Second mode. Figure 7. Variation of w(*10−5) with different X for constant a b = 1.5 and α = β1 = β2 = α1 = 0.0 and for different Y . Y = 0.20 Y = 0.60 Mode → I II I II X̄ ↓ T im e = 0. 0 × t p 0.0 0.0 0.0 0.0 0.0 0.2 130.43 40.12 431.01 15.93 0.4 307.69 6.96 1 424.07 28.12 0.6 307.69 6.96 1 424.07 28.12 0.8 130.43 40.12 431.01 15.93 1.0 0.0 0.0 0.0 0.0 T im e = 5. 0 × t p 0.0 0.0 0.0 0.0 0.0 0.2 59.13 1.93 190.34 0.83 0.4 136.32 0.43 627.65 1.17 0.6 136.32 0.43 627.65 1.17 0.8 59.13 1.93 190.34 0.83 1.0 0.0 0.0 0.0 0.0 Table 8. w(*10−5) for constant a b = 1.5 and α = β1 = β2 = 0.0, α1 = 0.4 and for different X and Y , see Figure 8. 306 vol. 64 no. 4/2024 Non-Homogeneity Effect on the Vibration of . . . 0 200 400 600 800 1000 1200 1400 1600 1800 0 0.2 0.4 0.6 0.8 1 First Mode Time=0.K Y=0.2 First Mode Time=0.K Y=0.6 First Mode Time=5.K Y=0.2 First Mode Time=5.K Y=0.6 (a). First mode. 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 Second Mode Time=0.K Y=0.2 Second Mode Time=0.K Y=0.6 Second Mode Time=5.K Y=0.2 Second Mode Time=5.K Y=0.6 (b). Second mode. Figure 8. Variation of w(*10−5) with different X for constant a b = 1.5 and α = β1 = β2 = 0.0, α1 = 0.4 and for different Y . Y = 0.20 Y = 0.60 Mode → I II I II X̄ ↓ T im e = 0. 0 × t p 0.0 0.0 0.0 0.0 0.0 0.2 233.72 40.12 3 196.18 15.22 0.4 654.68 6.94 10 753.11 27.83 0.6 654.68 6.94 10 753.11 27.83 0.8 233.72 40.12 3 196.18 15.22 1.0 0.0 0.0 0.0 0.0 T im e = 5. 0 × t p 0.0 0.0 0.0 0.0 0.0 0.2 96.12 1.13 1 290.07 0.47 0.4 258.24 0.21 4 209.84 0.69 0.6 258.24 0.21 4 209.84 0.69 0.8 96.12 1.13 1 290.07 0.47 1.0 0.0 0.0 0.0 0.0 Table 9. w(*10−5) for constant a b = 1.5 and α = 0.2, β1 = 0.3, β2 = α1 = 0.4 and for different X and Y , see Figure 9. 307 Sudhanshu Aggarwal Acta Polytechnica 0 2000 4000 6000 8000 10000 12000 14000 0 0.2 0.4 0.6 0.8 1 First Mode Time=0.K Y=0.2 First Mode Time=0.K Y=0.6 First Mode Time=5.K Y=0.2 First Mode Time=5.K Y=0.6 (a). First mode. 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 Second Mode Time=0.K Y=0.2 Second Mode Time=0.K Y=0.6 Second Mode Time=5.K Y=0.2 Second Mode Time=5.K Y=0.6 (b). Second mode. Figure 9. Variation of w(*10−5) with different X for constant a b = 1.5 and α = 0.2, β1 = 0.3, β2 = α1 = 0.4 and for different Y . Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II a b ↓ 0.5 20.93 14.02 14.34 3.63 1.0 63.12 32.41 38.67 4.84 1.5 126.32 38.87 55.12 1.38 2.0 233.96 37.37 61.42 0.14 2.5 398.28 36.09 53.21 0.003 Table 10. w(*10−5) for different a b and α = β1 = β2 = α1 = 0.0 and for X = Y = 0.2, see Figure 10. 308 vol. 64 no. 4/2024 Non-Homogeneity Effect on the Vibration of . . . -50 0 50 100 150 200 250 300 350 400 450 0.5 1 1.5 2 2.5 First Mode Time=0.K Second Mode Time=0.K First Mode Time=5.K Second Mode Time=5.K Figure 10. Variation of w(*10−5) for different a b and α = β1 = β2 = α1 = 0.0 and for X = Y = 0.2. Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II a b ↓ 0.5 24.23 15.33 17.68 4.37 1.0 66.84 33.97 41.32 5.60 1.5 130.43 40.12 59.13 1.93 2.0 238.27 39.11 64.92 0.21 2.5 403.03 37.28 57.16 0.006 Table 11. w(*10−5) for different a b and α = β1 = β2 = 0.0, α1 = 0.4 and for X = Y = 0.2, see Figure 11. -50 0 50 100 150 200 250 300 350 400 450 0.5 1 1.5 2 2.5 First Mode Time=0.K Second Mode Time=0.K First Mode Time=5.K Second Mode Time=5.K Figure 11. Variation of w(*10−5) for different a b and α = β1 = β2 = 0.0, α1 = 0.4 and for X = Y = 0.2. 309 Sudhanshu Aggarwal Acta Polytechnica Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II a b ↓ 0.5 28.03 15.73 21.53 3.64 1.0 96.67 33.78 58.45 4.34 1.5 235.72 40.31 98.27 1.29 2.0 501.18 38.92 121.02 0.099 2.5 935.23 37.35 109.33 0.00053 Table 12. w(*10−5) for different a b and α = β1 = 0.3, β2 = α1 = 0.4 and for X = Y = 0.2, see Figure 12. -200 0 200 400 600 800 1000 0.5 1 1.5 2 2.5 First Mode Time=0.K Second Mode Time=0.K First Mode Time=5.K Second Mode Time=5.K Figure 12. Variation of w(*10−5) for different a b and α = β1 = 0.3, β2 = α1 = 0.4 and for X = Y = 0.2. α1 = 0.0, α = 0.0, a b = 1.5 α1 = 0.4, α = 0.0, a b = 1.5 β1 = 0.0 β2 = 0.0 β1 = 0.4 β2 = 0.2 β1 = 0.0 β2 = 0.0 β1 = 0.4 β2 = 0.2 Mode → I II I II I II I II Present Study 6 462 1 623 5 324 1 332 6 897 1 728 5 543 1 433 [13] 6 462 1 623 5 324 1 332 6 883 1 724 5 531 1 428 Table 13. The comparative data of the time period with the reference [13]. α1 = α = β1 = β2 = 0.0 α1 = 0.4, α = β1 = β2 = 0.0 a b = 1.0 a b = 2.0 a b = 1.0 a b = 2.0 Mode → I II I II I II I II Present Study 10 411 2 638 3 724 905 10 860 2 747 4 082 961 [13] 10 411 2 638 3 724 905 10 847 2 741 4 073 958 Table 14. The comparative data of the time period with the reference [13]. two studies for the time function is shown in Tables 13– 15 while the comparison of the results of the deflection are shown in Tables 16–21. The Tables 13–15 show that the results of the period for the present study and the reference [13] are identical when the value of the constant of non-homogeneity (α1) is zero. These tables also show that the quadratic variation in the material density of the plate along the x-axis is more dominant as compared to that along the x-axis. Tables 16–18 show that the results of the deflection for the present study and the reference [13] are identical when the value of the constant of non-homogeneity (α1) is zero. The results shown in Tables 19–21 indicate that the quadratic variation in the material density of the 310 vol. 64 no. 4/2024 Non-Homogeneity Effect on the Vibration of . . . α1 = 0.0, α = 0.6, a b = 1.5 α1 = 0.4, α = 0.6, a b = 1.5 β1 = 0.0 β2 = 0.0 β1 = 0.4 β2 = 0.2 β1 = 0.0 β2 = 0.0 β1 = 0.4 β2 = 0.2 Mode → I II I II I II I II Present Study 7 737 2 002 6 213 1 603 8 497 2 176 6 626 1 759 [13] 7 737 2 002 6 213 1 603 8 486 2 173 6 617 1 756 Table 15. The comparative data of the time period with the reference [13]. Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II Present Study 126.32 38.87 55.12 1.38 [13] 126.32 38.87 55.12 1.38 Table 16. The comparative data of the deflection with the reference [13]; α = β1 = β2 = α1 = 0.0, a b = 1.5, X = Y = 0.20. Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II Present Study 233.96 37.37 61.42 0.14 [13] 233.96 37.37 61.42 0.14 Table 17. The comparative data of the deflection with the reference [13]; α = β1 = β2 = α1 = 0.0, a b = 2.0, X = Y = 0.20. Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II Present Study 427.32 14.52 187.31 0.53 [13] 427.32 14.52 187.31 0.53 Table 18. The comparative data of the deflection with the reference [13]; α = β1 = β2 = α1 = 0.0, a b = 1.5, X = 0.20, Y = 0.60. Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II Present Study 130.43 40.12 59.13 1.93 [13] 128.62 39.74 57.36 1.58 Table 19. The comparative data of the deflection with the reference [13]; α = β1 = β2 = α1 = 0.4, a b = 1.5, X = Y = 0.20. Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II Present Study 238.27 39.11 64.92 0.21 [13] 236.43 38.15 63.88 0.18 Table 20. The comparative data of the deflection with the reference [13]; α = β1 = β2 = α1 = 0.4, a b = 2.0, X = Y = 0.20. Time = 0.0 × tp Time = 5.0 × tp Mode → I II I II Present Study 431.01 15.93 190.34 0.83 [13] 428.52 15.02 188.82 0.64 Table 21. The comparative data of the deflection with the reference [13]; α = β1 = β2 = α1 = 0.4, a b = 1.5, X = 0.20, Y = 0.60. 311 Sudhanshu Aggarwal Acta Polytechnica plate along the x-axis is more dominant as compared to that along the x-axis. The comparative study of the present results with the reference [13] using Tables 13–21 ensure the accuracy and the validity of the derived formulas and the results of this study. 7. Conclusion The author effectively discussed the non-homogeneity effect on the vibration of the rectangular visco-elastic plate subjected to the linear temperature effect with both the directions quadratic thickness variation. 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Applied Sciences 11(13):6029, 2021. https://doi.org/10.3390/app11136029 313 https://doi.org/10.1016/j.amc.2005.11.068 https://doi.org/10.1016/j.matcom.2021.01.019 https://doi.org/10.1007/s11071-020-05892-0 https://doi.org/10.1177/1077546320926887 https://doi.org/10.3390/app11136029 Acta Polytechnica 64(4):297–313, 2024 1 Introduction 2 Assumptions of the study 3 Equation of the motion and the method of the analysis 4 Differential equation of the time function with its solution 5 Results and discussions 6 Comparison of the results of the present study with the previous study 7 Conclusion References