# Time Period of Thermal-Induced Vibration of Skew Plate with Two-Dimensional Circular Thickness

Authors: Amit Sharma; Reeta Bhardwaj; Neeraj Lather; Sudeshna Ghosh; Naveen Mani; Kamal Kumar
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, two-dimensional circular thickness effects on time period of skew plate made with nonhomogeneous material properties are computed in variable temperature environment. ClampedC, simply supported S, and free-edge F conditions are the various combinations considered in this work. One-dimensional circular density and two-dimensional parabolic temperature variation on the plate are considered due to the nonhomogeneous nature of the material and variable temperature environment. Rayleigh–Ritz method is used to compute the time period for the first two modes of vibration. A convergence study of frequency modes of skew plate on various edge conditions has also been performed. The present work has been compared with the frequency modes of parallelogram and rectangle plate on various edge conditions. The objective of the study is to present a numerical data of time period as well as to show how one can control the frequencies of plate by taking an appropriate variation in plate parameters.


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## Body

## 1. Introduction

A large number of structures/equipments are made up of plates, and this fact has prompted the researchers to do a detailed study of vibrational frequencies of plates. Almost, all structures while working undergo some vibrations. A large vibration is a kind of waste that can reduce the performance of structures/equipments. In order to increase the performance and reliability of structures/equipment, the vibration of the structure needs to be decreased. The various parameters such as variation in shape, size, tapering, nonhomogeneity, and temperature of the plate play a crucial role in minimizing the frequencies. Studies regarding plate’s vibration have been well documented in the literature. In the available literature, for the thickness of the plate, researchers focused on linear, parabolic variation (one dimensional, two dimensional), parabolic variation (one dimensional, two dimensional), and circular (only in one dimensional) variation. To the best of the knowledge of the authors, the effect of two-dimensional circular variation in thickness on time period of frequency modes has not been studied yet. This prompted the authors to carry out the research to study the effect of two-dimensional circular thickness impact on time period of parallelogram plate.Natural frequency of rectangular plates has been computed in [1] on various combinations of clamped, simply supported, and free-edge condition by applying Galerkin’s averaging technique. The first three modes of frequencies comprise the effect of foundation, density, thickness, and aspect ratio of nonhomogeneous orthotropic rectangle plate which have been computed in [2] using the Chebyshev collocation technique. The semianalytical method has been used in [3] to study the effect of skew angle, aspect ratio, orthotropy, and boundary conditions on the vibration of orthotropic skew plates. Rayleigh–Ritz and kinematic methods have been used in [4] to analyze the effects of material, number of layers, and fiber orientation on frequencies and mode shapes of anisotropic, simply supported plates of square planform. Natural frequencies and mode shapes of sector plate on simply supported edges have been computed in [5] using an analytical solution. Multiple scales method has been applied as in [6] to study the vibration of rectangle plate (cracked, fluid medium), and the effect of crack length, fluid level, fluid density, and immersed depth of the plate have been presented. High accurate solution (using accurate series) for rectangle plate with all possible combinations on edge conditions has been presented in [7]. A detailed comprehensive study of vibration of plates, beams, and shell has been studied in [8] using Rayleigh–Ritz method. Natural vibration of nonhomogeneous rectangle plate has been studied in [9] using Rayleigh–Ritz method with boundary characteristic orthogonal polynomials and presented the effect of various plate parameters on the frequency of the plate. Ritz method has been applied in [10] to investigate vibrational behavior (first three modes) of nonhomogeneous, nonuniform circular plates on clamped, simply supported, and free edges and presented the various plate parameters impact on frequencies. Effect of circular variation in density on time period of parallelogram plate having two-dimensional thickness and temperature on CCCC and CSCS edge condition has been computed in [11] using Rayleigh–Ritz method, where C and S stand for clamped and simply supported edge conditions, respectively. Modeling on time period of vibrational frequencies of nonuniform skew plate made up of nonhomogeneous material has been presented in [12] using Rayleigh–Ritz method, and effect of various plate parameters on time period has been computed. Rayleigh–Ritz method has been implemented in [13] to study the frequency behavior of skew plate with variable thickness on circular variation in Poisson’s ratio and linear variation in temperature. Vibration response of generally curved shell structures to different stacking sequences, material typologies, and boundary conditions has been analysed in [14] by using differential quadrature method. Generalized layer-wise (LW) along with generalized differential quadrature (GDQ) method has been used in [15] to analyze anisotropic double curved shells with an arbitrary shape.In this current study, the authors focused to show the impact of two-dimensional circular thickness on the time period of nonhomogeneous parallelogram plate on various boundary conditions under variable temperature field. The authors also computed one-dimensional circular density and two-dimensional parabolic thickness impact on time period. All the numerical data are presented in the form of tables. In order to authenticate the findings and to present the objectives, a comparative analysis of frequency modes of parallelogram and rectangle plate with the available published results is also presented in tabular form.


## 2. Analysis

A skew (parallelogram) plate made with nonhomogeneous material property with variable thicknessl having skew angle θ, length a, breadth b, density ρ, and Poisson’s ratio ν is considered referred to skew coordinates ζ=x−ytanθ,ψ=ysecθ (refer Figure 1).Figure 1
Parallelogram plate with skew angleθ.The equation for kinetic energyTs and strain energy Vs for natural transverse vibration of the nonuniform parallelogram is given by the expression as [16](1)Ts=12ω2cosθ∬ρlΦ2dζdψ,(2)Vs=12cos3θ∬D1∂2Φ∂ζ22−4sinθ∂2Φ∂ζ2∂2Φ∂ζ∂ψ+2sin2θ+νcos2θ∂2Φ∂ζ2∂2Φ∂ψ2+21+sin2θ−νcos2θ∂2Φ∂ζ∂ψ2−4sinθ∂2Φ∂ζ∂ψ∂2Φ∂ψ2+∂2Φ∂ψ22dζdψ,where D1=El3/121−ν2 is the flexural rigidity. Here, ν, E, and Φ are Poisson’s ratio, Young’s modulus, and deflection function, respectively.Rayleigh–Ritz method requires that maximum strain energy must be equal to maximum kinetic energy; that is,(3)J=δVs−Ts=0.On substituting (1) and (2) in (3), we get(4)J=12cos3θ∬D1∂2Φ∂ζ22−4sinθ∂2Φ∂ζ2∂2Φ∂ζ∂ψ+2sin2θ+νcos2θ∂2Φ∂ζ2∂2Φ∂ψ2+21+sin2θ−νcos2θ∂2Φ∂ζ∂ψ2−4sinθ∂2Φ∂ζ∂ψ∂2Φ∂ψ2+∂2Φ∂ψ22dζdψ−12ω2cosθ∬ρlΦ2dζdψ=0.The thicknessl of the skew plate is assumed to be circular in both dimensions (refer Figure 2), and density ρ is assumed to be circular in one dimension as(5)l=l01+ϖ11−1−ζ2a21+ϖ21−1−ψ2b2,ρ=ρ01−φ1−1−ζ2a2,where l0 and ρ0 are the thickness and density of the plate, respectively, at the origin. Also, ϖ1,ϖ20≤ϖ1,ϖ2≤1 and φ0≤φ<1 are taper parameters and nonhomogeneity parameters, respectively.Figure 2
Parallelogram plate having two-dimensional circular thickness.Two-dimensional steady-state temperature variations on the plate are considered to be parabolic and expressed as(6)η=η01−ζ2a21−ψ2b2,where τ and τ0 denote the temperature excess above the reference temperature on the plate at any point and the origin, respectively. The temperature dependence modulus of elasticity for engineering structures is given by(7)E=E01−γη,where E0 is the Young’s modulus at mentioned temperature (i.e., η=0), and γ is the slope of variation.Substituting (6) in (7), we get the following expression:(8)E=E01−κ1−ζ2a21−ψ2b2,where κ=γη0,0≤κ<1 is called temperature gradient.Using (5) and (8), (4) becomes(9)J=D02∫0a∫0b1−κ1−ζ2a21−ψ2b21+ϖ1Λ11+ϖ2Λ23∂2Φ∂ζ22−4Ysinθ∂2Φ∂ζ2∂2Φ∂ζ∂ψ+2Y2sin2θ+νcos2θ∂2Φ∂ζ2∂2Φ∂ψ2+2Y21+sin2θ−νcos2θ∂2Φ∂ζ∂ψ2−4Y3sinθ∂2Φ∂ζ∂ψ∂2Φ∂ψ2+Y4∂2Φ∂ψ22dψdζ−λ2cos4θ∫0a∫0b1−φΛ1,1+ϖ1Λ11+ϖ2Λ2Φ2dψdζ,where(10)D0=E0l03/121−ν2,Λ1=1−1−ζ2a2,Λ2=1−1−ψ2b2,λ2=ρ0ω2l0a4/D0andϒ=a/b.The two-term deflection function that satisfies all the edge conditions can be taken as(11)Φζ,ψ=ζaeψbf1−ζag1−ψbh×∑i=0NΩiζaψb1−ζa1−ψbi,which is the product of two functions. Here, the first function represents the boundary conditions depending on the value of e,f,g, and h, which can take different values depending upon the support edge condition. Values 0,1,2 are assigned for free edge, simply supported, and clamped edge, respectively. The second function represents the number of modes of frequencies, and Ωi,i=0,1,2,…,N represents arbitrary constants.In order to minimize the function given in (9), we require the following condition:(12)∂J∂Ωi=0,i=0,1,2,3…N.After simplifying (12), we get a homogeneous system of equations in Ωi whose nonzero solution gives the equation of frequency as(13)P−λ2Q=0,where P=pijN+1 and Q=qijN+1 are the square matrix of order n+1, i=0,1,2…N, and j=0,1,2…N.The following expression is used for calculating the time period:(14)K=2πλ,where λ is a frequency obtained from (13).


## 3. Numerical Results and Discussion

Time periodK of parallelogram plate on CCCC, CCCS, CCSS, CSSS, SSSS, CSCS, and SCSC edge conditions (refer Figure 3) is computed and shown the effect of various plate parameters (especially two-dimensional circular thickness impact) on the behavior of time period of frequency modes with the help of tables. The value of aspect ratio a/b=1.5, skew angle θ=30°, and Poisson’s ratio ν=0.345 is taken throughout the calculation.Figure 3
Parallelogram plate with various edge conditions.Table1 summarizes the time period K of frequency modes of parallelogram plate on CCCC, CCCS, CCSS, CSSS, SSSS, CSCS, and SCSC edge conditions corresponding to tapering parameters ϖ1,ϖ2 for a fixed value of thermal gradient κ=0.2 and nonhomogeneity φ=0.4. From Table 1, the following conclusions can be made:(i)
Time periodK decreases for both the increasing value of tapering parameters ϖ1,ϖ2 on each edge condition.(ii)
The rate of decrement in time periodK in case of tapering ϖ2 is more in comparison to the rate of decrement in time period K in case of tapering ϖ1 on all edge conditions.(iii)
The time periodK is higher on SCSC edge condition and lower on CCCS edge condition. But the rate of decrement in time period K is less in CSCS edge condition and higher on CSSS edge condition.(iv)
The order of increase in time periodK for different edge conditions is CCCS<CCSS<CSCS<CSSS<SSSS<CCCC<SCSC, corresponding to tapering parameters ϖ1,ϖ2.Table 1
Time period of parallelogram plate on various edge conditions vs tapering parametersϖ1, ϖ2 for a fixed value of aspect ratio a/b=1.5.
ϖ1κ=0.2,φ=0.4ϖ2=0.0ϖ2=0.2ϖ2=0.4ϖ2=0.6ϖ2=0.8ϖ2=1.0K1K2K1K2K1K2K1K2K1K2K1K2CCCC0.00.021 560.086 290.020 280.081 290.019 010.076 490.017 790.071 980.016 640.067 790.015 570.063 930.20.020 910.083 430.019 670.078 650.018 450.074 060.017 270.069 730.016 160.065 700.015 130.061 990.40.020 270.080 650.019 080.076 080.017 900.071 680.016 770.067 530.015 690.063 670.014 700.060 100.60.019 640.077 940.018 500.073 580.017 360.069 370.016 270.065 400.015 240.061 690.014 270.058 270.80.019 040.075 330.017 940.071 170.016 840.067 140.015 790.063 330.014 790.059 780.013 860.056 481.00.018 450.072 810.017 390.068 840.016 340.064 990.015 320.061 340.014 360.057 930.013 450.054 77CCCS0.00.014 100.105 800.012 350.099 740.010 950.094 140.009 820.088 960.008 890.084 170.008 110.079 760.20.013 660.101 660.011 960.095 830.010 610.090 430.009 510.085 430.008 610.080 820.007 850.076 560.40.013 230.097 680.011 590.092 070.010 280.086 860.009 210.082 050.008 340.077 610.007 610.073 510.60.012 820.093 890.011 230.088 480.009 960.083 470.008 930.078 830.008 080.074 550.007 370.070 600.80.012 430.090 270.010 880.085 070.009 650.080 240.008 650.075 770.007 830.071 640.007 140.067 841.00.012 050.086 840.010 550.081 830.009 350.077 180.008 380.072 870.007 590.068 890.006 920.065 23CCSS0.00.016 590.101 740.014 520.095 450.012 860.089 650.011 520.084 330.010 410.079 450.009 480.074 990.20.015 860.095 350.013 880.089 390.012 300.083 900.011 020.078 870.009 960.074 250.009 070.070 030.40.015 150.089 540.013 270.083 900.011 760.078 700.010 540.073 920.009 530.069 560.008 680.065 570.60.014 480.084 270.012 690.078 910.011 250.073 980.010 080.069 460.009 110.065 320.008 310.061 550.80.013 840.079 480.012 130.074 400.010 760.069 720.009 640.065 430.008 720.061 500.007 950.057 921.00.013 230.075 130.011 610.070 300.010 300.065 850.009 230.061 780.008 350.058 050.007 610.054 65CSSS0.00.021 250.150 760.019 850.143 650.018 470.136 750.017 160.130 120.015 940.123 800.014 820.117 830.20.020 290.140 950.018 950.134 080.017 630.127 440.016 390.121 080.015 230.115 050.014 170.109 360.40.019 350.131 990.018 080.125 380.016 830.119 000.015 640.112 930.014 540.107 190.013 530.101 790.60.018 460.123 840.017 240.117 500.016 060.111 400.014 930.105 610.013 880.100 150.012 920.095 030.80.017 610.116 450.016 460.110 380.015 330.104 560.014 250.099 040.013 260.093 850.012 340.088 991.00.016 810.109 750.015 710.103 940.014 640.098 390.013 610.093 130.012 670.088 200.011 790.083 59SSSS0.00.021 280.169 010.019 760.162 050.018 290.155 210.016 920.148 580.015 660.142 190.014 520.136 080.20.020 590.161 990.019 120.155 060.017 700.148 290.016 370.141 740.015 150.135 470.014 050.129 490.40.019 910.155 080.018 480.148 210.017 110.141 530.015 830.135 100.014 650.128 960.013 590.123 120.60.019 240.148 360.017 860.141 580.016 540.135 010.015 300.128 720.014 160.122 730.013 130.117 050.80.018 590.141 880.017 260.135 220.015 980.128 790.014 780.122 640.013 690.116 810.012 690.111 311.00.017 950.135 690.016 670.129 160.015 440.122 880.014 280.116 900.013 220.111 240.012 260.105 91CSCS0.00.018 360.155 140.017 040.148 710.015 770.142 400.014 590.136 270.013 500.130 380.012 510.124 750.20.017 810.149 340.016 530.143 040.015 290.136 870.014 140.130 900.013 080.125 180.012 130.119 720.40.017 270.143 630.016 030.137 490.014 830.131 480.013 710.125 690.012 680.120 130.011 760.114 850.60.016 760.138 090.015 550.132 100.014 380.126 260.013 290.120 640.012 300.115 270.011 390.110 160.80.016 260.132 730.015 080.126 900.013 950.121 250.012 890.115 810.011 920.110 610.011 050.105 671.00.015 780.127 570.014 630.121 920.013 530.116 440.012 500.111 180.011 560.106 160.010 710.101 40SCSC0.00.026 110.088 400.024 600.083 300.023 110.078 370.021 680.073 730.020 320.069 410.019 050.065 430.20.025 170.085 210.023 730.080 350.022 310.075 650.020 940.071 210.019 640.067 080.018 420.063 270.40.024 230.082 070.022 860.077 450.021 500.072 970.020 190.068 750.018 950.064 800.017 780.061 150.60.023 310.079 020.022 000.074 630.020 710.070 370.019 450.066 340.018 260.062 580.017 150.059 090.80.022 400.076 080.021 160.071 910.019 920.067 860.018 730.064 020.017 590.060 420.016 520.057 091.00.021 520.073 250.020 340.069 290.019 160.065 440.018 020.061 770.016 930.058 340.015 910.055 16Table2 represents the time period K of frequency modes of parallelogram plate on CCCC, CCCS, CCSS, CSSS, SSSS, CSCS, and SCSC edge conditions corresponding to thermal gradient κ for a fixed value of nonhomogeneity φ=0.4 and variable values of tapering parameters ϖ1,ϖ2 from 0.0 to 1.0. From Table 2, the following facts can be interpreted:(i)
Increase in the thermal gradientκ results in an increase in the time period K, but the increasing value of both tapering parameters ϖ1,ϖ2 results in the decrease in time period K, on all edge conditions.(ii)
The rate of increment in time periodK in case of thermal gradient κ is less than that in case of tapering parameters ϖ1, ϖ2, on all edge conditions.(iii)
Here, also the time periodK is higher on SCSC edge condition and lower on CCCS edge condition as shown in Table 1. But the rate of increment in time period K is less on CCSS edge condition and higher on SCSC edge condition.(iv)
The time periodK for different edge conditions corresponding to thermal gradient κ is in the following ascending order: CCCS<CCSS<CSCS<SSSS<CSSS<CCCC<SCSC.Table 2
Time period of parallelogram plate on various edge conditions vs thermal gradientκ for a fixed value of aspect ratio a/b=1.5.
κφ=0.4ϖ1=ϖ2=0.0ϖ1=ϖ2=0.2ϖ1=ϖ2=0.4ϖ1=ϖ2=0.6ϖ1=ϖ2=0.8ϖ1=ϖ2=1.0K1K2K1K2K1K2K1K2K1K2K1K2CCCC0.00.020 570.082 280.018 860.075 300.017 230.068 850.015 720.062 980.014 340.057 680.013 080.052 940.20.021 560.086 290.019 670.078 650.017 900.071 680.016 270.065 400.014 790.059 780.013 450.054 770.40.022 710.090 960.020 600.082 530.018 650.074 930.016 880.068 160.015 280.062 150.013 860.056 830.60.024 060.096 510.021 680.087 080.019 510.078 720.017 560.071 350.015 830.064 880.014 300.059 190.80.025 690.103 260.022 940.092 540.020 490.083 210.018 330.075 120.016 440.068 080.014 780.061 94CCCS0.00.013 940.101 590.011 840.092 230.010 180.083 790.008 850.076 190.007 770.069 370.006 870.063 270.20.014 100.105 800.011 960.095 830.010 280.086 860.008 930.078 830.007 830.071 640.006 920.065 230.40.014 270.110 600.012 090.099 900.010 380.090 320.009 000.081 770.007 890.074 160.006 970.067 380.60.014 450.116 150.012 220.104 560.010 480.094 230.009 080.085 070.007 960.076 960.007 030.069 770.80.014 630.122 670.012 360.109 950.010 580.098 710.009 170.088 820.008 020.080 110.007 080.072 44CCSS0.00.016 430.098 660.013 770.086 910.011 680.076 690.010 010.067 820.008 670.060 160.007 5750.053 550.20.016 590.101 740.013 880.089 390.011 760.078 700.010 080.069 460.008 720.061 500.007 6190.054 650.40.016 760.105 140.014 010.092 110.011 860.080 880.010 150.071 220.008 780.062 930.007 6630.055 830.60.016 940.108 930.014 130.095 120.011 950.083 270.010 220.073 130.008 830.064 480.007 7090.057 090.80.017 110.113 200.014 260.098 450.012 040.085 890.010 290.075 210.008 890.066 150.007 7550.058 44CSSS0.00.020 490.145 080.018 370.129 550.016 390.115 380.014 600.102 700.013 000.091 480.011 600.081 650.20.021 250.150 760.018 950.134 080.016 830.119 000.014 930.105 610.013 260.093 850.011 790.083 590.40.022 110.157 170.019 580.139 130.017 300.123 000.015 280.108 810.013 520.096 420.012 000.085 680.60.023 070.164 480.020 290.144 820.017 810.127 460.015 660.112 320.013 810.099 230.012 210.087 950.80.024 180.172 950.021 070.151 290.018 380.132 460.016 070.116 230.014 110.102 330.012 440.090 43SSSS0.00.020 370.160 300.018 420.147 750.016 580.135 430.014 890.123 650.013 370.112 600.012 010.102 400.20.021 280.169 010.019 120.155 060.017 110.141 530.015 300.128 720.013 690.116 810.012 260.105 910.40.022 320.179 310.019 900.163 580.017 710.148 540.015 750.134 480.014 030.121 550.012 530.109 820.60.023 530.191 760.020 790.173 690.018 360.156 720.016 240.141 110.014 400.126 940.012 810.114 220.80.024 960.207 240.021 820.185 960.019 100.166 440.016 780.148 860.014 800.133 140.013 110.119 22CSCS0.00.017 560.147 180.015 910.136 210.014 350.125 640.012 920.115 650.011 630.106 320.010 490.097 710.20.018 360.155 140.016 530.143 040.014 830.131 480.013 290.120 640.011 920.110 610.010 710.101 400.40.019 270.164 570.017 220.151 010.015 360.138 230.013 700.126 370.012 230.115 480.010 960.105 550.60.020 340.175 950.018 010.160 510.015 950.146 160.014 140.133 010.012 570.121 060.011 220.110 280.80.021 610.190 090.018 930.172 070.016 610.155 650.014 630.140 840.012 940.127 570.011 500.115 73SCSC0.00.024 930.084 350.022 780.076 990.020 730.070 170.018 830.063 970.017 090.058 390.015 510.053 400.20.026 110.088 400.023 730.080 350.021 500.072 970.019 450.066 340.017 590.060 420.015 910.055 160.40.027 470.093 120.024 820.084 210.022 370.076 180.020 140.069 040.018 140.062 710.016 350.057 130.60.029 060.098 700.026 070.088 730.023 340.079 900.020 900.072 140.018 730.065 340.016 820.059 380.80.030 970.105 480.027 520.094 130.024 450.084 290.021 750.075 770.019 390.068 390.017 340.061 98The time periodK of frequency modes of parallelogram plate on CCCC, CCCS, CCSS, CSSS, SSSS, CSCS, and SCSC edge conditions corresponding to nonhomogeneity φ for a fixed value of thermal gradient κ=0.4 and variable values of tapering parameters ϖ1,ϖ2 from 0.0 to 1.0 is summarized in Table 3. From Table 3, it can be concluded that(i)
Time periodK decreases with increasing value of nonhomogeneity φ as well as for the increasing value of both tapering parameters ϖ1,ϖ2, for each edge conditions.(ii)
The rate of decrement in time periodK in case of both the tapering parameters ϖ1, ϖ2 is higher when compared with a rate of decrement in time period K for nonhomogeneity φ, on all edge conditions.(iii)
Here, also the time periodK is higher on SCSC edge condition and lower on CCCS edge condition as shown in Tables 1 and 2. But the rate of decrement in time period K is lower on CCCS edge condition and higher on CSSS edge condition. The time period K for different edge conditions increases in the following sequence: CCCS<CCSS<CSCS<SSSS<CCCC<CSSS<SCSC, corresponding to nonhomogeneity φ.Table 3
Time period of parallelogram plate on various edge conditions vs nonhomogeneityφ for a fixed value of aspect ratio a/b=1.5.
φκ=0.4ϖ1=ϖ2=0.0ϖ1=ϖ2=0.2ϖ1=ϖ2=0.4ϖ1=ϖ2=0.6ϖ1=ϖ2=0.8ϖ1=ϖ2=1.0K1K2K1K2K1K2K1K2K1K2K1K2CCCC0.00.023 540.093 850.021 370.085 170.019 370.077 350.017 540.070 370.015 890.064 180.014 420.058 690.20.023 130.092 420.020 990.083 860.019 010.076 150.017 210.069 270.015 590.063 170.014 140.057 770.40.022 710.090 960.020 600.082 530.018 650.074 930.016 880.068 160.015 280.062 150.013 860.056 830.60.022 280.089 480.020 200.081 170.018 280.073 690.016 540.067 020.014 970.061 110.013 570.055 870.80.021 850.087 980.019 800.079 800.017 910.072 430.016 190.065 870.014 650.060 050.013 270.054 89CCCS0.00.014 760.114 080.012 510.103 070.010 740.093 210.009 320.084 410.008 170.076 580.007 230.069 600.20.014 520.112 350.012 300.101 500.010 560.091 780.009 170.083 100.008 030.075 380.007 100.068 500.40.014 270.110 600.012 090.099 900.010 380.090 320.009 000.081 770.007 890.074 160.006 970.067 380.60.014 020.108 820.011 880.098 270.010 190.088 830.008 840.080 410.007 750.072 920.006 840.066 250.80.013 770.107 010.011 660.096 620.010 000.087 330.008 670.079 040.007 600.071 660.006 710.065 09CCSS0.00.017 930.110 320.014 990.096 690.012 700.084 920.010 880.074 810.009 410.066 130.008 220.058 680.20.017 360.107 760.014 510.094 430.012 290.082 920.010 520.073 040.009 100.064 550.007 940.057 270.40.016 760.105 140.014 010.092 110.011 860.080 880.010 150.071 220.008 780.062 930.007 660.055 830.60.016 140.102 460.013 480.089 740.011 400.078 780.009 760.069 360.008 440.061 280.007 360.054 350.80.015 480.099 690.012 920.087 310.010 930.076 620.009 350.067 450.008 080.059 580.007 040.052 83CSSS0.00.023 680.165 140.021 000.146 250.018 560.129 350.016 410.114 450.014 530.101 460.012 900.090 180.20.022 910.161 200.020 300.142 740.017 950.126 220.015 860.111 670.014 040.098 970.012 460.087 960.40.022 110.157 170.019 580.139 130.017 300.123 000.015 280.108 810.013 520.096 420.012 000.085 680.60.021 260.153 030.018 820.135 430.016 620.119 710.014 680.105 870.012 980.093 800.011 510.083 340.80.020 370.148 780.018 020.131 630.015 900.116 320.014 030.102 850.012 410.091 110.011 000.080 94SSSS0.00.023 210.185 360.020 720.169 190.018 450.153 710.016 430.139 220.014 650.125 890.013 090.113 780.20.022 770.182 360.020 320.166 410.018 080.151 150.016 090.136 870.014 340.123 740.012 810.111 820.40.022 320.179 310.019 900.163 580.017 710.148 540.015 750.134 480.014 030.121 550.012 530.109 820.60.021 860.176 210.019 480.160 710.017 320.145 890.015 400.132 050.013 710.119 330.012 230.107 790.80.021 390.173 050.019 050.157 780.016 920.143 190.015 030.129 570.013 370.117 060.011 930.105 71CSCS0.00.019 930.169 780.017 820.155 850.015 900.142 700.014 190.130 490.012 680.119 270.011 360.109 050.20.019 600.167 190.017 530.153 450.015 630.140 480.013 940.128 440.012 460.117 390.011 160.107 320.40.019 270.164 570.017 220.151 010.015 360.138 230.013 700.126 370.012 230.115 480.010 960.105 550.60.018 930.161 900.016 910.148 540.015 080.135 950.013 440.124 260.012 000.113 530.010 750.103 760.80.018 590.159 190.016 600.146 020.014 790.133 620.013 180.122 110.011 770.111 550.010 530.101 94SCSC0.00.028 640.096 270.025 920.087 090.023 400.078 820.021 100.071 440.019 030.064 920.017 180.059 150.20.028 060.094 710.025 380.085 660.022 890.077 510.020 630.070 250.018 590.063 820.016 770.058 150.40.027 470.093 120.024 820.084 210.022 370.076 180.020 140.069 040.018 140.062 710.016 350.057 130.60.026 860.091 500.024 240.082 730.021 830.074 830.019 640.067 800.017 670.061 590.015 920.056 100.80.026 230.089 860.023 650.081 220.021 280.073 450.019 120.066 540.017 190.060 430.015 470.055 04


## 4. Convergence of Results

This section reports convergence study done on frequency modes of parallelogram plate onCCCC, CCCS, CCSS, CSSS, SSSS, CSCS, and SCSC edge condition, for the plate parameters in rage specified, that is, ϖ1=ϖ2=φ=κ=0.0,ν=0.345, and a/b=1.5. The results are presented in tabular form (refer Table 4). From Table 4, we can see that the frequency modes of parallelogram plate converge up to five decimal places in the fifth approximation.Table 4
Convergence study of frequency modes of parallelogram plate onCCCC, CCCS, CCSS, and CSSS edge conditions.
NCCCCCCCSCCSSCSSSSλ1λ2λ1λ2λ1λ2λ1λ2274.002 42294.603 5559.963 56435.889 8364.645 02373.445 9841.214 97286.236 83374.002 38285.645 1859.874 74276.028 2760.632 52232.215 4841.208 74208.260 38474.002 35285.557 7959.874 73268.009 5860.626 64228.436 8741.208 48207.067 62574.002 35285.557 7959.874 73268.009 5860.626 64228.436 8741.208 48207.067 62The convergence of frequency modes of parallelogram plate onSSSS,CSCS and SCSC edge conditions is presented in Table 5. Table 5 also concludes that the frequency modes of parallelogram plate converge upto five decimal places in the fifth approximation.Table 5
Convergence study of frequency modes of parallelogram plate onSSSS, CSCS, and SCSC edge conditions.
NSSSSCSCSSCSCλ1λ2λ1λ2λ1λ2237.917 040296.603 0641.381 87345.903 1572.037 93241.655 39337.916 401214.652 7941.379 22245.733 3772.037 32236.699 50437.916 246213.270 6541.379 04243.113 8172.036 57236.575 85537.916 246213.270 6541.379 04243.113 8172.036 57236.575 85


## 5. Results Comparison

In order to validate the findings of the present study as well as to present the objective of the study, a comparison of frequency modes obtained in the present study was done with the following:(i)
Frequency modes of parallelogram plate obtained in [11] on CCCC and CSCS edge condition corresponding to tapering parameters ϖ1,ϖ2 and aspect ratio a/b(ii)
Frequency modes of rectangle plate obtained in [17, 18] on CCCC, SSSS, and CSCS edge condition corresponding to tapering parameters ϖ1,ϖ2 and on CCCC, SSSS, CSSS, CSCS, and CCCS edge conditions corresponding to a thermal gradient κThe results are reported in tabular form (refer Tables6 and 7 for parallelogram plate and refer Tables 8 and 9 for rectangle plate). For comparison of frequency modes of rectangle plate and obtained in [17, 18], the authors exclude the nonhomogeneity φ in the present study as it was not considered in [17, 18].Table 6
Comparison of frequency modes of the present study (parallelogram plate) and obtained in [11] corresponding to tapering parameters ϖ1,ϖ2 for a fixed value of thermal gradient κ=0.0, nonhomogeneity φ=0.0, skew angle θ=30°, and aspect ratio a/b=1.5.
ϖ1ϖ2=0.0ϖ2=0.2ϖ2=0.4ϖ2=0.6ϖ2=0.8ϖ2=1.0λ1λ2λ1λ2λ1λ2λ1λ2λ1λ2λ1λ2CCCC0.0294.6074.00311.9978.29331.5282.97352.9787.97376.1293.22400.7698.69294.6074.00326.0281.85360.66590.44397.6999.557436.50109.04476.66118.790.2303.3876.45321.1580.842341.1285.631363.0690.74386.7596.12411.98101.71324.4781.52358.7590.10396.5899.50437.05109.47479.49119.85523.43130.520.4312.5479.01330.7083.50351.1288.39373.5893.62397.8499.12423.68104.84355.0289.24392.2398.58433.33108.80477.33119.65523.49130.95571.30142.570.6322.0481.68340.6186.26361.5191.26384.5096.61409.34102.24435.83108.09386.0997.12426.28107.22470.72118.28518.31130.03568.27142.26620.02154.850.8331.8784.43350.8689.11372.2594.23395.7999.70421.24105.45448.39111.44417.56105.12460.79116.00508.61127.91559.84140.57613.64153.75669.38167.321.0342.0087.27361.4392.05383.3297.28407.43102.87433.51108.75461.33114.88449.35113.21495.64124.88546.88137.65601.80151.23659.48165.37719.27179.93CSCS0.0345.9041.38370.4943.07398.3144.89428.8846.82461.8048.853496.6950.97345.9041.38383.7445.65426.8850.15473.8854.82523.6959.61575.6064.490.2356.1142.90381.5444.70410.3046.621441.9048.65475.9150.793511.9753.02380.8945.61422.8150.41470.5755.45522.5660.67577.6566.02635.0371.480.4366.6644.52392.9646.41422.6848.44455.3450.58490.4852.83527.7255.18416.5450.00462.6255.34515.0760.94572.1566.73632.6072.66695.5578.710.6377.5346.22404.7148.22435.4350.35469.1752.61505.4654.98543.9357.44452.7154.52503.0060.40560.2066.57622.4272.94688.3179.47756.9186.110.8388.6948.00416.7850.11448.5152.35483.3654.72520.8457.21560.5559.80489.2759.12543.8165.56605.8272.31673.2479.27744.6286.39818.9293.641.0400.1349.85429.1452.07461.9154.43497.8856.92536.5759.52577.5662.23526.1463.80584.9770.80651.8178.13724.4885.69801.3993.42881.44101.28Bold font values are obtained in [11].Table 7
Comparison of frequency modes of the present study (parallelogram plate) and obtained in [11] corresponding to aspect ratio a/b for variable values of tapering parameters ϖ1,ϖ2, a fixed value of thermal gradient κ=0.0, nonhomogeneity φ=0.0, and skew angle θ=30°.
a/bϖ1=ϖ2=0.0ϖ1=ϖ2=0.2ϖ1=ϖ2=0.4ϖ1=ϖ2=0.6ϖ1=ϖ2=0.8ϖ1=ϖ2=1.0λ1λ2λ1λ2λ1λ2λ1λ2λ1λ2λ1λ2CCCC0.251960.78478.172141.81522.492348.03571.792579.23625.742835.11684.093115.34746.681960.78478.172390.31582.592894.31704.393472.23843.224123.48998.744847.571170.670.50498.67122.88544.32134.25596.11146.85653.99160.59717.91175.42787.78191.28498.67122.88607.62149.66734.95180.83880.55216.281044.30255.951226.09299.750.75245.0962.31266.9568.07291.5574.42318.8881.32348.9188.727381.6396.61245.0962.31298.3675.86360.0991.57430.26109.40508.86129.31595.85151.261.0194.8150.56211.8355.23230.8860.38251.9565.95275.0471.92300.1278.27194.8150.56236.9861.54285.5974.25340.6288.65402.07104.72469.92122.441.25225.1557.63245.1162.96267.5268.83292.3875.20319.6882.039349.4089.31225.1557.63274.0370.16330.5684.68394.75101.14466.57119.53546.01139.801.50294.6074.00321.1580.84351.1288.39384.5096.61421.24105.45461.33114.88294.6074.00358.7590.10433.33108.80518.31130.03613.64153.75719.27179.93CSCS0.251543.47478.031686.39523.501850.25574.312035.23630.112241.40690.652468.80755.741543.47478.031882.13582.762280.66705.582738.84846.103256.501003.983833.431178.920.50396.13122.06432.69133.68474.45146.62521.45160.80573.72176.14631.26192.60396.13122.06482.83148.78584.46180.04701.02215.76832.46255.85978.76300.240.75208.2059.31227.6964.94249.9671.22275.0578.11302.9685.56333.7193.56208.2059.31253.8372.28307.4187.46368.94104.78438.38124.22515.74145.731.0195.5741.44214.8645.27237.0849.57262.2354.32290.2759.51321.1765.13195.5741.44238.8250.51290.2861.14349.8973.29417.5586.92493.20102.021.25253.5638.01279.3941.30309.2345.02343.0049.16380.6153.72421.9758.69253.5638.01309.8946.33377.3956.07455.8867.20545.2179.70645.2293.551.50345.9041.38381.5444.70422.6848.44469.1752.61520.8457.21577.5662.23345.9041.38422.8150.41515.0760.94622.4272.94744.6286.39881.44101.28Bold font values are obtained in [11].Table 8
Comparison of frequency modes of the present study (rectangle plate) and obtained in [17] corresponding to tapering parameters ϖ1,ϖ2 for a fixed value of thermal gradient κ=0.0 and aspect ratio a/b=1.5.
ϖ2ϖ1=0.0ϖ1=0.2ϖ1=0.4ϖ1=0.6ϖ1=0.8λ1λ2λ1λ2λ1λ2λ1λ2λ1λ2CCCC0.0217.5454.15223.9855.93230.7057.782237.6659.701244.8661.68240.4760.85271.4368.92287.8173.31305.2077.97323.4682.850.2230.5557.31237.3759.18244.4861.12251.8463.14259.4665.22277.7570.26294.1974.65311.8579.36330.5884.35350.2389.560.4245.1660.76252.4162.73259.9664.77267.7866.90275.8769.08302.3576.32320.1581.04339.2686.10359.5091.44380.7297.020.6261.2264.44268.9366.52276.9768.68285.2970.91293.9073.21329.4882.85348.7887.94369.4793.37391.3799.09414.32105.070.8278.5568.31286.7770.50295.3272.78304.1975.13313.3577.55358.7089.75379.6195.21402.00101.04425.68107.17450.47113.58SSSS0.0220.0326.66226.7927.69233.9428.78241.4529.95249.3031.18256.9134.25273.2036.62291.0839.22310.3442.01330.7744.970.2235.7827.69243.0028.77250.6229.92258.6331.13266.9832.42281.8936.63299.5739.17318.9541.97339.7944.98361.8748.170.4253.5828.80261.3129.92269.4731.12278.0332.40286.9633.75310.8539.22330.1341.97351.2244.99373.8648.24397.8351.690.6273.1529.96281.4431.142290.1832.40299.3533.746308.9135.16342.9742.01364.0444.97387.0248.24411.6551.76437.7055.490.8294.2131.18303.1132.42312.4933.74322.3235.15332.5636.64377.6144.96400.5948.15425.6151.68452.4055.48480.6959.51CSCS0.0256.8428.98264.3930.04272.1831.16280.2032.34288.4333.57288.8241.94305.1844.90322.4748.16340.5751.66359.3755.390.2275.2130.11283.2931.21291.6232.38300.2033.61309.0134.90317.8444.79335.7847.94354.7151.41374.5055.15395.0659.130.4295.9831.32304.6632.47313.6233.69322.8334.97332.2936.31351.6247.88371.3851.25392.2254.96413.9958.96436.5763.200.6318.8232.60328.1633.80337.7935.07347.7036.40357.8737.80389.1351.19410.9354.80433.8858.76457.8463.04482.6767.570.8343.4033.94353.4535.19363.8236.51374.4837.91385.4139.38429.5654.68453.5458.53478.7762.77505.0967.33532.3672.18Bold font values are obtained in [17].Table 9
Comparison of frequency modes of the present study (rectangle plate) and obtained in [18] corresponding to thermal gradient κ for a fixed value of tapering parameters ϖ1=ϖ2=0.0 and aspect ratio a/b=1.5.
κCCCCSSSSCSSSCSCSCCCSλ1λ2λ1λ2λ1λ2λ1λ2λ1λ20.0217.5454.15220.0326.66286.2341.21256.8428.98435.8859.96256.2064.83256.9234.25260.9739.75288.8141.94357.6555.240.4196.8448.92200.9626.66265.2838.04234.3025.96425.6655.07243.0561.50243.7332.50252.6038.37273.9939.79353.7053.120.8173.7043.01179.8720.80242.5234.58209.3422.52415.1949.65229.1557.98229.7930.64243.9536.92258.3237.51349.7150.91Bold font values are obtained in [18].Table6 presents the comparison of frequency modes of the present study (parallelogram plate) and the frequency obtained in [11] on CCCC and CSCS edge conditions corresponding to tapering parameters ϖ1,ϖ2, for a fixed value of thermal gradient HTML translation failed and nonhomogeneity φ=0.0, skew angle θ=30°, and aspect ratio a/b=1.5. Table 6 clearly shows that the frequency obtained in the present study (parallelogram plate) is less when compared with the frequency modes obtained in [11] with the increasing value of tapering parameters ϖ1,ϖ2 on both CCCC and CSCS edge conditions.A comparison of frequency modes of the present study (parallelogram plate) and the frequency modes obtained in [11] is presented in Table 7 on CCCC and CSCS edge conditions corresponding to aspect ratio a/b and tapering parameters ϖ1,ϖ2 for a fixed value of thermal gradient κ=0.0 and nonhomogeneity φ=0.0 and skew angle θ=30°. Here also, the frequency modes obtained in the present study (parallelogram plate) are less when compared with the frequency modes obtained in [11] with the increasing value of aspect ratio a/b and tapering parameters ϖ1,ϖ2 on both CCCC and CSCS edge conditions.Table8 shows the comparison of frequency modes of rectangle plate (by taking skew angle θ=0° in the present study) and obtained in [17] on CCCC, SSSS, and CSCS edge conditions corresponding to tapering parameters ϖ1,ϖ2 for a fixed value of κ=0.0 and aspect ratio a/b=1.5. Table 8 enlightens the fact that frequency modes of rectangle plate are very less in comparison to the frequency modes obtained in [17] on all the above edge conditions with the increasing value of tapering parameters ϖ1,ϖ2.A comparison of frequency modes of rectangle plate (by taking skew angleθ=0° in the present study) and obtained in [18] on CCCC, SSSS, CSSS, CSCS and CCCS edge conditions corresponding to thermal gradient κ for fixed value tapering parameters ϖ1=ϖ2=0.0 and aspect ratio a/b=1.5 is shown in Table 9. Here also, one can easily see that the frequency modes of the rectangle plate are very less in comparison to the frequency modes obtained in [18] on all the above edge conditions with the increasing value of thermal gradient κ.


## 6. Conclusion

The effect of various plate parameters (especially the effect of two-dimensional tapering parametersϖ1,ϖ2) on time period of the first two modes of frequency of parallelogram plate on various boundary conditions is computed. Based on the results obtained from numerical simulation and comparison done, the authors would like to conclude the following facts:(a)
Frequency modes of parallelogram plate in case of two-dimensional circular thickness in this study are less when compared with the frequency modes obtained in [11] in case of two-dimensional linear thickness on both CCCC and CSCS edge conditions. The frequency modes of the present study and obtained in [11] match at ϖ1=ϖ2=0.0 on both CCCC and CSCS edge conditions (refer Table 6).(b)
Frequency modes of parallelogram plate in case of aspect ratio’s impact on two-dimensional circular thickness in the current study provide less frequency when compared with frequency modes obtained in [11], in case of aspect ratio’s impact on the two-dimensional linear thickness on CCCC and CSCS edge conditions. The frequency modes of the present study obtained in [11] coincide at a/b=0.25 on CCCC and CSCS edge conditions (refer Table 7).(c)
Frequency modes of rectangle plate in case of two-dimensional circular thickness (present study) are very less in comparison to frequency modes obtained in [17] in case of two-dimensional linear thickness on CCCC, SSSS, and CSCS edge conditions (refer Table 8).(d)
In the current study, frequency modes of rectangle plate in case of two-dimensional parabolic temperature are very less as compared to frequency modes obtained in [18] in case of two-dimensional linear temperature on CCCC, SSSS, CSSS, CSCS, and CCCS edge conditions (refer Table 9).(e)
Tapering parameterϖ2 dominates the time period (rate of decrement) more than the tapering parameter ϖ1 (refer Table 1).(f)
Increase in both the tapering and nonhomogneity on the plate results in the decrease in time period (refer Tables1 and 3) while an increase in thermal gradient on plate results in an increase in the time period (refer Table 2).(g)
We can minimize the vibrational frequencies of the plate by choosing appropriate variation in plate parameters (refer points a to d). We can also control the variation (rate of increment/decrement) in time period of frequency modes by choosing appropriate variation in plate parameters (refer points e and f).(h)
A numerical data in the form of time period is calculated which shows the impact of various plate parameters.



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