Abstract
In this article, authors dealt with general solution of differential equation of orthotropic rectangular plate with clamped boundary conditions under biparabolic temperature variations. Rayleigh Ritz technique is adopted to solve the resultant equation and evaluate the frequencies for first four modes of vibration. The effect of circular thickness and density on frequencies of orthotropic plate are analyzed which is not done yet. The objective of the study is to optimize the frequency modes by choosing the appropriate variation in plate parameters. Th findings of the study complete the objective of the article. All the results are provided in tabular form.
1. Introduction
Orthotropic rectangular plate with various plate parameters is widely used in various engineering branches like aerospace, aircraft, automobiles etc. A significant work has been reported by many researchers on orthotropic rectangular plate along with different variations in plate parameters but few work has been reported to study the effect of circular variation in plate parameters (thickness and density).
Effect of nonhomogeneity on vibration of orthotropic rectangular plates of varying thickness resting on Pasternak foundation is discussed in [1]. Free vibration analysis of laminated composite plate with elastic point and line supports by using finite element method was presented by [2]. Buckling analysis of rectangular isotropic plate having simply supported boundary conditions under the influence of nonuniform inplane loading by using firstorder shear deformation theory (FSDT) has been implemented in [3] Free vibration of orthotropic rectangular plate with thickness and temperature variation is studied in [4] and [5] by using Rayleigh Ritz method. Theoretical analysis on time period of vibration of rectangular plate with different plate parameters was proposed by [6] and [7].
New straight forward benchmark solutions for bending and free vibration of clamped anisotropic rectangular thin plates by a double finite integral transform method is described in [8]. Natural vibration of skew plate on different set of boundary conditions with temperature gradient is analyzed in [9]. Free vibration analysis of isotropic and laminated composite plate [10] on elastic point supports using finite element method (FEM) is studied. Vibration frequencies of a rectangular plate with linear variation in thickness and circular variation in Poisson’s ratio was provided in [11]. Thermally induced vibrations of shallow functionally graded material arches is considered and analyzed in [12] by using classical theory of curved beams. Fast converging semi analytical method was developed by [13] for assessing the vibration effect on thin orthotropic skew (or parallelogram/oblique) plates. A mathematical model is presented in [14] to analyse the vibration of a tapered isotropic rectangular plate under thermal condition.
In this study, authors examined the behavior of frequencies of orthotropic rectangular plate having circular variations in thickness and density. Authors also discussed the frequency variation when temperature increased on the plate. The present study provided how we can optimize the frequency value by choosing circular variation in plate parameters which will be helpful to structural engineering to get ride from unwanted vibrations.
2. Problem geometry and analysis
Consider a nonhomogeneous orthotropic rectangle plate with sides $a$ and $b$, thickness $l$ and density $\rho $ as shown in Fig. (1).
Fig. 1Orthotropic rectangular plate with 2D circular thickness
The kinetic energy and strain energy for vibration of plate are expressed in the following manner as in [15]:
where $\mathrm{\Phi}$ deflection function, $\omega $ natural frequency, ${D}_{x}={E}_{x}{l}^{3}/12\left(1{\nu}_{x}{\nu}_{y}\right)$, ${D}_{y}={E}_{y}{l}^{3}/12\left(1{\nu}_{x}{\nu}_{y}\right)$ are flexural rigidities in $x$ and $y$ directions respectively and ${D}_{xy}={E}_{x}{l}^{3}/12\left(1{\nu}_{x}{\nu}_{y}\right)$ is torsional rigidity.
To solve frequency equation, Rayleigh Ritz technique is implemented which requires:
Using Eqs. (1), (2), we get functional equation:
Now, introducing nondimensional variable as:
together with two dimensional circular thickness, we get:
where ${l}_{0}$ is thickness at origin and ${\beta}_{1}$, ${\beta}_{2}(0\le {\beta}_{1},{\beta}_{2}\le 1)$ are tapering parameters.
For nonhomogeneity consideration, authors assumed one dimensional circular variation in density:
Two dimensional parabolic temperature distribution is taken as in [5]:
where $\tau $ and ${\tau}_{0}$ denotes the temperature on and at the origin respectively.
For orthotropic materials, modulus of elasticity is given by [17]:
where ${E}_{x}$ and ${E}_{y}$ are the Young’s modulus in $x$ and $y$ directions, ${G}_{xy}$ is shear modulus and $\gamma $ is called slope of variation. Using Eq. (7), Eq. (8) becomes:
${E}_{y}\left(x\right)={E}_{2}\left(1\alpha \left(1{\zeta}_{1}^{2}\right)\left(1{{\frac{{a}^{2}}{{b}^{2}}\psi}_{1}}^{2}\right)\right),$
${G}_{xy\left(x\right)}={G}_{0}\left(1\alpha \left(1{\zeta}_{1}^{2}\right)\left(1\frac{{a}^{2}}{{b}^{2}}{{\psi}_{1}}^{2}\right)\right),$
where $\alpha =\gamma {\tau}_{0}\left(0\le \alpha <1\right)$ is called thermal gradient. Using Eqs. (5), (6), (9) and non dimensional variable, the functional in Eq. (4) become:
where:
We choose deflection function which satisfies all the edge condition as taken in [16]:
where ${\mathrm{\Psi}}_{i}$, $i=\mathrm{0,1},2...n$ are unknowns and the value of $e$, $f$, $g$, $h$ can be 0, 1 and 2, corresponding to free, simply supported and clamped edge conditions respectively. To minimize Eq. (11), we impose the following condition:
Solving Eq. (11), we have the following frequency equation:
where $P={\left[{p}_{ij}\right]}_{i,j=\mathrm{0,1},..n}$ and $Q={\left[{q}_{ij}\right]}_{i,j=\mathrm{0,1},..n}$ are square matrix of order $(n+1)$.
3. Numerical results and discussion
Vibrational frequency of orthotropic rectangular plate having circular variation for first four modes under thermal effect with circular variation in thickness and density is discussed. Frequency equation is solved by using RayleighRitz technique with the help of Maple software. Frequency values for first four modes is calculated on different variations of plate parameters (i.e., thermal gradient, tapering parameters and nonhomogeneity). The results have been presented in the form of tables. In through out calculation the aspect ratio i.e., ratio of length to breadth of the plate is considered to be 1.5.
Tables 1 and 2 represents the frequencies for first four modes corresponding to tapering parameters${\beta}_{1}$, ${\beta}_{2}$ for three different values of thermal gradient, tapering paramter and nonhomogeneity $m$ i.e., $\alpha ={\beta}_{2}=m=\text{0.2}$, $\alpha ={\beta}_{2}=m=\text{0.4}$, $\alpha ={\beta}_{2}=m=\text{0.6.}$(In Table 1) i.e., $\alpha ={\beta}_{1}=m=\text{0.2}$, $\alpha ={\beta}_{1}=m=\text{0.4}$, $\alpha ={\beta}_{1}=m=\text{0.6}$ (In Table 2).
Table 1Tapering parameter β1 vs vibrational frequency λ
${\beta}_{1}$  $\alpha ={\beta}_{2}=m=0.2$  $\alpha ={\beta}_{2}=m=0.4$  $\alpha ={\beta}_{2}=m=0.6$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  
0.0  16.2250  62.5266  140.527  314.224  15.4920  59.2450  132.974  303.002  114.7460  55.8753  125.414  291.617 
0.2  16.9440  64.3744  145.547  331.268  16.2045  61.5334  137.954  319.324  15.4520  58.1060  130.414  306.785 
0.4  17.7186  67.3744  151.897  347.316  16.9703  63.9622  143.549  334.372  16.2086  60.4747  135.793  322.357 
0.6  18.5423  70.0079  156.897  366.204  17.7826  66.5125  149.182  353.426  17.0089  62.9527  141.414  341.070 
0.8  19.4086  72.7501  163.162  384.709  18.6352  69.1850  155.236  372.291  17.8470  65.5327  147.325  359.995 
1.0  20.3118  75.5910  169.577  405.599  19.5224  71.9365  161.609  391.622  18.7174  68.1935  153.484  380.389 
Table 2Tapering parameter β2 vs vibrational frequency λ
${\beta}_{2}$  $\alpha ={\beta}_{1}=m=0.2$  $\alpha ={\beta}_{1}=m=0.4$  $\alpha ={\beta}_{1}=m=0.6$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  
0.0  16.6318  63.8209  143.168  325.058  16.3257  61.8337  138.677  320.118  16.0163  59.7280  134.016  317.501 
0.2  16.9440  64.8730  145.547  331.223  16.6431  62.8843  141.117  326.682  16.3380  60.7803  136.374  325.163 
0.4  17.2665  65.9491  147.895  339.639  16.9703  63.9622  143.549  334.372  16.6691  61.8582  138.930  332.343 
0.6  17.5987  67.0584  150.517  345.112  17.3068  65.0711  146.010  342.215  17.0089  62.9517  141.463  340.544 
0.8  17.9401  68.1874  153.129  352.618  17.6521  66.2017  148.587  349.894  17.3571  64.0756  144.056  348.595 
1.0  18.2902  69.3457  155.639  361.626  18.0057  67.3523  151.159  359.151  17.7131  65.2175  146.720  357.118 
From Tables 1 and 2, the following facts can be interpreted:
1) Frequency increases in both the Tables 1 and 2 with the increasing in value of tapering parameters ${\beta}_{1}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}{\beta}_{2}$.
2) for fixed values of both the tapering parameters ${\beta}_{1}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}{\beta}_{2}$, frequency decreases in both the Tables 1 and 2.
3) Rate of decrement of frequencies reported in Table 2 is much slow as compared to Table 1.
Table 3 summarizes the frequency for first four modes corresponding to thermal gradient $\alpha $ for three different value of tapering parameters ${\beta}_{1}\text{,}$${\beta}_{2}$ and non homogeneity $m$ i.e., ${\beta}_{1}={\beta}_{2}=m=\text{0.2}$, ${\beta}_{1}={\beta}_{2}=m=\text{0.4}$, ${\beta}_{1}={\beta}_{2}=m=\text{0.6}$. Table 4 presents frequency for first four modes corresponding to nonhomogeneity $m$ for three different values of thermal gradient $\alpha $, tapering paramters ${\beta}_{1}$, ${\beta}_{2}$ i.e., ${\beta}_{1}={\beta}_{2}=\alpha =\text{0.2}$, ${\beta}_{1}={\beta}_{2}=\alpha =\text{0.4}$, ${\beta}_{1}={\beta}_{2}=\alpha =\text{0.6}$ respectively.
From Tables 3 and 4, it can be concluded that:
1) Increase in thermal gradient $\alpha $ and nonhomogeneity m results in decrease in frequency in both the tables 3 and 4.
2) Rate of decrement of frequency in nonhomogeneity is slower as compared to thermal gradient.
3) Here, behavior of both the Tables 3 and 4 is quite similar but in Table 4 have higher frequency values as compared to Table 3.
4) Without thermal effect, Table 3 have maximum frequency value then it decreases gradually with the increase in thermal gradient.
Table 3Thermal gradient α vs vibrational frequency λ
$\alpha $  ${\beta}_{1}={\beta}_{2}=m=0.2$  ${\beta}_{1}={\beta}_{2}=m=0.4$  ${\beta}_{1}={\beta}_{2}=m=0.6$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  
0.0  17.7264  68.0981  152.773  344.414  18.5608  70.5842  158.161  359.868  19.4501  73.1977  163.831  377.364 
0.2  16.9440  64.8730  145.547  331.268  17.7860  67.3676  151.009  347.406  18.6797  69.9838  156.690  365.602 
0.4  16.1196  61.4568  138.000  317.463  16.9703  63.9628  143.544  334.381  17.8688  66.5801  149.261  353.332 
0.6  15.2450  57.8227  129.915  303.204  16.1054  60.3435  135.542  321.048  17.0089  62.9563  141.409  340.651 
0.8  14.3085  53.9052  121.350  288.101  15.1795  56.4381  127.131  306.930  16.0884  59.0494  133.108  327.429 
1.0  13.2923  49.6287  112.057  272.273  14.1752  52.1774  118.015  292.304  15.0893  54.7852  124.157  313.803 
Table 4Nonhomogeneity m vs vibrational frequency λ
$m$  $\alpha ={\beta}_{1}={\beta}_{2}=0.2$  $\alpha ={\beta}_{1}={\beta}_{2}=0.4$  $\alpha ={\beta}_{1}={\beta}_{2}=0.6$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  
0.0  17.2013  65.9667  148.314  335.442  17.4856  66.1290  148.314  348.144  17.7821  66.1528  149.118  362.262 
0.2  16.9440  64.8663  145.634  331.105  17.2222  65.0210  145.943  341.903  17.5128  65.0309  146.362  355.401 
0.4  16.6979  63.8335  143.210  323.724  16.9703  63.9628  143.544  334.381  17.2554  63.9650  143.309  348.519 
0.6  16.4622  62.8448  140.797  318.704  16.7292  62.9605  141.119  328.422  17.0089  62.9572  141.309  341.897 
0.8  16.2363  61.8978  138.611  313.065  16.4980  62.0028  138.814  323.043  16.7728  61.9888  139.107  334.293 
1.0  16.0193  60.9914  136.613  306.693  16.2762  61.0890  136.707  316.661  16.5461  61.0639  136.860  329.374 
4. Conclusions
Present model provides frequencies of orthotropic rectangular plate with circular variations in thickness and density. from above results and discussion, authors concluded that increasing in tapering parameters ${\beta}_{1}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}{\beta}_{2}$, results the decrease in frequency as shown in tables 1 and 2. but increase in thermal gradient $\alpha $ and non homogeneity $m$, results the increase in frequency is illustrated in tables 3 and 4. the variation in frequency mode (increasing or decreasing) are very slow due to implementation of circular variation in plate parameters i.e., there is no sudden increment or decrement reported in frequencies.
References

R. Lal and Dhanpati, “Effect of Nonhomogeneity on Vibration of orthotropic rectangular plates of varying thickness resting on Pasternak foundation,” Journal of Vibration and Acoustics, Vol. 131, No. 1, Feb. 2009, https://doi.org/10.1115/1.2980399

S. Ghosh, S. Haldar, and S. Haldar, “Free vibration analysis of laminated composite plate with elastic point and line supports using finite element method,” Journal of the Institution of Engineers (India): Series C, pp. 1–12, Jan. 2022, https://doi.org/10.1007/s40032021007990

S. Das and P. Jana, “Analytical solution for buckling of rectangular plate subjected to nonuniform uniaxial compression using FSDT,” Lecture Notes in Mechanical Engineering, pp. 487–496, 2022, https://doi.org/10.1007/9789811664908_40

A. Sharma, A. K. Sharma, A. K. Raghav, and V. Kumar, “Effect of vibration on orthotropic viscoelastic rectangular plate with two dimensional temperature and thickness variation,” Indian Journal of Science and Technology, Vol. 9, No. 2, pp. 1–7, Jan. 2016, https://doi.org/10.17485/ijst/2016/v9i2/51314

S. K. Sharma and A. K. Sharma, “RayleighRitz method for analyzing free vibration of orthotropic rectangular plate with 2D thickness and temperature variation,” Journal of Vibroengineering, Vol. 17, No. 4, pp. 1989–2000, Jun. 2015.

A. Sharma, A. Kumar, N. Lather, R. Bhardwaj, and N. Mani, “Effect of linear variation in density and circular variation in Poisson’s ratio on time period of vibration of rectangular plate,” Vibroengineering PROCEDIA, Vol. 21, pp. 14–19, Dec. 2018, https://doi.org/10.21595/vp.2018.20367

N. Lather, A. Kumar, and A. Sharma, “Theoretical analysis of time period of rectangular plate with variable thickness and temperature,” Advances in Basic Science (ICABS 2019), Vol. 2142, No. 1, p. 110027, 2019, https://doi.org/10.1063/1.5122487

D. An, Z. Ni, D. Xu, and R. Li, “New straightforward benchmark solutions for bending and free vibration of clamped anisotropic rectangular thin plates,” Journal of Vibration and Acoustics, Vol. 144, No. 3, Jun. 2022, https://doi.org/10.1115/1.4053090

A. Sharma and N. Lather, “Natural vibration of skew plate on different set of boundary conditions with temperature gradient,” Vibroengineering Procedia, Vol. 22, pp. 74–80, Mar. 2019, https://doi.org/10.21595/vp.2019.20550

S. Ghosh and S. Haldar, “Free Vibration analysis of isotropic and laminated composite plate on elastic point supports using finite element method,” Lecture Notes in Mechanical Engineering, pp. 371–384, 2022, https://doi.org/10.1007/9789811664908_31

A. Sharma, “Vibration frequencies of a rectangular plate with linear variation in thickness and circular variation in Poisson’s ratio,” Journal of Theoretical and Applied Mechanics, Vol. 57, 2019.

M. M. Khalili, A. Keibolahi, Y. Kiani, and M. R. Eslami, “Application of ritz method to large amplitude rapid surface heating of FGM shallow arches,” Archive of Applied Mechanics, Vol. 92, No. 4, pp. 1287–1301, Apr. 2022, https://doi.org/10.1007/s00419022021064

A. M. Farag and A. S. Ashour, “Free vibration of orthotropic skew plates,” Journal of Vibration and Acoustics, Vol. 122, No. 3, pp. 313–317, Jul. 2000, https://doi.org/10.1115/1.1302085

A. Khanna and A. Singhal, “Effect of plates parameters on vibration of isotropic tapered rectangular plate with different boundary conditions.,” Journal of Low Frequency Noise, Vibration and Active Control, Vol. 35, No. 2, pp. 139–151, 2016.

A. W. Leissa, “Vibration of plates,” Scientific and Technical Information Division, National Aeronautics and Space Administration, 1969.

S. Chakarverty, Vibration of Plates. CRC Press: Boca Raton, 2008.

A. Sharma, R. Bhardwaj, N. Lather, S. Ghosh, N. Mani, and K. Kumar, “Time period of thermalinduced vibration of skew plate with twodimensional circular thickness,” Mathematical Problems in Engineering, Vol. 2022, pp. 1–12, Mar. 2022, https://doi.org/10.1155/2022/8368194