Abstract
The objective of this study is to investigate the natural vibration of a rectangular plate made of orthotropic material with circular thickness (two dimensions) and temperature variation on the plate is parabolic (two dimensions) in nature. The solution to the problem is obtained by utilizing the RayleighRitz technique and the first four frequency modes are obtained under clamped edge conditions. The study aims to provide numerical data that demonstrate how circular variation in tapering parameters of plate can effectively control and optimized vibrational frequencies of the plate. Orthotropic rectangular plate, thermal gradient, circular tapering, aspect ratio.
1. Introduction
To design structures or understand system characteristics, it becomes vital to investigate the vibrational properties of plates. Many systems and structures such as bridges, buildings, and aircraft wings consist of plates of various shapes. The vibration characteristics of a plate are influenced by plate parameters such as tapering, nonhomogeneity (in the case of nonhomogeneous materials), and thermal gradient. A considerable number of studies in the literature have focused on various values of plate parameters.
The approach outlined in [1] was utilized to amalgamate solutions for plates with different geometries (such as circular, annular, circular sector, and annular sector plates) under various boundary conditions. In [2], the wavebased method (WBM) was utilized to forecast the flexural vibrations of orthotropic plates. In [3], a solution based on twovariable refined plate theory of Levy type was developed for free vibration analysis of orthotropic plates. In [4], a new analytical solution utilizing a double finite sine integral transform technique was introduced for the vibration response of plates reinforced by orthogonal beams. In [5], the Rayleigh Ritz method was utilized to determine the frequency of an orthotropic rectangular plate, whereas in [6], the time period of transverse vibration of a skew plate with different edge conditions was assessed. In [7], the influence of temperature on the frequencies of a tapered plate was discussed, while [8] investigated a nonuniform triangular plate subjected to a twodimensional parabolic temperature distribution. The investigation of time period of rectangular plates with varying thickness and temperature was examined in [9]. Time period analysis of isotropic and orthotropic visco skew plate having circular variation in thickness and density at different edge conditions is discussed in [10] and [11].
It is noticeable from the literature that most of the authors have investigated either linear or parabolic variations in tapering parameters, but no one has focused on circular variation in tapering parameter. This study aims to fill this research gap by exploring the influence of two dimension circular thickness on the vibrational frequency of an orthotropic rectangular plate under a two dimension parabolic temperature profile. The circular variation examined in this paper results in a reduction in the variation in frequency modes, as shown in the numerical results section.
2. Problem geometry and analysis
Taking into account that the nonhomogeneous rectangular plate shown in Fig. 1 with sides $a$, $b$ and thickness $l$.
Fig. 1Orthotropic rectangular plate with 2D circular thickness
The formulation of the kinetic energy and strain energy for plate vibration is given below, similar to the approach presented in [12]:
where $\varphi $ is deflection function, $\omega $ is natural frequency, ${D}_{\zeta}={E}_{\zeta}{l}^{3}/12\left(1{\nu}_{\zeta}{\nu}_{\psi}\right)$, ${D}_{\psi}={E}_{\psi}{l}^{3}/12\left(1{\nu}_{\zeta}{\nu}_{\psi}\right)$, ${D}_{\zeta \psi}={E}_{\zeta}{l}^{3}/12\left(1{\nu}_{\zeta}{\nu}_{\psi}\right)$. Here, ${D}_{\zeta}$ and ${D}_{\psi}$ is flexural rigidity in $\zeta $ and $\psi $ directions respectively and ${D}_{\zeta \psi}$ is torsional rigidity.
In order to address the investigated problem, the RayleighRitz method is utilized, which necessitates:
Using Eqs. (1), (2), we have:
Proposing nondimensional variable as ${\zeta}_{1}=\zeta /a$, ${\psi}_{1}=\psi /a$ along with two dimension circular thickness as:
where ${l}_{0}$ is thickness at origin and ${\beta}_{1}$, ${\beta}_{2}\le 1$ are tapering parameters.
The twodimensional parabolic temperature distribution, as presented in Eq. (6):
where $\tau $ and ${\tau}_{0}$ represent the temperature at a given point and at the origin respectively.
For orthotropic materials, modulus of elasticity is evaluated by:
where ${E}_{\zeta}$ and ${E}_{\psi}$ are the Young’s modulus in $\zeta $ and $\psi $ directions, ${G}_{\zeta \psi}$ is shear modulus and $\gamma $ is called slope of variation.
Using Eq. (6), Eq. (7) becomes:
${G}_{\zeta \psi}={G}_{0}\left(1\alpha \left(1{\zeta}_{1}^{2}\right)\left(1{\frac{{a}^{2}{\psi}_{1}}{{b}^{2}}}^{2}\right)\right),$
where $\alpha =\gamma {\tau}_{0}\left(0\le \alpha <1\right)$ is called thermal gradient.
Using Eqs. (5), (8) and non dimensional variable, the functional in Eq. (4) become:
${\bullet \left(1+{\beta}_{2}\left\{1\sqrt{1\frac{{a}^{2}}{{b}^{2}}{{\psi}_{1}}^{2}}\right\}\right)}^{3}\left\{{\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}_{1}^{2}}\right)}^{2}+\frac{{E}_{2}}{{E}_{1}}{\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\psi}_{1}^{2}}\right)}^{2}+2{\nu}_{\zeta}\frac{{E}_{2}}{{E}_{1}}\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}_{1}^{2}}\right)\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\psi}_{1}^{2}}\right)\right.$
$+\left.\left.4\frac{{G}_{0}}{{E}_{1}}\left(1{\nu}_{\zeta}{\nu}_{\psi}\right){\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}_{1}\partial {\psi}_{1}}\right)}^{2}\right\}\right]d{\psi}_{1}d{\zeta}_{1}{\lambda}^{2}{\int}_{0}^{1}\mathrm{\u200d}{\int}_{0}^{\frac{b}{a}}\mathrm{\u200d}\left[\left(1+{\beta}_{1}\left\{1\sqrt{1{{\zeta}_{1}}^{2}}\right\}\right)\right.$
$\bullet \left.\left(1+{\beta}_{2}\left\{1\sqrt{1\frac{{a}^{2}}{{b}^{2}}{{\psi}_{1}}^{2}}\right\}\right)\right]{\mathrm{\Phi}}^{2}d{\psi}_{1}d{\zeta}_{1}=0,$
where ${D}_{0}=\frac{1}{2}\left(\frac{{E}_{1}{l}_{0}^{3}}{12\left(1{\nu}_{\zeta}{\nu}_{\psi}\right)}\right)$ and ${\lambda}^{2}=\frac{12{a}^{4}\rho {\omega}^{2}\left(1{\nu}_{\zeta}{\nu}_{\psi}\right)}{{E}_{1}{{h}_{0}}^{2}}$.
The deflection function that meets all the edge conditions is taken as in [13]:
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 given edge condition.
Eq. (10) can be minimized by imposing the following condition:
Solving Eq. (11), we have 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
In this study, the first four natural frequencies of a clamped orthotropic rectangular plate with two dimension circular thickness and two dimension parabolic temperature variations are investigated corresponding to various plate parameters aspect ratio $a/b$, tapering parameters ${\beta}_{1}$ and ${\beta}_{2}$, and thermal gradient $\alpha $. The numerical calculations are based on the subsequent parameter values:
Table 1Modes of frequency of clamped orthotropic rectangular plate corresponding to β1
${\beta}_{2}$  $\alpha =0.2$  $\alpha =0.4$  $\alpha =0.6$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  
0.0  17.002  65.265  146.471  335.761  17.085  64.734  145.409  344.526  17.173  64.058  144.568  353.913 
0.2  17.747  67.704  151.913  351.645  17.848  67.198  151.149  359.343  17.951  66.545  150.178  372.482 
0.4  18.548  70.301  157.829  367.708  18.665  69.815  156.993  378.568  18.782  69.169  156.353  390.070 
0.6  19.400  73.032  163.844  388.066  19.531  72.556  163.396  397.071  19.660  71.918  162.620  412.912 
0.8  20.295  75.877  170.500  405.951  20.439  75.414  169.982  417.520  20.579  74.769  169.561  432.019 
1.0  21.228  78.822  177.057  430.024  21.385  78.367  176.579  443.749  21.534  77.713  176.455  457.574 
Table 2Modes of frequency of clamped orthotropic rectangular plate corresponding to β2
$\alpha $  ${\beta}_{1}={\beta}_{2}=0.2$  ${\beta}_{1}={\beta}_{2}=0.4$  ${\beta}_{1}={\beta}_{2}=0.6$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  
0.0  18.539  70.998  159.217  364.788  20.306  76.727  172.199  403.590  22.244  82.942  186.104  449.270 
0.2  17.747  67.705  151.912  351.645  19.507  73.367  164.790  391.249  21.429  79.480  178.659  437.433 
0.4  16.912  64.226  144.209  338.017  18.665  69.816  156.984  378.585  20.571  75.821  170.826  425.369 
0.6  16.027  60.520  136.028  323.885  17.772  66.023  148.867  365.262  19.660  71.917  162.636  412.879 
0.8  15.079  56.527  127.337  309.053  16.816  61.942  140.183  351.597  18.684  67.710  153.955  400.094 
1.0  14.050  52.161  118.026  293.397  15.777  57.482  130.883  337.433  17.620  63.109  144.730  386.914 
Table 3Modes of frequency of clamped orthotropic rectangular plate corresponding to α
$\alpha $  ${\beta}_{1}={\beta}_{2}=0.2$  ${\beta}_{1}={\beta}_{2}=0.4$  ${\beta}_{1}={\beta}_{2}=0.6$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  
0.0  18.539  70.998  159.217  364.788  20.306  76.727  172.199  403.590  22.244  82.942  186.104  449.270 
0.2  17.747  67.705  151.912  351.645  19.507  73.367  164.790  391.249  21.429  79.480  178.659  437.433 
0.4  16.912  64.226  144.209  338.017  18.665  69.816  156.984  378.585  20.571  75.821  170.826  425.369 
0.6  16.027  60.520  136.028  323.885  17.772  66.023  148.867  365.262  19.660  71.917  162.636  412.879 
0.8  15.079  56.527  127.337  309.053  16.816  61.942  140.183  351.597  18.684  67.710  153.955  400.094 
Table 1 presents the modes of frequency (first four modes) corresponding to ${\beta}_{1}$. Specifically, the values of ${\beta}_{2}=\alpha $ chosen were 0.2, 0.4, and 0.6 respectively. Based on the results presented in Table 1, it can be inferred that:
1. The frequency modes increase in all four modes as ${\beta}_{1}$ rises from 0.0 to 1.0.
2. As both ${\beta}_{1}$ and $\alpha $ increase from 0.2 to 0.6 (i.e., ${\beta}_{2}=\alpha =\text{0.2}$ to ${\beta}_{2}=\alpha =\text{0.6}$), the modes of frequency also show an increment.
3. The modes of frequency (rate of increment) are predominantly influenced by ${\beta}_{1}$, as opposed to the $\alpha $ and ${\beta}_{2}$.
Table 2 presents the modes of frequency (first four modes) corresponding to ${\beta}_{2}$, with fixed values of tapering parameter ${\beta}_{1}$ and thermal gradient $\alpha $ i.e., ${\beta}_{1}=\alpha =\text{0.2}$, ${\beta}_{1}=\alpha =\text{0.4}$, and ${\beta}_{1}=\alpha =0.6$, respectively. Based on the findings in Table 2, it is evident that:
1. The modes of frequency exhibit an increase as ${\beta}_{2}$ varies from 0.0 to 1.0.
2. The modes of frequency decrease as ${\beta}_{1}$ and $\alpha $ increase from 0.2 to 0.6 (i.e., ${\beta}_{1}=\alpha =\text{0.2}$ to ${\beta}_{1}=\alpha =\text{0.6}$), when ${\beta}_{2}$ changes from 0.0 to 0.6. However, the modes of frequency increase as ${\beta}_{1}$ and $\alpha $ increase from 0.2 to 0.6 (i.e., ${\beta}_{1}=\alpha =\text{0.2}$ to ${\beta}_{1}=\alpha =\text{0.6}$), when ${\beta}_{2}$ varies from 0.8 to 1.0.
3. In terms of the rate of change in modes of frequency, ${\beta}_{2}$ exerts a stronger influence compared to $\alpha $ and ${\beta}_{1}$.
4. The tables 1 and 2 indicate that the modes of frequency are primarily influenced by ${\beta}_{2}$ as compared to ${\beta}_{1}$.
Table 3 presents the modes of frequency (first four modes) for various values of $\alpha $, while keeping both ${\beta}_{1}$ and ${\beta}_{2}$ fixed at 0.2, 0.4, and 0.6, respectively. By examining Table 3, the subsequent key findings can be observed:
1. The frequency modes also increase with an increase in ${\beta}_{1}$ and ${\beta}_{2}$ from $0.0$ to $1.0$, while the modes of frequency decrease with an increase in the value of $\alpha $ from $0.0$ to $1.0$.
2. Compared to ${\beta}_{1}$ and ${\beta}_{2}$, $\alpha $ has a greater influence on the modes of frequency (i.e., rate of change in the modes of frequency).
4. Conclusions
The above observation suggests that the plate parameters have a significant impact on the frequency modes of the plate. The selection of appropriate plate parameters can enable the manipulation of frequency modes and their variations as per the system requirements. Thus, it can be inferred that circular variation in plate parameters can be effective in minimizing and regulating frequency modes and their variations.
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About this article
The authors have not disclosed any funding.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
The authors declare that they have no conflict of interest.