Published: 21 April 2022

# Behavior of frequencies of orthotropic rectangular plate with circular variations in thickness and density

Neeraj Lather1
Amit Sharma2
1, 2Department of Mathematics, Amity School of Applied Sciences, Gurugram, India
Corresponding Author:
Amit Sharma
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#### Abstract

In this article, authors dealt with general solution of differential equation of orthotropic rectangular plate with clamped boundary conditions under bi-parabolic 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 non-uniform in-plane loading by using first-order 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]:

1
${T}_{s}=\frac{1}{2}{\omega }^{2}{\int }_{0}^{a}‍{\int }_{0}^{b}‍\rho l{\mathrm{\Phi }}^{2}d\psi d\zeta ,$
2
${V}_{s}=\frac{1}{2}{\int }_{0}^{a}‍{\int }_{0}^{b}‍\left[{\left[D}_{x}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)}^{2}+{D}_{y}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)}^{2}+2{\nu }_{x}{D}_{y}\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}+4{D}_{xy}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \partial \psi }\right)}^{2}\right]d\psi d\zeta ,$

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:

3
$L=\delta \left({V}_{s}-{T}_{s}\right)=0.$

Using Eqs. (1), (2), we get functional equation:

4
$L=\frac{1}{2}{\int }_{0}^{a}‍{\int }_{0}^{b}\left[{D}_{x}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\right)}^{2}+{D}_{y}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}\right)}^{2}+2{\nu }_{x}{D}_{y}\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }^{2}}\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }^{2}}+4{D}_{xy}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial \zeta \partial \psi }\right)}^{2}\right]‍d\psi d\zeta -\frac{1}{2}{\omega }^{2}{\int }_{0}^{a}‍{\int }_{0}^{b}‍\rho l{\mathrm{\Phi }}^{2}d\psi d\zeta =0.$

Now, introducing non-dimensional variable as:

${\zeta }_{1}=\frac{\zeta }{a},{\psi }_{1}=\frac{\psi }{a},$

together with two dimensional circular thickness, we get:

5
$l={l}_{0}\left(1+{\beta }_{1}\left\{1-\sqrt{1-{{\zeta }_{1}}^{2}}\right\}\right)\left(1+{\beta }_{2}\left\{1-\sqrt{1-{\frac{{a}^{2}}{{b}^{2}}{\psi }_{1}}^{2}}\right\}\right),$

where ${l}_{0}$ is thickness at origin and ${\beta }_{1}$, ${\beta }_{2}\left(0\le {\beta }_{1},{\beta }_{2}\le 1\right)$ are tapering parameters.

For non-homogeneity consideration, authors assumed one dimensional circular variation in density:

6
$\rho ={\rho }_{0}\left(1+{m}_{0}\left\{1-\sqrt{1-{{\zeta }_{1}}^{2}}\right\}\right).$

Two dimensional parabolic temperature distribution is taken as in [5]:

7
$\tau ={\tau }_{0}\left(1-{\zeta }_{1}^{2}\right)\left(1-\frac{{a}^{2}}{{b}^{2}}{{\psi }_{1}}^{2}\right),$

where $\tau$ and ${\tau }_{0}$ denotes the temperature on and at the origin respectively.

For orthotropic materials, modulus of elasticity is given by [17]:

8
${E}_{x}={E}_{1}\left(1-\gamma \tau \right),{E}_{y}={E}_{2}\left(1-\gamma \tau \right),{G}_{xy}={G}_{0}\left(1-\gamma \tau \right),$

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:

9
${E}_{x}\left(x\right)={E}_{1}\left(1-\alpha \left(1-{\zeta }_{1}^{2}\right)\left(1-\frac{{a}^{2}}{{b}^{2}}{{\psi }_{1}}^{2}\right)\right),$
${E}_{y}\left(x\right)={E}_{2}\left(1-\alpha \left(1-{\zeta }_{1}^{2}\right)\left(1-{{\frac{{a}^{2}}{{b}^{2}}\psi }_{1}}^{2}\right)\right),$
${G}_{xy\left(x\right)}={G}_{0}\left(1-\alpha \left(1-{\zeta }_{1}^{2}\right)\left(1-\frac{{a}^{2}}{{b}^{2}}{{\psi }_{1}}^{2}\right)\right),$

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:

10
$L=\frac{{D}_{0}}{2}{\int }_{0}^{1}‍{\int }_{0}^{\frac{b}{a}}‍\left[{\left(1-\alpha \left(1-{\zeta }_{1}^{2}\right)\left(1-\frac{{a}^{2}}{{b}^{2}}{{\psi }_{1}}^{2}\right)\right){\left(1+{\beta }_{1}\left\{1-\sqrt{1-{{\zeta }_{1}}^{2}}\right\}\right)}^{3}\left(1+{\beta }_{2}\left\{-\sqrt{1-\frac{{a}^{2}}{{b}^{2}}{{\psi }_{1}}^{2}}\right\}\right)}^{3}\left\{{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }_{1}^{2}}\right)}^{2}+\frac{{E}_{2}}{{E}_{1}}{\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }_{1}^{2}}\right)}^{2}+2{\nu }_{x}\frac{{E}_{2}}{{E}_{1}}\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }_{1}^{2}}\right)\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\psi }_{1}^{2}}\right)+4\frac{{G}_{0}}{{E}_{1}}\left(1-{\nu }_{x}{\nu }_{y}\right){\left(\frac{{\partial }^{2}\mathrm{\Phi }}{\partial {\zeta }_{1}\partial {\psi }_{1}}\right)}^{2}\right\}]\rightd{\psi }_{1}d{\zeta }_{1}{-\lambda }^{2}{\int }_{0}^{1}‍{\int }_{0}^{\frac{b}{a}}‍\left[\left(1+{m}_{0}\left\{1-\sqrt{1-{{\zeta }_{1}}^{2}}\right\}\right)\left(1+{\beta }_{2}\left\{1-\sqrt{1-{{\zeta }_{1}}^{2}}\right\}\right)\right\left(1+{\beta }_{2}\left\{1-\sqrt{1-{\frac{{a}^{2}}{{b}^{2}}{\psi }_{1}}^{2}}\right\}\right)]{\mathrm{\Phi }}^{2}d{\psi }_{1}d{\zeta }_{1}=0,$

where:

${D}_{0}=\frac{1}{2}\left(\frac{{E}_{1}{l}_{0}^{3}}{12\left(1-{\nu }_{x}{\nu }_{y}\right)}\right),{\lambda }^{2}=\frac{12{a}^{4}\rho {\omega }^{2}\left(1-{\nu }_{x}{\nu }_{y}\right)}{{E}_{1}{{h}_{0}}^{2}}.$

We choose deflection function which satisfies all the edge condition as taken in [16]:

11
$\mathrm{\Phi }\left(\zeta ,\psi \right)=\left[{\left({\zeta }_{1}\right)}^{e}{\left({\psi }_{1}\right)}^{f}{\left(1-{\zeta }_{1}\right)}^{g}{\left(1-\frac{a{\psi }_{1}}{b}\right)}^{h}\right]\left[\sum _{i=0}^{n}‍{\mathrm{\Psi }}_{i}{\left\{\left({\zeta }_{1}\right)\left({\psi }_{1}\right)\left(1-{\zeta }_{1}\right)\left(1-\frac{a{\psi }_{1}}{b}\right)\right\}}^{i}\right],$

where ${\mathrm{\Psi }}_{i}$, $i=0,1,2...n$ 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:

12
$\frac{\partial L}{\partial {\Psi }_{i}}=0,i=0,1,...n.$

Solving Eq. (11), we have the following frequency equation:

13
$\left|P-{\lambda }^{2}Q\right|=0,$

where $P={\left[{p}_{ij}\right]}_{i,j=0,1,..n}$ and $Q={\left[{q}_{ij}\right]}_{i,j=0,1,..n}$ are square matrix of order $\left(n+1\right)$.

## 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 Rayleigh-Ritz 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.

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