Abstract
In the current research, modes of frequency of isotropic tapered triangular plate having 1D (one dimensional) circular thickness and 1D (one dimensional) linear temperature profile for clamped boundary conditions are discussed. Authors implemented Rayleigh Ritz technique to solve the frequency equation of isotropic triangular plate and computed the first four modes with a distinct combination of plate parameters. Authors have performed the convergence study of modes of frequency of the isotropic triangular plate. Also, conducted comparative analysis of modes of frequency of the current study with available published papers and the results presented in tabular form. The aim of the present study is to show the impact of a one dimensional circular thickness and one dimensional linear temperature on modes of frequency of vibration of an isotropic tapered triangular plate.
1. Introduction
Now a days, study of vibration of nonuniform plates is very essential because vibration plays significantly role in many engineering applications i.e., nuclear reactor, aeronautical field, submarine etc. Study of vibration of triangular plates with variable thickness and temperature has been carried out by many researchers/scientists and has been reported in literature. but till date to the best of the knowledge of the authors, vibration of triangular plate with one dimensional circular thickness has not been considered yet.
Free vibration of cantilevered and completely free isosceles triangular plates based on exact threedimensional elasticity theory has been investigated in [1] and derived the eigen frequency equation by using Rayleigh Ritz method. Chebyshevs Ritz method is applied in [2] to the free in plane vibration of arbitrary shaped laminated triangular plates with elastic boundary conditions. Time period analysis of isotropic and orthotropic visco skew plate having circular variation in thickness and density at different edge conditions is discussed in [3] and [4]. Two dimensional temperature effect on the vibration is computed in [5] for the first time for a clamped triangular plate with two dimensional thickness by using the Rayleigh Ritz method. A unified formulation was proposed in [6] for the free inplane vibration of arbitrarily shaped straightsided quadrilateral and triangular plates with arbitrary boundary conditions by improved Fourier series method (IFSM). Fourier series method is used in [7] for free vibration of arbitrary shaped laminated triangular thin plates. A computationally efficient and accurate numerical model is presented in [8] for the study of free vibration behavior of anisotropic triangular plates with edges elastically restrained against rotation and translation. Free vibration of thick equilateral triangular plates with classical boundary conditions has been investigated in [9] based on a new shear deformation theory. Free vibration of circular and annular threedimensional graphene foam (3DGrF) plates under various boundary conditions is discussed in [10].
From the above literature, it is evident that till date to the best of the knowledge of the authors, none of the researchers have worked on triangular plate with one dimensional circular thickness and one dimensional linear temperature environment for clamped boundary conditions. Therefore, in this present study we aim to study the above mentioned problem and investigate the impact on frequency modes of the plate The main purpose of the present study to provide a mathematical model for analyzing the effect of 1D circular variation in thickness on frequency modes of triangular plate under 1D linear temperature variation, which had not been investigated earlier. All the numerical results in the form of modes of frequency are presented in tabular form.
2. Problem geometry and analysis
Consider a viscoelastic triangle plate having aspect ratio $\theta =b/c$ and $\mu =c/a$ and one dimensional thickness $l$ as shown in Fig. 1. Now transform the given triangle into rightangled triangle using the transformation $x=a\zeta +b\psi $ and $y=c\psi $ as shown in Fig. 2.
Fig. 1Triangle plate
Fig. 2Transformed triangle plate
The kinetic energy and strain energy for vibration of a triangle plate are taken as in [11]:
$\left.+\left(\frac{{b}^{2}}{{a}^{2}{c}^{2}}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}^{2}}+\frac{1}{{c}^{2}}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\psi}^{2}}\frac{2b}{a{c}^{2}}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial \zeta \partial \psi}\right)+2(1\nu ){\left(\frac{b}{{a}^{2}c}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}^{2}}+\frac{1}{a{c}^{2}}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial \zeta \partial \psi}\right)}^{2}\right]acd\psi d\zeta ,$
where $\mathrm{\Phi}$ is the deflection function and ${D}_{1}={E}^{3}/12\left(1{\nu}^{2}\right)$ is flexural rigidity.
The Rayleigh Ritz method requires:
Using Eqs. (1) and (2), we have:
$\left.+\left(\frac{{b}^{2}}{{a}^{2}{c}^{2}}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}^{2}}+\frac{1}{{c}^{2}}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\psi}^{2}}\frac{2b}{a{c}^{2}}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial \zeta \partial \psi}\right)+2(1\nu ){\left(\frac{b}{{a}^{2}c}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}^{2}}+\frac{1}{ac}\frac{{\partial}^{2}\mathrm{\Phi}}{\partial \zeta \partial \psi}\right)}^{2}\right]acd\psi d\zeta $
$\frac{1}{2}\rho {\omega}^{2}{\int}_{0}^{1}{\int}_{0}^{1\zeta}l{\mathrm{\Phi}}^{2}acd\psi d\zeta .$
Introducing one dimensional circular thickness as:
where ${l}_{0}$ are the thickness at origin. Also $\beta $ is tapering parameter.
One dimensional temperature on the plate is assumed to be linear as:
where $\tau $ and ${\tau}_{0}$ denote the temperature on and at the origin respectively.
The modulus of elasticity is given by:
where ${E}_{0}$ is the Young’s modulus at $\tau =0$, and $\gamma $ is called the slope of variation. Using Eq. (6) and Eq. (7) becomes:
where $\alpha =\gamma {\tau}_{0}$, $(0\le \alpha <1)$ is called thermal gradient. Using Eqs. (5) and (8), the functional in Eq. (4) becomes:
$+{\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\psi}^{2}}\right)}^{2}\frac{2\left(2{\theta}^{2}+1\nu \right)}{{\mu}^{2}}{\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial \zeta \partial \psi}\right)}^{2}+\frac{2\left(\nu +{\theta}^{2}\right)}{{\mu}^{2}}\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}^{2}}\right)\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\psi}^{2}}\right)$
$\left.\frac{4\theta \left(1+{\theta}^{2}\right)}{\mu}\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\zeta}^{2}}\right)\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial \zeta \partial \psi}\right)\frac{4\theta}{{\mu}^{3}}\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial {\psi}^{2}}\right)\left(\frac{{\partial}^{2}\mathrm{\Phi}}{\partial \zeta \partial \psi}\right)\right]acd\psi d\zeta $
$\frac{1}{2}\rho {\omega}^{2}{\int}_{0}^{1}{\int}_{0}^{1\zeta}\left(1+\beta \left\{1\sqrt{1{\zeta}^{2}}\right\}\right){\mathrm{\Phi}}^{2}acd\psi d\zeta .$
where ${D}_{0}={E}_{0}{l}_{0}^{3}/12\left(1{v}^{2}\right)$ and ${\lambda}^{2}=\rho {\omega}^{2}{l}_{0}{a}^{2}/{D}_{0}$.
The deflection function is taken as:
where ${\mathrm{\Psi}}_{i}$, $i=\mathrm{0,1},2\dots n$ are unknowns and the value of $e$, $f$, $g$ can be 0, 1 and 2 corresponding to a given edge condition.
To minimize Eq. (9), we have:
Solving Eq. (11), we have frequency equation:
where $P={\left[{p}_{ij}\right]}_{i,j=\mathrm{0,1},\dots n}$ and $Q={\left[{q}_{ij}\right]}_{i,j=\mathrm{0,1},\dots n}$ are the square matrix of order $(n+1)$.
3. Numerical results and discussion
In the current study, authors evaluated numerical data in the form of modes of frequency (first four modes) for right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate on clamped edge condition for the different value of plate parameters. Throughout the calculation the value of aspect ratio $a/b=$ 1.5, Poisson’s ratio $\nu =$ 0.345, ${E}_{0}=$ 2.80·10^{3 }N/M^{2} and $\rho =$ 2.80·10^{3}kg/M^{3} is taken into consideration. All the results are presented in tabular form (refer Tables 13). Table 1 presents the modes of frequency $\lambda $ for right angled isosceles triangular plate corresponding to tapering parameter $\beta $ for fixed value of $\theta =$ 0, $\mu =$ 1.0 and the variable value of thermal gradient $\alpha $ i.e., $\alpha =$ 0.2, 0.6. From the Table 1, it can be seen that the modes of frequency $\lambda $ decreases with the increasing value of tapering parameter $\beta $ for all the above mentioned value of thermal gradient $\alpha $. It is also observed that the value modes of frequency $\lambda $ decreases with the increasing value of thermal gradient $\alpha $, while the rate of decrement in modes of frequency $\lambda $ increases with the increasing value of thermal gradient $\alpha $.
Table 2 incorporates the modes of frequency $\lambda $ for right angled scalene triangular plate corresponding to tapering parameter $\beta $ for fixed value of $\theta =$ 0, $\mu =$ 1.5 and the variable value of thermal gradient $\alpha $ i.e., $\alpha =$ 0.2, 0.6. In table 2 also, modes of frequency $\lambda $ decreases with the increasing value of tapering parameter $\beta $ for all the above mentioned value of thermal gradient $\alpha $ as shown in Table 1. Like in Table 1, it is also observed in Table 2 that the value modes of frequency $\lambda $ decreases with the increasing value of thermal gradient $\alpha $, while the rate of decrement in modes of frequency $\lambda $ increases with the increasing value of thermal gradient $\alpha $.
Table 1Modes of frequency of right angle isosceles triangle plate corresponding to tapering parameter
$\theta =0$, $\mu =1.0$  
$\alpha =0.2$  $\alpha =0.6$  
$\beta $  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$ 
0.0  100.130  366.937  831.845  2127.902  84.6067  309.174  702.733  1815.61 
0.2  97.9199  360.068  816.224  2063.28  82.5137  302.794  687.724  1753.50 
0.4  95.7800  353.380  799.694  2024.049  80.4804  296.555  672.161  1712.71 
0.6  93.7147  346.888  785.481  1958.46  78.5112  290.473  658.629  1649.75 
0.8  91.7283  340.614  770.259  1916.75  76.6108  284.567  644.601  1606.46 
1.0  89.8245  334.544  755.453  1883.44  74.7835  278.878  630.245  1571.98 
Table 2Modes of frequency of right angle scalene triangle plate corresponding to tapering parameter
$\theta =$0, $\mu =1.5$  
$\alpha =0.2$  $\alpha =0.6$  
$\beta $  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$ 
0.0  90.3135  330.939  750.081  1917.22  76.1245  278.106  631.328  1626.59 
0.2  88.3716  324.966  736.576  1861.97  74.2734  272.520  618.272  1573.02 
0.4  86.4950  319.165  722.302  1829.51  72.4780  267.054  604.933  1538.14 
0.6  84.6875  313.532  710.242  1773.00  70.7424  261.745  593.284  1483.65 
0.8  82.9526  308.0911  697.482  1737.62  69.0705  256.601  581.116  1447.01 
1.0  81.2937  302.856  684.760  1710.59  67.4662  251.626  569.113  1417.35 
Table 3Modes of frequency of scalene triangle plate corresponding to tapering parameter
$\theta =1/\sqrt{3}$, $\mu =\sqrt{3}/2$  
$\alpha =0.2$  $\alpha =0.6$  
$\beta $  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{3}$  ${\lambda}_{4}$ 
0.0  78.6772  289.064  654.766  1661.231  64.3411  237.763  538.269  1352.55 
0.2  77.5677  285.1960  645.948  1632.17  63.2087  233.907  529.352  1321.08 
0.4  76.5167  281.485  636.874  1618.99  62.1304  230.217  519.905  1303.43 
0.6  75.5254  277.949  629.700  1584.65  61.1079  226.664  512.370  1268.33 
0.8  74.5946  274.599  621.852  1569.17  60.1430  223.287  504.310  1248.69 
1.0  73.7247  271.443  614.434  1554.99  59.2370  220.074  496.777  1230.15 
Table 3 provides the modes of frequency $\lambda $ for scalene triangular plate corresponding to tapering parameter $\beta $ for fixed value of $\theta =1/\sqrt{3}$, $\mu =\sqrt{3}/2$ and the variable value of thermal gradient $\alpha $ i.e., $\alpha =$ 0.2, 0.6. In table 3 also, modes of frequency $\lambda $ decreases with the increasing value of tapering parameter $\beta $ for all the above mentioned value of thermal gradient $\alpha $ as shown in Tables 1, 2. Like in Tables 1, 2, it is also reported in Table 3 that the value modes of frequency $\lambda $ decreases with the increasing value of thermal gradient $\alpha $, while the rate of decrement in modes of frequency $\lambda $ increases with the increasing value of thermal gradient $\alpha $.
4. Convergence study
In this section, authors shows the convergence study done on modes of frequency $\lambda $ (first two modes) of right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate at clamped edge condition for the plate parameters specified as $\alpha =\beta =0.0$, $\nu =\mathrm{}$0.345 and $a/b=\mathrm{}$1.5. The results are displayed in tabular form (refer Table 4). From the Table 4, one can concluded that modes of frequency for the above mentioned triangular plates converges up to three decimal place in fifth approximation.
Table 4Modes of frequency of scalene triangle plate corresponding to tapering parameter
$N$  $\theta =0.0$, $\mu =1.0$  $\theta =0.0$, $\mu =1.5$  $\theta =1/\sqrt{3}$, $\mu =\sqrt{3}/2$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{1}$  ${\lambda}_{2}$  
2  107.077  436.191  107.077  436.191  92.7314  377.753 
3  107.046  394.917  96.6275  356.480  92.7048  342.008 
4  107.045  392.630  96.6272  354.406  92.7045  340.018 
5  107.045  392.630  96.6272  354.406  92.7045  340.018 
Table 5Comparison of modes of frequency with [12] for right angled isosceles, right angled scalene and scalene triangular plate corresponding to tapering parameter
$\alpha =0.0$  
$\beta $  $\theta =0.0$, $\mu =1.0$  $\theta =0.0$, $\mu =1.5$  $\theta =1/\sqrt{3}$, $\mu =\sqrt{3}/2$  
${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{1}$  ${\lambda}_{2}$  ${\lambda}_{1}$  ${\lambda}_{2}$  
0.0  107.077  436.192  96.665  393.737  84.965  346.119 
107.077  436.192  79.2171  322.7010  70.8248  288.5140  
0.2  104.798  426.334  94.658  385.287  83.849  341.319 
98.813  401.130  74.1212  301.2510  66.7330  271.5330  
0.4  102.595  416.894  92.731  377.213  82.792  336.839 
91.069  368.577  69.4564  281.9920  63.0304  256.6090  
0.6  100.472  407.884  90.877  369.526  81.798  332.682 
83.983  339.162  65.3054  265.3940  59.7782  244.1390  
0.8  98.432  284.632  89.099  362.233  80.865  328.853 
77.713  313.628  61.7481  251.9720  57.0324  234.5460  
Bold values are obtained from [12] 
5. Results comparison
In this section, authors performed a comparative analysis of modes of frequency $\lambda $ (first two modes) obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) and modes of frequency $\lambda $ obtained in [12] at clamped edge condition and presented in tabular form (refer Table 5). In [12], authors assumed the thickness variations in both the direction but in the present study authors taken the thickness in one direction so authors compared the modes of frequency $\lambda $ of present study with modes of frequency $\lambda $ obtained in [12] when the value of second tapering parameter ${\beta}_{2}$ is 0.0 in [12]. Table 5 shows the comparison of modes of frequency $\lambda $ obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) and modes of frequency $\lambda $ obtained in [12] at clamped edge condition corresponding to tapering parameter $\beta $ for fixed value of thermal gradient $\alpha $ i.e., $\alpha =0.0$. From the Table 5, authors conclude that:
1) Modes of frequency $\lambda $ obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) are higher in comparison to modes of frequency $\lambda $ obtained in [12].
2) The rate of change in (decrement) in modes of frequency $\lambda $ obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) are smaller in comparison to modes of frequency $\lambda $ obtained in [12], at clamped edge condition for all the three above mentioned values of thermal gradient $\alpha $.
6. Conclusions
The effect of circular thickness on modes of frequency $\lambda $ of right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate under temperature environment at clamped edge condition is computed. Based on numerical discussions and results comparisons, authors would like to records the following facts:
1) The modes of frequency obtained in present study in case of circular thickness is higher than the modes of frequency obtained in [12] in case of linear thickness. The modes of frequency obtained in present study and modes of frequency obtained in [12] exactly match at $\beta =0.0$ (refer Table 5).
2) The variation in modes of frequency obtained in present study in case of circular thickness is less in comparison to modes of frequency obtained in [12] in case of linear variation in thickness (refer Table 5).
3) The modes of frequency obtained for the present study decreases (less rate of decrements) with the increasing value of tapering parameter and thermal gradient. (refer Tables 13).
4) As temperature increases on the plate, the modes of frequency decreases but the rate of change (decrement) in modes of frequency increases (refer Tables 13).
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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.