Vibrational frequency of triangular plate having circular thickness

. In the current research, modes of frequency of isotropic tapered triangular plate having 1-D (one dimensional) circular thickness and 1-D (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.


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 three-dimensional 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 in-plane vibration of arbitrarily shaped straight-sided 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 three-dimensional graphene foam (3D-GrF) 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 1-D circular variation in thickness on frequency modes of triangular plate under 1-D 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.

Problem geometry and analysis
Consider a viscoelastic triangle plate having aspect ratio  = / and  = / and one dimensional thickness  as shown in Fig. 1.Now transform the given triangle into right-angled triangle using the transformation  =  +  and  =  as shown in Fig. 2. The kinetic energy and strain energy for vibration of a triangle plate are taken as in [11]: where Φ is the deflection function and  =  /12(1 −  ) is flexural rigidity.The Rayleigh Ritz method requires: Using Eqs. ( 1) and ( 2), we have: Introducing one dimensional circular thickness as: where  are the thickness at origin.Also  is tapering parameter.One dimensional temperature on the plate is assumed to be linear as: where  and  denote the temperature on and at the origin respectively.The modulus of elasticity is given by: where  is the Young's modulus at  = 0, and  is called the slope of variation.Using Eq. ( 6) and Eq. ( 7) becomes: where  =  , (0 ≤  < 1) is called thermal gradient.Using Eqs. ( 5) and ( 8), the functional in Eq. ( 4) becomes: where  =   /12(1 −  ) and  =    / .The deflection function is taken as: where Ψ ,  = 0,1,2 …  are unknowns and the value of , ,  can be 0, 1 and 2 corresponding to a given edge condition.

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   ⁄ = 1.5, Poisson's ratio  = 0.345,  = 2.80•10 3 N/M 2 and  = 2.80•10 3 kg/M 3 is taken into consideration.All the results are presented in tabular form (refer Tables 1-3).Table 1 presents the modes of frequency  for right angled isosceles triangular plate corresponding to tapering parameter  for fixed value of  = 0,  = 1.0 and the variable value of thermal gradient  i.e.,  = 0.2, 0.6.From the Table 1, it can be seen that the modes of frequency  decreases with the increasing value of tapering parameter  for all the above mentioned value of thermal gradient .It is also observed that the value modes of frequency  decreases with the increasing value of thermal gradient , while the rate of decrement in modes of frequency  increases with the increasing value of thermal gradient .
Table 2 incorporates the modes of frequency  for right angled scalene triangular plate corresponding to tapering parameter  for fixed value of  = 0,  = 1.5 and the variable value of thermal gradient  i.e.,  = 0.2, 0.6.In table 2 also, modes of frequency  decreases with the increasing value of tapering parameter  for all the above mentioned value of thermal gradient  as shown in Table 1.Like in Table 1, it is also observed in Table 2 that the value modes of frequency  decreases with the increasing value of thermal gradient , while the rate of decrement in modes of frequency  increases with the increasing value of thermal gradient .⁄ ,  = √3 2 ⁄ and the variable value of thermal gradient  i.e.,  = 0.2, 0.6.In table 3 also, modes of frequency  decreases with the increasing value of tapering parameter  for all the above mentioned value of thermal gradient  as shown in Tables 1, 2. Like in Tables 1, 2, it is also reported in Table 3 that the value modes of frequency  decreases with the increasing value of thermal gradient , while the rate of decrement in modes of frequency  increases with the increasing value of thermal gradient .

Convergence study
In this section, authors shows the convergence study done on modes of frequency  (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  =  = 0.0,  = 0.345 and   ⁄ = 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.

Results comparison
In this section, authors performed a comparative analysis of modes of frequency  (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  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  of present study with modes of frequency  obtained in [12] when the value of second tapering parameter  is 0.0 in [12].Table 5 shows the comparison of modes of frequency  obtained in present study (right angled isosceles scalene triangular plate, right angled scalene triangular plate and scalene triangular plate) and modes of frequency  obtained in [12] at clamped edge condition corresponding to tapering parameter  for fixed value of thermal gradient  i.e.,  = 0.0.From the Table 5, authors conclude that: 1) Modes of frequency  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  obtained in [12].
2) The rate of change in (decrement) in modes of frequency  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  obtained in [12], at clamped edge condition for all the three above mentioned values of thermal gradient .

Conclusions
The effect of circular thickness on modes of frequency  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  = 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 1-3).
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 1-3).

Table 1 .
Modes of frequency of right angle isosceles triangle plate corresponding to tapering parameter

Table 2 .
Modes of frequency of right angle scalene triangle plate corresponding to tapering parameter

Table 3 .
Modes of frequency of scalene triangle plate corresponding to tapering parameter

Table 3
provides the modes of frequency  for scalene triangular plate corresponding to tapering parameter  for fixed value of  = 1 √3

Table 4 .
Modes of frequency of scalene triangle plate corresponding to tapering parameter

Table 5 .
[12]arison of modes of frequency with[12]for right angled isosceles, right angled scalene and scalene triangular plate corresponding to tapering parameter  = 0.0