Analyzing vibration modes in non-homogeneous parallelogram plates

. This study analyzes vibration modes in non-homogeneous orthotropic parallelogram plate with a one-dimensional circular thickness, focusing on SCCC edge condition, where C and S represent the clamped and simply supported edges of the plate, respectively. Circular Poisson’s ratio variation is considered, along with linear temperature changes. The study demonstrates the advantages of variable Poisson’s ratio over density parameter variation in obtaining shorter vibration time periods


Introduction
The analysis of vibrational modes in various plate configurations, including tapered ones, is vital in engineering, given their wide-ranging applications.Numerous studies have contributed to this area.
Di et al. [2] introduced a theory enhancing the accuracy of predicting the behavior of thick orthotropic plates compared to classical lamination and shear deformation theories.Gupta et al. [3] investigated transverse motion in an elastic plate with non-linear thickness and thermal gradient, utilizing the Rayleigh-Ritz technique.Gupta et al. [4] introduced a model for analyzing vibrations in parallelogram-shaped, viscoelastic, orthotropic plates with linear thickness variations in both directions.Khanna [6] used the Rayleigh Ritz method to study frequency modes of a viscoelastic isotropic rectangular plate, considering the impact of thermal gradient.The natural vibration of non-homogeneous tapered parallelogram plates with temperature variation was examined in [8].Vibrational frequencies in a non-uniformly thick parallelogram plate were mathematically analyzed, considering linear temperature effects by authors in [9].The influence of thermal effects on non-homogeneous parallelogram plates with two-dimensional circular variations in thickness was explored in a study referenced as [10].In [11], the authors focused on the vibration of orthotropic square plates with varying thickness, using the Rayleigh-Ritz method and MATLAB software.Sharma et al. [12] examined an orthotropic parallelogram plate with bilinear thickness variation and a parabolic temperature distribution, specifically focusing on the  edge condition.
This paper investigates the impact of one-dimensional circular tapering, Poisson's ratio, and linear temperature on vibrational modes in a non-homogeneous orthotropic parallelogram plate under  boundary conditions.

Geometry and analytical approach
Consider a nonhomogeneous parallelogram plate depicted in Fig. 1 with dimensions  and , thickness , and density .
The skew plate, represented in Fig. 1, has a circular thickness denoted as  in one dimension.Additionally, Poisson's ratio  is assumed to be circular in one dimension: where,  and  stand for the initial thickness and Poisson's ratio of the plate at the origin.Furthermore,  (ranging from 0 to 1) and  (ranging from 0 to 1) represent the taper and non-homogeneity parameters, respectively.We assume a two-dimensional steady-state temperature distribution on the plate, following the parabolic model introduced in [7]: where,  and  denote the temperature excess above the reference temperature at any point on the plate and at the origin, respectively.The temperature dependence modulus of elasticity for engineering structures follows [7]: where,  and  denote Young's moduli in the  and  directions, while  represents the shear modulus.The parameter  accounts for the slope variation of moduli with temperature.By substituting Eq. ( 2) in Eq. ( 3), we obtain the following expressions: where  =  , (0 ≤  < 1) is called temperature gradient.The flexural rigidities  ,  and torsional rigidity  of the plate are taken as in [12]: where  and  are Poisson's ratios.Using Eqs. ( 1), ( 3) and (4) in Eq. ( 5), we get: Now, introducing non-dimensional variable as: The equation for kinetic energy  and strain energy  for natural transverse vibration of non-uniform orthotropic parallelogram is taken as in [1]: Rayleigh-Ritz method requires that maximum strain energy must be equal to maximum kinetic energy i.e.,: From Eqs. ( 1), (6-9), we have: where The two term deflection function which satisfy all the edge conditions can be taken as in [7] The two term deflection function which satisfy all the edge conditions can be taken as in [7]: This expression results from two components: one defining boundary conditions (0 for free edge, 1 for simply supported, 2 for clamped edge), and the other representing mode frequencies with constants Ω for  = 0,1,2, ⋯ , .To minimize the functional in Eq. ( 11), the following condition is necessary: After simplifying Eq. ( 13), we get a homogeneous system of equations in Ω whose non zero solution gives equation of frequency as: where,  =  and  =  are square matrices of order ( + 1) with  = 0,1,2. . . and  = 0,1,2. . ..The time period is calculated using the expression: where  is a frequency obtained from Eq. ( 14).

Numerical results and discussion
This study investigated the impact of parameters such as tapering, thermal gradient, and nonhomogeneity on the vibration time period of an orthotropic parallelogram plate.The plate had a fixed aspect ratio of   ⁄ = 1.5, a skew angle of  = 30°, circular thickness, and Poisson's ratio.The analysis considered specific edge conditions and linear temperature effects, with material parameters sourced from [5]:  *  * ⁄ = 0.01,  *  * ⁄ = 0.3, G  * ⁄ = 0.0333,  *  ⁄ = 3.0×10 5 and  = 0.345.The results are presented in Tables 1-3.  1 presents time period data for an orthotropic parallelogram plate under the  edge condition and a constant thermal gradient ( = 0.2).The table covers a range of non-homogeneity parameter () and tapering parameter () values from 0.0 to 0.8.The results show that higher  values lead to a decrease in time period , similar to the effect of increasing  on reducing .
Table 2 exhibits time period () data for an orthotropic parallelogram plate under the  edge condition, with a constant non-homogeneity parameter ( = 0.2).The table encompasses  and  values from 0.0 to 0.8.Remarkably, higher thermal gradient () values result in elevated time period (), while increasing the tapering parameter () leads to a significant decrease in .
Table 3 presents time periods for an orthotropic parallelogram plate under the  edge condition, with variable non-homogeneity (), thermal gradient (), and tapering () parameters ranging from 0.0 to 0.8.Higher values of  and  lead to an increase in the time period (), while raising  results in a reduction of .

Conclusions
In terms of time periods, Table 1 indicates that  holds greater influence than  under the  edge condition.Additionally, in Tables 2 and 3,  demonstrates a more substantial effect on the time period's rate of change compared to  and , respectively.

Table 1 .
Time period of orthotropic parallelogram plate at  edge condition corresponding to non-homogeneity parameter

Table 2 .
Time period of orthotropic parallelogram plate at  edge condition corresponding to thermal gradient

Table 3 .
Time period of orthotropic parallelogram plate at  edge condition corresponding to tapering parameter