Vibration of CFFF Functionally Graded Plates with an Attached Point Mass at an Arbitrary Point in Thermal Environment

Abstract

Purpose

In many cases, working on models such as carrying a point attached mass or distributed attached mass provides a more realistic depiction of the problem. The main reason for the vibration studies of the structural elements with coupled mass, which constitute the main purpose of this study, is to see the changes in the resonance frequency due to the attached mass and to reduce the resonance frequency to a desired value. Although the vibration problems of mass-loaded rectangular plates are a very common problem in engineering applications, no study on functionally graded plates has been found in the literature. In this study, the effect of the variation of temperature-dependent material properties along the thickness according to a simple power law on the vibration behavior of point mass carrying functionally graded plates is investigated for the cantilever (CFFF) boundary condition. Numerical studies are performed for different mass ratio (M), different location of point mass on the plate region and throughout the x axis, volume fractions with p and side-to-side ratio (a/b) at nonlinear temperature distribution.

Methods

In this study, effect of the mass and temperature on free vibration of the functionally graded plate carrying a point mass at an arbitrary position is analysed with three-dimensional Ritz solution. Material properties of considered functionally graded plate are assumed to be temperature dependent and reinforcement in thickness direction according to a power law distribution and effective material properties are estimated using Mori-Tanaka homogenization method.

Results

Free vibration frequencies decrease with increasing p index and increasing temperature difference. The frequencies obtained in the case of with point mass are always smaller than the frequency values obtained in the case of without point mass. When the point mass is located on a nodal line, the mass does not move during these and so frequency remains constant as independent of the presence of mass. If the mass is located on the nodal lines, at these frequencies it will resonate at the natural frequency of the unperturbed plate.

Conclusion

The study shows that the effect of the presence of added mass, mass size and location on structures may not be negligible.

Introduction

Functionally graded materials (FGMs) are designed for the first time in 1984 by Japanese scientists [1, 2] as high temperature resistant materials, especially for structures such as aircraft, spacecraft and other engineering structures. FGMs are high-tech materials in which the desired thermal and mechanical properties can be produced by continuously changing the material properties in a thickness and/or in-plane direction. These advanced properties of FGMs have allowed them to be considered as materials for structures modeled as structural elements such as plates, beams, and shells. Free vibration analysis of structural elements with FGM-designed plate geometry has always attracted the attention of researchers.

A simple yet effective first order shear deformation plate theory integrated with meshfree moving Kriging method has been developed by Vu et al. [3] for the analysis of free vibration and static deflection of functionally graded ceramic–metal plates. Zhao et al. [4] presented a free vibration analysis of functionally graded rectangular and skew plates with used the element-free kp-Ritz method based on first order shear deformation theory under clamped, simply supported and cantilever boundary conditions. The free vibration and static analysis of square and rectangular functionally graded plates was presented, which is based on the higher order shear deformation theory with a new finite element model by Talha ve Singh [5]. In the study the systems of algebraic equations were derived using variational approach for the free vibration and static problem. Nguyen [6] proposed a higher order hyperbolic shear deformation plate model for bending, buckling and vibration analysis of functionally graded plates. Ferreira et al. [7] used the asymmetric collocation method with multiquadrics basis functions, and the FSDT and the TSDT to obtained natural frequencies of square functionally graded plates. Matsunaga [8] presented a 2-dimensional high-order theory in which the full effects of shear deformations, thickness changes and rotational inertia are taken into account to analyze the natural frequencies and buckling stresses of functionally graded plates. Exact free vibration analysis of moderately thick and thick functionally graded plates using two-dimensional higher order kinematic theories which considered both shear and normal deformation effects is investigated by used Levy method by Dozio [9].

A unified and accurate solution method has been developed by Jin et al. [10] to deal with the free vibration analysis of arbitrarily thick functionally graded plate with general boundary based on the linear, small-strain 3D elasticity theory. Vel and Batra [11] presented a three-dimensional exact solution for the vibration of simply supported rectangular thick functionally graded plates. In the study the effective material properties at a point were estimated from the local volume fractions and the material properties of the phases either by the Mori–Tanaka [12, 13] or the self-consistent [14] scheme. Uymaz and Aydoğdu [15] carried out the Ritz method with Chebyshev polynomials for the free vibration analysis of functionally graded plates based on three-dimensional elasticity.

Free vibration analysis of simply supported functionally graded plates is numerically studied using QUAD-8 shear flexible element with and without thermal environment based on first order shear deformation theory by Natarajan et al. [16]. The effective material properties are estimated using Mori–Tanaka homogenization method.

A theoretical method was developed to investigate vibration characteristics of initially stressed functionally graded rectangular platesmade up of metal and ceramic in thermal environment by Kim [17]. Free vibration of functionally graded material rectangular plates with simply supported and clamped edges in the thermal environment was studied based on the three-dimensional linear theory of elasticity by Li et al. [18].

However, in many cases, working on models such as carrying a point attached mass or distributed attached mass provides a more realistic depiction of the problem. The main reason for the vibration studies of the structural elements with coupled mass, which constitute the main purpose of this study, is to see the changes in the resonance frequency due to the attached mass and to reduce the resonance frequency to a desired value.

Gürgöze et al. [19] investigated free vibration of a cantilever euler–bernoulli beam carrying a tip mass with in-span support using the Dunkerley's procedure. Cha and Wong [20] presented a method to analyze free vibration of combined dynamical system which consist of a uniform cantilever Euler–Bernoulli beam carrying an undamped oscillator system using the Lagrange multiplier method and the Green function method.

The free vibration analysis of an isotropic simply supported plate carrying a uniformly distributed mass was investigated by Kopmaz and Telli [21]. The analysis was carried out using the Galerkin procedure, the equation of motion was reduced to a set of ordinary differential equations based on clasical plate theory. This polynomial equation was solved numerically. Wong [22] performed Ritz method based on classical plate theory for solved the free bending vibration of a simply supported rectangular plate carrying distributed mass. Free vibration of isotropic plate carrying distributed spring mass using Ritz-Galerkin method with Chebyshev polynomial series based on classical plate theory by Zhou and Ji [23]. Alibeigloo et al. [24] investigated the free vibration of simply supported angle-ply laminated plates carrying distributed attach mass using the Hamilton’s Principle by means of a double Fourier series based on a third-order shear deformation theory.

Chiba and Sugimoto [25] analyzed free vibration of a cantilever plate carrying spring-mass system using Rayleigh–Ritz method based on classical plate theory. Yu [26] presented analytical solutions for free and forced vibrations of cantilever plates carrying single attached mass using Gorman’s method of superposition and the modal summaton method. Ciancio et al. [27] presented the vibration problem for a cantilever anisotropic plate carrying a concentrated mass on of center using Ritz method with beam functions based on the classical plate theory. Vibration of symmetrically laminated composite plate carrying an attached mass on of center using Ritz method with simple polynomials based on the classical plate theory is analyzed by Aydoğdu and Filiz [28].

An experimental study on vibration response of an elastically point-supported isotropic plate carrying an attached point mass was presented by Watkins et al. [29]. These results are compared to frequencies and to modes shapes determined from the Rayleigh–Ritz method and a finite element analysis using COMSOL. Ritz method is performed with Orthogonal polynomials as admissible functions, and finite element analysis is based on Mindlin plate theory, adjusted for negligible transverse shear effects.

Khalili et al. [30] studied free vibration of simply supported laminated composite cylindrical shells with uniformly distributed attached mass using Galerkin method based on higher order shell theory including stiffness effect.

Free vibration of clamped thin elliptical plates carrying a concentrated mass at an arbitrary position using Ritz method by polynomial expressions as admissible functions was researched by Maiz et al. [31].

Although the vibration problems of mass-loaded rectangular plates are a very common problem in engineering applications, no study on functionally graded plates has been found in the literature. In this study, the effect of the variation of temperature-dependent material properties along the thickness according to a simple power law on the vibration behavior of point mass carrying functionally graded plates is investigated for the cantilever (CFFF) boundary condition. Numerical studies are performed for different mass ratio (M), different location of point mass on the plate region and throughout the x axis, volume fractions with p and side-to-side ratio (a/b) at nonlinear temperature distribution.

Problem Formulation

In this study, effect of the mass and temperature on free vibration of the functionally graded plate carrying a point mass at an arbitrary position is analyzed with three-dimensional Ritz solution. Material properties of considered functionally graded plate are assumed to be temperature dependent and reinforcement in thickness direction according to a power law distribution and effective material properties are estimated using Mori–Tanaka homogenization method.

Effective Material Properties of Functionally Graded Plates

The lower surface of the considered plate consists of the metal phase and the upper surface of the ceramic phase, and the ceramic phase varies according to a power law distribution in the thickness direction, and it is also assumed that material properties of both phases are temperature dependent. The sum of volume fraction of ceramic and metal phases is

$${V}_{\mathrm{c}}+{V}_{\mathrm{m}}=1,$$

(1)

and the ceramic volume fraction distribution in the thickness direction is as follows.

$${V}_{\mathrm{c}}\left(z\right)={\left(\frac{z}{h}+\frac{1}{2}\right)}^{p}.$$

(2)

Here, the p value, which is called the volume ratio exponent, takes values in the range of 0 ≤ p ≤ ∞ and shows the amount of ceramic volume ratio. As can be seen in Fig. 1(a), when the p value is 0, the material is full ceramic, when the p value is 1, the variation of the ceramic material in the thickness direction is linear, when p > 1 the amount of ceramic increases and when p < 1, the amount of ceramic increases with a decrease. In general, an increase in the p value means a decrease in the ceramic volume ratio.

Fig. 1

Volume fraction through the nondimensional thickness co-ordinate, and effective elastic moduli estimated by the Mori–Tanaka scheme with nonlinear temperature distribution

The Mori–Tanaka scheme assumes that isotropic particles are randomly dispersed within the isotropic matrix material. In this study, the matrix phase is metal and the particle phase is ceramic. The effective material properties Young modulus (E), poisson ratio (υ) and thermal coefficients (α) of functionally graded plate which defined according to the Mori–Tanaka homogenization scheme are given as follows,

$$E=\frac{9KG}{3K+G},$$

(3)

$$\nu =\frac{3K-2G}{2\left(3K+G\right)},$$

(4)

$$\frac{\alpha -{\alpha }_{\mathrm{m}}}{{\alpha }_{\mathrm{c}}-{\alpha }_{\mathrm{m}}}=\frac{\left(\frac{1}{\kappa }-\frac{1}{{\kappa }_{\mathrm{m}}}\right)}{\left(\frac{1}{{\kappa }_{\mathrm{c}}}-\frac{1}{{\kappa }_{\mathrm{m}}}\right)}.$$

(5)

In here the effective Bulk modulus K, the effective shear modulus G and the effective heat conductivity κ are defined as follows,

$$\frac{\kappa -{\kappa }_{\mathrm{m}}}{{\kappa }_{\mathrm{c}}-{\kappa }_{\mathrm{m}}}=\frac{{V}_{\mathrm{c}}}{1+\left(1-{V}_{\mathrm{c}}\right)\frac{\left({\kappa }_{\mathrm{c}}-{\kappa }_{\mathrm{m}}\right)}{2{\kappa }_{\mathrm{m}}}},$$

(6)

$$\frac{K-{K}_{\mathrm{m}}}{{K}_{c}-{K}_{\mathrm{m}}}=\frac{{V}_{\mathrm{c}}}{1+\left(1-{V}_{c}\right)\frac{3\left({K}_{\mathrm{c}}-{K}_{m}\right)}{3{K}_{\mathrm{m}}+4{G}_{\mathrm{m}}}},$$

(7)

$$\frac{G-{G}_{\mathrm{m}}}{{G}_{\mathrm{c}}-{G}_{\mathrm{m}}}=\frac{{V}_{\mathrm{c}}}{1+\left(1-{V}_{\mathrm{c}}\right)\frac{\left({G}_{\mathrm{c}}-{G}_{\mathrm{m}}\right)}{{G}_{\mathrm{m}}+{\mathrm{f}}_{1}}},$$

(8)

where

$${f}_{1}=\frac{{G}_{\mathrm{m}}\left(9{K}_{\mathrm{m}}+8{G}_{\mathrm{m}}\right)}{6\left({K}_{\mathrm{m}}+2{G}_{\mathrm{m}}\right)},$$

(9)

The effective mass density (ρ) of functionally graded plate which defined according to the rule of mixtures is given as follow,

$$\rho ={V}_{\mathrm{c}}\left({\rho }_{\mathrm{c}}-{\rho }_{\mathrm{m}}\right)+{\rho }_{m}.$$

(10)

The sub-indices c and m used in the equations given above represent ceramic and metal materials, respectively. Variation of the nondimensional effective elastic moduli throughout the nondimensional thickness co-ordinate estimated by the Mori–Tanaka scheme are given in Fig. 1(b)–(d) for various temperature in nonlinear thermal environment.

Stress–Strain Relations Based on Three-Dimensional Elasticity

The considered thin functionally graded plate in this paper is in the form of rectangular with length a, width b and thickness h. The origin of the co-ordinate system (x,y,z) is placed at the geometric center of the plate and the axes are parallel to the edges of the plate and the corresponding displacement components u, v and w along the x, y and z directions, respectively. For free vibration problem based on three-dimensional elasticity theory the displacement field is as follows,

$$ u(x,y,z;t)\, = \,U(x,y,z)e^{i\omega t} ; \,v(x,y,z;t)\, = \,V(x,y,z)e^{i\omega t} ;\, w(x,y,z;t)\, = \,W(x,y,z)e^{i\omega t} , $$

(11)

where ω corresponds the natural frequency of the plate and $i=\sqrt{-1}$. The strain components ε_ij (i,j = x,y,z) for small deformations are given as,

$$ \varepsilon_{{\text{x}}} = u_{{,{\text{x}}}} ;\, \varepsilon_{{\text{y}}} = v_{{,{\text{y}}}} ; \,\varepsilon_{{\text{z}}} = w_{{,{\text{z}}}} , $$

(12)

$$ \gamma_{{{\text{yz}}}} = v_{{,{\text{z}}}} + w_{{,{\text{y}}}} ; \,\,\gamma_{xz} = u_{{,{\text{z}}}} + w_{{,{\text{x}}}} ; \,\gamma_{{{\text{xy}}}} = u_{{,{\text{y}}}} + v_{{,{\text{x}}}} , $$

(13)

where${}_{,\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)}=\left(\frac{\partial }{\partial \mathrm{x}},\frac{\partial }{\partial \mathrm{y}},\frac{\partial }{\partial \mathrm{z}}\right)$. The stress–strain relations for a linear elastic isotropic material are given by the generalized Hooke’s law as follows,

$${\sigma }_{\mathrm{x}}={C}_{11}{\varepsilon }_{\mathrm{x}}+{C}_{12}{\varepsilon }_{\mathrm{y}}+{C}_{12}{\varepsilon }_{\mathrm{z}},$$

(14)

$${\sigma }_{\mathrm{y}}={C}_{12}{\varepsilon }_{\mathrm{x}}+{C}_{11}{\varepsilon }_{y}+{C}_{12}{\varepsilon }_{\mathrm{z}},$$

(15)

$${\sigma }_{\mathrm{z}}={C}_{12}{\varepsilon }_{\mathrm{x}}+{C}_{12}{\varepsilon }_{\mathrm{y}}+{C}_{11}{\varepsilon }_{\mathrm{z}},$$

(16)

$${\tau }_{yz}={C}_{66}{\gamma }_{\mathrm{yz}},$$

(17)

$${\tau }_{\mathrm{xz}}={C}_{66}{\gamma }_{\mathrm{xz}},$$

(18)

$${\tau }_{\mathrm{xy}}={C}_{66}{\gamma }_{\mathrm{xy}},$$

(19)

where [C] is stiffness matrix and its components are defined as follows,

$${\text{C}}_{11}\text{(z,T)=}\frac{E(\mathrm{z},\mathrm{T})(1-\upnu (\mathrm{z},\mathrm{T}))}{(1+\upnu (\mathrm{z},\mathrm{T}))(1-2\upnu (\mathrm{z},\mathrm{T}))},$$

(20)

$${\text{C}}_{12}\text{(z,T)=}\frac{E(\mathrm{z},\mathrm{T})\upnu (\mathrm{z},\mathrm{T})}{(1+\upnu (\mathrm{z},\mathrm{T}))(1-2\upnu (\mathrm{z},\mathrm{T}))},$$

(21)

$${\text{C}}_{66}\text{(z,T)=}\frac{E(\mathrm{z},\mathrm{T})}{2(1+\upnu (\mathrm{z},\mathrm{T}))}.$$

(22)

Thermal Analysis

In this study, the effect of a thermal environment on the free vibration behavior of the cantilever functionally graded plate which carrying a point mass is also investigated. In here the temperature distribution is considered as a nonlinear distribution that can be obtained by solving a steady-state heat transfer equation. The temperature distribution through the thickness is as follow [16],

$$ - \frac{{\text{d}}}{{{\text{dz}}}}\left[ {{\upkappa }\left( {\text{z}} \right)\frac{{{\text{dT}}}}{{{\text{dz}}}}} \right] = 0,{\text{ T}} = {\text{T}}_{{\text{m}}} {\text{ at z}} = - {\text{h}}/2;{ }\,{\text{ T}} = {\text{T}}_{{\text{c}}} {\text{ at z}} = {\text{h}}/2, $$

(23)

The solution of Eq. (23) is obtained using a polynomial series [16] as follow,

$$\mathrm{T}\left(\mathrm{z}\right)={\mathrm{T}}_{0}+\mathrm{\Delta T\xi }\left(\mathrm{z}\right),$$

(24)

$$ \xi \left( {\text{z}} \right) = \frac{1}{\delta }\left[ \begin{gathered} \left( {\frac{{2{\text{z}} + {\text{h}}}}{{2{\text{h}}}}} \right) - \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)}}{{\left( {{\text{p}} + 1} \right)\kappa _{{\text{m}}} }}\left( {\frac{{2{\text{z}} + {\text{h}}}}{{2{\text{h}}}}} \right)^{{{\text{p}} + 1}} + \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)^{2} }}{{\left( {2{\text{p}} + 1} \right)\kappa _{{\text{m}}} ^{2} }}\left( {\frac{{2{\text{z}} + {\text{h}}}}{{2{\text{h}}}}} \right)^{{2{\text{p}} + 1}} \hfill \\ - \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)^{3} }}{{\left( {3{\text{p}} + 1} \right)\kappa _{{\text{m}}} ^{3} }}\left( {\frac{{2{\text{z}} + {\text{h}}}}{{2{\text{h}}}}} \right)^{{3{\text{p}} + 1}} + \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)^{4} }}{{\left( {4{\text{p}} + 1} \right)\kappa _{{\text{m}}} ^{4} }}\left( {\frac{{2{\text{z}} + {\text{h}}}}{{2{\text{h}}}}} \right)^{{4{\text{p}} + 1}} - \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)^{5} }}{{\left( {5{\text{p}} + 1} \right)\kappa _{{\text{m}}} ^{5} }}\left( {\frac{{2{\text{z}} + {\text{h}}}}{{2{\text{h}}}}} \right)^{{5{\text{p}} + 1}} \hfill \\ \end{gathered} \right], $$

(25)

$$\begin{gathered} \delta = 1 - \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)}}{{\left( {{\text{p}} + 1} \right)\kappa _{{\text{m}}} }} + \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)^{2} }}{{\left( {2{\text{p}} + 1} \right)\kappa _{{\text{m}}} ^{2} }} - \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)^{3} }}{{\left( {3{\text{p}} + 1} \right)\kappa _{{\text{m}}} ^{3} }} \hfill \\ \,\,\, + \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)^{4} }}{{\left( {4{\text{p}} + 1} \right)\kappa _{{\text{m}}} ^{4} }} - \frac{{\left( {\kappa _{{\text{c}}} - \kappa _{{\text{m}}} } \right)^{5} }}{{\left( {5{\text{p}} + 1} \right)\kappa _{{\text{m}}} ^{5} }}, \hfill \\ \end{gathered} $$

(26)

In the comparison results, the uniform temperature distribution given as follow were used.

$$T={T}_{0}+\Delta T,$$

(27)

T₀ in Eqs. (24) and (27) is the room temperature with a value of 300 K and ΔT represents the temperature change. At nonlinear temperature distribution, the full metal bottom surface of the plate is assumed to be at T₀ temperature.

Thermal Stresses Based on Three-Dimensional Elasticity

The plate is initially stress-free at temperature T₀ and thermal stresses occur in the plate with temperature change. The initial stresses due to a temperature change of ΔT(z) are defined for a functionally graded plate as:

$${\upsigma }_{i}^{\mathrm{T}}=-\left({\mathrm{C}}_{11}\left(\mathrm{z},\mathrm{T}\right)\mathrm{\alpha }\left(\mathrm{z},\mathrm{T}\right)+{\mathrm{C}}_{12}\left(\mathrm{z},\mathrm{T}\right)\mathrm{\alpha }\left(\mathrm{z},\mathrm{T}\right)\right)\mathrm{\Delta T}\left(\mathrm{z}\right)\left(\mathrm{i}\hspace{0.17em}=\hspace{0.17em}\mathrm{x},\mathrm{y}\right),$$

(28)

Three-Dimensional Ritz Solution in Thermal Environment

The linear elastic strain potential energy U_s of the plate can be given as,

$$ {\text{U}}_{{\text{s}}} = \frac{1}{2}\int\limits_{{\text{V}}}^{{}} {\left[ {{\upsigma }_{{\text{x}}} {\upvarepsilon }_{{\text{x}}} + {\upsigma }_{y} {\upvarepsilon }_{y} + {\upsigma }_{{\text{z}}} {\upvarepsilon }_{z} + {\uptau }_{{{\text{yz}}}} {\upgamma }_{{{\text{yz}}}} + {\uptau }_{{{\text{xz}}}} {\upgamma }_{{{\text{xz}}}} + {\uptau }_{{{\text{xy}}}} {\upgamma }_{{{\text{xy}}}} } \right]{\text{dV}},} $$

(29)

The strain energy U_T from the initial stresses due to temperature rise can be given from Kim [17] as follow,

$$ {\text{U}}_{{\text{T}}} = \frac{1}{2}\int\limits_{{\text{V}}}^{{}} {\left[ {{\upsigma }_{{\text{x}}}^{{\text{T}}} {\text{d}}_{{{\text{xx}}}} + 2{\uptau }_{{{\text{xy}}}}^{{\text{T}}} {\text{d}}_{{{\text{xy}}}} + {\upsigma }_{{\text{y}}}^{{\text{T}}} {\text{d}}_{{{\text{yy}}}} } \right]{\text{dV}},} $$

(30)

$$ {\text{d}}_{{{\text{ij}}}} = {\text{u}}_{{,{\text{i}}}} {\text{u}}_{{,{\text{j}}}} + {\text{v}}_{{,{\text{i}}}} {\text{v}}_{{,{\text{j}}}} + {\text{w}}_{{,{\text{i}}}} {\text{w}}_{{,{\text{j}}}} \left( {{\text{i}},{\text{j}}\, = \,{\text{x}},{\text{y}}} \right). $$

(31)

The kinetic energy T_p of the plate can be given as:

$$ {\text{T}}_{{\text{p}}} = \frac{1}{2}\int\limits_{{\text{V}}}^{{}} {{\uprho }\left( {{\text{z}},{\text{T}}} \right)\left[ {\left( {\frac{{\partial {\text{u}}}}{{\partial {\text{t}}}}} \right)^{2} + \left( {\frac{{\partial {\text{v}}}}{{\partial {\text{t}}}}} \right)^{2} + \left( {\frac{{\partial {\text{w}}}}{{\partial {\text{t}}}}} \right)^{2} } \right]{\text{dV}}} + \frac{1}{2}\mathop \sum \limits_{{{\text{p}} = 1}}^{{\text{N}}} {\text{m}}_{{\text{p}}} \left( {\frac{{\partial {\text{w}}}}{{\partial {\text{t}}}}} \right)^{2} , $$

(32)

The nondimensionalization process is performed using the following nondimensionalized parameters:

$$ X = 2x/a;\, \, Y = 2y/b;\, \, Z = 2z/h, $$

(33)

According to thermal vibration problem the maximum energy functional Π of the elastic plate is defined as:

$$\Pi =\left({\mathrm{U}}_{\mathrm{smax}}+{\mathrm{U}}_{\mathrm{Tmax}}\right)-{\mathrm{T}}_{\mathrm{max}},$$

(34)

In here; U_smax is maximum of the nondimensionalized linear elastic strain potential energy, U_Tmax is maximum of the nondimensionalized thermal strain potential energy and T_max is maximum of the nondimensionalized kinetic energy and are obtained as follow:

$$ {\text{U}}_{{{\text{smax}}}} = \frac{1}{2}\int\limits_{{\text{V}}}^{{}} {\left[ {{\text{C}}_{11} \left( {{\upvarepsilon }_{{\text{x}}}^{2} + {\upvarepsilon }_{{\text{y}}}^{2} + {\upvarepsilon }_{{\text{z}}}^{2} } \right) + 2{\text{C}}_{12} \left( {{\upvarepsilon }_{{\text{x}}} {\upvarepsilon }_{{\text{y}}} + {\upvarepsilon }_{{\text{x}}} {\upvarepsilon }_{{\text{z}}} + {\upvarepsilon }_{{\text{y}}} {\upvarepsilon }_{{\text{z}}} } \right) + {\text{C}}_{66} \left( {{\upgamma }_{{{\text{yz}}}}^{2} + {\upgamma }_{{{\text{xz}}}}^{2} + {\upgamma }_{{{\text{xy}}}}^{2} } \right)} \right]{\text{dV}},} $$

(35)

$$ {\text{U}}_{{{\text{Tmax}}}} = - \frac{1}{2}\int\limits_{{\text{V}}}^{{}} {\left( {{\text{C}}_{11} + {\text{C}}_{12} } \right){\upalpha }\left( {{\text{z}},{\text{T}}} \right)\Delta {\text{T}}\left( {\text{z}} \right)\left[ {\left( {\frac{{\partial {\text{U}}}}{{\partial {\text{x}}}}} \right)^{2} + \left( {\frac{{\partial {\text{V}}}}{{\partial {\text{x}}}}} \right)^{2} + \left( {\frac{{\partial {\text{W}}}}{{\partial {\text{x}}}}} \right)^{2} + \left( {\frac{{\partial {\text{U}}}}{{\partial {\text{y}}}}} \right)^{2} + \left( {\frac{{\partial {\text{V}}}}{{\partial {\text{y}}}}} \right)^{2} + \left( {\frac{{\partial {\text{W}}}}{{\partial {\text{y}}}}} \right)^{2} } \right]{\text{dV}},} $$

(36)

$$ {\text{T}}_{{{\text{max}}}} = \frac{{{\upomega }^{2} }}{2}\int\limits_{{\text{V}}}^{{}} {{\uprho }\left( {{\text{z}},{\text{T}}} \right)\left[ {{\text{U}}^{2} + {\text{V}}^{2} + {\text{W}}^{2} } \right]{\text{dV}} + \frac{{{\upomega }^{2} }}{2}\mathop \sum \limits_{{{\text{p}} = 1}}^{{\text{N}}} {\text{m}}_{{\text{p}}} {\text{W}}^{2} .} $$

(37)

In this study, the Chebyshev polynomials are preferred which are the orthogonal polynomials reduced the computational effort [32]. In accordance with the Ritz method, each of the displacement amplitude functions is written as a triple series of Chebyshev polynomials, the displacement component of which is multiplied by a boundary function that satisfies the geometric boundary conditions of the plate. The displacement components are written in terms of nondimensionalized coordinates

$$\mathrm{U}\left(\mathrm{X},\mathrm{Y},\mathrm{Z}\right)={\mathrm{F}}_{\mathrm{u}}\left(\mathrm{X},\mathrm{Y}\right)\sum_{\mathrm{i}=1}^{\infty }\sum_{\mathrm{j}=1}^{\infty }\sum_{\mathrm{k}=1}^{\infty }{\mathrm{A}}_{\mathrm{ijk}}{\mathrm{P}}_{\mathrm{i}}\left(\mathrm{X}\right){\mathrm{P}}_{\mathrm{j}}\left(\mathrm{Y}\right){\mathrm{P}}_{\mathrm{k}}\left(\mathrm{Z}\right),$$

(38)

$$\mathrm{V}\left(\mathrm{X},\mathrm{Y},\mathrm{Z}\right)={\mathrm{F}}_{\mathrm{v}}\left(\mathrm{X},\mathrm{Y}\right)\sum_{\mathrm{l}=1}^{\infty }\sum_{\mathrm{m}=1}^{\infty }\sum_{\mathrm{n}=1}^{\infty }{\mathrm{B}}_{\mathrm{lmn}}{\mathrm{P}}_{\mathrm{l}}\left(\mathrm{X}\right){\mathrm{P}}_{\mathrm{m}}\left(\mathrm{Y}\right){\mathrm{P}}_{\mathrm{n}}\left(\mathrm{Z}\right),$$

(39)

$$\mathrm{W}\left(\mathrm{X},\mathrm{Y},\mathrm{Z}\right)={\mathrm{F}}_{\mathrm{w}}\left(\mathrm{X},\mathrm{Y}\right)\sum_{\mathrm{p}=1}^{\infty }\sum_{\mathrm{q}=1}^{\infty }\sum_{\mathrm{r}=1}^{\infty }{\mathrm{C}}_{\mathrm{pqr}}{\mathrm{P}}_{\mathrm{p}}\left(\mathrm{X}\right){\mathrm{P}}_{\mathrm{q}}\left(\mathrm{Y}\right){\mathrm{P}}_{\mathrm{r}}\left(\mathrm{Z}\right),$$

(40)

where ${\mathrm{P}}_{\mathrm{s}}\left(\upzeta \right)=\mathrm{cos}\left[\left(\mathrm{s}-1\right)\mathrm{arccos}\left(\upzeta \right)\right]$ (s = 1,2,3,…; ζ = X,Y,Z) is the sth order one-dimensional Chebyshev polynomial and ${\mathrm{F}}_{\mathrm{\alpha }}\left(\mathrm{X},\mathrm{Y}\right)={\mathrm{f}}_{\mathrm{\alpha }}^{1}\left(\mathrm{X},\mathrm{Y}\right){\mathrm{f}}_{\mathrm{\alpha }}^{2}\left(\mathrm{X},\mathrm{Y}\right)$ (α = U, V, W) is the boundary function satisfying the geometric boundary conditions, are as follows in terms of nondimensionalized coordinates and Chebyshev polynomials. The boundary functions used for boundary condition in this study are given in the Table 1.

Table 1 Boundary functions for considered boundary conditions

In accordance with the Ritz method, by substituting the displacement components given by Eq. (38)–(40) at the maximum energy values and substituting the maximum energy values in the maximum energy functional given by Eq. (34)–(37), the energy functional Π is obtained in terms of Chebyshev polynomials. Then the energy functional Π is minimized according to the unknown coefficients A_ijk, B_lmn and C_pqr.

$$ \frac{\partial \prod }{{\partial {\text{A}}_{{{\text{ijk}}}} }} = 0, $$

(41)

$$ \frac{\partial \prod }{{\partial {\text{B}}_{{{\text{lmn}}}} }} = 0, $$

(42)

$$ \frac{\partial \prod }{{\partial {\text{C}}_{{{\text{pqr}}}} }} = 0. $$

(43)

As a result of the Ritz procedure, the eigenvalue problem given below is obtained, and the solution of the system of equations gives the natural frequencies of the free vibration problem occurring in the thermal environment under the influence of temperature.

$$\left(\left[\begin{array}{ccc}\left[{\mathrm{K}}_{\mathrm{uu}}\right]& \left[{\mathrm{K}}_{\mathrm{uv}}\right]& \left[{\mathrm{K}}_{\mathrm{uw}}\right]\\ {\left[{\mathrm{K}}_{\mathrm{uv}}\right]}^{\mathrm{T}}& \left[{\mathrm{K}}_{\mathrm{vv}}\right]& \left[{\mathrm{K}}_{\mathrm{vw}}\right]\\ { \left[{\mathrm{K}}_{\mathrm{uw}}\right]}^{\mathrm{T}}& {\left[{\mathrm{K}}_{\mathrm{vw}}\right]}^{\mathrm{T}}& \left[{\mathrm{K}}_{\mathrm{ww}}\right]\end{array}\right]-{\Omega }^{2}\left[\begin{array}{ccc}\left[{\mathrm{M}}_{\mathrm{uu}}\right]& 0& 0\\ 0 & \left[{\mathrm{M}}_{\mathrm{vv}}\right]& 0\\ 0& 0& \left[{\mathrm{M}}_{\mathrm{ww}}\right]\end{array}\right]\right)\left\{\begin{array}{c}\left\{{\mathrm{A}}_{\mathrm{ijk}}\right\}\\ \left\{{\mathrm{B}}_{\mathrm{lmn}}\right\}\\ \left\{{\mathrm{C}}_{\mathrm{pqr}}\right\}\end{array}\right\}=\left\{\begin{array}{c}\left\{0\right\}\\ \left\{0\right\}\\ \left\{0\right\}\end{array}\right\},$$

(44)

where [K_ij] and [M_ij] (i,j = u,v,w) are the stiffness matrix and diagonal mass matrix, respectively. The dimensionless coefficients {A_ijk}, {B_lmn} and {C_pqr} corresponding to the eigenvectors in the eigenvalue problem, represent the amplitude. Also, Ω is the nondimensional frequnecy parameter and obtained as:

$$\Omega =\upomega \left({\mathrm{a}}^{2}/\mathrm{h}\right)\sqrt{{\uprho }_{\mathrm{m}0}/{\mathrm{E}}_{\mathrm{m}0}},$$

(45)

Here ω is the natural frequency and ρ_m0 and E_m0 are mass density per unit volume and Young modulus of metal at room temperature (T₀ = 300 K).

Numerical Results

In this study, silicon nitride (Si3N4) as ceramic phase and stainless steel (SUS304) as metal phase are chosen to be the constituent materials of the functionally graded plate. Mechanical properties of constituent materials are temperature dependent as follows:

$$\mathrm{P}\left(\mathrm{T}\right)={\mathrm{P}}_{0}\left({\mathrm{P}}_{-1}{\mathrm{T}}^{-1}+1+{\mathrm{P}}_{1}\mathrm{T}+{\mathrm{P}}_{2}{\mathrm{T}}^{2}+{\mathrm{P}}_{3}{\mathrm{T}}^{3}\right),$$

(46)

where P₀ is the material property of ceramic and metal at temperature T₀, respectively. P₀ and the coefficients of material properties depending on temperature, P_i (i = 0,1,2,3) are given in Table 2 [17, 18].

Table 2 Coefficients of the temperature dependent material properties of constituents of FGM

Convergence and Accuracy Studies

In this study, the natural frequencies are obtained by Ritz method. The number of terms of the Chebyshev polynomials used as admissible functions is decided by comparing the results for isotropic plate results which presented by Gorman [33] in the case of without mass and which presented by Chiba and Sugimoto [25] and Aydoğdu and Filiz [28] in the case of with mass. As the results were found to be in good agreement, it was concluded that it would be appropriate to determine the term number of Chebyshev polynomials in all directions equal number as 8 × 8 × 8 as seen in Table 3. However, computational optimization could be obtained using unequal number of series terms in all directions.

Table 3 Convergence and comparison of first five frequency parameters of cantilever isotropic plate without mass (a/h = 100, ΔT = 0, υ = 0.333)

Tables 4 and 5 show that the results for the clamped square functionally grade plates subjected the uniform temperature distribution and nonlinear temperature distribution, respectively are in good agreement as with presented by Kim [17] and Li et al. [18].

Table 4 Comparison of natural frequency parameters of clamped square functionally graded plates subjected to uniform temperature rise (p = 2, a/h = 10, a = 0.2)

Table 5 Comparison of natural frequency parameters of clamped functionally graded plates subjected to non-linear temperature rise (h/b = 0.05, a/b = 1, ΔT = 300 K)

Parametric Studies

The object of this study is determined to effects of the mass ratio and location of the point mass on frequency parameters that moderately thick (a/h = 10) functionally graded plates with and without thermal environment. In the results the values of side to side ratios are a/b = 1; 1.5 and 2, volume fraction exponents are p = 1, 2, 5 and 10 and temperature differences are ΔT = 0, 300 and 500 K are considered. The temperature distribution is considered as in the form of nonlinear. The results are presented as for with a point mass that mass ratio M = 0.1, 0.2, 0.5 and 1 and without mass. The location coordinates of point mass are presented as (X,Y) = (− 0.5,0); (0,0); (0.5,0); (1,0) in the tables. And to see the case of different locations according to y axis the location coordinates of point mass are presented as (X,Y) = (0,0); (1,0); (–0 .5,0.5); (0.5,− 0.5) in the mod shape figures. The location of the point mass was determined according to whether the coordinates were on the symmetry axis or not according to the boundary condition considered. Here, the (1,0) and (0,0) coordinates are above the symmetry axis according to the considered boundary condition. And also the (0,0) coordinates are correspond the geometric center of the plate. The (− 0.5,0.5) ve (0.5,− 0.5) coordinates are not above the axis that is symmetrical with respect to the considered boundary condition. Thus, it provides a better understanding of the effects of the point mass whether on the symmetry axis or not on the frequency parameters.

Tables 6, 7 and 8 presents the frequency parameters obtained for the cases of with and without mass depending on the volume fraction exponent p and temperature differences for a/b = 1, 1.5 and 2, respectively. For the cases with point mass, the location of point mass is considered as (1,0). According to these results free vibration frequencies decreases with increasing p index and temperature difference. The frequencies obtained according to the with point mass cases are always smaller than the frequency values obtained for the without point mass case. One can be seen that from Tables 6, 7 and 8, when the a/b = 1, the second, the fourth and the sixth frequencies remain constant as independent of the presence of mass. At the values of a/b = 1.5 and 2, it is seen that the second frequencies remain constant as independent from the presence of mass. It means that the mass does not move during these vibrations because of the mass is at a nodal line. However, when the a/b = 1,5 and 2, it is seen that at high frequencies, intermediate frequencies occur in the case of with mass. As a result of this, in the case of without mass the third and the fifth frequencies, respectively, appears to be equivalent in the case of with mass to the fourth and sixth frequencies. It means that the case of with point mass the nodal lines are displaced.

Table 6 Frequency parameters of cantilever functionally graded plates with various temperature differences (a/h = 10, a/b = 1)

Table 7 Frequency parameters of cantilever functionally graded plates with various temperature differences (a/h = 10, a/b = 1.5)

Table 8 Frequency parameters of cantilever functionally graded plates with various temperature differences (a/h = 10, a/b = 2)

Tables 9, 10 and 11 presents the frequency parameters obtained for in the case of for different location of the point mass throughout the x axis for different mass rate from 0.1 to 1, for a/b = 1, 1.5 and 2, respectively. It is seen that when the point mass is located at symmetry axis according the boundary condition which correspond Y = 0 and throughout the x axis; some frequencies remain constant independent of the increasing of mass ratios because of the mass is at a nodal line and the mass dos not move during these vibrations.

Table 9 Frequency parameters of cantilever functionally graded plates with various mass ratios and various mass locations (p = 2, a/h = 10, ΔT = 300, a/b = 1)

Table 10 Frequency parameters of cantilever functionally graded plates with various mass ratios and various mass locations (p = 2, a/h = 10, ΔT = 300, a/b = 1.5)

Table 11 Frequency parameters of cantilever functionally graded plates with various mass ratios and various mass locations (p = 2, a/h = 10, ΔT = 300, a/b = 2)

Variation of fundamental frequency parameter with volume fraction index for various temperature differences for cantilever functionally graded plate with point mass on plate center are given in Fig. 2. Generally the effect of temperature on frequency parameter higher at higher volume fraction index. Also, The effect of volume fraction index on frequency parameter higher decreasing with increasing of volume fraction index.

Fig. 2

Variation of fundamental frequency parameter with temperature differences for cantilever functionally graded plate with point mass (a/h = 10, (X,Y) = (0,0))

Effect of the location of point mass throguhout x axis for cantilever functionally graded plate for different point mass ratio and for various temperature differences on fundamental frequency parameter and on third frequency parameter are given in Figs. 3 and 4, respectively. For all considered mass ratios, the fundamental frequency remains almost unchanged when the point mass is near the clamped edge, while it decreases substantially as the point mass approaches the free edge. Finally, the fundamental frequency reaches its minimum value at the free edge for all considered mass ratios, temperature differences, p values and a/b ratios. The decreasing on fundamental frequency parameters are greater with increasing mass ratios. It can be seen that the change of the third mode at different mass ratios along the x-axis is in the waveform for a/b = 1. The frequency parameters reaches a maximum value on a point which shifts gradually outward with increasing mass ratios. For a/b = 2 ratio, it is seen that the change in the position of the point mass along x axis, the effect on the frequency remains constant locally and has a sharp effect on the frequency change at some critical points. It is seen that the frequency values obtained by considering the temperature effect are always lower than the frequency values obtained at room temperature, and the change always remains parallel to each other.

Fig. 3

Variation of fundamental frequency parameter with the location of point mass throguhout x axis for cantilever functionally graded plate with point mass (a/h = 10, Y = 0)

Fig. 4

Variation of third frequency parameter with the location of point mass throguhout x axis for cantilever functionally graded plate with point mass (a/h = 10, Y = 0)

The mod shapes for the first six frequencies of plate for p value is 2, side to side ratio a/b is 1.5 and mass ratio is 1 are presented with Figs. 5, 6, 7, 8 and 9. These figures are presented for in the case of without mass and drawn for in the case of with mass four different locations of point mass and the conditions where the temperature difference is 0 and 500 K, respectively. Whether the cases where the point mass is on the axis of symmetry, the mode shapes maintain their symmetry with respect to the axis. It is seen that the nodal lines changing at the points where the mass is outside the symmetry axis and some modes shifting. On the other hand, it is seen that the temperature to cause waves to change phase in some modes.

Fig. 5

First six mode shapes of cantilever functionally graded plate without mass (p = 2, a/b = 1.5, a/h = 10)

Fig. 6

First six mode shapes of cantilever functionally graded plate with point mass (p = 2, a/b = 1.5, a/h = 10, M = 1, (X,Y) = (0,0))

Fig. 7

First six mode shapes of cantilever functionally graded plate with point mass (p = 2, a/b = 1.5, a/h = 10, M = 1, (X,Y) = (1,0))

Fig. 8

First six mode shapes of cantilever functionally graded plate with point mass (p = 2, a/b = 1.5, a/h = 10, M = 1, (X,Y) = (− 0.5,0.5))

Fig. 9

First six mode shapes of cantilever functionally graded plate with point mass (p = 2, a/b = 1.5, a/h = 10, M = 1, (X,Y) = (0.5,− 0.5))

Conclusions

The free vibration has been performed on a cantilever functionally graded plate carrying a point mass on an arbitrary point using Ritz method based on three-dimensional elasticity under nonlinear temperature distribution. The effective material properties are estimated by Mori–Tanaka homogenization scheme. The following conclusions can be reached as a result of the analysis.

• Choosing the terms number of Chebyshev polynomials which used as admissible functions in Ritz method as 8 × 8 × 8, the results can be obtained in good agreement with literature and consitent.

• Free vibration frequencies decreases with increasing p index and increasing temperature difference.

• The frequencies obtained for in the case of with point mass are always smaller than the frequency values obtained for in the case of without point mass.

• When the point mass is located on a nodal line, the mass does not move during these and so frequency remain constant as independent of the presence of mass. It is understood from this that if the mass is located on the nodal lines, at these frequencies it will resonate at the natural frequency of the unperturbed plate: e.g. for a/b = 1, the second, the fourth and the sixth frequencies remain constant; for a/b = 1.5 and 2, the second mode remains constant.

• When a/b = 1.5 and 2, at high frequencies, intermediate frequencies occur in the case of with mass: e.g. it is seen that in the case of without mass the third and the fifth frequencies, respectively, equivalent in the case of with mass to the fourth and sixth frequencies.

• The p value and temperature are more effective at increasing a/b values in the case of with mass and accordingly the nodal lines are displaced.

• An increase in temperature always causes a decrease in the frequency value, regardless of the mass ratio and the position of the point mass, while this decrease becomes more temperature sensitive with increasing mass ratio.

• Increasing of the p value causes the frequency value to decrease regularly, in all mass ratio which considered and the location of the point mass.

• Temperature affects the vibration behavior in the form of the change of direction of the waves in some modes or the displacement of the mode shapes at some frequencies.

• The presence of point mass affects vibrational behavior in some modes by changing the direction of the waves or changing their mode shape or increasing the number of waves at some frequencies. In addition, in cases where the point mass is on the axis of symmetry, the vibration behavior occurs symmetrically with respect to the axis, while this symmetry is broken at points where the mass is outside the axis of symmetry.