Introduction {#s0005}
============

In recent years, differential quadrature method (DQM) has become increasingly popular in the numerical solution of initial and boundary value problems. The advantages of this method lie in its easy use and flexibility with regard to arbitrary grid spacing. Also, DQM method can yield accurate results with relatively much fewer grid points compared with the previous numerical techniques such as the finite element and finite difference methods. The present work aims to realize the ability of DQM to solve two complicated problems. The first one concerns with the free vibration of elastically supported cracked beams and the second problem concerns with the free vibration of irregular plates.

In general, there are two approaches to analyze the free vibration of the cracked beams. The first one employees the variational principles through a continuous model, see for example, those in [@b0005; @b0010]. The second approach employees the line spring models (LSMs) to simulate existence of the cracks. Shen and Pierre [@b0015] analyzed the free vibration of beams with pairs of symmetric open cracks. Yokoyama and Chen [@b0020] examined the vibration characteristics of a Bernoulli--Euler beam with an edge crack. Qian et al. [@b0025] explained the dynamic behavior and crack detection of a beam with a crack by using the finite element method. Gudmundson [@b0030; @b0035] discussed the dynamic model for beams with cross sectional crack and predicted the changes in resonance frequencies of a structure resulting from the crack. Rizos et al. [@b0040] analyzed the cracked structures by measuring the modal characteristics. All of these works [@b0015; @b0020; @b0025; @b0030; @b0035; @b0040] concerned with the cracked beams made of an isotropic materials.

Also, there are several publications concerned with the vibration analysis of plates. Leissa [@b0045] derived exact solutions for the free vibration problems of the rectangular plates. Xiang et al. [@b0050] used Ritz method to analyze the vibration of rectangular Mindlin plates resting on elastic edge supports. More recently, DQM is extensively applied for solving vibration problems. Bert and Malik [@b0055] introduced a review on the early stages of the method development and its applications. Also, they [@b0060] made the first attempt to apply DQM for vibration analysis of irregular plates. Liew et al. [@b0065] and Han and Liew [@b0070] also used a similar approach to analyze irregular quadrilateral thick plates. Lam [@b0075] introduced a mapping technique to apply the DQ method to conduction, torsion, and heat flow problems with arbitrary geometries.

Functionally graded materials (FGMs), a novel class of macroscopically nonhomogeneous composites with spatially continuous material properties, have attracted considerable research efforts over the past few years due to their increasing applications in many engineering fields. Numerous studies have been conducted on FGM beams and plates, dealing with a variety of subjects such as thermal elasticity [@b0080; @b0085], fracture mechanics [@b0090; @b0095], and vibration analysis [@b0100; @b0105; @b0110; @b0115; @b0120; @b0125].

The present work aims to extend the applications of DQM to solve two difficult problems. The first one concerns with the free vibration of an elastically supported cracked beam. The beam is made of a FGM and rests on a Winkler--Pasternak foundation. The line spring model is employed to simulate the crack actions. In the second problem, the DQM with a mapping technique are applied to analyze the free vibration of irregular plates. The obtained results are agreed with the previous similar ones. Further, a parametric study is introduced to explain the effects of elastic and geometric characteristics of the problem on the values of natural frequencies.

**Methodology** {#s0010}
===============

Free vibration analysis of cracked beams {#s0015}
----------------------------------------

Consider an elastically cantilever beam of length *L* and thickness *h*, containing an edge crack of depth *a* located at a distance *L*~1~ from the left end, as shown in [Fig. 1](#f0005){ref-type="fig"}. The beam is made of a FGM, such that shear modulus, Young's modulus, and mass density of the beam vary in the thickness direction only as follows [@b0140]:$$\mu(z) = \mu_{0}e^{\alpha z}\text{,}\quad E(z) = E_{0}e^{\alpha z}\text{,}\quad\rho(z) = \rho_{0}e^{\alpha z}\text{,}$$where *μ*~0~, *E*~0~*,* and *ρ*~0~ are shear modulus, Young's modulus, and mass density at the mid-plane, (*z* = 0), of the beam. *α* is a constant characterizing the beam material grading, *α* = ln (*E*~2~/*E*~1~)/*h*, where *E*~1~ and *E*~2~ are the values of Young's modulus at the lower and upper beam surfaces, respectively.

It is assumed that the crack is always open and its surfaces are free of traction, such that the beam can be treated as a two sub-beams connected by an elastic rotational spring at the cracked section which has no mass and no length. The bending stiffness of the cracked section, *K~T~*, is related to the flexibility *G* by:$$K_{T} = \frac{1}{G}$$Referring to Broek's approximation [@b0130], one can find that flexibility is governed by:$$\frac{\mathit{dG}}{\mathit{da}} = \frac{2w(1 - v^{2})K_{I}^{2}}{E(z)M_{1}^{2}}\text{,}$$where *v* is Poisson's ratio. *M*~1~ is bending moment at the cracked section. *K*~I~ is mode I stress intensity factor, which can be obtained as a special case of the results introduced by Erdogan and Wu [@b0135].

The governing equations, for the prescribed cracked FGM beam, can be written as [@b0140]:$$A_{11}\frac{\partial^{2}u_{i}}{\partial X^{2}} - B_{11}\frac{\partial^{3}w_{i}}{\partial X^{3}} = 0\quad(i = 1\text{,}2)\text{,}$$$$\left( {D_{11} - \frac{B_{11}^{2}}{A_{11}}} \right)\frac{\partial^{4}w_{i}}{\partial X^{4}} + \mathit{Kw}_{i} - k\frac{\partial^{2}w_{i}}{\partial X^{2}} + I_{1}\frac{\partial^{2}w_{i}}{\partial t^{2}} = 0\text{,}\quad(i = 1\text{,}2)\text{,}$$where the subscript *i* = 1 stands for the left sub-beam in ($0 \leqslant X \leqslant L_{1}$), while *i* = 2 holds true for the right sub-beam occupying ($L_{1} \leqslant X \leqslant L$), see [Fig. 1](#f0005){ref-type="fig"}.

*w~i~*, *u~i~* are the components of displacement vector in the directions of *z* and *x*, respectively. *t* is time. *K*, *k* are elastic and shear modules of the foundation reaction, respectively.$$(A_{11}\text{,}B_{11}\text{,}D_{11}) = \int_{- h/2}^{h/2}\frac{E(z)}{1 - v^{2}}(1\text{,}z\text{,}z^{2})\mathit{dz}\text{,}\quad I_{1} = \int_{- h/2}^{h/2}\rho(z)\mathit{dz}$$

The normal force ${\bar{N}}_{i}(X\text{,}t)$, bending moment ${\bar{M}}_{i}(X\text{,}t)$, and transverse shear force ${\bar{Q}}_{i}(X\text{,}t)$ are related to the displacement components as follows [@b0140]:$${\bar{N}}_{i}(X\text{,}t) = A_{11}\frac{\partial u_{i}}{\partial X} - B_{11}\frac{\partial^{2}w_{i}}{\partial X^{2}}\text{,}\quad(i = 1\text{,}2)\text{,}$$$${\bar{M}}_{i}(X\text{,}t) = B_{11}\frac{\partial u_{i}}{\partial X} - D_{11}\frac{\partial^{2}w_{i}}{\partial X^{2}}\text{,}\quad(i = 1\text{,}2)\text{,}$$$${\bar{Q}}_{i}(X\text{,}t) = B_{11}\frac{\partial^{2}u_{i}}{\partial X^{2}} - D_{11}\frac{\partial^{3}w_{i}}{\partial X^{3}}\text{,}\quad(i = 1\text{,}2)\text{.}$$Let *x* be a normalized parameter defined as:$$x = \left\{ \begin{array}{ll}
{X/L_{1}} & {i = 1} \\
{(X - L_{1})/L_{2}} & {i = 2} \\
\end{array} \right)\text{,}\quad L_{2} = L - L_{1}\text{.}$$

Also for free vibration analysis, let the prescribed field quantities can be expressed as:$$\begin{array}{ll}
{u_{i}(x\text{,}t) = U_{i}(x)\sin\omega t\text{,}} & {w_{i}(x\text{,}t) = W_{i}(x)\sin\omega t\text{,}} \\
{{\bar{N}}_{i}(x\text{,}t) = N_{i}(x)\sin\omega t\text{,}} & {{\bar{M}}_{i}(x\text{,}t) = M_{i}(x)\sin\omega t\text{,}} \\
{{\bar{Q}}_{i}(x\text{,}t) = Q_{i}(x)\sin\omega t\text{,}} & {(i = 1\text{,}2)\text{.}} \\
\end{array}$$where *ω* is the natural frequency of the cracked FG beam.

Then, by substituting from Eq. [(10)](#e0035){ref-type="disp-formula"} into Eqs. [(4), (5)](#e0160 e0165){ref-type="disp-formula"}, and [(7)--(9)](#e0170 e0175 e0180){ref-type="disp-formula"}, the problem can be reduced to a quasi-static one as follows:$$\frac{A_{11}}{L_{i}^{2}}\frac{d^{2}U_{i}}{\mathit{dx}^{2}} - \frac{B_{11}}{L_{i}^{3}}\frac{d^{3}W_{i}}{\mathit{dx}^{3}} = 0\text{,}\quad(i = 1\text{,}2)\text{,}$$$$\left( {D_{11} - \frac{B_{11}^{2}}{A_{11}}} \right)\frac{1}{L_{i}^{4}}\frac{d^{4}W_{i}}{\mathit{dx}^{4}} + \mathit{KW}_{i} - k\frac{1}{L_{i}^{2}}\frac{d^{2}W_{i}}{\mathit{dx}^{2}} = \omega^{2}I_{1}W_{i}\text{,}\quad(i = 1\text{,}2)\text{,}$$

The boundary conditions (at the clamped and free ends), can be written as:$$U_{1}(0) = W_{1}(0) = \left( \frac{\mathit{dW}_{1}}{\mathit{dx}} \right|_{x = 0} = 0\text{,}$$$$N_{2}(1) = Q_{2}(1) = M_{2}(1) = 0\text{,}$$

The boundary conditions, (at the cracked section), can be described through a line spring model as follows [@b0140]:$$U_{1}(1) = U_{2}(0)\text{,}\quad W_{1}(1) = W_{2}(0)\text{,}$$$$N_{1}(1) = N_{2}(0)\text{,}\quad M_{1}(1) = M_{2}(0)\quad Q_{1}(1) = Q_{2}(0)\text{,}$$$$\frac{1}{L_{1}}\frac{\mathit{dW}_{1}}{\mathit{dx}} - \frac{1}{L_{2}}\frac{\mathit{dW}_{2}}{\mathit{dx}} = \frac{1}{K_{T}}M_{1}(1)\text{,}$$

Differential quadrature solution {#s0020}
--------------------------------

The method of DQ requires to descretize the domain of the problem into *N* points. Then the derivatives at any point are approximated by a weighted linear summation of all the functional values along the descretized domain, as follows [@b0145; @b0150; @b0155; @b0160]:$$\left( \frac{d^{m}f(x)}{\mathit{dx}^{m}} \right|_{x = x_{i}} \approx \sum\limits_{j = 1}^{N}C_{\mathit{ij}}^{m}f(x_{j})\quad(i = 1\text{,}N)\text{,}\mspace{6mu}(m = 1\text{,}M)\text{,}$$where *M* is the order of the highest derivative appearing in the problem.

*f*(*x~i~*) are the values of the function at the sampling points *x~i~*, (*i* = 1, *N*); *N* \> *M*. $C_{\mathit{ij}}^{m}$, (*i*, *j* = 1, *N*), are the weighting coefficients relating the *m*th derivative to the functional values at *x~i~*. These coefficients can be determined by making use of Lagrange interpolation formula as follows [@b0145; @b0150]:$$f(x) = \sum\limits_{i = 1}^{N}\frac{L(x)}{(x - x_{i})L_{1}(x_{i})}f(x_{i})\text{,}$$where $L(x) = \prod_{j = 1}^{N}(x - x_{j})$, $L_{1}(x_{i}) = \prod_{j = 1}^{N}(x_{i} - x_{j})$, (*i* = 1, *N*).

On substitution from Eq. [(19)](#e0045){ref-type="disp-formula"} into [(18)](#e0040){ref-type="disp-formula"}, the weighting coefficients relating the 1st order derivative to the functional values at *x~i~* can be obtained as [@b0145; @b0150]:$$C_{\mathit{ij}}^{1} = \left\{ \begin{array}{ll}
{\frac{L_{1}(x_{i})}{(x_{j} - x_{i})L_{1}(x_{j})}\text{,}} & {\text{when}\mspace{6mu}(i \neq j)} \\
{- \sum\limits_{j = 1\text{,}j \neq i}^{N}C_{\mathit{ij}}^{1}\text{,}} & {\text{when}\mspace{6mu}(i = j)} \\
\end{array} \right)\quad(i\text{,}j = 1\text{,}N)\text{.}$$

Further, $C_{\mathit{ij}}^{m}$ relating the higher order derivatives can be obtained as [@b0145; @b0150]:$$C_{\mathit{ij}}^{m} = \sum\limits_{k = 1}^{N}C_{\mathit{ik}}^{1}C_{\mathit{kj}}^{m - 1}\text{,}\quad(m = 1\text{,}M)\text{,}\mspace{6mu}(i\text{,}j = 1\text{,}N)$$

The accuracy of DQ results, is affected by choosing of the number, *N*, and type of the sampling points, *x~i~*. It is found that the optimal selection of the sampling points in vibration problems, is the normalized Gauss--Chebyshev--Lobatto points [@b0145; @b0150; @b0165]:$$x_{i} = \frac{1}{2}\left\lbrack {1 - \cos\left( {\frac{i - 1}{N - 1}\pi} \right)} \right\rbrack\text{,}\quad(i = 1\text{,}N)\text{.}$$

On suitable substitution from Eqs. [(18)--(22)](#e0040 e0045 e0050 e0055 e0060){ref-type="disp-formula"} into [(11)--(17)](#e0185 e0190 e0195 e0200 e0205 e0210 e0215){ref-type="disp-formula"}, one can reduce the problem to the following system of linear algebraic equations:$$A_{11}\sum\limits_{j = 1}^{N}C_{\mathit{ij}}^{2}U_{1}(x_{j}) - \frac{B_{11}}{L_{1}}\sum\limits_{j = 1}^{N}C_{\mathit{ij}}^{3}W_{1}(x_{j}) = 0\text{,}\quad(i = 1\text{,}N)\text{,}$$$$A_{11}\sum\limits_{j = 1}^{N}C_{\mathit{ij}}^{2}U_{2}(x_{j}) - \frac{B_{11}}{L_{2}}\sum\limits_{j = 1}^{N}C_{\mathit{ij}}^{3}W_{2}(x_{j}) = 0\text{,}\quad(i = 1\text{,}N)\text{,}$$$$\sum\limits_{j = 1}^{N}\left\lbrack {\frac{C_{\mathit{ij}}^{4}}{L_{1}^{4}} - \frac{A_{11}k}{L_{1}^{2}(A_{11}D_{11} - B_{11}^{2})}C_{\mathit{ij}}^{2}} \right\rbrack\mspace{2mu} W_{1}(x_{j}) = \lambda_{1}^{4}\mspace{2mu} W_{1}(x_{i})\text{,}\quad(i = 1\text{,}N)\text{,}$$$$\sum\limits_{j = 1}^{N}\left\lbrack {\frac{C_{\mathit{ij}}^{4}}{L_{2}^{4}} - \frac{A_{11}k}{L_{2}^{2}(A_{11}D_{11} - B_{11}^{2})}C_{\mathit{ij}}^{2}} \right\rbrack W_{2}(x_{j}) = \lambda_{1}^{4}\mspace{2mu} W_{2}(x_{i})\text{,}\quad(i = 1\text{,}N)\text{,}$$where$$\lambda_{1}^{4} = \frac{A_{11}(I_{1}\omega^{2} - K)}{A_{11}D_{11} - B_{11}^{2}}$$

The boundary conditions can be rewritten using the DQM as follows:

At the clamped end:$$U_{1}(x_{1}) = 0\text{,}\quad W_{1}(x_{1}) = 0\text{,}$$$$\sum\limits_{j = 1}^{N}C_{1j}^{1}W_{1}(x_{j}) = 0\text{.}$$

At the free end:$$A_{11}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{1}U_{2}(x_{j}) - \frac{B_{11}}{L_{2}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{2}W_{2}(x_{j}) = 0\text{.}$$$$B_{11}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{1}U_{2}(x_{j}) - \frac{D_{11}}{L_{2}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{2}W_{2}(x_{j}) = 0\text{.}$$$$B_{11}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{2}U_{2}(x_{j}) - \frac{D_{11}}{L_{2}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{3}W_{2}(x_{j}) = 0\text{.}$$

At the crack location:$$U_{1}(x_{N}) = U_{2}(x_{1})\text{,}\qquad W_{1}(x_{N}) = W_{2}(x_{1})\text{,}$$$$\frac{A_{11}}{L_{1}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{1}U_{1}(x_{j}) - \frac{B_{11}}{L_{1}^{2}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{2}W_{1}(x_{j}) = \frac{A_{11}}{L_{2}}\sum\limits_{j = 1}^{N}C_{1j}^{1}U_{2}(x_{j}) - \frac{B_{11}}{L_{2}^{2}}\sum\limits_{j = 1}^{N}C_{1j}^{2}W_{2}(x_{j})\text{,}$$$$\frac{B_{11}}{L_{1}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{1}U_{1}(x_{j}) - \frac{D_{11}}{L_{1}^{2}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{2}W_{1}(x_{j}) = \frac{B_{11}}{L_{2}}\sum\limits_{j = 1}^{N}C_{1j}^{1}U_{2}(x_{j}) - \frac{D_{11}}{L_{2}^{2}}\sum\limits_{j = 1}^{N}C_{1j}^{2}W_{2}(x_{j})\text{,}$$$$\frac{B_{11}}{L_{1}^{2}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{2}U_{1}(x_{j}) - \frac{D_{11}}{L_{1}^{3}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{3}W_{1}(x_{j}) = \frac{B_{11}}{L_{2}^{2}}\sum\limits_{j = 1}^{N}C_{1j}^{2}U_{2}(x_{j}) - \frac{D_{11}}{L_{2}^{3}}\sum\limits_{j = 1}^{N}C_{1j}^{3}W_{2}(x_{j})\text{,}$$$$\frac{1}{L_{1}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{1}W_{1}(x_{j}) - \frac{1}{L_{2}}\sum\limits_{j = 1}^{N}C_{1j}^{1}W_{2}(x_{j}) = \frac{1}{K_{T}}\left\lbrack {\frac{B_{11}}{L_{1}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{1}U_{1}(x_{j}) - \frac{D_{11}}{L_{1}^{2}}\sum\limits_{j = 1}^{N}C_{\mathit{Nj}}^{2}W_{1}(x_{j})} \right\rbrack\text{,}$$On suitable substitution from Eqs. [(28)--(37)](#e0090 e0095 e0220 e0225 e0230 e0235 e0240 e0245 e0250 e0255){ref-type="disp-formula"} into [(23)--(26)](#e0065 e0070 e0075 e0080){ref-type="disp-formula"} and after some lengthy but straight forward manipulations, the problem can be reduced to an eigenvalue one of dimensions (4*N*-12). A Matlab program has been designed to solve the problem, such that the natural frequencies and the mode shapes of the cracked beam can be obtained.

Free vibration analysis of irregular plates {#s0025}
-------------------------------------------

Consider a curvilinear quadrilateral plate in Cartesian *x*--*y* plane, see [Fig. 2](#f0010){ref-type="fig"}a. A geometric mapping can be applied to transform this irregular plate into a rectangular one in *ξ* − *η* plane, as in [Fig. 2](#f0010){ref-type="fig"}. The following blending function may be applied to do for this mapping [@b0060; @b0170; @b0175]:$$s = \frac{1}{2}\lbrack(1 - \eta){\overline{s}}_{1}(\xi) + (1 + \xi){\overline{s}}_{2}(\eta) + (1 + \eta){\overline{s}}_{3}(\xi) + (1 - \xi){\overline{s}}_{4}(\eta)\rbrack - \frac{1}{4}\left\lbrack {(1 - \xi)(1 - \eta)s_{1}} \right)\left( {+ (1 + \xi)(1 - \eta)s_{2} + (1 + \xi)(1 + \eta)s_{3} + (1 - \xi)(1 + \eta)s_{4}} \right\rbrack\text{,}$$where *s* = *x*, *y*. ${\overline{x}}_{i}(\xi)\text{,}{\overline{x}}_{i}(\eta)\text{,}{\overline{y}}_{i}(\xi)\text{,}{\overline{y}}_{i}(\eta)\text{,}\mspace{6mu}(i = 1\text{,}4)$, are the parametric forms of the curvilinear boundaries. *x~i~*, *y~i~*, (*i* = 1, 4), are the Cartesian coordinates of the corner points of the physical domain, as shown in [Fig. 2](#f0010){ref-type="fig"}a.

It is noted that Eq. [(38)](#e0100){ref-type="disp-formula"} achieves exact geometric transformation for the boundaries of the curvilinear quadrilateral domain [@b0060; @b0175]. To apply the DQM, one must descretize the computational domain to a grid of dimensions *N~ξ~* × *N~η~*, where *N~ξ~* and *N~η~* are the number of sampling points in the *ξ* and *η* directions, respectively. The first order partial derivatives at a sampling point (*ξ~i~*, *η~j~*), (*i* = 1, *N~ξ~*, and *j* = 1, *N~η~*), can be written as [@b0145; @b0150; @b0155; @b0160]:$$\left( \frac{\partial f}{\partial\xi} \right|\begin{matrix}
{\xi_{i}\text{,}\eta_{j}} \\
\end{matrix} = \sum\limits_{k = 1}^{N_{\xi}}P_{\mathit{ik}}f_{\mathit{kj}}\text{,}\quad\left( \frac{\partial f}{\partial\eta} \right|\begin{matrix}
{\xi_{i}\text{,}\eta_{j}} \\
\end{matrix} = \sum\limits_{l = 1}^{N_{\eta}}R_{\mathit{jl}}f_{\mathit{il}}\text{,}$$where *f~ij~* are the functional values at (*ξ~i~*, *η~j~*). *P~ik~* and *R~jl~* are the weighting coefficients that can also be determined by making use of Lagrange interpolation formula as in Eqs. [(18)--(22)](#e0040 e0045 e0050 e0055 e0060){ref-type="disp-formula"}. By the Chain rule of differentiation,$$\frac{\partial f}{\partial x} = \frac{1}{|J|}\left\lbrack {\frac{\partial y}{\partial\eta}\frac{\partial f}{\partial\xi} - \frac{\partial y}{\partial\xi}\frac{\partial f}{\partial\eta}} \right\rbrack\text{,}\quad\frac{\partial f}{\partial y} = \frac{1}{|J|}\left\lbrack {- \frac{\partial x}{\partial\eta}\frac{\partial f}{\partial\xi} + \frac{\partial x}{\partial\xi}\frac{\partial f}{\partial\eta}} \right\rbrack\text{,}$$where \|*J*\| is the determinant of the Jacobian *J* = ∂ (*x*, *y*)/ ∂ (*ξ*, *η*).

Then by substituting from Eqs. [(39)](#e0105){ref-type="disp-formula"} into [(40)](#e0110){ref-type="disp-formula"}, one can find the first order partial derivatives of the function *f*(*x*, *y*) at a point *x~ij~* = *x*(*ξ~i~*, *η~j~*), *y~ij~* = *y*(*ξ~i~*, *η~j~*) of the mapped domain as:$$\left( \frac{\partial f}{\partial x} \right|_{\mathit{ij}} = \frac{1}{|J_{\mathit{ij}}|}\left\lbrack {\left( \frac{\partial y}{\partial\eta} \right)_{\mathit{ij}}\sum\limits_{k = 1}^{N_{\xi}}P_{\mathit{ik}}f_{\mathit{kj}} - \left( \frac{\partial y}{\partial\xi} \right)_{\mathit{ij}}\sum\limits_{l = 1}^{N_{\eta}}R_{\mathit{jl}}f_{\mathit{il}}} \right\rbrack\text{,}\mspace{6mu}\left( \frac{\partial f}{\partial y} \right|_{\mathit{ij}} = \frac{1}{|J_{\mathit{ij}}|}\left\lbrack {- \left( \frac{\partial x}{\partial\eta} \right)_{\mathit{ij}}\sum\limits_{k = 1}^{N_{\xi}}P_{\mathit{ik}}f_{\mathit{kj}} + \left( \frac{\partial x}{\partial\xi} \right)_{\mathit{ij}}\sum\limits_{l = 1}^{N_{\eta}}R_{\mathit{jl}}f_{\mathit{il}}} \right\rbrack\text{,}$$that can be rewritten as [@b0060; @b0175]:$$\left( \frac{\partial f}{\partial x} \right|_{m} = \sum\limits_{n = 1}^{N_{\xi\eta}}G_{\mathit{mn}}^{(1)}f_{n}\text{,}\quad\left( \frac{\partial f}{\partial y} \right|_{m} = \sum\limits_{n = 1}^{N_{\xi\eta}}H_{\mathit{mn}}^{(1)}f_{n}\text{,}$$where $N_{\xi\eta} = N_{\xi} \times N_{\eta}$. $m\text{,}n = (i - 1)N_{\eta} + j$, $(i = 1\text{,}N_{\xi}\mspace{6mu}\text{and}\mspace{6mu} j = 1\text{,}N_{\eta})$.

$G_{\mathit{mn}}^{(1)}$ and $H_{\mathit{mn}}^{(1)}$ are the weighting coefficients of the first order partial derivatives with respect to *x* and *y*, respectively. Simillarly, one can find$$\left( \frac{\partial^{r}f}{\partial x^{r}} \right|_{m} = \sum\limits_{n = 1}^{N_{\xi\eta}}G_{\mathit{mn}}^{(r)}f_{n}\text{,}\quad\left( \frac{\partial^{s}f}{\partial y^{s}} \right|_{m} = \sum\limits_{n = 1}^{N_{\xi\eta}}H_{\mathit{mn}}^{(s)}f_{n}\text{,}\quad\left( \frac{\partial^{r + s}f}{\partial x^{r}\partial y^{s}} \right|_{m} = \sum\limits_{n = 1}^{N_{\xi\eta}}V_{\mathit{mn}}^{(\mathit{rs})}f_{n}\text{,}$$in which the weighting coefficients can be obtained as [@b0060; @b0175]:$$\begin{array}{ll}
{\lbrack G^{(r)}\rbrack = \lbrack G^{(1)}\rbrack\lbrack G^{(r - 1)}\rbrack\text{,}} & {r \geqslant 2} \\
{\lbrack H^{(s)}\rbrack = \lbrack H^{(1)}\rbrack\lbrack H^{(s - 1)}\rbrack\text{,}} & {s \geqslant 2} \\
{\lbrack V^{(\mathit{rs})}\rbrack = \lbrack G^{(r)}\rbrack\lbrack H^{(s)}\rbrack\text{,}} & {r\text{,}s \geqslant 1\text{.}} \\
\end{array}$$

The equation of motion governing the free harmonic vibration of a thin isotropic plate in a dimensionless form, can be written as [@b0165; @b0175]:$$\frac{\partial^{4}W}{\partial X^{4}} + 2\frac{\partial^{4}W}{\partial X^{2}\partial Y^{2}} + \frac{\partial^{4}W}{\partial Y^{4}} - \Omega^{2}W = 0\text{,}$$where *X*, *Y* = *x*/*a*, *y*/*a* are dimensionless Cartesian coordinates in the plane of the mid-surface of the plate. *W* = *W*(*X*, *Y*) is the mode function of plate deflection. $\Omega = \omega a^{2}\sqrt{\rho h/D}$ is the dimensionless frequency, where *ω* is a natural frequency, *a* is a characteristic plate dimension, *ρ* is the plate material density, *h* is the plate thickness, and *D* = *Eh*^3^/12(1 − *ν*^2^), is the flexural rigidity of the plate, *E* and *ν* being Young's modulus and Poisson's ratio of the plate material, respectively.

The boundary conditions, (at the clamped edges), can be written as:$$W = 0\text{,}\quad\cos\theta\frac{\partial W}{\partial X} + \sin\theta\frac{\partial W}{\partial Y} = 0\text{,}$$

The boundary conditions, (at simply supported edges), can be written as:$$W = 0\text{,}\quad(\cos^{2}\theta + \nu\sin^{2}\theta)\frac{\partial^{2}W}{\partial X^{2}} + (\sin^{2}\theta + \nu\cos^{2}\theta)\frac{\partial^{2}W}{\partial Y^{2}} + ((1 - \nu)\sin 2\theta)\frac{\partial^{2}W}{\partial X\partial Y} = 0\text{,}$$where *θ* is the angle between normal to the plate boundary and the *x*-axis. Using the quadrature rules in Eq. [(42)](#e0120){ref-type="disp-formula"}, the equation of motion can be reduced to$$\sum\limits_{n = 1}^{N_{\xi\eta}}\lbrack G_{\mathit{mn}}^{4} + 2V_{\mathit{mn}}^{22} + H_{\mathit{mn}}^{4}\rbrack W_{n} = \Omega^{2}W_{m}\text{,}$$

At the clamped edges, the boundary conditions reduces to$$W_{m} = 0\text{,}\quad\sum\limits_{n = 1}^{N_{\xi\eta}}\lbrack(\cos\theta_{m})G_{\mathit{mn}}^{1} + (\sin\theta_{m})H_{\mathit{mn}}^{1}\rbrack W_{n} = 0\text{,}$$At simply supported edges, the boundary conditions reduces to$$W_{m} = 0\text{,}\quad\sum\limits_{n = 1}^{N_{\xi\eta}}\lbrack(\cos^{2}\theta_{m} + \nu\sin^{2}\theta_{m})G_{\mathit{mn}}^{2} + (\sin^{2}\theta_{m} + \nu\cos^{2}\theta_{m})H_{\mathit{mn}}^{2} + ((1 - \nu)\sin 2\theta_{m})V_{\mathit{mn}}^{11}\rbrack.W_{n} = 0\text{,}$$

On suitable substitutions from Eqs. [(48) and (49)](#e0150 e0155){ref-type="disp-formula"} into [(47)](#e0145){ref-type="disp-formula"}, the problem be reduced to eigenvalue one of dimensions (*N~ξ~* − 4)(*N~η~* − 4). A Matlab program has been designed to solve the problem.

Results and discussions {#s0030}
=======================

Cracked beam results {#s0035}
--------------------

For practical purposes the values of the natural frequencies of the concerned cracked FG beam are divided by that of the un-cracked isotropic beam, such that one can define the fundamental frequency ratio as: *ω~C~*~1~/*ω*~1~, where *ω~C~*~1~ is the fundamental frequency of the cracked FG beam. *ω*~1~ is that of the un-cracked isotropic beam. A parametric study is introduced to investigate the effects of crack location, crack depth, Young's modulus gradation ratio, (*E*~2~/*E*~1~), and the foundation moduli, (*K*, *k*), on the values of the fundamental frequency ratio, (*ω~C~*~1~/*ω*~1~) and the mode shapes. Eqs. [(23)--(37)](#e0065 e0070 e0075 e0080 e0085 e0090 e0095 e0220 e0225 e0230 e0235 e0240 e0245 e0250 e0255){ref-type="disp-formula"} are solved with *N* = (6, 20), It is observed that the results for *N* = 10 are the same as those corresponding to *N* = (11, 20). Therefore, the present results are implemented with *N* = 10.

To examine the validity of the obtained results, the problem of a clamped -- free cracked beam made of an isotropic material is considered. It corresponds to a special case of the present analysis when *E*~1~ = *E*~2~ and *K* = *k* = 0. This problem was previously solved by Yokoyama and Chen [@b0020], Yang and Chen [@b0140]. [Table 1](#t0005){ref-type="table"} shows the agreement between the present results and the previous ones in [@b0020; @b0140].

Further, [Figs. 3--5](#f0015 f0020 f0025){ref-type="fig"} explain the effects of geometric and elastic characteristics of the problem on the values of fundamental frequency ratio and the mode shapes. [Fig. 3](#f0015){ref-type="fig"} show that the values of natural frequencies increase with increasing both of foundation elastic modulus (*K*) and the distance of crack site from the clamped end (*L*~1~/*L*). While, these values decrease with increasing of foundation shear modulus (*k*), the crack depth (*a*/*h*), and the gradation of Young's modulus, (*E*~1~/*E*~2~) across the beam depth as shown. [Fig. 4](#f0020){ref-type="fig"}a and b shows the first three mode shapes of the instantaneous lateral and longitudinal displacements, (*w*(*L*~1~, *t*), *u*(*L*~1~, *t*)), at the crack tip when *L*~1~/*L* = 0.35. At the crack location, [Fig. 4](#f0020){ref-type="fig"}c shows that the values of instantaneous lateral fundamental amplitude increase with decreasing of the crack depth. [Fig. 5](#f0025){ref-type="fig"} report that existence of cracks affects on the local flexibility of the beam. Further, these figures show that the values of the lateral amplitude, (along the whole beam), increase with the decreasing both of foundation elastic modulus (*K*) and the distance of crack site from the clamped end (*L*~1~/*L*). While, these values are increased with increasing of Young's graded modulus, (*E*~2~/*E*~1~ ratio).

Irregular plate results {#s0040}
-----------------------

To examine the validity of DQ results, the free vibration problem of rectangular plate is considered. [Tables 2 and 3](#t0010 t0015){ref-type="table"} show the first five natural frequencies *ω* for different aspect ratios $\lambda = a/b = 2/5$, 2/3, 13/2, 5/2, where *a* and *b* are lengths of the rectangular plate. [Table 2](#t0010){ref-type="table"} also, shows the results corresponding to all of the plate edges are clamped (CCCC), while [Table 3](#t0015){ref-type="table"} shows the results corresponding to all of the plate edges are simply supported (SSSS). For a numerical scheme with *N~ξ~* = *N~η~* = 11, the DQ results agreed with the previous ones were obtained by Leissa [@b0045].

Further, DQM with a geometric mapping are applied to solve a free vibration problem of an irregular parabolic plate, as in [Fig. 6](#f0030){ref-type="fig"}. Referring to Eq. [(38)](#e0100){ref-type="disp-formula"}, one can find a blending function for such mapping as: $(x\text{,}y) = \left( {\frac{a(1 + \xi)}{2}\eta\sqrt{b^{2} - (b^{2} - c^{2})\frac{(1 + \xi)}{2}}} \right)$, $- 1 \leqslant \xi$, $\eta \leqslant 1$.

Furthermore, [Fig. 6](#f0030){ref-type="fig"} explain the effects of aspect ratio $\lambda = a/b$, vortex angle *θ* = 2 tan^−1^(2*ac*/(*b*^2^ − *c*^2^), and the flexural rigidity of the plate D on values of the fundamental frequency ratio $\omega_{1r}/\omega_{1}$, where *ω*~1~*~r~* is the fundamental frequency of the irregular plate while the *ω*~1~ is that of the rectangular one with *λ* = 1. [Fig. 6](#f0030){ref-type="fig"} show that (for such irregular parabolic plate), the values of *ω*~1~*~r~*/*ω*~1~ increase with increasing both of *λ* and *D*, while the converse is true with the vortex angle *θ*.

Conclusion {#s0045}
==========

The method of DQ is applied to analyze the free vibration of an elastically supported cracked beam. Also, the method of DQ with a geometric mapping are employed to solve the free vibration problem of an irregular plate. The new trends in this work are the method of solution (DQM), material of the beam (FGM), elastic foundation model (Winkler--Pasternak), and the irregular boundaries of the plate. So, this work can be considered as an extension for the applications of DQM. Further, the obtained results may be employed to detect, locate, and quantify the extent of the cracks or damages in FG beams.

![Cracked functionally graded beam on a Winkler--Pasternak foundation.](gr1){#f0005}

![Geometric mapping.](gr2){#f0010}

![Variation of the fundamental frequency ratio with elastic and geometric characteristics of the problem.](gr3){#f0015}

![Variation of the mode shapes with time and the crack depth.](gr4){#f0020}

![Fundamental amplitude variation with the crack location and elastic characteristics.](gr5){#f0025}

![Variation of the fundamental frequency of a clamped irregular parabolic plate with elastic and geometric characteristics.](gr6){#f0030}

###### 

Variation of the fundamental frequency ratio with the crack location. Case of a cracked cantilever made of an isotropic material: (*v *= 0.33, *L*/*h *= 4.0, *a*/*h *= 0.2).

        *ω~C~*~1~/*ω*~1~             
  ----- ------------------ --------- ----------
  0.2   0.94101            0.97833   0.978327
  0.4   0.96667            0.99107   0.991069
  0.6   0.99583            0.99776   0.997758

###### 

Natural frequencies of a clamped rectangular plate (*N~ξ~* = *N~η~* = 11)).

        Mode                                                                                      
  ----- --------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
  2/5   23.648    23.645    27.817    27.815    35.446    35.399    46.702    47.500    61.554    63.067
  2/3   27.010    27.007    41.716    41.706    66.143    66.108    66.552    66.354    79.850    79.831
  1.0   35.992    35.987    73.413    73.383    73.413    73.383    108.270   108.248   131.640   131.706
  3/2   60.772    60.764    93.860    93.840    148.820   148.743   149.740   149.296   179.660   179.619
  5/2   147.800   147.784   173.850   173.847   221.540   221.245   291.890   293.885   384.710   389.163

###### 

Natural frequencies of a simply supported rectangular plate (*N~ξ~* = *N~η~* = 11).

        Mode                                                                                             
  ----- --------- --------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
  2/5   11.4487   11.4487   16.1862    15.83103   24.0818    24.0813    35.1358    36.160     41.0576    41.0307
  2/3   14.2561   14.2561   27.4156    26.7598    43.8649    43.8395    49.3480    49.1034    57.0244    56.0732
  1.0   19.7361   19.7392   49.3480    49.325     49.3480    49.325     78.9568    78.3735    98.6960    98.6713
  3/2   32.0762   32.0762   61.6850    61.6674    98.6960    97.7441    111.0330   110.4827   128.3049   126.1394
  5/2   71.5564   71.5547   101.1634   101.1529   150.5115   150.1146   229.5987   230.1461   256.6097   255.6151
