y = 0 , y = b (2b)

2.2. Fields of Displacement

In the classical Kirchhoff model, the normal remains straight and is perpendicular to the average surface after deformation. Kirchhoff’s field of displacement is then written:

$\left\{\begin{array}{l}u\left(x,y,t\right)={u}_{0}\left(x,y\right)+z{\beta }_{x}\left(x,y,t\right)\\ v\left(x,y,t\right)={v}_{0}\left(x,y\right)+z{\beta }_{y}\left(x,y,t\right)\\ w\left(x,y,t\right)={w}_{0}\left(x,y\right)\end{array}$ (3)

Figure 1. Simply supported rectangular plate and caractéristics.

${u}_{0}$ and ${v}_{0}$ are displacements of the membrane in x and y direction and w the displacement according to oz or displacement of bending. On the other hand, the rotations are given by:

${\beta }_{x}={\theta }_{y}=-\frac{\partial w}{\partial x}$ ; (4a)

${\beta }_{y}=-{\theta }_{x}=-\frac{\partial w}{\partial y}$ ; (4b)

${\beta }_{z}=0=\frac{\partial w}{\partial z}$ ; (4c)

Which allows to write:

$\left\{\begin{array}{l}u\left(x,y,t\right)={u}_{0}\left(x,y\right)-z\frac{\partial w}{\partial x}\\ v\left(x,y,t\right)={v}_{0}\left(x,y\right)-z\frac{\partial w}{\partial y}\\ w\left(x,y,t\right)={w}_{0}\left(x,y\right)\end{array}$ (5)

2.3. Deformation

The state of deformation of a plate can be considered as the state of superposition of membrane deformations and flexural deformations. In small deformation, we know that:

${\epsilon }_{xx}=-z\frac{{\partial }^{2}w}{\partial {x}^{2}}$, ${\epsilon }_{yy}=-z\frac{{\partial }^{2}w}{\partial {y}^{2}}$, ${\gamma }_{xy}=-z\frac{{\partial }^{2}w}{\partial x\partial y}$ (6)

${\epsilon }_{zz}={\gamma }_{xz}={\gamma }_{xz}=0$

What causes these constraints:

$\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\tau }_{xy}\end{array}\right]=\frac{E}{1-{\nu }^{2}}\left[\begin{array}{ccc}1& v& 0\\ v& 1& 0\\ 0& 0& \frac{1-v}{2}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\epsilon }_{xy}\end{array}\right\}$ (7)

2.4. Expression of Bending Moments

The bending moments per unit length in the plate Mx, My and Mxy are obtained by integrating the constraints on the thickness of the plate.

$M={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\sigma z\text{d}z$ (8)

2.5. Definition of Efforts

Let Nx, Ny and Nxy be these efforts, Hooke allows to write: $N=\sigma S$. Proceeding by integration, we have:

$N={\int }_{-\frac{h}{2}}^{\frac{h}{2}}\sigma \text{d}z$ (9)

2.6. Expression of Deformation and Kinetic Energy

The deformation energy of the plate is calculated by integration on the volume of the deformation energy density.

$u={\int }_{v}\sigma \epsilon \text{d}v={\int }_{v}{\epsilon }_{ij}{c}_{ijkl}{\epsilon }_{kl}\text{d}v$ (10)

The kinetic energy T is calculated by integration of the volume element:

$T=\frac{1}{2}{\int }_{s}m{\stackrel{˙}{w}}^{2}\text{d}s$ (11)

2.7. Movement Equation

The vertical equilibrium of the plate element (dx, dy) gives rise to the relation:

$\sum \stackrel{\to }{Fext}=m\stackrel{\to }{a}$ (12)

We obtain the differential equation of Lagrange which is a partial derivative equation verified by the vertical displacement:

$D\left[\frac{{\partial }^{4}w}{\partial {x}^{4}}+\frac{{\partial }^{4}w}{\partial {y}^{4}}+2\frac{{\partial }^{4}w}{\partial {x}^{2}\partial {y}^{2}}\right]-p\left(x,y\right)+m\stackrel{¨}{w}=0$ (13)

Such vibrations are called free or natural transverse vibrations. As previously stated, the natural vibrations depend only on the properties of the material and the geometry of the plate, and are inherent properties of the elastic plate regardless of any charge. Thus, in the case of free or natural vibration (harmonic movement external load $p\left(x,y\right)$ is equal to zero), the only transverse forces acting on the plate are the inertial forces due to mass ρ per unit area, the equation above becomes:

$D{\nabla }^{4}w+\rho h\frac{{\partial }^{2}w\left(x,y,t\right)}{{\partial }^{2}t}=0$ (14)

Classical equation of plate theory, which for most technical applications is sufficient for the study of bending problems.

$D=\frac{E{h}^{3}}{12\left(1-{v}^{2}\right)}$ : bending stiffness;

h: thickness of the plate;

E: Young’s modulus;

v: coefficient of poisson;

$w\left(x,y,t\right)$ : transverse displacement must satisfy the conditions at the limits of fixity;

3. Analytical Study of the Rectangular Plate

The solution to this equation is obtained by looking for the transverse displacement $w\left(x,y,t\right)$ such that:

$w\left(x,y,t\right)=X\left(x\right)Y\left(y\right)T\left(t\right)=w\left(x,y\right){\text{e}}^{j\omega t}$ (15)

Which is a solution of the form function.

$X\left(x\right)Y\left(y\right)=w\left(x,y\right)$ describes the modes of vibration and some harmonic function of a time, $\omega$ is the natural frequency of the vibration of the plate which is related to the frequency and period of the vibration by the relation:

$\omega =2\pi f=\frac{2\pi }{T}$ (16)

By introducing these elements, Equation (14) becomes:

$\begin{array}{l}{X}^{\left(4\right)}\left(x\right)Y\left(y\right)T\left(t\right)+2{X}^{\left(2\right)}\left(x\right){Y}^{\left(2\right)}\left(y\right)T\left(t\right)+X\left(x\right){Y}^{\left(4\right)}\left(y\right)T\left(t\right)\\ =-\frac{\rho h}{D}\cdot X\left(x\right)Y\left(y\right){T}^{\left(2\right)}\left(t\right)\end{array}$ (17)

We put: $\frac{D}{\rho h}={\mu }^{2}$, and we consider unknown $\beta$

$\frac{{X}^{\left(4\right)}\left(x\right)}{X\left(x\right)}+2\frac{{X}^{\left(2\right)}\left(x\right)}{X\left(x\right)}\frac{{Y}^{\left(2\right)}\left(y\right)}{Y\left(y\right)}+\frac{{Y}^{\left(4\right)}\left(y\right)}{Y\left(y\right)}=-\frac{1}{{\mu }^{2}}\frac{{T}^{\left(2\right)}\left(t\right)}{T\left(t\right)}={\beta }^{4}$ (18a)

Which results in:

$\left\{\begin{array}{l}{T}^{\left(2\right)}\left(t\right)+{\mu }^{2}{\beta }^{4}T\left(t\right)=0\\ \frac{{X}^{\left(4\right)}\left(x\right)}{X\left(x\right)}+2\frac{{X}^{\left(2\right)}\left(x\right)}{X\left(x\right)}\frac{{Y}^{\left(2\right)}\left(y\right)}{Y\left(y\right)}+\frac{{Y}^{\left(4\right)}\left(y\right)}{Y\left(y\right)}-{\beta }^{4}=0\end{array}$ (18b)

Of these two equations, it follows from the equation of complete displacement of the plate simply supported on two opposite edges:

$w\left(x,y,t\right)={\sum }_{m=1}^{\infty }{\sum }_{n=1}^{\infty }\left[{A}_{mn}\mathrm{cos}{\omega }_{mn}t+{B}_{mn}\mathrm{sin}{\omega }_{mn}t\right]{\varphi }_{mn}\left(x,y\right)$ (19)

The coefficients ${A}_{mn}$ and ${B}_{mn}$ depending on the load and/or initial conditions. However, the eigen modes are given by:

${X}_{m}\left(x\right){Y}_{n}\left(y\right)={A}_{mn}\cdot {\varphi }_{mn}\left(x,y\right)$ (20)

$w\left(x,y,t\right)=0$, represents the modal lines The form function can be taken as:

$\begin{array}{c}w\left(x,y\right)={\sum }_{m=1}^{\infty }{\sum }_{n=1}^{\infty }{C}_{mn}\cdot {\varphi }_{mn}\left(x,y\right)\\ ={\sum }_{m=1}^{\infty }{\sum }_{n=1}^{\infty }{C}_{mn}\cdot \mathrm{sin}\frac{m\pi }{{l}_{x}}x\cdot \mathrm{sin}\frac{n\pi }{{l}_{y}}y\end{array}$ (21a)

${\varphi }_{mn}\left(x,y\right)=\mathrm{sin}\frac{m\pi }{{l}_{x}}x\cdot \mathrm{sin}\frac{n\pi }{{l}_{y}}y$ represents the modal deformed, the own deformed, of the nm mode satisfying the supported boundary conditions, a and b are respectively the length and the width of the plate, ${C}_{mn}$ are the modal coefficients, corresponding to the projection of the motion in the modal base. This is the vibration amplitude for each value of m and n.

By replacing the expression

$w\left(x,y\right)={\sum }_{m=1}^{\infty }{\sum }_{n=1}^{\infty }{A}_{mn}\mathrm{sin}\frac{m\pi }{{l}_{x}}x\cdot \mathrm{sin}\frac{n\pi }{{l}_{y}}y$ (21b)

In the main Equation (14), we obtain:

$\frac{{m}^{4}{\pi }^{4}}{{a}^{4}}+2\frac{{m}^{2}{\pi }^{2}}{{a}^{2}}\frac{{n}^{2}{\pi }^{2}}{{b}^{2}}+\frac{{n}^{4}{\pi }^{4}}{{b}^{4}}-{\omega }^{2}\frac{\rho h}{D}=0$ (22)

The resolution of this equation leads to a natural pulse ${\omega }_{mn}$ of the mode m and n such that:

${\omega }_{mn}=\pi \sqrt{\frac{D}{\rho h}}.\left[{\left(\frac{n}{{l}_{y}}\right)}^{2}+{\left(\frac{m}{{l}_{x}}\right)}^{2}\right]$ (23)

Eigen frequency: ${f}_{mn}=\frac{{\omega }_{mn}}{2\pi }$

In the case of the plate under study, we have:

$D=\frac{E{h}^{3}}{12\left(1-{v}^{2}\right)}=146520.15\text{\hspace{0.17em}}\text{N}\cdot {\text{m}}^{2}$,

The analysis of the first two modes gives us:

· if $m=1,n=1$, ${\omega }_{11}=\frac{{\pi }^{2}}{{L}^{2}}\left[{1}^{2}+{1}^{2}\right]\sqrt{\frac{D}{\rho h}}=\frac{{\pi }^{2}}{{L}^{2}}\left[2\right]\sqrt{\frac{D}{\rho h}}$, ${\omega }_{11}=603.02\text{\hspace{0.17em}}\text{rad}/\text{s}$ and ${f}_{11}=\frac{{\omega }_{11}}{2\pi }=95.97\text{\hspace{0.17em}}\text{Hz}$

· if $m=1,n=2$, ${\omega }_{21}={\omega }_{12}=\frac{{\pi }^{2}}{{L}^{2}}\left[5\right]\sqrt{\frac{D}{\rho h}}=1507.53\text{\hspace{0.17em}}\text{rad}/\text{s}$ and ${f}_{21}={f}_{12}=\frac{{\omega }_{21}}{2\pi }=239.93\text{\hspace{0.17em}}\text{Hz}$

The analysis of natural pulsations by the Rayleigh technique confirms the veracity of these results especially for the first vibratory mode.

4. Modeling of the Dynamic Behavior of Plates by FEM in Ansys

The finite element method is a widely used and powerful tool for the analysis of complex structures. It consists of a discretization of the element into a finite number of generally triangular or rectangular elements . This resolution technique involves the calculation of the matrix of mass and rigidity of the whole structure:

$\left[K\right]-{\omega }^{2}\left[M\right]=0$ (24)

Equation (24) will be solved to give the eigenfrequencies and the eigen mode or modal deformations of the structure using the ANSYS software for a simply supported plate.

Let us always consider our rectangular plate whose characteristics are defined in section 1 and represented under Ansys in Figure 2. The plate is modeled on a volume of 2, e + 007 mm3 and has a mass of 157 kg.

The dimensions of the plate illustrated in Figure 2 and Figure 3 are given in section 2. The simulations are made on the plate having a referential temperature of 22˚C and discretized in 2401 elements for 17,300 nodes as seen in Figure 3. The bar graph indicates the specific frequency of each calculated mode and whose values are given in Table 1 after modal analysis of the structure.

It is noted that the frequencies grow with vibratory modes as illustrated in Figure 4 and Figure 5. On the other hand the displacements of the plate as shown in Figure 6 do not follow this law, they oscillate around 6 mm.

Figure 2. Plate geometry under Ansys.

Figure 3. Mesh of the plate.

Table 1. Frequency values for different plate modes.

Figure 4. Frequency evolution.

Figure 5. Specific frequency of each mode.

Figure 6. Modal displacement of the plate.

As shown in Figure 6, the maximum displacement is observed when m = 3, n = 1 and is equal to 7.548 mm.

These frequencies are geometrically translated by the following modes of Figure 7 (from (a) to (l)):

Figure 7. Vibratory modes of a plate simply supported on two opposite edges (from (a) to (l)). (a) mode 1: m = 1, n = 1; (b) mode 2: m = 2, n = 1; (c) mode 3: m = 2, n = 1; (d) mode 4: m = 1, n = 2; (e) mode 5: m = 2, n = 2; (f) mode 6: m = 3, n = 1; (g) mode 7: m = 3, n = 2; (h) mode 8: m = 1, n = 3; (i) mode 9: m = 3, n = 2; (j) mode 10: m = 2, n = 3; (k) mode 11: m = 3’, n = 1; (l) mode 12: m = 3, n = 3.

5. Comparison of the Eigenfrequencies Analytic and Those Obtained by Finite Elements

Table 1 and Table 2 show the values of the eigenfrequencies obtained by two methods considered, analytical and finite element structure calculation, corresponding to the rectangular configuration. From these two tables and in accordance with Figure 8 and Figure 9, we use Equation (25) to calculate the error of the modes m = 1, n = 1; m = 1,n = 2;m = 2, n = 2; m = 3’, n = 1 and m = 3, n = 3:

$\epsilon \left(%\right)=\frac{fS-fC}{fC}$ (25)

Table 2. Comparison of the eigenfrequencies analytic and those obtained by finite elements.

Figure 8. Comparison between MEF and calculated frequency.

Figure 9. Error graph.

The error between the two methods use ın thıs case study is minimized and converges to zero according to the error graph illustrated in Figure 9.

6. Conclusion

In this paper, we highlight the dynamic analysis of a plate simply supported on two opposite edges in free vibration. Two techniques have been deployed to approach the fundamental eigenfrequencies of the plate to be studied. It is the method of separable variables based on the modeling of the transversal displacement from the characteristic functions of vibrations of the thin plates and the analysis by the method of the finite elements. The responses obtained analytically and numerically from the nonlinear equations developed for the calculation of the eigenvalues of the plate show us that the frequencies increase with the modal deformations. The conditions of fixity and their location impose a great influence on the behavior of the plate structure in vibration. A comparison between the frequencies calculated and those obtained by Ansys for the rectangular plates reveals a convergence of the two calculation techniques.

Cite this paper
Nkounhawa, P. , Ndapeu, D. , Kenmeugne, B. and Beda, T. (2020) Analysis of the Behavior of a Square Plate in Free Vibration by FEM in Ansys. World Journal of Mechanics, 10, 11-25. doi: 10.4236/wjm.2020.102002.
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