Author: Ralf Alexander Roßlan

Supervisor: Bettina Chocholaty

This project deals with the implementation of the Rayleigh-Ritz method for the free vibration problem of a conservative thick orthotropic rectangular plate with unconstrained boundary conditions. Furthermore, the accuracy of the results is validated by comparison with the results obtained through simulations using the Finite Element Method as well as results retrieved through experiments of an orthotropic plate.

Introduction

The vibration analysis of plates has been a topic of research for several decades; granting numerous methods for the dynamic analysis problem of plated structural elements. This is maybe due to the fact that In today’s world plate systems are widely used in various engineering applications.

Two main theories dominate the world of plates, the Kirchhoff plate theory for thin plates and the Reissner-Mindlin plate theory for moderately thick plates. The underlying difference between the two theories is the consideration of the normal hypothesis. In the deformed configuration, by assuming the orthogonality of the normal vector to the mid-plane of the plate; the Kirchhoff plate neglects the effect of shear deformation. Through the assumption of a shear rigid plate, the theory yields, for thick plates, an over prediction of the natural frequencies and an under prediction of the deflections. Thus, the benefit of the Reissner-Mindlin theory emerges, where first order shear deformation is considered due to the treatment of the bending angle as an independent degree of freedom.


Figure 1: Reissner-Mindlin plate theory illustrating the decoupling of the degrees of freedom [5]	Figure 2: Kirchhoff plate theory illustrating the normal hypothesis [5]

In order to overcome the over prediction of the natural frequencies and the under prediction of the deflections the Rayleigh-Ritz method is used. Through this systematic method, developed by Lord Rayleigh and later extended by Walter Ritz, the free vibration analysis of an orthotropic plate is performed. Thereby, shear deformation as well as the rotary inertia of the plate are considered.

Theoretical formulations

Generally, for thin plates the appropriate theory is the Kirchhoff/Classical plate theory however as the plate thickness increases the shear deformation needs to be considered where the Reissner-Mindlin plate theory is the desired theory for the analysis of the structural element.

For the analysis of a Reissner-Mindlin plate the following are assumed:

Elastic material behaviour.
Normal stress $\begin{array}{l}\sigma_z_z\end{array}$ disappears at each point of the plate.
Cross-sections remain plane. That is, a straight line which was normal to the plate middle plane before deformation remains straight in the deformed state but is no longer normal to the middle plane.
The thickness of the plate does not change during the deformation process.
Small deformations.

Description of model

For the implementation of the Rayleigh-Ritz method a rectangular plate is studied. The figure below displays the geometrical parameters with $\begin{array}{l}a\end{array}$ and $\begin{array}{l}b\end{array}$ denoting the lengths in x- and y-direction, respectively as well as $\begin{array}{l}h\end{array}$ the thickness dimension in the z-direction. Furthermore, the unconstrained boundary condition of the plate is examined, i.e. the plate is free on all four edges.


Figure 3: Model of a thick plate with unconstrained boundary conditions and illustrating the positive axes

Kinematic relations

With in the Reissner-Mindlin theory there are three independent degrees of freedom. The deflection $\begin{array}{l}(w)\end{array}$ , the rotation around x $\begin{array}{l}(\psi_x)\end{array}$ and the rotation around y $\begin{array}{l}(\psi_y)\end{array}$ . The displacement fields in accordance with first order shear deformation are shown in equations (1) to (3) below.

(1)	$\begin{array}{l}\displaystyle u(x, y, z, t) = z \cdot \psi_x (x,y,t)\end{array}$

(2)	$\begin{array}{l}\displaystyle v(x, y, z, t) = z \cdot \psi_y (x,y,t)\end{array}$

(3)	$\begin{array}{l}\displaystyle w(x,y,z,t)=w(x,y,t)\end{array}$

The normal, shear and transverse shear strains can be expressed as follows:

(4)

$\begin{array}{l}\left[\begin{matrix}\varepsilon_x_x \\ \varepsilon_y_y \\ \gamma_x_y \\ \gamma_x_z \\ \gamma_y_z \end{matrix}\right] = \left[\begin{matrix} z\cdot \frac{\partial \psi_x}{\partial x} \\ z\cdot \frac{\partial \psi_y}{\partial y} \\ z\cdot (\frac{\partial \psi_x}{\partial y}+\frac{\partial \psi_y}{\partial x}) \\ \psi_x+\frac{\partial w}{\partial x} \\ \psi_y +\frac{\partial w}{\partial y} \end{matrix}\right]\end{array}$

Constitutive law

Considering an orthogonal anisotropic plate and assuming elastic material behaviour. The mirror symmetry with respect to the diagonal of the plate stiffness (5) can be expressed using Young's moduli $\begin{array}{l}(E_x\; and \;E_y)\end{array}$ and Poisson's ration $\begin{array}{l}(\nu_x\; and \;\nu_y)\end{array}$ in two different directions as well as shear moduli $\begin{array}{l}(G_{xy}\; , \;G_{xz}\; and \; G_{yz})\end{array}$ to compensate for shear in the system.

(5)

$\begin{array}{l}[D] =\left[\begin{matrix} \frac{E_x}{1-\nu_x\cdot\nu_y} & \frac{\nu_x\cdot E_x}{1-\nu_x\cdot\nu_y} & 0 & 0 & 0 \\\frac{\nu_y\cdot E_y}{1-\nu_x\cdot\nu_y} & \frac{ E_y}{1-\nu_x\cdot\nu_y} & 0 & 0 & 0 \\ 0 & 0 & G_x_y & 0 & 0 \\ 0 & 0 & 0 & \kappa \cdot G_x_z & 0 \\ 0 & 0 & 0 & 0 & \kappa \cdot G_y_z \end{matrix}\right]\end{array}$

Using the strain vector (4) as well as the before mentioned plate stiffness (5), the generalised Hooke's law can be defined according to the linear elasticity theory.

(6)

$\begin{array}{l}\left[\begin{matrix}\sigma_x_x \\ \sigma_y_y \\ \tau_x_y \\ \tau_x_z \\ \tau_y_z \end{matrix}\right] =\left[\begin{matrix} \frac{E_x}{1-\nu_x\cdot\nu_y} & \frac{\nu_x\cdot E_x}{1-\nu_x\cdot\nu_y} & 0 & 0 & 0 \\\frac{\nu_y\cdot E_y}{1-\nu_x\cdot\nu_y} & \frac{ E_y}{1-\nu_x\cdot\nu_y} & 0 & 0 & 0 \\ 0 & 0 & G_x_y & 0 & 0 \\ 0 & 0 & 0 & \kappa \cdot G_x_z & 0 \\ 0 & 0 & 0 & 0 & \kappa \cdot G_y_z \end{matrix}\right] \cdot \left[\begin{matrix}\varepsilon_x_x \\ \varepsilon_y_y \\ \gamma_x_y \\ \gamma_x_z \\ \gamma_y_z \end{matrix}\right]\end{array}$

As seen in the equations above, a shear correction factor $\begin{array}{l}(\kappa)\end{array}$ is applied to the transverse shear modulus. This is done to compensate for the the strain energy since the shear stresses at the free edges should be equal to zero.

Energy expressions

The Rayleigh-Ritz method is an approach that was developed by Lord Rayleigh and was then later extended by Walter Ritz. Lord Rayleigh derived a solution to estimate the circular natural frequency by adjusting the shape of a chosen function to find the lowest possible value for the frequency [3]. Through this, he developed what we no today as the Rayleigh quotient. A number of years later Walter Ritz took this a step further by performing the minimisation of the difference of the strain energy expression and the kinetic energy expression.

The linear elastic strain energy is defined as:

(7)	$\begin{array}{l}\displaystyle U = \int_{V} (\sigma_x_x\cdot\varepsilon_x_x + \sigma_y_y\cdot\varepsilon_y_y + \tau_x_y\cdot\gamma_x_y + \tau_x_z\cdot\gamma_x_z + \tau_y_z\cdot\gamma_y_z)\; dV\end{array}$

Substituting the strain vector (equation (4)) and the generalised Hooke's equation (6) into the the linear elastic strain energy expression (7) yields:

(8)

$\begin{array}{l}\displaystyle U=\int_{0}^{a}\int_{0}^{b} \int_{\frac{-h}{2}}^{\frac{h}{2}}[D_1_1\cdot(\frac{\partial \psi_x}{\partial x})^2 + D_2_2\cdot(\frac{\partial \psi_y}{\partial y})^2 + (D_1_2+D_2_1)\cdot\frac{\partial \psi_x}{\partial x}\cdot\frac{\partial \psi_y}{\partial y} + D_3_3\cdot(\frac{\partial \psi_x}{\partial y}+\frac{\partial \psi_y}{\partial x})^2 + D_4_4\cdot(\psi_x + \frac{\partial w}{\partial x})^2 + D_5_5\cdot(\psi_y + \frac{\partial w}{\partial y})^2 ]dx\;dy\; dz\end{array}$

where:

$\begin{array}{l}\displaystyle D_1_1= \frac{E_x}{1-\nu_x\cdot\nu_y},\;\;\; D_1_2=\frac{\nu_x\cdot E_x}{1-\nu_x\cdot\nu_y},\;\;\; D_2_1=\frac{\nu_y\cdot E_y}{1-\nu_x\cdot\nu_y},\;\;\; D_2_2=\frac{E_y}{1-\nu_x\cdot\nu_y},\;\;\; D_3_3=G_x_y,\;\;\; C_4_4=\kappa \cdot G_y_z,\;\;\; C_5_5=\kappa \cdot G_z_x,\;\;\;\end{array}$

With the consideration of the continuous plate and taking into account the rotary inertia of the mass density $\begin{array}{l}(\rho)\end{array}$ , the kinetic energy is defined in equation (9).

(9)	$\begin{array}{l}\displaystyle T = \int_{0}^{a}\int_{0}^{b} \int_{\frac{-h}{2}}^{\frac{h}{2}} \rho\cdot[(\frac{\partial w}{\partial t})^2+(z\cdot\frac{\partial \psi_x}{\partial t})^2 + (z\cdot\frac{\partial \psi_y}{\partial t})^2]\;dx\;dy\;dz\end{array}$

Shape functions

A very important requirement for the implementation of the Rayleigh-Ritz method is the choice of displacement functions and thereby the shape functions. In order to formulate the energy expressions defined above, the displacement functions should satisfy the boundary conditions of the element. Furthermore, it is required that the functions be continuous over the domain of the element and lastly the displacement functions must be complete, whereby they must contain all expansion coefficients. Thus, one can express them as a linear combination of the systems vibration modes. Due to these requirements lower polynomials cannot be used since it is required that at least two-order derivatives of the shape functions exist and are continuous at any point on the plate and the higher polynomials become numerically unstable.

(10)	$\begin{array}{l}\displaystyle \psi_x (x,y)= \sum_{m\mathop=-2}^{\infty}\sum_{n\mathop=-2}^{\infty}\; A_m_n\cdot \varphi_m(x)\cdot\varphi_n(y)\end{array}$

(11)	$\begin{array}{l}\displaystyle \psi_y (x,y)= \sum_{m\mathop=-2}^{\infty}\sum_{n\mathop=-2}^{\infty}\; B_m_n\cdot \varphi_m(x)\cdot\varphi_n(y)\end{array}$

(12)	$\begin{array}{l}\displaystyle w (x,y)= \sum_{m\mathop=-2}^{\infty}\sum_{n\mathop=-2}^{\infty}\; C_m_n\cdot \varphi_m(x)\cdot\varphi_n(y)\end{array}$

This leads to cosine and sine functions, $\begin{array}{l}\varphi_m\end{array}$ and $\begin{array}{l}\varphi_n\end{array}$ . With the use of these transcendental functions, one creates a modified Fourier series expansion weighted by the unknown coefficients in order to approximate the true natural vibration mode. For a better convergence the series starts from $\begin{array}{l}m \; = \; n=-2\end{array}$ in which the sine function is brought forth to start the linear combination. Interestingly, the use of cosine and sine functions with their dependency on the number of series is an arbitrary set up for arbitrary boundary conditions.

(13)	$\begin{array}{l}\varphi_m(x)=\begin{cases}cos(\frac{m\cdot\pi\cdot x}{a}) \; \;\;\;for \;m \ge 0 \\sin(\frac{m\cdot\pi\cdot x}{a}) \; \;\;\;for \;m < 0 \end{cases}\end{array}$

(14)	$\begin{array}{l}\varphi_n(y)=\begin{cases}cos(\frac{n\cdot\pi\cdot y}{b}) \; \;\;\;for \;n \ge 0 \\sin(\frac{n\cdot\pi\cdot y}{b}) \; \;\;\;for \;n < 0 \end{cases}\end{array}$

Solution procedure

Using the extension provide by Ritz. One can formulate the Lagrangian $\begin{array}{l}(L=T-U)\end{array}$ by calculating the difference between the kinetic energy (Equation (9)) and the strain energy (Equation (8)). Through minimising the Lagrangian with respect to the unknown expansion coefficients , the determination of the eigenvalues as well as the corresponding eigenvectors is fulfilled.

$\begin{array}{l}\displaystyle \frac{\partial L}{\partial \Phi}=0\end{array}$

Where the unknown expansion coefficients can be express as:

(15)

$\begin{array}{l}[\Phi] =\left[\begin{matrix} A_{-2}_{-2} & A_{-2}_{-1} & \cdot \cdot \cdot & A_{0}_{0} & \cdot \cdot \cdot & A_{m}_{n}\\B_{-2}_{-2} & B_{-2}_{-1} & \cdot \cdot \cdot & B_{0}_{0} & \cdot \cdot \cdot & B_{m}_{n}\\C_{-2}_{-2} & C_{-2}_{-1} & \cdot \cdot \cdot & C_{0}_{0} & \cdot \cdot \cdot & C_{m}_{n}\end{matrix}\right]^T\end{array}$

The minimisation procedure of the Lagrangian can be generalised as an eigenvalue problem expressed in matrix form below.

(16)	$\begin{array}{l}\displaystyle ([K]-\omega^2\cdot [M])\cdot[\Phi]=[0]\end{array}$

The the stiffness matrix and mass matrix used in the Eigenvalue problem (equation (16)) can be expressed as follows:

(17)

$\begin{array}{l}[K] = \left[\begin{matrix} \bm{K_{1,1}} & \bm{K_{1,2}} & \bm{K_{1,3}}\\ \bm{K_{2,1}} & \bm{K_{2,2}} & \bm{K_{2,3}} \\ \bm{K_{3,1}} & \bm{K_{3,2}} & \bm{K_{3,3}}\end{matrix}\right],\;\;\; [M] = \left[\begin{matrix} \bm{M_{1,1}} & \bm{M_{1,2}} & \bm{M_{1,3}}\\ \bm{M_{2,1}} & \bm{M_{2,2}} & \bm{M_{2,3}} \\ \bm{M_{3,1}} & \bm{M_{3,2}} & \bm{M_{3,3}}\end{matrix}\right]\end{array}$

Furthermore, a simplified way of determining the stiffness matrix is through the numerical basis of the differentiation of the strain energy expressed in equation (8) with respect to the unknown expansion coefficients (15). This is displayed in equation (18). The mass matrix follows the same procedure but the kinetic energy is differentiated. This implementation creates an efficient way for the determination of the matrices with the help of numeric programming.

(18)	$\begin{array}{l}\displaystyle [K] = \frac{\partial^{2} U}{\partial \phi_i\partial \phi_j} = \int\limits_V \bm{B^T}\cdot \bm{D}\cdot\bm{B}\;\; dV\end{array}$

The B-Matrix is the strain matrix, whereby the matrix is calculated by differentiating the displacement vector, containing the shape functions, with respect to the reference directions, this is equivalent to equation (4).

(19)	$\begin{array}{l}\displaystyle [M] = \frac{\partial^{2} T}{\partial \phi_i\partial \phi_j} = \int\limits_V \rho\cdot\bm{N^T}\cdot \bm{N}\;\; dV\end{array}$

The N-Matrix is the displacement matrix which is composed of equations (1) - (3).

MATLAB Implementation

In this section the procedure of the Rayleigh-Ritz approach for the analysis of a thick orthotropic plate is described using the equations presented in the previous chapters.

The implementation of the Rayleigh-Ritz approach for thick orthotropic plates is a combination of two MATLAB function files for the calculation of the mass and stiffness matrix (Figure 4) and the main file where the functions are called and used for the determination of the eigenvalues and eigenvectors. Lastly, with the implementation of the eigenvectors in the displacement and rotation functions modal shapes of the rectangular plate can be visualised.

Figure 4: Flow procedure of the MATLAB implementation

Numerical results and discussion

In this section, the implementation of the Rayleigh-Ritz method is validated according to literature, Finite element simulations and free vibration experiments performed on an orthotropic plate.

Validation according to literature

In order to assess the accuracy as well as the validity of the implementation, systematic comparison of the results with literature is performed. Reference is made to the paper [1] by Wang, Q. and Shi, D., where a similar approach was used for the free vibration analysis of a thick orthotropic plate. The task of the validation in this section, due to the similarity of the implementation, is to certify the functionality of the program.

The following geometrical parameters and material properties are give as follows:

Table 1: Geometrical parameters and material properties for the validation according to literature

Geometrical parameters:
Length ratio	$\begin{array}{l}\frac {a} {b} = 2\end{array}$
Thickness to length ratio	$\begin{array}{l}\frac {h} {b} = 0,2\end{array}$
Material properties:
Elasticity moduli	Open $\begin{array}{l}\frac {E_x} {E_y} = 2,\; 5,\; 10\end{array}$
Shear moduli	$\begin{array}{l}G_x_y = G_x_z = 0,6\cdot E_y\end{array}$ $\begin{array}{l}G_y_z = 0,5\cdot E_y\end{array}$
Poisson’s ratio	$\begin{array}{l}\nu_y=0,25\end{array}$ using Betti's law: $\begin{array}{l}ν_x = ν_y\cdot \frac{E_y}{E_x}\end{array}$
Mass density	$\begin{array}{l}ρ = 200\end{array}$
Shear correction factor	$\begin{array}{l}κ = \frac {π²} {12}\end{array}$

Studied here is the influence of the Elasticity modulus quotient for values of 2, 5 and 10.

For the comparison a frequency parameter is defined using the bending rigidity of the plate. The frequency parameter is defined as:

$\begin{array}{l}\displaystyle \Omega = (\omega\cdot \frac{ b^2}{\pi^2})\cdot \ \sqrt{\frac{\rho\cdot h}{D}}\end{array}$

where:

$\begin{array}{l}\displaystyle D = \frac{E_y\cdot h^3}{12\cdot(1-\nu_x\cdot\nu_y)}\end{array}$

Table 2: Comparison of the mode 4 - 11 frequency parameters $\begin{array}{l}\Omega\end{array}$ for a thick orthotropic rectangular plate with different elasticity quotients

		Frequency parameters Ω
$\begin{array}{l}\frac{E_x}{E_y}\end{array}$	Mode number →	4	5	6	7	8	9	10	11
2	Result	0,7483	0,7840	1,6820	1,9236	2,0071	2,4371	2,8296	3,3381
	Literature	0,7594	0,7831	1,6816	1,9522	2,0156	2,4301	2,8340	3,3522
	Difference	-0,0147	0,0012	0,0002	-0,0146	-0,0042	0,0029	-0,0015	-0,0042
5	Result	0,8004	1,1480	1,8853	2,0151	2,3797	2,8910	3,4264	3,4899
	Literature	0,7990	1,1780	1,8884	2,0233	2,4236	2,8849	3,4479	3,4648
	Difference	0,0018	-0,0255	-0,0016	-0,0040	-0,0181	0,0021	-0,0062	0,0072
10	Result	0,8080	1,5410	2,0345	2,1280	2,4135	3,6108	3,6158	3,9726
	Literature	0,8062	1,5998	2,0269	2,1354	2,4384	3,5595	3,6281	4,0114
	Difference	0,0022	-0,0368	0,0038	-0,0035	-0,0102	0,0144	-0,0034	-0,0097

As three degrees of freedom $\begin{array}{l}(\psi_x,\;\psi_y\;and\;w)\end{array}$ are considered it is to be expected that there are three rigid body motions. Therefore, the analysis starts from the most relevant mode, the fourth vibration mode.

The values in the table can be better illustrated through the use of the figures below.


Figure 5: Comparison of results with literature for different Elasticity quotients for a thick orthotropic plate

The calculation follows for $\begin{array}{l}m \; = \; n=8\end{array}$ . Truncating the number of series at 8 shows a maximum difference discrepancy of 0,0368 for the 5th vibration mode with the elasticity quotient of 10. By increasing series parameter a nearer approximation can be achieved. As seen in Table 2 and Figure 5, as the elasticity quotient increases so does the frequency parameter increase. This is to be expected due to the proportionality of the frequency and the stiffness. Validating the implementation according to the paper by Wang et al. shows that the variation between the results is near negligible and the functionality of the implementation of the Rayleigh-Ritz method for orthotropic plates is confirmed.

Rayleigh-Ritz method vs Finite Element method

As a result of the confirmation of functionality, the implementation is further validated using results from Finite Element simulations in conjunction with experimental results performed using the geometrical parameters and material properties listed in Table 3.

Furthermore, the series expansion is studied where the convergence of the solution is examined. The results of the influence of the number of series is tabulated and graphed in Figure 6.

Lastly the vibration modes with use of the Rayleigh-Ritz method, the Finite Element method and experiments are visualised. As mentioned in the previous section, by considering three degrees of freedom for the orthotropic plate three rigid body modes are to be expected. Therefore the analysis starts with the fourth mode.

Table 3: Geometrical parameters and material properties for the validation with FE-Simulation and experiment

Geometrical parameters [m]:
Length in x-direction	$\begin{array}{l}a = 0,600\; m\end{array}$
Length in y-direction	$\begin{array}{l}b = 0,825\;m\end{array}$
Thickness	$\begin{array}{l}h=0,045\;m\end{array}$
Material properties:
Elasticity moduli	$\begin{array}{l}E_x = 1,6\cdot 10^{10} \; \frac{N}{m^2}\end{array}$ $\begin{array}{l}E_y = 2,42\cdot 10^{9}\; \frac{N}{m^2}\end{array}$
Shear moduli	$\begin{array}{l}G_x_y = 7,77\cdot 10^{8}\; \frac{N}{m^2}\end{array}$ $\begin{array}{l}G_y_z = G_x_z = 1,50\cdot 10^{8} \; \frac{N}{m^2}\end{array}$
Poisson’s ratio	$\begin{array}{l}\nu_x=0,05\end{array}$ using Betti's law: $\begin{array}{l}ν_y = ν_x\cdot \frac{E_x}{E_y}\end{array}$
Mass density	$\begin{array}{l}\rho=530 \;\frac {kg}{m^3}\end{array}$
Shear correction factor	$\begin{array}{l}κ = \frac {π²} {12}\end{array}$

Convergence

As the implementation of the Rayleigh-Ritz method makes use of the modified Fourier series expansion, the influence of the number of series is detrimental to the determination of the eigenvalues. For the calculation of higher eigenvalues the use of m and n must be large to retrieve a good approximation whereas, in this case, the lower eigenvalues were sought after thus an optimal amount of series is wanted for a good approximation of the natural frequencies. Table 4 and Figure 6 illustrate the convergence starting from $\begin{array}{l}m \; = \; n=1\end{array}$ until $\begin{array}{l}m \; = \; n=10\end{array}$ .

The values marked in red are the natural frequency values used for the comparison with Finite Element simulations as well as those determined in the experiment.

Table 4: Influence of the number of series $\begin{array}{l}(m \; = \; n)\end{array}$ on the Eigenfrequency (f) in Hertz

Mode	Eigenfrequency f [Hz]
	Number of Series $\begin{array}{l}(m \; = \; n)\end{array}$
	1	2	3	4	5	6	7	8	9	10
4	119,504	119,322	107,786	107,739	106,693	106,678	106,441	106,435	106,361	106,358
5	140,566	138,274	138,016	137,951	137,936	137,923	137,921	137,918	137,918	137,917
6	266,457	253,319	248,049	246,548	245,912	245,395	245,235	245,044	244,992	244,915
7	448,672	358,709	355,565	353,125	352,926	352,790	352,747	352,722	352,711	352,704
8	491,041	466,510	453,261	449,924	447,709	447,625	446,918	446,910	446,653	446,651
9	536,493	470,052	465,604	465,389	465,319	465,285	465,276	465,263	465,260	465,255
10	539,318	501,896	497,816	497,077	496,689	496,571	496,442	496,403	496,350	496,334
11	648,691	591,153	589,222	585,294	584,965	584,123	584,093	583,780	583,766	583,634
12	852,351	772,277	750,123	647,723	638,798	638,397	637,819	637,751	637,644	637,621
13	1.048,487	834,376	798,716	717,032	707,801	706,743	706,140	705,811	705,693	705,563


Figure 6: Convergence of the Eigenfrequencies of vibration modes 4 to 13

The calculation using the linear combination and thereby the use of high series expansion values can be numerically expensive, meaning the time in order to solve the problem increases with the each increment of the number of series. By examining Figure 6 the graph begins to plateau at the series parameter of 5. By truncating the number of series at 5 for the determination of the lower eigenvalues, as done in this case, one can retrieve a solid approximation of the natural frequency.

Vibration modes

Figures 7 - 16 visualise 10 vibration modes with the use of the Rayleigh-Ritz method implemented in MATLAB, the Finite Element method used in ANSYS and an experiment conducted by the Chair of Structural Mechanics. In Figure 17, with the use of a graph, the comparison of the natural frequency for each mode is visualised.

Rayleigh-Ritz Method	Finite Element Method	Experiment

Figure 7: Vibration mode shape 4 – RRM f = 106,36 Hz (left), FEM f = 106,33 Hz (middle) and Experiment f = 106,73 Hz (right)

Figure 8: Vibration mode shape 5 – RRM f = 137,92 Hz (left), FEM f = 140,35 Hz (middle) and Experiment f = 152,46 Hz (right)

Figure 9: Vibration mode shape 6 – RRM f = 244,92 Hz (left), FEM f = 245,25 Hz (middle) and Experiment f = 250,86 Hz (right)
		Image not found
Figure 10: Vibration mode shape 7 – RRM f = 352,70 Hz (left), FEM f = 358,75 Hz (middle) and Experiment f = 437,56 Hz (right)

Figure 11: Vibration mode shape 8 – RRM f = 446,65 Hz (left), FEM f = 448,53 Hz (middle) and Experiment f = 561,93 Hz (right)

Figure 12: Vibration mode shape 9 – RRM f = 465,26 Hz (left), FEM f = 467,86 Hz (middle) and Experiment f = 663,96 Hz (right)

Figure 13: Vibration mode shape 10 – RRM f = 496,33 Hz (left), FEM f = 495,93 Hz (middle) and Experiment f = 785,46 Hz (right)

Figure 14: Vibration mode shape 11 – RRM f = 583,63 Hz (left), FEM f = 581,78 Hz (middle) and Experiment f = 871,38 Hz (right)

Figure 15: Vibration mode shape 12 – RRM f = 637,62 Hz (left), FEM f = 638,70 Hz (middle) and Experiment f = 561,93 Hz (right)

Figure 16: Vibration mode shape 13 – RRM f = 705,56 Hz (left), FEM f = 710,43 Hz (middle) and Experiment f = 663,96 Hz (right)

With concerns to the vibration modes 12 and 13 (Figure 15 and 16) conducted through experiments, the mode shapes are formed by the the natural frequency of modes 8 and 9, respectively. This is due to the fact that at this frequency the vibration modes are superimposed in the experiments.

The results comparison show feasible and reliable results with an absolute difference discrepancy between the Rayleigh-Ritz method and Finite Element method between 0% and 2%, which is due to numerical simplifications in Finite element simulations.


Figure 17: Comparison of results with Finite Element simulations for a thick orthotropic plate

Conclusion

In conclusion, a free vibration analysis of a moderately thick orthotropic plate using the Rayleigh-Ritz method is presented. Where first order shear deformation and rotary inertia are considered. The analysis is performed by implementing a series expansion to approximate the displacement field for an unconstrained plate whereby the stiffness and mass matrix are calculated for the continuous system so that the generalised conservative eigenvalue problem can be solved.

The results were then verified and validated against functionality and accuracy according to literature, FE-simulations and experiments. The validation according to literature shows that the implementation is functional due to the comparison with a similar approach. Furthermore, the results show a fast convergence, where the series parameter may be truncated at 5 for an accurate approximation of the lower natural frequencies. For the determination of higher natural frequencies and a good approximation thereof, the larger the series parameter $\begin{array}{l}(m \; = \; n)\end{array}$ the better the convergence.

Overall, the Rayleigh-Ritz method yields a good approximation of the natural frequencies in comparison with literature and FE-simulations and a reasonable approximation compared with experiments. However, taking into consideration that a perfectly unconstrained experiment is impossible to perform as well as the true associated material properties of the thick orthotropic plate are simplified properties according to literature taking into consideration the imperfections of the material. Thus, a good approximation can be obtained through considering the induced stiffness of fulfilling the "unconstrained" boundary condition and conducting material tests for the true material properties of the orthotropic plate.

References

[1] Wang, Q., Shi, D., Shi, X. A modified solution for free vibration analysis of moderately thick orthotropic rectangular plates with general boundary conditions, internal line supports and resting on elastic foundation

[2] Shi, D., Zhuang, Z., Zhang, T. Free vibration analysis of orthotropic rectangular Mindlin Plates with general elastic boundary conditions

[3] Ilanko, S., Monterrubio, Luis. The Rayleigh-Ritz Method for Structural Analysis

[4] AA242B: Mechanical Vibrations: Approximation of continuous systems by Displacement Methods

[5] Mittelstedt, Chr. Theory of Plates and Shells

Seitenhierarchie

Rayleigh-Ritz Method for Orthotropic Plates