Free and forced vibration analysis of T-shaped plates with general elastic boundary supports

Abstract

This investigation proposes a series solution method for free and forced vibration analysis of T-shaped plate structure with general elastic supports, arbitrary coupling angles and elastically coupled conditions. The present solution framework can be readily utilized for various boundary or coupling conditions without modifying the solution algorithms or procedures. The general boundary and coupling conditions are considered with artificial spring method and can be easily achieved with assigning the restraining springs with specified values. In the current approach, the displacement functions for in-plane and bending vibration are expressed as a new form of trigonometric series expansion. The sine terms are introduced to eliminate the discontinuity along the boundary edge or coupling edge. Rayleigh–Ritz method is employed to determine the series expansion coefficients, which are treated as the generalized coordinates. Numerous numerical examples are carried out to verify the accuracy and reliability of the present method. The present method can be directly extended to more complicated structures with any number of plates.

Keywords

T-shaped plate vibration general elastic supports spectro-geometric method (SGM)

Introduction

T-shaped plate structures, which are composed with two rectangular plates joined at edges, are widely used in various engineering structures, such as car bodies, building structures, aerospace structures, ship hulls and marine structures. Determining the vibration of such structures is the critical step for reducing the vibration level in the design phase of developing these kinds of product. Therefore, it is of great importance to investigate the vibration characteristic of T-shaped plate structure with various boundary and coupling conditions.

Because of the wide engineering applications of the plate assemblies, its vibration problems have attracted considerable researcher’s interest and many methods for investigating their dynamical characteristics have been promoted. A theoretical model was developed by Boisson et al.¹ to investigate the vibration energy transmission in finite coupled plates forming L, T, and cross junctions. Farag and Pan² presented a theoretical model for the coupling of two plates at an arbitrary angle. They also calculated the vibration response and power flow at the coupling edge and any cross section. Kessissoglou³ studied the power flow propagation in the low and high frequency for L-shaped plate. Park et al.⁴ investigated the longitudinal and in-plane shear waves of thin plates with power flow model. Chen and Sheng⁵ employed a mobility power flow method to study the vibration characteristics of L-shaped plate. Liu et al.⁶ utilized a wave approach to determine the wave and vibration power propagation in a finite L-shaped Mindlin plate. Lin et al.⁷ proposed a closed form solution for the vibration response of an L-shaped plate. A point force or a moment excitation was considered in their study. Chen et al.⁸ investigated the vibration characteristics of a box-type built-up structure and energy transmission through the structure with improved Fourier series method (IFSM). In their study, the general coupling and boundary conditions are considered. An energy flow analysis was employed by Ma et al.⁹ to investigate the mid-frequency vibration characteristics of built-up plate structures. Wang¹⁰ used the method of reverberation-ray matrix (MRRM) to develop an analytical finite coupled plate model, which was modeled as two Mindlin plates connected at an arbitrary angle. Samet et al.¹¹ studied the problem of vibration sources identification in coupled thin plates at high frequency range.

From the review of the literature above, it appears that most of the previous studies on the coupled plate are limited to classical boundary conditions and rigid coupling conditions. Only few researches are carried out for vibration analysis of T-shaped plates and plate assemblies with elastic boundary and coupling conditions. Du et al.¹² presented an analytical model for free vibrations of two rectangular plates elastically coupled together at an arbitrary angle based on IFSM. Xu et al.¹³ presented Fourier spectral element method for the dynamic analysis of plate structures, which are modeled with any number of arbitrarily oriented rectangular plates. Recently, Jiang et al.¹⁴ presented a spectro-geometric method (SGM) for vibration analysis of built-up structures with elastic boundary conditions. This method is subsequently exploited to the free transverse vibration analysis of truncated conical shells¹⁵ and acoustic cavity.¹⁶ Therefore, the current study can be considered as an extension of the SGM and attempts to provide a general solution method for free and forced vibration analysis of T-shaped plates with general elastic supports, arbitrary coupling angles and elastically coupled conditions. In the current solution framework, the SGM together with the Rayleigh–Ritz method and the artificial spring technique are utilized to derive the theoretical formulation. The arbitrary boundary and coupling conditions of the T-shaped plates are realized by applying the artificial spring boundary and coupling technique. The displacement solutions are invariantly expressed in the new form of series expansions regardless of the boundary conditions. The unknown series expansions coefficients are then obtained with Rayleigh–Ritz procedures. The reliability and accuracy of the proposed solution technique are validated extensively through numerical examples.

Theoretical formulations

Descriptions of the model

Consider T-shaped coupling plate with general elastic boundary supports, arbitrary coupling angles and elastically coupled conditions, as shown in Figure 1. The coupled plate structure are consisted with plate 1 (of length a₁ width b, and thickness h₁) and plate 2 (of length a₂ width b, and thickness h₂). The geometry and dimensions are defined in the Cartesian coordinate system (x₁, y₁, z₁) and (x₂, y₂, z₂), respectively. It can be seen from Figure 1 that the model can deal with arbitrary boundary supports, arbitrary connected angles and elastically coupled condition.

Figure 1.

Structure model of the coupling plate with elastic boundary condition.

Three translational springs (k_n, k_p, k_w) and one rotational springs (K_w) are arranged along each edge of the coupled plate structure. Thus, the given boundary conditions can be readily achieved by assigning the translational springs and rotational springs with proper stiffness. For instance, the simply supported boundary condition of the analytical model is approximated by specifying the infinite stiffness (10¹²) for the translational springs, and zero stiffness for the rotational springs.

It can be seen from Figure 1 that the second plate is connected to plate 1 at the coupling edge x₁ = x_c. The boundary conditions for all the other “free” edges of plate 2 are free boundary supports. Suppose the x₁–y₁–z₁ is treated as the global coordinate system, the coupling angle α can be defined as the relative position between these two plates. The counter-clockwise is predefined as the positive direction of the coupling angle, which is variable in the range from −π to π.

The compatibility conditions along the plates’ conjunctions are generally described in terms of three-dimensional elastic couplers with both translational and rotational stiffnesses. The three-dimensional elastic coupler is generally described by four distributed springs (K_c, k_cw, k_cu and k_cv) along the coupling common edge, which are utilized to simulate the transverse shearing forces, bending moments, in-plane longitudinal forces and in-plane shearing forces separately. Different coupling conditions can be readily realized by changing the stiffness of the corresponding springs.

Series representations of the displacement functions

In order to construct a unified model, which can be utilized for arbitrary boundary supports and elastic coupling conditions, the admissible displacement functions for plates are unchangeably expressed as an accelerated trigonometric series expansions with SGM.^17–22 The displacement function for the transverse vibration of plate i can be described as

w_{i} (x, y) = \sum_{m = - 4}^{\infty} \sum_{n = - 4}^{\infty} A_{i, m n} φ_{a_{i} m} (x) φ_{b n} (y)

(1)

The in-plane displacement function in x- and y- direction is separately described as

u_{i} (x, y) = \sum_{m = - 2}^{\infty} \sum_{n = - 2}^{\infty} B_{i, m n} φ_{a_{i} m} (x) φ_{b n} (y)

(2)

v_{i} (x, y) = \sum_{m = - 2}^{\infty} \sum_{n = - 2}^{\infty} C_{i, m n} φ_{a_{i} m} (x) φ_{b n} (y)

(3)

where A_i,mn, B_i,mn, C_i,mn denote the series expansion coefficients, and

φ_{a_{i} m} (x) = {\begin{matrix} \begin{matrix} \cos λ_{a_{i} m} x & m \geq 0 \end{matrix} \\ \begin{matrix} \sin λ_{a_{i} m} x & m < 0 \end{matrix} \end{matrix} \begin{matrix} λ_{a_{i} m} = m π / a_{i} \end{matrix}

(4)

φ_{b n} (x) = {\begin{matrix} \begin{matrix} \cos λ_{b n} x & n \geq 0 \end{matrix} \\ \begin{matrix} \sin λ_{b n} x & n < 0 \end{matrix} \end{matrix} \begin{matrix} λ_{b n} = n π / b \end{matrix}

(5)

The use of SGM is to ensure the residual displacement function and its relevant derivatives will not have any potential discontinuities along any edge. Mathematically, the series expansion equation (1) (or equations (2) to (3)) can be able to expand and uniformly converge to any function f(x_i, y_i)∈C³ (f(x_i, y_i)∈C¹)for ∀ (x_i, y_i)∈ ([0, a_i]⊗[0, b]). Therefore, an exact transverse displacement solution can be obtained as a particular function w_i(x_i, y_i)∈C³ for ∀ (x_i, y_i)∈ ([0, a_i]⊗[0, b]) which satisfies the governing differential equation exactly at every field point and the boundary conditions exactly at every boundary point. The displacement function is constructed sufficiently smooth in this investigation, the weak solution determined using Rayleigh–Ritz method is mathematically equivalent to the strong solution (exact solution). In order to model complex built-up structures, the weak solution is utilized in this study.

Energy description

The Lagrangian’s function for the coupled plate system can be generally expressed as

L = V - T - W_{ext}

(6)

where V and T denote the total potential energy and the total kinetic energy, respectively; W_ext represents the work done by the external excitation. With the assumption of thin plate theory, V and T can be concisely written as

V = V_{1 b e n d i n g} + V_{1 i n - p l a n e} + V_{2 b e n d i n g} + V_{2 i n - p l a n e} + V_{c o u p i n g}

(7)

T = T_{1 b e n d i n g} + T_{1 i n - p l a n e} + T_{2 b e n d i n g} + T_{2 i n - p l a n e}

(8)

where V_ibending (i = 1,2) represents the total potential energy associated with the bending vibration of the ith plate, including the strain energies and the potential energies stored in the bending-related boundary springs; V_iin-plane (i = 1,2) indicates the total potential energy associated with the in-plane vibrations of the ith plate, including the strain energies due to the in-plane motions and potential energies stored in the in-plane-related boundary springs; T_ibending and T_iin-plane are the kinetic energies, respectively, corresponding to the bending and in-plane vibrations of the ith plate; and V_couping denotes the potential energy associated with the three-dimensional elastic coupler.

Take plate 1 for example, the potential and kinetic energies can be expressed as

\begin{matrix} V_{1 b e n d i n g} = \frac{D_{1}}{2} \int_{0}^{a_{1}} \int_{0}^{b} [{(\frac{\partial^{2} w_{1}}{\partial x_{1}^{2}})}^{2} + {(\frac{\partial^{2} w_{1}}{\partial y_{1}^{2}})}^{2} + 2 μ_{1} \frac{\partial^{2} w_{1}}{\partial x_{1}^{2}} \frac{\partial^{2} w_{1}}{\partial y_{1}^{2}} + 2 (1 - μ_{1}) {(\frac{\partial^{2} w_{1}}{\partial x_{1} \partial y_{1}})}^{2}] d x_{1} d y_{1} \\ + \frac{1}{2} {\int_{0}^{b} [k_{w x 10} w_{1}^{2} + K_{w x 10} {(\frac{\partial w_{1}}{\partial x_{1}})}^{2}]}_{x_{1} = 0} d y_{1} + \frac{1}{2} {\int_{0}^{b} [k_{w x 11} w_{1}^{2} + K_{w x 11} {(\frac{\partial w_{1}}{\partial x_{1}})}^{2}]}_{x_{1} = a_{1}} d y_{1} \\ + \frac{1}{2} {\int_{0}^{a_{1}} [k_{w y 10} w_{1}^{2} + K_{w y 10} {(\frac{\partial w_{1}}{\partial y_{1}})}^{2}]}_{y_{1} = 0} d x_{1} + \frac{1}{2} {\int_{0}^{a_{1}} [k_{w y 11} w_{1}^{2} + K_{w y 11} {(\frac{\partial w_{1}}{\partial y_{1}})}^{2}]}_{y_{1} = b} d x_{1} \end{matrix}

(9)

T_{1 b e n d i n g} = \frac{1}{2} \int_{0}^{a_{1}} \int_{0}^{b} ρ_{1} h_{1} {(\frac{\partial w_{1}}{\partial t})}^{2} d x_{1} d y_{1} = \frac{1}{2} \int_{0}^{a_{1}} \int_{0}^{b} ρ_{1} h_{1} ω^{2} w_{1}^{2} d x_{1} d y_{1}

(10)

\begin{matrix} V_{1 i n - p l a n e} = \frac{G_{1}}{2} \int_{0}^{a_{1}} \int_{0}^{b} [{(\frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial v_{1}}{\partial y_{1}})}^{2} - 2 (1 - μ_{1}) \frac{\partial u_{1}}{\partial x_{1}} \frac{\partial v_{1}}{\partial y_{1}} + \frac{(1 - μ_{1})}{2} {(\frac{\partial u_{1}}{\partial y_{1}} + \frac{\partial v_{i}}{\partial x_{1}})}^{2}] d x_{1} d y_{1} \\ + \frac{1}{2} {\int_{0}^{b} [k_{n x 10} u_{1}^{2} + k_{p x 10} v_{1}^{2}]}_{x_{1} = 0} d y_{1} + \frac{1}{2} {\int_{0}^{b} [k_{n x 11} u_{1}^{2} + k_{p x 11} v_{1}^{2}]}_{x_{1} = a_{1}} d y_{1} \\ + \frac{1}{2} {\int_{0}^{a_{1}} [k_{n x 10} v_{1}^{2} + k_{p x 10} u_{1}^{2}]}_{y_{1} = 0} d x_{1} + \frac{1}{2} {\int_{0}^{a_{1}} [k_{n x 11} v_{1}^{2} + k_{p x 11} u_{1}^{2}]}_{y_{1} = b} d x_{1} \end{matrix}

(11)

and

T_{1 i n - p l a n e} = \frac{1}{2} \int_{0}^{a_{1}} \int_{0}^{b} ρ_{1} h_{1} {(\frac{\partial u_{1}}{\partial t} + \frac{\partial v_{1}}{\partial t})}^{2} d x_{1} d y_{1} = \frac{1}{2} \int_{0}^{a_{1}} \int_{0}^{b} ρ_{1} h_{1} ω^{2} (u_{1}^{2} + v_{1}^{2}) d x_{1} d y_{1}

(12)

where w is the transverse displacement in the z-direction, u and v are the in-plane displacements in the x- and y-direction, respectively; ω is the angular frequency, D, G, mu, ρ and h denote, respectively, the flexural rigidity, extensional rigidity, Poisson’s ratio, mass density and the thickness of plates. The symbols (k_wx1₀, K_wx1₀) separately mean the linear and the torsional boundary springs for transverse vibration of plate 1 at the edge x₁ = 0, while symbols (k_nx₁₀, k_px₁₀) separately denote the normal and the parallel linear springs for the in-plane vibration at the edge x₁ = 0. The other boundary springs given above are similarly defined. In this paper, the uniform material parameters for all the plates are handled.

The presence of the three-dimensional elastic coupler can be accounted by the potential energies

V_{c o u p i b g} = \frac{1}{2} {\int_{0}^{b} [k_{c w} (w_{1} |_{x_{1} = x_{c}} - w_{2} |_{x_{2} = a_{2}} \cos α + u_{2} |_{x_{2} = a_{2}} \sin α)}^{2} + K_{c u} {(u_{1} |_{x_{1} = x_{c}} - u_{2} |_{x_{2} = a_{2}} \cos α - w_{2} |_{x_{2} = a_{2}} \sin α)}^{2} + K_{c v} {(v_{1} |_{x_{1} = x_{c}} - v_{2} |_{x_{2} = a_{2}})}^{2} + K_{c} {(\partial w_{1} / \partial x_{1} |_{x_{1} = x_{c}} - \partial w_{2} / \partial x_{2} |_{x_{2} = a_{2}})}^{2} d y

(13)

where the K_c, k_cw, k_cu and k_cv denote the stiffnesses of the coupling springs (refer to Figure 1) and α is the coupling angle between the two plates.

The work done by external force can be given as

W_{ext} = \int_{0}^{a_{1}} \int_{0}^{b} f (x_{1}, y_{1}) w_{1} (x_{1}, y_{1}) d x_{1} d y_{1}

(14)

where f(x₁, y₁) is the external force function acted on the plate.

For a point force, the force function can be expressed as

f (x_{1}, y_{1}) = F δ (x - x_{e}) δ (y - y_{e})

(15)

where F is the amplitude of the point force, δ() is the Dirac delta function and (x_e, y_e) is the position where the point force is applied.

For simplicity, boundary conditions and coupling conditions are all assumed to be uniform along each edge. But the present method can also be easily extended to the most general boundary and coupling conditions with non-uniform stiffness (varying with the spatial coordinates continuously, discontinuously or discretely). For the non-uniform cases, the stiffnesses in equations (9), (11), and (13) can be expressed as a function of the spatial coordinates.

Solution scheme

Substituting equations (7) to (15) into equation (6) and minimizing the Lagrangian with respect to all the unknown series expansion coefficients, with R as the unknown series coefficients and L as the Lagrangian, the Lagrange equation for the whole system can be obtained as

\frac{d}{d t} (\frac{\partial L}{\partial R}) - \frac{\partial L}{\partial R} = 0

(16)

Make a differential calculation in equation (16), a group of linear equations can be derived in the matrix form

(K - ω^{2} M) E = F

(17)

The unknown coefficients in the displacement expressions can be expressed in the vector form as E.

E = {\begin{matrix} A_{- 4, - 4}^{1} & A_{- 4, - 3}^{1} & \dots & A_{m^{'}, 0}^{1} & A_{m^{'}, 1}^{1} & \dots & A_{m^{'}, n'}^{1} & \dots & A_{M, N}^{1} \\ B_{- 2, - 2}^{1} & B_{- 2, - 1}^{1} & \dots & B_{m^{'}, 0}^{1} & B_{m^{'}, 1}^{1} & \dots & B_{m^{'}, n'}^{1} & \dots & B_{M, N}^{1} \\ C_{- 2, - 2}^{1} & C_{- 2, - 1}^{1} & \dots & C_{m^{'}, 0}^{1} & C_{m^{'}, 1}^{1} & \dots & C_{m^{'}, n'}^{1} & \dots & C_{M, N}^{1} \\ A_{- 4, - 4}^{2} & A_{- 4, - 3}^{2} & \dots & A_{m^{'}, 0}^{2} & A_{m^{'}, 1}^{2} & \dots & A_{m^{'}, n'}^{2} & \dots & A_{M, N}^{2} \\ B_{- 2, - 2}^{2} & B_{- 2, - 1}^{2} & \dots & B_{m^{'}, 0}^{2} & B_{m^{'}, 1}^{2} & \dots & B_{m^{'}, n'}^{2} & \dots & B_{M, N}^{2} \\ C_{- 2, - 2}^{2} & C_{- 2, - 1}^{2} & \dots & C_{m^{'}, 0}^{2} & C_{m^{'}, 1}^{2} & \dots & C_{m^{'}, n'}^{2} & \dots & C_{M, N}^{2} \end{matrix}}^{T}

(18)

In equation (17), K is the stiffness matrix for the T-shaped structure, M is the mass matrix and F is the load vector . For conciseness, the detailed expressions for stiffness and mass matrices will not be shown here. They can be expressed separately as

K = [\begin{matrix} K_{1 - 1} & K_{1 - 2} & K_{1 - 3} & K_{1 - 4} & K_{1 - 5} & K_{1 - 6} \\ K_{2 - 1} & K_{2 - 2} & K_{2 - 3} & K_{2 - 4} & K_{2 - 5} & K_{2 - 6} \\ K_{3 - 1} & K_{3 - 2} & K_{3 - 3} & K_{3 - 4} & K_{3 - 5} & K_{3 - 6} \\ K_{4 - 1} & K_{4 - 2} & K_{4 - 3} & K_{4 - 4} & K_{4 - 5} & K_{4 - 6} \\ K_{5 - 1} & K_{5 - 2} & K_{5 - 3} & K_{5 - 4} & K_{5 - 5} & K_{5 - 6} \\ K_{6 - 1} & K_{6 - 2} & K_{6 - 3} & K_{6 - 4} & K_{6 - 5} & K_{6 - 6} \end{matrix}]

(19)

M = [\begin{matrix} M_{1 - 1} & M_{1 - 2} & M_{1 - 3} & M_{1 - 4} & M_{1 - 5} & M_{1 - 6} \\ M_{2 - 1} & M_{2 - 2} & M_{2 - 3} & M_{2 - 4} & M_{2 - 5} & M_{2 - 6} \\ M_{3 - 1} & M_{3 - 2} & M_{3 - 3} & M_{3 - 4} & M_{3 - 5} & M_{3 - 6} \\ M_{4 - 1} & M_{4 - 2} & M_{4 - 3} & M_{4 - 4} & M_{4 - 5} & M_{4 - 6} \\ M_{5 - 1} & M_{5 - 2} & M_{5 - 3} & M_{5 - 4} & M_{5 - 5} & M_{5 - 6} \\ M_{6 - 1} & M_{6 - 2} & M_{6 - 3} & M_{6 - 4} & M_{6 - 5} & M_{6 - 6} \end{matrix}]

(20)

F = F φ_{a_{1} m} (x_{e}) φ_{b n} (y_{e})

(21)

For the analysis of free vibration characteristics, the load vector F is set to zero. The natural frequencies and eigenvectors can now be readily obtained by solving a standard matrix eigen problem. Since the components of each eigenvector are actually the expansion coefficients of the modified trigonometric series, the corresponding mode shape can be directly determined from equations (1) to (3).

For a given excitation frequency ω, the response vector E can be directly determined as

E = {(K - ω^{2} M)}^{- 1} F

(22)

To overcome the numerical instability and singularity encountered at modal resonances, the damping of the structure is introduced in the form of complex Young’s modulus.

\tilde{E} = E (1 + j η)

(23)

where η is the damping coefficient and

j = \sqrt{- 1}

After the series expansion coefficient vector E and the displacement are determined, the mobility at any position on the plate can be calculated from

Y_{i j} = \frac{j ω w_{j}}{F_{i}}

(24)

where the Y_ij, w_j and F_i denote, respectively, structural mobility from point i to point j, transverse vibration displacement in point j and the amplitude of the force on the point i. If the i = j, the Y_ij represent the origin structural mobility.

Numerical examples and discussion

In this section, a systematic comparison between the current solutions and theoretical results published by other researchers or finite element analysis (FEA) results is carried out to validate the excellent accuracy, reliability and feasibility of the SGM. The discussion is arranged as: Firstly, the convergence of the SGM solution is checked. In addition, influence of the stiffness of boundary spring components is studied. Secondly, the free vibration of the T-shaped plate structure with general boundary condition, arbitrary coupling angles and elastically coupled condition is examined. Finally, the forced vibration of the T-shaped plate structure with general boundary condition, arbitrary coupling angles and elastically coupled condition is also studied.

Free vibration analysis of T-shaped plates

Several examples with different coupling configurations (see Figure 2) will be considered in this section. The geometric parameters are taken as: a₁ = 1.5 m, a₂ = 1.0 m, b = 1 m, h₁ = h₂ = h = 0.005 m. The material properties for both plates are given as: E = 7.1 × 10¹⁰ N/m², ρ = 2700 kg/m³ and mu = 0.3.

Figure 2.

Four coupling configurations with different angles at the structural junction: SS-simply supported for bending component, C- clamped for in-plane component.

As the first example, T-shaped plate structure (α = π/2 and x_c = a₁/2) as shown in Figure 2(a) is taken as the research object. The simply supported (or clamped) boundary conditions are specified for bending (or in-plane) vibration displacements on plate 2 along the free edges (x₂ = 0, y₂ = 0 and y₂ = b). The boundary conditions for plate 1 are kept the same with plate 2. The stiffness values of the three-dimensional elastic coupler are given as: K_c = 10⁵ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m and k_cv = 10⁴ N/m. These stiffness values are close to the nominal bending rigidity of the plate, Eh³/12(1−mu²) = 10⁴. The elastic supports or coupling conditions tend to play an important role on the vibration characteristic of the plate(s). The first 10 natural frequency parameters of the coupled system are listed in Table 1 for different truncation schemes, M = N = 6,7,8,9,10,11,12. The FEA results calculated with ABAQUS model are also given there for comparisons. In the FEA model, the element sizes are uniformly specified as 0.01 m $\times$ 0.01 m which is considered fine enough to accurately capture the spatial variations of these modes. It can be seen from Table 1 that the current approach has fast convergence behavior. The maximum discrepancy in the worst case between the 10 × 10 truncated configuration and the 11 × 11 one is less than 0.087%. It is also shown that the results converge at M = N = 12 for the given five-digit precision. Therefore, in the following examples, all the trigonometric series expansions are truncated into M = N = 12. In addition, it is clear that the results of the present approach with just, 1478 DOFs(M × N = 12 × 12) can predict the vibration characteristics accurately. Most of them are identical to those obtained from FEA (FEA-based ABAQUS) with 149,200 DOFs (S4R: 0.01 m × 0.01 m). That is to say, it needs only 1.0% DOFs compared with FEA to obtain the same precision solutions for the considered case. On the same hardware (Intel i7–3.9 GHz), the computing time of the present formulation for the solution (M × N = 12 × 12) implemented in optimized MATLAB scripts is about 21.263 s, whereas the finite element solution consumes 347.4983 s, i.e. at least 15 times more CPU time than the present method for the same problem.

Table 1.

Natural frequencies for an T-shaped plate structure with the coupling springs: K_c = 10⁵ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m and k_cv = 10⁴ N/m (Hz).

M, N	Model sequence
M, N	1	2	3	4	5	6	7	8	9	10
M = N = 6	16.311	18.295	34.286	40.987	52.003	54.397	61.145	71.150	76.709	85.075
M = N = 7	16.308	18.295	34.284	40.883	51.992	54.397	61.145	71.143	76.666	85.019
M = N = 8	16.308	18.295	34.284	40.883	51.992	54.397	61.145	71.143	76.666	85.019
M = N = 9	16.305	18.295	34.283	40.804	51.983	54.397	61.145	71.138	76.608	84.901
M = N = 10	16.305	18.295	34.283	40.804	51.983	54.397	61.145	71.138	76.608	84.901
M = N = 11	16.304	18.295	34.283	40.765	51.978	54.397	61.145	71.135	76.565	84.852
M = N = 12	16.304	18.295	34.283	40.765	51.978	54.397	61.145	71.135	76.565	84.852
FEA	16.273	18.269	34.227	40.467	51.939	54.380	61.146	71.126	76.549	84.846

The corresponding mode shapes can be readily calculated by substituting the series coefficients into the displacement expressions, equations (1) to (3). Plotted in Figure 3 are the first six modes for the T-shaped plate structure. It can be seen that due to the elastic coupling, the displacements are no longer continuous across the junction. Now, consider these two plates are rigidly connected together. The condition of rigid connection can be easily simulated by setting the stiffness for elastic coupler to be infinitely large. The corresponding natural frequencies are given in Table 2 together with the FEA results. It can be seen that current results agree well with those obtained from FEA. The first six mode shapes are plotted in Figure 4. Compared with the results plotted in Figure 3, the results for rigid coupling conditions do not have the displacement gaps at the junction. Meanwhile, the mode shape activities can also cover the entire plate structure.

Figure 3.

The first six mode shapes for an elastically coupled T-shaped plate structure (x_c = a₁/2, K_c = 10⁵ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m and k_cv = 10⁴ N/m.).: (a) 1^st mode shape, (b) 2^nd mode shape, (c) 3^rd mode shape, (d) 4^th mode shape, (e) 5^th mode shape, (f) 6^th mode shape

Table 2.

Natural frequencies for two rectangular plates rigidly coupled in an T-shaped configuration.

Model sequence	Present (Hz)	FEA (Hz)	Difference (%)
1	26.914	26.931	0.062
2	37.813	37.824	0.028
3	43.677	43.682	0.013
4	62.664	62.697	0.053
5	69.158	69.186	0.041
6	72.967	72.988	0.028
7	77.017	77.154	0.178
8	103.941	104.120	0.172
9	104.482	104.647	0.158
10	120.547	120.629	0.068

Figure 4.

The first six mode shapes for the rigidly coupled T-shaped plate structure (x_c = a₁/2).: (a) 1^st mode shape, (b) 2^nd mode shape, (c) 3^rd mode shape, (d) 4^th mode shape, (e) 5^th mode shape, (f) 6^th mode shape

As mentioned earlier, all the general boundary conditions and elastically coupled condition can be conveniently viewed as special cases when stiffness for boundary springs and coupling springs become zero and/or infinitely large. Thus, the effects of the stiffness of boundary springs (k_w, K_w, k_n, k_p) and coupling springs (K_c, k_cw, k_cu, k_cv) on the modal characteristics of the T-shaped plate structure should be investigated. Since there are various possible varying conditions of boundary springs and coupling springs, for simplicity, just a type springs stiffnesses is considered to be changed, while the other type springs stiffnesses are all set as infinitely large numbers. The non-dimensional stiffness $\bar{K} = K * a / D$ is defined and used here, in which a = (a₁+a₂)/2 and D = Eh³/12(1−mu²) are the flexural rigidity of plate structure. As shown in Figure 5(a) to (d), the first and the fifth frequency parameters are separately obtained by varying the non-dimensional stiffness of one group of the boundary springs from extremely large (10⁸) to extremely small (10⁻³) while assigning the other group of the springs with infinite stiffness (10⁸). It can be found in Figure 5(a) that the frequency parameter almost keeps at a level when the non-dimensional stiffness of the bending translation springs is larger than 10⁶ or small than 10°. In Figure 5(b), the influences of the bending rotation springs on frequency parameters are given. It is shown that the frequency curves change greatly within the stiffness range (10⁻¹ to 10³); while out of this range, the frequency curves separately keep the level. In Figure 5(c) and (d), the influences of the in-plane springs on frequency parameters are given. It is shown that the frequency curves almost change when the stiffness changes in the whole range. Figure 6 illustrates the variation of the first and the fifth modal frequencies with different coupling conditions. It can be seen in Figure 6(a) that when the non-dimensional rotational stiffness coefficient is smaller than 10⁻², its variation almost has no effect on modal parameters of the coupled plates. As the stiffness parameter further increases, a sensitive effect zone can be observed, in which the modal frequency increases with different rate depending on the coupling stiffness. When the coupling factor is beyond 10³, the modal parameter approaches the upper limit basically corresponding to the rigid interaction case. In the same way, in Figure 6(b) (or Figure 6(c)), the influences of the bending rotation springs on frequency parameters are given. It is shown that the frequency curves change greatly within the stiffness range 10° to 10³ (or 10¹ to 10⁴); while out of this range, the frequency curves separately keep the level. In Figure 6(d), the influences of the coupling springs k_cv on frequency parameters are given. It is shown that the frequency curves almost changes when the stiffness changes in the whole range. Thus, it is suitable to use non-dimensional stiffness 10⁶ (dimensional stiffness 10¹²) to simulate the infinite stiffness value of the boundary spring and coupling spring in the previous examples and in the following examples.

Figure 5.

The effect of boundary spring stiffness on the natural frequencies: (a) k_w, (b) K_w, (c) k_n, (d) k_p.

Figure 6.

The effect of coupling spring stiffness on the natural frequencies (a) K_c, (b) k_cw, (c) k_cu, (d) k_cv.

In order to further investigate the influence of the boundary spring and coupling spring on the coupled plates system, the effect on the first 8 modes for T-shaped plate structure with different boundary rotational and coupling rotational stiffness coefficients are shown in Tables 3 and 4. Similar influence trend of the boundary and coupling conditions can be found for the other structural modes. In fact, this boundary and coupling stiffness effect behavior can be also understood by observing the evolution of spatial distribution pattern of the coupled structural mode, as given in Figures 7 and 8 for the second mode cases. From Tables 3 and 4 and Figures 7 and 8, it is clearly shown that how the transfer from separate modes into a joined mode occurs within the coupling stiffness sensitive zone.

Figure 7.

Evolution of the second mode shape of the T-shaped plate with the variation of boundary conditions: (a) ${\bar{K}}_{b x} = 1,$ (b) ${\bar{K}}_{b x} = 1,$ (c) ${\bar{K}}_{b x} = 1,$ (d) ${\bar{K}}_{b x} = 1,$ (e) ${\bar{K}}_{b x} = 1,$ (f) ${\bar{K}}_{b x} = 10^{4}$

Figure 8.

Evolution of the second mode shape of the T-shaped plate with the variation of coupling conditions:(a) ${\bar{K}}_{c} = 1,$ (b) ${\bar{K}}_{c} = 1,$ (c) ${\bar{K}}_{c} = 1,$ (d) ${\bar{K}}_{c} = 1,$ (e) ${\bar{K}}_{c} = 1,$ (f) ${\bar{K}}_{c} = 10^{4}$

Table 3.

The first eighth modal frequencies of the T-shaped plate with various non-dimensional rotational boundary spring stiffness ${\bar{K}}_{w} = K_{w} × a / D$ (Hz).

${\bar{K}}_{w}$	Model sequence
${\bar{K}}_{w}$	1	2	3	4	5	6	7	8
10⁻³	26.937	37.873	43.701	62.689	69.298	73.018	77.063	104.072
10⁻²	26.950	37.887	43.716	62.705	69.309	73.034	77.079	104.086
10⁻¹	27.082	38.024	43.856	62.863	69.417	73.192	77.239	104.218
10⁰	28.276	39.286	45.159	64.347	70.434	74.681	78.738	105.478
10⁰	(28.284^a)	(39.291)	(45.168)	(64.355)	(70.438)	(74.702)	(78.769)	(105.497)
10¹	34.346	46.115	52.380	73.337	76.688	83.883	88.058	113.482
10¹	(34.348^a)	(46.164)	(52.395)	(73.345)	(76.704)	(83.891)	(88.082)	(113.497)
10²	40.704	53.948	60.957	85.410	86.009	97.387	101.858	125.971
10²	(47.721^a)	(53.957)	(60.962)	(85.412)	(86.104)	(97.385)	(101.869)	(125.994)
10³	42.029	55.661	62.863	87.521	89.159	100.857	105.423	129.328
10³	(42.038^a)	(55.674)	(62.872)	(87.538)	(89.257)	(100.884)	(105.438)	(129.387)
10⁴	42.176	55.855	63.078	87.762	89.521	101.261	105.837	129.717
10⁴	(42.184^a)	(55.874)	(63.092)	(87.781)	(89.543)	(101.284)	(105.839)	(129.757)
10⁵	42.191	55.875	63.099	87.787	89.558	101.302	105.879	129.757
10⁶	42.193	55.876	63.102	87.789	89.562	101.305	105.962	129.758
10⁷	42.193	55.877	63.102	87.790	89.562	101.306	105.884	129.761
10⁸	42.193	55.876	63.102	87.790	89.562	101.306	105.884	129.761

^aResults in parentheses are taken from FEA.

Table 4.

The first eighth modal frequencies of the T-shaped plate with various non-dimensional rotational coupling spring stiffness ${\bar{K}}_{c} = K_{c} × a / D$ (Hz).

${\bar{K}}_{c}$	Model sequence
${\bar{K}}_{c}$	1	2	3	4	5	6	7	8
10⁻³	39.302	51.501	63.102	78.205	87.771	98.554	105.884	124.467
10⁻²	39.306	51.505	63.102	78.212	87.773	98.556	105.884	124.471
10⁻¹	39.347	51.544	63.102	78.282	87.789	98.574	105.884	124.514
10⁰	39.701	51.899	63.102	78.929	87.937	98.745	105.884	124.914
10⁰	(39.705^a)	(51.924)	(63.141)	(78.948)	(87.955)	(98.769)	(105.897)	(124.928)
10¹	41.113	53.756	63.102	82.642	88.683	99.748	105.884	127.367
10¹	(41.124^a)	(53.793)	(63.178)	(82.687)	(88.724)	(99.757)	(105.912)	(127.386)
10²	42.035	55.517	63.102	86.841	89.407	100.997	105.884	129.397
10²	(43.058^a)	(55.529)	(63.178)	(86.859)	(89.429)	(101.104)	(105.947)	(129.426)
10³	42.176	55.838	63.102	87.687	89.545	101.272	105.884	129.722
10³	(42.179^a)	(55.847)	(63.116)	(87.764)	(89.581)	(101.354)	(105.892)	(129.748)
10⁴	42.192	55.880	63.102	87.816	89.563	101.302	105.884	129.588
10⁴	(42.198^a)	(55.974)	(63.158)	(87.829)	(89.591)	(101.347)	(105.924)	(129.607)
10⁵	42.193	55.876	63.102	87.789	89.562	101.306	105.884	129.760
10⁶	42.193	55.877	63.102	87.789	89.562	101.306	105.884	129.761
10⁷	42.193	55.877	63.102	87.790	89.562	101.306	105.884	129.761
10⁸	42.193	55.877	63.102	87.790	89.562	101.306	105.884	129.761

^aResults in parentheses are taken from FEA.

Now, change the coupling angle to 45° which corresponds to the second configuration as shown in Figure 2(b). In addition, the boundary conditions are modified; that is, both the flexural and in-plane displacements on each plate are now clamped at y = b, and all other edges are completely free. In Table 5, the first 10 natural frequencies are presented for the elastic coupling condition: K_c = 10⁷ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m and k_cv = 10⁵ N/m. The first six modes are shown in Figure 9. They clearly exhibit a global characteristic as compared with those modes given in Figure 4 for the T-shaped plate structure case. When the coupling angle is different from 90°, the flexural vibrations on both plates will be directly coupled together through the transverse spring k_cw. In the next example, we will change the coupling location x_c = a₁/2 to x_c = a₁, as in shown Figure 2(c). Figure 2(b) and (c) has the same boundary conditions. The first 10 natural frequencies are given in Table 6 for the elastic coupling condition: K_c = 10⁵ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m and k_cv = 10³ N/m and α = 90°, the FEA results are also listed in Table 6 as a reference. The first six modes are shown in Figure 10. It can be clearly seen that the comparison is extremely good, which implies that the current analysis model is able to make correct predictions for the L-shaped plate structure. Based on these analysis, Table 7 given the first 6 natural frequencies under different coupling location (x_c = 0 m, 0.53 m, 0.83 m) and coupling angle (α = 30°, 60°, 120°), the FEA results calculated using ABAQUS are also presented as a reference. Again, good agreement can be observed. Through Table 7, it is found that the current method is also suitable to solve the T-shaped plate structure with the arbitrary coupling angles (α) and coupling location (x_c).

Figure 9.

The mode shapes for two plates elastically coupled at the angle of 45-degree with the following coupling springs: K_c = 10⁷ Nm/rad, k_cu = 10⁴ N/m, k_cv = 10⁴ N/m and k_cw = 10⁵ N/m.: (a) 1^st mode shape, (b) 2^nd mode shape, (c) 3^rd mode shape, (d) 4^th mode shape, (e) 5^th mode shape, (f) 6^th mode shape

Table 5.

Natural frequencies for two plates coupled at 45° with the coupling springs: K_c = 10⁷ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m and k_cv = 10⁵ N/m.

Model sequence	Present (Hz)	FEA (Hz)	Difference (%)
1	4.773	4.791	0.383
2	6.090	6.102	0.194
3	8.833	8.847	0.162
4	13.664	13.673	0.066
5	18.438	18.449	0.062
6	27.050	27.094	0.163
7	27.440	27.452	0.043
8	32.587	32.596	0.028
9	37.169	37.194	0.069
10	39.479	39.488	0.023

Table 6.

Natural frequencies for L-type plates (α = 90°) with the coupling springs: K_c = 10⁵ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m and k_cv = 10³ N/m.

Model sequence	Present (Hz)	FEA (Hz)	Difference (%)
1	4.995	5.014	0.382
2	5.650	5.691	0.724
3	9.675	9.724	0.512
4	13.266	13.297	0.237
5	20.352	20.382	0.150
6	27.091	27.098	0.025
7	27.302	27.322	0.074
8	32.820	32.874	0.163
9	36.608	36.621	0.036
10	39.301	39.335	0.087

Figure 10.

The first six mode shapes for an elastically coupled L-shaped plate structure: K_c = 10⁵ Nm/rad, k_cu = 10⁴ N/m, k_cv = 10⁴ N/m and k_cw = 10³ N/m.: (a) 1^st mode shape, (b) 2^nd mode shape, (c) 3^rd mode shape, (d) 4^th mode shape, (e) 5^th mode shape, (f) 6^th mode shape

Table 7.

Natural frequencies for two plates under different coupling angle and coupling location (Hz).

x _c (m)	α(°)	Model sequence
x _c (m)	α(°)	1	2	3	4	5	6
0	30	18.273	26.499	36.494	54.548	60.938	62.245
	30	(18.256^a)	(26.468)	(36.460)	(54.537)	(60.900)	(62.200)
	60	18.274	26.505	36.507	54.550	60.938	62.254
	60	(18.256^a)	(26.471)	(36.472)	(54.538)	(60.895)	(62.206)
	120	18.274	26.505	36.509	54.550	60.930	62.255
	120	(18.255^a)	(26.470)	(36.473)	(54.537)	(60.883)	(62.203)
0.5	30	25.917	29.475	60.932	61.858	64.059	73.279
	30	(25.857^a)	(29.154)	(60.887)	(61.784)	(63.743)	(72.290)
	60	25.917	29.486	60.937	61.858	64.075	73.364
	60	(25.857^a)	(29.165)	(60.891)	(61.784)	(63.758)	(72.366)
	120	25.915	29.488	60.937	61.855	64.078	73.384
	120	(25.855^a)	(29.168)	(60.891)	(61.781)	(63.762)	(72.385)
0.8	30	26.900	36.457	45.723	62.658	69.282	71.258
	30	(26.749^a)	(36.145)	(45.542)	(62.474)	(68.462)	(70.952)
	60	26.904	36.458	45.755	62.665	69.323	71.261
	60	(26.752^a)	(36.146)	(45.576)	(62.480)	(68.502)	(70.955)
	120	26.905	36.456	45.756	62.665	69.328	71.258
	120	(26.753^a)	(36.144)	(45.576)	(62.481)	(68.507)	(70.953)

^aResults in parentheses are taken from FEA.

Lastly, a flat configuration as illustrated in Figure 2(d) will be considered for both elastic and rigid coupling conditions. In this example, the boundary conditions will be changed back to the ones as previously specified for the T-shaped plate structure. When the coupling angle is equal to π, the bending and in-plane vibrations will essentially become decoupled, and can be determined separately. For the elastic connection, the coupling springs are taken as K_c = 10⁵ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m, and k_cv = 10³ N/m. In Table 8, the calculated natural frequencies are compared with those calculated from an ABAQUS model. The first six mode shapes are plotted in Figure 11. It is seen that the weak coupling actually allows the plates to move almost independently.

Table 8.

Natural frequencies for the π-angle coupled plate structure with the coupling springs: K_c = 10⁵ Nm/rad, k_cw = 10⁴ N/m, k_cu = 10⁴ N/m and k_cv = 10³ N/m (Hz).

Model sequence	Present (Hz)	FEA (Hz)	Difference (%)
1	13.962	13.968	0.046
2	16.567	16.574	0.043
3	24.442	24.461	0.076
4	37.591	37.614	0.062
5	44.753	44.784	0.069
6	50.176	50.183	0.015
7	51.840	51.847	0.013
8	60.654	60.716	0.103
9	73.373	73.394	0.028
10	74.776	74.786	0.014

Figure 11.

The mode shapes for two plates elastically coupled in a flat configuration with the following coupling springs (K_c = 10⁵ Nm/rad, k_cu = 10⁴ N/m, k_cv = 10⁴ N/m and k_cw = 10³ N/m).: (a) 1^st mode shape, (b) 2^nd mode shape, (c) 3^rd mode shape, (d) 4^th mode shape, (e) 5^th mode shape, (f) 6^th mode shape

The rigid connection condition is obtained simply by increasing stiffness values of the coupling springs to infinity. In this case, the two plate structure simply degenerates to a single plate, which has a combined length of a₁+a₂. Thus, the calculated natural frequencies and mode shapes can be directly compared with the analytical results for a single simply supported plate. A comparison of the natural frequencies is made in Table 9. The first six mode shapes are shown in Figure 12. The minor differences for the calculated natural frequencies are believed due to the combination of the numerical truncation of the series expansions and the actual finiteness (instead of infinity) of the stiffness values for the boundary and coupling springs. This example, although it may appear to be trivial, has some very important implications, for instance it says that a continuous plate component can be arbitrarily divided into a number of pieces to avoid non uniformities/irregularities related to geometry or materials.

Table 9.

A comparison of the first 10 natural frequencies for the two plates rigidly coupled in a flat configuration.

Model sequence	Present (Hz)	FEA (Hz)	Difference (%)
1	14.138	14.149	0.079
2	19.988	20.014	0.131
3	29.738	29.871	0.449
4	43.388	43.423	0.080
5	50.701	50.712	0.021
6	56.551	56.568	0.029
7	60.938	60.942	0.006
8	66.302	66.308	0.009
9	79.952	79.984	0.040
10	82.389	82.418	0.035

Figure 12.

The mode shapes for a rigidly coupled plate structure in a π-degree configuration.: (a) 1^st mode shape, (b) 2^nd mode shape, (c) 3^rd mode shape, (d) 4^th mode shape, (e) 5^th mode shape, (f) 6^th mode shape

Forced vibration analysis of T-shaped plates

This section concentrates on the forced vibration problems of T-shaped plates structure with general boundary condition, arbitrary coupling angles and elastically coupled condition are also studied. In this section, the structural parameter and the material parameter are the same as those in ‘Free vibration analysis of T-shaped plates’ section. The model has the same boundary condition and coupling angle as that in Figure 2(a), and the stiffness is rigid coupled connection. The harmonic external lateral exciting force with 1 N amplitude acts on point A (0.6 m, 0.5 m) on plate 1 in the coupled system. The mobility characteristics of point A on plate 1 and point B (0.4 m, 0.5 m) on plate 2 are shown in Figure 13. At the same time, the results obtained with FEA software ABAQUS are also utilized to verify the validity and reliability of the methods and computational models in this study. The element size for the FEM model is 0.01 m × 0.01 m, which can accurately catch the vibration response characteristics of the structural system. It can be seen from Figure 13 that the computational methods of this study can accurately predict the mobility characteristics of the coupling plate structure system. The mobility characteristics change with the frequency, and the frequency value of the resonance peak stays the same as the inherent frequency of the system.

Figure 13.

The mobility characteristics of the rigid coupling plate structure system.: (a) point A, (b) point B

When the coupling connecting spring stiffness is changed, the coupling plate structural system will turn into elastic coupling plate structural system. The mobility characteristics for coupled system with elastic coupler (K_c = 10³ Nm/rad, k_cu = 10⁴ N/m, k_cv = 10⁵ N/m and k_cw = 10⁶ N/m) are shown in Figure 14. It can be seen that the elastic coupler obviously affects the mobility characteristics of the coupling plate structural system, and the mobility characteristics of the rigidly coupled plate structural system are apparently higher than those with elastic coupler system.

Figure 14.

The mobility characteristics of the elastic coupling plate structure system.: (a) point A, (b) point B

Conclusions

A general method is presented for the free and forced vibration analysis of T-shaped plate structure with general elastic supports, arbitrary coupling angles and elastically coupled conditions. Unlike the existing solution framework, each “free” edge for two plates can be elastically restrained by a set of elastic springs with arbitrary stiffness. The bending and in-plane vibration displacement functions for each plate are described as an improved trigonometric series expansion with SGM. These displacement functions can deal with arbitrary boundary conditions and coupling conditions. The unknown series expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure.

The accuracy and reliability of the proposed solution method have been illustrated through numerical examples which include various boundary conditions and coupling configurations. From the numerical computation, the present method derived here is a numerically efficient and accurate solution applicable to a wide range of vibration problems of T-shaped plate structure. Since the free and forced vibration analyses of T-shaped plate structure with general elastic supports, arbitrary coupling angles and elastically coupled conditions have not been reported previously, it is believed that the numerical results presented in this paper will be useful to future researchers. In addition, compared to other solving technique such as FEA and other analytical solution, the following advantages may be drawn out: (1)no mesh is used, then there is no need for mesh refinement like the FEA does as the analysis frequency increases; (2) a fast convergence rate is observed, which implies that the efficient calculation can be achieved; (3) the current method does not involve much any modification to the solution algorithm or procedure when the boundary or coupling condition is changed. The current model can also be used to investigate the effect of a boundary or coupling condition on the dynamic behavior of a built-up structure and on the power flows through any junctions and structural components.