Energetics of In-plane Motions in Coupled Plate Structures

4.1 Exact Solution of Coupled Plate Structures

In the exact solution of a coupled plate for which all the outer boundaries are simply supported, the displacement can be calculated in the form of a Levy series that satisfies the y-direction boundary condition equations, i.e., Eqs. (7)–(10). The displacement in the x- and y-directions in the i^th region can be expressed using Eqs. (11)–(13).

Here, k_m = mπ/L_y and U_m,i(x_i), V_m,i(x_i), and W_m,i(x_i) represent the wave solutions in the x_i direction that satisfy the equation of motion. The following equation can be obtained by substituting the out-of-plane displacement expressed by Eq. (13) into Eq.(5), i.e., the equation of motion (Park et al., 2001).

Here, kf,i=(ρihiω2)/Dc,i4) represents the flexural wavenumber of the plate in the i^th plate section, and ρ_i and h_i represent the density and thickness of the plate, respectively. Because Eq. (14) is a fourth-order homogeneous ordinary differential equation, the wave solution of the displacement w in the x_i direction can be obtained as follows:

where αm,i2=km2+kf,i2 and βm,i2=km2+kf,i2. Although such an exact solution of the flexural wave, which is an out-of-plane displacement, was used to validate the energy flow model for the flexural wave of a plate, the exact solution of the in-plane displacement and the energy flow solution were not directly compared in previous studies (Park et al., 2001). Park and Hong (2008) derived an exact out-of-plane motion solution from the equation of motion implemented as the simultaneous differential equations for the Mindlin plate. The matrix equation can be obtained as follows by substituting Eqs. (11)–(12) into the simultaneous differential Eqs. (1)–(2) to obtain the wave solution in the x_i direction of the in-plane motion of a plate in a similar manner (Park, 2018).

Here, the matrices {M_m,i} and [Nm_,i] are given by Eqs. (17) and (18), respectively.

The general solution of the above differential-equation system can be obtained as follows:

where C_m,i,n is the complex coefficient of the n^th-order mode, and H→_m,i,n and λ_m,i,n represent the n^th eigenvector and eigenvalue of matrices [N_m,i(ω)], respectively. The number of unknowns for each plate boundary of the plate structure in Fig. 1 is 4 flexural waves and 4 in-plane waves, requiring a total of 24 boundary conditions. The boundary conditions at x₁ = 0 are expressed as follows.

The boundary conditions at the excitation-point locations of x₁ = x₀ and x₂ =− L_x1 +x₀ are expressed as follows.

Here,

The boundary conditions at x₂ = x₃ = 0, where the two plates are coupled at an arbitrary angle, are expressed as follows.

Lastly, the four boundary conditions at x₃ = L_x2 are expressed as follows (similar to the boundary conditions at x₁ = 0).

If the displacement solution is obtained using the foregoing boundary conditions, the energy density and vibration intensity of the flexural wave and the in-plane wave of the plate can be obtained using the following equations:

Here,

where G_c,i = E_c,i/{2(1+ν_i)} and Kc,i=Ec,ihi/(1−νi2)

4.2 Energy Flow Solution of Coupled-plate Structures

The energy flow solution of the coupled plate can be obtained as in previous studies (Park et al., 2001). The Levy series solution that satisfies the y-direction boundary condition of the energy governing equation in Eqs. (3), (4), and (6) was expressed, and the energy flow solution was calculated using the boundary condition of the wave solution in the x_i direction.

Here, the wave solution of the energy density solution for each n-type wave is given by Eq. (45).

Here, Am,n,i+ represents the m^th-order mode coefficient of the wave solution traveling in the n-type ± direction in the i-plate region, and Am,n,i+.

4.3 Energy Density Comparison of Coupled-plate Structures

It was assumed that the size of the plate structure in Fig. 1 was L_x1 = L_x2 = L_y =1 m and that the coupled angle of the plate was 30°. The material of the plate was aluminum (E = 7.1 ×10¹⁰ N/m² and ρ = 2700 kg/m³), and the structural damping loss factor of the plate was set as η = 0.01 to express the spatial damping to a certain degree. All the plate thicknesses were h = 0.001 m. The vertical excitation force was applied to the center of the first plate position (L_x1, Ly) = (0.5 m, 0.5 m), and the size was 10 N. The number of modes for obtaining an approximate solution was set as 200 to improve the accuracy of the analysis. The analysis results for an exact solution at an excitation frequency of 10 kHz are shown in Figs. 2–5. The energy density of the flexural wave was the highest at the excitation point because the plate was excited with a vertical load, and it tended to decrease as the distance from the excitation point increased. Additionally, the plate had the highest energy density at the boundary where the plates were coupled at an arbitrary angle, because the in-plane wave was generated at the position where the two plates were coupled at an arbitrary angle (in contrast to the flexural wave), and the energy density decreased owing to the increased structural damping toward the inside of each plate. The propagation tendency of the vibration energy can be observed in the excitation part of the vibrational intensity distribution, as shown in Figs. 4 and 5. The modal density of the plate vibration mode with respect to the frequency is presented in Table 1. Although the theoretical modal density of a flexural wave is constant with respect to the frequency, the in-plane wave tended to increase in proportion to the frequency.

The effectiveness of the EFA, which is a statistical approach, increased in the high frequency range with a high modal density. The energy density distributions of in-plane waves for exact solutions at excitation frequencies of 30 and 100 kHz are shown in Figs. 6 and 7, respectively. The modal density increased with the frequency, increasing the mode superposition.

The comparison results for the energy density of the exact solution and the energy flow solution at different frequencies, with L_y = = 0.5 m, are shown in Figs. 8–10.

Compared with the flexural wave, the modal density was relatively low for the in-plane wave at the same frequency, as indicated by the foregoing analysis results. Therefore, although the exact solution and energy flow solution for the flexural wave in the previous studies had similar energy density distributions owing to mode superposition even in a relatively low frequency range, the spatial distributions of the energy density of the exact solution and the energy flow solution became similar at a significantly high frequency. However, the in-plane energy flow solution successfully showed the average distribution of the exact solution even in a relatively low frequency range, indicating that the EFA is extremely effective for predicting the overall vibration energy level in the middle–high frequency broadband. The size of the plate was doubled (L_x1 = L_x2 = L_y = 2 m) to examine the effects on the plate size, plate coupled angle, and structural damping loss factor, as shown in Figs. 11–13, which present comparisons of the energy density between the exact solution and the EFA solution of the model with a plate coupled angle of 45° and a structural damping loss factor η = 0.001 for the plate material. Overall, the results were similar to those for the first example, even when the plate size, coupled angle, and structural damping loss factor were different.