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Abstract

The classical laminate and lattice sandwich plate structure can be simplified into a multilayer plate system, wherein the plate components of the system are continuously joined along the transverse direction by elastic layers and can have different combinations of boundary conditions. A symplectic analytical wave propagation approach is developed for the forced vibration of a system of multiple elastically connected thin plates considering the Kirchhoff thin plate theory. The proposed method overcomes the limitation of the traditional analytical method, wherein the exact vibration field function only exists for a system with all edges of the plate components simply supported. First, the coupled partial differential equations governing the vibration of the multi-plate system are decoupled using a technique based on matrix theory; for decoupled equations, a general “vibration” state is innovatively introduced into the symplectic dual system. Next, the general “vibration” state can be analytically described in symplectic space by solving the symplectic eigenproblem and utilizing wave propagation theory. Finally, by using these analytical wave shapes and satisfying the physical boundary conditions of the system, the forced responses can be analytically calculated. In the numerical examples, the forced transverse vibrations of the double- and three-plate systems are investigated, and the cases with various combinations of boundary conditions are considered. The effectiveness of the present method is validated by comparing the present results with the results from the literature and those calculated using the finite element method. The influence of the elastic layer stiffness and number of plate components on the vibration is also investigated.

Keywords

Elastically connected plates forced vibration wave propagation symplectic method analytical solution

Introduction

The multiple-layered plate structures have broad applications in reality, such as floating-slab railway tracks, track-bridge system, sandwich plates composed of an interconnected layer, vibration absorbers, and sandwich type composite road and airport pavements. Dynamic analysis of multiple-layered plate structures is of significance to these various fields.^1–3 Therefore, it is critical to develop efficient calculation methods for vibration characteristics of a multi-plate system. Oniszczuk^4,5 formulated exact theoretical solutions describing free and forced transverse vibrations of a rectangular simply supported double-plate system by applying the classical Navier and modal expansion methods. Hedrih⁶ provided analytical solutions for the free and forced vibration of a double-plate system with a discontinuity in the elastic bonding layer of the Winker type using Bernoulli’s particular integral method and Lagrange’s method of variation constants, which were also used in their previous work.⁷ De Rosa and Lippiello⁸ calculated the free vibrations of a system consisting of a double simply supported rectangular plate using the classical Navier method. Jeong and Kang⁹ presented a theoretical dynamic model to calculate the natural frequencies of multiple clamped-clamped-free-free rectangular plates submerged in a liquid using the Rayleigh–Ritz method based on orthogonal polynomial functions generated using the Gram–Schmidt process.¹⁰ Stojanović et al.,¹¹ provided an analytical solution for the vibration analysis of systems of elastically connected thin and thick isotropic plates utilizing the trigonometric method, which was introduced by Stojanović et al.,¹² previously. Wang and Lin¹³ analyzed the influence of the elastic modulus and geometry of the elastic medium in a plate–plate system on the damping capability. Cao et al.,¹⁴ introduced a multilayer moving plate method for the dynamic analysis of viscoelastically connected infinitely long double-plate systems subjected to moving loads. Kim et al.,¹⁵ studied the free vibration of a composite laminated double-plate system using the mesh-free method, wherein all of the displacement functions, including the boundary conditions, are approximated by the mesh-free shape function. Zhang et al.,¹⁶ investigated the free vibration of a laminated composite double-plate system based on the improved Fourier series methods.

Thus, various analytical and numerical methods have been developed for the analysis of the forced vibration response of multi-plate systems, particularly for the double-plate system. A common procedure adopted in these studies requires the natural frequencies and mode shapes to be obtained initially through various methods. Subsequently, the orthogonality of the obtained mode shapes is derived and used to decouple the motion differential equations. Finally, the forced response is determined using the modal expansion method based on the natural frequencies and mode shapes obtained from the free vibration analysis. However, for the displacement function, the exact expressions only exist for the multi-plate system with all plate edges simply supported. For other combinations of plate boundary conditions, a series with refined scales or short support lengths should be chosen as the global expansion functions in the traditional methods, because the adopted displacement function cannot satisfy the plate boundary conditions accurately. This inevitably leads to an increase in the computational cost and even to a problem that is difficult to converge to the solution. In this study, we attempted to overcome this limitation by developing a new analytical wave propagation method. Additionally, to the best of our knowledge, the wave propagation method has not been reported for the forced vibration analysis of multi-plate systems built using parallel plates connected by elastic layers. In a recent study, Karlicic et al.,¹⁷ investigated the band structure in an array of vertically aligned and elastically connected structural elements such as beams, strings, plates, and other slender structures based on the Galerkin approximation and Floquet–Bloch theorem. The wave propagation method has been widely used to predict the free and forced vibrations of traditional periodic structures owing to its high efficiency and accuracy.^18–20 The wave and finite element method^21,22 has been extensively used as a popular numerical wave propagation methods based on the finite element method (FEM). However, the finite element model of the structure can be large; therefore, model reduction is essential to reduce the computational cost and avoid potential numerical issues.²² As one of the most popular analytical wave propagation methods, the method proposed by Ma et al.,²³ based on the symplectic method^23–30 can treat the boundary condition more conveniently and accurately. Compared with the wave and finite element method, the symplectic analytical wave propagation method is based on analytical waves that provide efficient and accurate solutions. In the latest work by the authors, the symplectic method was applied to the vibration analysis of a multi-beam system,³⁰ and the high accuracy and efficiency of the proposed method were validated. In this study, the symplectic analytical wave propagation method is applied for the first time for a multi-plate system built using parallel plates connected by elastic layers. To utilize the symplectic analytical wave propagation method, the coupled equations governing the vibrations of multiple plates must first be decoupled. Zhou et al.,²⁸ introduced rigorous analytical symplectic method into the free vibration of a rectangular double-layered orthotropic nanoplate system. In their study, the system governing equations were decoupled using a proper change in variables that is used frequently for double-beam system.^31–33 However, it is challenging to decouple a multi-plate system with several plate components and consider the case where the plate components have different boundary conditions.

In this study, the differential equations governing the vibration of a multi-plate system are decoupled based on the method proposed by Kelly.³⁴ A general “vibration” state, established based on decoupled differential equations, is then introduced innovatively into the symplectic dual system. By solving the derived symplectic eigenvalue problem, the wave propagation parameters and wave shapes can be analytically obtained. Subsequently, using these wave modes, the physical vibration problem can be efficiently solved in terms of a general “vibration” state in the wave space. In the numerical examples, the forced vibrations of several multi-plate structures with various combinations of boundary conditions were investigated. To validate the effectiveness of the present method, these results were compared with those from the literature and FEM. The influence of the stiffness of the elastic layer and number of plate components on the vibration response were also investigated.

Governing equations of transverse vibration

The classical laminate and lattice sandwich plate structure can be simplified into a multilayer plate system, wherein the plate components of the system are continuously joined along the transverse direction by elastic layers, as shown in Figure 1. The system of $N$ elastically connected rectangular thin plates is considered in this paper. The elastic layer connecting two adjacent plate components is modeled by uniformly distributed springs for simplicity. Without loss of generality, the classical Kirchhoff thin plate theory is adopted for the plate components. All plate components have the same width and length. The transverse vibrations of the multi-plate system are governed by the following differential equations⁵

D_{n} ∆^{2} w_{n} + m_{n} \partial^{2} w_{n} / \partial t^{2} + K_{n - 1} (w_{n} - w_{n - 1}) + K_{n} (w_{n} - w_{n + 1}) = f_{n}

(1)

for

n = 1, 2, \dots, N

, where

w_{n}

is the transverse displacement of the

n

th plate component and

w_{0} = w_{N + 1} = 0

K_{0} = K_{N} = 0

D_{n} = E_{n} h_{n}^{3} / 12 (1 - ν_{n}^{2})

is the flexural rigidity,

E_{n}

is the Young’s modulus,

h_{n}

is the thickness,

ν_{n}

is the Poisson ratio,

m_{n} = ρ_{n} h_{n}

and

ρ_{n}

is the density,

K_{n}

is the stiffness of the elastic layer between the

n

th and

n + 1

th plate component,

f_{n}

is the external force, and

∆^{2} (\cdot) = \frac{\partial^{4} (∙)}{\partial x^{4}} + 2 \frac{\partial^{4} (∙)}{\partial x^{2} \partial y^{2}} + \frac{\partial^{4} (∙)}{\partial x^{4}}

Figure 1.

Schematic of $N$ elastically connected plates system.

By assuming $f_{n} (x, y, t) = F_{n} (x, y) e^{i ω t}$ and $w_{n} (x, y, t) = W_{n} (x, y) e^{i ω t}$ in a steady vibration analysis, equation (1) becomes

D_{n} ∆^{2} W_{n} - m_{n} ω^{2} W_{n} + K_{n - 1} (W_{n} - W_{n - 1}) + k_{n} (W_{n} - W_{n + 1}) = F_{n}

(2)

where

W_{n}

and

F_{n}

are complex amplitudes,

i = \sqrt{- 1}

ω

is circular frequency.

The matrix form of equation (2) is

D ∆^{2} W + K W - ω^{2} M W = F

(3)

where

W = {\begin{array}{c} W_{1} & W_{2} & \dots & W_{N} \end{array}}^{T}

∆^{2} W = {\begin{array}{c} ∆^{2} W_{1} & ∆^{2} W_{2} & \dots & ∆^{2} W_{N} \end{array}}^{T}

D = d i a g {D_{1}, D_{2}, \dots, D_{N}}

M = d i a g {m_{1}, m_{2}, \dots, m_{N}}

, and

F = {\begin{array}{c} F_{1} & F_{2} & \dots & F_{N} \end{array}}^{T}

, the superscript “

T

” means transpose, and

K = [\begin{array}{c} K_{1} & - K_{1} & 0 & 0 & 0 \\ - K_{1} & K_{1} + K_{2} & 0 & 0 & 0 \\ 0 & 0 & ⋱ & 0 & 0 \\ 0 & 0 & 0 & K_{N - 2} + K_{N - 1} & - K_{N - 1} \\ 0 & 0 & 0 & - K_{N - 1} & K_{N - 1} \end{array}]

(4)

The differential equations governing the vibration of the multi-plate system, that is, equation (3), are coupled and difficult to be solved directly. By utilizing the method proposed by Kelly,³⁴ equation (3) can be transformed into a set of decoupled equations as

α ∆^{2} \bar{W} + Λ \bar{W} - ω^{2} \bar{W} = \bar{F} o r α ∆^{2} {\bar{W}}_{i} - α β_{i}^{4} {\bar{W}}_{i} = {\bar{F}}_{i}

(5)

where

α = D_{i} / m_{i}

β_{i} = {(\frac{{ω^{2} - λ}_{i}}{α})}^{1 / 4}

Λ = d i a g {λ_{i}}

is the eigenvalue of

\bar{K}

\bar{K} = M^{- \frac{1}{2}} K M^{- \frac{1}{2}}

\bar{W} = U^{T} M^{\frac{1}{2}} W

\bar{F} = {{\bar{F}}_{i}} = U^{T} M^{- \frac{1}{2}} F

, and

U = {U_{i}}

U_{i}

is the

i

th eigenvector corresponding to

λ_{i}, i = 1 \sim N

. Detail derivations can refer to Ref.³⁰ and is not given here for simplicity.

Symplectic analytical waves

Figure 2 shows the displacement and force of a plate component. For the $n t h$ plate component, the bending moment is $M_{x, n} = D_{n} (\frac{\partial^{2} W_{n}}{\partial x^{2}} + ν_{n} \frac{\partial^{2} W_{n}}{\partial y^{2}}), M_{y, n} = - D_{n} (\frac{\partial^{2} W_{n}}{\partial y^{2}} + ν_{n} \frac{\partial^{2} W_{n}}{\partial x^{2}})$ , the torsional moment is $M_{x y, n} = M_{y x, n} = - D_{n} (1 - v_{n}) \frac{\partial^{2} W_{n}}{\partial x \partial y}$ , the equivalent shear force is $F_{x, n} = D_{n} (\frac{\partial^{3} W_{n}}{\partial x^{3}} + (2 - ν_{n}) \frac{\partial^{3} W_{n}}{\partial x \partial y^{2}}), F_{y, n} = D_{n} (\frac{\partial^{3} W_{n}}{\partial y^{3}} + (2 - ν_{n}) \frac{\partial^{3} W_{n}}{\partial x^{2} \partial y})$ , and the angle displacement is $θ_{x, n} = \frac{\partial W_{n}}{\partial x}$ and $θ_{y, n} = \frac{\partial W_{n}}{\partial y}$ . By considering $W = Q \bar{W}$ and $Q = M^{- \frac{1}{2}} U$ , we can have $M_{x} = {M_{x, n}} = α M^{\frac{1}{2}} U (\frac{\partial^{2} \bar{W}}{\partial x^{2}} + ν \frac{\partial^{2} \bar{W}}{\partial y^{2}}), M_{y} = {M_{y, n}} = - α M^{\frac{1}{2}} U (\frac{\partial^{2} \bar{W}}{\partial y^{2}} + ν \frac{\partial^{2} \bar{W}}{\partial x^{2}})$ , $M_{x y} = {M_{x y, n}} = M_{x y} = {M_{y x, n}} = - α M^{\frac{1}{2}} U (I - ν) \frac{\partial^{2} \bar{W}}{\partial x \partial y}, F_{x} = {F_{x, n}} = α M^{\frac{1}{2}} U (\frac{\partial^{3} \bar{W}}{\partial x^{3}} + (2 I - ν) \frac{\partial^{3} \bar{W}}{\partial x \partial y^{2}}), F_{y} = {F_{y, n}} = α M^{\frac{1}{2}} U (\frac{\partial^{3} \bar{W}}{\partial y^{3}} + (2 I - ν) \frac{\partial^{3} \bar{W}}{\partial x^{2} \partial y}), Θ_{x} = {θ_{x, n}} = Q \frac{\partial \bar{W}}{\partial x}, Θ_{y} = {θ_{y, n}} = Q \frac{\partial \bar{W}}{\partial y}$ , where $ν = {v, v, \dots v}$ which means the Poisson ratio of each plate component is considered to be the same in this paper. Therefore, we can define a general “vibration” state with general transverse displacements $\bar{W} = {{\bar{W}}_{i}}$ , general angle displacements ${\bar{θ}}_{x} = {{\bar{θ}}_{x, i}}$ and ${\bar{θ}}_{y} = {{\bar{θ}}_{y, i}}$ , general moments ${\bar{M}}_{x} = {{\bar{M}}_{x, i}}$ , ${\bar{M}}_{y} = {{\bar{M}}_{y, i}}$ and ${\bar{M}}_{x y} = {{\bar{M}}_{x y, i}}$ , general shear forces ${\bar{F}}_{x} = {{\bar{F}}_{x, i}}$ , ${\bar{F}}_{y} = {{\bar{F}}_{y, i}}$ , ${\bar{Q}}_{x} = {{\bar{Q}}_{y, i}}$ , and ${\bar{Q}}_{x} = {{\bar{Q}}_{y, i}}$ , where ${\bar{θ}}_{x, i} = \frac{\partial {\bar{W}}_{i}}{\partial x}$ and ${\bar{θ}}_{y, i} = \frac{\partial {\bar{W}}_{i}}{\partial y}$ , ${\bar{M}}_{x, i} = α (\frac{\partial^{2} {\bar{W}}_{i}}{\partial x^{2}} + ν_{i} \frac{\partial^{2} {\bar{W}}_{i}}{\partial y^{2}})$ , ${\bar{M}}_{y, i} = - α (\frac{\partial^{2} {\bar{W}}_{i}}{\partial y^{2}} + ν_{i} \frac{\partial^{2} {\bar{W}}_{i}}{\partial x^{2}})$ , ${\bar{M}}_{x y, i} = {\bar{M}}_{y x, i} = - α (1 - ν_{i}) \frac{\partial^{2} {\bar{W}}_{i}}{\partial x \partial y}$ , ${\bar{F}}_{x, i} = {\bar{Q}}_{x, i} - \frac{\partial {\bar{M}}_{x y, i}}{\partial y}$ , ${\bar{F}}_{y, i} = {\bar{Q}}_{y, i} - \frac{\partial {\bar{M}}_{x y, i}}{\partial x}$ , ${\bar{Q}}_{x, i}$ and ${\bar{Q}}_{y, i}$ can be obtained from $\frac{\partial {\bar{M}}_{x, i}}{\partial x} - \frac{\partial {\bar{M}}_{y x, i}}{\partial y} - {\bar{Q}}_{x, i} = 0$ and $\frac{\partial {\bar{M}}_{y, i}}{\partial y} + \frac{\partial {\bar{M}}_{x y, i}}{\partial x} + {\bar{Q}}_{y, i} = 0$ . Since the free vibration case is considered in this section, that is, ${\bar{F}}_{i} = F_{n} = 0$ , equation (5) can be rewritten as $\frac{\partial {\bar{Q}}_{x, i}}{\partial x} + \frac{\partial {\bar{Q}}_{y, i}}{\partial y} - α β_{i}^{4} {\bar{W}}_{i} = 0$ , which means that the decoupled equations equation (5) is the governing equation of the general “vibration” state defined here.

Figure 2.

Schematic of the $n t h$ plate component and associated coordinate system.

From the relationships between the general displacements and general forces, the general “vibration” state corresponding to the $i$ th decoupled equation equation (5) can be transformed into the symplectic dual system as $\frac{\partial z_{i}}{\partial y} = H_{i} z_{i}$ , where the state vector $z_{i} = {\begin{array}{c} {\bar{W}}_{i} & {\bar{θ}}_{y, i} & {\bar{F}}_{y, i} & {\bar{M}}_{y, i} \end{array}}^{T}$ , and Hamiltonian matrix operator $H_{i}$ is given by

H_{i} = [\begin{array}{c} 0 & 1 & 0 & 0 \\ - ν_{i} \frac{\partial^{2}}{\partial x^{2}} & 0 & 0 & - \frac{1}{α} \\ - α (1 - ν_{i}^{2}) \frac{\partial^{4}}{\partial x^{4}} + α β_{i}^{4} & 0 & 0 & ν_{i} \frac{\partial^{2}}{\partial x^{2}} \\ 0 & 2 α (1 - ν_{i}) \frac{\partial^{2}}{\partial x^{2}} & - 1 & 0 \end{array}]

(6)

At this point, equation (5) has been translated to the symplectic space. Therefore, the general “forced” response of the general “vibration” state defined here can be obtained by using the wave propagation method, as the solution of single plate system described in physical space.²³ The required wave modes, that is, wave parameters $μ_{y, i}$ and wave shapes $η_{i} (x)$ , are obtained analytically through the symplectic method. Brief solution procedure is given in the Appendix, and detail procedure can refer to Ref.²³ for simplicity. It should be noted that the wave modes will now be determined by satisfying the general “boundary conditions” at the two opposite edges $x = 0, a$ . For the multi-plate system considered in this paper, the plate components can have different boundary conditions with each other on the same side along the $x$ -direction. Along the $y$ -direction, the boundary conditions of all the plates on the same side are required to be the same with each other. The boundary condition along each of the edges $x = 0$ and $x = a$ is taken as one of the following: for a clamped edge (C), ${W_{n} |}_{x = 0 o r a} = 0$ , ${\frac{\partial W_{n}}{\partial x} |}_{x = 0 o r a} = 0$ ; for a simply supported edge (S), ${W_{n} |}_{x = 0 o r a} = 0$ , ${\frac{\partial^{2} W_{n}}{\partial x^{2}} |}_{x = 0 o r a} = 0$ ; for a free edge (F), ${D_{n} (\frac{\partial^{2} W_{n}}{\partial x^{2}} + ν_{n} \frac{\partial^{2} W_{n}}{\partial y^{2}}) |}_{x = 0 o r a} = 0$ , ${D_{n} (\frac{\partial^{3} W_{n}}{\partial x^{3}} + (2 - ν_{n}) \frac{\partial^{3} W_{n}}{\partial x \partial y^{2}}) |}_{x = 0 o r a} = 0$ . By considering $W = Q \bar{W}$ and $Q$ is linear independent, the general “boundary conditions” along the two edges $x = 0, a$ can be obtained in terms of the general coordinate $\bar{W}$ as:

For a clamped edge (C)

{{\bar{W}}_{i} |}_{x = 0 o r a} = 0, {\frac{\partial {\bar{W}}_{i}}{\partial x} |}_{x = 0 o r a} = 0

(7)

for a simply supported edge (S)

{{\bar{W}}_{i} |}_{x = 0 o r a} = 0, {\frac{\partial^{2} {\bar{W}}_{i}}{\partial x^{2}} |}_{x = 0 o r a} = 0

(8)

for a free edge (F)

{(\frac{\partial^{2} {\bar{W}}_{i}}{\partial x^{2}} + ν_{i} \frac{\partial^{2} {\bar{W}}_{i}}{\partial y^{2}}) |}_{x = 0 o r a} = 0, {(\frac{\partial^{3} {\bar{W}}_{i}}{\partial x^{3}} + (2 - ν_{i}) \frac{\partial^{3} {\bar{W}}_{i}}{\partial x \partial y^{2}}) |}_{x = 0 o r a} = 0

(9)

Once the general “boundary conditions” is determined, the wave propagation parameter $μ_{y, i}$ and wave shapes $η_{i} (x)$ can be obtained accordingly following the procedure given in Appendix or the detail procedure given in Ref. ²³ Then the general “vibration” state can be described in terms of waves traveling in the positive and negative directions as

z_{i} = A_{i}^{+} c_{i}^{+} + A_{i}^{-} c_{i}^{-} = A_{i} c_{i}

(10)

where

A_{i} = [\begin{array}{c} A_{i}^{+} & A_{i}^{-} \end{array}] = [\begin{array}{c} η_{i, 1} (x) & η_{i, 2} (x) & \dots & η_{i, 2 g} (x) \end{array}]

is the wave shape matrix,

g

is the number of positively traveling waves, and

c_{i} = {[\begin{array}{c} c_{i}^{+} & c_{i}^{-} \end{array}]}^{T}

are the wave amplitudes. The wave shape matrix satisfies the adjoint symplectic orthogonal relations as

\int_{0}^{a} A_{i}^{T} J_{2} A_{i} d x = J_{g}

(11)

where

J_{2} = [\begin{array}{c} 0 & I_{2} \\ - I_{2} & 0 \end{array}]

J_{g} = [\begin{array}{c} 0 & I_{g} \\ - I_{g} & 0 \end{array}]

, and

I_{2}

I_{g}

are identity matrices of size two and

g

Forced response in terms of wave propagation

The forced “vibration” response of the general “vibration” state described by equation (5) can also be expressed in terms of waves, following the procedure used in Ref.²³ as shown in Figure 3. By using the wave propagation theory, equation (10) can be rewritten as

z_{i} (x, y_{r e s, i}) = A_{i} T_{i} (y_{r e s, i}) a_{i} + A_{i} {\hat{T}}_{i} (y_{r e s, i}) e_{i}, y_{r e s, i} \leq y_{e, i} z_{i} (x, y_{r e s, i}) = A_{i} T_{i} (y_{r e s, i}) a_{i} + A_{i} T_{i} (y_{r e s, i}) e_{i}, \dots y_{r e s, i} > y_{e, i}

(12)

where

T_{i} = d i a g {\tilde{T_{i}} (y), \tilde{T_{i}} (b - y)}, {\tilde{T}}_{i} (y) = d i a g {e^{μ_{y, i, 1} y}, e^{μ_{y, i, 2} y}}, {e^{μ_{y, i, g} y}}, a_{i} = {[\begin{array}{c} a_{i, 0}^{+ T} & a_{i, b}^{- T} \end{array}]}^{T}

a_{i, 0}^{+}

and

a_{i, b}^{-}

are the amplitudes of the positively traveling waves at

y = 0

and of the negatively traveling waves at

y = b

, respectively.

e_{i} = {\begin{array}{c} e_{i}^{+ T} & e_{i}^{- T} \end{array}}^{T}

is the directly excited waves,

y_{e, i}

is the

y

-coordinate of the excitation

f_{i}^{e}

{\hat{T}}_{i} = d i a g {0, {\tilde{T}}_{i} (y_{e, i} - y_{r e s, i})}

T_{i} = d i a g {{\tilde{T}}_{i} (y_{r e s, i} - y_{e, i}), 0}

, where the sub-matrix

0

has the same order as

{\tilde{T}}_{i}

. The directly excited waves

e_{i}

can be obtained by considering the continuity of general displacement and balance of general force at the excitation location²³ as

{\begin{array}{c} e_{i}^{+} \\ - e_{i}^{-} \end{array}} = - J_{g} \int_{0}^{a} A_{i}^{T} J_{2} {\begin{array}{c} 0 \\ f_{i}^{e} \end{array}} d x

(13)

where

f_{i}^{e} = {[\begin{array}{c} {\bar{F}}_{i} & 0 \end{array}]}^{T}

Figure 3.

Schematic of wave propagation description for the general forced response.

Wave amplitudes $a_{i}$ are obtained by considering the physical boundary conditions at $y = 0, b$ . For the multiple connected plates system investigated in this paper, the boundary conditions of each plate component on the same side at $y = 0$ or $y = b$ can be different. By considering $W = Q \bar{W}$ , $θ_{y} = Q {\bar{θ}}_{y}$ , $F_{y} = S {\bar{F}}_{y}$ , $M_{y} = S {\bar{M}}_{y}$ , and $S = M^{\frac{1}{2}} U$ , the physical boundary conditions of the $n$ th plate at $y = 0$ and $y = b$ can be expressed in terms of wave as

E_{n, l} {\hat{Q}}_{n} (P_{l} a + {\tilde{P}}_{l} e) = 0, = 0 E_{n, r} {\hat{Q}}_{n} (P_{r} a + {\tilde{P}}_{r} e) = 0, y = b

(14)

where

{\hat{Q}}_{n} = [Q_{n, 1}, Q_{n, 2}, \dots, Q_{n, N}]

Q_{n, i} = d i a g {Q_{n, i}, Q_{n, i}, S_{n, i}, S_{n, i}}

, the subscripts “

l

” and “

r

” mean the left and right end of each plate component,

a = {[a_{1}^{T}, a_{2}^{T}, \dots, a_{N}^{T}]}^{T}

P_{l} = d i a g {A_{1} T_{1} (0), A_{2} T_{2} (0), \dots, A_{N} T_{N} (0)}

{\tilde{P}}_{l} = d i a g {A_{1} {\hat{T}}_{1} (0), A_{2} {\hat{T}}_{2} (0), \dots, A_{N} {\hat{T}}_{N} (0)}

P_{r} = d i a g {A_{1} T_{1} (b), A_{2} T_{2} (b), \dots, A_{N} T_{N} (b)}

e = {[e_{1}^{T}, e_{2}^{T}, \dots, e_{N}^{T}]}^{T}

{\tilde{P}}_{r} = d i a g {A_{1} T_{1} (b), A_{2} T_{2} (b), \dots, A_{N} T_{N} (b)}

, and

E_{n, l}

and

E_{n, r}

are the indexes of boundary condition on the left and right edge of the

n

th plate component which can be different for each plate. For the clamped, simply supported, and free cases, there are

E_{n, l} = d i a g {1,1,0,0}

E_{n, l} = d i a g {1,0,0,1}

, and

E_{n, l} = d i a g {0,0,1,1}

, respectively. By utilizing equation (11), pre-multiplying both sides of equation (14) by

A_{n}^{+ T} J

and integrating from

0

a

leads to

L_{n, l} \hat{X} a = - L_{n, l} \hat{V} e, L_{n, r} X a = - L_{n, r} V e

(15)

where

L_{n, l} = \int_{0}^{a} A_{n}^{+ T} J E_{n, l} {\hat{Q}}_{n} \hat{A} d x

L_{n, r} = \int_{0}^{a} A_{n}^{+ T} J E_{n, r} {\hat{Q}}_{n} \hat{A} d x

\hat{A} = d i a g {A_{1}, A_{2}, \dots, A_{N}}

\hat{X} = d i a g {T_{1} (0), T_{2} (0), \dots, T_{N} (0)}

X = d i a g {T_{1} (b), T_{2} (b), \dots, T_{N} (b)}

\hat{V} = d i a g {{\hat{T}}_{1} (0), {\hat{T}}_{2} (0), \dots, {\hat{T}}_{N} (0)}

, and

V = d i a g {T_{1} (b), T_{2} (b), \dots, T_{N} (b)}

. By considering the boundary conditions along the two edges at

y = 0

and

y = b

for all the

N

plate components, from equation (15), we have

C a = - \tilde{C} e

(16)

where

C = {[{(L_{1, l} \hat{X})}^{T}, {(L_{2, l} \hat{X})}^{T}, \dots, {(L_{N, l} \hat{X})}^{T}, {(L_{1, r} X)}^{T}, {(L_{2, r} X)}^{T}, \dots, {(L_{N, r} X)}^{T}]}^{T}

\tilde{C} = {[{(L_{1, l} \hat{V})}^{T}, {(L_{2, l} \hat{V})}^{T}, \dots, {(L_{N, l} \hat{V})}^{T}, {(L_{1, r} V)}^{T}, {(L_{2, r} V)}^{T}, \dots, {(L_{N, r} V)}^{T}]}^{T}

. Then, the wave amplitude

a

can be obtained from equation (16) as

a = - C^{- 1} \tilde{C} e

(17)

Once the wave amplitude $a$ is obtained, the forced response can be obtained by considering equation (12) and $W = Q \bar{W}$ , $θ_{y} = Q {\bar{θ}}_{y}$ , $F_{y} = S {\bar{F}}_{y}$ and $M_{y} = S {\bar{M}}_{y}$ . It is also noted that by searching peak response, the natural frequencies can be found. Considering free vibration with $e = 0$ and substituting the natural frequencies into equation (16) lead to the wave amplitude $a$ which can be used to calculate the correspongding modal shape.

Numerical examples

The forced vibrations of an elastically connected double-plate system and a three-plate system were investigated to validate the proposed method. An external unit point force is applied at $x_{e, 1} = 0.5 a$ and $y_{e, 1} = 0.5 b$ of the first plate component, this is, $F_{1} = 1$ , $F_{n \neq 1} = 0$ . The displacement response at the input point was calculated. The parameters of the double-plate system used in Refs.^4,5 were used here for each plate component: $E = 1 \times 10^{10}$ N m², $ν = 0.3$ , $h = 1 \times 10^{- 2}$ m, $ρ = 5 \times 10^{3}$ kg m⁻³, and $K_{n} = k = 0.6 \times 10^{5}$ N m⁻³. For the double-plate system, $a = 1$ m, $b = 2$ m, and simply supported (S) boundary conditions were considered for all plate edges. The frequency range of the excitation was uniformly distributed from 1 to 140 Hz with a frequency interval 1 Hz. For the three-plate system, $a = 0.2$ m, $b = 0.8$ m, and two types of combinations of boundary conditions were considered for the edges at $y = 0, b$ , this is, SSS-SSS and SCS-FSC; the edges of all the plate components at $x = 0, a$ were simply supported.

Double-plate system

The eigenvalues of $\bar{K}$ of the double-plate system can be obtaiend as $λ_{1,2} = 0,2 k / m$ , and the corresponding eigenvectors $U_{1,2} = \frac{1}{\sqrt{2}} {\begin{array}{c} 1 \\ \pm 1 \end{array}}$ . To validate the present method, the FEM results using ABAQUS were obtained, where the shell element S4R and mode superposition method were used. The stiffness of each spring is $k a b / N_{s}$ , where $N_{s}$ is the number of springs that are uniformly distributed and used to model the elastic layer between two adjacent plate components. Convergence analysis is carried out to determine the numbers of spring, element, and truncated modes used in the FEM method. Figure 4 shows the magnitude of the input point displacement calculated using the proposed method with different wave numbers. From the convergence analysis, the results calculated by the FEM with mesh sizes of 0.006 m, 800 modes, and 105 springs can be seen as a convergent result, and the result calculated by the present method with 80 waves is convergent, as shown in Figure 5; therefore, the results from the proposed method are in agreement with the FEM results. Discrepancies occurred only at certain resonant frequencies because damping was not considered in the plate system. Consequently, a minor difference between the natural frequencies obtained from the present analytical method and those obtained from the FEM will induce apparent discrepancies in the forced vibration response around the resonant frequencies. Damping can be considered directly in the present method in terms of the complex stiffness, that is, $E_{n} = E_{0} (1 + i η)$ , where $η$ is the damping loss factor. Figure 6 shows the magnitude of the transverse deformation considering the damping loss factor $η = 0.01$ . From Figures 5 and 6, consideration of damping reduces the difference between the present results and FEM results at the resonance points. However, the apparent discrepancies still appeared at approximately 12 and 23 Hz; this could be attributed to the frequency interval adopted in the analysis being large. Figure 7 shows the forced response obtained by the proposed method with a smaller frequency interval of 0.05 Hz. From Figure 7, the first four natural frequencies can be obtained by searching for a peak response. Table 1 gives the natural frequencies calculated by the present method, the FEM method, and from Figure 6 of Ref.⁵ As can be seen, the natural frequency obtained by the present method agrees accurately with the results shown in Figure 6 of Ref.⁵ For the FEM results, compared with the first and third natural frequencies, which are synchronous modes, the second and fourth natural frequencies, which are asynchronous modes, present a relatively evident deviation from the exact results given by the present results and Ref.⁵; this leads to a pronounced discrepancy in the vibration response around these frequencies as shown in Figures 5 to 7. For the first and third modes, the natural frequencies are not dependent on the elastic layers because each plate component vibrates in the same direction with the same deformation, and the boundary conditions at the left end of each plate are the same as those at the right end. For the second and fourth modes, the natural frequencies are dependent on elastic layers and influenced by the spring distribution in the FE model. It is believed that more accurate results can be obtained by increasing the number of springs used in the finite element model.

Figure 4.

Magnitude of the input point displacement calculated by the present method with different wave numbers.

Figure 5.

Magnitude of the input point displacement without damping from the FEM with mesh size being 0.006 m, 800 modes and 105 springs, and from the present method with 80 waves.

Figure 6.

Magnitude of the input point displacement with damping loss factor $η = 0.01$ from the FEM with mesh size being 0.006 m, 800 modes and 105 springs, and from the present method with 80 waves.

Figure 7.

Magnitude of the input point displacement obtained by the present method and ABAQUS with frequency interval of 0.05 Hz.

Table 1.

The first four natural frequencies obtained by the present method and FEM, and obtained from Figure 6 of Ref.⁵

Method	Natural frequencies (rad/s)
Method	1	2	3	4
Present	52.78	71.94	137.29	145.77
Ref⁵	52.8	72.0	137.3	145.8
FEM	52.787	75.575	137.2	147.48

Three-plate system

The eigenvalues of $\bar{K}$ of the three-plate system can be obtained as $λ_{1,2,3} = 0$ , $K / m$ , and $3 K / m$ , and the corresponding eigenvectors are

U_{1} = \frac{1}{\sqrt{3}} {\begin{array}{c} 1 \\ 1 \\ 1 \end{array}}, U_{2} = \frac{1}{\sqrt{2}} {\begin{array}{c} 1 \\ 0 \\ 1 \end{array}}, U_{3} = \frac{1}{\sqrt{6}} {\begin{array}{c} 1 \\ - 2 \\ 1 \end{array}}

(18)

SSS-SSS

Figure 8 shows the magnitude of the input point displacement obtained by the proposed method and FEM without damping. In the present method, the results calculated using 40 waves were convergent. The number of waves used here was significantly reduced compared with that in the double-plate example structure. Because the length/width ratio of the plate in the three-plate system is larger than that of the double-plate system, a part of the near-field wave components can be ignored. In the FEM results, 105 springs were used to model each elastic layer, 200 modes were used in the mode superposition method, and the mesh size was 0.0025 m. The analysis frequency range was from 1 to 500 Hz with a frequency interval of 1 Hz. As shown in Figure 8, the present results are in good agreement with the FEM results. However, discrepancies were observed at some frequencies; the reason for this had been discussed in Section 6.1. To carry out an in-depth exploration, a more precise discretization of the analysis frequency is considered. Figure 9 shows the magnitude of the input point displacement obtained by the proposed method and FEM with a frequency interval of 0.01 Hz. As shown in Figure 8, more peaks appeared when the frequency interval was reduced. The first peak from the FEM and proposed method in Figure 9(a) and (b) correspond to the synchronous natural modes. In the in-phase motion, as the springs are undeformed, the frequencies and mode shapes are the same as those of a single isolated plate. For these natural modes, exact solutions exist for the natural frequencies, that is, $ω_{m n} = π^{2} \sqrt{\frac{D}{ρ h}} {{(\frac{m}{a})}^{2} + {(\frac{n}{b})}^{2}}$ , where $m$ and $n$ are the number of half-waves within width $a$ and length $b$ , respectively. The exact result for the natural mode $(m = 1, n = 1)$ , which is the first synchronous mode, is 178.56 Hz, and that for the third synchronous natural mode $(m = 1, n = 3)$ is 262.59 Hz. Therefore, the results of the present method agree with the exact results. The natural frequencies calculated by the FEM had minor discrepancies with the results of the present method. The two mode shapes obtained using these two methods are listed in Table 2. For the present results, only the mode shape of the first plate component is provided for simplicity because the other two plates have the same deformation in these modal shapes. Therefore, the mode shapes from the proposed method and FEM are in agreement with each other. As can also be noted that the natural frequencies are locally densified along the frequency axis. Table 3 gives the mode shapes corresponding to the three dense natural frequencies clustered around 177.8 Hz and 260.8 Hz, respectively. As can be seen, for the SSS-SSS case, the mode shapes exhibit a clear regularity. The first mode of each clustered natural frequencies is synchronous. For the second mode, the first and third plates are asynchronous, and the middle plate, that is, the second plate, keeps motionless. For the third mode, the first and third plates are synchronous, and the middle plate vibrates in the opposite direction.

Figure 8.

Magnitude of input point displacement obtained by the present method with 40 waves and FEM with 105 springs, 200 modes and mesh size 0.0025 m.

Figure 9.

Magnitude of the input point displacement obtained by the present method and FEM with frequency interval of 0.01 Hz.

Table 2.

Mode shape obtained by the present method and FEM for the 1st and 3rd synchronous mode.

	1st synchronous mode	3rd synchronous mode
FEM
FEM	177.7 Hz	260.8 Hz
Present $W_{1}$
Present $W_{1}$	178.56 Hz	262.59 Hz

Table 3.

Mode shapes corresponding to the three dense natural frequencies clustered around 177.8 Hz and 260.8 Hz, respectively.


177.74 Hz	177.84 Hz	178.05 Hz

260.78 Hz	260.85 Hz	260.99 Hz

SCS-FSC

Figure 10 gives the geometry configuration and boundary conditions of the SCS-FSC case, the blue edges indicate simply supported boundaries, the red edges indicate clamped boundaries, the purple lines represent the springs. Figure 11 shows the magnitude of the input point displacement obtained by the proposed method and FEM for the SCS-FSC case. In the proposed method, 40 waves were used. In the FEM results, 105 springs were used to model each elastic layer, 200 modes were used in the mode superposition method, and mesh size was 0.0025 m. The analysis frequency range was from 1 to 500 Hz with a frequency interval of 1 Hz. As shown in Figure 10, the present results agree well with the FEM results. However, there were discrepancies at approximately 270 and 450 Hz. To investigate these differences, the frequency interval 0.01 Hz was adopted. Figure 12 shows the forced response at approximately 270 Hz obtained using the present method and FEM. As can be seen, compared with Figure 11, more peaks appeared when the frequency interval was reduced. In contrast to the previous case of SSS-SSS where boundary conditions of each plate were the same, the boundary conditions of each plate were different for the SCS-FSC case. Therefore, there were no synchronous modes in the SCS-FSC case, and the elastic layer affected the natural frequencies of each mode. Table 4 lists the mode shape corresponding to the first peak, shown in Figure 12, obtained from the present method and FEM. Therefore, the mode shapes from the proposed method and FEM are in good agreement with each other. Table 5 shows the mode shapes of the first nine natural frequencies of the SCS-FSC case. The regularity is significantly different from that for the SSS-SSS case shown in Table 3, mainly because that the right end of the first plate here is free.

Figure 10.

Geometry configuration and boundary conditions of the SCS-FSC case, the blue edges indicate simply supported boundaries, the red edges indicate clamped boundaries, and the purple lines represent the springs (refer to the electronic version for color images).

Figure 11.

Magnitude of input point displacement obtained by the present method with 40 waves and FEM with 105 springs, 200 modes and mesh size 0.0025 m.

Figure 12.

Magnitude of the input point displacement obtained by the present method and FEM with frequency interval of 0.01 Hz.

Table 4.

Mode shape obtained by the present method and FEM for the mode corresponding to the first peak shown in Figure 12.

	FEM (269.82 Hz)	Present (271.97 Hz)
$W_{1}$
$W_{2}$
$W_{3}$

Table 5.

Mode shapes of the first nine natural frequencies of the SCS-FSC case.


168.56 Hz	179.00 Hz	179.23 Hz

188.56 Hz	213.45 Hz	213.64 Hz

227.90 Hz	269.82 Hz	269.97 Hz

Parameter analysis

The influence of the stiffness of the elastic layer on the vibration response is investigated in this section. The damping loss factor $η =$ 0.01 is adopted. Structures consisting of only one plate component and two-plate components can be obtained directly by deleting two plates and one plate in the three-plate system of SSS-SSS, respectively. The stiffness of the elastic layer was changed to $K_{n} = 10000 k = 0.6 \times 10^{9}$ N m⁻³ Figure 13 shows the magnitude of the transverse deformation of the first plate component of the three structures calculated using the proposed method. As can be seen, with the increasing of the number of plate components, the vibration around the resonant frequencies of the one-plate system can be significantly reduced. Compared with the one-plate system, more secondary resonant peaks appear in structures with more plate components owing to the multimode character of the multi-plate system. Compared with the two-plate system, the three-plate system causes a significant reduction in the vibration around the primary resonant peaks, such as at 179, 263, 431, and 683 Hz; however, it shows a narrower frequency band of lower vibration before 744 Hz.

Figure 13.

Magnitude of the input point displacement obtained by the present method for the one-, two-, and three-plate system.

The influence of stiffness $K_{n}$ of the elastic layer between two adjacent plate components on the vibration response of the first plate component of the two-plate system was now investigated. Figure 14 shows the magnitude of the transverse deformation of the first plate component calculated with different stiffness, $K_{n}$ , for the two-plate system using the proposed method. Therefore, with the increasing of the stiffness $K_{n}$ , the secondary resonant peaks of the two-plate system move to the right along the frequency axis. That is, on the one hand, the multi-plate system can effectively reduce the resonant peaks of the one-plate system, and on the other hand, by adjusting the stiffness, the negative effect of the secondary resonant peaks on the vibration reduction of the multi-plate system can be effectively reduced to reduce the vibration in a wider frequency band. For example, the vibration response of a one-plate system is significantly reduced by a two-plate system with stiffness $K_{n} = 10000 k$ before approximately 748 Hz, which is a wide frequency band. A similar trend can be concluded in Figure 15, which shows the magnitude of the transverse deformation of the first plate calculated with different stiffness for the three-plate system using the present method. And then, we considered different stiffness for the two elastic layers of the three-plate system. Assuming $K_{2} = ϵ K_{1} = 10000 ϵ k$ , Figure 16 gives the magnitude of the transverse deformation of the first plate calculated with different $ϵ$ for the three-plate system using the present method. As can be seen, with the increasing of the stiffness $K_{2}$ and keeping $K_{1}$ unchanged, the secondary resonant peaks of the three-plate system move to the right along the frequency axis.

Figure 14.

Magnitude of the transverse deformation of the first plate calculated with different stiffness $K_{n}$ for the two-plate system by the present method.

Figure 15.

Magnitude of the transverse deformation of the first plate calculated with different stiffness $K_{n}$ for the three-plate system by the present method.

Figure 16.

Magnitude of the transverse deformation of the first plate calculated with different $ϵ$ for the three-plate system by the present method.

Conclusions

A symplectic analytical wave-based method was developed for the forced vibration of an elastically connected multi-plate system. Using the decoupling technique based on the eigensolution of the elastic coupling matrix, a series of independent equations were obtained and introduced into the symplectic dual system. The solution of the forced responses of the multi-plate system in the physical space was subsequently transformed into wave space. By searching for the peak response, the natural frequencies and mode shapes could also be obtained using the proposed method. In the numerical examples, the forced responses of the double- and three-plate systems with various combinations of boundary conditions were calculated. Comparisons of the present results with the results from the literature and FEM validated the high accuracy of the present method. The influence of the stiffness of the elastic layer and number of plate components on the vibration response was also investigated. With an increase in the number of plate components, the vibration around the primary resonant peaks of the first plate significantly reduced; however, more secondary resonant peaks appeared. With an increase in the stiffness of the elastic layer, the secondary resonant peaks due to the addition of the plate component moved to larger frequencies, which further increased the vibration isolation effect of the multi-plate system.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number: 12072280).

ORCID iD

Qingfeng Cheng

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Symplectic analytical solution for forced vibration of a multilayer plate system

Abstract

Keywords

Introduction

Governing equations of transverse vibration

Symplectic analytical waves

Forced response in terms of wave propagation

Numerical examples

Double-plate system

Three-plate system

SSS-SSS

SCS-FSC

Parameter analysis

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

Appendix

References