Stochastic Optimal Bounded Parametric Control of Periodic Viscoelastomer Sandwich Plate with Supported Mass Based on Dynamical Programming Principle

Abstract

The sandwich plate (SP) with supported mass can model structural systems such as platform or floor with installed vibration-sensitive apparatus under random loading. The stochastic optimal control (in time domain) of periodic (in space) viscoelastomer (VE) SP with supported mass subjected to random excitation is an important research subject, which can fully use VE controllability, but it is a challenging problem on optimal bounded parametric control (OBPC). In this paper, a stochastic OBPC for periodic VESP with supported mass subjected to random base loading is proposed according to the stochastic dynamical programming (SDP) principle. Response-reduction capability using the proposed OBPC is studied to demonstrate further control effectiveness of periodic SP via SDP. Controllable VE core modulus of SP is distributed periodically in space. Differential equations for coupling vibration of periodic SP with supported mass are derived and transformed into multi-dimensional system equations with parameters as nonlinear functions of bounded control. The OBPC problem is established by the system equations and performance index with bound constraint. Then, an SDP equation is derived according to the SDP principle. The OBPC law is obtained from the SDP equation under bound constraint. Optimally controlled responses are calculated and compared with passively controlled responses to evaluate control effectiveness. Numerical results on responses and statistics of SP via the proposed OBPC show further remarkable control effectiveness.

1. Introduction

Stochastic vibrance mitigation has great significance for engineering structures such as buildings under strong wind or earthquake loading [1,2,3,4]. For instance, a vibration-sensitive apparatus is installed on a platform or floor under random excitations, and its vibration needs to be effectively controlled. The beam or plate with supported mass has been proposed to represent the structural system [5]. Stochastic vibration of the supported mass can be suppressed by controlling the floor and building using properties-adjustable devices or smart buildings. Presently, the floor vibration control is considered for the mass vibration mitigation. A composite structure was designed for its vibration suppression. Properties-adjustable materials, e.g., magnetorheological (MR) viscoelastomer (VE), have been developed to use as the cores of composite structures. The structural properties can rapidly change by MRVE under applied magnetic fields [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. The vibration-mitigation effectiveness of sandwich beams with MRVE cores has been studied [7,8,9,10,11,12]. Sandwich plates (SPs) with MRVE cores have also been studied in regard to their characteristics [13,14,15,16,17], random responses [18], core position optimization [19,20], and stability [21,22]. In the reported research, MRVE was considered to have fully uniform mechanical properties or locally uniform mechanical properties in plate plane, and it was used only as a passive control material (e.g., applied magnetic fields and then mechanical properties were unvaried with time in space).

A periodic structure (having parameters distributed periodically in space) has particular characteristics, and the periodicity design is applicable to improve structural dynamics [23,24]. A periodic composite structure with controllable VE cores has dynamic properties that are adjustable by only external action (e.g., applied magnetic fields for MRVE) in space, and its construction keeps unchanged. For example, a periodic SP with controllable VE core under random excitations was analyzed, and it was observed that their dynamics characteristics are adjusted greatly via periodic parameter variation [25,26]. In the reported research, the VE was considered only as spatially adjustable, and its dynamic properties were still unvaried with time. The spatial adjustability of VE sandwich structures has a certain limit. To further improve composite structure dynamics and vibration control, the VE needs to be considered as dynamic properties varying simultaneously with time in space for full use of its controllability. The stochastic optimal control (in time domain) of periodic (in space) VESP under random excitation has not been studied and will be a challenging problem in regard to optimal bounded parametric control (OBPC). The analytical solution to the problem is an alternative for stochastic response estimation of periodic composite structures [27,28,29,30,31].

For a controllable VE, the control variable is its material constants, such as shear modulus. Then, a periodic VE sandwich structure has its parameters in vibration equations containing the control variable (in time domain), and the parameters depend nonlinearly on control [32]; that is, the stochastic vibration suppression for periodic VE sandwich structure results in a parametric nonlinear control problem for a multi-dimensional structure system after space discretization. The stochastic dynamical programming (SDP) principle is applicable to determine an optimal control [33,34,35,36,37,38,39,40]. Structural controls as external actions (not in parameters) have been studied extensively. Parametric controls are different from external controls. For instance, the linear quadratic Gaussian control for the latter cannot satisfy the SDP equation for the former. The optimal parametric control of uniform MRVE sandwich beams subjected to random excitation was studied [32]. However, the stochastic OBPC for a periodic VE composite structure has not been studied. A parametric control such as the VE modulus has a certain bound due to the limit of dynamic properties (e.g., magnetic–mechanical saturation for MRVE), and the bound will be incorporated in control design [6,14,32,35,40].

This paper studies the stochastic OBPC of periodic VESP with supported mass subjected to random base loading and evaluates the response reduction effectiveness of SP using the optimal control by comparing with passive control. First, consider the VE modulus of SP as distributed periodically in space. Then, derive the differential equations for coupling vibration of periodic SP and supported mass and transform the equations into multi-dimensional system equations with parameters dependent nonlinearly on bounded control. Second, apply the SDP principle to the OBPC problem described by the equations and performance index with bound constraint to obtain an SDP equation. Determine the OBPC law in the time domain by the SDP equation under bound constraint. After the substitution of OBPC, calculate and estimate the optimally controlled SP responses to evaluate the control effectiveness. Finally, the numerical results on responses and statistics of SP via the proposed OBPC and passive control (i.e., constant control value) are shown and discussed.

2. Differential Equations for Controlled Periodic SP and Supported Mass

A VESP with supported mass is considered to model a composite platform or floor with installed apparatus under random base loading, as shown in Figure 1. The modulus of the controllable VE core layer is firstly designed as a spatial periodic function. Then, the modulus amplitude, as a control variable, is adjusted within a certain bound in the time domain. The stochastic vibration control of the periodic VESP with supported mass is a parametric nonlinear control problem. The SDP principle is applied to determine a stochastic OBPC.

The rectangular plate has a length of a and width of b. The two face layers have an equal Young’s modulus of E₁, Poisson’s ratio of μ, density of ρ₁, and thickness of h₁. The middle layer has a density of ρ₂ and thickness of h₂. The plate has mass per area of ρh_t = 2ρ₁h₁ + ρ₂h₂, where h_t = 2h₁ + h₂, and supports a concentrated mass of m₀ × ab. The boundary base has vertical displacement of w₀, which is random excitation, due to environmental disturbances. The fundamental assumptions on SP include the following: (1) surface-layer material is isotropic, and middle material is transversely isotropic; (2) normal stresses of middle layer are omitted; (3) z-axis normal stresses of surface layers are omitted; (4) z-axis displacements are equal; (5) cross-sections of every layers remain planar; (6) inertias, except for the vertical, are omitted; and (7) interfaces between surface and middle layers are continuous [18,26,41,42].

The dynamics characteristics of the VE core are controllable (e.g., MRVE stiffness with damping controllable via applied magnetic field). The Young’s modulus of the middle layer is omitted because it is very low compared with the face layers. The shear deformation of the middle layer is high compared with the face layers and taken account. According to the viscoelastic dynamic stress–strain relation [43], the shear stresses, τ_2xz and τ_2yz, of the middle layer expressed via the corresponding shear strains, γ_2xz and γ_2yz, are

$${\tau }_{2xz}=G_{21}{\gamma }_{2xz}+G_{c1}{\dot{\gamma }}_{2xz},$$

(1)

$${\tau }_{2yz}=G_{21}{\gamma }_{2yz}+G_{c1}{\dot{\gamma }}_{2yz},$$

(2)

where the overdot, “·”, is the operation of time derivative, G₂₁ is the static shear modulus, and G_c1 is the dynamic shear constant. The modulus constants are adjustable by external action and, thus, are designed firstly as the spatial period and secondly as varying with time. As periodic functions of coordinates x and y, they are as follows [25,26]

$$G_{21}=e_{a1}(1+\beta \cos {\displaystyle \frac{2k_a\pi x}{a}}\cos {\displaystyle \frac{2k_b\pi y}{b}}),$$

(3)

$$G_{c1}=\xi G_{21},$$

(4)

where e_a1 is the non-periodic modulus component, β is the ratio of periodic part amplitude to non-periodic part, k_a and k_b are the wave numbers of the periodic component, and ξ is the ratio of G_c1 to G₂₁. The non-periodic modulus component, e_a1, is controllable and, thus, is considered to be the control variable, e_a1(t) (t is time). The other parameters are invariant with time. The control, e_a1(t), is bounded due to material saturation properties.

For the SP, vertical displacement relative to the boundary support base is w = w(x,y,t). Horizontal displacements (x-axis and y-axis directions) of face layers (top and bottom) are represented by

$$u_1(x,y,z_1,t)=u_{10}(x,y,t)-z_1{\displaystyle \frac{\partial w}{\partial x}},$$

(5)

$$v_1(x,y,z_1,t)=v_{10}(x,y,t)-z_1{\displaystyle \frac{\partial w}{\partial y}},$$

(6)

$$u_3(x,y,z_3,t)=u_{30}(x,y,t)-z_3{\displaystyle \frac{\partial w}{\partial x}},$$

(7)

$$v_3(x,y,z_3,t)=v_{30}(x,y,t)-z_3{\displaystyle \frac{\partial w}{\partial y}},$$

(8)

where u₁₀, v₁₀, u₃₀, and v₃₀ are middle-layer displacements of top and bottom face layers; and z₁ and z₃ are the local vertical coordinates of the face layers. Based on interfacial displacements of the SP, the shear strains of the middle layer are given by

$${\gamma }_{2xz}={\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial w}{\partial x}}+{\displaystyle \frac{u_{10}-u_{30}}{h_2}},$$

(9)

$${\gamma }_{2yz}={\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{v_{10}-v_{30}}{h_2}},$$

(10)

where h_a = h₁ + h₂. By using strains (9) and (10), the shear stresses (1) and (2) of the middle layer become

$${\tau }_{2xz}=G_{21}[{\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial w}{\partial x}}+{\displaystyle \frac{u_{10}-u_{30}}{h_2}}]+G_{c1}[{\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial \dot{w}}{\partial x}}+{\displaystyle \frac{{\dot{u}}_{10}-{\dot{u}}_{30}}{h_2}}],$$

(11)

$${\tau }_{2yz}=G_{21}[{\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{v_{10}-v_{30}}{h_2}}]+G_{c1}[{\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial \dot{w}}{\partial y}}+{\displaystyle \frac{{\dot{v}}_{10}-{\dot{v}}_{30}}{h_2}}].$$

(12)

Based on displacements (5)–(8), the horizontal normal strains and shear strains of face layers can be derived. Then, corresponding normal stresses and shear stresses are obtained. According to equilibrium relations (x-axis and y-axis directions), shear stresses τ_1xz, τ_1yz, τ_3xz, and τ_3yz of face layers are derived with the following results:

$$\begin{array}{l}{\tau }_{1xz}=-{\displaystyle \frac{E_1}{1-{\mu }^2}}\{{\displaystyle \frac{{\partial }^2{u}_{10}}{\partial x^2}}(z_1-{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial x^3}}({\displaystyle \frac{h_1^2}{8}}-{\displaystyle \frac{z_1^2}{2}})+\mu [{\displaystyle \frac{{\partial }^2{v}_{10}}{\partial y\partial x}}(z_1-{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial y^2\partial x}}({\displaystyle \frac{h_1^2}{8}}-{\displaystyle \frac{z_1^2}{2}})]\}\\ -{\displaystyle \frac{E_1}{2(1+\mu )}}[{\displaystyle \frac{{\partial }^2{u}_{10}}{\partial y^2}}(z_1-{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^2{v}_{10}}{\partial y\partial x}}(z_1-{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial y^2\partial x}}({\displaystyle \frac{h_1^2}{4}}-z_1^2)]\end{array},$$

(13)

$$\begin{array}{l}{\tau }_{1yz}=-{\displaystyle \frac{E_1}{1-{\mu }^2}}\{{\displaystyle \frac{{\partial }^2{v}_{10}}{\partial y^2}}(z_1-{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial y^3}}({\displaystyle \frac{h_1^2}{8}}-{\displaystyle \frac{z_1^2}{2}})+\mu [{\displaystyle \frac{{\partial }^2{u}_{10}}{\partial y\partial x}}(z_1-{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial y}}({\displaystyle \frac{h_1^2}{8}}-{\displaystyle \frac{z_1^2}{2}})]\}\\ -{\displaystyle \frac{E_1}{2(1+\mu )}}[{\displaystyle \frac{{\partial }^2{v}_{10}}{\partial x^2}}(z_1-{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^2{u}_{10}}{\partial y\partial x}}(z_1-{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial y}}({\displaystyle \frac{h_1^2}{4}}-z_1^2)]\end{array},$$

(14)

$$\begin{array}{l}{\tau }_{3xz}=-{\displaystyle \frac{E_1}{1-{\mu }^2}}\{{\displaystyle \frac{{\partial }^2{u}_{30}}{\partial x^2}}(z_3+{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial x^3}}({\displaystyle \frac{h_1^2}{8}}-{\displaystyle \frac{z_3^2}{2}})+\mu [{\displaystyle \frac{{\partial }^2{v}_{30}}{\partial y\partial x}}(z_3+{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial y^2\partial x}}({\displaystyle \frac{h_1^2}{8}}-{\displaystyle \frac{z_3^2}{2}})]\}\\ -{\displaystyle \frac{E_1}{2(1+\mu )}}[{\displaystyle \frac{{\partial }^2{u}_{30}}{\partial y^2}}(z_3+{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^2{v}_{30}}{\partial y\partial x}}(z_3+{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial y^2\partial x}}({\displaystyle \frac{h_1^2}{4}}-z_3^2)]\end{array},$$

(15)

$$\begin{array}{l}{\tau }_{3yz}=-{\displaystyle \frac{E_1}{1-{\mu }^2}}\{{\displaystyle \frac{{\partial }^2{v}_{30}}{\partial y^2}}(z_3+{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial y^3}}({\displaystyle \frac{h_1^2}{8}}-{\displaystyle \frac{z_3^2}{2}})+\mu [{\displaystyle \frac{{\partial }^2{u}_{30}}{\partial y\partial x}}(z_3+{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial y}}({\displaystyle \frac{h_1^2}{8}}-{\displaystyle \frac{z_3^2}{2}})]\}\\ -{\displaystyle \frac{E_1}{2(1+\mu )}}[{\displaystyle \frac{{\partial }^2{v}_{30}}{\partial x^2}}(z_3+{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^2{u}_{10}}{\partial y\partial x}}(z_3+{\displaystyle \frac{h_1}{2}})+{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial y}}({\displaystyle \frac{h_1^2}{4}}-z_3^2)]\end{array}.$$

(16)

Using continuity relations of interfacial shear stresses yields differential equations for SP horizontal displacements:

$${\displaystyle \frac{E_1{h}_1}{1-{\mu }^2}}({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+\mu {\displaystyle \frac{{\partial }^{2}v}{\partial y\partial x}})+{\displaystyle \frac{E_1{h}_1}{2(1+\mu )}}({\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}v}{\partial y\partial x}})=G_{21}({\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial w}{\partial x}}+{\displaystyle \frac{2}{h_2}}u)+G_{c1}({\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial \dot{w}}{\partial x}}+{\displaystyle \frac{2}{h_2}}\dot{u}),$$

(17)

$${\displaystyle \frac{E_1{h}_1}{1-{\mu }^2}}({\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}+\mu {\displaystyle \frac{{\partial }^{2}u}{\partial y\partial x}})+{\displaystyle \frac{E_1{h}_1}{2(1+\mu )}}({\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y\partial x}})=G_{21}({\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{2}{h_2}}v)+G_{c1}({\displaystyle \frac{h_a}{h_2}}{\displaystyle \frac{\partial \dot{w}}{\partial y}}+{\displaystyle \frac{2}{h_2}}\dot{v}),$$

(18)

where u = u₁₀ = −u₃₀, and v = v₁₀ = −v₃₀. The dynamic equation for an element of SP with mass (z-axis direction) is given by

$${\displaystyle \sum _{i=1}^3{\displaystyle \underset{h_i}{\int }(\frac{\partial {\tau }_{ixz}}{\partial x}+\frac{\partial {\tau }_{iyz}}{\partial y})}}\mathrm{d}{z}_i-[\rho h_t+m_{0}ab\delta (x-x_0)\delta (y-y_0)](\ddot{w}+{\ddot{w}}_0)=0,$$

(19)

where δ(·) is impulse function, and (x₀, y₀) are concentrated mass coordinates. By using stresses (11)–(16), the differential equation for SP vertical displacement is obtained from Equation (19) as follows:

$$\begin{array}{l}[\rho h_t+m_{0}ab\delta (x-x_0)\delta (y-y_0)]\ddot{w}+D_1{\displaystyle \frac{\partial }{\partial x}}[h_1^3({\displaystyle \frac{{\partial }^{3}w}{\partial x^3}}+{\displaystyle \frac{{\partial }^{3}w}{\partial y^2\partial x}})]\\ +D_1{\displaystyle \frac{\partial }{\partial y}}[h_1^3({\displaystyle \frac{\partial w^3}{\partial y^3}}+{\displaystyle \frac{{\partial }^{3}w}{\partial x^2\partial y}})]-{\displaystyle \frac{\partial }{\partial x}}[G_{21}{\displaystyle \frac{h_a^2}{h_2}}(h_a{\displaystyle \frac{\partial w}{\partial x}}+{\displaystyle \frac{2}{h_a}}u)+G_{c1}{\displaystyle \frac{h_a^2}{h_2}}({\displaystyle \frac{\partial \dot{w}}{\partial x}}+{\displaystyle \frac{2}{h_a}}\dot{u})]\\ -{\displaystyle \frac{\partial }{\partial y}}[G_{21}{\displaystyle \frac{h_a^2}{h_2}}({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{2}{h_a}}v)+G_{c1}{\displaystyle \frac{h_a^2}{h_2}}({\displaystyle \frac{\partial \dot{w}}{\partial y}}+{\displaystyle \frac{2}{h_a}}\dot{v})]\\ =-[\rho h_t+m_{0}ab\delta (x-x_0)\delta (y-y_0)]{\ddot{w}}_0\end{array},$$

(20)

where D₁ = E₁/6(1 − μ²). Equations (17), (18), and (20) are for the coupled vibration of periodic VESP with supported mass subjected to random base loading. Parameters G₂₁ and G_c1 vary periodically in space, as given by Equations (3) and (4). Simultaneously, they are functions of bounded control, e_a1(t); thus, the temporal control of SP is a stochastic bounded parametric control problem.

SP boundary conditions can be obtained via displacement and force constraints. Simply supported conditions of rectangular plate are given by the following [18]:

$$\begin{gathered} w{|}_{x=\pm a/2}=0, w{|}_{y=\pm b/2}=0, {\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}{|}_{x=\pm a/2}=0, {\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}{|}_{y=\pm b/2}=0, \\ ({\displaystyle \frac{\partial u}{\partial x}}+\mu {\displaystyle \frac{\partial v}{\partial y}}){|}_{x=\pm a/2}=0, ({\displaystyle \frac{\partial v}{\partial y}}+\mu {\displaystyle \frac{\partial u}{\partial x}}){|}_{y=\pm b/2}=0. \end{gathered}$$

(21)

Furthermore, coordinates and displacements are replaced by the nondimensional (ND) variables:

$$\overline{x}={\displaystyle \frac{x}{a}}, \overline{y}={\displaystyle \frac{y}{b}}, \overline{u}={\displaystyle \frac{u}{w_a}}, \overline{v}={\displaystyle \frac{v}{w_a}}, \overline{w}={\displaystyle \frac{w}{w_a}}, {\overline{w}}_0={\displaystyle \frac{w_0}{w_a}},$$

(22)

where w_a is the amplitude of the base displacement. Equations (17), (18), and (20) of the controlled SP system with boundary conditions (21) are rewritten as follows:

$$\begin{array}{l}[\rho h_t+m_0\delta (\overline{x}-{\overline{x}}_0)\delta (\overline{y}-{\overline{y}}_0)]\ddot{\overline{w}}+D_1{\displaystyle \frac{\partial }{\partial \overline{x}}}[{\displaystyle \frac{h_1^3}{a^4}}({\displaystyle \frac{{\partial }^3\overline{w}}{\partial {\overline{x}}^3}}+{\displaystyle \frac{a^2}{b^2}}{\displaystyle \frac{{\partial }^3\overline{w}}{\partial {\overline{y}}^2\partial \overline{x}}})]\\ +D_1{\displaystyle \frac{\partial }{\partial \overline{y}}}[{\displaystyle \frac{h_1^3}{b^4}}({\displaystyle \frac{\partial {\overline{w}}^3}{\partial {\overline{y}}^3}}+{\displaystyle \frac{b^2}{a^2}}{\displaystyle \frac{{\partial }^3\overline{w}}{\partial {\overline{x}}^2\partial \overline{y}}})]-{\displaystyle \frac{\partial }{\partial \overline{x}}}[G_{21}{\displaystyle \frac{h_a^2}{a^2{h}_2}}({\displaystyle \frac{\partial \overline{w}}{\partial \overline{x}}}+{\displaystyle \frac{2a}{h_a}}\overline{u})+G_{c1}{\displaystyle \frac{h_a^2}{a^2{h}_2}}({\displaystyle \frac{\partial \dot{\overline{w}}}{\partial \overline{x}}}+{\displaystyle \frac{2a}{h_a}}\dot{\overline{u}})]\\ -{\displaystyle \frac{\partial }{\partial \overline{y}}}[G_{21}{\displaystyle \frac{h_a^2}{b^2{h}_2}}({\displaystyle \frac{\partial \overline{w}}{\partial \overline{y}}}+{\displaystyle \frac{2b}{h_a}}\overline{v})+G_{c1}{\displaystyle \frac{h_a^2}{b^2{h}_2}}({\displaystyle \frac{\partial \dot{\overline{w}}}{\partial \overline{y}}}+{\displaystyle \frac{2b}{h_a}}\dot{\overline{v}})]\\ =-[\rho h_t+m_0\delta (\overline{x}-{\overline{x}}_0)\delta (\overline{y}-{\overline{y}}_0)]{\ddot{\overline{w}}}_0\end{array},$$

(23)

$$6D_1{h}_1[{\displaystyle \frac{1}{a^2}}{\displaystyle \frac{{\partial }^2\overline{u}}{\partial {\overline{x}}^2}}+(1-\mu ){\displaystyle \frac{1}{2b^2}}{\displaystyle \frac{{\partial }^2\overline{u}}{\partial {\overline{y}}^2}}+(1+\mu ){\displaystyle \frac{1}{2ab}}{\displaystyle \frac{{\partial }^2\overline{v}}{\partial \overline{y}\partial \overline{x}}}]=G_{21}({\displaystyle \frac{h_a}{a{h}_2}}{\displaystyle \frac{\partial \overline{w}}{\partial \overline{x}}}+{\displaystyle \frac{2}{h_2}}\overline{u})+G_{c1}({\displaystyle \frac{h_a}{a{h}_2}}{\displaystyle \frac{\partial \dot{\overline{w}}}{\partial \overline{x}}}+{\displaystyle \frac{2}{h_2}}\dot{\overline{u}}),$$

(24)

$$6D_1{h}_1[{\displaystyle \frac{1}{b^2}}{\displaystyle \frac{{\partial }^2\overline{v}}{\partial {\overline{y}}^2}}+(1-\mu ){\displaystyle \frac{1}{2a^2}}{\displaystyle \frac{{\partial }^2\overline{v}}{\partial {\overline{x}}^2}}+(1+\mu ){\displaystyle \frac{1}{2ab}}{\displaystyle \frac{{\partial }^2\overline{u}}{\partial \overline{y}\partial \overline{x}}}]=G_{21}({\displaystyle \frac{h_a}{b{h}_2}}{\displaystyle \frac{\partial \overline{w}}{\partial \overline{y}}}+{\displaystyle \frac{2}{h_2}}\overline{v})+G_{c1}({\displaystyle \frac{h_a}{b{h}_2}}{\displaystyle \frac{\partial \dot{\overline{w}}}{\partial \overline{y}}}+{\displaystyle \frac{2}{h_2}}\dot{\overline{v}}),$$

(25)

$$\begin{gathered} \overline{w}{|}_{\overline{x}=\pm 1/2}=0, \overline{w}{|}_{\overline{y}=\pm 1/2}=0, {\displaystyle \frac{{\partial }^2\overline{w}}{\partial {\overline{x}}^2}}{|}_{\overline{x}=\pm 1/2}=0, {\displaystyle \frac{{\partial }^2\overline{w}}{\partial {\overline{y}}^2}}{|}_{\overline{y}=\pm 1/2}=0, \\ ({\displaystyle \frac{\partial \overline{u}}{a\partial \overline{x}}}+\mu {\displaystyle \frac{\partial \overline{v}}{b\partial \overline{y}}}){|}_{\overline{x}=\pm 1/2}=0, ({\displaystyle \frac{\partial \overline{v}}{b\partial \overline{y}}}+\mu {\displaystyle \frac{\partial \overline{u}}{a\partial \overline{x}}}){|}_{\overline{y}=\pm 1/2}=0. \end{gathered}$$

(26)

3. Multi-Dimensional System Equations with Parameters Dependent Nonlinearly on Control

Using constraint (26), the ND displacements of periodic SP are expressed in the series:

$$\overline{u}={\displaystyle \sum _{i=1}^{N_1}{\displaystyle \sum _{j=1}^{N_2}{r}_{ij}}}(t)\sin [(2i-1)\pi \overline{x}]\cos [(2j-1)\pi \overline{y}],$$

(27)

$$\overline{v}={\displaystyle \sum _{i=1}^{N_1}{\displaystyle \sum _{j=1}^{N_2}{s}_{ij}}}(t)\cos [(2i-1)\pi \overline{x}]\sin [(2j-1)\pi \overline{y}],$$

(28)

$$\overline{w}={\displaystyle \sum _{i=1}^{N_1}{\displaystyle \sum _{j=1}^{N_2}{q}_{ij}}}(t)\cos [(2i-1)\pi \overline{x}]\cos [(2j-1)\pi \overline{y}],$$

(29)

where r_ij(t), s_ij(t), and q_ij(t) are generalized displacements; and N₁ and N₂ are term numbers. By substituting expressions (27)–(29) into Equations (23)–(25) and using the Galerkin method, differential equations for q_ij, r_ij, and s_ij can be obtained. Note that horizontal velocities are little relative to the others. After eliminating r_ij and s_ij, coupling vibration equations for generalized displacements, q_ij, are derived and expressed as

$$M\ddot{Q}+C_1(e_{a1})\dot{Q}+(K_0+K_1(e_{a1}))Q=F(t),$$

(30)

where generalized displacement vector $Q={[\begin{array}{cccc}{Q}_1^{\mathrm{T}}& Q_2^{\mathrm{T}}& \cdots & Q_{N_2}^{\mathrm{T}}\end{array}]}^{\mathrm{T}}$, $Q_j={[\begin{array}{cccc}{q}_{1j}& q_{2j}& \cdots & q_{N_{1}j}\end{array}]}^{\mathrm{T}}$ (j = 1, 2, …, N₂), and generalized excitation vector $F(t)=-{\ddot{\overline{w}}}_0{F}_C$. The generalized mass matrix, M; damping matrix, C₁; stiffness matrices, K₀ and K₁; and vector, F_C are given in Appendix A. The generalized stiffness matrix (K₁) and damping matrix (C₁) are nonlinear functions of the control variable, e_a1(t), due to the involvement of generalized horizontal displacements, r_ij and s_ij.

For the simply supported rectangular plate, a set of harmonic functions can be used as base functions of the Galerkin method to discretize plate displacements in space (27)–(29). Therefore, the Galerkin method can simply transform partial differential equations for coupling vibration of periodic SP (23)–(25) into multi-dimensional ordinary differential Equations (30) with parameters as nonlinear functions of the bounded control (e_a1). The dynamics equations in the time domain will be used conveniently to derive SDP equation according to the SDP principle. Then, the analytical expression of OBPC can be determined by the SDP equation. The Galerkin method makes the calculation of stochastic responses and OBPC of the parametric control system simpler and more efficient.

Equation (30) describes a multi-dimensional control system of periodic VESP and supported mass subjected to random base loading, which has stiffness, K₁(e_a1), and damping, C₁(e_a1), nonlinearly dependent on the bounded control, e_a1(t). The stochastic vibration mitigation of the SP system is a stochastic bounded nonlinear parametric control problem. The SDP principle is applied to determine an OBPC. For application, Equation (30) is rewritten as

$$\dot{Z}=AZ-B(e_{a1})Z+W(t),$$

(31)

where state vector, Z; parameter matrices, A and B; and excitation vector, W, are

$$\begin{gathered} Z=\left[\begin{array}{c}Q\\ \dot{Q}\end{array}\right], A=\left[\begin{array}{cc}0& I\\ -M^{-1}{K}_0& 0\end{array}\right], \\ B(e_{a1})=\left[\begin{array}{cc}0& 0\\ M^{-1}{K}_1(e_{a1})& M^{-1}{C}_1(e_{a1})\end{array}\right], W=\left[\begin{array}{c}0\\ M^{-1}F\end{array}\right], \end{gathered}$$

(32)

in which I is the identity matrix. Parameter matrix B is a nonlinear function of control e_a1(t) due to stiffness, K₁(e_a1), and damping, C₁(e_a1). The control e_a1(t) is bounded due to material saturation properties. The bounded control constraint is given by

$$e_l\le e_{a1}\le e_h,$$

(33)

where e_l and e_h are positive constants which are determined by VE properties and required control. The random base excitation can be modeled by a filtering Gaussian white noise, and then excitation, W, is assumed as a Gaussian white-noise vector with zero means.

4. OBPC Law and Controlled Responses of SP

The stochastic OBPC (e_a1) of SP system (31) can be determined by the SDP principle. The OBPC is to minimize a performance index related to system response, which is expressed as

$$J(e_{a1})=E\{{\displaystyle {\int }_0^{t_f}{Z}^{\mathrm{T}}SZ}\mathrm{d}t+\Psi (Z(t_f))\},$$

(34)

where E{·} is the expectation operation, S is the semipositive definite symmetric matrix, Ψ(Z(t_f)) is the final cost function, and t_f is the final time. By applying the SDP principle [33,34] to system (31) and performance index (34) with control bound (33), the SDP equation is obtained as

$${\displaystyle \frac{\partial V}{\partial t}}+\underset{e_{a1}}{\min }\{{\displaystyle \frac{1}{2}}\mathrm{tr}(D_w{\displaystyle \frac{{\partial }^{2}V}{\partial Z^2}})+{[AZ-B(e_{a1})Z]}^{\mathrm{T}}{\displaystyle \frac{\partial V}{\partial Z}}+Z^{\mathrm{T}}SZ\}=0,$$

(35)

where V(Z, t) is the value function, tr(·) is the trace operation, and D_w is the intensity matrix of random excitation, W.

By minimizing the left side of Equation (35) or maximizing [B(e_a1)Z]^T∂V/∂Z, the OBPC law is derived and expressed in the following form:

$$e_{a1}^{*}=e_{a1}{|}_{\max \{{[B(e_{a1})Z]}^{\mathrm{T}}\partial V/\partial Z\},e_{a1}\in [e_l,e_h]}.$$

(36)

Parameter matrix B(e_a1) depends nonlinearly on control e_a1, and, thus, the optimal control e_a1^* corresponding to the maximum of [B(e_a1)Z]^T∂V/∂Z may not be bound e_l or e_h. Substituting OBPC (36) into Equation (35) yields a differential equation for the value function:

$${\displaystyle \frac{\partial V}{\partial t}}+{\displaystyle \frac{1}{2}}\mathrm{tr}(D_w{\displaystyle \frac{{\partial }^{2}V}{\partial Z^2}})+{[AZ-B(e_{a1}^{*})Z]}^{\mathrm{T}}{\displaystyle \frac{\partial V}{\partial Z}}+Z^{\mathrm{T}}SZ=0.$$

(37)

The value function, V, is obtained from Equation (37) and used to determine [B(e_a1)Z]^T∂V/∂Z in Equation (36). Equation (37) has a quadratic stationary solution, V = Z^TPZ, in which positive definite symmetric matrix, P, is obtained from the following equation:

$$S+{[\mathbf{A}-B(e_{a1}^{*})]}^{\mathrm{T}}P+P[\mathbf{A}-B(e_{a1}^{*})]=0.$$

(38)

The OBPC e_a1^* is obtained finally from Equations (36) and (38). By using control e_a1^*, the optimally controlled system Equation (31) becomes

$$\dot{Z}=AZ-B(e_{a1}^{*})Z+W.$$

(39)

Controlled system responses, Z, are obtained from Equation (39) by numerical integration, and then ND displacement responses of controlled SP subjected to random base loading can be calculated by Equation (29). For example, mass responses or displacement responses of the plate middle point are obtained by the results with $\overline{x}$ = 0 and $\overline{y}$ = 0. Response statistics such as the standard deviation (SD) of the controlled SP are estimated by using the results, which can be used for evaluating control effectiveness by comparing with those of uncontrolled and passively controlled SP.

5. Results and Discussions

To show the proposed stochastic OBPC’s effectiveness, consider a controllable period-parametric VESP with a supported mass subjected to random base loading. Its parameter values are a = 8 m, b = 8 m, ρ₁ = 6000 kg/m³, ρ₂ = 1200 kg/m³, E₁ = 0.1 GPa, μ = 0.3, k_a = 1.5, k_b = 0.5, β = 0.5, e_l = 0.2 MPa, e_h = 1.0 MPa, ξ = 0.01 s, h₁ = 0.015 m, h₂ = 0.6 m, w_a = 1, x₀ = y₀ = 0, and m₀ = 120 kg/m², unless otherwise specified. The numbers of expansion terms in (27)–(29) are N₁ = 5 and N₂ = 5. Figure 2 shows a sample of ND white-noise excitation with unit intensity. The control weight matrix is S = diag [0, 0, …, 0, 1, 1, …, 1]. The passively controlled system is the SP system, with a constant control e_a1, which is used for comparison. The passive parametric control is e_a1 = 0.6 MPa as the mean value of bounds e_l and e_h, while the optimal parametric control is e_a1(t)∊[e_l, e_h], determined based on Equation (36). The numerical results on the effectiveness of the proposed OBPC for ND stochastic responses of SP at middle point or the mass were obtained by using MATLAB 2016b and are shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18.

5.1. Optimal Control of SP with Uniform Parameters

First, consider the SP with uniform parameters (β = 0). Figure 3 shows optimally controlled ND displacement responses ($\overline{w}$) of SP compared with passively controlled responses. Stochastic responses are reduced effectively via the proposed OBPC. The maximum ND displacement decreases from passively controlled 3.2629 to optimally controlled 1.3062. The response SD decreases from passively controlled, 1.2484, to optimally controlled, 0.4865. The relative reduction in the optimally controlled response SD (i.e., ratio of the difference between passively and optimally controlled response SDs to passively controlled response SD) is 61.03%. Figure 4 shows a sample of the corresponding OBPC e_a1*. Its mean value is 0.57 MPa, which is smaller than that of the passive control, 0.6 MPa. Therefore, the proposed OBPC can obtain better effectiveness and then better use of the controllable VE than the passive control for the stochastic vibration mitigation of the SP system.

The influence of the control bounds (e_l and e_h) on the responses (displacements and velocities) of the optimally controlled SP is shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. Figure 5 shows optimally controlled ND displacement responses of SP for various control upper bounds (e_h). As the bound e_h increases from 0.4 to 4.6 MPa, the maximum ND displacement decreases from 1.7663 to 1.2018. Accordingly, the response SD decreases from 0.6878 to 0.4080, and the relative reduction in SD increases from 44.9% to 67.3%. The mean values of optimal controls are 0.29 and 2.09 MPa, corresponding to bounds e_h = 0.4 and 4.6 MPa, respectively. Figure 6 shows that optimally controlled response (displacements and velocities) SDs decrease nonlinearly as bound e_h increases (e_l = 0.2 MPa). The response SD changes largely when bound e_h is little, but little when bound e_h is large. Figure 7 shows relative reductions in response SDs increasing with bound e_h. A further increase in the relative reductions requires a greater increment of bound e_h. The mean value of the corresponding optimal control, e_a1^*, increases linearly with bound e_h, as shown in Figure 8.

Figure 9 shows the SDs of optimally controlled ND responses (displacements and velocities) varying with the control lower bound, e_l (e_h = 3.0 MPa). Figure 10 shows corresponding relative reductions in response SDs. Different from the e_h effect, the system-response SDs decrease, but relative response reductions increase as the lower bound, e_l, decreases. The lower-bound reduction enlarges the control domain of e_a1; meanwhile, the mean value of the optimal control, e_a1*, has a certain reduction.

5.2. Optimal Control of SP with Periodic Parameters

Furthermore, consider the SP with periodic parameters (β = 0.5). Figure 11 shows the optimally controlled ND displacement responses ($\overline{w}$) of periodic SP compared with passively controlled responses. Stochastic responses are reduced remarkably via the proposed OBPC. The maximum ND displacement decreases from passively controlled 2.4138 to optimally controlled 1.3182. The response SD decreases from passively controlled 0.9244 to optimally controlled 0.4857. The relative reduction in the optimally controlled response SD is 47.46%. Figure 12 shows a sample of the corresponding OBPC e_a1^*. Its mean value is 0.55 MPa, which is smaller than that of the passive control, 0.6 MPa. The optimally controlled response SD of SP with periodic parameters is close to that of SP with uniform parameters, but the mean value of optimal control of periodic SP is smaller than that of uniform SP, because spatial periodic distribution has reduced certain stochastic responses of SP. Therefore, the proposed OBPC can obtain further better effectiveness and better use of controllable VE than passive control (i.e., spatial periodic distribution of VE modulus unchanged with time) for stochastic vibration suppression of SP.

The influence of control bounds (e_l and e_h) on responses (displacements and velocities) of optimally controlled SP is shown in Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18. Figure 13 shows optimally controlled ND displacement responses of SP for different control upper bounds (e_h). As the bound e_h increases from 0.4 to 4.6 MPa, the maximum ND displacement decreases from 1.6905 to 1.1910. Accordingly, the response SD decreases from 0.6678 to 0.4078, and the relative reduction in SD increases from 27.8% to 55.9%. The mean values of optimal controls are 0.29 and 1.99 MPa, corresponding to bounds e_h = 0.4 and 4.6 MPa, respectively. Figure 14 shows optimally controlled response (displacements and velocities) SDs decreasing nonlinearly as bound e_h increases (e_l = 0.2 MPa). Similar to uniform SP, the system-response SD of periodic SP changes largely when bound e_h is little, but little when bound e_h is large. The system parameters, such as stiffness and damping, are nonlinear functions of control e_a1 (see Equation (30)), and thus the controlled responses nonlinearly depend on the control. As control bound e_h increases, the mean value and SD of OBPC increase, and then the optimally controlled responses or response SDs decrease nonlinearly. Figure 15 shows relative reductions in response (displacements and velocities) SDs increasing with bound e_h. A further increase in the relative response reductions requires a greater increment of bound e_h. The mean value of the corresponding optimal control e_a1^* increases linearly with bound e_h, as shown in Figure 16.

Figure 17 shows optimally controlled ND response (displacements and velocities) SDs varying with control lower bound, e_l (e_h = 3.0 MPa). Figure 18 shows corresponding relative reductions in response SDs. Similar to uniform SP, system-response SDs decrease, but relative response reductions increase as the lower bound, e_l, decreases. The lower-bound reduction enlarges the control domain of e_a1; meanwhile, the mean value of the optimal control e_a1* has a certain reduction. Thus, the control bound optimization first raises the upper bound, e_h, and then suitably reduces the lower bound, e_l.

6. Conclusions

The stochastic OBPC for periodic VESP with supported mass subjected to random base loading is proposed. The response reduction capability via the proposed OBPC was studied to illustrate the further control effectiveness of periodic SP by SDP in time than periodic distribution in space. The differential equations containing bounded nonlinear parametric control for periodic SP were derived and converted into multi-dimensional system equations according to the Galerkin method. The SDP equation for the control system and performance index with bound constraint was established according to the SDP principle, and then the OBPC law was obtained. The optimally controlled responses of SP with periodic and uniform parameters under random loading were calculated, and the control effectiveness was evaluated by comparing with passively controlled responses.

The following results were obtained: (1) The stochastic responses of SP with periodic and uniform parameters subjected to random loading can be reduced greatly via the proposed OBPC. (2) The OBPC effectiveness of SP is better than the corresponding passive control effectiveness; thus, the VE controllable characteristics can be used fully by temporal optimal adjustment for spatial periodic SP. (3) The OBPC effectiveness can be improved further via increasing the upper bound and suitably reducing the lower bound of the control variable or controllable VE modulus. Therefore, the proposed stochastic OBPC has potential for being applied to periodic (or non-periodic) VESP systems and smart building structures under random excitations, as well as the controllable VE design based on application. The experiment with simulation validation on SP vibration control will be carried out in the future.