Boundary optimal control of structural vibrations in an annular plate

doi:10.1016/j.jfranklin.2004.11.001

Journal of the Franklin Institute

Volume 342, Issue 3, May 2005, Pages 295-309

https://doi.org/10.1016/j.jfranklin.2004.11.001 Get rights and content

Abstract

A maximum principle is formulated and validated for the vibration control of an annulus plate with the control forces acting on the boundary. In addition, the maximum principle can be applied to plates with multiply connected domains. The performance index is specified as a quadratic functional of displacement and velocity along with a suitable penalty term involving the control forces. Using this index an explicit control law is derived with the help of an adjoint variable satisfying the adjoint differential equation and certain terminal conditions together with the proposed maximum principle. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by a numerical example.

Introduction

Plates are commonly used as elements in the construction of a large number of structures such as spacecraft, large space structures and robots. All these structures have flexible extensions and/or platforms which are made as light and slender as possible due to heavy penalties attached to excessive weight. Such slender elements lack the necessary damping properties to be able to function effectively under dynamic loads and may require an active control mechanism to damp out excessive vibrations and to improve their performance characteristics. Moreover, the destabilizing effects of dynamic loads need to be counteracted to avoid failure under operational conditions.

An economical and convenient way of applying control on these mechanical systems is by means of actuators placed on the boundary of the domain as opposed to actuators acting on any point on the structure [1]. There has been extensive research on stabilization, controllability, optimal control and other related problems of hyperbolic systems with non-dynamical boundary control involving a dissipation mechanism (see [2], [3], [4], [5], [6], [7]). Hybrid systems with dynamical boundary control have been the focus of research with applications to stabilization and active control of large elastic structures [8], [9], [10], [11], [12].

Before a boundary control mechanism can be implemented, the applicable optimal control laws have to be derived in order to prescribe the behavior of force actuators. Recent work in this area involves the optimal boundary control of a hyperbolic equation in which a maximum principle has been formulated to derive the control laws for one-dimensional [13], [14] and two-dimensional structures [15], [16]. The present study deals with the formulation and proof of a maximum principle for the optimal boundary control of vibrating plates along with a solution approach and numerical application which extends the results of Ref. [16]. This will provide a generalization of the boundary control results obtained for rods [13] and elastic beams [14], and give explicit solutions for annular plates subject to asymmetrical initial conditions. A large number of exact boundary controllability/stabilizability results for various plate models have appeared in the literature (see [10], [17], [18], [19], [20], [21], [22]). However, there are relatively few results on the optimal boundary control of vibrating plates [15], [23]. Boundary control results for one-dimensional vibrating structures include [24], [25], [26] where the optimality conditions were obtained by various techniques.

In the annular plate problem studied in the present paper, the control is applied through torque actuators placed along the boundaries of the plate. The application of these controls to a plate configuration is often more easily implemented than distributed surface controls. The present study obtains the control functions which will govern the motion of these actuators in an optimal manner. The control objective is the minimization of a quadratic functional involving the displacement and velocity of the plate, as well as the control forces. By introducing an adjoint variable and using the proposed maximum principle, an explicit expression is obtained for the boundary control functions.

The results are applied to an annulus in order to illustrate the method of solution and to investigate the effectiveness of the proposed control mechanism. Numerical results indicate that the optimal boundary control is quite effective in damping out the vibrations of the plate.

Section snippets

Optimal Control Problem Formulation

Consider an annular plate in the plane described by the polar coordinates $(r, θ)$ , centered at the origin having inner and outer radii denoted by $r = a$ and b, respectively, as shown in Fig. 1. Let the points $(r, θ)$ of the plate be described by the set $Ω = (a, b) \times [0, 2 π]$ . The physical properties of the plate are described by the density ρ, thickness h, Young's modulus E and Poisson's ratio $μ$ .

The equation of motion for the transverse vibration $w (r, θ, t)$ of a classical annular plate is given by [27] $L [w (r, θ, t$

Statement of the maximum principle

In order to state the maximum principle, it is necessary to consider, corresponding to the state variable problem for $w (r, θ, t)$ given in Eqs. (1), (3), (4), the following adjoint problem for the adjoint variable $v (r, θ, t)$ : $L [v (r, θ, t)] = ρ h \partial_{t}^{2} v + D Δ^{2} v = 0, on Ω \times (0, t_{f})$ with boundary conditions $v (a, θ, t) = 0, \partial_{r} v (a, θ, t) = 0, v (b, θ, t) = 0, κ μ \partial_{r} v (b, θ, t) + \partial_{r}^{2} v (b, θ, t) = 0 .$ In order to state the maximum principle, consider the functional $H [θ, t; v, P] = D_{0} \partial_{r} v (b, θ, t) P (θ, t) - μ_{3} P^{2} (θ, t)$ where $D_{0} = D / ρ h$ .

The maximum principle is then stated in

Solution method

In this section, we develop a solution method for the control problem (6) by applying the results of the previous theorem. The solution of the control problem (6) can be outlined as follows.

Step 1: The solution of the adjoint system of Eqs. (7), (8) is obtained by using eigenfunction expansions, that is, assume $v^{o} (r, θ, t) = \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} v_{mn}^{o} (t) ϕ_{mn} (r, θ),$ where the eigenfunctions $ϕ_{mn} (r, θ)$ are given by $ϕ_{mn} (r, θ) = y_{mn} (r) h_{n} (θ),$ in which $y_{mn} (r) = α_{mn} J_{n} (λ_{mn} r) + β_{mn} J_{n} (i λ_{mn} r) + γ_{mn} Y_{n} (λ_{mn} r) + δ_{mn} Y_{n} (i λ_{mn} r), h_{n} (θ) = C_{1 n} \cos n θ + C_{2 n}$

Numerical results

As a numerical example consider the initial conditions (2) to be of the form $f (r, θ) = 0, g (r, θ) = ϕ_{00} (r, θ)$ and thus $\begin{matrix} f_{mn} = 0, m \geq 0, n \geq 0 \\ g_{00} = 1, g_{mn} = 0, m \geq 1, n \geq 1 . \end{matrix}$

An approximate solution of Eqs. (45), (46) was obtained using the symbolic, as well as the numerical, features of the software program MAPLE V in the special case when a=1, b=2, $t_{f} = 5.0$ , ( $μ_{1}$ , $μ_{2}$ , $μ_{3}$ )=(1.0, 1.0, 0.1), $μ = 0.25$ and $D_{0} = 1$ . For the initial data given, $a_{00} = - 0.20395 \times 1 0^{- 2}$ , $b_{00} = - 0.23543 \times 1 0^{- 3}$ and all other a_mn's and b_mn's are essentially zero. This

Conclusions

A maximum principle relating the boundary control functions to an adjoint variable has been formulated for an annular vibrating plate. This approach leads to explicit expressions for the computation of optimal control laws and to a method of solution by which the displacements and velocities of the controlled structure can be obtained. Techniques of variational calculus and mathematical analysis are used in the proof of the maximum principle. It is shown that the optimal function obtained using

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View full text

Boundary optimal control of structural vibrations in an annular plate

Abstract

Introduction

Section snippets

Optimal Control Problem Formulation

Statement of the maximum principle

Solution method

Numerical results

Conclusions

A new approach to modeling of distributed structures for control

J. Franklin Inst.

Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping

Proc. R. Soc. Edinburgh Sect. A

Feedback stabilization of distributed semilinear controls

Appl. Math. Opt.

Energy decay estimates and exact boundary control of the wave equation in a bounded domain

J. Math. Pure Appl.

Decay of solutions of wave equations in a bounded region with boundary dissipation

J. Differential Equations

Dirichlet boundary stabilization of the wave equation with damping feedback

J. Math. Anal. Appl.

On boundary controllability of a vibrating plate