James M. Sloss

Optimal piezo-actuator locations/lengths and applied voltage for shape control of beams

Shape control of beams under general loading conditions is implemented using piezoceramic actuators to provide the control forces. The objective of the shape-control is to minimize the maximum deflection of the beam to obtain a min-max deflection configuration with respect to loading and piezo-actuators. In practice, the loading on a beam is a variable quantity with respect to its magnitude, and this aspect can be handled easily by optimizing the magnitude of the applied voltage to achieve the min-max deflection. This property of the smart materials technology overcomes the problem of one-off conventional optimal designs which become suboptimal when the loading magnitude changes. In addition to the magnitude of the applied voltage, the optimal values for the locations and the lengths of the piezo-actuators are determined to achieve the min-max deflection. Due to the complexity of the governing equations involving finite length piezo patches, the numerical results are obtained by the finite-difference method. The analysis of the problem shows the effect of the actuator locations, lengths and the applied voltage on the maximum deflection. The optimal values for the actuator locations and the voltage are determined as functions of the load locations and load magnitudes, respectively. The effect of the actuator length on the min-max deflection is investigated and it is observed that the optimal length depends on the applied voltage. Finally, it is shown that using multiple actuators are more effective than a single actuator in the cases of complicated loading.

Maximum principle for optimal boundary control of vibrating structures with applications to beams

A maximum principle is derived for open-loop boundary control of one dimensional structures undergoing transverse vibrations. The optimal control law is obtained using a maximum principle and the applicability of the results to the boundary control of vibrating beams is demonstrated. The method of solution involves the transformation of the original problem into one with homogeneous boundary conditions for a general set of boundary forces and torques. An adjoint variable is introduced and used in the formulation of a Hamiltonian function which in turn leads to the derivation of the maximum principle. The effectiveness of the proposed control mechanism is illustrated numerically and it is shown that the implementation of the optimal boundary control using one force actuator can lead to substantial decrease in the dynamic response of a vibrating beam.

Piezoelectric patch control using an integral equation approach

Closed-loop displacement feedback control for a beam using bonded piezoelectric patch sensors and actuators is considered. It is shown that there is an equivalence between the eigensolutions of the differential equation formulation of the problem and the eigensolutions of a certain integral equation. It is also shown that a pair of aligned oriented sensors and actuators generate an orthogonal set of eigenfunctions. The natural frequencies are found numerically for a cantilevered beam using the integral equation.

Optimal design and control of a cross-ply laminate for maximum frequency and minimum dynamic response

Abstract Optimal thickness design and optimal closed-loop and open-loop distributed control functions are determined for a symmetric, cross-ply laminate. The design/control problem is formulated as a multiobjective optimization problem by taking a performance index which comprises a weighted sum of the design and control objectives and a penalty functional of the control force. The design objective is the maximization of the fundamental frequency. The control objective is the minimization of the dynamic response of the plate, which is expressed in terms of the energy of the structure. The design/control problem is solved using two different approaches, namely closed-loop (feedback) control and the open-loop control. In the former approach, displacement and velocity feedback controls are employed. In the latter approach, the open-loop control involves an unknown control function which is determined optimally using a maximum principle. The design variables are determined by direct minimization of the design and control objectives. Numerical results are presented for a rectangular laminate made of an advanced composite material. Comparisons are given for controlled and uncontrolled laminates as well as for optimally designed and non-optimal laminates.

Vibration damping in beams via piezo actuation using optimal boundary control

Open-loop optimal control theory is formulated and applied to damp out the vibrations of a beam where the control action is implemented using piezoceramic actuators. The optimal control law is derived by using a maximum principle developed for one-dimensional structures where the control function appears in the boundary conditions in the form of a moment. The objective function is specified as a weighted quadratic functional of the displacement and velocity which is to be minimized at a specified terminal time using continuous piezoelectric actuators. The expenditure of control force is included in the objective functional as a penalty term. The explicit solution of the problem is developed for cantilever beams using eigenfunction expansions of the state and adjoint variables. The effectiveness of the proposed control mechanism is assessed by plotting the displacement and velocity against time. It is shown that both quantities are damped out substantially as compared to an uncontrolled beam and this reduction depends on the magnitude of the control moment. The capabilities of piezo actuation are also investigated by means of control moment versus piezo and beam thickness graphs which indicate the required minimum level of voltage to be applied on piezo materials in relation to geometric dimensions of the combined active/passive structure. The graphs show the magnitude of the control moment which can be achieved using piezoceramics in terms of problem inputs such as voltage, piezo and beam thicknesses.

Optimal control of a Timoshenko beam by distributed forces

The present paper considers the problem of optimally controlling the deflections and/or velocities of a damped Timoshenko beam subject to various types of boundary conditions by means of a distributed applied force and moment. An analytic solution is obtained by employing a maximum principle.

Maximum principle for the optimal control of a hyperbolic equation in one space dimension, part 1: theory

A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression.

Modified bang-bang piezoelectric control of vibrating beams

The converse piezoelectric effect is used to suppress the vibrations of a beam stiffened with piezoceramic actuators. The control problem involves the minimization of the dynamic response of the beam by using the voltage applied to the piezoactuators as a control variable. The dynamic response is defined as the vibrational energy of the beam, which is used as the cost functional of the control problem. The piezoactuators are bonded on the opposite surfaces of the beam and placed symmetrically with respect to the middle plane. The control moments are activated by applying out-of-phase voltages. The control voltage is subject to a maximum value constraint and is defined as a modified bang-bang type which can take this voltage as a plus or minus value as well as the zero value. It is found that the optimal active control takes the form of a piecewise constant alternating voltage with varying switch-over intervals.

Structural control to minimize the dynamic response of Mindlin—Timoshenko plates

Abstract The problem of damping out the vibrations of a thick plate is solved using the optimal control theory of distributed parameter systems. The plate is modelled as a Mindlin- Timoshenko plate to include shear effects and may exhibit viscous damping. The dynamic response of the structure comprises the displacement and velocity components which are combined with the amount of force expended in controlling the motion in a multiobjective cost functional. This functional is minimized with respect to distributed controls. A maximum principle is formulated to relate the control forces to adjoint variables, the use of which leads to the explicit solution of the title problem. The control over the plate is exercised by distributed moment and transverse forces which are in practice applied by means of torque and force actuators.

Seepage from a trapezoidal and a rectangular channel using variational inequalities

Abstract A variational inequality method has been applied to the problem of two-dimensional seepage from trapezoidal channels and a rectangular channel into permeable soil underlain at a finite depth by a drain. Location of the free surface, shape of the seepage region and seepage flow rate all appear as solutions through the utilization of this method. The numerical portion of the problem is solved using successive overrelaxation (SOR) methods with projection. Both finite-difference and finite-element methods are used. A discussion of the computer approach is given along with comparisons of the results.