Ibrahim Sadek

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.

A computational method for solving optimal control of a system of parallel beams using Legendre wavelets

The optimal control of transverse vibration of two Euler-Bernoulli beams coupled in parallel by discrete springs is considered. An index of performance is formulated which consists of a modified energy functional of two coupled structures at a specified time and penalty functions involving the point control forces. The minimization of the performance index over these forces is subject to the equation of motion governing the structural vibrations, the imposed initial condition as well as the boundary conditions. By use of the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of a linear time-invariant lumped-parameter systems. A computationally attractive method based on Legendre wavelets in time domain for solving the optimal control of the lumped parameter systems for any finite interval is proposed. Legendre wavelet integral operational matrix and the properties of a Kronecker product are used to find the approximated optimal trajectory and optimal law of the linear systems with respect to a quadratic cost function by only solving a linear system of algebraic equations. This method provides a straightforward and convenient approach for digital computation. A numerical example is provided to demonstrate the applicability and effectiveness of the proposed method.

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.

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.

Variational principles for multiwalled carbon nanotubes undergoing vibrations based on nonlocal timoshenko beam theory

Variational principles are derived for multiwalled carbon nanotubes undergoing linear vibrations using the semi-inverse method with the governing equations based on nonlocal Timoshenko beam theory which takes small scale effects and shear deformation into account. Physical models based on the nonlocal theory approximate the nanoscale phenomenon more accurately than the local theories by taking small scale phenomenon into account. Variational formulation is used to derive the natural and geometric boundary conditions which give a set of coupled boundary conditions in the case of free boundaries which become uncoupled in the case of the local theory. Hamiltons principle applicable to this case is also given.

An efficient computational method for the optimal control problem for the Burgers equation

Pointwise control of the viscous Burgers equation in one spatial dimension is studied with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the Burgers equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the Burgers equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parameterization by using the Runge-Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions.

Optimal Boundary Control of the Longitudinal Vibrations of a Rod Using a Maximum Principle

A maximum principle is derived for the determination of the optimal boundary control of the lon gitudinal vibration of a variable cross-section rod in which the index of performance is a quadratic function. The control is expressed explicitly in terms of an adjoint variable and can be applied by displacement or force control, or a combination of both. The adjoint variable and the state variable are related through termi nal conditions. The results are applied to a specific problem, and numerical results are given that reveal the effectiveness of the proposed control mechanism. The numerical results are obtained using MAPLE V.

Integral Equation Approach for Beams with Multi-Patch Piezo Sensors and Actuators

An integral equation approach is developed for the solution of an adaptive beam problem with the beam undergoing free vibrations. The beam is controlled by a closed-loop system consisting of multiple patches of sensors and actuators which are bonded to the bottom and top surfaces of the beam. The coupling between sensors and actuators can be chosen arbitrarily and the control is exercised by displacement feedback. The integral equation governing the vibrations of the beam/piezo-patch system is derived by converting the corresponding differential equation, which is non-standard as a result of the discontinuities caused by the piezo patches. The mathematical formulation involves Heaviside and distribution functions in a differential setting, while the integral equation avoids these difficulties and is expressed in terms of a smooth kernel which is developed using a Greens function approach based on suitable patch functions. The numerical results are obtained for various locations of patches, gain factors and coupling configurations, and the first three eigenfrequencies and eigenfunctions of the beam/piezo system are given in table and graph forms.

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.

Optimal control of a beam with Kelvin-Voigt damping subject to forced vibrations using a piezoelectric patch actuator

A maximum principle is derived for the optimal control of a beam with Kelvin–Voigt damping subject to an external excitation with the control exercised by means of piezoelectric patch actuators. The objective functional is defined as a weighted quadratic functional of the displacement and velocity which is to be minimized at a given terminal time. A penalty term is also part of the objective functional defined as the control voltage used in the control process. The maximum principle makes use of a Hamiltonian defined in terms of an adjoint variable and the control function. The optimal control problem is expressed as a coupled system of partial differential equations in terms of state, adjoint and control variables subject to boundary, initial and terminal conditions. The solution is obtained by expanding the state and adjoint variables in terms of eigenfunctions and determining the optimal control using the maximum principle. Numerical examples are given to demonstrate the applicability and the efficiency of the proposed method.