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Abstract: Optimal induced strain actuator sizes for an L-shaped laminated frame are determined under deterministic and uncertain loads with the objective of minimizing the tip deflection. The bending moments generated by the induced strain actuators control the deflection in order to minimize the maximum deflection. The frame is elastically restrained at the tip point and is subjected to a tip load which is specified for the deterministic case and lies in an uncertainty domain in the uncertain case. The specific uncertain loading studied in the present paper involves a load of given magnitude but of unknown direction which should be determined to induce the highest tip deflection caused by the least favorable loading. The effects of the elastic restraints and actuator locations on the deflection behaviour are investigated. It is shown that the optimal actuator size depends on the magnitude of the moments provided by the actuators. Moreover, it is observed that a deterministic design under an uncertain load leads to excessive deflections indicating the necessity of a robust design. [Copyright &y& Elsevier]

Abstract: The Euler–Bernoulli model of transverse vibrations of a cantilever beam is extended to include viscous and Kelvin–Voigt (strain rate) damping to study active vibration control under damping. The controls are of feedback type with displacement and velocity feedback controls provided by full length or patch piezoelectric actuators and sensors bonded to the top and the bottom of the beam. The resulting partial differential equation which contains discontinuities in a differential equation setting due to patch actuators is solved by formulating the problem in an integral equation setting which eliminates the singularities. The solution is reduced to the solution of a linear system of equations by using an orthonormal set of the eigenfunctions of the freely vibrating beam which provide an accurate approximation to the controlled vibration case. Control effectiveness is investigated in terms of changes in the damped natural frequencies for different gains, damping coefficients, and patch locations. The effect of the displacement and velocity feedback controls and problem parameters on the real and complex parts of the fundamental, second and third order frequencies is discussed with a view towards establishing the damped and oscillatory behaviour of the piezo controlled beams. [Copyright &y& Elsevier]