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In this paper, the deflection and natural frequency of a microbeam under combined electrostatic and piezoelectric actuations is obtained for the first time. The microbeam has been assumed as a composite Euler–Bernoulli beam with a piezoelectric layer deposited on it. The nonlinear equation of motion has been derived using the Hamilton principle, and solved using the Galerkin method. It is assumed that the neutral axis of bending is stretched when the beam is deflected. The axial load or pre-stress has been considered in derivations of the motion equations, and the boundary conditions in transverse vibration are assumed as clamped–clamped. It has been shown that the pull-in voltage, natural frequency and deflection of the system depend on the value of electrostatic actuation and the location, thickness and applied voltage of the piezoelectric layer. Thus the deflection and natural frequency of an electrostatic actuated microbeam may be tuned to a suitable value by altering the applied voltage to the piezoelectric layer, which is more convenient than altering the axial load, as used in previous methods. Also, this shows that a new sensor–actuator system may be constructed which is actuated by applying the voltage to the piezoelectric layer, and this actuation is sensed by the value of the output electric current induced from movement of the polarized microbeam.

In this article, the non-linear vibrations of a piezoelectrically actuated microresonator is studied. The microresonator is assumed as a clamped—clamped Bernouli—Euler microbeam. In contrast to previous researches in which the piezoelectric layer has been deposited on the integral length of the microbeam, here it is assumed that the piezoelectric layer is deposited on a part of microbeam length with equal distance from two ends. The microbeam is actuated by an AC voltage between upper and lower sides of the piezoelectric layer. Also, an electrostatic actuation is applied between the microbeam and an electrode plate, for the first time. The non-linear equation of motion has been derived by using the Hamilton principle by stretching the neutral axis assumption. The obtained equations are solved using the Galerkin, Rayleigh—Ritz, and multiple scale perturbation methods. It is shown that the sensitivity and natural frequency of the piezoelectrically actuated microresonator may be altered and controlled conveniently by applying an electrostatic actuation to the microresonator. Also, it has been shown that the system shows softening or hardening behaviour depending on the value of piezoelectric actuation, electrostatic actuation, axial load, thickness, length, and elasticity module of piezoelectric layer; thickness of microbeam; and its distance from the electrode plate. It is shown that non-linear behaviour of the piezoelectrically actuated microbeam may be changed to a linear behaviour by applying a suitable electrostatic actuation to the microbeam.

In this paper, the effect of nonlinearity on vibration of a rotating shaft passing through critical speed excited by nonideal energy source is investigated. Here, the interaction between a nonlinear gyroscopic continuous system (i.e. rotating shaft) and the energy source is considered. In the shaft model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. Firstly, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with nonconstant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the nonstationary vibration of the nonideal system, multiple-scale method is directly applied to the equations expressed in complex coordinates. Three analytical expressions that describe variation of amplitude, phase, and angular acceleration during passage through critical speed are derived. It is shown that Sommerfeld effect in specific range of driving torque occurs. Finally, effect of damping and nonlinearity on occurrence of Sommerfeld effect is investigated. It is shown that the linear model predicts the range of Sommerfeld effect occurrence inaccurately and, therefore, nonlinear analysis is necessary in the present problem.

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