Effect of displacement feedback control on the frequencies of cantilevered beams with tip mass and axial load using piezo actuators.

Abstract

This work provides a study of the natural frequencies of a cantilevered beam with tip mass and axial load. Displacement feedback control is applied using piezo actuators attached to the top and bottom of the beam. The center of gravity of the mass and its rotary inertia are accounted for in the solution. The analysis of flexible components is essential to provide for the successful design of various engineering structures. This study provides an analytical solution to the dynamic behavior of a cantilevered beam carrying a mass at the free end, while being subjected to constant axial load. The structure is modeled using the Euler-Bernoulli theory and the contributions of the mass, thickness and stiffness of the piezoelectric actuators to the structure are taken into account. The effects of the piezo input voltage polarity is also taken into account. The natural frequencies of the beam can be altered by applying a voltage in the desired polarity and thereby causing an extension or contraction in the piezo actuator. This mechanical response alters the frequencies of the piezoelectric beam. The piezoelectric effect causes a compression or extension strain when a voltage is applied along the direction of polarization. The strain in the piezoelectric beam causes a moment at the free end, which directly affects the natural frequencies. By applying a voltage in the same or opposite direction of the poling of the piezo, the result is a compression or extension perpendicular to the poling. An applied voltage in the same direction can be considered positive and reduces the frequencies, whilst in the opposite direction negative and increases the natural frequencies. In this investigation the piezo layer thickness is varied, which in turn allows for a variable voltage input. For a thicker layer, the voltage can be increased and the actuation strain increased. The frequency content of the dynamically varying forces applied to a structure has the potential to excite the structure at one or more of its natural frequencies. Using piezo actuators, the natural frequencies and the natural frequency gaps can be maximized. Maximizing the natural frequencies is useful to avoid resonance when the external excitation frequency is less than the natural frequency.