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Title: Dynamic modeling and response analysis of a planar piezoelectric-based mechanism
Authors: Vahid Hassani
Keywords: DRNTU::Engineering
Issue Date: 2014
Abstract: Piezoelectric-based mechanisms and piezoelectric motors have been extensively used over the recent decades because of their wide applications in several industries. Some studies and researches on this type of mechanism focus on two interests, namely (i) the development of mechanisms with efficient working principles and (ii) the nonlinearities that are apparent in the corresponding elements. Based on the operating mode, two main types of mechanism depending on its frequency-related-working principle are distinguished. If the mechanism operates at relatively low frequency, which is referred to as the stepping mode, the mechanism is called a step motor. In this mode, an accurate positioning task is possible to be achieved. On the other hand, when it operates at its natural frequencies, high speed positioning can be performed. The corresponding mode is referred to as the resonant mode. This research focuses on merging the two modes in one single mechanism. Therefore, the proposed mechanism can operate at high speed yet high positioning accuracy. However, the design requirement for the two modes seems conflicting. The former requires a stiff structure, while the latter requires flexible structure. As mentioned above, piezoelectric material is known to suffer from a severe nonlinearity that arises from the hysteresis relationship between input and output variables in such devices. The hysteresis nonlinearity brings difficulties in achieving an ideal controlled system that causes performance degradation, for instance, in micropositioning applications. In order to mitigate the problem, such hysteresis nonlinearity has to be well-characterized in advance. In this study, the hysteresis nonlinearity is characterized and modeled in a typical stacked-type piezoelectric actuator under load-free and preloaded circumstances incorporating the inertial effect of the system. For this purpose, the piezoelectric actuator is modeled as mass-spring-damper system, which is utilized with the stop operator as one of the hysteresis operators of the Prandtl-Ishlinskii model. Merging the structural model with the nonlinear hysteresis characteristics and applying the basic equation of motion for this system, we observe that the modeling results will reveal better correspondence to the measured output compared to that of the classical PI model, in particular, as the input frequency and the input amplitude are increasing. In the next step, a test platform, namely the 2-DOF triangular-shaped piezo-driven mechanism is utilized to investigate the effect of hysteresis nonlinearity existing in the piezoelectric actuator in the dynamic response of the mechanism. This mechanism is basically designed to be periodically excited while in contact with a driven surface. Two piezoelectric elements separated by the angle of 90o are exploited to create elliptical motion on the end effector of the mechanism depending on the applied phase difference. In order to evaluate the nonlinear hysteretic output of the actuator in correlation with any types of structures, e.g. structure of the mechanism, the basic linear constitutive equations of piezoelectric actuators are integrated with the hysteretic response of the piezoelectric actuator so as to have more realistic estimation of actuator’s response. Then, the hysteretic effect of the piezoelectric actuator is taken into consideration to characterize and model the dynamic response of the designated structure. The trajectory output of the structure is simulated using finite element approach while the excitation input to the model, incorporating the hysteresis properties, is predicted based on the proposed formulation. It is expected that the simulation results will exhibit good agreement to the measured output when the hysteretic signal is applied to the mechanism as compared to the linear pure sinusoidal input. Finally, a 3-DOF piezo-driven mechanism, which is called as pyramidal-shaped piezo-driven mechanism, is proposed utilizing three piezoelectric actuators. The design procedure is performed based on the modal analyses to ensure that the pyramidal-shaped piezo-driven mechanism is able to operate at the desired resonant mode. In the next stage, the dynamic modeling of the mechanism is presented to show that the designated structure is able to provide different types of planar motions at different sets of phase differences applied to the piezoelectric elements. Through dynamic modeling, the response at the tip of the mechanism (end effector) is estimated with respect to two types of displacement inputs imposed by piezoceramics at the contact point where the head of the piezoelectric actuators meets the structure of the mechanism. These inputs, known as the ideal linear pure sinusoidal input and the real nonlinear hysteretic input are applied to the contact point in order to compare the response at the tip of the mechanism to both linear and nonlinear signals. Finally, we observe that the modeling results demonstrates better agreement to the measured output as the real nonlinear hysteretic input is applied to the contact point of the mechanism as compared to the real linear pure sinusoidal input.
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