This paper introduces the Modified Positive Velocity Feedback (MPVF) controller as an alternative to the conventional Positive Position Feedback (PPF) controller, with the goal of suppressing unwanted resonant vibrations in smart structures. The MPVF controller uses two parallel feedback compensators working on the fundamental modes of the structure. The vibration velocity is measured by a sensor or state estimator and is fed back to the controller as the input. To control n-modes, n sets of parallel compensators are required. MPVF controller gain selection in multimode cases highly affects the control results. This problem is resolved using the Linear Quadratic Regulator (LQR) and the M-norm optimization method, which are selected to form the desired performance of the MPVF controller. First, the controller is simulated for the two optimization approaches, and then, experimental investigation of the vibration suppression is performed. The LQR-optimized MPVF provides a better suppression in terms of vibration displacement. The M-normoptimized MPVF controller focuses on modes with higher magnitudes of velocity and provides a higher level of vibration velocity suppression than LQR-optimized method. Vibration velocity attenuation can be very important in preventing fatigue failures due to the fact that velocity can be directly related to stress.
The nonlinear vibrations of the tapping-mode atomic force microscopy probe are investigated due to both nonlinearity in tip–sample contact force and curvature of the microcantilever probe. The nonlinear equations of motion for vibrations of the probe are obtained using Hamilton’s principle. In this work, the contact force is considered to be more dominant while previous works only consider Van der Waals force. The nonlinear contact force is expanded using a Taylor series to provide a polynomial with coefficients that are functions of the tip–sample distance. The outcome of this work allows the proper distance to be chosen before scanning to avoid instability in the response. Instability regions must be avoided for accurate imaging. The results show that initial tip–sample distance has a major effect on the stability of the frequency response and force response curves. For analytical investigation, the mode shapes of the atomic force microscopy probe are derived based on the presence of the nonlinear contact force as a boundary condition at the free end of the probe. The frequency response curve is obtained using the method of multiple scales. The results show that the effects of the nonlinearities due to probe geometry and contact force can be minimized. Minimizing the effects of nonlinearities allows for less cumbersome and calculation intensive software packages for atomic force microscopies. This research shows that one possible method of decreasing the nonlinearity effect is increasing the excitation force; however, this can increase the contact region and is not the best solution for canceling the nonlinearity effect. The superior method which is the major contribution of this paper is to find the optimal initial tip–sample distance and excitation force that minimize the nonlinearity effect. It is shown that at a certain tip–sample distance the quadratic and cubic nonlinearities cancel each other and the system responds linearly.
The equations of motion for a piezoelectric microcantilever are derived for a nonlinear contact force. The analytical expressions for natural frequencies and mode shapes are obtained. Then, the method of multiple scales is used to analyze the analytical frequency response of the piezoelectric probe. The effects of nonlinear excitation force on the microcantilever beam's frequency and amplitude are analytically studied. The results show a frequency shift in the response resulting from the force nonlinearities. This frequency shift during contact mode is an important consideration in the modeling of AFM mechanics for generation of more accurate imaging. Also, a sensitivity analysis of the system parameters on the nonlinearity effect is performed. The results of a sensitivity analysis show that it is possible to choose parameters such that the frequency shift minimizes. Certain parameters such as tip radius, microcantilever beam dimensions, and modulus of elasticity have more influence on the nonlinearity of the system than other parameters. By changing only three parameters—tip radius, thickness, and modulus of elasticity of the microbeam—a more than 70% reduction in nonlinearity effect was achieved.
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