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.
Atomic Force Microscopy (AFM) uses a scanning process performed by a microcantilever beam to create a three dimensional image of a nano-scale physical surface. AFM includes a microcantilever probe with a tip at the end that is controlled in order to keep the force between the tip and the surface constant by changing the distance of the microcantilever from the surface. Some microcantilevers have a layer of piezoelectric material on one side of the microcantilever for actuation purpose. An accurate understanding of the microcantilever motion and tip-sample force is needed to generate accurate imaging. In this paper, the equations of motion for an AFM piezoelectric microcantilever probe are derived for a nonlinear contact force. The analytical expressions for natural frequencies and mode shapes are determined. Then, the analytical frequency response of the piezoelectric probe is found using the method of multiple scales. The effects of nonlinear excitation force on the microcantilever probe’s frequency and amplitude have been analytically studied. The force nonlinearities lead to a frequency shift in the response. Accurately modeling this frequency shift during contact mode of the AFM probe is a significant consideration for the generation of more accurate imaging.