When measuring with an atomic force microscope (AFM), the probe would move to the prescribed position via the platform and its vibration would occur. To achieve precision positioning for the follow-up scanning and high-speed measurement, in this paper, mathematical modeling and control of the probe is focused to avoid the damage incurred by the collision between the probe and the sample and to obtain the high measurement in the scanning step. The Hamilton’s principle is firstly employed to derive the equation of motion and its boundary conditions. Next, the summation method, Lagrangian equation and Laplace transform are applied to obtain natural frequencies, a dynamic model of the AFM probe, and to work out the transfer function of the open-loop system. The proposed model is compared with two conventional AFM probe models: point–mass model and conventional cantilever beam model - where one of the ends of the cantilever and basic platform are assumed to be fixed. Finally, collocated control is exercised to designate the positions for actuator and sensor and the root locus method cooperated with proportional-integral-derivative controllers to simulate the performance of the control system. The result shows that the controller can ensure the stability of this continuous system and perform effective control.

This paper is devoted to a theoretical investigation on the wave amplitude enhancement of surface sea water waves with under-sea periodic arrays of cylinders. A two-dimensional shallow water wave equation is derived and solved by using the plane-wave expansion method. The lattice types studied here include triangular, square and hexagonal lattices. These under-sea structures alter the sea bottom topography and induce constructive interference on the surface water waves. Given that the wave potential energy is dependent on the square of the wave amplitude, this mechanism can thus be used to increase the potential energy. It is shown that the enhancement factor depends on two geometric parameters and the maximum wave amplitude can be found by adjusting the two geometric parameters. Among the lattice types, the triangular and square lattice structures can induce more wave amplitude enhancement (and thus potential energy) than the hexagonal structures. Guided by numerical simulations, we have performed a reduced-scale water tank experiment to demonstrate the feasibility of the proposed idea. Preliminary experimental results show promising evidence of the predicted wave amplitude enhancement, suggesting perspective real-scale nearshore deployment and test.