Application of DT method to bifurcation analysis of microcandilevers with proportional-plus-derivative control

This paper studies the bifurcation behavior of the probe tip of an atomic force microscope with a proportional-plus-derivative (PD) feedback control using the DT (differential transformation) method. The dynamic behavior of the probe tip with PD control law is characterized by reference to maximum Lyapunov exponent plots produced using the time-series data obtained from differential transformation method. Furthermore, the detailed transitions in the dynamic response of the probe tip are examined using bifurcation diagrams of the tip displacement and the tip velocity, respectively. The results indicate that the probe tip behavior is significantly dependent on the magnitude of the proportional and derivative control gain. Specifically, the probe tip motion includes T-, 2T-, 3T-, 4T-, multi-periodic, and chaotic motion. Numerical results show that the dynamic behavior will leave chaotic motion to periodic motion at K_p=-0.45 in the steady state by changing the control loop gain Kv from -0.1 to -1.0. Furthermore, it is demonstrated that the differential transformation method is in good agreement for the considered system.