Stabilization of Driven Pendulum with Periodic Linear Forces

Using Kapitza method of averaging for arbitrary periodic forces, the pendulum driven by different forms of periodic piecewise linear forces is stabilized. These periodic piecewise linear forces are selected in the range to establish an exact comparison with harmonic forces. In this contest, the rectangular force was found to be the best, but this force is more effective when it has a time-dependent structure. This time-dependent structure is found by defining a parametric control on some other periodic piecewise linear forces. 1. Introduction A pendulum with fixed suspension has only one stable point, while a pendulum whose suspension has fast oscillation can have more stable points (can oscillate). Such phenomena were first studied by Stephenson in 1908 [1–3]. In 1951, Kapitza presented this problem in a different way [4], so-called Kapitza pendulum. In 1960,Landau et al. studied the stability of such a system driven by harmonic force [5]. Then, its rapid growing applications started such as trapping of particles by laser [6–8], control of robotic devices [9, 10], effect on price equilibrium [11], and control by lasers in cybernetics [12]. Next in place of harmonic force Ahmad and Borisenok (2009) used periodic kicking forces, modifying Kapitza method for arbitrary periodic forces [13]. Also, Ahmad used symmetric forces and stabilized the system with comparatively low frequency of fast oscillation [14]. 2. Kapitza Method for Arbitrary Periodic Forces A classical particle of mass is moving in time-independent potential field and a fast oscillating control field. For simplicity, consider one-dimensional motion. Then, the force due to time-independent potential is and a periodic fast oscillating force with zero mean in Fourier series is This fast oscillation has frequency . Here, is the frequency of motion due to . The mean value of a function is denoted by bar and is defined as Also, the Fourier coefficient is From (3) and (4), it follows that In (2), and are the Fourier coefficients given as Due to (1) and (2), the equation of motion is Here at a time two motions are observed: one along a smooth path due to and the other small oscillations due to . So the path can be written as (see Figure 1). Here, represents small oscillations. Figure 1: Path of the particle. By averaging procedure, the effective potential energy function is [13] The pendulum driven by a periodic force is stabilized by minimizing (8). These forces are chosen in the range to establish an exact comparison with harmonic forces. Next, an -parametric control is developed with one of the