On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

Abstract

The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov-Holmes-Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge-Kutta algorithms for certain cases to verify the analysis.

Stability and chaotic behavior of a PID controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory

“…Taking into account Eqs (44) and (45), the variables of the system in normal form y 1 (t), y -1 (t) and y 0 (t) can be obtained and can Ft is close to zero or 2π, the phase difference between x 1 (t) and y 1a (t) is small and thus the dynamical behavior of the PID controlled pendulum can be predicted from the analysis of the simulation results. Figure 6 To corroborate the previous conclusions, the system has been simulated by using Eqs (9) and (19) On the other hand, Eqs (57) and (58) allow to deduce the values of x 0 (t), x 1 (t) and the conjugate complex of x 1 (t) (i.e. x -1 (t)) once the deviation variables z' 1 (t) i = 1,2,3 are known through the simulation of Eqs (9) and (19).…”

“…Figure 6 To corroborate the previous conclusions, the system has been simulated by using Eqs (9) and (19) On the other hand, Eqs (57) and (58) allow to deduce the values of x 0 (t), x 1 (t) and the conjugate complex of x 1 (t) (i.e. x -1 (t)) once the deviation variables z' 1 (t) i = 1,2,3 are known through the simulation of Eqs (9) and (19). Consequently, Eqs (44)-(45) allow to calculate y 1 (t) and y 0 (t) and compare them with the analytical results obtained from Eqs (52)-(53), as it is shown in Fig 8. It should be noted that the values of y 1 (t) and y 0 (t) are very close to y 1a (t) and y 0a (t) respectively, in accordance with the previous considerations.…”

“…Next we choose a value for τ d to obtain an appropriate value for the equivalent damping coefficient given by x -1 (t), x 0 (t) and y 1 (t), y -1 (t), y 0 (t). The values of z' i (t) can be obtained through the simulation of Eqs (9) and (19). On the other hand, the variables x 1 (t), x -1 (t) and x 0 (t) can be obtained as functions of z' i (t) (i =1,2,3) from the inverse normalizing transformation, which is deduced from Eqs (25), (27), (41) and (42) …”

“…Another interesting verification of the analytical and numerical computations can be carried out taking into account the following reasoning. From Eqs (19) and (25) it is deduced that:…”

“…However, the use of a simple control law to obtain chaotic behavior has been less used. This is probably due to the difficulty in determining whether a pendulum with harmonic base excitations can exhibit chaotic dynamics [16][17][18][19][20].…”