Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain

Abstract

In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for unilateral double-collision will be discussed in high-dimensional non-smooth asymmetric system. Firstly, the impact laws are classified in order to detect the existence of periodic solutions of the system. The impact recovery matrix is introduced to transform the impact laws of high-dimensional system into matrix. Furthermore, by use of modal analysis and matrix theory, an invertible transformation is constructed to obtain the parameter conditions for the existence of the impact periodic solution, which simplifies the calculation and can be easily extended to high-dimensional non-smooth system. Hence, the range of physical parameters and the restitution coefficients is calculated theoretically and non-smooth analytic expression of the periodic solution is given, which provides ideas for the study of approximate analytical solutions of high-dimensional non-smooth system. Finally, numerical simulation is carried out to obtain the impact periodic solution of the system with small angle motion.

Subharmonic Bifurcation for a Non-smooth Double Pendulum with Unilateral Impact

“…However, the decoupling transformation method of the smooth double pendulum can not be directly applied to the decoupling problem of the the non-smooth double pendulum with unilateral impact. During the previous research, Guo et al [16] solved the problem of coupling of impact conditions by directly replacing the boundary conditions of the original system, but the calculation was slightly cumbersome, which was not conducive to the practical application of engineering. Therefore, the key problem is to solve the coupling problem of the nonlinear and non-smooth factor.…”

Real Time Swinging Up and Stabilizing a Double Inverted Pendulum Using PID-LQR

“…We used two control strategies to move the pendulum from its descending position to the vertical upright position and keep it there: one to swing up the pendulum and the other to balance the pendulum parts with a minimal amount of error for a rapid attainment of the dead position [8]. The state-space equation for the stance control system used was 𝓏̇= ℋ • 𝓏 + ℳ • 𝑢 + ℱ, where the variables ℋ and ℳ are the system matrix and ℋ is a concentrated disturbance that contains a disturbance due to an oscillating connection of the pendulum [9].…”

Control Theory Application for Swing Up and Stabilisation of Rotating Inverted Pendulum

“…Mathematical modelling should be possible by utilizing the Euler-Lagrange strategy. In order to write the differential equations of the motion of the pendulum [31], taking into consideration the symmetry between two links, we started from the general form of the Lagrange equations of the second type.…”