Design and modelling of nonlinear dynamical system

Abstract

The dynamic behavior of double and triple link pendula considering both the lumped and distributed mass has been studied. The system of governing differential equations for double pendulum (DP) and triple pendulum (TP) is derived using the Euler- Lagrange (EL) approach. Effect of damping of pivots is also considered in the model. The governing equations are also derived in the terms of momenta and angular velocity without considering the damping to confirm the results obtained by the E-L approach. Chaotic behavior of these systems was investigated using time series plot, Poincare map and Lyapunov exponent. It is observed that bottom pendulum is most sensitive than other pendula. Further, tendency of chaotic behavior increases with degree of freedom (dof) for the same initial conditions. Nature of Poincare maps of the TP becomes more intricate in comparison to the DP. Interestingly, damping of pivots results in regular motion of the pendulum system. In other words, introduction of damping in the multiple pendula reduces the tendency of chaos. Moreover, there is no qualitative difference between the lumped and distributed pendulum systems as far as the dynamical behavior is concerned. All the observations and results are validated experimentally. The LQR technique was used to control inverted triple pendulum. The control technique was found to be effective. This is also valid for double pendulum newline