Factors of length ratio and mass at chaos of double pendulum system

Abstract

The double pendulum consists of two pendulums attached with the pivot of the second pendulum located at the end of the first pendulum by a rigid bar, massless and considers the effect of gravity. The double pendulum system has strange and unpredictable behaviour. Chaos in a double pendulum system is usually used as an example of chaos in mechanical systems. The chaotic system is characterized by a sensitive system to the initial condition. It means that giving the initial condition with a small difference will give a different solution. Lyapunov Exponent is used to calculating the range of two trajectories with a small difference of initial condition, where the positive exponent value shows two different solutions close at the beginning then more diverge away. In this paper, we determine parameters that will cause the double pendulum system to reach chaos easily. To facilitate this, the equations of the double pendulum motion are simplified by =gl1,β=gl2,δ=m2m2. In particular, we focus on determining the mass ratio and length ratio from the double pendulum system. In some simulations with simulation program that have been made in Maple, the program has resulted in the greater value of δ at the specific problem shows that the system needs the value of θ1,θ2,θ˙1,θ˙2, as first initial condition smaller to reaches chaos easily.