Subsequent behaviour of the oscillations of the double pendulum under various initial conditions

In this study, we analyzed the pendulum case which is a good model for several oscillating phenomena. Also we investigated nonlinearities and sensitivity to initial conditions in pendulum systems. It is known that a simple pendulum that consists of a bob of mass m attached to the free end of a massless rod of length b, performing simple harmonic motion is widely used in physics. There are various kinds of pendula such as double, triple, inverted, conical or mathematical pendulum. Although it exhibits a nonlinear behaviour its motion can be approximated to be linear for relatively small angles. The pendulum systems are one of the primary topics in mechanical structures for nonlinear dynamics and chaos researches. In this thesis, we have examined double and triple pendula. Our aim is to examine and find the motional structure of the pendulum according to different initial values. There are two basic parameters on the pendulum, m being the mass at the end of the rope and b being the length of the massless rod. m_1, m_2, b_1, b_2 are for the double pendulum, m_1, m_2, m_3, b_1, b_2, b_3 are for the triple pendulum. In our study, pendulum motion was examined for different values of mass and massless rod lengths and a behaviour model was derived. In addition, after solving the equations on the fixed pendulum, horizontal and vertical oscillations were superimposed to the fixed pendulum application and the equations of motion of the pendulum were derived again and its motion was examined. All of our equation solutions, variation of model parameters and motion structures at the initial values in the specified intervals in each different model setup were examined by the help of Lagrange formalism and solved by computer programs. We examined the motion behaviours of double and triple pendula under various physical conditions and critical motion regions on the different initial conditions. By subjecting system variables to analyses using phase portraits, bifurcation diagrams and Lyapunov Exponents, we examined the motion behaviour models.