Mathematical pendulum
This program can be used to calculate the movement of a mathematical pendulum. The pendulum consists of a point mass at the end of a massless rod. The program solves the non-linear differential equation for this pendulum and therefore allows the study of various effects caused by the non-linearity. The linearized differential equation can be used for comparison. The pendulum can also be driven by a periodic excitation. With suitable parameters, this results in chaotic behavior. The representations in phase space and as a Poincaré section are helpful here.
Operation
The initial conditions for the movement are entered at the top left of the control panel. The angle in degrees and the angular velocity in rad/s. The parameters of the pendulum length l in meters and damping constant gamma in 1/s are entered below. The acceleration due to gravity is always 9.81m/s2. Pressing the Start button starts the movement of the pendulum in real time. At the same time, the deflection is displayed as a function of time (the time is plotted to the right and the angle upwards). By ticking the "Linearize" box, you can switch to the solution of the linearized differential equation. This makes it easy to make comparisons. If "Periodic force" is ticked, the pendulum is driven by a cosine force. The angular frequency omega and the amplitude can be entered at the bottom left. The pendulum oscillation is stopped with the "Stop" button. Click "Clear" to delete all representations (function phi(t), phase space omega(phi) and Poincaré section). Otherwise, several curves can be drawn on top of each other for comparison.
Display options
The phase space is a representation in which the deflection (angle) is plotted upwards and the angular velocity to the right. The phase space for the driven pendulum is actually 3-dimensional and contains time as the third dimension. In this sense, the representation is a projection onto the two dimensions mentioned above. The Poincaré section avoids this problem of representation by only drawing points after integer periods of excitation in the plane of angle (upwards) and angular velocity (to the right).
Example of chaos
With suitable parameters, e.g. l = 10, damping = 0.1, omega = 1 and amplitude of excitation = 3 (non-linear with periodic excitation), a chaotic movement can be started. After a short time, this pendulum with dissipation moves towards a strange attractor, which becomes visible in the Poincaré section (set of all red dots). This attractor in the Poncaré section, which has a dimension between 1 and 2, is neither a line nor a surface. If the parameters are varied slightly, the shape of the attractor changes. If the movement of the pendulum is periodic and not chaotic, then only one point is obtained in the Poincaré section.