In the following experiments you should study the phase space diagrams in parallel with the xy- diagram.

E 1:  Choose free oscillator. Change coordinate range xmax, step width and initial values. Superimpose several graphs by selecting back, and start after changing parameters.

E 2:  Check that the period of the free oscillator is π. Change the formula to get other periods (frequencies). Derivate from the differential equation the quadratic dependency (remember sin´´(ax) = - a2 sin(ax)). Superimpose oscillations with multiple frequencies..

E 3:  Experiment with the non-resonant non-damped driven oscillator. How does the beat frequency depend on free oscillator and drive frequencies? Study the step function in the formula (jump from 0 to 1 at a given variable x), which is broadly applicable. Change the position of the step.

E 4:  For the resonantly driven oscillator without damping the linear increase in amplitude is recognizable in the linear blue spiral of the phase space diagram. What does it mean, that the red curve is no longer linear after the step? Determine the zero crossings of y(x) (when you click at a point; its exact coordinates appear in the lower left corner of the coordinate system).

E 5:  Study the behavior with damping. The blue phase space curve now is an exponentially damped spiral. The red curve shifts to a new linear line, indicating transition to a different, constant frequency. (The transient is best observed at low speed )

E 6:  Interpret the results of E4 and E5 from the view of Fourier analysis!

E 7:  Study the non-resonantly driven oscillator with damping.

E 8: As for E 7. In the formula first delete the term of damping, then that of drive. Compare the results. Experiment with other formulas by inventing additional terms (more than one drive frequency, more complicated forms of damping, etc.).

E 9: Study the seconds pendulum with small amplitude. Change the initial value y (starting angle) and measure the period; study the phase space diagrams. In precision clocks the amplitude is limited to a few degrees; why? Study the phase space diagrams for large amplitudes.

E 10: Study the pendulum just before and after looping. The predefined calculation step is minimal to get good results even very close to the unstable point. Increase the step width and study the resulting artifacts.

E 11:  Choose the highest possible resolution and try to come as near as possible to the unstable equilibrium point by choosing proper initial conditions. You may have to extend xmax.