How can I create the Mathematica model of the inverted pendulum with a moving vertically base?

To dos For the described system do the following: (a) Prepare the Mathematica model of the inverted pendulum with a moving vertically base (1): i. define a function which returns the pendulum base position, ii. define a function which returns the mass m position, iii. define the differential equation 1. Analyze the model behavior for an initial value equal to 0 and 0.2, and the steady base (y=0). Plot the joint angle as a function of time. Animate the pendulum movement. Investigate the influence of the dumping coefficient on the pendulum behavior. (b) Equip the pendulum with the oscillator (2): i. define the rule which introduces the controller (2), ii. apply the defined rule to the system. Examine the correctness of the provided solution. (c) Simulate the system behavior for both the initial values with different values of oscil- lations amplitude and frequency. Try A = 0.2, w = 4; A = 0.2, w = 40; A = 0.2, w = 400; A = 0.02, w 400; A= 0.002, w = 400; A 0.002, w 4000. Try to derive some conclusions about the influence of the oscillator amplitude and its frequency on the system behaviors. = =