Oscillations and Waves > Forced and damped oscillations > Problem 3.5.3

Statement

Under the conditions of problem 3.5.1, the pendulum had a velocity and a coordinate at the initial moment (). What will be the amplitude of oscillations after impacts if the first one occurred at the initial moment? Plot the phase portrait.

*(Note: in the answer section of the problem book, the transferred momentum is denoted as , and not as in the condition of problem 3.5.1. For consistency with the author's answer, the notation is used below).*

Solution

From problem 3.5.2, it is known that simple harmonic motion on the phase plane with coordinates corresponds to motion along a circle. The radius of this circle is numerically equal to the oscillation amplitude .
By the Pythagorean theorem, the amplitude for any state is expressed as:

It is important to note that at the moment of a brief impact, the coordinate of the pendulum does not have time to change, but its momentum increases abruptly by the value of the transferred momentum .

Let us consider two cases for the impact frequency.

Case 1: Impacts follow each other after a period

Over a time equal to the period of free oscillations , the representative point on the phase portrait makes a full revolution (a rotation by ) and returns to the same coordinates.

1. At the initial moment (), the initial momentum is . Immediately after the first impact, the momentum becomes:
   
2. By the time of the second impact (after a time ), the pendulum completes a full oscillation and returns to the point with coordinate , and its momentum is again equal to . The second impact instantly adds another momentum :
   
3. Since every period the system returns to its initial position , the momenta simply accumulate algebraically. Immediately after the -th impact, the total momentum at this point will be:
   
4. Substituting the coordinate and the accumulated momentum into the amplitude formula, we obtain:
   

On the phase portrait, this motion looks like an unwinding spiral consisting of concentric circular arcs of increasing radius.

Case 2: Impacts follow each other after half a period

Over the time , the point on the phase plane makes half a revolution (a rotation by ). In this process, both coordinates (both position and momentum) change their sign to the opposite.

1. First impact (, odd): Occurs at . The momentum abruptly becomes . The amplitude after the first impact is:
   
2. Free oscillations until the second impact: After the time , both coordinates change sign. The position becomes , and the momentum becomes .
3. Second impact (, even): An impact is delivered, adding a momentum . We calculate the new momentum:
   
   Let us calculate the amplitude for this state (taking into account the coordinate ):
   
   The pendulum "sheds" the energy of the impact and transitions to an orbit of a smaller radius (the amplitude decreases).
4. Free oscillations until the third impact: After another (a full period has passed in total), the coordinates change sign again: and . The system has returned exactly to its initial state (which was before the first impact).
5. Third impact (, odd): Once again adds a momentum to the initial . The state becomes completely identical to the state after the first impact, therefore, .

Thus, for all odd , the amplitude is equal to , and for all even , it is equal to . On the phase portrait, the representative point infinitely jumps between two closed semicircles (outer and inner).

Answer

If the impacts follow each other at time intervals , then the amplitude is:

If at intervals , then the amplitude is:


where .