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I have an underdamped pendulum. This is just what you'd expect to happen to a real pendulum if you pick it up then let it go: It slowly oscillates down to nothing. An equation describing this is:

[tex]y^{''}+0.1y^{'}+4y=0\ \ \ \ \ \ \ \ \ \ \ \ (1)[/tex]

with y(0)=1 and y'(0)=0

The first plot shows this behavior: it just winds down.

However, I'd like to "tap" the pendulum at the other side just when it reaches it's maximum angular displacement and to do it with just enough force so that it comes right back to me so I can tap it again and so forth.

An equation to model this is:

[tex]y^{''}+0.1y^{'}+4y=C\delta(y-y_{max})[/tex]

Where [itex]\delta(x)[/itex] is the Dirac Delta function.

With C to be determined so that the pendulum comes back to the same spot.

I've analyzed this for just the first half-swing and determined that c[itex]\approx[/itex]1.15. That is:

[tex]y^{''}+0.1y^{'}+4y=1.15\delta(t-t1_{min})[/tex]

Where I've calculated the first minimum point by solving (1) and using some trig relations. The second plot shows this result. Note at the bottom of the first valley, 1.15[itex]\delta(t-t1_{min})[/itex] is applied. Note the pendulum reaches back to 1, thus it will fall back again to where I tapped it.

Any ideas how to express this in a periodic fashion so that the once underdamped pendulum is turned into an harmonic oscillator?

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# A pendulum revisited

Can you offer guidance or do you also need help?

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