Revisiting Pendulum Dynamics - Tap to Re-oscillate

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In summary: The equation to model this behavior is y^{''}+0.1y^{'}+4y=C\delta(t-nT), where T is the period of the taps and n is an integer representing the number of taps. By adjusting the value of C and the timing of the taps, you can achieve a periodic motion of the pendulum. In summary, it is possible to turn an underdamped pendulum into a harmonic oscillator with periodic taps, by calculating the necessary energy for each tap and applying it at regular intervals.
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saltydog
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Regarding an earlier post:

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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  • #2
I've read some papers on this subject but I'm still puzzled.The answer to your question is yes, it is possible to turn an underdamped pendulum into a harmonic oscillator with periodic taps. To do this, you need to calculate the exact amount of energy required for each tap so that the pendulum reaches its maximum displacement, then apply that amount of energy at regular intervals. This will cause the pendulum to oscillate harmonically instead of winding down.
 
  • #3



It is interesting to see how tapping the pendulum at just the right time and with the right amount of force can re-oscillate it and turn it into a harmonic oscillator. Your analysis and calculation of the value of C is impressive and shows a good understanding of the dynamics of the pendulum.

As for expressing this in a periodic fashion, one idea could be to use a periodic function such as a sine or cosine wave to represent the tapping force. By adjusting the amplitude and frequency of the wave, you may be able to achieve the desired effect of re-oscillating the pendulum continuously. Additionally, you could also explore using different types of forcing functions, such as a square wave or sawtooth wave, to see how they affect the behavior of the pendulum.

Overall, your approach and analysis of the pendulum dynamics is commendable and I believe with further exploration and experimentation, you can find a periodic solution to turn your underdamped pendulum into a harmonic oscillator.
 

Related to Revisiting Pendulum Dynamics - Tap to Re-oscillate

1. What is the purpose of revisiting pendulum dynamics and tap to re-oscillate?

The purpose is to gain a deeper understanding of pendulum dynamics and explore the concept of tapping to re-oscillate, which has potential applications in energy harvesting and synchronization.

2. How does tapping affect the oscillation of a pendulum?

Tapping introduces additional energy into the system, causing the pendulum to re-oscillate at a higher amplitude or frequency. This can be seen as a form of external forcing on the pendulum's natural motion.

3. Are there any practical applications for tapping to re-oscillate in pendulum dynamics?

Yes, tapping to re-oscillate can potentially be used in energy harvesting systems to convert the kinetic energy of the pendulum into electrical energy. It can also be used in synchronization of multiple pendulums, such as in clocks or metronomes.

4. What factors can affect the success of tapping to re-oscillate in pendulum dynamics?

The success of tapping to re-oscillate depends on various factors including the amplitude and frequency of the tapping, the length and weight of the pendulum, and the damping and friction in the system. The angle at which the pendulum is tapped also plays a role.

5. Can tapping to re-oscillate be applied to other systems besides pendulums?

Yes, the concept of tapping to re-oscillate can be applied to other systems with oscillatory motion, such as springs or vibrating strings. However, the specific effects and applications may vary depending on the properties of the system.

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