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Following is the equation of motion for a physical pendulum which is damped and driven by a force of frequency $f$:

$$\frac{d^2 \theta}{dt^2} + b \frac{d\theta}{dt} + sin(\theta) = Tsin(2\pi ft)$$

In this equation $\theta$ is angle made by pendulum with the vertical. $t$ is time.

This equation is given on page 406 of Patrick Hamill's book "Intermediate Dynamics".

Question 1:

Dimension of 1st term on LHS is $radian/second^2$.

So dimension of 2nd term on LHS must be $radian/second^2$. So dimension of $b$ is $second^{-1}$.

But 3rd term on LHS has no dimension. Is this then physically valid equation?

Question 2:

This is non-linear equation. How to solve it (after correcting for dimension)?

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  • $\begingroup$ Are you saying that Hamill writes this equation without any explanation? And that he says nothing about its solution? I suspect that the answers to your questions are contained in the book, if you study it. As this eqn appears on p 406 I guess it is the brief answer to an exercise, but the assumptions made are probably explained in the question or the chapter in which it appears. Given that the book has around 350 pages of text, it ought to be quite detailed in its explanations. $\endgroup$ – sammy gerbil Jun 23 '16 at 10:31
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  1. No this is not a physically valid equation. It is a mathematical description of the pendulum in which the variables do not have units. The corresponding physical equation would be:

$$I \frac{d^2\theta}{dt^2} + b \frac{d\theta}{dt} + c \sin(\theta) = T \sin(2\pi ft),$$

where $$[I]=kgm^2, [b] = Nms, [c] = [T] = Nm$$

  1. For small angles $$ sin(\theta) = \theta$$ This makes it linear again.
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