I was watching the video: Equations of Motion for the Multi Degree of Freedom (MDOF) Problem Using LaGrange's Equations from Good Vibrations with Freeball

However, I didn't understand how to obtain these 2 terms (in yellow)

Can someone explain it?

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1 Answer 1


The first one $x_2- x_1= 3r\cdot \theta$ is very easy. This is based on the fact that for a circle R the arc described by an angle $\theta$ is equal to $R\cdot \theta$. In this case the external radius is 3r.

To understand it easier, it helps me to try and understand intuitively what the equation says for the problem.

So, in this case $x_2 - x_1$ the length of the rope between the wheel and mass 3m. If you translate the wheel --without rotating it-- then the length of the rope between the wheel and the 3m mass will not change.

Or if you keep O fixed in space and rotate the wheel by $\theta$, then the length of the hanging rope will change by $3r \theta$ because the external radius of the wheel is 3r.

Regarding the other term $\left(\frac{1}{2} (2k) \left(x_3 -(x_1-r\theta)\right)^2\right)$ the term in the parenthesis is the extension of the spring marked as 2k (above mass m). If the quantity $x_3 -(x_1-r\theta)$ is :

  • positive: the spring marked 2k will be in extension
  • negative: the spring marked 2k will be in compression

In simple cases such as this, I usually do the following thought experiment. I change only one degree of freedom, and see the effect on the element of interest. E.g.

  • if mass m moves downwards (and the wheel does not rotate and its center O does not move) then the spring will be extended by (positive) $x_3$
  • if the the center of the wheel O moves downwards by (positive) $x_1$ (and mass m remains fixed, and wheel does not rotate) then the spring 2k state will be in compression (i.e $-x_1$)
  • if the wheel rotates by an angle $\theta$ CW (and mass m, and the wheel do not translate), then again the spring will be in extension.
  • $\begingroup$ Got it, thanks for the answer $\endgroup$
    – user35114
    Feb 12, 2022 at 11:04

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