# Equations of Motion for the Multi Degree of Freedom (MDOF)

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?

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.
• Got it, thanks for the answer
– user35114
Feb 12, 2022 at 11:04