Coordinate References

Heavily supported by the notes here
If we have a spring mass system that is angled theta from the horizontal axis (See Week 5 Notes), the general equation for that system is
$m^{{}^{\prime \prime }}-k\left({\delta }_{st}+x\right)+mg\sin \left(\varphi \right)$
By choosing the equilibrium position of the spring mass system as a point of reference, you can set the following
$mg\sin \varphi -k{\delta }_{st}=0$
Therefore, for a spring, mass system example, if we write the equation of motion based on the equilibrium position as the coordinate reference, the equation of motion will be
$m{x}^{{}^{\prime \prime }}+kx=0$
without angles!

Equilibrium Position

It is easiest to set up equations relative to the equilibrium position

Equation of Motion of Spring Mass Systems

You should be able to start with an equation like
$m{x}^{{}^{\prime \prime }}(t)+kx(t)=0$
Which you can then take to the LaPlace domain
$m\left(s^{2}X\left(s\right)-sx\left(0\right)-x^{\prime }\left(0\right)\right)+kX\left(s\right)$
Then you can solve for…
$X\left(s\right)=\frac{sx\left(0\right)+x^{\prime }\left(0\right)}{s^2+\frac{k}{m}}$ Which eventually yields
$x\left(t\right)=x\left(0\right)\cos \left(w_n{t}_{}\right)u\left(t\right)+\frac{x^{\prime }\left(0\right)}{w_n}\sin \left(w_{n}t\right)u\left(t\right)$
$x\left(t\right)=A\sin \left(w_{n}t+\varphi \right)$

Acceleration

Acceleration is proportional to displacement with an opposite sign, this is called harmonic motion

Using the representation above
$x\left(t\right)=A\sin \left(w_{n}t+\varphi \right)$ $x^{{}^{\prime }}\left(t\right)=A{w}_n\cos \left(w_{n}t+\varphi \right)=A{w}_n\cos \left(w_{n}t+\varphi +\frac{\pi }{2}\right)$ $x^{{}^{{}^{{}^{\prime \prime }}}}\left(t\right)=-A{w}_n^2\sin \left(w_{n}t+\varphi \right)=A{w}_n^2\sin \left(w_{n}t+\varphi +\pi \right)$
Acceleration is proportional to displace. ent (with opposite sign), this motion is a simple harmonic motion.

Summary

To summarize the point of this entire note: If we make the assumption that x and theta are measured from equilibrium, the weight (mg) and static spring deflections OF SPRING MASS SYSTEMS WHERE THE MASS IS HANGING FROM A SPRING don’t appear in the equations.