derive an equation for this mass spring damper

Question

derive an equation to represent this mass spring damper in terms of input fore $F$ and relates to output displacement $(x)$ when springs $K_1=3$ , $K_2=5$ damper $C=6$ and mass $M=1$ , $F$ is a step of $10$

Answer 1

You just need to use Newtons law $\sum F=ma$. You have the forces:

• $F$ pulling the mass to the right.
• $F_{k_1}$ and $F_{k_2}$ of the springs acting against the movement of $m$.
• $F_c$ which is the force of the damper.

The corresponding forces of a spring and a damper are $F_k=-kx$ and $F_c=-c\dot x$. So you have

$$ m\ddot x=-(k_1+k_2)x-c\dot x+F. $$

Finally you obtain the usual equation

$$ m\ddot x+c\dot x+(k_1+k_2)x=F. $$

Tip: although I assume you are just learning this topic, I suggest you first read any book on elementary mechanics (this ideology can be extended to any field of study). This is a classical example and many books treat this problem. It is also always good to show that you have tried something in advance and not let people just answer for you. It is always good to help, but it is always better to help knowing that you've already given it a try.