Dampened Spring Systems

A damping element resists relative velocity across it. In other words, it exerts a damping force:

$$f=cv=c\dot{x}$$

where $c$ is the damping coefficient.

In the context of a dampened mass-spring system, we then have:

$$m\ddot{x}=f-kx-c\dot{x}$$

or

$$f=m\ddot{x}+c\dot{x}+kx$$

The same concept applies to rotational springs as well.

Example

Derive the equations of motion of the two-mass system shown below.

We assume that $x_2>x_1$, and ${\dot{x}}_2>{\dot{x}}_1$. From this, we can construct a diagram of forces acting on each mass, shown in (b) above.

From the diagram of forces, we obtain:

$$\begin{array}{rl}{m}_1{\ddot{x}}_1& =-k_1{x}_1+k_2(x_2-x_1)-c_1{\dot{x}}_1+c_2({\dot{x}}_2-{\dot{x}}_1)\\ m_2{\ddot{x}}_2& =f-k_2(x_2-x_1)-c_2({\dot{x}}_2-{\dot{x}}_1)\end{array}$$