Introduction
Bodies collide e.g. when play is present in the design. This collision causes elastic or plastic deformations to the colliding bodies.
![Impact force](https://www.jpe-innovations.com/wp-content/uploads/impact-force-1.png)
Collision Stiffness
If both are ‘rigid’ take the local Hertzian stiffness.
$c=\frac{C_1C_2}{C_1+C_2}$
Max. Collision Force
$F=v_0\sqrt{mc}$
Max. Approach
$u=v_0\sqrt{\frac{m}{c}}$
Deceleration Time
$t=\frac{\pi}{2}\sqrt{\frac{m}{c}} $
Deceleration to $v=0$
${a_{max}=-v}_0\sqrt{\frac{c}{m}}$
$a_{ave}=-\frac{v_0}{t}=-\frac{2}{\pi}v_0\sqrt{\frac{c}{m}}$
Collision of bodies
Differential Equation:
$m\ddot{x}+cx=0$
$x\left(t\right)=k_1\cos{\left(\sqrt{\frac{c}{m}}t\right)}+k_2\sin{\left(\sqrt{\frac{c}{m}}t\right)}$
Coefficients:
$x\left(0\right)=k_1=0$
$v\left(0\right)=k_2\sqrt{\frac{c}{m}}=v_0$ with $k_2=v_0\sqrt{\frac{m}{c}}$
Distance, velocity, and acceleration:
$x\left(t\right)=v_0\sqrt{\frac{m}{c}}\sin{\left(\sqrt{\frac{c}{m}}t\right)} $
$v\left(t\right)=v_0\cos{\left(\sqrt{\frac{c}{m}}t\right)}$
$a\left(t\right)={-v}_0\sqrt{\frac{c}{m}}\sin{\left(\sqrt{\frac{c}{m}}t\right)}$