#### Introduction

Bodies collide e.g. when play is present in the design. This collision causes elastic or plastic deformations to the colliding bodies.

#### 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)} $

$v\left(t\right)=v_0\cos{\left(\sqrt{\frac{c}{m}}t\right)}$

$a\left(t\right)={-v}_0\sqrt{\frac{c}{m}}\sin{\left(\sqrt{\frac{c}{m}}t\right)}$