# Predicting contact time and COR (coeff of restitution) for a cricket ball and pitch

For real world application, COR --> Bounce produced by pitch, for both spinners & seamers; Contact time --> grip provided by pitch, mainly for spinners

Assuming cricket ball to be a sphere and cricket pitch to be a flat. 1. Would it be appropriate to assume cricket pitch as elastic-plastic flat and cricket ball as elastic-perfectly plastic sphere for modelling purposes? 2. Does there exist any relationship (empirical or physical), relating coefficient of restitution and contact time between a sphere and a flat to their respective material properties and incident velocity.

• The mechanics are WAY more complicated than what you can put into a COR. Look at a tennis ball hit at 6000fps (youtube.com/watch?v=VHV1YbeznCo). – ja72 Jan 26 '16 at 19:37
• You might to explain what a pitch, spinner etc is. – ja72 Jan 27 '16 at 18:16
• The words, 'spinner', 'seamer' and pitch are irrelevant to this problem. It's simply about impact dynamics of a leather ball hitting a hardened soil flat. – Akash Malhotra Jan 31 '16 at 14:19
• It is because in mechanics it is all about geometry and without a sketch people have no idea what you are talking about. – ja72 Jan 31 '16 at 15:43
• COR is empirically derived, unless you want to do the quantum mechanics of molecular bond strain. – ja72 Jan 31 '16 at 16:17

So this is an oversimplification of the impact process, but if you model it as a spring damper system, then the deflection response is

$$x(t) = X \exp(-\zeta \omega_n t) \sin (\omega_n t \sqrt{1-\zeta^2})$$

where

\begin{align} \omega_n & =\sqrt{ \frac{k}{m}} && \mbox{: natural frequency, rad/s} \\ \zeta &= \frac{c}{2 m \omega_n} && \mbox{: damping ratio}\\ k & & &\mbox{: linear stiffness, N/m} \\ c & & &\mbox{: linear damping, N/(m/s)} \\ m & & &\mbox{: mass, kg} \end{align}

This means the impact speed is $$v_0= \lim_{t=0} \frac{{\rm d}x(t)}{{\rm d}t} = X \omega_n \sqrt{1-\zeta^2}$$

After half-cycle the rebound occurs at $t_1=\frac{\pi}{\omega_n \sqrt{1-\zeta^2}}$ with rebound velocity $$v_1 = \lim_{t=t_1}\frac{{\rm d}x(t)}{{\rm d}t} = -X \omega_n \sqrt{1-\zeta^2}\exp\left(- \frac{\pi \zeta}{\sqrt{1-\zeta^2}}\right)$$

The coefficient of restitution (COE), $\epsilon$, is defined as $v_1 =-\epsilon v_0$ or in this case $$\boxed{ \epsilon = \exp \left(- \frac{\pi \zeta}{\sqrt{1-\zeta^2}}\right)}$$

So when the damping ratio is 0 the COR is 1 and vise versa. The problem with this is that neither the damping ratio or the COR can be determined from first principals. They have to be derived experimentally.