# Dynamics of a bouncing capsule

I am working on a research project investigating the bounce of a capsule. I have tried to model it using conservation of energy, where we considered torque by normal force and friction during the bounce, which will result in the rotational kinetic energy of the capsule after the bounce. We modelled the rotation using angular impulse $$\omega=\frac{m(u+v)l\sin\theta}{I}$$, where $$u$$,$$v$$,$$\theta$$,$$l$$ and $$I$$ are the impact velocity, rebounding velocity, angle of contact, length of moment arm and moment of inertia respectively.

However, this method does not take into account of dissipation of energy. Thus I have decided to take on a different approach.

I am currently trying to model the bounce of the capsule using a spring-mass-damper model: $$m\ddot{x}+kx+c\dot{x}+mg=0$$

One problem with this model is that the normal force $$kx+c\dot{x}$$ will always be a constant value no matter what is the contact angle. However, in our conservation of energy model, the impulse by normal force, which we modelled as $$m(u+v)$$, would be clearly different for different contact angles $$\theta$$ as different $$\theta$$ will result in different conversions between RKE and KE, thus affecting $$v$$, which is the rebounding velocity.

Please refer to the free body diagram shown above. During the collision, the spring will come into contact with the ground. The centre of mass of the capsule will continue to move down due to inertia, and thus resulting in a pivoting motion about the point of contact. At the same time, the spring will be compressed. The capsule may also slip at a certain critical angle.

We have worked out the tilting motion of the capsule, but that is without considering the spring-mass-damper model. The non slip condition which we have worked out is as such: $$\mu_s \geq \frac{mgl^2\sin\phi \cos \phi-Il\omega^2 \sin \phi}{-l^2 mg \sin^2 \phi-Il\omega^2 \cos \phi +Ig}$$

Equations of motion with static friction: $$\alpha=\frac{mgl}{I}\sin\phi$$

Equations of motion with kinetic friction: $$\alpha=\frac{ml(g-l\omega^2 \cos \phi)(\sin\phi - \mu_k \cos\phi)}{I+ml^2\sin\phi(\sin\phi-\mu_k \cos \phi)}$$

May I know if there is any way to combine the friction model with the spring mass damper model that we have to obtain a more comprehensive model that can accurately predict the motion of the capsule during collision. Thank you.

• Are you in a simulation environment, and thus just calculation of force and torques is sufficient, or are you trying to come up with analytical solutions? – John Alexiou Dec 7 '20 at 4:59
• Would you be ok with an impulsive method, where energy loss is modeled via a coefficient of restitution (elastic-plastic impact), with friction? The spring damper method works well in simulations, but the non-linearities mean analytical solutions are impossible. The tilting while deflecting changes the effective mass the spring sees, and thus the response of the spring damper type of system. – John Alexiou Dec 7 '20 at 5:07
• The coefficient of restitution may not be a constant value (unlike dropping a spherical ball) due to the asymmetric nature of the capsule. I don't mind solving the equations numerically. – bob the legend Dec 7 '20 at 5:14
• Please read this article in its entirety to understand the modeling issues with collisions. – John Alexiou Dec 7 '20 at 6:33
• I have read through the article. May I know how we can apply that to the capsule? It seems like they have only used symmetrical shapes like squares and circles. The spring model at the back is interesting although they did not include the equations that they used to simulate it. Another thing is that there are both static friction and kinetic friction during the bounce of the capsule which is not included in their simulation. I think it is crucial to take these into considerations as well when simulating the bounce of a capsule-shaped object. – bob the legend Dec 7 '20 at 7:04