# Rolling Pendulum: Solving the Dynamics Equations

• Andy Froncioni
In summary, the conversation discusses a system consisting of a wheel with a rubber tire and a pendulum attached to it. The wheel's rotation and the pendulum's motion are coupled, and the system experiences energy loss due to rolling resistance. The goal is to find the dynamical equations of motion for the system, and one of the equations mentioned is for the normal force on the wheel axle. The system's amplitude decreases over time, and the question is posed of whether the amplitude of swing is small or large. The answer is that it is a large amplitude swing, ranging from -pi/2 to +pi/2.
Andy Froncioni
(This is NOT a homework problem. It's an engineering problem I'm trying to crack.)

A wheel with a rubber tire (friction) can roll on a suspended rail. Attached to it is a pendulum that's rigidly mounted on the axle of the wheel with a mass that can hand down and swing. (The wheel's rotation and the pendulum's are coupled.)

The system swings freely and due to the energy lost to rolling resistance of the tire against the rail ( F = N*Crr ), the system's amplitude decreases over time.

I am trying to find the dynamical equations of motion of this system.

d (theta)/dt = alpha

d(alpha)/dt = f(Crr,M,L,R,m,...)Can anyone help me? I know the equation for the normal force on the wheel axle is given by something like:

N - M*g - m*v^2/L cos(theta) = 0, where
N = normal force
M is the total mass of the system
m is the effective mass of the pendulum portion
v is the velodity of the COM of the pendulum portion
theta is the angle of the pendulum to the vertical
Crr is the coeff of rolling resistance of the tire

Small amplitude or large amplitude of swing ?

Nidum said:
Small amplitude or large amplitude of swing ?
Large amplitude. -pi/2 to +pi/2

I don't want the solution. I want to understand the governing equations.

## 1. What is a rolling pendulum?

A rolling pendulum is a physical system where a pendulum is attached to a mass that is free to roll along a surface.

## 2. What are the dynamics equations for a rolling pendulum?

The dynamics equations for a rolling pendulum involve the equations of motion for both the pendulum and the rolling mass, as well as the constraint equations that relate the motion of the two components.

## 3. How are the dynamics equations solved?

The dynamics equations for a rolling pendulum can be solved using various mathematical techniques, such as numerical integration or analytical methods like Lagrangian mechanics.

## 4. What factors affect the motion of a rolling pendulum?

The motion of a rolling pendulum is affected by factors such as the length and mass of the pendulum, the radius and mass of the rolling mass, and the surface on which it is rolling.

## 5. What are the practical applications of studying rolling pendulums?

Studying rolling pendulums can have practical applications in fields such as robotics and mechanical engineering, as well as providing insight into the dynamics of complex systems.

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