Ball and beam dynamics

Ball and Beam Dynamics
Introduction
This report does not omit any mathematical steps in order to reflect upon details of the mechanical operation of the process.

1. The Process
The ball and beam model consists of a horizontal beam and a DC motor mechanically attached at the centre of the beam. The angle of the beam is controlled by the motor. The angle in turn influences the position of the ball.

2. Mathematical Model Composition
The process model can be broken down into two separate models a. Model of the angle process with respect to the motor voltage H φ(s) b. Model of the ball position with respect to the beam angle Hx(s)
The total transfer function from the input voltage to the voltage that indicates the ball position is then
[pic]

3. Mathematical Model Derivation a. Model of beam angle vs. input voltage
The relationship between the input voltage and the angle of the beam is defined by the DC motor transfer function. The DC motor, used for angle control application may be thought of as the ‘dirty integrator’ or the integrator with a filter action as shown in figure 1 on the next page.
[pic]
Figure 1: General DC motor block diagram for angle control application
The K is the motor constant and the tau is the motor time constant. The actual model of the motor used for the project is shown in figure 2.

[pic]
Figure 2: Actual block diagram of the DC motor used

The measured constants are summarized below:
[pic]

Therefore the DC motor transfer function becomes
[pic]

b. Model of ball position vs. beam angle
Consider the following sketch
[pic]
Figure 3: Rolling ball free-body diagram
The inclination is considered the x-coordinate.
Let acceleration of the ball be denoted as [pic]
The force due to translational motion is then
[pic]
The torque developed through ball rotation is determined by the force at the edge of the ball multiplied by the radius which can be further expressed as:
[pic]
where
J = moment of