Ingeniero
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Publicado: 24 de febrero de 2013
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 andbeam 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 theball.
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 theball 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
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 controlapplication may be thought of as the ‘dirty integrator’ or the integrator with a filter action as shown in figure 1 on the next page.
Figure 1: General DC motor block diagram for angle controlapplication
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.
Figure 2: Actual block diagram of the DC motorused
The measured constants are summarized below:
Therefore the DC motor transfer function becomes
b. Model of ball position vs. beam angle
Consider the following sketch
Figure 3:Rolling ball free-body diagram
The inclination is considered the x-coordinate.
Let acceleration of the ball be denoted as
The force due to translational motion is then
The torque developedthrough ball rotation is determined by the force at the edge of the ball multiplied by the radius which can be further expressed as:
where
J = moment of inertia (for solid ball defined by...
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 andbeam 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 theball.
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 theball 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
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 controlapplication may be thought of as the ‘dirty integrator’ or the integrator with a filter action as shown in figure 1 on the next page.
Figure 1: General DC motor block diagram for angle controlapplication
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.
Figure 2: Actual block diagram of the DC motorused
The measured constants are summarized below:
Therefore the DC motor transfer function becomes
b. Model of ball position vs. beam angle
Consider the following sketch
Figure 3:Rolling ball free-body diagram
The inclination is considered the x-coordinate.
Let acceleration of the ball be denoted as
The force due to translational motion is then
The torque developedthrough ball rotation is determined by the force at the edge of the ball multiplied by the radius which can be further expressed as:
where
J = moment of inertia (for solid ball defined by...
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