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Example: Modeling a Bus Suspension System using Transfer Function
Zidarta - jimfackyou@hotmail.com

Physical setup
Photo courtesy: Eisenhower Center

Designing an automatic suspension system for a bus turns out to be an interesting control problem. When the suspension system is designed, a 1/4 bus model (one of the four wheels) is used to simplify the problem to a one dimensional spring-damper system. A diagram of this system is shown below: Where: * body mass (m1) = 2500 kg, * suspension mass (m2) = 320 kg, * spring constant of suspension system(k1) = 80,000 N/m, * spring constant of wheel and tire(k2) = 500,000 N/m, * damping constant of suspension system(b1) = 350 Ns/m. * damping constant of wheel and tire(b2) = 15,020 Ns/m. * control force (u) = force from the controller we are going to design.

Design requirements:
A good bus suspension system should have satisfactory road holding ability, while still providing comfort when riding over bumps and holes in the road. When the bus is experiencing any road disturbance (i.e. pot holes, cracks, and uneven pavement),the bus body should not have large oscillations, and the oscillations should dissipate quickly. Since the distance X1-W is very difficult to measure, and the deformation of the tire (X2-W) is negligible, we will use the distance X1-X2 instead of X1-W as the output in our problem. Keep in mind that this is an estimation. The road disturbance (W) in this problem will be simulated by a step input. This step could represent the bus coming out of a pothole. We want to design a feedback controller so that the output (X1-X2) has an overshoot less than 5% and a settling time shorter than 5 seconds. For example, when the bus runs onto a 10 cm high step, the bus body will oscillate within a range of +/- 5 mm and return to a smooth ride within 5 seconds.

Equations of motion:
From the picture above and Newton's law, we can obtain the dynamic equations as the following:

Transfer Function Equation:

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