This is done below for our train system.įrom Newton's second law, we know that the sum of the forces acting on a body is equal to the product of the mass of the bodyĪnd its acceleration. The first step in deriving the mathematical equations that govern a physical system is to draw the free-body diagram(s) representing The force represents the force generated between the wheels of the engine and the track, while represents the coefficient of rolling friction.įree-body diagram and Newton's second law In other words, the coupling is modeled as a spring with a spring constant. Furthermore, the engine and car are connected via a coupling with stiffness. The mass of the engine and the car will be represented by and, respectively. It can track a constant speed command with minimal error in steady state. Assuming that the train only travels in oneĭimension (along the track), we want to apply control to the train so that it starts and comes to rest smoothly, and so that In this example, we will consider a toy train consisting of an engine and a car.
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