Question

2. In many mechanical positioning systems there is flexibility between one part of the system and another. The figure below depicts such a situation, where a force u(t) is applied to the mass M, and another mass m is connected to it. The coupling between the objects is often modeled by a spring constant k with a damping coefficient b, although the actual situation is usually much more complicated than this. y(t) m M ut) no friction no friction a) In this part, consider the input signal to be u(t), which is not the unit-step function, but a general signal. Consider there to be two outputs, x(t) and y(t). Therefore, write two (coupled) differential equations that describe the system dynamics. b) In this part. we consider the input signal to be u(t) but consider the output to be only y(t). We want a full dynamic description in a single differential equation of the input- output relationship between u(t) and y(t). In other words, you need to substitute one of your equations in part (a) into the other equation to eliminate the variable x(t). You may use Laplace transforms to verify your math if you wish, but do all of your derivations in the time domain.