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Question: 5 linear timeinvariant mechanical system 90 points the cartoon in...

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5. Linear Time-Invariant Mechanical System (90 points). The cartoon in Fig. 1 shows a mechanical system composed of a mass with constant m, a spring with constant k and a damper wit h constant c. The output of the system is the position of the mass, y. Here, the left distal ends of the spring and the damper are connected together by a vertical bar, whose horizontal position u, is arbitrarily externally determined by an experimenter. The inertial frame of coordinates is defined so that when u = y = 0, the spring is at its natural length. (a) Using Newtons laws write the equation of motion that describes the dynamics of the system. (b) Using your answer for (a), explicitly show that the systems equation of motion can be rearranged as TI (c) Clearly define the concept of transfer function. (d) What is the relationship between the notions of transfer function and impulse response? (e) Defining f t1 + u, and assuming zero initial conditions, find the transfer function that maps F(s)-Cf(t))to Y (s) Cy(t)), G1(s). (f) Considering the definition of the unilateral Laplace transform, explain the notion of region of convergence (ROC) (g) What is the region of convergence (ROC) of Gi(s)? Clearly explain your answer (h) Assuming zero initial conditions, find the transfer function that maps U(s)-Lu(t))to Y(s)-lu)),G2ls) (i) What is the region of convergence (ROC) of G2(s)? Clearly explain your answer tu IT2 No Friction Figure 1: Two-Mass Mechanical System.
(j) Defining f- u as the input, y as the output and the state as y y ]T и + find a two-dimensional state-space realization describing the system, .e., find matrices A, B,C,D) for the equations (t) = Az(t) + Bu(t). (t)-Ca(t) +Du(t). (Ck) Defining u as the input, y as the output and the state as ru ]T, show that a state-space realization of the system is given by 1) For m-1, k1 and 1, find the impulse response gi(t) associated with the transfer (m) For m (n) Find limit oon(t) in the time domain. function Gi(s) in (e). 1, k 1 and c= 1, find the unit-step response si (t) associated with the transfer function Gi(s) in (e). (o) In this case, can you use the Final Value Theorem (FVT) to find lms(t)? Clearly explain your answer. (p) If the answer for (o) is yes, using the FVT find limi·o si (t) in the Laplace domain. )For m-1, k 1 andc-1, find the impulse response 92(t) associated with the transfer funetion G2(s) in (h). (r) For m = 1, k = 1 and c = 1, find the unit-step response s2(t) associated with the transfer function Ga(s) in (h).
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