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  3. figure 5 shows a translational system with three input variables...

Question: figure 5 shows a translational system with three input variables...

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Figure 5 shows a translational system with three input variables. The system consists of a block of mass m, a spring of sti↵ness k, and two dampers with damping coecients B1 and B2.

Vs(t) Figure 5: A translational system with three input variables 6. Figure 5 shows a translational system with three input variables. The system consists of a block of mass m, a spring of stiffness k, and two dampers with damping coefficients Bi and B2. The three input sources include a velocity input Vs(t) at point A, a force input Fsi (t) at point C (i.e., the block), and a second force input Fs2(t) at point B. Engineer A selects the velocity of the block vm(t) and the spring force fk(t) as the two state variables and derives the following state equation d m dt Vs (t) Fs1 (t) fk Answer the following questions (a) Derive the ordinary differential equation governing the state variable vm (b) Engineer B wants to know the displacement of the block m(t), and, therefore, chooses fk(t), m(t), and m(t) as the state variables. Derive the state equation using fk(t) Um(t), and rm(t) as the state variables. Please write the state equation in a matrix form. [Hint: If you cannot work out this problem right away, try part (c) to score some points.] (c) Engineer B is also interested in knowing displacement rB(t) of point B and acceleration am (t) of the block. Use B(t) and am(t) as output variables, derive the output equation. Note that the output equation should have fk(t), Vm(t), and m(t) as the state variables and V,(t), Fs1(t), and Fs2(t) as input variables. Please write the output equation in a matrix form.

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