illustrates an active vibration control scheme for a

Chapter , Problem 11.34

(choose chapter or problem)

Figure P.11.34 illustrates an active vibration control scheme for a two-mass system. An electrohydraulic actuator between the two masses provides a force that acts on both and is under feedback control. The system model is

                                       \(\begin{aligned}& m_1 \ddot{x}_1=k_1\left(y-x_1\right)-k_2\left(x_1-x_2\right)-c\left(\dot{x}_1-\dot{x}_2\right)-f \\& m_2 \ddot{x}_2=k_2\left(x_1-x_2\right)+c\left(\dot{x}_1-\dot{x}_2\right)+f\end{aligned}\)

The given parameter values are \(m_1=50 \ \mathrm{~kg}, m_2=250 \ \mathrm{~kg}, k_1=1.5 \times 10^5 \ \mathrm{~N} / \mathrm{m}, k_2=1.2 \times 10^4 \ \mathrm{~N} / \mathrm{m}\), and \(c=100 \ \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}\).

a. Put the model into state variable form.

b. Assume that we can measure all four state variables, and use P action with state-variable feedback. In the original passive system, \(k_2=1.6 \times 10^4 \mathrm{~N} / \mathrm{m}\) and \(c=98 \mathrm{~N} \cdot \mathrm{m} / \mathrm{s}\), which resulted in characteristic roots at \(s=-1.397 \pm 69.94 j, s=-0.168 \pm 7.779 j\). Compute the values of the feedback gains so that the closed-loop roots will be near those of the passive system.