Consider the system shown in Figure 4-64. The system is at rest for \( t<0 \). Assume that the displacement \( x \) is the output of the system and is measured from the equilibrium position. At \( t=0 \), the cart is given initial conditions \( x(0)=x_{o} \) and \( \dot{x}(0)=v_{o} \). Obtain the output motion \( x(t) \). Assume that \( m=10 \mathrm{~kg}, b_{1}=50 \mathrm{~N}-\mathrm{s} / \mathrm{m}, b_{2}=70 \mathrm{~N}-\mathrm{s} / \mathrm{m}, k_{1}= \) \( 400 \mathrm{~N} / \mathrm{m} \), and \( k_{2}=600 \mathrm{~N} / \mathrm{m} \).
Figure 4-64 Mechanical system.