Figure shows a 1-DOF mass-damper idealized form of engineering mechanical system. Displacement x is measured from an equilibrium position where the damper is at the “neutral” position. External force fa(t) =Acos3t is applied directly to mass m.

1. Derive the mathematical model of the mechanical system with position x as the dynamic variable.

2. Derive the mathematical model with velocity v(t)= x(t) as the dynamic variable.

3. Use MATLAB’s command to solve the differential equation of velocity.

4. Use MATLAB’s command to plot the evolution of velocity with time t in the interval [0, 20]. b = 0. 3N/m/s and m= 200kg, A=2.

5. Use MATLAB’s command to plot the evolution of velocity with time t

6. Use MATLAB’s command to plot in the same graph the evolution of position x when External force =0 and

when A= 10N

7. Analyze your result