he mechanical system shown in Figure 1 is initially at rest. The displacement x of mass m is measured from the rest position. With m = 10 kg, b = 20 N-s/m, and k = 500 N/m, the system is set into motion at t=0 by an impulsive force whose strength is 20 N. Obtain the initial velocity *(0,) of the mass due to the impulsive force.

A 1.5 kg mass is attached to a 1.6 m long rigid rod of negligible mass that is anchored to the ceiling. The mass is pulled back to an initial angle of 45°and released with an initial speed of 3.0 m/s. Note: The mass is initially moving down not up! 2. L m (a) [Hint: The height of the mass above its lowest point is L(1 – cos 0).] How fast is the mass moving at the bottoms of its swing, when 0 = 0°?

Problem 2. F Consider the Mass-Spring-Damper system shown in Fig. 2 (left), where m is the mass of the cart, k is the spring constant, and b is the damper constant. The input of the system is u(t), which represents the external force acting on the cart. The displacement of the cart is denoted as x(t). = x(t). 1) (10 points) Consider the velocity (t) as the output of the system, i.e., y(t) Assuming the zero initial conditions, write the transfer function G(s) from the input u(t) to the output y(t). 2) (15 points) If a proportional controller is used to control the system, then the block diagram of the closed-loop system is given by Fig. 2 (right). Assume that m = 1 kg, b = 4 N.s/m, and k = 3N/m. Sketch the Bode plot of G(s) for K = 1. 3) (15 points) Sketch the Nyquist plot based on the Bode plot. Determine the range of K for which the system is stable. Consider both positive and negative values of K. b x(t) J u(t) m R K G(s) Y Figure 2: Mass-Spring-Damper system in Problem 2.

3: A 24 lb weight stretches a spring 6 feet. The weight hangs vertically from the spring and a damping force numerically equal to 2√√3 times the instantaneous velocity acts on the system. The weight is released from 3 feet above the equilibrium position with a downward velocity of 14 ft/s. (a) Determine the time (in seconds) at which the mass passes through the equilibrium position. (b) Find the time (in seconds) at which the mass attains its extreme displacement from the equilibrium position.

Fig. P4.6 4.22. Derive the differential equation of motion of the simple vehicle model shown in Fig. P4.7. The vehicle is assumed to travel over the rough surface with a constant vehicle speed v. Obtain also the steady state response of the vehicle. For this system, let m = 10kg, k = 4 x 103 N/m, c = 150N · s/m, i = 2 m, and the amplitude Yo = 0.1 m. Determine the maximum vertical displacement of the mass and the corresponding vehicle speed. Determine also the maximum dynamic force transmitted to the mass at the resonant speed. m 2πη y = Y, sin y Fig. P4.7

Figure 4 shows a trolley of mass 6.12 kg passing through point A with a velocity of 6 m/s. It slides down a rail to point C. From A to C, 250 J of energy is lost due to friction. What is the equation used to find the velocity of the trolley at point C ?

If the rope on the wall suddenly snaps, what will be the acceleration vector and velocity vectors of these masses at 1=1s after the break? N.B.: the strings are assumed to be massless, flexible and inelastic.

HW Air flows through a C-d notcle having a throat of 0.05m². The Pressure and temperature at the threat section are MokPa and 333 K, respectively At Section 2, the pressure is to kla. 2 Find Az and in m 1 Air throat 1270 * Pa Pt=140kPa T=333K