Consider the mechanical vibratory system shown in Figure 3. Assume that the displacement x is measured from the equilibrium position in the absence of the sinusoidal excitation force. For zero initial condition and assuming that m = 10 kg, bị = 10 N-s/m, b2= 20 N-s/m, k = 400 N/m, k2 = 600 N/m, P = 10 N and w = 2 rad/s, what is the steady-state output x(t)? ki b1 p(t) = P sin wt %3D kz b2

Given an oscillator of mass 2.0kg and spring constant of 180N/m, what is the period without damping?  Use numerical methods to model this oscillator with an additional friction force equal to where c is a positive damping constant.  Using c=5.0, what is the new period of oscillation.  What about for c=10? Assume initial position is 0.2m and initial velocity is zero. Please find the period using the position versus time plot and use the first full cycle of the motion.

Consider the following system mä(t) + cả (t) + kæ(t) = Ê18(t) + Ê 28(t – t1) N Assume zero initial conditions. The units are in Newtons. If m = 1 kg, c = 3 Ns/m, k = 100 N/m, F1 = 1 N.s, F2 = 3 N.s and tj = 0.01 s, the displacement of the system at t2 = 0.047 s is 72.46 mm 434.76 mm 869.52 mm 115.94 mm 144.92 mm O O O O O

a) A vehicle circulates on a road as shown in Figure Q2a. The road profile can be modelled as the input u(t). The vehicle is modelled as a quarter car of mass m, and the suspension has a spring stiffness coefficient k = 2 Nm and a damper of coefficient c = 2 Nm s. Find the position of the mass, y(t) and any time t if the road profile is a unit step. m 一 Road Figure Q2a b) In Figure Q2b, a disk flywheel J of mass m 32 kg and radius r = 0.5 m is driven by an electric motor that when it is working produces an oscillating torque of Tin = 10sin(wt) N- m. The shaft bearings may be modelled as viscous rotary dampers with a damping coefficient of BR = 0.4 N-m-s/rad and stiffness coefficient KR = 2 Nm If the flywheel is at rest at t 0 and the power is suddenly applied to the motor, do the following [Hint:J = mr²/2]: (i) Find the natural frequency of oscillation of the disk expressed in Hz. (ii) Find the damping ratio for this system. (iii) Describe in no more than 3-4 lines the behaviour of the…