In periodic motion, it is often interesting to investigate the harmonic motion of an object. The decrease in amplitude caused by dissipative force is called damping, and the corresponding motion is called damped oscillation. Damped oscillation is governed by the following equation, where is the angular frequency of oscillation, A is the initial amplitude of the oscillation, m is the mass of an object, b is a constant that describes the strength of the damping force, t is time in seconds, h is the ratio between a force constant k and the mass m: where x(t) = Ae If = 60°, A = 4, v = b 2m @=h sin(at + p) b² 4m² b = 0.25, create a plot x(t) for 0 ≤t≤ 15, with h = 4. Label all the axes, 2m include gridlines and provide a title in your plot.

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.

please be specific and explain will up vote if correct

2. Consider a car suspension, modeled as a mass/spring/damper system (mass m, stiffness k, damping b). Suppose the height of the chassis is lo at rest, the height of the terrain below the driver varies as h(t), and the height of the chassis is denoted lo + y(t). (i.e., spring deflection away from rest is y(t) – h(t)). 2 (a) Give the transfer function G(s) = H(s) · = (b) Suppose the ground follows an oscillatory profile h(t) A cos(wx (t)) with magnitude A (in meters) and frequency w (measured in radians per meter). Suppose the car is traveling at a constant forward speed v. Using a frequency response analysis strategy, give the amplitude of oscillations experienced by the driver at steady state as a function of m, k, b, A, w, and v. Hint: You can't simply consider |G(iw)| to get the amplification in this case. (c) Suppose the ground varies by A = 5cm, w = 2 rad/m, and you are driving at v = 15 m/s. Using your answer to part (b), what amplitude of oscillation is felt by the driver when m…