Consider the double mass/double spring system shown below. -click to expand. Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of Ffrictionca' (friction proportional to velocity). We can write the resulting system of second-order DEs as a first-order system, t'(t) = Au(t), with = (₁, ₁, ₂, For values of k= 4, m= 1 and c= 1, the resulting eigenvalues and eigenvectors of A are -0.039-0.248 0.813 A₁,2 -0.53.21, ₁- 0.024 +0.153 -0.502 -0.134-0.3021 0.409 -0.2160.4891 0.661 (a) Find a set of initial displacements (0), 2(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities will be zero. ((0), ₂(0)) Enter your answer using angle braces, (and). A3,4 -0.5 1.134. = and (b) At what frequency will the masses be oscillating in this mode? Frequency= rad/s

3 Problem Consider the following model of a mechanical system: The system has a mass m, a linear spring with stiffness k, and two identical dampers with damping constant b. The left wall generates an input motion xin (t) that causes the mass to undergo a displacement x(t) from its equilibrium position. The initial position and velocity are zero. • Find the ODE describing the motion of the system by drawing a free-body diagram and applying Newton's 2nd Law. Your answer should be in terms of the following variables: Xin Xin, X, X, x, b,k, m Show that the transfer function is G(s) = - X(s) Xin(s) = 2b m k s+ m k s²+ + m When taking the Laplace transform L[xin(t)] you may assume the initial (input) condition Xin(0) = 0.

A force of 20 newton stretches a spring 1 meter. A 5 kg mass is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 10 times the instantaneous velocity.   1) Let x denote the downward displacement of the mass from its equilibrium position. [Note that x>0 when the mass is below the equilibrium position. ] Assume the mass is initially released from rest from a point 3 meters above the equilibrium position. Write the differential equation and the initial conditions for the function x(t) 2)  Solve the initial value problem that you wrote above. 3)Find the exact time at which the mass passes through the equilibrium position for the first time heading downward. (Do not approximate.) 4)Find the exact time at which the mass reaches the lowest position. The "lowest position" means the largest value of x

The quarter-car model of a vehicle suspension and its free body diagram are shown in Figure 1. In this simplified model, the masses of the wheel, tire, and axle are neglected, and the mass m represents one-fourth of the vehicle mass. The spring constant k models the elasticity of both the tire and the suspension spring. The damping constant c models the shock absorber. The equilibrium position of m when y=0 is x=0. The road surface displacement y(t) can be derived from the road surface profile and the car’s speed. Draw free body diagram (FBD) and derive the equation of motion of m with y(t) as the input, and obtain the transfer function. If assume: m=300 kg k=20000, 40000, 60000 N/m c=1000, 3000, 5000 N.s/m Plot magnification ratio vs frequency ratio (r=0-4) diagrams for the parameters given above (you can draw the three curves in one diagram for three different k values and do the same for the three c values as well). Use the derived transfer function to model the system and plot…