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 F friction We can write the resulting system of second-order DEs as a first-order system, w ´(1) = Aw(1), with w = (x₁, x₁, x₂, x₂) ™ For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are 21.2 -0.5±3.2i, V₁ = -0.039-0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i 0.409 23.4 = -0.5 ± 1.13i, V3 = -0.216 - 0.489i 0.661 and - cx (friction proportional to velocity). (a) Find a set of initial displacements x₁ (0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. (0), x₂(0)) = Enter your answer using angle braces, (and). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/s

For the mechanical system shown, a- Find the equivalent mass, equivalent spring stiffness and equivalent damping coefficient when x (the displacement of the 2 kg block) is used as the generalized coordinate. b- Derive the equation of motion of the system in terms of equivalent system parameters (equivalent mass, equivalent spring stiffness and equivalent damping coefficient). 3000 N/m 2 kg 200 N.s/m /=0.04 kg-m² r = 10 cm 1000 N/m 1 kg I' 400 N.s/m ial

you are given the mass, damping, andspring constants of an undriven spring-mass systemmy'' + μy + ky = 0.You are also given initial conditions. Use a numerical solverto(i) provide separate plots of the position versus time (y vs.t) and the velocity versus time (v vs. t),(ii) provide a combined plot of both position and velocity versus time, and(iii) provide a plot of the velocity versus position (v vs. y) inthe yv phase plane. a. m = 1 kg, μ = 0 kg/s, k = 9 kg/s2, y(0) = 3 m,y'(0) = 2 m/s b.m = 4 kg, μ = 4 kg/s, k = 1 kg/s2, y(0) = 3 m,y'(0) = 1 m/s

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 Ffriction -cz' (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, 21, 22, 2₂) I For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are -0.039-0.248i 0.813 A₁2=-0.5±3.2i, v₁ = 0.024 +0.153i -0.502 -0.134-0.302i 0.409 -0.2160.489 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 wil be zero. A3,4 -0.5± 1.13i, z = and (2₁ (0), ₂(0)) = Enter your answer using angle braces, (and). (b) At what frequency will the masses be oscillating in this mode? Frequency rad/s