Consider the damped harmonic oscillator shown below and modeled by the system: k - ()*+ (-) m (1) with mass m = 1 kg, spring stiffness k = 100 N/m, and initial conditions x(to) = 0.1 m and x(to) = 0, where x(t) is the displacement of the mass from the nominal (unstretched) spring position. Four dashpots are being considered for this system: D1, D2, D3 and D4 have damping coefficients b₁ = 3, b₂ = 10, b3 = 20, and b4 = 50 N/(m/s), respectively. Pneumatic Dashpots: D1 D2 end *+ D3 D4 x=0 Assume: Horizontal Plane (no gravity) x(t) bu m www k You are tasked with selecting the correct dashpot that renders the system critically damped and to determine the response of the system over a two second interval from the initial condition. To perform your analysis, include the following: • A single plot with all five curves showing the response of the system to the initial conditions for each dashpot over a two second interval, and for the case of no dashpot b = 0. Each curve should be plotted with a different color with time on the abscissa and the displace- ment x(t) on the ordinate axis. Label your axes and include a legend. • Determine the damping ratio (numerical value) for Case 0 (no dashpot) and for Cases 1-4 with dashpots D1, D2, D3, and D4. For each case specify whether it as critically damped, overdamped, undamped, or underdamped. • Make your final recommendation on which dashpot to choose MATLAB Hints (optional): • To avoid repeating the same code for each of the five cases you are encouraged to encapsu- late it in a function of the form below. function xhist = simulate MassSpring Damper (tvals,x0, xdo to, k, m, b) % simulate system (your code here) This function can then be called five times in your main script to simulate each case (with a different input value for b).

A 2 DOF mass-spring system shown below with n = 1, k = 8, and m initial conditions 2₁ (0) = 1, 2₂(0) = 0, ₁ (0) = 0, and ₂(0) = 1. (a) Find the natural frequencies and normalized modal matrix [X] for the system. T x₁ (1) (b) Using modal analysis and the Cartesian solution form, find the free vibration re- sponse of the system x₁(t) and x2(t). Ţ x2₂ (1) www. 00000 = m m₁ = 00000 k₁ = k 00000 k₂ = nk m₂ = m k3= k = 2 DOF mass-spring system. 2 with the

t. Design the following application case The mechanical vibration system is shown in Figure (a). When a step input with 3N amplitude is subjected to this system, the movement rule of the mass m is shown in Figure (b). Evaluate the values of mass m, damping constant c and spring constant k. m (a) F-3N XX 1.0 x(1)(cm) 0 2 (b) M,-0.095 Ⓒ

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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.

damping, c Sliding Crate with Bumper Stop A crate of mass m enters a lubricated slide at a loading dock with a velocity v, to the right and an initial position x(0) = x, in the diagram shown above. The sliding motion is lubricated, characterized by a linear damping coefficient C. At a distance x = L the crate contacts a bumper stop characterized by a spring constant k and a damping coefficient c. Numerical values for the system parameters are: C = 8 N-s/m k = 1000 N/m m = 50 kg X = 0 Vo = 5 m/s L= 20 m Ca is to be experimentally determined in completing this assignment Bumper: Assignment 1) Given these parameters, determine the governing dynamic equation for the crate position. 2) Convert the governing equation to state variable form. This requires two first-order equations: a. dv/dt f:(x,v) b. dx/dt = f.(x,v) where f, is a function of the states, x and v. where f, is a different function of the states, x and v. Make two sets of the above equations, one each depending on whether there…

For a mass-spring oscillator, Newton's second law implies that the position y(t) of the mass is governed by the second-order differential equation my''(t) +by' (t) + ky(t) = 0. (a) Find the equation of motion for the vibrating spring with damping if m= 20 kg, b = 80 kg/sec, k = 260 kg/sec², y(0) = 0.3 m, and y'(0) = -0.3 m/sec. (b) After how many seconds will the mass in part (a) first cross the equilibrium point? (c) Find the frequency of oscillation for the spring system of part (a). (d) The corresponding undamped system has a frequency of oscillation of approximately 0.574 cycles per second. What effect does the damping have on the frequency of oscillation? What other effects does it have on the solution? (a) y(t) = plz answer a-d

k 0.5m 1m The figure shows a slender rod with a mass of 6 kg. Attached to the rod is a spring of stiffness k = 250 N/m.Amoveable point mass m1 = 1 kg is position at a distance X from the pivot point. You can assume small angle oscillations from the equilibrium position shown in the figure. Calculate the required position, X, of the moveable point mass m1 needed to give the system a natural frequency of 5.3 rad/s. Give your answer to two decimal places.

Show this complete solution for this force driven without damping equation:

t. Design the following application case m is The mechanical vibration system is shown in Figure (a). When a step input with shown in Figure (b). Evaluate the values of mass m, damping constant c and spring constant k. B (a) yu P=2N X(t)/cm4 0.1095 0.1 0 2 (b) 3 4