4>() x (0) = 3 cm kg x (0) = N 20 4 X 10 AN/m 750M-5/m 12.5

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.

For the system shown in the figure, set up the equation of motion for the mass, m, and solve for the steady state amplitude and phase angle.

Q3/ For the system shown in figure if the damping ratio is 0.3 derive the equation of motion using Newton's second law and find: a- The natural frequency. b- The damping constant. c- The ratio of any consecutive cycles. k=2 x 10 N/m m = 4.2 kg 10 cm 40 cm 60 cm ITTTI

4. Problem 2: Consider the block diagram shown in the figure below. Determine the values of K and k such that the system has a damping ratio & = 0.7 and an undamped natural frequency wn = 4 rad/s. Please, use the block diagram algebra and the second

Derive the governing differential equation for each system with the chosen generalized coordinate. a. The springs both have the coefficient of k. (see figure below). Solve completely. Answers that will be provided COPIED/PLAGIARIZED from the other sources will be reported.

Equation of motion of a suspension system is given as: Mä(t) + Cx(t) + ax² (t) + bx(t) = F(t), where the spring force is given with a non-linear function as K(x) = ax²(t) + bx(t). %3D a. Find the linearized equation of motion of the system for the motion that it makes around steady state equilibrium point x, under the effect of constant F, force. b. Find the natural frequency and damping ratio of the linearized system. - c. Find the step response of the system ( Numerical values: a=2, b=5, M=1kg, C=3Ns/m, Fo=1N, xo=0.05m

Forcing Function Spring Constant (2)4 Mass k Friction Constant b m Mass Displacement y(t) 3. (20 pts.) Consider the following spring-mass-damper mechanical system (it is placed sideways, so that you won't need to consider gravity). The input is given by f(t), and the output is y(t). Find an equation in frequency domain (s domain) that defines the relationship between the input F(s) and the output Y(s).

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

1) Guess the form and solve for the particular solution of a driven oscillator described by the equation: mx" + cx' + kx = Fosin(wt) 2) A mass of 4 kg on a spring with k = 4 N/m and a damping constant c = 1Ns/m is driven by a force following the function Focos(at); where FO = 2 N. Find the driving frequency that causes practical resonance and find the amplitude. 3) Suppose there is no damping in a mass and spring system with m = 5, k = 20, and Fo= 5. Suppose is chosen to be precisely the resonance frequency. a) Find a b) Find the amplitude of the oscillations at time = 100, given that the system is at rest at / = 0.