Vibrations

2- An electric motor has an eccentric mass of 10 kg (10% of the total mass of 100 kg) and is set on two identical springs (k=3200 N/m). The motor runs at 1750 rpm, and the mass eccentricity is 100 mm from the center. The springs are mounted 250 mm apart with the motor shaft in the center. Neglect damping and determine the amplitude of the vertical vibration. (15 Marks)

5- Consider an automobile traveling over a rough road at a speed of v km/hr. The suspension system has a spring constant of 40 kN/m and a damping ratio of = 0.1. The road surface varies sinusoidally with an amplitude of Y = 0.05 m and a wavelength of 6 m. Write a MATLAB program to find the displacement amplitude of the automobile for the following conditions: (a) mass of the automobile = 600 kg (empty), 1000 kg (loaded), (b) velocity of the automobile (v) = 10 km/h, 50 km/h, 100 km/h. (15 Marks)

6 Use Simulink to compute and plot the solution for 0<t<5 of the following nonlinear equation of motion of a simple pendulum with viscous damping and an applied moment M(t), which is square wave with an amplitude of 2 and a frequency of 1Hz: (15 Marks) 100 + 158 + 20 sin 8 = M(t) 8(0) = 8 (0) = 0

Problem 1 The stiffness and damping properties of a mass-spring system with friction damping are to be determined by a free vibration test. In this test, a mass, m = 0.5 lb*s²/in is displaced 0.5 in by a jack and then suddenly released. At the end of 2 complete cycles, the time is 0.9 s and the amplitude is 0.06 in. Determine the stiffness and friction coefficients. Plot the free vibration response, u(t) from t = 0 to 2 s. At what time does the motion stop and what is the residual displacement?

Problem 2 Implement the central difference method (CDM) to solve vibration problems for linearly elastic viscously-damped SDF systems following Module 3, Slide 11. Chopra, Table 5.3.1 outlines the step-by-step procedure you may follow. Make sure your spreadsheet/code can plot the force as well as response quantities such as displacement, velocity, etc. To validate your code, solve Chopra, Example 5.2 and compare your results with Chopra, Table E5.2. Your output, ti vs. u should be identical. Check the three requirements per Module 3, Slide 10!

1. Consider two masses m, connected to each other and to two walls by three springs, as shown in the figure. Both masses feel a damping force-bv. The three springs have the same spring constant k. Find the general solution for the positions of the masses as functions of time. Assume underdamping.

For the system shown in the figure a) determine the equation of motion of the equivalent linear system (at the mass "m" location) b) find the natural frequencies (wn and wa). c) the frequency ratio r. d) obtain the term X/ost using the graph (state the used points on the graph). Take: k=(1000+Z) N/m, where Z is the last digit of your PMU ID no. m=10 kg, c=380 N s/m, r1=0.2 m, r2=0.4 m. Jpulley = 1.0 kg m², forced vibration w=12 rad/sec

For the system shown in the figure a) determine the equation of motion of the equivalent linear system (at the mass "m" location) b) find the natural frequencies (ωn and ωd), c) the frequency ratio r. d) obtain the term X/δst using the graph (state the used points on the graph).

A cylinder of known mass m and known mass moment of inertia Jo= mr²/2 is free to roll without slipping but is restrained by a known damper c and two springs of known stiffness k, and k; as shown. a) Determine the equation of motion in terms of the rotation angle 8. b) Determine the value of a that maximizes the natural frequency of the system.

The system shown has a natural frequency, fn, of 5Hz, m-10kg, J.-5kg-m², r=0.10m, and r=-0.25m. If the mass is displaced 0.01m to the right and released from rest, the amplitude of the resulting free vibration reduces by 80% in exactly 10 cycles. Gravity may be taken as being into the page. a) Determine the equation of motion in terms of the motion of the mass x(t). b) Determine the value of the stiffness k. c) Determine the value of the damping coefficient c.