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)

4- A single-cylinder air compressor of mass 100 kg is mounted on rubber mounts. The stiffness and damping constants of the rubber mounts are given by 106 N/m and 2000 N-s/m, respectively. If the unbalance of the compressor is equivalent to a mass 0.1 kg located at the end of the crank (point A), determine the response of the compressor at a crank speed of 3000 rpm. Assume r = 10 cm and /= 40 cm.

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

Lab experiment 1 Procedure

A sprung table has been designed to support a measurement device to isolate the table from the surroundings.Before the measurement device is added,tests are made on the table to determine is vibration and damping characteristics.It can be assumed that the table has a mass of 100kg,and is supported by springs that deflect 4 cm under the static loading.A damper is fitted to the table,and in a free vibration test,it is found that the amplitude of the oscillation decays to 10%of the original value after 1 complete oscillation. Calculate the spring stiffness (assume the total spring stiffness)and the natural frequency of the table,k= N/m,@n= rad/s Determine also: The logarithmic decrement,& The damping ratio,= and the damping coefficient,c= Ns/m

2 The simply-supported beam shown on the left supports a weight, w = 10 kip at mid-span with, L = 10 ft. E= 29,000 ksi and I = 200 in. The beam is displaced downward to a distance of, u(0) = 2.0 in and suddenly released with an additional upward velocity of, ù (0)=-400.0 in/s. Calculate the (a) natural period of vibration, (b) displacement amplitude, (c) velocity amplitude, (d) acceleration amplitude, and (e) phase angle. Derive the functions describing displacement, velocity, and acceleration and plot them from 0 to 0.5 s. Use a time increment, At = 0.001 s. Label (a) through (d) in your plots.

Problem 1 A falling weight deflectometer (see figure) is applied to a bridge at mid-span to initiate vibrations. After initial disturbance, the oscillations, which were measured using an accelerometer, were found to decay exponentially from an amplitude of 1.2g to 0.4g after five cycles of free vibration. Determine the damping ratio for the bridge girder. State your assumptions.

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 3 A 500 lb machine is supported by a four-element spring-damper system. The static vertical deflection of the supporting system due to the weight of the machine is determined as 0.1 in. The viscous dampers are designed to reduce the amplitude of vertical vibration to 2% of the initial amplitude after three complete cycles of free vibration. Perform the following: (a) Estimate the damping ratio, (b) Compute the natural frequency of damped vibration, fo (c) Plot the free vibration response, u(t) for an initial condition, u(0) = 0.2 inch from t = 0 to 0.5 s. In addition to u(t), also plot the envelope functions. Pick a suitable time increment, At. Label pertinent features in the plot.