Vibration HW

1

HW #4

ME 3323 Mechanical Vibrations, Summer 2015, UTSA

Assigned: 07/07/2015 Due: 07/14/2015

Max points: 100+20

Q1. The dynamic amplification or attenuation of a single degree of freedom system is

defined as the ratio of the steady-state magnitude of the displacement response to its

static displacement 𝐹0/𝑘, where the mass is being driven by a harmonic force of magnitude 𝐹0. Find the dynamic amplification or attenuation of a single degree-of- freedom system that is being excited at 100 rad/s and has the following system

parameters: m=100 kg, k=20 kN/m, and c=6000 N.s/m.

Q2. A motor of mass m is mounted at the end of a cantilever beam and it is found that the

beam deflects 10 mm. When the motor is running at 1800 rpm, an unbalanced force of

100 N is measured. If the beam damping is negligible and its mass can be neglected,

then what speed should the motor operate at so that the amplitude of the dynamic

response is less than 𝑎0𝑚.

Q3. The damped single degree of freedom mass-spring system shown below has a mass

m=20 kg, and a spring stiffness coefficient k=2400 N/m. Determine the damping

coefficient of the system, if it is given that the mass exhibits a response with an

amplitude of 0.02 m when the support is harmonically excited at the natural frequency

of the system with an amplitude 𝑌0=0.007 m. In addition, determine the amplitude of the dynamic force transmitted to the support.

Points: 25

Points: 25

Points: 25

2

Q4: Torsional oscillations of a vibratory system is governed by the following equation

𝐽0�̈� + 𝑐𝑡�̇� + 𝑘𝑡𝜙 = 𝑀1 cos(𝜔1𝑡) + 𝑀2cos⁡(𝜔2𝑡)

𝑤ℎ𝑒𝑟𝑒, 𝐽0=20 kgm 2, 𝑘𝑡=20 Nm/rad, 𝑐𝑡=20 Nm/(rad/s)

𝑀1 = 10Nm, 𝑀2=20 Nm, 𝜔1= 1.0 rad/s, 𝜔2= 2.0 rad/s

Determine the steady state response of the system

Q5: Extra Credits:

Determine an expression for the output of an accelerometer with the damping factor 𝜁 and natural frequency𝜔𝑛, when it is mounted on a system executing periodic displacement motions of the form

𝑦 = 𝐴1𝑠𝑖𝑛𝜔1𝑡 + 𝐴2𝑠𝑖𝑛𝜔2𝑡

Points: 25

Points: 20