Philip Griffin - MABE Department, UL. 1

ME4117 - Vibration Analysis - Homework Sheet 3

This assignment is worth 3% of the total grade for the module. Marks are shown

beside each question. Marks will be lost for work that is not clear, illegible or if

incorrect units are used. Submit your completed assignment as a handwritten

document with a signed coursework declaration to the MABE Department Office (L1-

034) by 3pm Friday 25th October 2013.

1. A viscously damped spring-mass system is excited by a harmonic force of constant

amplitude 0F but varying frequency . if the amplitude of the steady-state motion

is observed to decrease by a factor of 8 as the frequency ratio is varied from 1 to 2.

Determine the viscous damping factor for the system.

[10 Marks]

2. A viscously damped spring-mass system is forced harmonically at its undamped

natural frequency. If the damping ratio is doubled from 0.1 to 0.2, compute the

percentage reduction in the steady-state amplitude. Compare with the result

obtained when the driving frequency is twice the undamped natural frequency.

[10 Marks]

3. A single cylinder four-stroke petrol engine with a mass of 90 kg is mounted on four

stiff spring pads, each with a stiffness of 30 MN/m and is designed to run at 3600

rev/min. The mounting system is equipped with viscous dampers which have a

large enough combined damping coefficient c so that the system is critically

damped when it is given a vertical displacement and then released while not

running. When the engine is running, it fires every other revolution, causing a

periodic vertical displacement modelled by 1.2cost mm with t in seconds.

Determine the magnification factor and overall damping coefficient c

[15 Marks]

4. A 20 kg variable speed motorised unit is restrained in the horizontal direction by

two springs each of which has a stiffness of 2.1 kN/m. Each of the two dashpots

have a viscous damping coefficient c = 58 N-s/m. In what range of speeds can the

motor be run for the magnification factor not to exceed 2?

[20 Marks]

Problem 4

ME4117 – Vibration Analysis Homework Sheet 3

Philip Griffin - MABE Department, UL. 2

5. A yacht’s mast of height 20 metres and diameter 0.5 metres is made of solid Sitka

spruce. It has a constant cylindrical cross section. Determine the wind velocity

flowing around the mast that will induce transverse vibration in the direction of

airflow.

Also comment briefly on how the use of stays and shrouds could used to alter the

vibration characteristics of the mast.

Use 11.2 GPa for the Young’s Modulus and 450 kg/m3 for the density of Sitka

spruce.

[25 Marks]

6. A single cylinder air compressor of mass 100 kg is mounted on rubber mounts as

shown below. The stiffness and damping constants of the rubber mounts are given by 106 N/m and 2000 N-s/m respectively. If the imbalance 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 l = 40 cm.

[20 Marks]

Problem 6