Vibrations MatLab Excercise
From: …………………..
Due: December 5, 2017
Machine Dynamics and Vibrations
Dr. Mehdi Sarmast
Abstract:
Vibration is typically represented most simply as a mass, spring and damper system and
can be easily modeled to serve as a mathematical tool for predicting the behavior of real world
systems to a certain degree. Rotating machinery is a very common place where oscillations can
be found and easily found in the real world from a ceiling fan, tires and your washer and dryer.
However, it is not uncommon to find that imbalances can cause vibration to the system, and
resonance effects can be dangerous. Fortunately, even this induced, almost erratic vibration
can be modeled using dependent masses. This project covers the mathematical modeling of
such a system using MATLAB to run the simulations, plot the desired values, and then compare
them to textbook models.
Table of Contents:
Title
1
Abstract
2
Table of Contents
3
Introduction
4
Program
4
Results
6
Conclusion
7
Appendix
8
Introduction:
Question 3.120 from the Mechanical Vibrations 6th ed.:
𝑀𝑋
Plot the graphs of 𝑚𝑒 versus r and ϕ versus r for a damped system under rotating
unbalance for the damping ratios ζ=0, 0.2, 0.4, 0.6, 0.8 and 1 using MATLAB.
The equation below can be used for modelling a damped system under rotating unbalance:
𝑀𝑋
=
𝑚𝑒
𝑟2
1 =𝑟
[(1 − 𝑟 2 )2 + (2𝜁𝑟)2 ]2
2
⃒𝐻(𝑖𝜔)⃒
2𝜁𝑟
𝜙 = tan−1 (
)
1 − 𝑟2
𝜔
Where: 𝑟 = (𝜔𝑛)
Program:
The program was written in MATLAB and uses 1 input to interpolate and output data. As
the equations are written such that they are in respect to the damping ratio, a damping ratio
input is unnecessary. Given such the flow of the program can be seen below:
Vibrationsproblem
3120
Input S (Damping
Ratio)
Interpolate
Mx/me
Output Figure with
axis plotting Mx/me
vs r
Interpolate
Phase Angle
Output Figure with
axis plotting Phase
angle (Radians) vs r
To operate the program, run the program and the user will be prompted to enter a value,
or a series of values for the damping ratio. For the purposes of this assignment, the damping
ratios specified are 0, 0.2,0.4, 0.6, 0.8, and 1. Multiple numbers can be fed as an input, but
Brackets [ ] must be used to enclose the values, but no comma is necessary to separate the
values. The range for the frequency ratio is set as 0 to 5, such that the closest comparison to
the book can be utilized (example 3.17). After submitting all 6 damping ratios, the program will
output 2 separate graphs, as pictured in the Results.
Results:
The Program serves as a useful tool for demonstrating the points of resonance and
gives plots that allow a user to visually compare the varying effects of various damping ratios. In
addition to the motion, the phase angle can also be depicted versus the frequency ratio.
Conclusion:
From figure 3.17 of the textbook, a variation of the (MX/me) over the frequency ratio r
can be found.
The plots produced by the program can be compared to this plot and the conclusion drawn that
as the 2-approach unity, the natural and forced frequencies of the approach tangency to the
𝑀𝑥
∆
𝑀𝑥
𝑚𝑒
phase angle. Regarding 𝑚𝑒, it appears to reach its maximum at unity, with the greatest ∆𝑟
occurring before unity.
Appendix:
Forced Vibrations, www.roymech.co.uk/Useful_Tables/Vibrations/res_curves.html.
Rao, Singiresu S., and Philip Griffin. “Chapter 3 Harmonically Excited Vibration.” Mechanical
Vibrations, Pearson, 2018.
MATLAB from MathWorks