ME 440
Intermediate Vibrations
Tu, January 27, 2009
Sections 1.10 & 1.11
© Dan Negrut, 2009
ME440, UW-Madison
Before we get started…
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Last Time:
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Discussed two examples of how to determine equivalent spring
Discussed the concept of linear system and how to linearize a
function
Covered material out of 1.8, 1.9
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Equivalent mass, damping elements
Today:
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HW Assigned: 1.34 and 1.66 out of the text
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HW due in one week
Covering material out of 1.10, maybe start 1.11
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Periodic Functions and Fourier Series Expansion
2
General Concepts
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Periodic Motion: motion that repeats itself after an interval of time 
  is called the period of the function
f

t
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Harmonic Motion: a particular form of periodic motion represented by a sine or
cosine function
Very Important Observation: Periodic functions can be resolved into a series of
sine and cosine functions of shorter and shorter periods (more to come, see
Fourier series expansion):
3
Sinusoidal Wave
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The motion with no friction of the system below (mass-spring system)
leads to a harmonic oscillation
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Formally discussed in Chapter 2
Plot below shows time evolution of function
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Nomenclature:
4
Harmonic Motion (Cntd)
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If displacement x(t) represented by a harmonic function, same holds true
for the velocity and acceleration:
Quick remarks:
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Velocity and acceleration are also harmonic with the same frequency of oscillation, but
lead the displacement by /2 and  radians, respectively
For high frequency oscillation ( large), the kinetic energy, since it depends on
,
stands to be very large (unless the mass and/or A is very small…)
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That’s why it’s not likely in engineering apps to see large A associated with large 
5
Exercise
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Show that you can always represent a generic harmonic function as the
sum of two other harmonic functions of the same frequency 
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Specifically, for any X and  in
, find A and B such that
Note that  in equation above is arbitrary
Alternatively, the formula above can be reformulated as (showing only
cosine functions on the right side)
6
Review of Complex Algebra
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The need for complex numbers
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Solve “characteristic equation” (concept to be introduced later):
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Roots:
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To make life simpler, use notation
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Using notation, roots above become:
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Incidentally, the following hold:
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Complex Numbers:
From Algebraic Representation
to
Geometric Representation
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Representation of complex number z=a+bj provided below
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Note that
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Therefore,
8
Dwelling on the Construct
(Euler’s Formula)
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Use Taylor expansion for sine and cosine
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Sum up and interleave terms to get:
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In other words, we got Euler’s formula:
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It follows that our complex number z can be expressed as
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Algebra of Complex Numbers
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Multiplication
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Division
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Integer powers
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Roots of order n
10
A Brief Excursion
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Suppose I have a complex number that changes in time. That is, the
real and imaginary part change in time:
The first two time derivatives of this function of time are:
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The important observations are as follows (commutability):
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This will be used in conjunction with Fourier Series Expansion
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A Brief Excursion
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Why is this important?
Imagine that you are interested in a quantity
the real part of a complex number
.
which happens to be
Then, if you want to find out the value of
then simply look at
the real part of
and the real part of its derivatives:
The same general remark holds for the Fourier series expansion
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(Cntd)
If you want to find the Fourier series expansion of
then get the Fourier
series expansion of
. The real part of this expansion is going to be the
expansion of
.
Begin Section 1.11
(Harmonic Analysis)
13
Intro to Fourier Series
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Joseph Fourier - French mathematician (1768 – 1830)
Fourier’s doctoral adviser was Lagrange, whose doctoral adviser was
Euler, whose doctoral adviser was Bernoulli, whose adviser was
another Bernoulli, whose adviser was Leibniz. The latter had no
adviser, he invented Calculus (at the same time as Newton).
Fourier’s doctoral students included Dirichlet, who later was the
adviser of Kroneker, who later was the adviser of Cantor.
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Key Result (Fourier)
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Any periodic function of period  can be represented by a series of
sin and cosine which are harmonically related
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Here
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When is a function f(t) periodic though?
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A function is periodic if there is a positive, constant, and finite
Note:
 is called the period of the function
 such that
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Periodic Functions, Examples
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Fourier Expansion, Definition
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Getting Odd, Getting Even
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A function is odd provided
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A function is even provided
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Example 1.
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Determine the Fourier expansion of the following periodic function:
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