The Art of Problem Solving

1
Complex Variables
Applications
2
PREFACE

• Traditionally, the variable I or i is used to
  indicate current. Physicists and engineers
  (particularly electrical engineers) all have to
  deal with current flow in some aspect of their
  fields. Clearly this is in direct conflict with
  complex variable nomenclature. For this reason,
  it is common to use the variable j to indicate
  the complex number. From this point onward, we
  will use this definition to avoid confusion since
  it is also the nomenclature that will be used in
  the rest of your engineering courses.

3
INTRODUCTION

• In many situations, engineers are interested in
  the response of a system to an oscillatory or
  sinusoidal type of input.
• Quite often such inputs can cause unwanted
  response in a system (large movements, noise,
  etc.). Some common examples are a bumpy car ride,
  unwanted vibrations in sensitive hardware or
  machines in a plant and acoustical feed back
  causing noise in sound systems.
• The spectacular failure of the Tacoma Narrows
  suspension bridge was due to oscillatory
  vibration in the suspension cables which were
  excited by the wind.

4
Spring Mass Damper System
As an example of the application of complex
variables to engineering, we will consider spring
mass damper systems (force displacement -
velocity- acceleration) Spring
k
k spring constant Units are force/unit length
(e.g. N/m, Lb/in)
x
The force generated by the spring in the x
direction, Fk, is Fk kx
5
Damper
Damper A damper or shock absorber can be
represented schematically by the piston-cylinder
arrangement shown
Neglecting friction between the piston and
cylinder wall, the damper is characterized by its
damping coefficient, B Units are force/unit
velocity e.g.
6
Damper - MASS
Force produced by the damper in the x direction,
Fb, is Fb B (dx/dt)
Mass
7
Complex Impedances
If we subject any of the preceding components to
an oscillatory or sinusoidal type input of
displacement (or force) of frequency, , then
the resistance or impedance to the motion of each
component, Z, can be represented as a complex
number that is frequency dependent. That is, if
we think of the motion as a rotating complex
vector function of time expressed using Eulers
formula, we have
This enables us to convert the derivative
expressions in the previous component equations
to complex impedances
8
Complex Impedances
Spring
(complex impedance of a spring is identical to
its real counterpart)
Damper
Mass
(a real number but frequency dependent)
Note that there are electrical equivalents
Capacitor, inductor and resistor.
9
Spring-Mass-Damper-System Application
Consider the interconnected spring-mass-damper
system shown
Assume that the friction in the wheel bearings of
the cart is negligible
10
Spring-Mass-Damper-System Application
Free body diagram of the cart
x1
Note
(no motion in vertical direction)
11
Spring-Mass-Damper-System Application
Replacing derivatives to get complex impedances
OR
We now have a relationship, a ratio, of the
output displacement x2, to the input
displacement, x1, as a function of an input
sinusoidal frequency,
12
Spring-Mass-Damper-System Application

• Relationships like these turn out to be very
  important tools for predicting dynamic
  performance and for the design of mechanical,
  structural and electrical systems.
• Recall that the product of complex numbers gives
  a new vector or complex number whose amplitude is
  the product of amplitudes of the previous
  vectors, and the phase of the new vector is the
  sum of the previous phase angles.
• Therefore, the phase angle of a quotient will be
  the difference between the phase angle of the
  numerator and the phase angle of the denominator.

13
Spring-Mass-Damper-System Application
The amplitude of the output/input ratio for our
example is
And the phase angle between the input and output
signals becomes
14
Spring-Mass-Damper-System Application
Thus, given values for M, B and k, we can make a
plot of and determine the
magnitude ratio or the magnitude of the
output , given a value for , over a wide
range of frequency as well as the phase angle
between the input and output signals
The most interesting observation and one of huge
practical significance is what happens to the
magnitude ratio when the system damping, B,
becomes vary small (i.e. B 0)
15
Spring-Mass-Damper-System Application
Note the special case where
We have
This frequency is referred to as the natural
frequency,
What this tells us it that if we excite an
undamped or very lightly damped system at its
natural frequency ( ), the
resulting output oscillations (x2) will be huge
and potentially destructive (Tacoma Narrows
Bridge)
16
Spring-Mass-Damper-System Application
In these situations, engineers either try to
design to avoid the system being excited at its
natural frequency or introduce damping, B, to
reduce the magnitude ratio (typically lt1)
Consider our system equation which can be
rearranged to the form
To examine the effect of increasing damping, B,
on the magnitude ratio, we will let
and plot the frequency response for various
ratios of B/k. We will also examine the effect
on the phase angle which can be written as
17
Note that when , the real part
becomes negative and the complex number moves
into quadrant II so our angle expression becomes
18
It is also common to construct Z and
frequency on log paper. Typically the Z plots
are on log-log paper. The reason for this has to
do with the limit of the variables. Consider
For values of such that
And for values of such that
The plot is typically on rectangular- log
paper
19
Log-Log Plot Mod(X2/X1)
Where,
B/k values of 0.02, 0.05, 0.50 and 5.00 M/k 1
20
Regular-Log Plot ARG(X2/X1)
Where,
B/k values of 0.02, 0.05, 0.50 and 5.00 M/k 1