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Background
One of the most straightforward means of describing the behavior of a
system is the differential equation. Applications of the differential equation
can extend to different fields that are not limited to physics and engineering.
Many facets of biology, sociology and economics can be described using the
differential equation (e.g. carbon dating, population growth, compound interest
on a financial loan). This blog entry will focus primarily on second order
differential equation review with a mini-recap of first order equation theory.
First Order Differential
Equation Review
The general solution of a first order differential equation can be
demonstrated through a simple example using the variables y and t. There exists
for a function y = F(t) a solution which satisfies the condition:
The solution can be attained by simply isolating the differentials and
integrating both sides, as follows:
The above
function must satisfy the condition of equivalence to its own derivative. A
common elementary solution to the differential equation is:
in which C is a constant with the value
. If there
are initial conditions given, plugging them in the general solution for y and t
yields a solution to the more specific case under observation.
Second Order Differential Equations
Second order differential equations are applicable throughout engineering
and science and are used in applications where there is either electrical or
mechanical oscillatory motion to be modeled. A second order differential
equation takes the general form:
This is called the characteristic equation of a second order differential
equation. In order to solve equations of this nature, one can split out the
differential terms into variables and solve for them in the same manner they
would the roots of a polynomial equation:
In a second order differential equation, there are three different cases
of roots that can occur, and the root types are conditional based upon the square
root value in the numerator. These cases represent something called damping, a
phenomenon which reduces the amplitude of waveforms in a system. The pink
sinusoidal waveform does not appear to taper off in the image below. The other
three waveforms are relevant to the cases that will be discussed in this entry.
Case #1: Real and Distinct
Roots
The first case is where both roots to the polynomial are real and
distinct, and results when:
Since the value here is positive, this guarantees that no complex numbers
will appear in the solution. This case is called the over-damped case.
Constructive Over-damping
Over-damping can be a positive in certain cases. One example would be the
need to completely avoid overshooting of the steady state value within the
system. An over-damped response will slowly reach its steady state in an
exponentially decaying fashion, thus avoiding problems associated with excessive
motion or current flow, especially in systems that are sensitive to sudden and
rapid change. Temperature is an example: some things such as pressure cookers
and kitchenware need to slowly be brought up to the correct temperature lest
the material will fracture under excessive initial heat.
Destructive Over-damping
Over-damping can be destructive in that it takes excessive amounts of
time for the system to reach its steady state, especially in cases where
transient system effects should dissipate much sooner. Though there are not
generally oscillations associated with over-damped motion, the strain
associated with it can be stressful on system equipment if sudden surges or
forces take place. Over-damped automotive systems tend to be excessively stiff
and resistant to bumps in the road, thus causing an extremely rough ride.
Case #2: Repeated Roots
The second case is where instead of two real and distinct roots, there is
a single real repeated root. This case results when:
This case is called the critically damped case.
We have found a single solution for y(t). However,
according to the general solution of a differential equation, there should be
another solution and it should be linearly independent of the above y(t). The next
choice for a solution would be a solution of the form:
This makes sense as the function is assumed
to be the product of y and some unknown t. One can prove the above solution by
taking the derivative of
and substituting it into the characteristic
equation for a second order differential equation. This leads us to our final
form of the critically damped case, which is:
Constructive Critical Damping
Critical damping is usually the ideal choice when designing a system and
this can be seen especially in the automotive department with shock absorption.
In an over-damped system, a small bump in the road would have a large effect on
the system. The critically damped case ensures the fastest return of the system
to the steady state along with eliminating any oscillatory motion.
Destructive Critical Damping
While critical damping often yields the best performance in systems,
sometimes it is not the preferred method for the case under observation. In systems
where electromagnetic interference poses a threat, the over-damped case can be
preferred to the critically damped case. This can help cushion the transition
between signals and kill some of the high frequency material. In excess, this degrades
the system performance.
Case #3: Complex Roots
The final case
under discussion is where the roots to the
polynomial are complex numbers. This results when:
Taking the square root of a negative number often yields a solution with
both a real and imaginary component (e.g. a + bj and a - bj). This case is
called the under-damped case.
Using Euler's Formula and
the form a + bj and a-bj from above, a new variation of the second order solution emerges for complex
numbers:
Constructive Under-damping
Under-damping can be useful in some applications as opposed to critical
damping. If the input to a system changes rapidly, selecting a slightly under
damped response would be wise as it helps gain the fastest initial response. If
initial speed is the most important part of the design, the under damped case
is ideal.
Destructive Under-damping
While under-damping yields the quickest initial results in a system, the
loss of functionality associated with overshooting the ideal output is a large
trade off. For example: in an electric circuit, under-damping of the power
supply may cause the output is allowed to swing past the final allowed voltage.
In this case, damage to the equipment could occur. Significant amounts of
under-damping in an automobile will cause the vehicle to bounce repeatedly
after hitting one small bump in the road.
References:
http://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx
http://www.sosmath.com/diffeq/second/second.html
http://en.wikipedia.org
Images:
http://www.gender.org.uk/about/06encrn/61_cntrl.htm
http://en.wikipedia.org/wiki/Damping
http://homepages.which.net/~paul.hills/Circuits/PowerServo/PowerServo.html
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