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ORDINARY DIFFERENTIAL EQUATIONS
Lecture 15
Physical Systems Modelled by Linear Homogeneous SecondOrder ODEs
(Revised 22 March, 2009 @ 18:00)
Professor Stephen H Saperstone
Department of Mathematical Sciences
George Mason University
Fairfax, VA 22030
email: sap@gmu.edu
Copyright © 2009 by Stephen H Saperstone
All rights reserved
The simplest way to understand the nature of solutions to linear homogeneous second-order ODEs is via a real
system with whose behavior we are familiar. Consider the damped mass-spring system in Figure 15.1.
Figure 15.1 Unforced Mechanical System
The ODE
models the motion of the block. Here denotes the mass of the block,
is the displacement of the block from it's
is the damping coefficient of the shock absorber, and is the spring constant. Divide Eqn.
rest position at time
(15.1) by to get
For algebraic convenience (which we shall see shortly), set
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Then Eqn.(15.2) transforms to
The parameters and
are always nonnegative as they must be to model a physical system. As is the coefficient
measure the
of friction and is the spring constant (see the Example 1.06 in Lecture 1), the parameters and
following:
There are two cases: 1. undamped motion (
), and 2. damped motion (
):
15.1 NO DAMPING - SIMPLE HARMONIC MOTION (
friction)
no damping or
In the absense of damping (friction), once the block is set in motion, it continues to stay in motion and executes
simple harmonic motion.
Figure 15.2 Unforced Undamped Mechanical System
The ODE for this motion is given by
Because the characteristic polynomial for Eqn. (15.5) is
by
with characteristic roots
an FSS is given
so for SHM every solution has the form
for some constants
and
that are uniquely determined by initial conditions.
for all
It is seems reasonable that
is periodic , i.e.,
of
This is such an important result that we state AND prove it here.
where
is the period. What is the value
Theorem 15.1 (Simple Harmonic Motion):
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The general solution
to the ODE
is periodic with period
Proof: Since sine and cosine are periodic with period
Hence replace with
it follows that
everywhere in Eqn. (15.6):
Substitute Eqns. (15.7) in the right side of Eqn. (15.8) to obtain
which is equivalent to the statement
Thus
is periodic with period
Q.E.D.
The parameter
plays the role of an angular frequency in Eqn. (15.6). The frequency with which the block
oscillates depends on the spring constant . In fact, according to Eqn. (15.3)
No matter what initial conditions are imposed on the ODE
the block ALWAYS oscillates with angular
frequency
It is for this reason that
is called the natural frequency of the ODE. Moreover
is related to the
period through the relation
Eqn (15.6) lends itself to even further interpretation. Let
triangle below in Figure 15.3.
Thus
represents the hypotenuse in the
Figure 15.3
Since
then we rewrite Eqn (15.6) as
We recognize the last expression to be the
cosine of the difference of the angles
since cosine is an even function; that is,
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and
Thus
Thus Equation (26) can be represented by
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Thus we are led to a new and important definition
Theorem 15.2 (Phase-Amplitude Form):
The general solution
to the ODE
can be expressed in the phase-amplitude form
where the parameters
and
are defined by
Moreover, the motion is periodic with period
Figure 15.4 illustrates these new terms.
Figure 15.4: Simple Harmonic Motion
Example 15.3
Compute the amplitude, period and phase shift of
Solution: The amplitude is
illustrates these computations.
. Since
then the period is
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The phase shift is
Figure 15.5
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Figure 15.5 - Period-12 Simple Harmonic Moion
End of Example 15.3
Example 15.4
Compute the amplitude, period and phase shift of
Solution: The amplitude is
express the function in the form
. Since
then the period is
Write
To calculate the phase shift, we must first
where we have used the trigonometric identity
(the cosine is the left-translation of the sine by
that the phase shift is
Figure 15.6 illustrates these computations.
Figure 15.6 - Phase Shift of Period-
). It follows
Simple Harmonic Moion
End of Example 15.4
An important example that exhibits SHM is the undamped linearized pendulum, a schematic of which is illustrated
in Figure 15.7.
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Figure 15.7 - Simple Undamped Pendulum
The ODE that models the undamped and unforced pendulum is
This is a nonlinear ODE which cannot be solve for an explicit solution. On the other hand, in practical applications,
the angle remains small (e.g., a grandfather or pendulum clock, a tuning fork). When remains very close to zero,
we have that
When we replace
with
in Eqn. (15.10), we linearize the ODE.
Example 15.5 [Undamped Linearized Pendulum]
The ODE that models the undamped linearized pendulum is
(Observe that neither model - nonlinear or linear - includes the mass
behind the model.) Show that for the initial values
of the pendulum bob. This is a consequence of the physics
the unique solution to the IVP is
Solution: (Click me)
End of Example 15.5
15.2 DAMPED HARMONIC MOTION (
damping or friction present)
The ODE for this motion is given by
Because the characteristic polynomial for Eqn. (15.12) is
with characteristic roots
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a FSS depends on whether or not the characteristic roots are real or complex. Thus we get three cases:
(a) Overdamped:
(b) Critically Damped:
(c) Underdamped:
Case (a) Overdamped:
Set
It is obvious that
It is not so obvious that
As
is less than
as well. To prove this note that
It follows that by subtracting
from
hence
that
Thus
Thus we have shown that both characteristic roots are negative with
Now an FSS is given by
and a general solution has the form
In view of the fact that both
and
are negative, then all solutions must tend to zero as
Overdamping occurs when the damping (frictional) force overwhelms the restoring force of the spring so no
oscillations can occur.
Example 15.6 (Overdamping)
The ODE for the charge
on the capacitor at time for the
circuit
is given by the ODE
When
Henrys,
ohms, and
and plot the results.
Solution:
Substitute the values of
Farads, determine the solution for
and
when the initial conditions are
into the ODE and normalize to obtain
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The characteristic equation is
This polynomial factors readily:
so we get the distinct real roots
From Table 15.1 we get the FSS
so that a general solution is given by
The calculation of
and
is left to the reader. We get
The important thing to get from this example is the rapid decay to zero of
as
Figure 15.8 - Overdamped Motion:
with
End of Example 15.6
Case (b) Critical Damping:
Set
The roots of the characteristic polynomial are repeated with common value
Thus a FSS is given by
and a general solution has the form
L'Hospital's rule implies that all solutions must tend to zero as
Critical damping occurs when the damping (frictional) force balances the restoring force of the spring. No
oscillations can occur, although an "overshoot" of the equilibrium
can occur.
Example 15.7 (Critical Damping)
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Solve and graph the solution of the ODE
for each of the two initial data sets
Solution:
The characteristic equation is
with the repeated real roots
From Table 15.1 we get the FSS
so that a general solution is given by
(a) Initial Data Set
Set
in Eqn. (15.14):
Next, differentiate Eqn. (15.14) with respect to
and set
in Eqn. (15.16):
Solve Eqns. (15.15) and (15.17) for
and
:
Solution to the initial data set
(b) Initial data set
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Figure 15.9 - Critically Damped Motion:
for
and
End of Example 15.7
Case (c) Underdamping:
The characteristic roots are
Since the roots are complex, it follows that a general solution to Eqn. (15.12) has the form
We can write Eqn. (15.19) in phase-amplitude form
The term
tends to zero as
The term
is periodic and represents SHM. Therefore Eqn.
(15.20) represents an oscillatory motion in which the oscillations generated by the cosine function have the timedependent amplitude
We illustrate this behavior in Figure 15.5(a) with the graph of the solution to the IVP
The graph of the solution is bounded by the graphs of the functions
envelope that "contains" the oscillations.
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and
These functions form an
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Figure 15.10:
Although Eqn. (15.20) is not a periodic function, we can consider the IVP motion to be periodic in some sense
because the cosine term has period
This periodic-like behavior leads us to call the damped motion quasiperiodic with quasiperiod
Set
and call the (angular) quasifrequency. The motion is called underdamped because the damping is not sufficient to
prevent the system from oscillating ( is too small).
The introduction of underdamping to simple harmonic motion causes more than just exponential decay of the
oscillations. Two other effects occur:
1. The oscillations slow down; that is, the quasiperiod of the underdamped motion is larger than the the period of the
undamped motion. This is because
2. The motion is delayed by an amount equal to the difference between the phase angles of the underdamped and
undamped systems, namely
Figure 15.5(b) compares the solutions of the underdamped
The underdamped IVP is solved below in Example 15.6.
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and undamped
versions of Eqn. (15.21).
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Figure 15.11 Comparison of Simple & Under Damped Harmonic Motion
Example 15.8 (Underdamping)
Solve and graph the solution of the IVP
Solution:
The characteristic equation is
has roots
so that a general solution is given by
Next, calculate values of
and
to satisfy the initial conditions. We get
. (Check the details!)
The amplitude of the motion is given by
The phase angle
can be computed from the relation
The quasifrequency is given by
Thus the phase-amplitude form of the solution is
with phase shift
End of Example 15.8
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VISUAL SUMMARY OF DAMPED MOTION
The transition from SHM to underdamped motion to overdamped motion can be visualized in Figure 15.12. Here we
plotted a family of solutions to the IVP
When
there is no damping, so the motion is purely oscillatory, as Figure 15.6 suggests. As increases from
through the critical value and beyond, we see that the oscillations slowly fade into the exponential decay of
overdamped systems.
Figure 15.12: Transition from Simple Harmonic Motion to Over Damped Motion
Animation of Figure 15.12 (R. Devaney)
Example 15.9 (Critical damping of MacPherson Strut)
A MacPherson strut supports a portion of the weight of a car. Under a load of 640 lb, the strut is compressed 3 in. to its rest position.
Find the value of the damping coefficient in the ODE
for critical damping. [See Example 10.1 with external force
Solution: At critical damping we must have
The value of the spring constant
compressed while at rest. This is expressed in the relationship
which gives us
Since
lb and
depends on the amount the strut is
ft,
End of Example 15.9
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15.3 HARMONICALLY FORCED MOTION OF PHYSICAL SYSTEMS
Simple mechanical and electrical systems best illustrate the value of the UC method. Now we add forcing so that the
governing ODE has the form
when
and
are nonnegative. and
represents an external force.
In the case of the horizontal damped mass-spring system subject to a periodic forcing function
extend the to the forced one
we
The parameter is called the input (angular) frequency. Also as in the unforced case in Lecture 12, we examine
both undamped motion (
), and damped motion (
).
Figure 15.13: Forced Mechanical System
Undamped Harmonically-Forced Motion:
Contrary to the fact that actual systems exhibit some level of damping or friction, undamped motion is just an
idealization of a system with negligible damping. It is instructive to study such an ideal system if only because the
mathematics is relatively simple and the solution sheds light on the actual (negligibly) damped system. This motion
is given by the following theorem. Note that we have taken the system to be initially at rest. There is no loss in
generality in taking
and
because the forcing function gets the system in motion. The solution, Eqn.
(15.23) to follow, was derived in Example 13.8 earlier.
Theorem 15.10 [Undamped Periodically-Forced Motion]
When
the solution to the IVP
is
Proof: See the derivation in Example 13.8 .
Eqn. (15.23) indicates that two motions occur; one with the natural angular frequency
unforced ODE another with the angular frequency of the input
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from the corresponding
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Beats
When the input frequency
trigonometric identity
in Eqn. (15.23) with
is close in value to the natural frequency
and
, the phenomenom of beats occurs. Use the
to obtain
Think of Eqn. (13.14) as a representation of the solution in phase-amplitude form, namely
where
Eqn. (15.26) represents a periodic output with (angular) frequency
and time-varying amplitude
which
itself is periodic with the smaller (angular) frequency
In engineering terminology we say that the periodic
signal with higher (angular) frequency
is modulated by another periodic signal with lower (angular)
frequency
Figure 15.14 illustrates this idea when the difference is small but not zero.
Figure 15.14:
The rapidly varying signal has period
whereas the slowly varying "envelope" has period
Beats occurs when the frequency of a periodic input is close to the natural frequency of the system. Piano tuners use
beats in their craft: When a tuning fork is struck simultaneously with a piano string at nearly the same frequency, the
beats allow the piano tuner to discern the difference in frequencies. The piano tuner adjusts the tension in the piano
wire so as to slow down the beat frequency and eventually eliminate it.
As
we see from Eqn. (15.23) that the amplitude of the undamped oscillations becomes unbounded. When
in Eqn. (15.22) the ODE for this motion is given by
Java Applet: Beats (MIT Open CourseWare)
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Theorem 15.11 [Undamped Periodically-Forced Motion: Resonance]
When
the solution to the IVP
is
Proof: A short and elegant proof follows from an application of L'Hôpital's rule to the case of Eqn. (15.22) when
Start with rewriting Eqn. (15.23)
and let
. As the quotient above has the indeterminate form
when
we get
Q.E.D.
Observe that the input
is a solution to the corresponding homogeneous ODE,
The UC method suggests that a trial particular solution has the form
in Theorem 15.9.
Pure Resonance
The motion described by Eqn. (15.29) becomes unbounded as
(simple harmonic motion)
of frequency
is modulated by
oscillations at the angular frequency
Figure 15.15 illustrates this behavior. The SHM
The periodic input reinforces the natural
Figure 15.15:
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Pure resonance can have serious consequences for some structures and mechanical systems . Not all occurrences of
resonance are destructive, though. A seismograph relies on resonance to detect earth tremors. AM radios "tune in" to
a fixed carrier frequency
by the adjustment of a variable capacitor, which in turn alters the frequency of a simple
circuit so as to agree with
We can visualize the transition to pure resonance by examining the solution to the IVP
We leave it to the reader to show that
As
we expect the solutions to exhibit increasingly larger swings in amplitude. Plots of Eqn. (15.30) for
and
are displayed in Figure 15.16.
Figure 15.16
for
and
Animation of Figure 15.16 (R. Devaney)
Example 15.12 (Seismic Vibration)
A typical undamped seismic vibration-measuring instrument records the relative motion of a suspended body with respect to its
housing as depicted below.
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Let denote the displacement of the body from its rest position, and let denote the displacement of the base of the instrument cage
from its rest position. Determine the steady-state displacement of the body relative to the cage when the function that acts on the
cage is harmonic, i.e.,
Assume the body has mass and the spring has constant
Solution:
Define the displacement of the body relative to the cage by
Because the restoring force on the body is
or equivalently expressing
Since
then
the ODE for the motion is
in terms of the dependent variable
so that the ODE for becomes
WE leave it to the reader to derive the solution
End of Example 15.12
More generally, pure resonance occurs in an undamped ODE when a bounded forcing function produces an
unbounded solution.
Example 15.13 (Pure Resonance)
Compute all frequencies
Solution:
at which the solution exhibits pure resonance occurs for the ODE
First observe that the forcing function
is bounded; i.e.,
for all
Although it is tempting to say
that
produces pure resonance, we would be wrong. Let us proceed to calculate a particular solution to determine what values of
if any, produce an unbounded solution.
Next we note that a FSS is given by
so that the complementary solution is
Although
is a UC function (it is the product of the UC functions
combination of simple functions by the trigonometric identity:
and
we can easily reduce
to a linear
Thus we can rewrite the ODE as
The superposition principle tells us that the solution to the ODE is the sum of the solutions to each of the following ODEs:
For what we want to do, we don't have to calculate the coefficients for the particular solution. Thus a particular solution to Eqn. (1)
has the form
and the particular solution to Eqn (2). has the form
Thus a particular solution to the original ODE has the form
It seems reasonable that if we choose
then the forcing functions
follows that a particular solution to Eqn. (2) must have the form
No matter what values we calculate for and (note - not both
solution. See Theorem 15.11 for confirmation of this behavior.
End of Example 15.13
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and
and
become
and
respectively. It
can be zero), the result is an unbounded (particular)
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Damped Periodically-Forced Motion:
All real physical systems exhibit some form of damping or resistance (loss of energy over time), no matter how slight
it is. Thus
. the ODE for the periodically forced system
has the general solution given by the following theorem. Unlike the undamped system of Eqn. (15.28), the case when
doesn't require a separate analysis. We continue using the notation regarding unforced damped motion (with
replaced by in Eqn. (15.31)).
Theorem 15.14 [Damped Periodically-Forced Motion]
A particular solution to the ODE
is
where
Proof: Under Construction.
Remark 15.15 [Damped Periodically-Forced Motion]
1. The general solution to Eqn. (15.32) is
where
is a FSS for the corresponding homogeneous ODE.
2. Depending on the the relationship between and , the complementary solution to the corresponding
homogeneous ODE
is either overdamped, critically damped, or underdamped. In any
event
Because the complementary solution
is asymptotic to zero as
we say that
is a transient solution.
3. Because the particular solution
is periodic and hence persists as
we say that
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is a steady-state solution to Eqn. (15.32).
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Example 15.16 (Transient & Steady-State Solutions)
Compute the transient & steady-state solutions to the IVP
and sketch graphs of these solutions, along with the forcing function on the same set of axes.
Solution:
We leave it to the reader to show that a FSS is given by
With
and
(where
in Eqn. (15.33) we obtain transient solution
, and the steady-state solution
Figure 15.17: Input, Output, Transient, & Steady State Response of
Comment - click here
End of Example 15.16
Practical Resonance
The amplitude of the particular solution depends on the input frequency
have from Eqn. (15.33) that
An application of max-min theory from calculus shows that if
then
is maximized at
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Letting
, then
denote this dependence, we
is maximized at
If
The maximum amplitude is
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The frequency
is called the resonance frequency of the system; it is the input frequency that maximizes the
amplitude of the steady-state response of the solution. When the system is driven at this frequency, we say that the
system is at resonance. Note that
the quasifrequency for unforced underdamped motion.
Figure 15.18 illustrates the graphs of
for different balues of the damping parameter We have chosen
and
for convenience. As decreases, the peaks of the graphs increase and
approaches the value of the
natural frequency,
Observe that as increases beyond some threshold,
, resonance fails to occur. This is
consistent with what we have just explained. The graph of
is called the frequency response curve or
resonance curve for the ODE.
Figure 15.18: Frequency response curve for various
Java Applets
Damped Forced Vibration (MIT Open CourseWare)
Forced Oscillator Java Applet (Walter Fendt) - Slinky\ Motion
Interactive Differential Equations (West, Strogatz, McDill, Cantwell)
Visual Differential Equations (P. Falstad)
Tacoma Narrows Bridge - 1940 (Galloping Gertie)
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