Damped Mechanical Oscillations
Equipment
• Driven harmonic motion apparatus
• Assortment of slotted masses
• Assortment red springs
• Calipers
• Digital balance
Preparation
Review the mathematics of the damped harmonic oscillator.
Goal of the Experiment
To understand and measure the behaviour of a forced harmonic
oscillator system with damping. To gain experience with dynamical
systems.
Theory
A mass hanging from an idealized vertical sprng forms a harmonic
oscillator. The mass moves with simple harmonic motion. Simple harmonic motion never ceases. The mass is expected to oscillate forever.
However, in a real physical system, the mass bobbing at the end of
the spring eventually comes to rest. The amplitude of the oscillations
gradually diminishes to zero. This happens because energy is removed
from the system. The physical spring gradually converts mechanical
energy into thermal energy that is then lost to the environment. Other
processes such as air resistance also contribute. All of these energy
losses damp out the oscillations. A more accurate model of a physical
spring and mass system must therefore include energy losses due to
damping.
Another feature of the idealized spring mass system is that all energy is input into the system at the start. No further energy is added
later. The energy in the mass and spring system is introduced by
stretching the spring before release (adding potential energy), or by
Fs = −k ( x − y − L) .
(1)
2
drive cord
spring
y
FS
guide
and
sensor
zero
point
0
index bar
giving the mass an initial velocity (adding kinetic energy), or both.
Once the mass is released, the energy content of the system does not
decrease or increase. A system that only has energy input at time zero
is called an unforced or undriven system. Suppose however that the
support the spring hangs from were moved up and down while the
mass is moving. Here, energy can be added to the system on a continuous basis. Such a system is called a forced or driven system. The
equation describing how the spring support moves with time is called
the forcing function. This experiment examines a driven damped harmonic oscillator where the forcing function is a sine wave at a single
frequency.
Figure 1 shows a forced damped harmonic oscillator. Masses and
a damping bar are supported by the spring. The upper end of the
spring is not fixed. Instead, it is connected to a driver that changes
the position of the upper end of the spring. The spring obeys Hooke’s
Law with spring constant k. The damping bar extends between two
permanent magnets. As the bar oscillates, eddy currents are generated
in the bar that produce heat, thus removing energy from theoscillator.
This damping device is called an eddy current damper. Using the
index bar, the guide and sensor can measure the system position.
Coordinates must be assigned to the system of Figure 1 in order to
derive the equation of motion. Define the zero point as the origin. Let
the distance of the index point from the zero point be the position, x.
Define the positive direction to be down. The position of the system is
then determined by how far the index point is from the sensor guide
zero point, and whether it is above (negative) or below (positive) the
sensor guide zero point.
Four forces are present in this system. The mass hanging from the
end of the spring supplies the force due to gravity, mg. For the system
of Figure 1, the effective total mass is given by the mass of the damping
bar, the index bar, the extra applied masses, and one third the mass of
the spring all added together.
The spring force is determined from Hooke’s law. The force of the
driving wheel is taken into account by including the displacement of
the top of the spring when determining the spring force Fs . When the
system is at rest the spring will be stretched by the weight of the index
bar, damping bar, and extra masses. Let the distance from the top of
the stretched spring to the index point be L. Define the position of the
upper end of the spring to be y. This is determined by the position
of the drive wheel. The position of the index point is x. Since the
spring can only pull, the spring force is always directed upwards and
therefore must be negative. The spring force therefore is
masses
damping
bar
drive cord
length
adjustment
x
index
point
ωd
A
mg
Ff
drive
wheel
damping magnets
Figure 1: Forced damped harmonic oscillator
The last force to be accounted for is the damping or frictional force
F f . Under normal circumstances, it can be assumed that the damping
force is proportional to the velocity of the mass bar. Therefore
dx
.
(2)
dt
The constant, B, is called the damping coefficient. The negative
sign asserts that the damping force always acts in the direction opposite to the mass bar velocity.
Inserting each of these forces into Newton’s second law yields
Ff = − B
m
d2 x
= mg + Fs + Ff .
dt2
(3)
Substituting Equations 1 and 2 into Equation 3 gives
m
d2 x
dx
mg +
B
+
kx
=
k
y
+
L
+
.
dt
k
dt2
(4)
The drive cord and drive wheel are capable of generating a forcing
function of the form
y = P + Asin (ωd t + φ) .
(5)
Equation 5 can then be substituted into Equation 4 to obtain the equation of motion. However, this can be simplified considerably. Choose
the phase, φ, to be zero by rotating the drive wheel to the zero position. Next, adjust the length of the drive cord so that the index point
coincides with the zero point at equilibrium. This sets the constant P
to be
mg
P = −L −
.
(6)
k
This yields the final form of the equation of motion to be
m
d2 x
dx
+ B + kx = kAsin (ωd t) .
2
dt
dt
(7)
Equation 7 is the most general form of the equation for harmonic
motion. Equation 7 is a second order differential equation. The solution of this equation will give the position of the system as a function
of time, x(t). Once the position is known, the velocity and acceleration
can be found by differentiation. Since the equation is of second order,
two initial conditions are needed to fully specify the solution. For this
system the initial conditions are the position, X0 , and the velocity, V0 .
To start the system oscillating, the index bar must be displacedfrom the
zero point (this gives the system an initial X0 ), pushed from the zero
point (this gives an initial V0 ), or both. The equation also contains five
system parameters, which are the mass m, the damping coefficient B,
the spring constant k, the driving amplitude A, and the driving frequency ωd . Changing any one of them can alter the behaviour of the
3
system because the solution form depends on the value of the system
parameters. Essentially, the solutions of Equation 7 have four qualitatively different sorts of behaviour depending upon the value of the
damping coefficient.
When
When
When
When
B
B
B
B
= 0√
< 2√km
= 2√km
> 2 km
the system is said to be undamped.
the system is said to be underdamped.
the system is said to be critically damped
the system is said to be overdamped
In addition, the solutions of Equation 7 have two qualitatively different
sorts of behaviour depending upon the driving amplitude.
When A = 0
When A 6= 0
the system is said to be undriven
the system is said to be driven
These different solutions are described in more detail below.
1. Undriven and Undamped, see Figure 2
s
X02
x (t) =
+
V0
ω0
s
C=
X02 +
C
x0
2
sin (ω0 t + θ )
(8a)
x(t)
V0
ω0
t
2
(8b)
2π/ω0
r
ω0 =
θ = tan−1
k
m
ω 0 X0
V0
(8c)
Figure 2:
damped
Undriven and Un-
(8d)
This is the simple harmonic oscillator. The frequency of oscillation, ω0 ,
is called the natural frequency.
C
B
2. Undriven and Underdamped, see Figure 3
x0
B
x (t) = Ce−( 2m )t sin (ω0 t + θ )
s
C=
X02 +
s
ω0 =
V0
BX0
+
ω0
2mω0
k
−
m
B
2m
(9a)
Ce−( 2m )t
x(t)
t
B
-Ce−( 2m )t
2
(9b)
2
(9c)
4
-C
2π/ω0
Figure 3: Undriven and Underdamped
!
ω 0 X0
θ = tan−1
(9d)
BX0
2m
V0 +
Here, the mass oscillates with decreasing amplitude at the natural
frequency.
3. Undriven and Critically Damped, see Figure 4
x (t) = ( X0 + Ct)
C> 0, V0 > 0
C< 0, V0 > 0
x(t)
−B
e 2m t
(10a)
t
C< 0, V0 < 0
x0
C< 0, V0 = 0
X0 B
C = V0 +
2m
(10b)
Figure 4: Undriven and Critically Damped
Here, the mass does not oscillate. Instead it smoothly moves towards the equilibrium position and comes to rest. The initial position
(intercept at t=0) and the initial velocity (the slope at t=0) determine
the actual shape of the curve.0
V0 > 0
4. Undriven and Overdamped, see Figure 5
x (t) = C1 e
−γ1 t
+ C2 e
V0 = 0
x0
−γ2 t
(11a)
x(t)
s
ω0 =
B
2m
2
−
k
m
(11b)
t
V0 < 0
x0 =C1 +C2
V0 << 0
−B
+ ω0
γ1 =
2m
(11c)
γ2 =
−B
− ω0
2m
(11d)
C1 =
V0 − γ2 X0
2ω0
(11e)
C2 =
γ1 X0 − V0
2ω0
(11f)
Here, the mass also moves to zero, but slower than the critically
damped case. The initial Figure 5 position and initial velocity determine the final shape of the curve.
In all the solutions shown above, observe that the motion eventually decays to zero when damping is present. Also, note that the
natural frequency of the oscillations is changed by the presence of
damping. The final form of these solutions is set by the intial conditions. These specify the slope and intercept of the solution curve at
t=0.
5
Figure 5: Undriven and Overdamped
The solutions for Equation 7 when a forcing function is present are
made up of two parts. The transient or natural solution is given by
one of the solutions above in Figures 2 through 5. In addition, a second
term appears that is due to the presence of the forcing function. This is
called the particular integral or steady-state solution. Here, where the
forcing function is a sinusoid, the steady-state solution does not decay.
The full answer to the forced and damped harmonic oscillator is the
sum of the natural and steady-state solutions. As with the unforced
case, the presence or absence of damping changes the form of the
solution.
The frequency of the forcing function also alters the form of the
solution. At certain driving frequencies, the extent of the motion of
the system greatly exceeds the amplitude of the driving function. This
is the condition known as resonance. Resonance can only occur when
the damping coefficient is below a critical value. The driven cases are
When B ≥
√
2km
√
When o ≤ B ≤ 2km
√
When B = 0, and ωd 6= 2km
√
When B = 0, and ωd = 2km
no resonance is possible. q
resonance occurs for ωd = mk −
B2
2m2
system generating beat frequencies.
system in undamped resonance.
These four cases of driven solutions are described in further
detail below.
√
5. Drive, B ≥ 2km, see Figure 6
full solution
x (t) = [ Natural Solutions f orA = 0] + f (ωd )kAsin (ωd t − θ ) (12a)
f ( ωd ) = q
1
k − mωd2
"
θ = tan
−1
2
(12b)
natural solution
x(t)
t
+ B2 ωd2
Bωd
k − mωd2
steady-state solution
#
(12c)
x0
Figure 6: Drive, B ≥
The amplitude of the steady-state solution is dependent on the
damping coefficient and on the driving frequency. This dependence
is described by the function f(ωd ), which is called the resonance curve
of the system. Here, where B is above the critical value, the resonance
curve has a single maximum at zero driving frequency. No resonance
can occur.
6
√
2km
6. Driven, 0 > B >
√
2km, see Figure 7
x (t) = [ Natural Solutions f orA = 0] + f (ωd )kAsin (ωd t − θ ) (13a)
f ( ωd ) = q
full solution
1
(13b)
2 2
k − mωd + B2 ωd2
"
#
Bωd
−1
θ = tan
k − mωd2
r
B2
k
−
.
ωr =
m 2m2
(13c)
forcing function
x(t)
t
x0
(13d)
natural solution
Here, resonance can happen. The amplitude of the steady-state solution is dependent on the damping coefficient and on the driving frequency. The steady-state solution can have an amplitude much larger
than the driving amplitude. This occurs because the damping coefficient is small enough that the resonance curve has a second maximum
when the driving frequency, ωd , is at the resonant frequency, ωr .
Note that the resonant frequency, the undamped natural frequency,
and the damped natural frequency are all different when damping is
present and are all equal without damping.
q
7. Driven, B = 0, ωd 6= ωr , ωr = mk , see Figure 8
Figure 7: Driven, 0 > B >
√
2km
beat frequency
x0
x (t) =
q
C2 + X02 sin (ω0 t + θ ) +
C=
kA
sin (ωd t + θ )
ω02 − ωd2
V0
ω
kA
− d
ω0
ω0 ω02 − ωd2
X0
θ = tan−1
C
r
ω0 = ωr =
(14a)
x(t)
t
(14b)
system motion
(14c)
k
m
(14d)
In this case, the system motion is a linear combination of simple
harmonic motion at two differrent frequencies. The two frequencies
are the driving frequency and the natural frequency. When the amplitudes of these two motions are roughly equal, the total motion appears
as a single higher frequency with an amplitude that varies at a lower
7
Figure 8: q
Driven, B = 0, ωd 6=
ωr , ωr = mk
frequency. This is the phenomenon known as beats. The beat frequency is proportional to the frequency difference between the driving frequency and the natural frequency.
This form of motion is also called quasi-periodic motion because
the sum of two harmonic motions is not periodic, except when the
two frequencies are rational multiples of each other. When the two
frequencies are equal, the above solution does not exist. Instead, the
motion takes on the rather different form shown next.
q
8. Driven, B = 0, ωd = ωr = mk , see Figure 9
x (t) =
q
C2 + X02 sin (ω0 t + θ ) −
kA
tcos (ω0 t)
2ω0
V0
kA
+
ω0
2ω02
− 1 X0
θ = tan
C
r
k
ω0 = ωr =
m
C=
(15a)
(15b)
x(t)
t
x0
(15c)
(15d)
The system oscillates with ever wilder amplitudes. In physical systems this motion usually does not last for long. The spring either
breaks, or the system deforms so that Equation 7 no longer applies.
This experiment will examine a driven and damped harmonic oscillator operating in several of the modes described above. The apparatus permits measurements of the system position so that the theoretical
equations of motion can be checked.
Figure 10 shows the damped harmonic motion analyzer (DHMA)
used in this experiment. There is a direct correspondence between
the instrument shown in Figure 10 and the diagram of Figure 1. The
DHMA includes digital readouts necessary for direct measurements of
the period, extent of travel, and phase of the index bar. A sinusoidal
forcing function is provided with readout and adjustment of the driving frequency. Also included, is a manual adjustment for setting and
reading the forcing function amplitude.
The DHMA power switch is located at the rear of the unit. The forcing function driving wheel is also located there. As shown in Figure 1,
a manual slide with a scale permits the operator to read and adjust the
driving amplitude. The forcing function is engaged and disengaged by
ωd
the front panel DRIVE switch. The drive frequency fd = 2π
is adjusted
with the front panel FREQUENCY control. When the readout selector
is switched to FREQ., the digital readout gives the forcing frequency,
fd , in Hertz.
8
Figureq9: Driven, B = 0, ωd =
ωr = mk
Figure 10: Damped Harmonic
Oscillator Apparatus
9
Observe that the index bar has a scale on one side and a black
rectangle on the other side. The black rectangle should be on the right
as seen from the front of the DHMA. The blackened and transparent
areas on this index bar are used to interrupt a light beam inside the
mass guide sensor. The red light on the front of the mass guide is
lit whenever the black rectangle is below the guide and off when this
rectangle breaks the light beam. The top edge of the black rectangle
is the index point as shown in Figure 1. This is considered to be the
origin of the coordinate system used by the DHMA. When the mass
is oscillating, the flashing phase set LED is timed by internal circuitry
to give a digital readout of the period. The black scale marks on the
left side of the index bar are counted as they pass the optical sensor
inside the mass guide. The total count of scale marks in each period
is a digital measurement of the travel distance of the index bar in
one period. Lastly, the time difference between when the index point
passes the optical sensor and when the forcing function drive wheel
passes its zero point gives a digital readout of the phase.
In order for the system to perform properly and conform to Equation 7, the DHMA must be correctly adjusted. The index bar should
move through the mass guide and sensor without rubbing. This is
accomplished by the levelling screws at the base of the unit and the
spade lug connecting the spring to the index bar. Figure 11 gives the
details.
The next adjustment to be made sets the forcing function phase to
0deg. Move the index bar up and down. Notice that when the phase
set LED flashes, so does a lamp inside the phase angle display at the
front of the DHMA. Manually turn the driving wheel at the rear of the
DHMA until this phase display reads 0deg. This adjustment aligns the
phase display with the index bar coordinate system.
Lastly, the length of the driving cord needs to be adjusted to conform with the requirements of Equation 6. This effectively sets the
origin of the DHMA coordinate system to be at the index point. Use
the drive cord coarse adjust to position the top of the black rectangle
even with the phase set LED when the system is at rest. At this position small oscillations of the index bar will cause the phase set LED to
turn on and off. Then adjust the drive cord fine adjust at the top of the
DHMA to until the amplitude of the index bar oscillations that make
the phase set LED flash are as small as possible.
Once these adjustments are completed, the system motion theoretically matches the derivation leading to Equation 7. These adjustments
should be rechecked every time the spring is changed or masses are
added to the index bar. It remains to be experimentally determined
whether the actual dynamics of the system corresponds to the solutions of the forced damped harmonic oscillator equation as described
10
above.
Mass guide and sensor
Figure 11: Alignment of damping bar
Index bar
(Top View)
Index bar is correctly aligned
Index bar is incorrectly aligned
DHMA should be levelled
Index bar is incorrectly aligned
Index bar should be rotated
Experimental Procedure
1. Align the DHMA as described above. A realignment should be carried out any time an extra mass is added or the spring is changed. If
the DHMA is moved or bumped the alignment should be rechecked.
2. The system parameters m, k, and B need to be measured. Using
the digital balance, determine the mass of the spring and the objects hanging from it (the index bar, damping bar, and all additional
masses). If the mass of the spring is not negligible, then the effective mass of the system is the sum of the masses hanging from the
spring and one third of the spring mass.
3. The spring constant, k, is found from Hooke’s law. Add 0 to 100
grams to the index bar in steps of 5 to 20 grams for at least 8 points.
Plotting applied force versus extension gives a line with slope k.
Note that the spring is already extended by the index and damping
bars hanging from it. The first data point will then be a spring
extension of zero with an applied force of mg Newtons, where m is
the mass of the index and damping bars. Use the scale on the index
bar to measure the amount of extension.
4. Determine the damping coefficient B. For the DHMA, the system
is always underdamped when magnetic damping is used regardless of the position of the magnets. Adjust the magnet positions for
minimum damping (magnets as far away from the damping bar as
possible). Measure the magnet spacing with calipers. In the undriven underdamped case Figure 9 shows that the peak amplitudes
of the motion lie on an exponential curve. If the period between
11
peaks is known (to be determined below), a graph of logarithm of
peak amplitude versus time gives a line with slope -B/2m. Set the
FUNCTION switch to AMPL and DRIVE off. Pull the damping bar
down and release it. Watch the digital readout and observe that the
peak amplitudes decrease with time as expected. Obtain data of
amplitude versus oscillation number for 50 to 100 cycles. Note the
amplitude every fourth or fifth cycle.
5. Measure the natural frequency, ω0 of the system. Set the FUNCTION switch to PERIOD. Pull and release the damping bar. The
digital display should stabilize to a roughly constant reading for
the period of the system. Determine the period for at least five
different masses including no extra mass.
6. Move the magnets as close together as possible without rubbing
the damping bar to get maximum damping. Measure the magnet
spacing and then repeat steps 4 and 5. Since the oscillations may
damp out quickly, the peak amplitude has to be noted down more
often than every fourth or fifth cycle.
7. Determine the resonance curve of the system. Set the amplitude of
the forcing function to 1mm. Any larger and the system will move
out of range at resonance. Adjust the damping magnets for minimum damping and leave no extra masses on the bar. For frequencies from 0.5 Hz to 3.0 Hz in increments of about 0.2 Hz, determine
the steady state amplitude. Set the FUNCTION switch to FREQ
and DRIVE on. Adjust the FREQUENCY control until the digital
display reads the desired driving frequency. Then set the FUNCTION switch to AMPL and measure the peak amplitude. Remember to let the amplitude reading stabilize so that the steady state
solution is measured and not the natural solution. Near resonance
the frequencies should be stepped at smaller intervals. Determine
the frequency that gives maximum response.
8. Perform any one of the following investigations.
(a) Show that all the undriven solutions decay to zero if damping is
present. Show that without an initial displacement or initial velocity the undriven system remains motionless for all time. For
an undriven system that is undamped or underdamped, is it possible to make the mass oscillate at a higher frequency by pulling
harder on the mass to start it moving? Why or why not? Design
and implement an experiment to test this. Present your results.
(b) In the driven underdamped case, the form of the steady state
solution suggests that the steady state amplitude is directly pro12
portional to the driving amplitude. Design and implement an
experiment to test this. Present your results.
(c) Design and implement an experiment to examine how the amplitude at resonance changes with different amounts of damping.
Present your results.
(d) If the damping magnets are removed completely, the DHMA can
roughly simulate an undamped system. Use this fact to design
and implement an experiment to examine beat frequencies. Also
use the fact that the effect of damping on the natural frequency
is minimized when the mass m is large.
(e) Design and implement an experiment that examines how the
damping constant varies with the spacing of the magnets. Present
your results.
(f) Plot the phase difference between the mass bar and the driving
wheel as a function of frequency and compare your results with
the theoretical formula.
(g) It is possible to compute the phase for which maximum power is
delivered to the mass bar. Use the power formula
dx
,
dt
where the driving force, F, is of the form
P=F
(16)
F = F0 cos (ωt + θ ) ,
(17)
and the position of the system, x, has the form
x = A0 cos (ωt) .
(18)
Integrate P over the time of one full oscillation to find the energy
per oscillation with respect to the phase angle θ. Find the value
ωmax for θ which maximizes the energy per oscillation. Near
which frequency would you expect maximum power to be delivered to the mass bar? In the driven underdamped case examine
the formula for the phase angle θ. Which value of ωd gives a
phase angle of θmax ? Is this the resonant frequency? Why or why
not?
(h) Derive the actual equation of the forcing function from the geometry of the drive wheel and cam. Show that it approximates
Equation 5 under the conditions of the experiment.
Error Analysis
The major source of error in this experiment is the deviation of the
mass spring system from the ideal. The largest error is motion of the
13
spring and masses in other directions than the vertical, which cannot
be readily quantified with this apparatus. The masses as found on
the digital balance can be considered to be exact. The frequency and
period measurements made by the DHMA can also be assumed to be
exact. The amplitude measurements are made by the instrument internally and electronically. Since the exact measurement process is not
known and the error not documented a reasonable error cannot be determined. However, the DHMA measurement errors are smaller than
perturbations from nonvertical motions of the mass spring system itself. This leaves the error in measuring the spring extension which
is half the smallest division on the index bar. Since most of the errors are small or unquantified, error analysis is not required for this
experiment.
To be handed in to the laboratory instructor
Prelab
1. List all the different types of solutions to Equation 7. Which ones
can be observed with the DHMA apparatus?
2. Derive the formula for the resonant frequency in the driven underdamped case.
Data Requirements
3. Data table and graph for the spring constant together with an equation for the fitted line and a value for the spring constant k.
4. Data table and two graphs for the damped amplitudes together with
equations for the lines of best fit. Present values for the damping
constant B at the two extreme magnet spacings.
5. Data table and graph of ω02 versus 1/m.
6. Data table and graph of the measured resonance curve. On the
same graph and table plot the theoretical resonance curve using
your measured values for m, k, and B.
7. A value for the resonant frequency found from the resonance curve
in part 6. A value for the theoretical resonant frequency of part 2.
Values of the measured natural frequencies from part 5.
Discussion
8. Discussion of the graph of ω02 versus 1/m. Is this graph linear?
What information does the equation of the best fit line provide?
14
Should this graph be linear?
9. Comparison of the predicted and measured resonance curves.
10. Comparison of the resonant frequency found from the resonance
curve with the theoretical resonant frequency and with the appropriate measured natural frequency. Does the resonant frequency
match the natural frequency? Should the resonant frequency match
the natural frequency?
11. Results and conclusions from the chosen optional investigation.
12. Is the system examined in this experiment a damped harmonic oscillator?
15