Janssen - Investigations of Parametric Excitation in Physical Systems - 2005.pdf

Janssen - Investigations of Parametric Excitation in Physical Systems - 2005.pdf
NAVAL
POSTGRADUATE
SCHOOL
MONTEREY, CALIFORNIA
THESIS
INVESTIGATIONS OF PARAMETRIC EXCITATION IN
PHYSICAL SYSTEMS
by
Michael T. Janssen
June 2005
Thesis Advisor:
Co-Advisor:
Bruce C. Denardo
Thomas Hofler
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Master’s Thesis
4. TITLE AND SUBTITLE: Investigations of Parametric Excitation in 5. FUNDING NUMBERS
Physical Systems
6. AUTHOR(S) Michael T Janssen
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13. ABSTRACT (maximum 200 words)
Parametric excitation can occur when the value of a parameter of an oscillator is modulated at twice the natural
frequency of the oscillator. The response grows exponentially and is only limited by a nonlinearity of the system, so
large response amplitudes typically occur. However, there is no response unless the parametric drive amplitude is
above a threshold value that is dictated by the damping. We investigate parametric excitation in three physical systems.
The first involves an acoustic standing wave in a pipe that is driven by a piston at one end. An analysis shows that
parametric excitation is not feasible in this system unless one uses a very large-excursion piston (for example, from an
aircraft engine). The second system is an inductor-capacitor circuit which can undergo oscillations of the current. An
analysis of capacitance modulation with a bank of alternate rotating and stationary parallel plates shows that parametric
excitation would be very difficult to achieve. Finally, we describe the construction of a torsional oscillator whose length
is modulated. Parametric excitation is successfully demonstrated in this system. A comparison of data to predictions of
the standard theory of parametric excitation reveals significant deviations.
14. SUBJECT TERMS Parametric excitation, torsional oscillator, capacitance modulation, and
length modulation.
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Approved for public release; distribution is unlimited
INVESTIGATIONS OF PARAMETRIC EXCITATION IN PHYSICAL SYSTEMS
Michael T. Janssen
Ensign, United States Navy
B.S., Rensselaer Polytechnic Institute, 2004
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN ENGINEERING ACOUSTICS
from the
NAVAL POSTGRADUATE SCHOOL
June 2005
Author:
Michael T. Janssen
Approved by:
Bruce C. Denardo
Thesis Advisor
Thomas Holfer
Thesis Co-Advisor
Kevin Smith
Chair, Engineering Acoustics Academic Committee
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iv
ABSTRACT
Parametric excitation can occur when the value of a parameter of an
oscillator is modulated at twice the natural frequency of the oscillator.
The
response grows exponentially and is only limited by a nonlinearity of the system,
so large response amplitudes typically occur. However, there is no response
unless the parametric drive amplitude is above a threshold value that is dictated
by the damping. We investigate parametric excitation in three physical systems.
The first involves an acoustic standing wave in a pipe that is driven by a piston at
one end. An analysis shows that parametric excitation is not feasible in this
system unless one uses a very large-excursion piston (for example, from an
aircraft engine). The second system is an inductor-capacitor circuit which can
undergo oscillations of the current. An analysis of capacitance modulation with a
bank of alternate rotating and stationary parallel plates shows that parametric
excitation would be very difficult to achieve. Finally, we describe the construction
of a torsional oscillator whose length is modulated.
Parametric excitation is
successfully demonstrated in this system. A comparison of data to predictions of
the standard theory of parametric excitation reveals significant deviations.
v
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vi
TABLE OF CONTENTS
I.
INTRODUCTION............................................................................................. 1
A.
PARAMETRIC EXCITATION............................................................... 1
B.
MOTIVATIONS AND OBJECTIVES .................................................... 4
II.
PISTON-DRIVEN PIPE ................................................................................... 7
A.
PARAMETRIC DRIVE THRESHOLD .................................................. 7
B.
INITIAL DESIGN CONCEPT.............................................................. 10
C.
REFINED DESIGN CONCEPT .......................................................... 17
III.
CAPACITANCE-MODULATED CIRCUIT..................................................... 23
A.
HISTORY AND MOTIVATION ........................................................... 23
B.
THEORY OF A CAPACITANCE-MODULATED CIRCUIT ................ 24
C.
FEASIBILITY OF A CAPACITANCE-MODULATED CIRCUIT ......... 27
D.
APPARATUS DESIGN ...................................................................... 29
IV.
TORSIONAL OSCILLATOR......................................................................... 37
A.
CONCEPT AND MOTIVATION.......................................................... 37
B.
THEORY ............................................................................................ 39
C.
CONSTRUCTION OF THE APPARATUS ......................................... 43
D.
DEMONSTRATIONS AND EXPERIMENTS ...................................... 48
V.
CONCLUSIONS AND FUTURE WORK ....................................................... 57
A.
CONCLUSIONS ................................................................................. 57
B.
FUTURE WORK................................................................................. 57
LIST OF REFERENCES.......................................................................................... 59
INITIAL DISTRIBUTION LIST ................................................................................. 61
vii
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viii
LIST OF FIGURES
Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Figure 8.
Figure 9.
Figure 10.
Figure 11.
Figure 12.
Figure 13.
Figure 14.
Figure 15.
Figure 16.
Threshold graph. This graph shows the relationship between drive
frequency and the stroke length required to achieve threshold.
Different piston diameters are considered. ......................................... 11
Rendition of initial design concept for parametric acoustic standing
wave generator................................................................................... 12
Conceptual drawing used to demonstrate how the ratio between the
radius and the rod length affect motion. ............................................. 13
Conceptual drawing used to demonstrate stroke length relative to
fly wheel radius................................................................................... 14
Conceptual drawing of slotted fly wheel.
The drawing
demonstrated how having a slotted fly wheel allows for a variable
stroke.................................................................................................. 15
Conceptual drawing of lubrication system. This demonstrates how
a oil reservoir and tapered hole can be used to lubricate the
system. ............................................................................................... 16
Commercial Off The Shelf (COTS) system. This system is more
simple than the initial design concept because only three main
components (the engine, motor and pipe) need to be connected....... 18
Connection of piston cylinder and pipe............................................... 19
Connection between the motor and engine. ....................................... 21
Capacitance-modulated LC circuit. Each plate in the variable
capacitor is shown on the left. In the apparatus, every other plate
rotates, as shown. The capacitance of the system thus periodically
varies from a minimum value of approximately C0 when the plates
do not overlap, to a maximum value of C0 + Cmax when the plates
completely overlap.............................................................................. 25
Variation of the modulated capacitance as a function of time. The
variation is a symmetrical triangular wave with peak amplitude ∆C.... 27
Misalignment of rotating plate leading to contact with stationary
plate. View of capacitor as seen perpendicular to axle...................... 32
Length-modulated torsional oscillator. The motor (top) and linkage
cause the double-roller to vertically oscillate, and to thereby
modulate the length and thus the frequency of the torsional
oscillator. ............................................................................................ 38
Uniform twisting of a strip of material in a torsional oscillator. On
the left is a front perspective view. The bottom segment of the
strip, where the body (not shown) is attached, rises a height h due
to the twisting. On the right is the top view. The bottom segment
rotates an angle θ............................................................................... 41
Length-modulated torsional oscillator apparatus. ............................... 44
Frontal view of “U” support and ribbon clamp..................................... 45
ix
Figure 17.
Figure 18.
Figure 19.
Figure 20.
Figure 21.
Figure 22.
Figure 23.
Figure 24.
Frontal view of drive system. .............................................................. 46
Top view of drive system. ................................................................... 47
Movable masses and controlled release device. ................................ 48
Free-decay data. The natural linear frequency of oscillator is
determined from the zero velocity intercept........................................ 50
Free-decay plot of velocity verses time. The damping parameter β
is calculated from slope. ..................................................................... 51
Velocity amplitude squared verses frequency. The slope is used to
calculate the nonlinear coefficient α. .................................................. 52
Steady-state response. The curves are theoretical and the points
are experimental. The dimensionless parametric drive amplitude η
=0.146 . .............................................................................................. 53
Drive plane plot of η verses f. The curve is theoretical, and the
points are experimental. ..................................................................... 54
x
LIST OF TABLES
Table 1.
Resistance of various inductors.......................................................... 34
xi
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xii
LIST OF SYMBOLS
a
A
Cmax
C0
Cth
∆C
d
D
E
fmod
f0
frot
h
L
Lth
n
N
P
Pr
Q
Qrad
Qvisc
Qtherm
r
R
t
x
radius
amplitude, or area
maximum capacitance
ambient capacitance
threshold capacitance
change in capacitance
distance between plates
plate diameter
energy
frequency of modulation
natural frequency
frequency of rotation
plate thickness
inductance or length
threshold length
number of plates
number of sectors cut out of plate
power
Prandtl number
overall quality factor
quality factor due to radiation losses
quality factor due to viscous losses
quality factor due to thermal losses
plate radius
resistance
time
displacement from equilibrium
xiii
α
β
δtherm
δ visc
nonlinear coefficient
linear damping coefficient
thermal penetration depth
viscous penetration depth
ε0
η
Ι
λ
κ
π
γ
ρ
τ
θ
Τ
ω
electric permittivity constant
shear viscosity
moment of inertia
wavelength
torsional constant
pi, constant 3.1416
ratio of specific heats
density
torque
angular displacement from equilibrium
ramp-up time
angular acceleration or frequency
xiv
ACKNOWLEDGEMENTS
I would be incredible remiss if I did not thank my wife first. She has been
more supportive then I could ever ask, both now and always. She has endured
the busy nights, and the long days. She never complained and always greeted
me with a smile and a kiss when ever I came home, no matter what hour. To you
Tiffany I will always be grateful. To my son Brynan, you taught me that playtime
is very important and not to forget it. To my daughter Katelynne, you showed me
that good things do come in small packages, and to keep it simple.
I would also like to extend my gratitude to Professor Denardo. I know that
at time I was frustrating, but for me it was all a learning process and I appreciate
your patience. I also want to thank you for the classes you taught, it is clear that
you care about your students.
To LT Lindberg, LT Sundem, and ENS Jones, thanks for the much need
stress relief.
Finally, I would like to thank God. Some would say that science and
religion do not go together. I believe in both and I do not think that they can be
separated.
xv
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xvi
I.
INTRODUCTION
The basic theoretical features of parametric excitation and other important
background material is presented in Sec. A. Motivations and objectives of this
thesis are explained in Sec. B.
A.
PARAMETRIC EXCITATION
A linearly-damped oscillator is described by
d2 x
dx
+ 2β
+ ω02 x = 0 ,
2
dt
dt
(1. .1)
where x(t) is the displacement of the mass from equilibrium, β is the damping
parameter, and ω0 is the natural frequency of the oscillator. The behavior of this
oscillator
is
exponentially-damped
simple
x ( t ) = Ae−βt cos ( ω0 t − δ ) ,
harmonic
motion:
(1.A.1)
where the values of the amplitude A and phase δ depend upon initial conditions.
That the amplitude decays exponentially with decay constant β offers a means of
experimentally determining β for an oscillator.
There are three types of driving mechanisms that can excite and maintain
the motion of an oscillator: (i) direct, (ii) maintained, and (iii) parametric. The
mathematically simplest case is (i), where an oscillatory force is exerted on the
mass of the oscillator. An inhomogeneous term is thus added to the equation of
motion (1.A.1). Case (ii) is the most common (for example, voice, whistling, and
woodwind musical instruments), although this case is mathematically the most
difficult.
The drive is typically described as a negative damping term in the
1
equation of motion (1.A.1).
It is the derivation of this term for any specific
physical system that is typically difficult.
Case (iii) is the subject of this thesis. Parametric excitation occurs when
an external agent modulates a parameter upon which the natural frequency of an
oscillator depends.
The standard equation of motion is the linearly-damped
Mathieu equation:
d2 x
dx
+ 2β
+ ω02 ⎡⎣1 + η cos ( ωt ) ⎤⎦ x = 0 ,
2
dt
dt
(1.A.2)
where η is the dimensionless parametric drive amplitude and ω is the parametric
drive frequency.
Note that the parametric drive is represented by a
homogeneous term in the equation of motion (1.A.3).
We will deal with the
principal parametric resonance, which occurs when ω is near 2ω0. It can be
shown that a parametric drive causes an exponential growth of the response
(Denardo and Larraza, 2004), and thus competes with the exponential decay
caused by linear damping. Hence, if η exceeds the parametric drive threshold,
the response grows without limit in the linear case (1.A.3). For weak damping,
weak parametric drive, and ω near 2ω0, the threshold curve due to Eq. (1.A.3) is
η2th =
4 ⎡
2
ω − 2ω0 ) + 4β2 ⎤ ,
2 ⎣(
⎦
ω0
(1.A.3)
which is an hyperbola in the drive plane of η vs. ω (Denardo and Larraza, 2004).
Parametric excitation occurs if η > ηth.
The minimum drive threshold
corresponds to ω = 2ω0 and has the value
( ηth )min
2
=
2
,
Q
(1.A.4)
where the quality factor of the oscillator is
Q =
ω0
.
2β
If excitation occurs, the response grows without bound.
(1.A.5)
Because the
response of any actual oscillator cannot be infinite, it is guaranteed that
nonlinearity must emerge to limit the growth. We add a cubic nonlinearity in the
displacement to Eq. (1.A.3), because this nonlinearity is the simplest, and
because it corresponds to an oscillator with a symmetric potential energy, which
models the case of a length-modulated torsional oscillator (Ch. IV). The equation
of motion is thus
d2 x
dx
+ 2β
+ ω02 ⎡⎣1 + η cos ( ωt ) ⎤⎦ x = αx 3 ,
2
dt
dt
(1.A.6)
where the nonlinear coefficient α can be positive or negative.
The nonlinear coefficient describes how the frequency of undamped
undriven oscillations depends upon amplitude. For β = η = 0 in Eq. (1.A.7), we
set x = Acos(ωt) + {higher harmonics}, where ω is the response frequency, and
where the amplitudes of the higher harmonics are small compared to A for
weakly nonlinear motion. Solving approximately for ω then yields (Denardo and
Larraza, 2004)
ω = ω0 −
3α 2
A .
8ω0
(1.A.7)
The oscillations soften for α >0 and harden for α < 0. For the purposes of
dealing with damped driven response curves of steady-state motion, it is
convenient to view Eq. (1.A.8) as yielding the backbone curve of A vs. ω. The
natural linear frequency ω0 can be experimentally determined as the abscissa (A
3
= 0) intercept of the backbone curve. That A2 vs. ω is a straight line with slope =
8ω0/(3α) then offers a means of experimentally determining the nonlinear
coefficient α.
For weak damping, weak drive, and weak nonlinearity, the steady-state
response solution of the equation of motion (1.A.8) is (Denardo and Larraza,
2004)
A2 =
)
(
4ω0
2ω0 − ω ± 2 η2ω02 − 16β2 .
3α
(1.A.8)
The upper branch (+√ for α > 0 and –√ for α < 0) is stable, and the lower branch
is unstable.
B.
MOTIVATIONS AND OBJECTIVES
The original goal of this thesis research was to build an acoustical
resonator to observe parametric excitation of a mode.
The previous thesis
research of Varnadore (2001) and Smith (2003) suggested that the minimum
threshold value (1.A.5) could be readily exceeded by a piston-driven pipe. Such
an experiment would be important because it could lead to the use of parametric
drives in acoustic compressors, pumps, and refrigerators, which require highamplitude standing waves in order to operate. These devices are of commercial
and naval interest because they have very few moving parts and are
environmentally friendly.
In Ch. II, we present a detailed feasibility analysis of acoustical parametric
excitation by length-modulation of a piston-driven pipe. Our conclusion is that
the construction of such an apparatus that would exceed the minimum threshold
value (1.A.5) presents a serious engineering challenge involving high-speed
long-stroke-length pistons of large area.
experiment to possible future research.
4
We thus decided to relegate the
We next examined the feasibility of an inductor-capacitor (LC) circuit in
which the capacitance is modulated. An LC circuit oscillates analogously to a
mechanical oscillator. Because the frequency depends upon the capacitance, a
sufficiently large modulation of the capacitance at twice the natural frequency of
the circuit should lead to parametric excitation. Our motivation to build such an
apparatus was two-fold. First, the apparatus would be interesting and useful as a
demonstration of parametric excitation. The demonstration would be performed
in various classes at NPS and elsewhere, and would be especially important for
the NPS course PH4459 (Nonlinear Oscillations and Waves). A textbook for this
course is currently being written by NPS Profs. Denardo and Larraza (2004).
Second, as we explain in Ch. III, there is in the literature only one report of the
achievement of parametric excitation by capacitance-modulation of an LC circuit,
and this report substantially fails to give sufficient information so that the results
can be reproduced.
The importance of the experiment is evidenced by its
mention in books on nonlinear oscillations. Our second objective was thus to
properly perform and report an experiment.
However, a detailed feasibility
analysis in Ch. III shows that the construction of such an apparatus would again
present a serious engineering challenge.
We then turned our attention to a simpler parametric excitation apparatus,
although one that does not appear to have been reported in the literature. This
system is a torsional oscillator in which a double-roller assembly modulates the
length of a twisting strip, and thus modulates the natural frequency of the
oscillator.
One motivation for constructing this oscillator is its use as a
demonstration, as explained above in the case of the LC circuit.
Another
motivation, as we report in Ch. IV, is that the system is interesting as a case that
is well beyond the standard perturbative theory that is used to describe
parametric excitation. As described in Ch. IV, we successfully built and tested an
apparatus.
5
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6
II.
PISTON-DRIVEN PIPE
In this chapter, we discuss the theory and feasibility of parametric
excitation of an open-ended piston-driven pipe apparatus. In the design, we
must ensure that the stroke of the piston drive exceeds the parametric threshold
condition (I.A.5), which is ∆Lth/L = 1/Q, where ∆L is the peak displacement
amplitude of the length of the pipe, L is the ambient length, and Q is the quality
factor of the acoustic mode. The quality factor is a result of viscous, thermal, and
radiation losses. We first calculate the individual quality factors, from which we
determine the overall quality factor and the parametric drive threshold. We then
present parametric threshold curves that allow us to choose parameters that
optimize the performance of the apparatus. Next, we discuss the feasibility of
constructing an apparatus. Our conclusion is that this is possible but extremely
difficult.
A.
PARAMETRIC DRIVE THRESHOLD
The viscous penetration depth for oscillatory flow near a boundary is
(Kinsler et al., 2000)
δ visc =
2η
,
ρω
(2.A.1)
where ρ is the ambient density, η is the shear viscosity, and ω is the angular
frequency of the mode. For standing waves in a pipe of radius a, where δvisc <<
a, the quality factor due to viscous losses is Qvisc = a/δvisc, so
Qvisc = a
ρω
.
2η
The thermal penetration depth is (Kinsler et al., 2000)
7
(2.A.2)
2κ
,
c pρω
δtherm =
(2.A.3)
where κ is the thermal conductivity and cp is the specific heat at constant
pressure. The quality factor due to viscous losses is Qtherm = a/[δtherm(γ – 1)],
where γ is the ratio of specific heats cp/cv. Furthermore, the thermal and viscous
penetration depths are related by δvisc/δtherm = (Pr)1/2, where Pr is the Prandtl
number. We thus find that the quality factor due to thermal losses is
Q therm = a
ρω Pr
.
2η γ − 1
(2.A.4)
When the wavelength of the radiation from an open end of a pipe is much
larger than the radius of the pipe, the quality factor due to the radiation is (Kinsler
et al., 2000)
Lλ
,
A
Qrad =
(2.A.5)
where L is the length of the pipe, λ is the wavelength of the acoustic field, and A
is the cross-sectional area of the pipe. Substituting λ = c/f = 2πc/ω and A = πa2
into Eq. (2.A.5) yields
Qrad =
2cL
.
a2 ω
(2.A.6)
As long as a quality factor is not small (specifically, of the order of unity or
less), the quality factor is given by 2π times the energy E of the mode divided by
the energy loss ∆E per cycle:
Q = 2π
8
E
.
∆E
(2.A.7)
If more than one source of dissipation is present, the energy loss is the sum of
the individual energy losses. The quality factors thus add reciprocally, so the
overall quality factor Q in our case is given by
1
1
1
1
=
+
+
.
Q
Q visc
Q therm
Qrad
(2.A.8)
The parametric threshold condition ∆Lth/L = 1/Q is thus
∆L th
1
1
1
=
+
+
.
L
Qvisc
Qtherm
Qrad
(2.A.9)
Substitution of the expressions (2.A.2), (2.A.4), and (2.A.6) for the individual
quality factors yields
∆L th =
L 2η ⎛
γ − 1⎞
a2ω
1
+
+
.
⎟
a ρω ⎜⎝
2c
Pr ⎠
(2.A.10)
We utilize the following convenient parameters. The diameter is twice the
radius: D = 2a. The threshold stroke of the piston is twice the peak threshold
displacement: Sth = 2∆Lth. The parametric drive frequency is twice the natural
frequency of the mode: fd = 2fn, where the natural frequency of the nth mode is
approximately fn = (2n – 1)c/4L. These frequency values can be improved by
including the effective length due to the opening, and the adjusted speed of
sound due to the thermoviscous boundary layer (Kinsler et al., 2000).
The
approximate effective length for an unflanged opening is L + 0.6a. The adjusted
speed of sound is
δ + δ therm ⎞
⎛
c adj = c ⎜ 1 − visc
⎟ .
2a
⎝
⎠
(2.A.11)
However, because δvisc << a and δtherm << a in our case, we take cadj = c. The
natural frequency of the nth mode is thus
9
fn =
where n = 1, 2, 3, ….
( 2n − 1) c
4 (L + 0.6a )
.
(2.A.12)
The parametric drive frequency is twice the modal
frequency:
fd =
( 2n − 1) c
2 (L + 0.3D )
where we have also substituted D = 2a.
,
(2.A.13)
The threshold condition (2.A.10)
becomes
Sth
πD2 fd
4L 2η ⎛
γ − 1⎞
=
1 +
⎟ + 4c ,
D πρfd ⎜⎝
Pr ⎠
(2.A.14)
where the drive frequency fd is given by Eq. (2.A.13). We will deal only with the
fundamental (n = 1) mode, so the drive frequency is
fd =
c
.
2 (L + 0.3D )
(2.A.15)
For the feasibility analysis (next section), we neglect the relatively small end
correction term 0.3D in the drive frequency expression (2.A.15), so fd = c/2L. The
threshold stroke length thus becomes
S th
B.
8L ηL
=
D πρc
γ − 1⎞
πD2
⎛
⎜1 +
⎟ + 8L .
Pr ⎠
⎝
(2.A.16)
INITIAL DESIGN CONCEPT
From the theory we can look at the factors over which we have control
and then optimize our system design based on our
10
choices. The factors are stroke length and piston size. Figure (1) shows the
needed stroke length to achieve threshold at different frequencies for different
piston sizes.
Drive Frequency (RPM)
-12
Diameter
of Pipe
1 in
2 in
10
Stroke Length (in)
33330 16665 11110 8333 6666 5555 4761 4166 3703 3333
12
8
10
8
3 in
4 in
6
6
4
4
2
2
0
0
1
2
3
4
5
6
7
8
9
0
10
Pipe Length (ft)
Figure 1. Threshold graph. This graph shows the relationship between
drive frequency and the stroke length required to achieve threshold.
Different piston diameters are considered.
It is clear from the graph that the higher the drive frequency the lower the
required drive amplitude is. For all practical purposes, most pistons will only
operate at less than 6000 RPM. On the other hand if the size of the piston is
increase, the required drive amplitude is decreased. To be well above threshold
and to be at a low enough RPM, would require a very large piston. Though this
is not entirely impossible, for the scope of this thesis it is impractical. However,
11
previous work had lead me to believe that this system might work, so I
proceeded with the design process.
motor
fly wheel
Figure 2.
piston rod
piston
pipe
piston cylinder
Rendition of initial design concept for parametric acoustic
standing wave generator.
Figure (2) is a rendition of my design of the system. It consists of four
main parts: a pipe, piston/rod combination, fly wheel, and motor. The pipe will
have two parts. The longest part is to be made out of PVC; this is due to the
lightness, inexpensiveness, and ease of machining.
The second part is the
section that the piston travels in. This part will have to be made out of some type
of metal that transfers heat. This is important because heating could change the
flow within the pipe and thus change the acoustic properties of the pipe.
The piston/rod combination would have to be made or ordered.
Regardless of which is done the metal portion of the pipe will most likely have to
be bored-out in order to ensure an exact fit. Even then a small gap must be
introduced between the pipe and the piston. This is done so that only the piston
rings rub against the walls of the pipe. This must be a highly precise fit, for two
reasons. One is that the piston will rub on the pipe which will introduce friction
12
that could lead to excess heat or binding and then failure.
This could be
dangerous as the motor would still try to turn and something could break and fly
off. The second possibility is that there would not be a good enough seal and air
would leak past. This would mean that the piston would not be pushing all of the
air and would have less of an effect if any. The piston rod, if purchased, might
have to be lengthened. This would be necessary if a longer stroke length was
needed than what the piston and rod was designed for. The reason for this is
that in order to have simple harmonic motion for a piston, the distance from the
center of the crank to the place that the rod meets the fly wheel must be much
smaller than the length of the rod. See Figure (3). Lengthening the rod or using
a different stroke length than originally designed for can introduce problems that
will be discussed later.
rod length
Figure 3.
Conceptual drawing used to demonstrate how the ratio
between the radius and the rod length affect motion.
13
The flywheel is a key element of the system. Where it connects to the
piston rod determines the length of the stroke. From Figure (4) it can be seen
that the stroke length is twice the distance from the center of the crank to where
the flywheel and the piston rod connect.
stroke length
stroke length
Figure 4.
Conceptual drawing used to demonstrate stroke length
relative to fly wheel radius.
The other important function performed by the flywheel is to balance the
system. Since the piston rod is only on one side of the flywheel the force that it
creates is unbalance with respect to the crank. In order to balance this system a
weight must be added to the other end of the flywheel. One added benefit to
having a system like this is that the stroke length can be made variable. This can
be accomplished by either drilling holes in the in the flywheel so that the rod can
14
be moved from point to point or a slot could be cut out so that the rod could be
loosened and slid to a desired point and then tightened in place. See Figure (5).
.
Figure 5. Conceptual drawing of slotted fly wheel. The drawing
demonstrated how having a slotted fly wheel allows for a variable
stroke.
The last component of the system is the motor. This is probably the most
open-ended part in the system. This is due to the fact that we must meet all of
the other system requirements before we can select the appropriate motor to
use. The kind of motor I have in mind for this project is an AC variable speed
motor with shaft encoder. The shaft encoder will allow me to know at exactly
what RPM the shaft is turning and thus the drive frequency.
This is very
important because I will be varying the speed in order to find the exact frequency
where parametric excitation occurs. The power of the motor is important as well,
however, I must know the specifications of the rest of the system before I can
make an estimate of the horsepower needed.
A final part of the system that must be considered is how the piston will be
lubricated. The standard way in which pistons are lubricated is by oil. In our
system this could be problematic. Since the oil will be all over the piston and
15
cylinder, small particles of the oil will also be in the air which could change the
localized acoustic properties of the air. Another method that might be used is to
replace the metal piston rings with Teflon piston rings. This would allow for no
liquid particles to be introduce to the air, however, at the speed the system will be
operating the Teflon will not hold up. So in this particular situation we must
accept the small influence the oil might present. Since this is not an off-the-shelf
system the oil lubrication system must be designed. The most basic method of
achieving this is by developing some type of drip system. The easiest way to do
this is by creating an oil pan above the piston travel region and drilling a small
connecting hole. The size of the hole will determine the flow rate of the oil. Also
note that the size of the hole must be bigger than the width of the piston rings.
This is because if there is a difference in pressure between one side of the rings
and the other side, and the hole is around the size of the piston ring, then as the
ring passes over the hole suction will be introduced. See Figure (6).
oil reservoir
tapered hole
piston rod
piston
piston ring
pipe wall
Figure 6. Conceptual drawing of lubrication system. This demonstrates
how a oil reservoir and tapered hole can be used to lubricate the
system.
16
Although this system may seem very simplistic, there are several
considerations that must be made in order to ensure that it will be safe. One very
important consideration is whether it is possible to design a piston and piston rod.
This is very complicated and unless it is operating at very low speeds it can
cause extreme damage should it break. This is something that would have to be
done by a professional with experience in piston design and manufacturing.
Another possibility is to purchase a piston and rod that were designed for
something else but meet the needed of this project. I have found however, that
most pistons do not have a long enough stroke. As mention previously it is
possible to extend the length of the piston rod, however, this would have to be
done very carefully. If it were misaligned, or the joint not strong enough, or
imbalanced, the assembly could fail at high speeds. Another safety precaution
that would be needed is some kind of protective box that would have to be built
around the piston and motor. This would be to prevent material from flying off if
the system did fail.
C.
REFINED DESIGN CONCEPT
Due to the problems of the before mentioned system it was decided that a
commercial off-the-shelf engine would be used in place of the designed piston
assembly. When this portion of the design process was under taken, it was
perceived that a 2 inch piston with a 2 inch stroke would be optimal.
An
investigation of what type of engines had these parameters was begun. To keep
this system simple, an additional requirement that the engine be single-piston
was added. This was done to reduce the weight of the engine and to decrease
the size of motor needed to turn the engine. By doing this, automobile and
motorcycle engines were excluded from the possibilities. It was found that hobby
engines, used for remote control airplanes and boats were too small. Most only
had a maximum diameter of 1.5 inches and a similar size stroke. This reduced
the selection of engines to those used for utility purposes. The most common
and readily thought of was the standard lawn mower engine. These engines
could exceed over 2 inches in piston diameter and stroke length, as well as meet
the need for only one piston. For this reason it was decided that a lawn mower
17
engine would be used for the experiment. See Figure (7) for a rendition of the
system.
Detailed labeling of each section will follow in other figures.
engine body
engine cylinder
and piston
pipe
table
motor
Figure 7. Commercial Off The Shelf (COTS) system. This system is
more simple than the initial design concept because only three main
components (the engine, motor and pipe) need to be connected.
Although the system may appear to be more complicated than the system
from the previous section, it is simpler to construct and operate.
Since the
engine is preassembled very little needs to be modified in order to use it for the
experiment. The following is a description of the function of each component and
what modifications if any need to be performed in order to make the system
operable.
After having examined several lawn mower engines and selecting the one
with the largest piston size and stroke, the process of figuring out how to attach
the pipe to the piston cylinder was the next line of action. Upon removing all
extraneous parts from the engine (pull cord, head, blade, carburetor and so on),
18
it was determined that it would be necessary to disable the intake and exhaust
valves. This can be accomplished by simply removing the internal gearing that
make the valves move. Once this is done a template of the piston head can be
made so that an adapter can be machined to connect the piston cylinder and the
pipe. Figure (8) shows how the three components connect. The adaptor would
most likely be made from plastic so that it can easily be glued to the end of the
PVC pipe.
The adaptor would then be connected to the head of the piston
cylinder using the preexisting bolt holes.
piston cylinder
adaptor
pipe
bolts
Figure 8.
Connection of piston cylinder and pipe.
It is important to note that any jagged edge could cause turbulence in the
air. The main location where an edge could disrupt the air flow would be at the
junction of the adaptor and pipe, or the adaptor and the piston cylinder. Since
the pipe and adaptor are both PVC and will be glued together, they can be bored
out at the place they meet so that the transition area is smooth. The place where
adaptor and piston cylinder meet is much harder to smooth. The best way to
19
approach the problem would be to machine the adaptor as precisely as possible.
It might also be possible to take the engine apart and to fill any gaps between the
piston cylinder and adaptor by hand with filling agent and then sand it smooth by
hand. However, this could possible introduce more problems than it would fix.
Another consideration is air leaks which could reduce the efficiency of the
system. The piston rings ensure that no air will leak pass the piston. The joint
between the adaptor and pipe will be sealed with glue, which will prevent any air
from escaping. The only other place to consider is the joint between the adaptor
and the piston cylinder. Since the joint only exist due to compression from bolts,
it is possible that a gap could exist and thus air leakage could occur. To prevent
this, a paper or rubber gasket needs to be placed between the adaptor and
piston cylinder.
Since the engine is off-the-self and the pipe has already been connected,
all that is left to consider is how to attach the engine to the table and connect the
motor drive shaft with that of the engine. Figure (9) illustrates how this can be
done. Connecting the engine to the table is a fairly simple process. The engine
already has mounting holes so once an appropriately sized hole has been drilled
into the table the engine can simply be bolted to it. A rubber gasket might be
placed between the engine and the table to damp vibration. Before the motor
and the engine shafts can be connected, the motor must first be mounted to the
table in a secure fashion. Unfortunately until the appropriate motor is selected
the means of mounting is unknown. A shaft coupler can be used to connect the
motor and engine shafts. These devices are common and can be found easily.
The main function of the shaft coupler is to join to shafts together, but allowing
them to be slightly misaligned. This is important because two shafts cannot be
perfectly aligned. The coupler allows the shafts to wobble while not introducing
any lag in the direction of rotation.
20
engine
bolts
shaft coupler
motor
Figure 9.
Connection between the motor and engine.
Finally, some comments should be made about motor selection and
safety. There are many types and brands of motors. What is important is that
the motor be high quality and easy to control. In this particular situation an ac
motor is most likely the better choice over a dc motor. The reason for this is size
and price. Ac motors tend to be about half the size and cheaper than dc motors.
Also, at the operating values of the RPM there is very little difference in
performance. Since a fairly large engine will be used, the motor will most likely
have to operate on three-phase power.
This makes the experiment difficult
because that kind of power is not available in most places. For safety reasons
the engine and motor will also have to be encased in a protective box. This is
absolutely necessary to protect against parts of the system that might fly off
should the system fail.
21
THIS PAGE INTENTIONALLY LEFT BLANK
22
III.
CAPACITANCE-MODULATED CIRCUIT
In this chapter, we consider parametric excitation of an inductor-capacitor
(LC)
A.
circuit
in
which
the
capacitance
is
modulated.
HISTORY AND MOTIVATION
In 1935 a Russian paper was published by L. Mandelstam, N. Papalexi, A.
Andronov, S. Chaikin and A. Witt. The title of the paper was “Report on Recent
Research on Nonlinear Oscillations”. In this paper, the authors discuss many
types of nonlinear oscillations. One chapter is dedicated to parametric excitation
of nonlinear circuits. I came across references to this paper two different times.
Once was when professor Denardo mentioned that one of his books referenced
it.
The other time was when I was doing a Google search for parametric
excitation. The website, http://www.cheniere.org/misc/moscowuniv.htm (29 APR
05) seemed unreliable due to its content, which mostly consisted of claims for
free energy without proof. I nevertheless decided to investigate the reference. I
found a few more references to the paper on-line but no directions where to find
it. After consulting with NPS librarian Michaele Huygen and searching many
online resources of the library, I found a copy of the paper in English. The paper
had been translated from Russian to French to English. The copy that I had
found was produced by NASA Technical Translation.
The paper’s discussion of parametric excitation is limited mostly to theory
and a general description of the experimental apparatus that the authors
constructed.
There is also a brief introduction of the history of parametric
excitation of electric oscillations.
They state, “although the possibility of
parametric excitation of electric oscillations has been known for a long time
(Rayleigh, Poincare, Brillouin, and later van der Pol), it is only in the last few
years that this phenomenon was realized for its full value and its systematic
study was undertaken”.
I proceeded to investigate all of the valid references
23
from the paper but failed to find any of the papers in English. After thoroughly
reading the chapter on parametric excitation I found that the authors had not
properly documented the design, construction, or experimentation of the
apparatus they had built. This makes it very difficult to reproduce their work to
verify their findings. It turns out that several books have referenced this paper
but no one has attempted to reproduce the experiment to verify the findings. For
this reason it is very important that this experiment is attempted again so that it is
well documented for future reference. Another motivation for the construction of
a parametrically excited LC circuit is use it as a demonstration in various physics
courses.
B.
THEORY OF A CAPACITANCE-MODULATED CIRCUIT
To modulate the capacitance of an LC circuit, we consider a bank of n
parallel sectored plates, where every other plate is electrically connected, and
where one set of plates is rotated (Fig. 10). This configuration amounts to n – 1
identical
variable
capacitors
in
parallel
24
(n
=
10
in
Fig.
10).
d
A
D
frot
C(t)
C0
L
R
Figure 10. Capacitance-modulated LC circuit. Each plate in the variable
capacitor is shown on the left. In the apparatus, every other plate
rotates, as shown. The capacitance of the system thus periodically
varies from a minimum value of approximately C0 when the plates do
not overlap, to a maximum value of C0 + Cmax when the plates
completely overlap.
The capacitance of the circuit varies from approximately C0 for
nonoverlapping plates, to C0 + Cmax for completely overlapping plates. Because
the parallel-plate capacitance is small, we will find that we must choose C0 to be
substantially larger in order to avoid a prohibitively large rotational frequency.
Hence, Cmax << C0, so the average capacitance is C0 + Cmax/2 ≈ C0, and the
natural frequency of the circuit is
25
f0 =
1
2π LC0
.
(3.B.1)
Because high rotational frequencies are required, it is important to up-shift
the modulation frequency compared to the rotational frequency by arranging the
plates to have a number of alternate sectors, as shown in the left diagram in Fig.
10. This effect is utilized in optical choppers. If there are N sectors cut out of
each plate with N remaining sectors, where all sectors are approximately the
same (N = 4 in Fig. 10), then the capacitance is modulated N times for every
rotation, so the relationship between the rotation and modulation frequencies is
frot = fmod/N. For optimum parametric excitation, the modulation frequency should
be twice the natural frequency (3.B.1) of the circuit. Hence, frot = 2f0/N or
frot =
1
Nπ LC0
.
(3.B.2)
We now determine the maximum value Cmax of the variable capacitance.
We assume that the plates of the variable capacitor are closely spaced a uniform
distance d. The maximum area of overlap of a pair of plates is approximately the
area of a sectored plate:
A = πD2/8, where D is the plate diameter.
The
maximum capacitance of a neighboring pair of plates is C1 = ε0A/d, and there are
n – 1 of these capacitances in parallel. The maximum capacitance amplitude is
then Cmax = (n – 1)C1 or
Cmax =
(n − 1) πε0D2 .
8d
(3.B.3)
The peak parametric drive amplitude is ∆C = Cmax/2, and the quality factor is Q =
(L/C0)1/2/R, where L is the inductance and R is the resistance of the inductor.
The standard parametric drive amplitude threshold condition ∆C/C0 > 2/Q here is
not correct because the modulation of the capacitance is triangular rather than
sinusoidal in time (Fig. 11). We can correct for this by using the Fourier series
26
coefficient 8/π2 of the fundamental frequency of a symmetric triangular wave of
unit peak amplitude. The effective amplitude is then (8/π2)∆C. The parametric
drive amplitude threshold condition is thus ∆C/C0 > π2/(4Q), which can be
expressed as Cmax > Cth, where the threshold capacitance amplitude is Cth =
π2C0/(2Q) or
Cth
π2RC0
=
2
C0
.
L
(3.B.4)
Parametric excitation will occur if the modulation frequency fmod is
approximately twice the natural frequency (3.B.1), which occurs for the rotational
frequency (3.B.2), and if the drive amplitude threshold condition Cmax > Cth is met
for the maximum capacitance (3.B.3) and the threshold capacitance amplitude
(3.B.4).
variable
capacitance
C
Cmax
∆C = Cmax/2
Cmax/2
0
1/fmod
time t
Figure 11. Variation of the modulated capacitance as a function of time.
The variation is a symmetrical triangular wave with peak amplitude
∆C.
C.
FEASIBILITY OF A CAPACITANCE-MODULATED CIRCUIT
To have a rotational frequency (3.B.2) that is not large, we consider a
large value of the inductance: L = 1 H. A large fixed capacitance C0 also lowers
the requisite rotational frequency, but causes a diminished dimensionless drive
27
amplitude ∆C/C0. Trial-and-error has lead to the following value: C0 = 1 µF = 106
F. The rotational frequency (3.B.2) for N = 10 sectors of removed material is
then
frot =
1
Nπ LC0
≈
1
10π 1× 10−6
≈ 32 Hz ≈ 2000 rpm .
(3.C.1)
This frequency is not inordinately difficult to obtain for a rotational apparatus,
even for one that is used in a lecture demonstration. Note that the modulation
frequency is 10 times the rotational frequency. Because the rotational frequency
can be directly measured, the modulation frequency is then conveniently 10
times this value.
We desire to make the capacitance amplitude Cmax as large as
conveniently possible in order to maximize the parametric drive amplitude.
Reasonable values of the plate diameter, number of plates, and plate separation
distance for a lecture demonstration apparatus are D = 8 in ≈ 0.20 m, n = 100,
and d = 0.5 mm = 5 x 10–4 m. The maximum capacitance (3.B.3) is then roughly
Cmax =
(n − 1) πε0D2
8d
(3.C.2)
≈
99π × 8.85 × 10 −12 × ( 0.20 )
2
8 × 5 × 10−4
≈ 2.8 × 10 −8 F = 28 nF .
To determine the threshold value (3.B.4) of the capacitance amplitude for
parametric excitation, we assume an inductor resistance of R ≈ 2 Ω:
Cth
π2RC0
=
2
C0
L
(3.C.3)
π2 × 2 × 10−6 10 −6
=
≈ 9.9 × 10 −9 ≈ 10 nF .
2
1
28
The maximum capacitance (3.B.6) is a factor of 3 greater than the threshold
capacitance amplitude (3.B.7). We conclude that this capacitance-modulated LC
circuit is feasible although somewhat difficult to construct as a lecture
demonstration.
There are several possible means of increasing Cmax or decreasing Cth, so
that the parametric threshold condition Cmax > Cth might be enhanced:
•
Cool the inductor with liquid nitrogen, which would lower the
resistance R of the inductor, and would thus lower Cth. Although
requiring some effort, use of liquid nitrogen in lecture
demonstrations is always popular with an audience.
•
Decrease the fixed capacitance C0, which would decrease Cth.
However, note that a greater rotational frequency would be
required. This problem could be alleviated by adding inductors in
series in order to increase the inductance and thus lower the
frequency. It should be noted, however, that Cth is proportional to
L/R1/2, so two inductors in series raises Cth by 21/2.
•
Enclose the capacitor in a container filled with a gas of high
dielectric constant.
•
Have a bank of variable capacitors connected in parallel, which
would increase Cmax. Note that the rotations would all have to be
ganged so that they are all in phase.
•
D.
APPARATUS DESIGN
To better assess the feasibility of building this system a more detailed
consideration of the design, manufacturing, assembly, and operation of the
system is required.
There are three main components to the system: the
capacitor, inductor, and motor. Since the inductor and motor will be obtained
commercially there is very little to consider about them other than the
specification they must meet.
The capacitor on the other hand has several
issues that must be addressed.
The most basic component of the capacitor are the plates. From a design
standpoint it may seem a simple object to create. However, the design of the
plates is more complicated then it seems. The plates will be undergoing stresses
and strains as they spin. This must be accounted for in the design by using fillets
29
instead of sharp corners. The width of sectors and how far they are from the axle
are of major importance as well. If the narrowest part of a sector is too small
then that piece might fail and separate causing damage. If the gaps between the
sectors extend too far, making the outer rim small, it could have a similar effect.
These factors will be affected by what material the plates are made of. Stronger
materials allow the plates to be made thinner, but at the same time they make
the plate heavier which affects the size of motor needed.
The manufacturing of the plates must also be considered. Selecting the
supplier of the raw material is important. In order to have high quality sheets of
the material it is often wise to require mil-spec material when purchasing. The
reason it is necessary to have sheets of such high quality is that flaws would
create problems later. For example, if the sheet had a slight bow to it when it
was being cut the plate would then be bowed as well. A bowed plate, especially
with the plate separations used in this system, could rub against the plate
adjacent it. At a minimum this would cause a short circuit, at worst it could cause
the failure and damage. Other quality issues are density, and uniform thickness,
both of which would affect the balance of a plate and thus the balance of the
system.
Once a supplier is selected the plates must be cut by a machinist. There
available are several methods of cutting plates. These include laser, water jet,
chemical milling and milling. Milling is inexpensive but less accurate than the
other two. Laser cutting, although highly accurate, can warp the plates due to
heat produced during cutting.
Chemical milling uses a photolithography and
chemical etching technique which does not heat the metal. However, it is a
slower or more expensive process. The water jet method has accuracy close to
that of a laser but will not warp the plates. For this reason it is the best method
for our case.
After manufacturing, the next step is assembly. Some of the difficulties in
assembling the parallel plate capacitor are spacing, alignment, and balance. No
matter how the plates are attached to the axle, each plate must be placed
30
individually. This will be a very time consuming process. There are three main
ways to attach the plates to the axle. One is to heat the plate, allowing it to
thermally expand and thus increase the diameter of the inner hole. Once the
hole is sufficiently large the plate can then be slid down the axle to its position. It
must then rest there until it has cooled enough to be a tight compression fit. This
method is convenient because it does not involve any adhesives.
The
disadvantages are that the heating must be uniform, the attachment is
permanent, and the heat and then moving of the plate can introduce bending.
The second method involves using metal conductive adhesive. This has the
advantage that it can be undone. The disadvantages are that each plate must
dry before the next plate can be placed, adhesives can be difficult to work with,
and the plate separation distances used in this system are on the same
magnitude as a drop of adhesive which make unwanted connections hard to
prevent. The third method of connection is to make the axle have two flat sides
and to have a similar shape for the plates inside hole. The plates are then slid
onto the axle. The flat sides prevent the plates from spinning relative to the axle.
Placing a washer between each plate would guarantee uniform spacing. To
keep the plates from sliding along the axle a nut would be placed on the end of
the axial at the location of the last plate and tightened. This would create a
compression fit. This method is the best choice for this system.
When the plates are being placed on the axle, it is important that they be
aligned as closely to perpendicular as possible.
The reason for this is
demonstrated in Fig. 12. A small misalignment could introduce contact between
plates. With the small separation distance that is being using, the plates would
rub if they are misaligned by as little as 3 degrees.
31
stationary
plates
supports
rotating
plate
axial
Figure 12. Misalignment of rotating plate leading to contact with
stationary plate. View of capacitor as seen perpendicular to axle.
Balance of the capacitor is also important. If the capacitor is not balanced
properly then it will wobble.
This could lead to mechanical failure.
Static
balancing would not be sufficient. Dynamic balancing would be necessary due to
the speeds the capacitor would be operating at. It is not clear where to ship the
assembly to have the balancing done. Imbalance also introduces problems if the
capacitor had a plate that was damaged during operation. Since the plate would
have to be replaced the capacitor would have to be dynamically balanced again.
Once the parallel plate capacitor is assembled, the next line of action is to
mount it to a supporting structure. There are two main methods of supporting the
capacitor. One method is to mount it with the axle horizontal to the ground. This
method allows the capacitor to be supported by two ball bearings, one on each
end of the axial. The bearings are secured to brackets that are secured to the
table. This provides for a very strong supporting structure. However, it also
leads to a potential problem. Since the weight of the plates is resting on the axle,
32
bowing of the axle is introduced. This will cause the axle to wobble, possible
causing the rotating plates to hit the stationary plates. The way to avoid this
problem is by implementing the second method of mounting. This is achieved by
mounting the axle vertically. There is a problem associated with this method as
well. It is difficult to mount the axle accurately vertically. This could cause the
capacitor to naturally lean. The solution to this problem is to build a large support
structure around the capacitor. This would reduce the visibility of the capacitor,
diminishing its effect as a demonstration. However, it is the better of the two
mounting methods.
A final consideration for a cause of mechanical failure is turbulence. As
the plates spin past each other the air will be disturbed and the resulting effects
on the plates are unknown. The effect could cause the plates to flex and thus
cause mechanical failure of the capacitor. Once the capacitor is mounted and
the system is connected the next obvious consideration to investigate is
operational safety. Due to all of the causes for mechanical failure previously
mentioned, it would be prudent to encase the capacitor in a clear polycarbonate
safety box similar to the one described for the acoustical apparatus.
Although for this project it not necessary to build or design an inductor, it is
important to determine some realistic values of inductance and resistance. For
this purpose I gathered a variety of inductors from the physics department. A
summery of my findings is listed in table (1). From the table it is obvious that the
initial assumption of a 1 H inductor with 2 Ω resistances is unrealistic. Changing
the feasibility calculations to reflect this reduces the quality factor and thus
increases the required drive amplitude.
33
Table 1.
Resistance of various inductors.
inductance
(at 100 Hz)
Coil inductor
Powerstat inductor
Variac inductor
Toroidial inductor
621
Toroidial inductor
601
Toroidial inductor
616
ac resistance
(at 100 Hz)
dc resistance
321 mH
964 mH
740 mH
118 ohms
188 ohms
195 ohms
6.09 ohms
2.18 ohms
1.51 ohms
2.84 H
43.0 ohms
17.5 ohms
2.95 H
79.4 ohms
18.4 ohms
2.61 H
58.9 ohms
18.3 ohms
The final component to consider is the motor. In order to determine what
motor to use the power required to rotate the capacitor is needed. In calculating
the power it is assumed that the plates are solid, homogenous, and identical.
Friction is also neglected. First the mass of an individual plate is found. A
density of 2700 kg/m3 is used for aluminum. The radius and thickness are r =
4
in = 0.1016 m and h = 1/8 in = 0.003175 m, respectively. The mass is thus
ρπr 2h = 2700 × π × (0.1016)2 × 0.003175 = 0.325 kg .
(3.D.1)
Once the mass is calculated the next step is to find the rotational inertia
due to the n = 50 plates that will be rotating. The moment of inertia is
Ι=
n × mass × r 2 50 × 0.325 × 0.10162
=
= 0.826 kg × m2 .
2
2
Next the torque must be calculated.
(3.D.2)
In order to do this constant angular
acceleration is assumed. The angular velocity ω is assumed to be the following:
ω=
2 × π × RPM 2 × π × 2000
rad
=
= 209.4
.
60
60
s
34
(3.D.3)
It is also assumed that the ramp-up time Τ is 15 seconds. The torque is thus
Ι × ω 0.826 × 209.4
kg × m2
τ=
=
= 11.536
.
Τ
15
s2
(3.D.4)
Now that we have the torque the power can be calculated:
P = τ × ω = 11.536 × 209.4 ≅ 2416 W.
(3.D.5)
It is wise to convert the power into units of horsepower since most manufacturers
specifications are listed that way. The conversion factor is 1 hp = 746 watts, so
P=
2416 watts
× 1 hp = 3.24 hp .
746 watts
(3.D.6)
This motor is within a reasonable size however, if the required size motor
increased by 1/2 hp then three phase power would most likely be required.
Taking into consideration all of the potential problems it has been decided
that it is too large of an engineering undertake for the potential gain. We thus
moved to the third and final parametrically excited system.
35
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36
IV.
TORSIONAL OSCILLATOR
In this chapter, we investigate parametric excitation of a torsional oscillator
in which the length of the oscillator is modulated. The concept and motivation for
this work are stated in Sec. A, the theory is developed in Sec. B, the construction
of the apparatus is detailed in Sec. C, and demonstrations and experiments are
discussed in Sec. D.
A.
CONCEPT AND MOTIVATION
A torsional oscillator consists of a length of flexible material with one end
fixed and the other attached to a body, where the system performs angular
oscillations about the central axis. The frequency depends upon the length of the
material; specifically, the frequency increases for shorter lengths and decreases
for longer lengths.
Hence, by modulating the length at twice the natural
frequency of the oscillations, we should be able to parametrically excite torsional
oscillations if the modulation length is sufficiently large.
A preliminary sketch of how such a system could be constructed is shown
in Fig. 13. The torsional oscillator consists of a vertical strip of material with a
dumbbell attached to the lower end. The location of the upper end of the active
segment of the strip is vertically oscillated by a double-roller assembly driven by
a dc motor. It is interesting to consider the case where the material is not a strip
but is a cord or wire of circular cross section.
It is doubtful that parametric
excitation can be achieved in this case, but an experiment to resolve this issue is
left for future work.
37
Figure 13. Length-modulated torsional oscillator. The motor (top) and
linkage cause the double-roller to vertically oscillate, and to thereby
modulate the length and thus the frequency of the torsional
oscillator.
One of our motivations is to construct a lecture demonstration apparatus.
An important advantage of such an apparatus is that it typically has low
frequency (on the order of 1 Hz), which allows a high-amplitude drive to be
readily built. In addition, the damping is typically not high. We are thus almost
assured that the threshold for parametric excitation can be easily exceeded, in
contrast to our other attempts of a parametrically excited acoustic resonator (Ch.
II) and inductor-capacitor circuit (Ch. III). Another motivation is to compare the
experimental data to theoretical predictions. The theory is based on weak linear
damping, weak nonlinearity, and a drive frequency near twice the natural
frequency.
It may be that an actual system substantially deviates from this
regime.
38
B.
THEORY
We consider a torsional oscillator with linear torsional constant κ (torque
per unit angular displacement) and moment of inertia I of the attached body
about the axis of rotation. If θ is the angular displacement from equilibrium,
Newton’s second law for rotational motion (torque equals moment of inertia
multiplied by angular acceleration) yields the equation of simple harmonic motion
for small-amplitude oscillations:
&&
θ + ω02 θ = 0,
(4.B.1)
where the natural angular frequency is ω0 = (κ/I)1/2. The linear torsional constant
is inversely proportional to the length of the material that undergoes twisting, just
as the spring constant of a spring is inversely proportional to the length of the
spring. The torsional constant is thus κ = α/L(t), where L(t) is the length and α is
a constant that in general depends upon the material and the cross-sectional
geometry. If the length varies as L(t) = L0 – ∆Lcos(ωt), where ∆L << L0, the
equation of motion (4.B.1) approximately becomes Mathieu’s equation
&&
θ + ω02 ⎡⎣1 + η cos ( ωt ) ⎤⎦ θ = 0,
(4.B.2)
where the ambient natural frequency is
ω0 =
α
=
IL0
κ0
.
I
(4.B.3)
In Eq. (4.B.2), the ambient linear torsional constant is κ0 = α/L0, and the
dimensionless parametric drive amplitude is
39
η =
∆L
.
L0
(4.B.4)
To better quantify the system, we should include damping and
nonlinearity.
The damping dictates the threshold value of the dimensionless
parametric drive amplitude (4.B.4) for which parametric excitation from rest will
occur. The nonlinearity dictates the steady-state response amplitude, in which
the damping also plays a role. (As discussed in Ch. I, only a nonlinearity can
limit the growth in parametric excitation.)
In contrast to acoustic parametric excitation of an acoustic resonator (Ch.
II) and an inductor-capacitor circuit (Ch. III), the damping in a typical
demonstration-sized torsional oscillator is expected to be approximately
quadratic in the velocity rather than linear. Linear damping may only occur for
very small amplitudes. The standard theory of parametric excitation may thus
not apply.
Although a theory may be achievable for this type of nonlinear
damping, it would not be straightforward.
In addition to the nonlinear damping, there is another theoretical
complication. The restoring torque for a suspended torsional oscillator as in Fig.
13 arises from two different sources: the shear modulus of the material, and
gravity. The gravitational restoring torque is purely geometrical, and is due to
fact that the body rises in the gravitational field as the material is twisted. The
gravitational potential energy thus increases, which implies a restoring torque.
Although the shear torque for small amplitudes can be calculated, it is difficult to
determine the torque for finite amplitudes; specifically, the nonlinear coefficient of
the cube of the angular displacement. (There is no quadratic nonlinearity due to
symmetry.) However, because the gravitational torque is purely geometrical, this
nonlinear coefficient can be readily calculated, as we now show.
The geometry is shown in Fig. 14. A body (not shown) of mass m is
attached to the bottom segment of the strip. Because the body rises due to the
twisting of the strip, the increase in gravitational potential energy is
40
U = mgh ,
(4.B.5)
where h is the height that the body rises, which is the same height that the
bottom segment of the strip rises. We require to determine h in terms of the
angle θ for small but finite angular displacements, because the gravitational
restoring torque is
N( θ) = −
dU
,
dθ
(4.B.6)
from which we can then determine the linear restoring torque and cubic-nonlinear
correction to this torque.
z
y
2a
θ
g
L
x
y
x
h
Figure 14. Uniform twisting of a strip of material in a torsional oscillator.
On the left is a front perspective view. The bottom segment of the
strip, where the body (not shown) is attached, rises a height h due to
the twisting. On the right is the top view. The bottom segment
rotates an angle θ.
In Fig. 14, to determine the height h of the bottom segment of strip as a
function of the angular displacement θ, we note that the length of a long edge of
41
the strip has a constant value L0. The distance between the anchored point (x, y,
z) = (a, 0, L0) of an edge and the lower point (a cos(θ), a sin(θ), z) must therefore
also be L0, so
L20 = ⎡⎣a − acos ( θ ) ⎤⎦
2
+ a2 sin2 ( θ ) +
(L 0
− h) .
2
(4.B.7)
This expression holds for –π ≤ θ ≤ π; for greater angular displacements, the strip
touches itself, which invalidates the expression. Performing the square in Eq.
(4.B.7), simplifying, using the identity 1 – cos(θ) = 2sin2(θ/2), and solving for h
yields
⎡
h ( θ ) = L0 ⎢1 −
⎢⎣
1−
⎤
4a2
2⎛θ⎞
sin
⎥
⎜2⎟ m .
L20
⎝ ⎠ ⎥⎦
(4.B.8)
The power-series expansion of Eq. (4.B.8) in θ to fourth order must be done with
care so that all of the contributions are included. The expansion is
h ( θ)
⎛1
a2 ⎡ 2
a2 ⎞ 4 ⎤
=
−
⎢θ − ⎜
⎟θ ⎥ .
2L0 ⎣⎢
4L20 ⎠ ⎦⎥
⎝3
(4.B.9)
The torque is given by Eq. (4.B.6), where the potential energy is given by Eq.
(4.B.5). Substitution of Eq. (4.B.9) yields
N( θ) = −
⎛4
mga2 ⎡
a2 ⎞ ⎤
− 2 ⎟ θ3 ⎥ .
⎢θ − ⎜
L0 ⎣⎢
L0 ⎠ ⎦⎥
⎝3
42
(4.B.10)
This expression gives the restoring torque in a torsional oscillator with a perfectly
flexible strip, to fourth order in the angular displacement. We see that the linear
torsional constant is
mga2
,
L0
κ =
(4.B.11)
κ = mga2/L0. The experimental value of the linear torsional constant of any strip
being tested in the apparatus can be determined from the linear frequency
(4.B.3) by computing the moment of inertia and then timing a number of cycles of
small-amplitude motion, where the linear period is T0 = 2π/ω0. This value of the
torsional constant can then be compared to the theoretical value (4.B.11) for zero
shear modulus. This method offers a way of comparing the relative importance
of the shear and gravitational torques.
The nonlinear coefficient in Eq. (4.B.10) reveals that the system softens if
L0 > (3/4)1/2a, which holds in the typical case L0 >> a. The nonlinear coefficient
allows the steady-state amplitude to be predicted if the damping were linear.
C.
CONSTRUCTION OF THE APPARATUS
An overall picture of the system is shown in Fig. 15. The construction of
the apparatus was implemented in three phases.
Phase one consisted of
constructing the support structure, which is the foundation for the system. There
are two key features of the support.
One is the leveling system.
A two-
dimensional level was installed so that the user could determine if the apparatus
was level or not. If it was not level then the user could adjust one of three bolts
on the bottom, two in front and one in back. This provides for an easy method of
leveling. The second feature is the acrylic supports for the motor, and rollers.
The two pieces of acrylic both provide support and allow a clear view of the
system while it is functioning. The support system laid the foundation for the next
phase, which was the roller guide and ribbon clamp.
43
Figure 15. Length-modulated torsional oscillator apparatus.
The rollers are made of PVC with ball bearings in the ends to provide for
smooth rolling over the ribbon (Fig. 16). The rollers were machined exactly the
same so that there would be no nonuniformities. The rollers are hung by hinges
from a traveling “U” shaped support. The support travels along two cylindrical
guides that are attached to the ribbon clamp which in turn is attached to the
overall support structure. The “U” shaped support is driven by a single cylindrical
rod that is in turn driven by the drive system.
44
drive rod
ribbon
clamp
cylindrical
support
“U” support
rollers
Figure 16. Frontal view of “U” support and ribbon clamp.
The final phase of construction involved creating the drive system. Due to
the importance of finishing the construction in time to collect data, and the
reduced amount of time left, it was decided to use what was available for
construction instead of performing an analysis to determine the best suited
equipment. A motor that had been previously attached to a robotic arm was
used for the driving motor. This was thought to be a wise decision because the
motor has a high-accuracy encoder built on it, so it could be used later to
determine very accurate data. Since the motor that was chosen was low torque
it was decided to use a high gear ratio gear box to increase the torque and slow
the motor down. The gear ratio is 60:1. The motor is not directly coupled to the
gear box. It is mounted instead at a ninety degree angle so that the apparatus
would have a cleaner look and so that all the working parts would be easily
45
visible. The drive wheel is attached to the opposite side of the gear box. It has a
slot cut out of it for the purpose of attaching the drive armature. Grooves were
cut into the drive wheel and drive armature so that when the drive armature is in
its most vertical position the grooves would line up, which allows the distance
from the center of the bearing to the center of the drive wheel be determined.
This allows the user to set the modulation length. See Fig. 17 and Fig. 18 for
pictures of the components.
drive wheel
motor
drive armature
Figure 17. Frontal view of drive system.
46
gearbox
Figure 18. Top view of drive system.
A few added design features must be mentioned about the system. First
is the adjustable positioning of the masses on the dumbbell (Fig. 19). This allows
the user to change the natural frequency of the system with out having to change
ribbon length or material. The user can do this by removing the pin that goes
through the mass, slide the mass to the next hole and replace the pin. The next
feature that was added is a controlled release mechanism (Fig. 19), which allows
the user to release the pendulum from a specific displacement in a controlled and
repeatable fashion. This is done by placing the device in the desired release
position, raising the inner cylinder, and then placing a pin through the outer
cylinder into the inner cylinder to hold it in place. Once it is positioned exactly
where the user wants it, the pin is then removed, which drops the inner cylinder
47
and releases the pendulum. The final addition to the system is the photogates
and data collection system, which will be described in more detail in Sec. D. See
Fig. 19 for images of added design features.
Figure 19. Movable masses and controlled release device.
D.
DEMONSTRATIONS AND EXPERIMENTS
The data for the experiment is collected using photogates and a data
collection interface that interprets the signal from the photogates into usable
information. This information is then sent to the computer where the user can
plot it. The data gathered by the photogates is interrupt time information. This is
used by the computer software to determine the velocity and period information.
The velocity that is important in this experiment is the peak instantaneous
velocity.
However, the information gathered by the photogate is average
velocity. This is because the software takes the diameter of the pin at the end of
48
the dumbbell and divides it by the time it takes to pass the detector. Since the
pin has a finite diameter the velocity calculated is actually an average velocity.
However, because the diameter of the pin is small (1/16”) the average velocity
and instantaneous velocity will differ only slightly, so for calculation purposes the
average velocity will be treated as an instantaneous velocity. The period for the
dumbbell is calculated by the time between two interrupts. Due to the way that
the photogate triggers and the timers are activated only one-half of an oscillation
can be measured for period information, so the measured time must be doubled.
The period of the drive system is calculated in the same way as an encoder
would.
Period data is then translated into frequency data by the computer
software.
In order to describe this system there are a few necessary parameters that
need to be determined experimentally. The first is the natural frequency of the
torsional oscillator ωo.
This can be determined from the free-decay of the
velocity as a function of frequency. As velocity goes to zero the frequency of the
oscillator goes to the natural frequency, see Fig. 20. From this information it was
determined that the natural frequency is ωo = 0.132 Hz.
As important fact from Fig. (20) is that the system hardens; that is, the
natural frequency increases for greater amplitudes. The theory of a perfectly
flexible strip (Sec. B) yields softening due to gravitation. We thus conclude that
the shear modulus of the material dominates the gravitational effects
49
Velocity Amplitude (m/s)
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.35
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
Frequency (Hz)
Figure 20. Free-decay data. The natural linear frequency of oscillator is
determined from the zero velocity intercept.
The next parameter that needs to be determined is the linear damping
coefficient β.
This can be determined from a free-decay plot of the velocity
verses time. When the system is oscillating at low amplitude the decay of the
velocity verses time should be linear on a semi-log plot. The slope of this line is
the linear damping coefficient β. From the data β is determined to be 4.1x10-3
s-1. See Fig. 21.
50
Velocity Amplitude (m/s)
v1=v0e-βt
1 v0
1
0.080
β = ln =
ln
t v1 725 0.0040
0.1
= 4.1x10-3 s-1
0.01
0
100
200
300
400
500
600
Time (s)
Figure 21. Free-decay plot of velocity verses time. The damping
parameter β is calculated from slope.
The last parameter that needs to be determined is the nonlinear coefficient
α. This is calculated from the slope of the small-amplitude region of the velocity
amplitude squared verse frequency plot, see Fig. 22.
The displacement
amplitude is approximately
A=
v0
.
2πf 0
(4.D.1)
Substituting this into equation (1.A.8), the velocity amplitude squared as a
function of frequency can be determined. This equation then takes the form of a
linear equation with
51
−128π4 f03
slope =
.
3α
(4.D.2)
From this α is determined to be -17.38 Hz4s2/m2.
2
2
Velocity Amplitude Squared (m /s )
0.08
0.06
0.04
0.02
0.00
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
Frequency (Hz)
Figure 22. Velocity amplitude squared verses frequency. The slope is
used to calculate the nonlinear coefficient α.
Now that those three parameters have been determined the system can
be described by its steady state response and excitation response. The plots
associated with these are tuning curves and the drive plane of dimensionless
parametric drive amplitude η verses drive frequency f. See Fig. 23 and Fig. 24.
The equations used to create the theoretical curves are equations (1.A.9) and
52
(1.A.4), respectively, where the appropriate substitutions were made so that they
would be in terms of frequency and velocity amplitude. In the tuning curve graph
the triangles indicate data taken for decreasing frequency after the response has
fallen off the tuning curve.
Velocity Amplitude (m/s)
0.5
0.4
0.3
theoretical
upper branch
(stable)
0.2
0.1
theoretical
lower branch
(unstable)
0.0
0.20
0.25
0.30
0.35
0.40
0.45
Drive Frequency f (Hz)
Figure 23. Steady-state response. The curves are theoretical and the
points are experimental. The dimensionless parametric drive
amplitude η =0.146 .
53
Dimensionless Drive Amplitude η
0.25
0.26
0.27
0.28
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.25
0.26
0.27
0.28
Drive Frequency f (Hz)
Figure 24. Drive plane plot of η verses f. The curve is theoretical, and the
points are experimental.
From the graphs it is clear that the system is behaving differently than the
theory predicts. This could be due to strong nonlinearities or possibly it is related
to the level of accuracy that the data was taken at. Since the experiment was put
together expeditiously, high precision equipment was not available. In Chapter V
I will discuss recommendations for any continued efforts with this experiment.
What can be learned from the data is exactly how the system behaves
and thus what to expect when operating it. This is essential for demonstration
purposes.
This apparatus can be used to demonstrate, in a controlled,
reproducible method, excitation from rest.
It can also demonstrate the
characteristics of the tuning curve. By this it is meant that it can be shown to
increase in response velocity amplitude as frequency is increase to the point
54
where the system becomes detuned and thus velocity amplitude drops to zero.
The hysteresis effect can then be demonstrated by lowering the drive frequency
to the point where the system starts responding again.
The steady-state amplitude graph in Fig. 23 and the threshold data in Fig.
24 both show substantial deviations between the experimental data and the
theory. However, it should be noted that there are many approximations in the
theory: (i) The theory assumes weak linear dissipation, whereas the apparatus
exhibits strong nonlinear dissipation at larger amplitudes, as shown in Fig. 21.
(ii) The theory assumes a weak cubic nonlinearity, whereas a very strong
deviation from this is evident in Fig. 20. (iii) The theory assumes a weak drive
amplitude (η << 1). The typical values of η for the apparatus are 0.1. (iv) The
theory assumes that the drive frequency is near twice the natural linear
frequency. In the experiment, the average deviation of the drive frequency from
twice the linear natural frequency is roughly 4%. Of all the deviations, it appears
that only the final approximation (iv) is met by the apparatus. Approximation (iii)
may be roughly met. Due to the other deviations, however, we should not expect
good agreement of the steady-state amplitudes in Fig. 23. The threshold data in
Fig. 25 should not depend upon nonlinearity, either in regard to the cubic
nonlinearity or the nonlinear damping. An understanding of the discrepancies is
challenging, and more data are required. This is a subject of future work.
55
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56
V.
A.
CONCLUSIONS AND FUTURE WORK
CONCLUSIONS
One inherent difficultly of parametric excitation is that, without a detailed
examination of a system, it is not obvious how difficult it would be to excite the
system. That is why research of this nature is important. Without investigating
different types of systems, it is unclear what might be possible. From these
investigations we have learned that it will require either a significant engineering
undertaking or a discovery of a new parameter to vary in order to excite
acoustical standing waves in a resonator. We have also learned that parametric
excitation of an LC circuit by capacitance modulation would also require a
significant engineering undertaking. However, due to our calculations and the
claims previously made, it seems that it might be possible. Finally, in regard to a
length-modulated torsional oscillator, we have demonstrated that it is possible to
design and build a controlled, quantifiable, and reproducible demonstration
apparatus for the explicit purpose of educating students on the topic of
parametric excitation.
B.
FUTURE WORK
Several projects can be undertaken as future work.
Although several
attempts at parametric excitation of acoustic standing waves have failed, I am of
the firm belief that it is possible. However, the concept of modulating the length
of a resonance tube has been exhausted, unless a serious engineering challenge
is accepted. In order for excitation to readily occur a new parameter or method
of modulation must be uncovered.
The LC circuit has potential but it is probably worth investigating
inductance modulation before pursuing capacitance modulation any further. I
say this because of how large of an undertaking it would be to construct the
parallel plate capacitor and because in the paper referenced in Ch. III the author
had higher levels of excitation in the inductance-modulated circuit.
57
Finally, more data needs to be taken on the length-modulated torsional
oscillator so that a more accurate description is available to compare to the
theory. Future experiments might include using different ribbon material and
lengths of ribbon and drive amplitudes.
Also, as mentioned in Ch. IV, an
experiment should be done on materials of cylindrical shape such as wire, or
cord. Some system modifications that need to be made are: (i) Obtain a better
motor. The one currently in use is underpowered. (ii) Attach an encoder to the
shaft with a readout of some kind, that way the audience can see the frequency
at which the system is operating. (iii) The apparatus could be made smaller so
that it is more portable.
58
LIST OF REFERENCES
Denardo, B., and Larraza, A., Nonlinear Oscillations and Waves: An
Essential Introduction with Demonstrations (Naval Postgraduate School,
Department of Physics, 2004). This text is used for the NPS course PH4459
(Nonlinear Oscillations and Waves).
http://www.cheniere.org/misc/moscowuniv.htm (last accessed 29 APR 05)
Mandelstam L., Papalexi N., Andronov A., Chaikin S., and Witt A.,
“Report on Recent Research on Nonlinear Oscillations” NASA Technical
Translation, NASA TT F-12, 678. November 1969.
Smith, D., “Parametric Excitation of an Acoustic Standing Wave,” M.S. thesis,
Department of Physics, Naval Postgraduate School, June 2003.
Varnadore, P., “Feasibility Investigations of Parametric Excitation of
Acoustic Resonators,” M.S. thesis, Department of Physics, Naval Postgraduate
School, June 2001.
Wright, W. B. and Swift, G. W., “Parametrically driven variable-reluctance
generator,” J. Acoust. Soc. Am., vol. 88, no. 2, pp. 609-615 (1990).
59
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60
INITIAL DISTRIBUTION LIST
1.
Defense Technical Information Center
Ft. Belvoir, Virginia
2.
Dudley Knox Library
Naval Postgraduate School
Monterey, California
3.
Physics Department
Naval Postgraduate School
Monterey, California
4.
Professor Bruce Denardo
Department of Physics
Naval Postgraduate School
Monterey, California
5.
Professor Thomas Hofler
Department of Physics
Naval Postgraduate School
Monterey, California
6.
Kevin B. Smith
Chair, Engineering Acoustics Academic Committee
Naval Postgraduate School
Monterey, California
61
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