Butikov - Parametric Excitation of a Linear Oscillator - 2004.pdf

Butikov - Parametric Excitation of a Linear Oscillator - 2004.pdf
INSTITUTE OF PHYSICS PUBLISHING
Eur. J. Phys. 25 (2004) 535–554
EUROPEAN JOURNAL OF PHYSICS
PII: S0143-0807(04)76708-2
Parametric excitation of a linear
oscillator
Eugene I Butikov
St Petersburg State University, St Petersburg, Russia
E-mail: [email protected]
Received 25 February 2004
Published 28 May 2004
Online at stacks.iop.org/EJP/25/535
DOI: 10.1088/0143-0807/25/4/009
Abstract
The phenomenon of parametric resonance is explained and investigated both
analytically and with the help of a computer simulation. Parametric excitation
is studied for the example of the rotary oscillations of a simple linear system—
mechanical torsion spring pendulum excited by periodic variations of its
moment of inertia. Conditions and characteristics of parametric resonance and
regeneration are found and discussed in detail. Ranges of frequencies within
which parametric excitation is possible are determined. Stationary oscillations
at the boundaries of these ranges are investigated. The simulation experiments
aid greatly an understanding of basic principles and peculiarities of parametric
excitation and complement the analytical study of the subject in a manner that
is mutually reinforcing.
1. Introduction: the investigated physical system
A physical system undergoes a parametric forcing if one of its parameters is modulated
periodically with time. A common familiar example of parametric excitation of oscillations is
given by the playground swing on which most people have played in childhood (see, e.g., [1]).
The swing can be treated as a physical pendulum whose reduced length changes periodically
as the child squats at the extreme points, and straightens when the swing passes through
the equilibrium position. It is easy to illustrate this phenomenon in the classroom with the
following simple experiment. Let a thread with a bob hanging from its one end pass through a
little ring fixed in a support. You can pull by some small length the other end of the thread that
you are holding in your hand each time the swinging bob passes through the middle position
and release the thread to its previous length each time the bob reaches the maximum deflection.
These periodic variations of the pendulum length with the frequency twice the frequency of
natural oscillation cause the amplitude to increase progressively. Another canonical example
of parametric pumping is given by a pendulum whose support oscillates vertically (see [2]).
c 2004 IOP Publishing Ltd Printed in the UK
0143-0807/04/040535+20$30.00 535
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E I Butikov
Figure 1. Schematic image of the torsion spring oscillator with a balanced rotor whose moment
of inertia is forced to vary periodically (left), and an analogous LCR-circuit with a coil whose
inductance is modulated (right).
However, such systems do not perfectly suit the initial acquaintance with parametric
excitation because the ordinary pendulum is a nonlinear physical system: the restoring torque
of the gravitational force is proportional to the sine of the deflection angle. That is why we
suggest studying the basics of parametric resonance by using the simplest linear mechanical
system in which the phenomenon is possible, namely, the torsion spring oscillator, similar to
the balance device of a mechanical watch. An educational computer program that simulates
such a system has been developed by the author (see [3]).
The left-hand side of figure 1 shows a schematic image of the apparatus. It consists of a
rigid rod which can rotate about an axis that passes through its centre. Two identical weights
are balanced on the rod. An elastic spiral spring is attached to the rod. The other end of
the spring is fixed. When the rod is turned about its axis, the spring flexes. The restoring
torque −Dϕ of the spring is proportional to the angular displacement ϕ of the rotor from the
equilibrium position. After a disturbance, the rotor executes a harmonic torsional oscillation.
To provide modulation of a system parameter, we assume that the weights can be shifted
simultaneously along the rod in opposite directions to other symmetrical positions so that the
rotor as a whole remains balanced. However, its moment of inertia J is changed by such
displacements of the weights. When the weights are shifted towards or away from the axis,
the moment of inertia decreases or increases, respectively. Thus, the moment of inertia of the
rotor is the parameter to be modulated in the investigated physical
system. As the moment of
√
inertia J is changed, so also is the natural frequency ω0 = D/J of the torsional oscillations
of the rotor. Periodic modulation of the moment of inertia can cause, under certain conditions,
a growth of (initially small) natural rotary oscillations of the rod.
This physical system, being useless for any practical application, is nevertheless ideal
for gaining conceptual knowledge about the basics of parametric resonance and has several
advantages in an educational context because it gives a very clear example of the phenomenon
in a linear mechanical system. All peculiarities of parametric excitation in this linear
system can be completely explained and exhaustively investigated by modest means even
quantitatively.
Parametric excitation is also possible in an electromagnetic analogue of the spring
oscillator, namely in a series LCR-circuit containing a capacitor, an inductor (a coil) and
a resistor (right-hand side of figure 1). An oscillating current in the circuit can be excited
by periodic changes in the capacitance if we periodically move the plates closer together and
farther apart, or by changes in the inductance of the coil if we periodically move an iron
core in and out of the coil. Such periodic changes in the inductance are quite similar to the
changes in the moment of inertia in the mechanical system considered above. However, the
Parametric excitation of a linear oscillator
537
mechanical system has certain spectacular didactic advantages primarily because its motion is
easily represented on the computer screen, and it is possible to see directly what is happening
[3]. Such a visualization makes the simulation experiments very convincing and easy to
understand, aiding a great deal in developing our physical intuition.
2. Peculiarities of parametric resonance
The causes and characteristics of parametric resonance are considerably different from those
of the resonance occurring when the oscillator responds to a periodic external force exerted
directly on the system. Specifically, the resonant relationship between the frequency of
modulation of a parameter and the mean natural frequency of oscillation of the system is
different from the relationship between the driving frequency and the natural frequency for
the usual resonance in forced oscillations. Parametric excitation can occur only if at least
weak natural oscillations already exist in the system, and if there is friction, the amplitude of
modulation of the parameter must exceed a certain threshold value in order to cause parametric
resonance.
To understand how a change in the moment of inertia can increase or decrease the angular
velocity of the rotor, let us imagine for a while that the spiral spring is absent. Then the angular
momentum of the system would remain constant as the weights are being moved along the
rod. Thus the resulting reduction in the moment of inertia is accompanied by an increment
in the angular velocity, and the rotor acquires additional energy. The system is similar in
some sense to a spinning figure skater, whose rotation accelerates as she moves her initially
stretched arms closer to her body.
The greater the initial angular velocity, the greater the increment in the velocity and the
energy. This additional energy is supplied to the rotor by the source that moves the weights
along the rod. On the other hand, if the weights are moved apart along the rotating rod, the
angular velocity and the energy of the rotor diminish. The decrease in energy is transmitted
back to the source.
In order that increments in energy occur regularly and exceed the amounts of energy
returned, i.e., in order that, as a whole, the modulation of the moment of inertia regularly
feeds the oscillator with energy, the period and the phase of modulation must satisfy certain
conditions. Figure 2 shows the graphs of parametric oscillations of the torsion pendulum
excited by a constrained sinusoidal motion of the weights along the rod with a period which
is equal to one-half of the natural period. The graphs are obtained with the help of a computer
simulation program included in the educational software package [3].
To provide a growth of energy by modulation of the moment of inertia, the motion of the
weights towards the axis of rotation must occur while the angular velocity of the rotor is on
average greater in magnitude than it is when the weights are moved apart to the ends of the
rod. The graphs in figure 2 correspond to this case: we see clearly that during the intervals of
negative values of v, the angular velocity ϕ̇ is greater in magnitude than during the intervals of
positive v. Otherwise the modulation of the moment of inertia aids the damping of the natural
oscillations.
Parametric excitation is possible only if one of the energy-storing parameters, D or J
(C or L in the case of a LCR-circuit), is modulated. Modulation of the resistance R (or of the
damping constant γ in the mechanical system) can affect only the character of the damping
of oscillations. It cannot generate an increase in their amplitude.
The strongest parametric oscillations are excited when the cycle of modulation is repeated
twice during one period T0 of natural oscillations in the system, i.e., when the frequency ω
of parametric modulation is twice the natural frequency ω0 of the system. But the delivery
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E I Butikov
Figure 2. Graphs of the angular displacement and velocity of the rotor and the phase trajectory in
conditions of the principal parametric resonance.
of energy is also possible when the parameter changes once during one period, twice during
three periods and so on. That is, parametric resonance is possible when one of the following
conditions for the frequency ω of modulation (or for the period of modulation T = 2π/ω) is
fulfilled:
ω = 2ω0 /n,
T = nT0 /2,
(1)
where n = 1, 2, . . . . For a given amplitude of modulation of the parameter, the higher the
order n of parametric resonance, the less (in general) the amount of energy delivered to the
oscillating system during one period.
One of the most interesting characteristics of parametric resonance is the possibility of
exciting increasing oscillations not only at the frequencies ωn given in equation (1), but also
in intervals of frequencies lying on either side of the values ωn (in the ranges of instability.)
These intervals become wider as the range of parametric variation is extended, that is, as the
depth of modulation is increased.
An important distinction between parametric excitation and forced oscillations is related
to the dependence of the growth of energy on the energy already stored in the system. While
for a direct forced excitation, the increment of energy during one period is proportional to the
amplitude of oscillations, i.e., to the square root of the energy, at parametric resonance the
increment of energy is proportional to the energy stored in the system.
Energy losses caused by friction (unavoidable in any real system) are also proportional to
the energy already stored. In the case of direct forced excitation, an arbitrarily small external
force gives rise to resonance. However, energy losses restrict the growth of the amplitude
because these losses increase with the energy faster than does the investment of energy arising
from the work done by the external force.
In the case of parametric resonance, both the investment of energy caused by the
modulation of a parameter and the frictional losses are proportional to the energy stored
Parametric excitation of a linear oscillator
539
(to the square of the amplitude), and so their ratio does not depend on the amplitude. Therefore,
parametric resonance is possible only when a threshold is exceeded, that is, when the increment
of energy during a period (caused by the parametric variation) is larger than the amount of
energy dissipated during the same time. To satisfy this requirement, the range of the parametric
variation (the depth of modulation) must exceed some critical value. This threshold value of
the depth of modulation depends on friction. However, if the threshold is exceeded, the
frictional losses of energy cannot restrict the growth of the amplitude. In a linear system the
amplitude of parametrically excited oscillations must grow indefinitely.
In a nonlinear system the natural period depends on the amplitude of oscillations. If
conditions for parametric resonance are fulfilled at small oscillations and the amplitude begins
to grow, the conditions of resonance become violated at large amplitudes. In a real system the
growth of the amplitude over the threshold is restricted by nonlinear effects.
3. Parametric resonance and the threshold of parametric excitation
To explain the behaviour of the parametrically pumped oscillator, first we make use of the
conservation of energy. At resonance, additional energy must be transmitted to the rotor by
the source that makes the weights move periodically along the rod. Therefore, we calculate
the work done by the source during one period of oscillation and find those conditions under
which this work is positive.
In this model we assume the forced motion of the weights along the rod to be exactly
sinusoidal, and so their distance l from the axis of rotation varies with time according to the
following expression:
l(t) = l0 (1 + m sin ωt).
(2)
Here l0 is the mean distance of the weights from the axis of rotation, and m is the dimensionless
(fractional) amplitude of their harmonic motion along the rod (m < 1). For simplicity, we let
the rod be very light compared to the weights. We note that m is the modulation depth of the
distance l(t), while the modulation depth mJ of the moment of inertia J (t) is approximately
twice as great (mJ ≈ 2m if m 1), because the moment of inertia is proportional to the
square of the distance of the weights from the axis of rotation.
From equation (2), we find that a weight moves along the rod with a velocity and
acceleration (relative to the rod) which change with time as cos ωt and −sin ωt, respectively:
v(t) = dl/dt = ωl0 m cos ωt,
ar (t) = dv/dt = −ω2 l0 m sin ωt.
(3)
In order to find the force F exerted on the weight by the device that makes it move along
the rod, we use a noninertial reference frame rotating with the rod. Applying Newton’s second
law to the motion of the weight in this rotating frame of reference, we must take into account
the centrifugal pseudo force of inertia exerted on the weight, M ϕ̇ 2 (t)l(t), where M is the mass
of the weight and ϕ̇(t) is the angular velocity of the rod:
Mar (t) = F (t) + M ϕ̇ 2 (t)l(t).
(4)
We are interested in the work of this force F (t) done during one period of oscillation.
The amount of this work (for both weights) equals the change in the energy of oscillations
during one period. For the infinitesimal element of work dW done during a time interval dt
(during which the weight is displaced along the rod a distance dl = v(t) dt), we can write
dW = F (t) dl = F (t)v(t) dt = [Mar (t) − M ϕ̇ 2 (t)l(t)]v(t) dt.
(5)
As we see from equation (3), the radial velocity v(t) of the weight in equation (5) is
proportional to the dimensionless amplitude m of its forced motion along the rod. If we
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E I Butikov
Figure 3. The phase trajectory of stationary oscillations at the threshold conditions (the mode of
parametric regeneration at m ≈ 1/Q), and the time-dependent graphs of the rotor angular velocity
and of the radial velocity of the weights.
restrict our calculations to the first order of the small parameter m, we need to keep only
the second term in square brackets in equation (5), and we can substitute for l(t) from
equation (2) only its mean value l0 :
dW ≈ −M ϕ̇ 2 (t)l0 v(t) dt = −M ϕ̇ 2 (t)l02 ωm cos ωt.
(6)
As we noted above, the most favourable condition for the parametric excitation of the
rotor occurs if the weights execute two full cycles of the forced motion during one mean period
of the natural oscillation. In other words, the frequency ω in equations (2) and (6) must be
approximately twice the mean natural frequency ω0 = 2π/T0 of oscillation of the rotor. (Here
ω0 is the frequency of free oscillations of the rotor with the weights fixed at their average
distance l0 from the axis). For small values of the dimensionless amplitude m, the frequency
of modulation ω = 2ω0 in equation (2) corresponds to exact tuning to the principal resonance
(n = 1).
In addition, it is necessary that a certain phase relation between the forced motion of the
weights and the torsional oscillations of the rotor be satisfied: namely, the weights must move
with maximal relative velocity towards the axis of rotation at moments when the oscillating
rod moves with its greatest angular velocity. This phase relation is satisfied for the motion
of the weights described by equations (2) and (3) provided we assume the following time
dependence for the torsional oscillations of the rotor:
ϕ(t) = ϕm cos ω0 t,
ϕ̇(t) = −ϕm ω0 sin ω0 t.
(7)
These are only approximate expressions because, strictly speaking, the torsional oscillation of
the rotor is not harmonic (see figure 3). Deviations from a sinusoidal oscillation are caused
by the motion of the weights since this motion influences the moment of inertia and hence the
angular velocity of the rotor.
After the substitution of ω = 2ω0 and ϕ̇(t) from equation (7) into equation (6), we can
integrate dW given by (6) over a period T0 = 2π/ω0 , taking into account that the mean value
of cos2 ω0 t is 1/2. Finally we find that (up to terms of the first order in the small value m), the
work W of the force F (t) done during a period T0 is given by the following expression:
W = 12 Mϕm2 ω02 l02 · 2π m.
(8)
Parametric excitation of a linear oscillator
541
The same expression is valid for the second weight, and so as a whole the forces exerted
on the weights perform positive work (W > 0) during a period and increase the energy of the
oscillator by the amount 2W :
E = 2W = Mϕm2 ω02 l02 · 2π m.
(9)
Since we assume the rod to be very light compared to the weights, we can consider
the total kinetic energy of the rotor to be the kinetic energy of these massive weights. The
total energy E of the oscillator is equal to the maximal value of its kinetic energy, which is
attained at the instants when the oscillating rotor moves near its equilibrium position and has
its greatest angular velocity ω0 ϕm . Therefore E = Mϕm2 ω02 l02 . We do not take into account
here the kinetic energy of the weights in their radial motion along the rod, because this energy
is proportional to the square of small parameter m. Comparing this expression with the righthand side of equation (9), we see the most essential feature of parametric resonance, namely
that the investment of energy E due to modulation of a parameter is proportional to the
energy E already stored in the oscillator:
E = 2π mE.
(10)
Equation (10) means that at parametric resonance the total energy E of oscillations,
averaged over a period T0 = 2π/ω0 of oscillation, grows exponentially with time:
dE
= mω0 E,
E(t) = E0 exp(2αt),
where 2α = mω0 .
(11)
dt
This result is valid in the absence of friction. Dissipation of the mean energy E due to
viscous friction is also described by an exponential function:
dE
= −2γ E,
E(t) = E0 exp(−2γ t).
(12)
dt
At the threshold of parametric resonance, these energy losses are just compensated for by
the delivery of energy arising from the forced periodic motion of the weights. In this instance,
γ = α. Thus we can find the minimal value of m (for a given value of γ or of the quality
factor Q) which makes parametric excitation possible:
2γ
1
= .
mmin =
(13)
ω0
Q
Equivalently, the threshold condition can be expressed in terms of the maximal value of the
damping constant γ (or the minimal quality factor Q) for a given value m of the amplitude
in (2):
1
ω0
1
(14)
Qmin =
= .
γmax = mω0 ,
2
2γmax
m
These results concerning the threshold of parametric excitation are approximate and are
valid only for small values of the dimensionless amplitude m of the forced motion of the
weights along the rod. The simulation program [3] executes numerical integration of the
differential equation of motion. This integration is not restricted to small values of m. Thus
the simulation allows us to find the threshold conditions experimentally (by trial and error)
with greater accuracy.
Steady oscillations occurring at the threshold are called parametric regeneration. They
are shown in figure 3. These graphs should be compared with those shown in figure 2, which
displays plots of resonant oscillations occurring above the threshold, where the amplitude
grows exponentially in spite of the friction.
When the depth of modulation exceeds the threshold value, the (averaged over the period)
energy of oscillations increases exponentially with time. The growth of the energy again is
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E I Butikov
described by equation (11). However, now the index of growth 2α is determined by the amount
by which the energy delivered through parametric modulation exceeds the simultaneous losses
of energy caused by friction: 2α = mω0 − 2γ . The energy of oscillations is proportional to the
square of the amplitude. Therefore, the amplitude of parametrically excited oscillations also
increases exponentially with time (see figure 2): a(t) = a0 exp(αt) with the index α (one-half
the index 2α of the growth in energy). For the principal resonance we have α = mω0 /2 − γ .
4. Differential equation for sinusoidal motion of the weights
In the assumed model, we consider the rod itself to be very light, so that the moment of inertia
J of the rotor is due principally to the weights: J = 2Ml 2 (t). The angular momentum J ϕ̇(t)
changes with time according to the equation:
d
(J ϕ̇) = −Dϕ,
(15)
dt
where −Dϕ is the restoring torque of the spring. Substituting into equation (15) l(t) from
equation (2) and taking into account that ω02 = D/J0 (J0 = 2Ml02 is the moment of inertia
with the weights in their mean positions), we obtain finally:
d
[(1 + m sin ωt)2 ϕ̇] = −ω02 ϕ − 2γ ϕ̇.
(16)
dt
We have added the drag torque of viscous friction to the right-hand side of equation (16).
This equation is solved numerically in the computer program [3] in real time during the
simulation.
We note that the harmonic motion of the weights along the rod described by equation (2)
does not mean that the moment of inertia J (t) is harmonically modulated. Indeed, J is
proportional to the square of the distance l(t) rather than to its first power. The time
dependence of J (t) includes the second harmonic of the frequency ω. Only for small values
of the amplitude m (when m 1) can we consider the modulation of the moment of inertia
to be approximately sinusoidal:
J (t) = 2Ml 2 (t) = 2Ml02 (1 + m sin ωt)2 ≈ 2Ml02 (1 + 2m sin ωt) = J0 (1 + mJ sin ωt),
(17)
where J0 = 2Ml02 is the mean value of the moment of inertia, and mJ = 2m is the
depth of its modulation. (We note that the value of mJ is approximately twice the value
of m.) If we are interested only in an approximate solution valid up to terms of the first
order in the small parameter m, then instead of the exact differential equation of motion,
equation (16), we can solve the following approximate equation:
ϕ̈ + 2γ ϕ̇ + ω02 (1 − 2m sin ωt)ϕ = 0.
(18)
We ignore here the modulation of the coefficient of ϕ̇ because for parametric resonance, the
variation of only those parameters which store energy (the moment of inertia and the torsion
spring constant) is essential. Modulation of the damping constant γ cannot excite oscillations.
When γ = 0, equation (18) is called Mathieu’s equation. The theory of Mathieu’s
equation has been fully developed, and all significant properties of its solutions are well
known (see, for example, [4]). A complete mathematical analysis of Mathieu’s equation is
rather complicated and gives little insight into the physics of parametric excitation. This
analysis is usually restricted to the determination of the frequency intervals within which the
state of rest in the equilibrium position becomes unstable, so that at arbitrarily small deviations
from the state of rest the amplitude of incipient small oscillations increases progressively with
time. The boundaries of these intervals of instability depend on the depth of modulation m.
Parametric excitation of a linear oscillator
543
We emphasize that the application of the theory of Mathieu’s equation to the simulated
system is restricted to the linear order in m. For finite values of the depth of modulation m, the
resonant frequencies and the boundaries of the intervals of instability for the simulated system
differ from those predicted by Mathieu’s equation. We shall see this point in the following
section, in which we avoid struggling with Mathieu’s equation or Floquet theory and develop
a rather simple theory of the simulated system up to the terms of the second order in m by
using the approach described in [5].
5. The principal interval of parametric instability
In the vicinity of the principal resonance, the frequency of modulation is approximately twice
the natural frequency (ω ≈ 2ω0 ), and we can express ω in the form ω = 2ω0 + ε, where ε is a
small detuning from resonance (|ε| ω0 ). We then propose that an approximate solution ϕ(t)
to equation (16) represents a nearly harmonic motion with the frequency ω̃ = ω/2 = ω0 + ε/2.
We let the amplitude and phase of the trial function ϕ(t) slowly vary with time:
ϕ(t) = p(t) cos ω̃t + q(t) sin ω̃t.
(19)
Here p(t) and q(t) are functions of time that vary slowly relative to the oscillating sine and
cosine functions. In the exact solution to equation (16) there are also higher harmonics with
the frequencies 3ω̃, 5ω̃, . . . , but their contribution is proportional to higher powers of the small
parameter m 1. We do not include these higher harmonics in the approximate solution
expressed by equation (19).
The time variation of the amplitudes p(t) and q(t) is caused by the modulation of the
moment of inertia, and so the time derivatives of functions p(t) and q(t) are also proportional
to the small quantity m. Substituting ϕ from equation (19) into the differential equation,
equation (16), we can express the products of the sine and cosine functions in the following
way:
sin 2ω̃t cos ω̃t = (sin ω̃t + sin 3ω̃t)/2,
sin 2ω̃t sin ω̃t = (cos ω̃t − cos 3ω̃t)/2,
and omit in the equation the higher harmonics with the frequency 3ω̃. Thus for the unknown
functions p(t) and q(t) we obtain the following system of differential equations of the first
order:
2ω̃q̇ − ω̃2 − ω02 p + 2γ ω̃ − mω02 q = 0,
(20)
−2ω̃ṗ − 2γ ω̃ + mω02 p − ω̃2 − ω02 q = 0.
We have omitted here the terms 2γ ṗ and 2γ q̇ since parametric excitation is possible only if
friction is small enough (from equation (14) we see that 2γ < mω0 ). The contribution of
these omitted terms to equation (20) is of the order m2 .
According to general rules, we can search for a solution to these equations in the form
exp αt. The condition for the existence of a nontrivial (nonzero) solution to this system of
homogeneous equations gives the following expression for α:
(21)
α ≈ 12 (mω0 )2 − ε2 − γ .
Here we have taken into account that ω̃2 ≈ ω02 + ω0 ε. If there is an exact tuning
to resonance, the deviation in frequency ε vanishes (ε = 0), and equation (21) gives the
following value for the index α that determines the exponential growth in the amplitude of
parametrically excited oscillations:
α ≈ mω0 /2 − γ .
(22)
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E I Butikov
Figure 4. The phase trajectory of stationary oscillations occurring at the left boundary of the
principal instability interval (left), and the time-dependent graphs of the angular velocity and of
the radial velocity of the weights (right).
The amplitude of oscillation grows if α > 0. Therefore, for the threshold of parametric
resonance we obtain m = 2γ /ω0 = 1/Q. The same value for the threshold of parametric
excitation under conditions of exact tuning to resonance is obtained above by using the
conservation of energy (see equation (13)). For zero friction, the index of the exponential
resonant growth in the amplitude is proportional to the depth of modulation: α = mω0 /2.
For the case in which friction is absent (γ = 0), and for a given value m of the
depth of modulation, we find from equation (21) that the linearized differential equation,
equation (16), yields resonant solutions (i.e. solutions whose amplitude is growing with time)
if the frequency of modulation belongs to some interval which extends by ω on either side
of the resonant value ωres = 2ω0 . The half-width of the interval ω = mω0 is proportional
to the amplitude m of the forced oscillation of the weights. For a value ω of the frequency
of modulation lying somewhere within the interval, the amplitude of parametrically excited
oscillations grows exponentially with time as exp(αt), where the index α of the growth is
given by equation (21) with γ = 0:
(23)
α = 12 (mω0 )2 − (ω − ωres )2 .
(for |ω − ωres | mω0 ). The value of α is zero at the boundaries ω± of the interval of
instability: ω± = ωres ± mω0 . At these boundaries, stationary oscillations of a constant
amplitude are possible. An example of such oscillations is shown in figure 4.
The symmetric shape of these graphs shows clearly that on average there is no energy
transfer to the frictionless oscillator: the energy gained during one-half-cycle of modulation
is returned back during the next half-cycle.
In order to obtain more precise values for the frequencies of modulation ω± which
correspond to the boundaries of the instability interval, we need to include higher harmonics
in the trial function ϕ(t) for an approximate solution of equation (16). Their frequency
3ω̃, 5ω̃, . . . is an odd-number multiple of the fundamental frequency ω̃ = ω/2 ≈ ω0 .
Restricting the calculation up to the second order in m, we hold only the first and third
harmonics in the trial function:
ϕ(t) = C1 cos ω̃t + S1 sin ω̃t + C3 cos 3ω̃t + S3 sin 3ω̃t.
(24)
If we are interested only in the boundaries ω± of the interval of instability, at which the
oscillations are stationary and their amplitude does not vary with time, we can assume the
coefficients C1 , S1 , C3 and S3 to be constant.
Parametric excitation of a linear oscillator
545
Figure 5. Principal instability interval (a) and the diagram showing the boundaries of the first
three intervals (b). Thin curves which deviate slightly at large m values from the boundaries of the
principal interval are plotted according to the approximate expression (25).
Substituting equation (24) into equation (16), we can omit the terms with frequency 5ω̃.
In the terms with frequency ω̃ we need to keep quantities up to the first and second order in m,
while in the terms with frequency 3ω̃ we need to keep only the terms of the first order. Finally
we arrive at a system of homogeneous equations for C1 , S1 , and C3 , S3 . The condition for
the existence of a nontrivial solution to the system gives us approximate expressions for the
desired boundaries ω± :
ω± = 2ω0 1 ± 12 m2 − (1/Q)2 + 11
(25)
m2 .
16
The term of the second order in m has the same value for both boundaries of the interval.
It does not influence the width of the interval, shifting it as a whole by a value proportional
to m2 .
The structure of the principal interval of parametric instability is shown in figure 5(a) for
the absence of friction (thick bounding curves), for Q = 20, and Q = 10 (thin inner curves).
It is more convenient to express the boundaries using not the frequency ω of the parametric
modulation, but rather the period T = 2π/ω. This convention is usually used in presenting
the stability map for Mathieu-type systems by the so-called Incze–Strutt diagrams. We also
use it in the simulation program [3] and for all figures in this paper. The shaded regions in
figure 5(b) show the first three intervals of parametric instability in one T–m diagram1 .
In the presence of viscous friction the principal interval reduces as the friction is increased
and disappears if Q < 1/m: its boundaries merge at the threshold. Equation (25) gives, for
the threshold, the value mmin = 1/Q which has been found above, equation (13), from
considerations based on the energy conservation.
An example of steady oscillations occurring at the left boundary of the principal instability
interval is shown in figure 4. The upper part of figure 6 shows the phase diagram and the
graphs of steady oscillations at the right boundary of the interval in the absence of friction.
We note the departure of the shape of these graphs from a sine curve, which is caused by the
contribution of higher harmonics (mainly of the third harmonic with the frequency 3ω̃ = 32 ω).
The ratio of the amplitude of the third harmonic to the amplitude of the
fundamental harmonic
is approximately the same for both boundaries |C3 /C1 | ≈ 38 m . The difference in the
1 Actually the curves in figure 5 are plotted with the help of somewhat more complicated formulae than
equation (25) (not cited in this paper), which are obtained by holding several harmonic components in the trial
function ϕ(t).
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E I Butikov
Figure 6. The phase trajectory of stationary oscillations occurring at the right boundary of the
principal instability interval (left), and the time-dependent graphs of the angular velocity of the
rotor and of the radial velocity of the weights (right).
patterns of oscillations at the left and right boundaries (compare the graphs in figures 4
and 6) is explained by the different phase shift of the third harmonic with respect to the
fundamental one.
The lower part of figure 6 corresponds to the right boundary in the presence of friction.
From the asymmetry of the graph, it is clear that in this case the energy received by the
oscillator is greater than the energy returned: during the intervals of negative values of v
(while the weights are moving towards the axis) the angular velocity ϕ̇ is greater in magnitude.
The energy excess compensates for the frictional losses, providing the stationary oscillations.
Outside the instability interval, the modulation of the moment of inertia causes only a few
changes in the shape of those decaying natural oscillations which may have been excited.
The simulations show that stationary oscillations at the boundaries of the principal
resonance include also the fifth and even seventh harmonic components with frequencies
5
ω and 72 ω, respectively. To find the boundaries with greater precision, we should include
2
these high harmonics into the trial function ϕ(t), equation (24). For the frictionless oscillator it
is more convenient to choose the time origin in such a way that the motion of the weights along
the rod is described in equation (2) by l(t) = l0 (1 + m cos ωt) instead of the sine function.
In this case the sine and cosine harmonics do not mix, that is, the stationary oscillations at
the left boundary of the interval include only harmonics of the cosine type, and at the right
boundary—of the sine type.
The final analytical expressions for the frequencies (and periods) of modulation and for the
relative contributions of high harmonics (as functions of m) at the boundaries of the instability
interval are complicated and hence not cited in this paper. However, they show a very good
agreement with the simulations. We cite here the calculated values for a certain modulation
depth m = 0.3 (30%). The corresponding experimental values (obtained in the simulation)
are shown in parentheses:
• Left (cosine-type) boundary: period T /T0 = 0.4066 (0.4066);
C3 /C1 = −0.103 (−0.101); C5 /C1 = 0.015 (0.016); C7 /C1 = 0.002 (0.001).
• Right (sine-type) boundary: period T /T0 = 0.5528 (0.5528);
S3 /S1 = −0.129 (−0.129); S5 /S1 = 0.020 (0.020); S7 /S1 = 0.003 (0.003).
Parametric excitation of a linear oscillator
547
Figure 7. The phase trajectory of oscillations in conditions of the second parametric resonance
(left) and graphs of the angular velocity and of the radial velocity of the weights (right).
For arbitrary values of the modulation depth m, the calculated boundaries of the principal
instability interval are shown by the first ‘tongue’ of the T–m diagram in figure 5.
6. Resonance of the second order
In contrast to the principal resonance, for which the energy supply due to the parameter
modulation occurs even if we assume the torsional oscillations to be purely sinusoidal (see
equation (7)), for the second resonance a positive net energy delivery is possible only by
virtue of the asymmetric distortions in the shape of the oscillations. These distortions are
clearly seen in figure 7. They provide the motion of the weights towards the axis of rotation
(v > 0) to happen on average at a greater (in magnitude) angular velocity ϕ̇ than the backward
motion. The distortions can be described by the second harmonic component (frequency 2ω),
whose contribution is proportional to the depth of modulation m. Hence the amount of energy
delivered by modulation in conditions of the second parametric resonance is proportional not
to m (as at the principal resonance, see equation (10)), but only to m2 .
In order to find the boundaries of the second interval of parametric instability with n = 2,
for which ω ≈ ω0 (or T ≈ T0 ), we look for a periodic solution of equation (18) near the
value ω = ω0 . Considering terms up to the second order in the modulation depth m, we
should include in this approximate solution the sinusoidal oscillations with the fundamental
frequency2 ω = ω0 + ε (the frequency of modulation) and the second harmonic with the
frequency 2ω:
ϕ(t) = C2 cos ωt + S2 sin ωt + C4 cos 2ωt + S4 sin 2ωt.
(26)
An example of such stationary oscillations is shown in figure 8.
Substituting ϕ(t) into equation (18), we transform there the products of sine and cosine
functions into sums, keeping the terms with the frequencies ω and 2ω. Thus, for the coefficients
2 However, it may be convenient to consider the fundamental frequency of parametrically excited stationary
oscillations to be always equal to one-half of the frequency of modulation. Then the spectrum of oscillations in
the case of resonance of an odd order includes only odd harmonics. The spectrum of stationary oscillations for
resonance of an even order includes only even harmonics (the amplitude of the fundamental harmonic being zero).
We follow this convention introducing relevant notation for the coefficients of harmonics in equation (26).
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E I Butikov
Figure 8. The phase trajectory of stationary oscillations occurring at the left boundary of the
second instability interval (left) and graphs of the angular velocity and of the radial velocity of the
weights (right).
C2 , S2 , and C4 , S4 we obtain the following system of homogeneous equations:
ω02
3
2γ
1 − 2 C2 + m2 C2 + 2mS4 −
S2 = 0,
ω
4
ω0
ω2
1
2γ
C2 = 0,
1 − 02 S2 + m2 S2 + 2mC4 +
ω
4
ω0
3S4 + 2mC2 = 0.
3C4 − 2mS2 = 0,
(27)
The last two equations of the system give us the expressions for the amplitudes C4 and S4
of the second harmonic in ϕ(t) in terms of the depth of modulation m and the amplitudes C2
and S2 of the principal harmonic:
C4 = 23 mS2 ,
S4 = − 23 mC2 .
(28)
These relations essentially mean that the amplitude of the second harmonic in the
stationary oscillations equals 23 m times the amplitude of the principal harmonic. The ratio of
the amplitudes of these harmonics is the same for both boundaries of the interval. However,
for the left and right boundaries these harmonics add with different relative phases, creating
different shapes of resulting oscillations. Graphs of oscillations occurring at the right boundary
of the second instability interval are shown in figure 9.
Substituting C4 and S4 from equation (28) into the first two equations of the system,
equation (27), and taking into account that ω2 = (ω0 + ε)2 ≈ ω02 + 2ω0 ε, we obtain the system
of two homogeneous equations for C2 and S2 :
2ε
7
2γ
− m 2 C2 −
S2 = 0,
ω0
12
ω0
(29)
2ε
13 2
2γ
C2 +
− m S2 = 0.
ω0
ω0
12
Nontrivial solution to this system exists if its determinant equals zero. This condition
determines the values of ε = ω − ω0 which correspond to the boundaries ω± of the second
interval of instability:
5
m2 ± 18 m4 − (4/Q)2 ω0 .
(30)
ω± = 1 + 12
Parametric excitation of a linear oscillator
549
Figure 9. The phase trajectory of stationary oscillations occurring at the right boundary of the
second instability interval (left) and graphs of the angular velocity and of the radial velocity of the
weights (right).
Figure 10. The phase trajectory of stationary oscillations occurring at the threshold of the second
instability interval (left) and the graphs of the angular velocity and of the radial velocity of the
weights (right).
We note that even the lower boundary is displaced to a higher frequency from the value
ω0 . The boundaries of the interval merge at the threshold. From equation (30), we find the
threshold conditions for the second parametric resonance:
2
4
5 2
ωres = 1 + m ω0 .
(31)
mmin = √ ,
Qmin = 2 ,
m
12
Q
Stationary oscillations occurring at the threshold of the second parametric resonance are
illustrated in figure 10.
In order to observe the mode of parametric regeneration (stationary oscillations at the
threshold of the second parametric resonance) for a given modulation depth m in the simulation
experiment, we should choose the period of modulation and the quality factor according to
equation (31), and set the initial conditions properly. For the threshold equations (29) give
S2 = C2 . Therefore,
ϕ̇(0) = ω0 C2 1 − 43 m .
(32)
ϕ(0) = C2 1 + 23 m ,
To produce stationary oscillations, we can choose arbitrarily an initial angular
displacement ϕ(0), and enter an initial angular velocity ϕ̇(0) = ω0 ϕ(0)(1 − 2m), as follows
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E I Butikov
from equation (32). Or, equivalently, we can choose an arbitrary initial velocity ϕ̇(0), and
enter an initial displacement ϕ(0) = ϕ̇(0)(1 + 2m)/ω0 .
In the absence of friction, the width of the second interval of instability is proportional to
the square of the depth of modulation: ω+ − ω− = m2 ω0 /4. Equation (30) gives the following
boundaries of the interval for zero friction:
7
ω+ = 1 + 13
m2 ω0 ,
ω− = 1 + 24
m2 ω0 .
(33)
24
To find the frequencies corresponding to these boundaries with a greater precision, we
should include more harmonics into the trial function ϕ(t), equation (26). In the absence
of friction it is more convenient to assume that the motion of the weights along the rod is
described in equation (2) by l(t) = l0 (1 + m cos ωt) instead of the sine function. In this case,
the stationary oscillations at the left boundary of the interval include only harmonics of the
cosine type, and at the right boundary—of the sine type.
The final (rather complicated) expressions for the periods of modulation and for the
relative contributions of high harmonics at the boundaries of the instability interval show
a very good agreement with the simulations. Below we cite the calculated values for the
modulation depth m = 0.3 (30%). The corresponding experimental values are shown in the
parentheses:
• Left (cosine-type) boundary: period T /T0 = 0.9502 (0.9502);
C4 /C2 = −0.203 (−0.202); C6 /C2 = 0.038 (0.039).
• Right (sine-type) boundary: period T /T0 = 0.9727 (0.9727);
S4 /S2 = −0.207 (−0.207); S6 /S2 = 0.039 (0.039).
For arbitrary values of the modulation depth m the calculated boundaries of this instability
interval are shown by the second ‘tongue’ of the T–m diagram in figure 5.
In order to observe stationary oscillations in the simulation experiment for the case when
friction is zero, we should choose the period of modulation corresponding to one of these
boundaries, and set the initial conditions properly. If l(t) = l0 (1 + m cos ωt), for the left
boundary we can choose arbitrarily an initial angular displacement ϕ(0) and zero initial
angular velocity. For the right boundary, vice versa, we choose arbitrarily an initial angular
velocity ϕ̇(0) and zero initial displacement.
7. Resonances of the third and higher orders
Oscillations that occur in conditions of any parametric resonance have a mean period which
is rather close to the natural one. To compensate for or to overcome the frictional losses by
modulation of the moment of inertia, two cycles of modulation must be complete during an
integer number n of (almost natural) oscillations of the rotor: 2T ≈ nT0 . The width T of
high order resonance bands (of the intervals of parametric instability) diminishes very quickly
as the order n of resonance is increased—as mn . The index α of the rate of the amplitude
growth also diminishes as fast as T does with the increase in n. Both these properties make
an experimental observation of parametric resonances of high orders (n > 1) at moderate
values of m very difficult. High-order instability intervals disappear in the presence of very
small friction.
Stationary oscillations at the threshold of parametric resonance of the third order are
shown in figure 11.
In order to find the boundaries of the third instability interval in the absence of friction, we
assume that the weights move according to l(t) = l0 (1 + m cos ωt), and use the trial function
ϕ(t) that includes the fundamental harmonic of the frequency 12 ω and several high oddnumbered harmonics of frequencies 32 ω, 52 ω, . . . . Stationary oscillations at the left boundary
Parametric excitation of a linear oscillator
551
Figure 11. The phase trajectory and the time-dependent graphs of stationary oscillations at the
threshold of the third parametric resonance.
comprise only harmonics of cosine type, and at the right boundary—of sine type. After
substituting the trial function into the differential equation
d
d
(1 + m cos ωt)2 ϕ + ω02 ϕ = 0,
(34)
dt
dt
we equate to zero the coefficients of cosine (or sine) functions with frequencies 12 ω, 32 ω,
5
ω, . . . , and thus get a system of homogeneous equations for the coefficients C1 , C3 , . . .
2
(or S1 , S3 , . . .) of harmonic components in the trial function. The condition of existence of a
nontrivial solution to this system yields an equation for the desired boundaries. This equation
is the same as for the boundaries of the principal instability interval, but this time we look for
its approximate solution in the vicinity of 32 T0 (instead of 12 T0 ). The third harmonic component
(frequency 32 ω) dominates the spectrum.
To increase precision, more harmonics should be included in the trial function ϕ(t). We
cite below the values of the period and of the relative contributions of different harmonics at
stationary oscillations for the modulation depth m = 0.3, obtained by a calculation in which
harmonics up to 13th order were included. (We use the Mathematica-4 package by Wolfram
Research Inc.) The corresponding experimental values are shown in parentheses:
• Left (cosine-type) boundary: period T /T0 = 1.4336 (1.4336);
C1 /C3 = 0.107 (0.110); C5 /C3 = −0.289 (−0.288); C7 /C3 = 0.065 (0.067).
• Right (sine-type) boundary: period T /T0 = 1.4369 (1.4369);
S1 /S3 = 0.135 (0.136); S5 /S3 = −0.291 (−0.292); S7 /S3 = 0.066 (0.066).
Stationary oscillations at the boundaries of the third interval of parametric instability are
shown in figure 12.
Similar calculations allow us to find the periods of modulation at which resonances of
higher orders occur. The corresponding ranges of parametric instability are very narrow, that
is, both their boundaries very nearly coincide. We cite below the calculated values of the
modulation periods and the spectral composition of stationary oscillations for the boundaries
of the fourth and fifth resonances (at m = 0.3):
• Left (cosine-type) boundary of fourth resonance: period T /T0 = 1.9107 (1.9107);
C2 /C4 = 0.219 (0.220); C6 /C4 = −0.377 (−0.374); C8 /C4 = 0.100 (0.102);
C10 /C4 = −0.023 (−0.021).
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E I Butikov
Figure 12. The phase trajectories and the time-dependent graphs of stationary oscillations at the
boundaries of the third interval of parametric instability.
• Right (sine-type) boundary of fourth resonance: period T /T0 = 1.9112 (1.9112);
S2 /S4 = 0.222 (0.222); S6 /S4 = −0.377 (−0.377); S8 /S4 = 0.100 (0.100); S10 /S4 =
−0.023 (−0.023).
• Left (cosine-type) boundary of fifth resonance: period T /T0 = 2.3872 (2.3872);
C1 /C5 = 0.017 (0.019); C3 /C5 = 0.319 (0.321); C7 /C5 = −0.459 (−0.466); C9 /C5 =
0.124 (0.146).
• Right (sine-type) boundary of fifth resonance: period T /T0 = 2.3873 (2.3873);
S1 /S5 = 0.020 (0.020); S3 /S5 = 0.321 (0.321); S7 /S5 = −0.468 (−0.468); S9 /S5 =
0.142 (0.144).
The spectral composition, phase trajectories and time-dependent graphs of stationary
oscillations at the boundaries of the fourth and fifth intervals of parametric instability are
shown in figures 13 and 14, respectively.
Almost exact coincidence of both boundaries for the high-order intervals means that
at the period of modulation corresponding to one of the intervals, we can actually observe
(in the absence of friction) not a resonant growth but rather stationary oscillations of a constant
(arbitrarily large) amplitude. From the graphs in figures 13 and 14 we can conclude that at
exact tuning to n-order resonance, the oscillator completes just the whole number n of natural
oscillations (of varying period and amplitude) exactly during two cycles of modulation. The
process is periodic at arbitrary initial conditions, in contrast to the boundaries of low orders,
for which special initial conditions are required to provide periodic oscillations.
Parametric excitation of a linear oscillator
553
Figure 13. The spectrum, phase trajectory and time-dependent graphs of stationary oscillations at
the right boundary of the fourth interval of parametric instability.
Figure 14. The spectrum, phase trajectory and time-dependent graphs of stationary oscillations at
the left boundary of the fifth interval of parametric instability.
This behaviour can be explained in terms of the familiar phenomenon of frequency
modulation. For parametric resonances of high orders, the weights move along the rod rather
slowly compared to the natural torsional oscillations of the rotor (T T0 ). A slow periodic
variation of the moment of inertia means that the current natural frequency of the oscillator is
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E I Butikov
slowly modulated. We see clearly in figures 13 and 14 how oscillations slow down when the
weights are moved towards the ends of the rotor, and vice versa. Hence we can consider the
motion of the rotor in conditions of high-order parametric resonance as a frequency modulated
oscillation, in which the natural oscillation—the dominating harmonic component—plays the
role of a carrier.
The spectral composition shown in figures 13 and 14 gives convincing evidence of this
interpretation. The harmonic component with the frequency nω/2 ≈ ω0 has the greatest
amplitude (the carrier). The coefficients Cn−2 and Cn+2 of lateral spectral components with
frequencies (nω/2) ± ω have opposite signs and (for n 1) are nearly equal in magnitude.
This spectrum is characteristic of the frequency modulation.
8. Concluding remarks
We have developed in this paper a theoretical approach to the phenomenon of parametric
resonance complemented by a computerized experimental investigation.
A simple
mathematical model of the physical system (based on a linear differential equation) is used.
The model allows a complete quantitative description of the parametric excitation, which
can be verified by the simulations [3]. The visualization of motion, along with the plot of
the graphs of different variables and phase trajectories, makes the simulation experiments
very convincing and comprehensible. This investigation provides a good background for the
study of more complicated nonlinear parametric systems such as a pendulum whose length is
periodically changed (a model of the playground swing), or a pendulum with the suspension
point driven periodically in the vertical direction [2].
References
[1] Case W 1996 The pumping of a swing from the standing position Am. J. Phys. 64 215–20
[2] Butikov E I 2001 On the dynamic stabilization of an inverted pendulum Am. J. Phys. 69 755–68
[3] Butikov E I 1996 Educational software package Physics of Oscillations ed J S Risley and R W Brehme (New
York: AIP)
[4] Phelps F M and Hunter J H Jr 1965 An analytic solution of the inverted pendulum Am. J. Phys. 33 285–95
Phelps F M and Hunter J H Jr 1966 An analytic solution of the inverted pendulum Am. J. Phys. 34 215–20,
533–35
[5] Landau L D and Lifschitz E M 1958 Mechanics (Moscow: Fizmatlit)
Landau L D and Lifschitz E M 1976 Mechanics (New York: Pergamon) pp 93–5 (Engl. Transl.)
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