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Transcript
Audio Engineering Society
Convention Paper
Presented at the 122nd Convention
2007 May 5–8 Vienna, Austria
The papers at this Convention have been selected on the basis of a submitted abstract and extended precise that have been peer
reviewed by at least two qualified anonymous reviewers. This convention paper has been reproduced from the author's advance
manuscript, without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents.
Additional papers may be obtained by sending request and remittance to Audio Engineering Society, 60 East 42 nd Street, New
York, New York 10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof,
is not permitted without direct permission from the Journal of the Audio Engineering Society.
Chaotic state in an electrodynamic
loudspeaker
Danijel Djurek1, Ivan Djurek2, and Antonio Petosic2
1
AVAC – Alessandro Volta Applied Ceramics, Laboratory for Nonlinear Dynamics, Kesten brijeg 5,
Remete, Zagreb, Croatia
[email protected]
2 Faculty
of Electrical Engineering and Computing, Dept. of Electroacoustics, Unska 3, Zagreb, Croatia
[email protected]; [email protected]
ABSTRACT
An electrodynamic loudspeaker has been operated in nonlinear regime when k-factor strongly increases with
displacements. For driving AC currents up to 2 A the vibration spectrum contains high frequency harmonics of
classic von Kármán type, for currents in the range 2.6 to 4 A doubling of driving period appears, and for currents in
the range from 4 to 4.8 A multiple sequences of subharmonic vibrations begin with 1/4f and 3/4f. An application of
currents higher than 4.8 A results in a white noise spectrum, which is a characteristic of chaotic state.
1.
INTRODUCTION
Chaotic state in dynamical systems [1,2] opposes to
generally accepted concept of deterministic evolution of
particular segments in these systems, and most
important property is unpredictability of the trajectories
in momentum-displacement diagrams. An infinitesimal
change of initial conditions may result in dramatic
change of evolution steps which contrasts the
experience gathered by an observation of non-chaotic
systems when small error in initial conditions results in
the comparable error of trajectories.
Chaotic evolution of dynamic system resides on the
fulfillment of two fundamental conditions; firstly
system must be described by at least three independent
dynamical variables, and second some nonlinear term(s)
must be involved in dynamic equations.
Dynamic system driven with period T can be described
[3] set of equations:
dxi
 Fi x1 ,..., x N , i  1,..., N
dt
(1 )
The necessary condition for appearance of the chaotic
state in the dynamic system is N  3 N.
Djurek et al.
Chaotic state in an electrodynamic loudspeaker
Variables xi are assumed to be periodic in time with the
same period T. By an increase of driving force the
system launches sequence of n trajectories each of them
having revolving time T  2 n  T , that is the first
trajectory must revolve with double period 2·T.
electromotive force induced in the voice coil of the wire
length l vibrating in magnetic field B.
The set of nonlinearly coupled equations (1) is given by:
dx
v
dt
The equations (1) are mutually coupled by nonlinear
coefficients, and in most practical cases they are solved
numerically.
dv
 B  e i    v   02  k
dt
Linsay performed [4] one the first experiments with the
chaotic state realized in an electric LRC circuit
represented as damped forced oscillator described by an
equation of motion:
L
d 2I
dI 1
R
  I  f 0  cost 
2
dt C
dt
d

dt
Assuming that the magnetic field B in (3) is
homogeneous in the range of membrane displacements
all remaining terms k, M, XS, RS and RM are dependent
on the amplitude of driving current I0, and dependence
of the natural frequency 0 on driving force is generally
accepted as a measure of nonlinearity [5]. Nonlinearity
of k –factor is coupled to gas impedances RS and XS
since these quantities depend on the frequency. The
mass nonlinearity follows from the membrane internal
degrees of freedom, since for low vibration amplitudes
only central part of the membrane is involved in the
motion until full translation motion is accomplished at
higher vibration amplitudes for driving currents I0 > 120
mA. By further increase of driving current the
(2 )
LRC circuit was driven by an external oscillatory force
with period T, and nonlinear term in the system was
ensured by voltage dependent electric capacity C. By a
gradual increase of driving voltage V period doubling
2  T  4  was observed, and this was manifested by
sequence of subharmonics. By next discrete value of
driving voltage new period T* = 4·T appeared, and
further continuation of the procedure resulted in a white
noise spectrum which is characteristic of the chaotic
state.
X

 M  S


dx
 d 2x
 2  R S  R M  R BEF   k  x  BI 0 l  cost 
dt
 dt
RS  RM  R BEF

X
M S

k
M
Vibration system of an electrodynamic loudspeaker
(EDL) provides the next opportunity to realize chaotic
state in an anharmonic forced oscillator. Eq. 2 applied to
vibrating EDL when electric quantities L and 1/C are
replaced respectively by mass of the membrane M and k
-factor is shown in Eq. 4 where  is natural frequency
of the vibrating system.
RS and XS are respectively real and imaginary parts of
acoustic impedance Z = RS + XS, RM is the friction of the
vibrating membrane and RBEF  B 2 l 2 R  R g is back


(3 )
XS

  02
(4)
BI 0 l

X
M S

nonlinearity of the vibration system is dominated by the
displacement dependent k-factor, and in this work only
this regime of extreme nonlinearity will be considered,
when Eq.4 reduces to:


d 2x
dx

 k  x  1    x 2    cost 
dt
dt 2
2.
(5)
EXPERIMENTAL
Experiments were performed on the loudspeaker with
natural frequency f0= 45 Hz, diameter 2·r = 16 cm, cone
AES 122nd Convention, Vienna, Austria, 2007 May 5–8
Page 2 of 6
Djurek et al.
Chaotic state in an electrodynamic loudspeaker
angle 120° and resistance r=8 . The measured physical
quantity was electric impedance with real and imaginary
parts given, in harmonic approximation by:
impedance resonance curve is already visible at I0=100
mA (Fig.3a). When driving current increases up to
I0=1A strong narrowing and deformation of the
resonance curve takes place, which is shown in Fig. 3b.
For I0=2 A more pronounced cut-off of impedance
appears. While the amplitudes of resonance curves are
little changed with increasing driving current,
corresponding change of the phases (dashed lines) is
more faster and transition to cut-off state is
accompanied by reduction of the phase by factor two.
In addition, it is important to note that the phase is
independent on frequency at frequencies higher than
cut-off frequency. This phenomenon may be attributed
to the phase locking, when vibrating system oscillates
with phase  which is related to the phase of driving
force 0 as ratio of two integers /0=m/q.
Z re  R 
B 2 l 2  R 2
k  M 
Z im  L 
2 2
 R 2 2


(6)
B l  k  M 
2 2
2
k  M 
2 2
 R 2
2
The resonance frequency of the vibrating system was
evaluated from the zero value of the Zim and its
dependence on the driving current is shown in Fig. 1.
An initial decrease with increasing current is result of
the mass nonlinearity, and subsequent strong increase is
due to the nonlinear k-factor.
It is obvious that the lowest state of phase locking is for
m=1, and q=2, and this is accomplished in listed
experiments.
Figure 1 Resonance frequency as function of driving
current
Amplitude cut-off precedes the chaotic state in the
highly nonlinear regime (I0>120 mA), when the
resonance curve of the real part of electric impedance
deforms, with increasing driving current. Vibration in
air, which is characterized by an enhanced mass M due
to the imaginary part of acoustic radiation impedance
results in resonance curves shown in Fig. 2 a,b,c. For
I0=1 A resonance curve has more symmetric shape, and
for I0=2 A an asymmetric dependence on frequency
starts, until for I0=4 A amplitude suddenly drops to the
lower value. Deformation of the impedance resonance
curve is more pronounced when loudspeaker vibrates in
an evacuated space, since in this case vibration
amplitude for given current is higher than that in air,
and lower values of driving currents are required to
induce the same effects. Slight asymmetry of the
Figure 2 Resonance curves for different currents
recorded in air
AES 122nd Convention, Vienna, Austria, 2007 May 5–8
Page 3 of 6
Djurek et al.
Chaotic state in an electrodynamic loudspeaker
When the vibrating system leaves the chaotic state by
reduction of the frequency (Fig. 5) the cut-off frequency
obeys the hysteretic behavior of the width 10 Hz.
Figure 3 Resonance curves recorded in vacuo for
different driving currents
Subharmonic sequences were generated by gradual
increase of driving current keeping the selected driving
frequency constant (45 < f < 55 Hz). The sequences
appear at particular discrete values of I0, and the process
recorded in air is shown in Fig. 4. The resonant mode
(f=45 Hz) is clearly visible in all segments of
development of sequences, and in Fig. 4a, recorded
slightly before appearance of cut-off, it is shown the
resonant mode with the corresponding sequence of
higher von Kármán type harmonics [6]. A gradual
increase of driving current above cut-off results in
appearance (Fig. 4b) of new harmonic at f/2, the
phenomenon known as doubling of driving period, and
the corresponding sequence of harmonics which are
multiples of f/2 is also visible. The new subharmonic
sequences starting at f/4 and 3f/4 are shown in Fig 4c,
and finally, at driving current I0=4.8 A the subharmonic
sequences disappear (Fig. 4d) and spectrum resembles
white noise.
Figure 4 Vibration spectrum recorded in air for different
driving currents
AES 122nd Convention, Vienna, Austria, 2007 May 5–8
Page 4 of 6
Djurek et al.
Chaotic state in an electrodynamic loudspeaker
(d /dA=0) the system is unstable and amplitude is
jumping between two discrete values. Morse derived the
critical driving force Fcr necessary for appearance of the
amplitude cut-off, which in our notation reads:
Fcr  1.4  d   02  M  Q

3
2
Q>>1
(8)
In these experiments M=17.5 gram, Q=2.2, =283
rad/sec, which gives Fcr=3.5 N. The B·l value of the
loudspeaker is 5.5 T·m, and necessary driving current
I0=0.64 A, which fits the data shown in Fig. 3.
Figure 5 Hysteresis of impedance measured in vacuo
near cut-off frequency
3.
DISCUSSION
Eq. 5 is nonlinear ordinary differential equation and it
may be solved by application of perturbation method,
which provides the correct result up to cut-off
displacements, at least qualitatively.
Perturbation method for solving Eq. 5 was published in
a number of textbooks and most known are those of
Morse [7] and Landau [5]. The Morse approach is
applied in this work since it uses the normalization of
displacements, which properly matches our vibrating
system. It was introduced the static displacement
xst=F/k, and parameter  in Eq. 5 is written as =(xsr/d)2,
d being the characteristic displacement at which the
nonlinear force equals the linear one. In our vibrating
system, characteristic displacement corresponds to the
driving current at minimum in Fig. 1 (approximately,
d=4.2 mm). Following the method in reference [7] the
amplitude of the trial solution x=A·cos( t) of Eq. 5
satisfies equation:
2
 
3


   A 2  1   2  P  A 2   A 2  1
Q
4


 
Figure 6 Vibration amplitude according to Eq. 7
4.
CONCLUSION
An electrodynamic loudspeaker with strongly nonlinear
elastic suspension and corresponding k-factor dependent
on driving current I 0  cos2ft  obeys the amplitude
cut-off near the resonance frequency. The first
subharmonic sequence appears with period doubling 2/f,
which is followed by further subharmonics starting at
f/4 and 3f/4. Finally, at highest driving currents the
white noise spectrum is visible and chaotic state
appears.
2
 M 
P

 k d 
(7)
5.
REFERENCES
[1] Henry Abarbanel D.I., “The Analysis of Observed
Chaotic Data in Physical Systems”, Rev.
Mod.Phys. 65 (1993) 1331.
2
[2] E. Ott, Chaos in Dynamical Systems, 2nd Edition,
Cambridge University Press 2002.
Q stands for Q-factor.
Eq. 7 has 3 physically acceptable solutions for A, and
frequency dependence is visualized in Fig. 6. In the
region bounded with the slopes of A being infinite
AES 122nd Convention, Vienna, Austria, 2007 May 5–8
Page 5 of 6
Djurek et al.
Chaotic state in an electrodynamic loudspeaker
[3] G.L. Baker and J.P. Gollub, Chaotic Dynamics an
Introduction, 2nd Edition, Cambridge University
Press 1998.
[6] Th. Von Kármán, “The Engineer Grapples with
Nonlinear Problems”, Bulletin of American
Mathematical Society”, 46 (1940) 615.
[4] P. S. Linsay, “Period Doubling and Chaotic
Behavior in a Driven Anharmonic Oscillator”,Phys.
Rev. Lett. 47 (1981) 1349.
[7] P.M. Morse and K.U. Ingard, Theoretical
Acoustics, Princeton University Press 1968, p. 847855.
[5] L. D. Landau and E. M. Lifshitz, Mechanics, 3rd
Edition, Course of Theoretical Physics, Volume 1,
Elsevier, 2005, p. 87-93.
AES 122nd Convention, Vienna, Austria, 2007 May 5–8
Page 6 of 6