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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 cost 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 cost 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 cost 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 cos2ft 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