AES
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.
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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.
Ivan Djurek 1 , Antonio Petosic 2 , and Danijel Djurek 3
1 Faculty of Electrical Engineering and Computing, Dept. of Electroacoustics, Unska 3, Zagreb, Croatia ivan.djurek@zg.t-com.hr; antonio.petosic@fer.hr
2 AVAC – Alessandro Volta Applied Ceramics, Laboratory for Nonlinear Dynamics, Kesten brijeg 5,
Remete, Zagreb, Croatia danijel.djurek@zg.t-com.hr
ABSTRACT
Vibration of the loudspeaker membrane was analyzed in the regime of comparatively low driving currents ( I
0
< 100 mA) in terms of mass nonlinearity M eff
and friction R
M
. Latter contributes to damping term of the differential equation of motion and depends on the elongation of vibration. R
M
is the sum of intrinsic friction R i
of the membrane and friction R v
coming from air viscosity on its surface. Independent measurements of flexural strength of the membrane were performed and correlated to experimental observations of the vibrating system. Experiments were also performed on the membrane additionally reinforced by application of materials with higher Young modulus. air radiation impedance should be added to mass M in the form M

X
S

.
1.
INTRODUCTION
An electrodynamic loudspeaker is represented as a driven harmonic oscillator described [1] by an ordinary differential equation of motion: d dt
2
2 x

R
M dx dt
 k
M
 x
 f
M
0
 cos
 
(1 )
The friction term in (1) consists of three contributions
R=R
S
+R
M
+R
BEF
. R
S is real part of the air radiation impedance, R
M
membrane friction and R
BEF
is real part of the electromotive force induced in the voice coil.
Amplitude of driving force is expressed as f
0
= B·I
0
·l , B and l being respectively magnetic field and length of voice coil wire.
M is the mass of oscillating parts which include the membrane and voice coil. The imaginary part X
S
of the
Harmonic oscillations are preserved by an independency of coefficients R , M and k -factor (usually referred as to

Djurek, et al. Mass nonlinearity and intrinsic friction the Hooke constant k ) on displacements, that is, on the strength of driving current I
0
. The solution of equation
(1) gives for the vibration amplitude:
A
2 
 k

B
M

2
2
2 2
I

0
2 l

R
2

2
(2 ) expressed as M eff
/ M = r eff
2 / r 2 is the main cause of an anomalous dependence of the natural frequency on I
0
(Fig. 1), when

0
initially decreases with increasing I
0 until effective area is saturated and initial decrease is followed by an increase predominantly due to displacement dependent k -factor.
Electric impedance Z of the vibrating loudspeaker is obtained by multiplication of equation (1) by dx / dt , dividing the result by I
0
2 and by adding electrical part of loudspeaker's coil, electrical resistance R and inductance

L . Real and imaginary parts read out to be:
47
46
45
Z real

R

 k

B 2 l
M

2
2
2


R


2
R 2

2
Z imag
 
L

 k
B

2 l 2

M

 k
2

2
M


R
2
2




2
(3 )
44
43
0 20 40 60 80
Current (mA)
100 120 140
Harmonic oscillation is an idealized representation which also assumes an ideal sound reproduction, without distortions usually manifested as higher harmonics and/or subharmonics. However, such an idealistic picture contradicts the real application situations and equation (1) should be considered with components dependent on the amplitude of driving current I
0
. One measure of nonlinearity is dependence of natural frequency

2
0
 k M of vibrating system on driving current I
0
.
Nonlinearities entering equation (1) can be divided in two groups: a) Nonlinear electromechanical system consists of vibrating membrane and voice coil elastically suspended on the loudspeaker frame. In harmonic approximation of equation (1) mass M is assumed to be mathematical mass point suspended on an ideal mass-less spring which is perturbed and damped from outside. This presumes that membrane is perfectly rigid with no internal vibration degrees of freedom. However, this is a crude oversimplification since, instead of translation motion, the central part of the membrane, for low amplitude oscillations, is more deflected than the perimeter [2], and mass M measured in an independent experiment must be multiplied by an effective area a eff
 r
2 eff
 
, which increases with driving current I
0
and saturates at true geometric area r
2  
of the membrane for some critical value I
0C
. The mass nonlinearity
Figure 1 Resonant frequency measured in vacuo versus driving current.
The intrinsic friction R i
can be evaluated by vibration in vacuo and represents dissipative term in the membrane coming from the internal vibration degrees of freedom.
It manifests as the hysteresis of the strain dependent on the stress acting on the membrane surface. Hysteresis encircles an area in stress-strain diagram and results in an irreversible release of the heat. Like in the case of the vibrating mass the amount of intrinsic friction is also proportional to the effective area of the membrane being involved in deformation during the vibration cycle. This results in an independency of Q-factor

Q

M

R
M
 on the driving current I
0
, which is in a fair agreement with the measurements in vacuo on commercially available loudspeakers. The strength of nonlinearity may be divided in two levels. Medium level nonlinearity transforms the harmonic and symmetric resonance curve (2) to the asymmetric form as it is shown in Fig. 2a, but the change of Q-factor is limited extending up to 7 – 8 percents. The high level nonlinearity starts for driving currents higher than critical I
0C
when initial decrease of the resonance frequency (Fig. 1) is followed by an increase which results in strong resonance asymmetry (Fig. 2b) and finally in amplitude cut-off [3] shown in Fig. 2c.
Experiments described in this paper deal only with medium level nonlinearities when the change of Qfactor is within the tolerable limit.
AES 122nd Convention, Vienna, Austria, 2007 May 5 –8
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Djurek, et al. Mass nonlinearity and intrinsic friction b) Nonlinear contribution of vibrating air above the membrane is described [4] by Navier-Stokes equation:
 p
  d v
  v
 
 v
    v dt
(4 ) and it appeals an attention for the boundary layer problem close to the membrane surface.
The scope of this paper is to separate and investigate both types of listed nonlinearities of the loudspeaker operating in an evacuated chamber and in air.
24
22
20
18
16
14
12
10 where v is velocity of vibrating air,

is air density and

dynamic viscosity.
 p corresponds to pressure gradient in the gas system.
35
30
25
20
15
10 a
I =100 mA
15
10
5
0
-5
-10
-15
I b
=1 A
15
10
5
0
-5
-10
-15
2.
EXPERIMENTAL
Experiments described in this paper deal with two types of loudspeaker membranes, with standard commercial membrane ( S -membrane) and membrane reinforced by foamed phenolic resin ( R -membrane).
1000
900
800
700
600
500
400
300
-4 -2
R - membrane
S - membrane
0
Amplitude (mm)
2 4
Figure 3 k – factor of S and R membrane versus amplitude.
17
16
15
14
13
12
11
10
20 30 40 50
Frequency (Hz)
60 c
I =2 A
8
6
4
2
0
-2
-4
-6
-8
70
Figure 2 Loudspeaker impedance, measured in vacuo , versus frequency for selected driving currents.
The last term in equation (4) is Stokes viscosity and it is mathematically less tractable than first two terms. In addition, this term represents no acoustic radiation. Air viscosity

is added to intrinsic membrane friction R i
,
k-factor and effective mass were measured by the use of DC current in the voice, and constant
B·l
of the loudspeaker was evaluated in an independent experiment by the use of calibrated weights and the load was equalized with the force from DC current for the same displacement of the membrane. Fig. 3 shows dependence of k-factor on displacement, and it is apparent the asymmetry to the origin, the main reason for this asymmetry being the conic shape of the membrane. In addition to asymmetry it is also present a zero off-set in direction of positive displacement. The amplitude dependent effective mass evaluated by vibration in vacuo and calculated from equation (6) is shown in Fig. 4, for both S- and R- membrane.
Dependence of Q-factor for both S- and R- membrane on the vibration amplitude recorded in vacuo and air is shown in Fig. 5 and 6. Impedance measurements were performed with a relatively large value resistor placed in series with loudspeaker, in order that back electromotive force induced in the voice coil could be neglected.
AES 122nd Convention, Vienna, Austria, 2007 May 5 –8
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Djurek, et al.
11
10
9
8
7
15
14
13
12
0
S-membrane
R-membrane
1 2 3
Amplitude (mm)
4 5
7
Figure 4 Effective mass of S- and R-membrane measured in vacuo .
6
Mass nonlinearity and intrinsic friction
Stress-strain hysteresis was evaluated in an experimental set-up shown in Fig. 7. S-membrane was detached from the loudspeaker frame and elastic suspension was removed. The perimeter was then clamped between two metal rings and the load of calibrated weights W was used on the bottom of the membrane. Displacements were measured by the use of the laser triangulation method operating with relative accuracy of 2 microns.
The evaluated stress-strain diagram is shown in Fig. 8.
By decreasing stress there is visible hysteretic behavior of the membrane displacement and amount of dissipated energy is given by the area described within stress-strain curves. For the membrane of radius r = 8 cm and cone angle
 = 120° the dissipated energy evaluated from diagram is 3.4· 10 -3 Joules.
LASER
DISTANCE
METER
LASER
DISTANCE
METER
6
5
vacuum
air
4
3
S-membrane
2
0 1 2 3
Amplitude (mm)
4 5 6
Figure 5 Q-factor of S-membrane measured in vacuo and air.
6,5
6,0
5,5
5,0
vacuum
air
4,5
4,0
R-membrane
3,5
0,0 0,5 1,0 1,5 2,0
Amplitude (mm)
2,5 3,0 3,5
Figure 6 Q-factor of R-membrane measured in vacuo and air. w
Figure 7 Setup for stress-strain measurements. w
Figure 8 Stress-strain diagram of the S-membrane.
Friction term R
M
can be evaluated from the vibration resonance curve (Eq. 2) or from impedance curve (Eq.
AES 122nd Convention, Vienna, Austria, 2007 May 5 –8
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Djurek, et al. Mass nonlinearity and intrinsic friction
3). At resonance frequency vibration amplitude in air is given by:
A
0


0

B
R
S

I
0

 l
R
M
 (5 )
1,2
1,0 R
M
vacuum
air
Here an alternative procedure was used based on expression for natural frequency in air:
0,8
R i
0,6

0
2 
M eff k eff

X

S
(6 )
This expression is also valid in nonlinear regime, with only assumption that physical quantities like elongation, velocity and acceleration are periodic in time. The next step is the measurement of k -factor by the methods described above. The M eff

X
S

is than calculated from the natural frequency dependent on driving current
(Fig. 1). Finally, R
M
is calculated from Q-factor of loudspeaker's impedance resonance curves (Eq.3).
0,4
0,0 0,5 1,0 1,5 2,0
Amplitude (mm)
2,5 3,0
Figure 10 R
M
and R i
measured on R -membrane.
3,5
0,7
0,6
0,5
R-membrane
S-membrane
0,4
1,2
vacuum
air
0,3
1,0
0,8
R
M
0,2
0,0 0,5 1,0 1,5 2,0
Amplitude (mm)
2,5 3,0
Figure 11 Friction R v
versus amplitude.
0,6
R i
0,4
3.
DISCUSSION
0
Figure 9 R
M
1
and R i
2 3
Amplitude (mm)
4 5
measured on S -membrane.
6
R
M
is the sum of two contributions; intrinsic membrane friction R i
, recorded in vacuo and contribution of air R v
.
Fig. 9 shows R i
and R
M
values measured on S -membrane and dependent on the vibration amplitude. Intrinsic friction R i
is less dependent on the vibration amplitude, which contrasts the fairly strong dependence evaluated in air. This difference is less pronounced in Rmembrane (Fig. 10). The friction R
V

R
M

R i
is plotted in Fig. 11 and it is apparent similarity for two types of membranes, both in magnitude of friction and dependence on the vibration amplitude.
As it is shown in Fig. 3, k -factor of the loudspeaker differs for S- and R- membranes. Since elastic suspension of the membrane is unaffected by reinforcement such a result may be surprising, and only explanation resides on the contribution of the membrane to total elasticity. The k -factor can be considered as a complex function of two contributions k = f ( E
M
,k s
); k s being the elastic constant of the membrane suspension and represents single contribution to k when membrane is assumed to be perfectly rigid. S -membrane is far from perfect rigidity, and above listed arguments for mass M and intrinsic friction R i
can also be applied to the kfactor, that is, parts of the membrane which are virtually not involved in the vibration for small driving currents
(< 100 mA) contribute to total elasticity, and Young modulus E
M
of the material, being different for S - and
R - membranes, must be respected. This role could also
AES 122nd Convention, Vienna, Austria, 2007 May 5 –8
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Djurek, et al. Mass nonlinearity and intrinsic friction be recognized in the asymmetric result of the kfactor when membrane is displaced by the use of DC current of opposite sign. Due to the conic shape of the membrane constants k differ, which should not be the case when only elastic suspension affects the kfactor. distance z [4] up to some value v
 at infinity obeying the simple law: v
 v


1
 e
 z


 


 
(9 )
When loudspeaker vibrates in vacuo it is possible to evaluate R i
by equating friction power in equation (1) integrated over half time period to area described by hysteresis in Fig. 8. An independent experiment was used to measure deformations (strain) of center versus perimeter of the S-membrane and showed that vibration amplitude which corresponds to the maximum strain of the membrane itself (90 microns in Fig. 8) is 6.1 mm.
R i

T

0
2


 dx dt


2 dt

A hyst

3 .
4

10

3
J (7 )
In air

= 1.98·10 -5 kg/m·s,

= 1.18 kg/m 3 , and for

=317 Hz

= 0.23 mm.
The part of friction R v
obtained by subtraction of R i from the value R
M
measured in air (Fig. 11), nearly being the same by magnitude and dependence on the amplitude for both S - and R - membrane, additionally legitimates the role of viscous friction, since the thickness of boundary layer is unaffected by type of the membrane material.
4.
CONCLUSION
For A = 6.1 mm and f= 40 Hz R i
=0.65 kg/s.
Agreement with R i
obtained from Q-factor (Fig. 9) is within a half order of magnitude.
The role of Navier-Stokes equation (4) in the vibration dynamics of the loudspeaker is partly resolved by Morse expressions [5] for real R
S
and imaginary X
S
of acoustic impedance of air which are given by:
R
S
X
S



0 j

 c

S  1
0
 c
J
1
 
 r

S


H
2
1

2

2 r
2




(8 )
Where

is air density, c is velocity of sound, S is membrane area,

is wave number and r is membrane's radius.
An attempt has been made to elucidate the physical picture of friction dissipative term R
M
in equation of motion, and it was found that R
M
is the sum of two parts; R i
- the intrinsic friction evaluated in the loudspeaker vibrating in an evacuated chamber and R v
- the friction coming from the Stokes viscosity near the membrane surface. At frequencies about 50 Hz nonradiation term R i
is nearly an order magnitude higher than real part R s
of acoustic radiation impedance calculated by Morse [5]. R i
is dependent on displacements and correlates the effective mass M eff
of the membrane leaving Q-factor nearly independent on the amplitude of driving force. When air at 1 bar is added, the total friction R
M
is dependent on vibration amplitude.
5.
REFERENCES
[1] M. Tenenbaum and H. Pollard, Ordinary
Differential Equations, Dover, 1963.
Typical value of real part of impedance in air for loudspeaker with radius of the membrane of 8 cm at frequency f =47 Hz is R
S
=0.04 kg/s. Expressions (8) can be inhered from Navier-Stokes equation (4). The first term on the right side is inertial and contributes to X
S
, while the second partly (with turbulence excluded) contributes to R
S
. The last, viscous term, has no radiation significance, and is usually omitted in the evaluation of radiation impedance. Its importance is accentuated only within the narrow boundary layer

on the membrane surface where the air velocity component along the surface is zero at surface and increases with
[2] I. Djurek, D. Djurek, A. Petosic and N. Demoli,
Nonlinear stiffness of the loudspeaker measured in an evacuated space, 121 st AES Convention, October
5-8, 2006, S. Francisco, CA, Convention paper
6940
[3] L.D. Landau and E.M. Lifshitz, Mechanics, 3 rd
Edition, Course of Theoretical Physics, Volume 1,
Elsevier, 2005, p. 87-93
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Djurek, et al. Mass nonlinearity and intrinsic friction
[4] L. D. Landau and E. M. Lifshitz, Fluid Mechanics,
2 nd Edition, Course of Theoretical Physics, Volume
6, Elsevier, 2004.
[5] P.M. Morse, Vibration and Sound, Acoustical
Society of America, 5 th printing, 1995.
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