JAES: Loudspeaker Suspensions Modeled by Fractional Derivatives

Motivation
A lot of materials used in loudspeaker suspensions exhibit significant frequency dependence of damping and compliance due to their various viscoelastic properties. Few empirical models have been proposed to take into account these variations, the most used one is the LOG model proposed by Knudsen and Jensen. On the other hand, the fractional calculus has been successfully applied to characterize the rheological properties of viscoelastic materials including their frequency dependence. This study shows that the loudspeaker suspension can be successfully modeled using fractional derivatives.
Fractional Viscoelastic Element
The use of fractional derivatives, is nowadays accepted in many fields of physics and engineering. The fractional derivatives are based on the generalization of the differential operator $\frac{d^n}{dt^n}$, in which $n$ is a positive integer number, to $\frac{d^\beta}{dt^\beta}$ where $\beta$ is any real number. A typical application of fractional derivatives is a rheological fractional element in which the forcedisplacement relation is given by
$$\textbf{F}(t) = \eta \frac{d^\beta \textbf{x}(t)}{dt^\beta},$$
where the constant $\eta$ is called a viscoelastic coefficient.
Application to Loudspeaker
Adding the fractional element to the traditional model of a loudspeaker, we obtain a new proposed model depicted in figure below. The relation between the force and the displacement can be defined in the timedomain as
$$\textbf{F} = M_{ms}\frac{d^2\textbf{x}}{dt^2} + R_{v}\frac{d\textbf{x}}{dt} + \eta\frac{d^\beta \textbf{x}}{dt^\beta} + \frac{1}{C_{0}}\textbf{x},$$
or in frequency domain, defining $F(\omega)$ and $X(\omega)$ as the Fourier transforms of $\textbf{F}(t)$ and $\textbf{x}(t)$, as
$$\frac{F}{X} = \omega^2 M_{ms} + j \omega R_{v} + (j \omega)^\beta \eta + \frac{1}{C_{0}}.$$
In this modified model $R_v$ represents viscous losses and $C_0$ represents the static compliance. The mechanical impedance $Z_m(\omega)$, defined as the force over the velocity, of such a system is then defined as
$$Z_m = j\omega M_{ms} + R_{v} + (j \omega)^{(\beta1)} \eta + \frac{1}{j \omega C_{0}}.$$
Results
Four loudspeakers, each of them equipped with a different type of surround (see figure below) have been measured. (a) Peerless PLS75F25AL0208, a 3” compact full range driver equipped with an anodized aluminium cone, a large roll SBR lowdamping rubber surround, and a Nomex spider, (b) Peerless PLS75F25AL0408, an almost identical copy of the first loudspeaker (3” full range driver) except that this one is equipped with an inverted rubber surround, (c) PHL 1520 is a 6.5” midrange driver with a highstrength cellulose fiber cone impregnated and coated on both sides with damped resins (the suspension consists of a cup spider made of Nomex and a highspeed flat damped surround made of a lightweight strip of polyethylene foam, and (d) Pioneer W16FU9051DT is a 6.5” woofer with a woven aramid/carbon composite shell cone suspended by a corrugated surround and a cup spider made of Nomex.
While both Peerless speakers gave very similar results, the other two speakers differs a little in both storage compliance and losses. Following figure shows both parameters (compliance and the losses) measured as a function of frequency. While these two variables are usually considered being constant (ThieleSmall model) they are obviously not (even for small signals). The model with a fractional element fit both curves well for all measured speakers.
Conclusion
The paper shows that the mechanical models based on fractional derivatives may prove useful for loudspeaker suspensions. In view of their ability to model viscoelastic phenomena, they provide a suitable method of describing dynamical properties of viscoelastic materials used in loudspeaker manufacture.
The results of four studied loudspeakers with different types of suspension, presented in the paper, show that the proposed model with a single fractional element provides very low rms error when compared to the measured data for all loudspeakers measured in standard atmosphere as well as in vacuum.
This paper shows that mechanical models of the materials used in loudspeaker suspensions may be more accurate by incorporating fractional derivatives in the model. In conventional differential equations, the differentiating order is a real integer, whereas for fractional derivatives the order is only constrained to be a real number with a fractional piece. The results of four loudspeakers with different types of suspension show that the proposed model with a single fractional element provides very low RMS error when compared to the measured data for all loudspeakers measured in standard atmosphere as well as in vacuum. Many materials used in loudspeaker suspensions exhibit significant frequency dependence of damping and compliance due to their various viscoelastic properties. Many physical processes, including the viscoelastic materials, exhibit fractional order behavior. In addition there exists a physical interpretation of the fractional derivatives, which makes them more compelling that a purely empirical model.
@article{novak2016fractional,
author={Novak, Antonin},
title={Modeling Viscoelastic Properties of Loudspeaker Suspensions Using Fractional Derivatives},
journal={J. Audio Eng. Soc},
volume={61},
number={1},
pages={3544},
year={2016},
publisher={Audio Engineering Society}
}