### How Advanced CAE-based Simulations Predict the Structural Dynamics of a Loudspeaker’s Vibration System Efficiently

**December 18, 2015:**

**A key technology for higher engineering efficiency and innovation in the development of loudspeakers and loudspeaker systems is computer-aided engineering (CAE). Today, we would like to demonstrate how advanced CAE methods can predict the structural dynamics of a loudspeaker’s vibration system efficiently.**

**Use of Finite Element Models for the analysis of vibration systems**

Nowadays most loudspeakers can be treated as axisymmetric systems, and thus simplified 2D models are being applied. However, their multidimensional vibration pattern, which has a significant impact on the acoustic performance is then not taken into account. Depending on the final application, e. g. non-axisymmetric enclosures, non-axisymmetric behavior can be important, and advanced 3D models need to be used. Thus, finite element models for detailed vibration system design and optimization are highly valuable and efficient design tools.

For system or subsystem level simulations (without the goal of designing a vibration system) 1D lumped models are highly efficient. Below we will describe the basic differences between 1D, 2D and 3D models.

**Theory: Mathematical and physical background of structural dynamics**

A loudspeaker is based on a voice coil that is located in the field of a magnet. As soon as a signal passes through the coil, an electromagnetic field is being built. As the magnet of the loudspeaker is fixed, the coil is moving due to the interaction. This mechanical movement is transferred to the membrane. Hence, a loudspeaker is driven by a time-harmonic voltage, V = V_{0}exp(iωt).

Below we are describing mathematical background relating purely on the mechanical domain with focus on linear effects. We don’t consider the electromagnetic analysis of the voice coil and the driving force that this current generates. For more details on the relation between the driving voltage and the force exciting the vibration system, we refer to our technical paper “Advanced CAE-based Simulations of Motor System”.

The governing equation for the mechanical vibrations in the frequency domain, discretized by means of FEA (* F*inite

*lement*

__E__*nalysis), can be written as follows:*

__A__At a first glance there seems to be only a little difference in the governing equations by matrix methods and by lumped parameter models. However, the big difference is the dimension of the system.** **In the finite element governing equation stiffness, mass and damping are being described via matrices. K^{m}is the stiffness matrix, D^{m}is the damping matrix and M^{m}is the mass matrix. Furthermore, u^{m}is the vector of displacements and f^{m}is the vector of mechanical forces exciting the system. ω is the angular frequency.

Typically the dimension is of several of thousands degrees of freedom. In fact the governing equation is a system of equations describing the mechanical vibrations with respect to a detailed definition of the geometry (CAD model) discretized via finite elements. Thus it is possible to use these models for the whole audible frequency range which is typically from 20 Hz up to 20 kHz where a lot of non-pistonic and non-axisymmetric motion patterns occur.

For ω = 0 we get a static solution, and thus the lumped stiffness of the system, typically referred to as K_{ms}, can be calculated, or its inverse, the compliance C_{ms}.

**Simulation Model Setup of Vibration Systems**

**2D Model Setup**

Setting up a model is straight forward and relatively simple, at least in 2D. Starting point is a 2D (cleaned-up) cross section as given in the following figure:

**2D Solutions**

By applying a force of 1 [N] at the voice coil and assuming ω = 0 (i.e. a static solution) we get the following displacement pattern of the vibration system:

The displacement at the voice coil is 0.576 [mm]. Thus the lumped stiffness K_{ms}= 1 [N] / 0.576 [mm] = 1.74 [N/mm], which is in good agreement with a measured value of 1.67 [N/mm].

By performing a dynamic eigenvalue analysis, we get the first natural frequency (or eigenfrequency) in vacuum (i.e. without the influence of the surrounding air) at 37.0 [Hz]. This again is in excellent agreement with a measured value of 36.9 [Hz].

The following figure shows the displacement pattern of the 1^{st}eigenfrequency, the so-called piston mode of a loudspeaker.

In a further step, a forced response analysis is performed, where a constant force of 1 [N] in the frequency domain is being applied.

By evaluating the displacement at a point of the voice coil we actually get a description of the lumped dynamic stiffness as a function of frequency (see the following figure).

Below 1 [kHz] we see a pretty smooth variation, showing very prominent the 1^{st}natural frequency. However, above 1 [kHz] we see a couple of significant variations. The first variation is often referred to as the break-up frequency. The name actually comes from the fact that a lumped parameter solution breaks-up at that frequency, i.e. it simply delivers wrong results.

The following figure shows the operational deflection shape of the vibration system at 1,400 [Hz]:

The bending in the cone is dominating a disturbance in the frequency response.

Figure 5 shows the on-axis SPL in [dB] in a distance of 1 [m] for simulation and measurement as well. Please note that these results are based on a fully mutliphysical model, whereas details are not given in this paper.

**3D Solutions**

In a similar way a 3D model can be used to derive additional, non-axisymmetric, results. However, simply generating a full 3D geometry based on a sweep of the 2D cross-section model does not lead to satisfying models. Moreover, a surface based (instead of 2D solid based) model using shell finite elements leads to highly efficient 3D simulation models

Here the starting point is a (cleaned-up) CAD based surface model of the vibration system as given in the following figure.

The big advantage of 3D modeling is that we get additional results in terms of non-axisymmetric deflection shapes that are typically caused by non-axisymmetric enclosures (which is usually the case).

Most important is the so-called “rocking” of loudspeakers, showed in the figure 7. This effect leads to strong variations in radiated sound pressure, and can lead to heavy distortions in the extreme case when the voice coil hits the magnet.

**Prediction of the structural dynamics of vibration systems thanks to CAE methods **

The above mentioned descriptions illustrate by evidence that advanced CAE methods can predict the structural dynamics of the vibration system of loudspeakers efficiently. The analysis demonstrates that also complex simulations of the vibration system of loudspeakers in the early development stage – where no physical prototypes exist – show realistic results.

**Prime example of Virtual Product Development (VPD)**

The audio industry, as well as most industries, is constantly driven to improve engineering efficiency and continuously develop innovations, thereunder to develop loudspeakers of excellent sound quality as quickly as possible. A key technology for higher efficiency and innovation in the development of loudspeakers and loudspeaker systems is computer-aided engineering (CAE).

Computer-Aided Engineering (CAE) based on simulation and analysis of the functional performance of products has already played a key role for more than two decades. CAE methodologies are today typically used at every stage of the development cycle, from first concept studies up to detailed engineering for final product development to be released to the market place (including modeling of the manufacturing processes as well). In this context, the term frontloaded Virtual Product Development (VPD) is often used.

The above mentioned CAE-based simulation of the vibration system of loudspeakers is another prime example of a successful virtual product development. It shows that CAE-based simulations can support the development engineers in the design phase with realistic results in a highly complex environment. Hence, the engineers achieve more freedom in making design decisions and development changes produce less costs and time.

You can find further details of CAE-based simulations on vibration systems in the common technical paperof JJR Acoustics, LLC, Moca Audio and Konzpet-X (MVOID) (MVOID), presented on the occasion of the 139^{th}AES Convention in New York.