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2.4.3

FEM Simulation of loudspeakers and loudspeaker components Leonhard Kreitmeier Harman/Becker automotive systems Straubing, Germany

Summary The electrodynamic principle is widely used for Loudspeaker application due to the fact that the result is a robust type of transducer.This Fact is especially useful in car application where there is massive environmental testing done. FEM Simulations of these transducers helps defining the design of the components without a large amount of samples. Also the components are defined and optimised with respect to the sound pressure level frequency response SPL as the final target of the transducer. In this Paper we want to show some of the special problems arising with FEM simulation of these transducers. More specifically dealing with the problem of defining the material parameters.

Keywords Loudspeaker, Material Parameter, Correlation, ANSYS MECHANICAL/EMAG,

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

1

1.

Introduction

The investigation is done to an electrodynamic loudspeaker. (Fig. 1).

Figure 1: electrodynamic loudspeaker This speaker consists of the following main parts which are • a vibrating mechanical structure (membrane, dust cup, voice coil, voice coil former, surround and spider) which is clamped or glued to a basket (Fig. 2). • a fixing structure (basket) (Fig. 3). • a motor unit (Magnet structure) which is also fixed to the basket. (Fig. 4).

Figure 2: vibrat.structure

Figure 4: motor unit

Figure 3: basket

This structure is assembled that way that the voice coil, which is part of the vibrating structure, is placed in the radial air gap of the permanent magnet structure. The electrical signal is now transferred via Lorenzforces acting on the voice coil, due to the interaction of the current in the voice coil and the Magnet field in the air gap of the magnet system. Thus an axial movement (vibration) of the structure is created (electro dynamic interaction). The vibrating structure (membrane) then creates air waves in the audio frequency range. (Fig. 1). The FEM Simulation of this structure and of its parts has the target to create a constant sound pressure level over the operating audio frequency range. Also the limitation of this vibration (motion) in a defined manner and level is part of this simulations. Denomination of certain parts of the vibrating structure (Fig. 5).

surround

Dust cup

membran spider

former Figure 5: vibrating structure

Voice coil 20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

2

2.

Simulation of electrodynamic Loudspeakers and their components

Some examples for this type of calculations are magnet calculations, none linear force excursion, Force Factor Bl versus excursion calculation for magnet-voice coil configuration and frequency response calculation for the complete loudspeaker. •

magnet simulation (optimisation) Fig. 6a,b Flux Density B in the air gap Created are Design spaces of geometric data versus the Flux Density B[T] in the air gap. The variables to be optimised in this calculations, are iron part thicknesses, magnet material type and dimension. Principle design space plots are preferred versus single optimisation results (f.e. random optimisation because of changing optimisation target functions.

Figure 6a: magnet •

Figure 6b: Design Space

magnet simulation (large signal) Fig. 7a,b Force Factor BL versus excursion x

The Designspace consists of the configuration type of magnet system and voice coil system. The target is to design this excursion function of the Force Factor Bl [ Tm] in a way that the distortion of the transducer is minimised. Target function is created by special large signal measurement software.

Figure 7a: •

Figure 7b: Bl vers. Excursion ( voice coil)

suspension simulation ( large signal ) Force versus excursion

The Force – Excursion curve for the components (spider,surround) or the complete speaker, is evaluated by a none linear calculation. The figure 8b show results for a spider and the variation of its roll height. This allows geometric design with respect to excursion limits. Fig. 8a,b

Roll height spider Figure 8a:

Figure 8b: Force-Excursion ( spider ) 20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

3

•

frequency response Topology Correlation Sound Pressure Level (SPL) versus frequency ( dome tweeter ) Fig. 9a,b

The sound pressure frequency response is used two ways - as a target function of an existing sample for material parameter definition - optimisation for the geometric design data of the transducer. In this example the frequency response curve for a dome tweeter is calculated from 8kHz up to 30kHz. The results are used to determine the material parameters of the dome tweeter components by correlation with the measured response.Sensitivity analysis of the parameters allow to detect the influence of the parameter on part of the response curve and how it is typically changed.So certain specifics of the frequency response should be reproduced.

Figure 9a:

3.

Figure 9b: Sound Pressure Level (SPL) frequency response

Special requirements of an acoustic frequency response calculation

Now we want to look at certain specifics of calculating sound pressure level frequency response curve by FEM simulation means with ANSYS/MECHANICAL. What is of interest here are certain requirements concerning frequency calculation of the complete loudspeaker over the audio frequency range. 3.1.

Initial considerations

specifics on loudspeaker frequency response calculation should be considered here. Initial considerations • • • •

Axisymetric calculation in 2D Air space is reduced to 0.37 m Exact FEM modelling of geometry, wave guide, inner air spaces and glue is necessary sensitivity of the frequency response result is very high to geometric variations, so the geometric data must be defined very exactly

• Results data of the FEM simulation ( Frequency resonse,basic resonance fo,excursion x ) are used in different ways to correlate with measured data of a sample to determine material parameters to make design optimisation of geometric data in case the material parameters are defined. The calculation is now performed as a harmonic calculation (Small Signal Domain)

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

4

3.2.

special requirements of acoustic calculations

Specialities of a acoustic frequency response calculation The specifics for such an acoustic calculations are 3.2.1 Frequency bandwidth is large The frequency bandwidth for this transducers is very high . Fig. 10 SPL •

The frequency bandwidth covers the whole audio bandwidth from 20 Hz to 20 kHz.

•

Measured frequency range for the devices is 0 Hz up to 30 kHz.

8 kHz

20 Hz

30 000 Hz

Figure 10: frequency response bandwidth

f [Hz]

3.2.2 multi physic coupling There is a series of energy conversion from electrical signal to the final sound pressure wave in air. Fig. 10

Sound Pressure Level SPL

•

the fluid-structure coupling the mechanical vibration creates an acoustic pressure wave.

•

the electrodynamic coupling electrical signal is converted to mechanical vibration of the structure.

Lorenz Force F Figure 11: multi physic coupling 3.2.3 number of DOF´s is extremely high the number of DOF´s becomes under certain conditions very high dependent on dimension and upper frequency limit Fig.12a,b,c • the main contribution due to air alements • size of the elements is defined by upper bandwidth limit

Abbildung 12a: structure Elements : 906 DOF´s : 1812 Calc.time (25 frequ.) :

Figure 12b: Figure 12c: Structur+air space 0.375 m Elements : 14040 DOF´s : 14946 Calc.time (25 frequ.) : 7 min

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

Structur+air space 0.375 m Elements : 74646 DOF´s : 75552 Calc.time (25 frequ.) : -3 Std. October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

5

3.2.4 difference of 2D and 3D calculation (radiating modes) A 2D calculation is sufficient because only radial modes are contributing to the radiation. Only in special cases a 3D calculation is necessary (Oval transducers, asymmetric overlap of glued parts,etc,). High amount of elements and thus an extended calculation time is achieved by solving a 3D Problem and keeping the resolution at the measurement point the same as in the 2D axisymetric case. Fig. 13a,b,c

Figure 13b:

Figure 13a: Structur+air space 0.375 m Elements : 14040 DOF´s : 14946 Calc.time (25 frequ.) : 7 min

Figure 13c:

Structur+air space 0.375 m Elements : 560.000 DOF´s : 560.000 Calc.time (25 frequ.) : -8 days

Structur+air space 0.375 m Elements : 74646 DOF´s : 75552 Calc.time (25 frequ.) : -3 hours.

3.2.5 material parameters are frequency dependent Material Parameters for most materials (paper,polymers, glues) are frequency dependent over this extended frequency range.Measurements have been E-Module done on this area and show a clear dependency on frequency and temperature over the audio frequency 5 °C range. •

Measurements on Polyvinylchlorid results from Becker and Oberst are shown in Fig. 14. [1] [2]

•

The frequency range is from 10 Hz to 10 kHz

•

The temperature range from 5 °C to 120 °C

•

The Module is changing from 1.0e7 Pa to 5.0e9 Pa.

120 °C

Figure 14: E-Module of Polyvinylchlorid 3.2.6 to measure frequency parameter Most available material parameter measurement tools are only for static measurements. The consequence is for the determination of material parameters values at higher frequencies there is a need to relay on correlation with result values ( frequency response, etc). • • • • • •

measurement technique Push/Pull measurement Rotary-vibrating measurement ( Dreh/Schwingversuch ) DMTA Dynamic Mechanical Thermal Analysis Modal Analysis measurement (by Laser) Modal Correlation Software (LMS,IDEAS) Special Rotary-Vibration measurement ( Mastering Technique )

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

frequency range 0Hz 2Hz

temperature range -40-300 °C -40-300 °C

1 - 200Hz 1Hz - 5kHz

-40-300 °C -40-300 °C

1Hz – 5 kHz 1Hz – 10 kHz

-40-300 °C -40-300 °C

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

6

3.2.7 Material parameter correlation For the entire frequency range correlation’s with result values like frequency response are the only way to determine material parameter values. In the case of the frequency response the correlation is done reproducing the special characteristics of the response curve. If there are none then samples are created with different configuration (shapes, wave guide, etc ) or reduced amount of materials. There are certain categories ( force-excursion, resonances, response curve) used for correlation • correlation measurement component frequency range • static force-excursion measurement surround/spider 0 Hz • static force-3D deformation measurement cone 0 Hz • Basic resonance measurement surround/spider 50 Hz – 2 kHz • surround resonance Measurement surround 1 kHz • cone resonance measurement cone 15 kHz • Frequency response of SPL (Topology) loudspeaker 10 Hz – 30 kHz • consistent multi response correlation loudspeaker 10 Hz – 30 kHz • general sensitivity analysis • special samples with reduced amount of materials 3.2.8 large amount of materials The loudspeaker consists of a large amount of materials which are connected to each other. Mostly These parts are glued but there is also the possibility of clamping. They all interact creating the frequency response curve. So only those parameter formulations (damping models) are available which can be assigned to a lot of materials (material dependent damping ß i). • minimum 2 materials ( membrane, glue) • maximum up to 14 materials 3.2.9 restriction on material models to be used (due to large amount of materials) Only those parameter formulations are available in ANSYS/MECHANICAL which can be assigned to a lot of materials ( material dependent damping ß i ) • large amount of materials • only certain damping mechanisms can be used (material dependent ß damping and element damping) • parameter of the damping model is used as adjustment parameter Figure 16: damping mechanisms in ANSYS in an energetic view. Global mass damping element damping Global structural damping

Global constant damping

Material depending structural damping 3.2.10 MACRO in APDL To incorporate these aspects into the calculation and using frequency dependent material parameters an external Macro in APDL is created for these calculations. This external Macro in APDL allows to formulate the material parameters as frequency dependent functions as well as to use simply material dependent constant damping. Iteration process of external macro (Ansys APDL language) allows for - application of material parameter functions (redefining and remeshing) - constant damping ratio for different materials - to choose linear or logarithmic frequency spacing

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

7

4.

Examples

We want to go into details showing 2 examples of determining material parameters by correlation. 4.1.

material parameter correlation of cone membrane

In this example the frequency dependent material parameter of different paper cones is determined by use of correlation. • •

•

Theoretical response curve known from measurements. [1] [2] ( see point 4.1.1 ) in the static case correlation with force-excursion measurements are performed (3D deformation of Cone res. 3. cone ) 2. ( see point 4.1.2 ) In the upper frequency range 3kHz to 10 kHz 1. theory Force-excurs. resonances of the cone are used for correlation. ( in this frequency range the cone does not vibrate as a rigid poston – cone break up region ) ( see point 4.1.3 ) Figure 16: frequ.depending E of paper cone

4.1.1 theoretical response curve Theoretic response curve from measurements by Becker and Oberst on Polyvinylchlorid [1] [2] Fig. 17 The absolute values of the E-module function depends on the softening temperature of the material used. In principle there is a continuos raising function with temperature.

Abbildung 17: E-Module Polyvinylchlorid 4.1.2 static case force-excursion correlation In the static case correlation with force-excursion measurements are performed.(3D deformation of cone Fig. 18a,b,c,d • in the static case the Youngs modulus E is determined by correlation with force-excursion measurements • this correlation is a replacement for static push/pull measurement done with special tailored, flat samples. • the advantage of the correlation is that the initial sample is used.

Figure 18a: measurement

Figure 18b: modelling

Figure 18c: simulation

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

Figure 18d: correlation

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

8

4.1.3 dynamic case cone resonance correlation The definition of the cone material parameter in the dynamic case at higher frequencies ( 3 kHz to 10 kHz )is done by correlation with cone resonances. For this case a special sample is created with removed surround and the dust cup replaced by a massive part. So the relevant materials remained are the cone and the glue to the voice coil. In this upper frequency range the cone does not vibrate as a rigid piston. So a multitude of resonances are created ( cone break up modes ). Fig. 19a,b No surround.

Figure 19a: special sample loudspeaker (with no surround )

Figure 19b: cone resonances

4.1.3 comparison with DMTA measurement Comparison of the correlation results(Fig. 20a) of two cone paper materials with DMTA measurements Fig. 20b show a clear correspondence of the static values (0Hz or 1Hz) at a temperature of 27 °C. Also the tendency of the frequency dependency towards higher frequency is represented in a corresponding behaviour towards low temperatures in the DMTA measurement.

1.01 GPa

2.9 GPa

2.2 GPa

0.56 GPa

0.54 GPa

Figure 20a: Determine E-Modulus by correlation 4.2.

0.95 GPa

Figure 20b: DMTA measurement of E-modulus at 1Hz

dome tweeter frequency response calculation

Dome Tweeter loudspeaker for high frequency reproduction require a very exact modelling due to the fact that the sound pressure frequency response SPL shows high sensitivity concerning a variation of geometric data. Two procedures are possible either to use one frequency response and detect the frequency range where the specific parameter acts upon or to use multiple sample type results using the same materials (consistent set type of parameter). 4.1.3 Single frequency response topology sensitivity analysis of material parameters on the frequency response. Fig. 21a,b

Figure 11a: dome

Figure 21b: single frequency

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

9

4.2.2 multi frequency response topology Sensitivity analysis of material parameter on a multitude of frequency responses of different sample types using the same materials. The changed geometry shapes of the dome membrane and the changed acoustic wave guide in front of the transducer create a completely different frequency response behaviour. The effects of the material parameters on those responses is also different.These different target response curves allow to identify the correct set of parameters Tweeter with wave guide 1. Fig. 22a,b Tweeter with wave guide 2. Fig. 23a,b Tweeter without wave guide. Fig. 24a,b

Figure 22a: Figure 22b:

Figure 23a: Figure 23b:

Figure 24a: Figure 24b: Conclusions The Problems arising with the design of Loudspeakers can be solved with Ansys FEM simulations. Also the specific formulations of frequency dependent material parameters can be implemented in Ansys to adapt to the requirements of loudspeaker Simulations. The definition and correlation of material parameters is the most time consuming part of the simulations. Once these parameters are defined the performance of the loudspeaker, the geometric design, the material choice, tolerance considerations, the weight and the related costs could be optimised (minimised). This is a much more time saving and cost optimised procedure of adoption to the required performance than to build a huge amount of samples. References [1] [2]

Becker G.W., Oberst H..: "Frequency dependent material parameter of Polyvinylchlorid”,Kolloid Zeitschrift 148, 1956, pp. 6 Cremer L., Heckel M.: "Körperschall", Springer Verlag, 1996, pp 224

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

10

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FEM Simulation of loudspeakers and loudspeaker components Leonhard Kreitmeier Harman/Becker automotive systems Straubing, Germany

Summary The electrodynamic principle is widely used for Loudspeaker application due to the fact that the result is a robust type of transducer.This Fact is especially useful in car application where there is massive environmental testing done. FEM Simulations of these transducers helps defining the design of the components without a large amount of samples. Also the components are defined and optimised with respect to the sound pressure level frequency response SPL as the final target of the transducer. In this Paper we want to show some of the special problems arising with FEM simulation of these transducers. More specifically dealing with the problem of defining the material parameters.

Keywords Loudspeaker, Material Parameter, Correlation, ANSYS MECHANICAL/EMAG,

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

1

1.

Introduction

The investigation is done to an electrodynamic loudspeaker. (Fig. 1).

Figure 1: electrodynamic loudspeaker This speaker consists of the following main parts which are • a vibrating mechanical structure (membrane, dust cup, voice coil, voice coil former, surround and spider) which is clamped or glued to a basket (Fig. 2). • a fixing structure (basket) (Fig. 3). • a motor unit (Magnet structure) which is also fixed to the basket. (Fig. 4).

Figure 2: vibrat.structure

Figure 4: motor unit

Figure 3: basket

This structure is assembled that way that the voice coil, which is part of the vibrating structure, is placed in the radial air gap of the permanent magnet structure. The electrical signal is now transferred via Lorenzforces acting on the voice coil, due to the interaction of the current in the voice coil and the Magnet field in the air gap of the magnet system. Thus an axial movement (vibration) of the structure is created (electro dynamic interaction). The vibrating structure (membrane) then creates air waves in the audio frequency range. (Fig. 1). The FEM Simulation of this structure and of its parts has the target to create a constant sound pressure level over the operating audio frequency range. Also the limitation of this vibration (motion) in a defined manner and level is part of this simulations. Denomination of certain parts of the vibrating structure (Fig. 5).

surround

Dust cup

membran spider

former Figure 5: vibrating structure

Voice coil 20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

2

2.

Simulation of electrodynamic Loudspeakers and their components

Some examples for this type of calculations are magnet calculations, none linear force excursion, Force Factor Bl versus excursion calculation for magnet-voice coil configuration and frequency response calculation for the complete loudspeaker. •

magnet simulation (optimisation) Fig. 6a,b Flux Density B in the air gap Created are Design spaces of geometric data versus the Flux Density B[T] in the air gap. The variables to be optimised in this calculations, are iron part thicknesses, magnet material type and dimension. Principle design space plots are preferred versus single optimisation results (f.e. random optimisation because of changing optimisation target functions.

Figure 6a: magnet •

Figure 6b: Design Space

magnet simulation (large signal) Fig. 7a,b Force Factor BL versus excursion x

The Designspace consists of the configuration type of magnet system and voice coil system. The target is to design this excursion function of the Force Factor Bl [ Tm] in a way that the distortion of the transducer is minimised. Target function is created by special large signal measurement software.

Figure 7a: •

Figure 7b: Bl vers. Excursion ( voice coil)

suspension simulation ( large signal ) Force versus excursion

The Force – Excursion curve for the components (spider,surround) or the complete speaker, is evaluated by a none linear calculation. The figure 8b show results for a spider and the variation of its roll height. This allows geometric design with respect to excursion limits. Fig. 8a,b

Roll height spider Figure 8a:

Figure 8b: Force-Excursion ( spider ) 20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

3

•

frequency response Topology Correlation Sound Pressure Level (SPL) versus frequency ( dome tweeter ) Fig. 9a,b

The sound pressure frequency response is used two ways - as a target function of an existing sample for material parameter definition - optimisation for the geometric design data of the transducer. In this example the frequency response curve for a dome tweeter is calculated from 8kHz up to 30kHz. The results are used to determine the material parameters of the dome tweeter components by correlation with the measured response.Sensitivity analysis of the parameters allow to detect the influence of the parameter on part of the response curve and how it is typically changed.So certain specifics of the frequency response should be reproduced.

Figure 9a:

3.

Figure 9b: Sound Pressure Level (SPL) frequency response

Special requirements of an acoustic frequency response calculation

Now we want to look at certain specifics of calculating sound pressure level frequency response curve by FEM simulation means with ANSYS/MECHANICAL. What is of interest here are certain requirements concerning frequency calculation of the complete loudspeaker over the audio frequency range. 3.1.

Initial considerations

specifics on loudspeaker frequency response calculation should be considered here. Initial considerations • • • •

Axisymetric calculation in 2D Air space is reduced to 0.37 m Exact FEM modelling of geometry, wave guide, inner air spaces and glue is necessary sensitivity of the frequency response result is very high to geometric variations, so the geometric data must be defined very exactly

• Results data of the FEM simulation ( Frequency resonse,basic resonance fo,excursion x ) are used in different ways to correlate with measured data of a sample to determine material parameters to make design optimisation of geometric data in case the material parameters are defined. The calculation is now performed as a harmonic calculation (Small Signal Domain)

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

4

3.2.

special requirements of acoustic calculations

Specialities of a acoustic frequency response calculation The specifics for such an acoustic calculations are 3.2.1 Frequency bandwidth is large The frequency bandwidth for this transducers is very high . Fig. 10 SPL •

The frequency bandwidth covers the whole audio bandwidth from 20 Hz to 20 kHz.

•

Measured frequency range for the devices is 0 Hz up to 30 kHz.

8 kHz

20 Hz

30 000 Hz

Figure 10: frequency response bandwidth

f [Hz]

3.2.2 multi physic coupling There is a series of energy conversion from electrical signal to the final sound pressure wave in air. Fig. 10

Sound Pressure Level SPL

•

the fluid-structure coupling the mechanical vibration creates an acoustic pressure wave.

•

the electrodynamic coupling electrical signal is converted to mechanical vibration of the structure.

Lorenz Force F Figure 11: multi physic coupling 3.2.3 number of DOF´s is extremely high the number of DOF´s becomes under certain conditions very high dependent on dimension and upper frequency limit Fig.12a,b,c • the main contribution due to air alements • size of the elements is defined by upper bandwidth limit

Abbildung 12a: structure Elements : 906 DOF´s : 1812 Calc.time (25 frequ.) :

Figure 12b: Figure 12c: Structur+air space 0.375 m Elements : 14040 DOF´s : 14946 Calc.time (25 frequ.) : 7 min

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

Structur+air space 0.375 m Elements : 74646 DOF´s : 75552 Calc.time (25 frequ.) : -3 Std. October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

5

3.2.4 difference of 2D and 3D calculation (radiating modes) A 2D calculation is sufficient because only radial modes are contributing to the radiation. Only in special cases a 3D calculation is necessary (Oval transducers, asymmetric overlap of glued parts,etc,). High amount of elements and thus an extended calculation time is achieved by solving a 3D Problem and keeping the resolution at the measurement point the same as in the 2D axisymetric case. Fig. 13a,b,c

Figure 13b:

Figure 13a: Structur+air space 0.375 m Elements : 14040 DOF´s : 14946 Calc.time (25 frequ.) : 7 min

Figure 13c:

Structur+air space 0.375 m Elements : 560.000 DOF´s : 560.000 Calc.time (25 frequ.) : -8 days

Structur+air space 0.375 m Elements : 74646 DOF´s : 75552 Calc.time (25 frequ.) : -3 hours.

3.2.5 material parameters are frequency dependent Material Parameters for most materials (paper,polymers, glues) are frequency dependent over this extended frequency range.Measurements have been E-Module done on this area and show a clear dependency on frequency and temperature over the audio frequency 5 °C range. •

Measurements on Polyvinylchlorid results from Becker and Oberst are shown in Fig. 14. [1] [2]

•

The frequency range is from 10 Hz to 10 kHz

•

The temperature range from 5 °C to 120 °C

•

The Module is changing from 1.0e7 Pa to 5.0e9 Pa.

120 °C

Figure 14: E-Module of Polyvinylchlorid 3.2.6 to measure frequency parameter Most available material parameter measurement tools are only for static measurements. The consequence is for the determination of material parameters values at higher frequencies there is a need to relay on correlation with result values ( frequency response, etc). • • • • • •

measurement technique Push/Pull measurement Rotary-vibrating measurement ( Dreh/Schwingversuch ) DMTA Dynamic Mechanical Thermal Analysis Modal Analysis measurement (by Laser) Modal Correlation Software (LMS,IDEAS) Special Rotary-Vibration measurement ( Mastering Technique )

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

frequency range 0Hz 2Hz

temperature range -40-300 °C -40-300 °C

1 - 200Hz 1Hz - 5kHz

-40-300 °C -40-300 °C

1Hz – 5 kHz 1Hz – 10 kHz

-40-300 °C -40-300 °C

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

6

3.2.7 Material parameter correlation For the entire frequency range correlation’s with result values like frequency response are the only way to determine material parameter values. In the case of the frequency response the correlation is done reproducing the special characteristics of the response curve. If there are none then samples are created with different configuration (shapes, wave guide, etc ) or reduced amount of materials. There are certain categories ( force-excursion, resonances, response curve) used for correlation • correlation measurement component frequency range • static force-excursion measurement surround/spider 0 Hz • static force-3D deformation measurement cone 0 Hz • Basic resonance measurement surround/spider 50 Hz – 2 kHz • surround resonance Measurement surround 1 kHz • cone resonance measurement cone 15 kHz • Frequency response of SPL (Topology) loudspeaker 10 Hz – 30 kHz • consistent multi response correlation loudspeaker 10 Hz – 30 kHz • general sensitivity analysis • special samples with reduced amount of materials 3.2.8 large amount of materials The loudspeaker consists of a large amount of materials which are connected to each other. Mostly These parts are glued but there is also the possibility of clamping. They all interact creating the frequency response curve. So only those parameter formulations (damping models) are available which can be assigned to a lot of materials (material dependent damping ß i). • minimum 2 materials ( membrane, glue) • maximum up to 14 materials 3.2.9 restriction on material models to be used (due to large amount of materials) Only those parameter formulations are available in ANSYS/MECHANICAL which can be assigned to a lot of materials ( material dependent damping ß i ) • large amount of materials • only certain damping mechanisms can be used (material dependent ß damping and element damping) • parameter of the damping model is used as adjustment parameter Figure 16: damping mechanisms in ANSYS in an energetic view. Global mass damping element damping Global structural damping

Global constant damping

Material depending structural damping 3.2.10 MACRO in APDL To incorporate these aspects into the calculation and using frequency dependent material parameters an external Macro in APDL is created for these calculations. This external Macro in APDL allows to formulate the material parameters as frequency dependent functions as well as to use simply material dependent constant damping. Iteration process of external macro (Ansys APDL language) allows for - application of material parameter functions (redefining and remeshing) - constant damping ratio for different materials - to choose linear or logarithmic frequency spacing

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

7

4.

Examples

We want to go into details showing 2 examples of determining material parameters by correlation. 4.1.

material parameter correlation of cone membrane

In this example the frequency dependent material parameter of different paper cones is determined by use of correlation. • •

•

Theoretical response curve known from measurements. [1] [2] ( see point 4.1.1 ) in the static case correlation with force-excursion measurements are performed (3D deformation of Cone res. 3. cone ) 2. ( see point 4.1.2 ) In the upper frequency range 3kHz to 10 kHz 1. theory Force-excurs. resonances of the cone are used for correlation. ( in this frequency range the cone does not vibrate as a rigid poston – cone break up region ) ( see point 4.1.3 ) Figure 16: frequ.depending E of paper cone

4.1.1 theoretical response curve Theoretic response curve from measurements by Becker and Oberst on Polyvinylchlorid [1] [2] Fig. 17 The absolute values of the E-module function depends on the softening temperature of the material used. In principle there is a continuos raising function with temperature.

Abbildung 17: E-Module Polyvinylchlorid 4.1.2 static case force-excursion correlation In the static case correlation with force-excursion measurements are performed.(3D deformation of cone Fig. 18a,b,c,d • in the static case the Youngs modulus E is determined by correlation with force-excursion measurements • this correlation is a replacement for static push/pull measurement done with special tailored, flat samples. • the advantage of the correlation is that the initial sample is used.

Figure 18a: measurement

Figure 18b: modelling

Figure 18c: simulation

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

Figure 18d: correlation

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

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4.1.3 dynamic case cone resonance correlation The definition of the cone material parameter in the dynamic case at higher frequencies ( 3 kHz to 10 kHz )is done by correlation with cone resonances. For this case a special sample is created with removed surround and the dust cup replaced by a massive part. So the relevant materials remained are the cone and the glue to the voice coil. In this upper frequency range the cone does not vibrate as a rigid piston. So a multitude of resonances are created ( cone break up modes ). Fig. 19a,b No surround.

Figure 19a: special sample loudspeaker (with no surround )

Figure 19b: cone resonances

4.1.3 comparison with DMTA measurement Comparison of the correlation results(Fig. 20a) of two cone paper materials with DMTA measurements Fig. 20b show a clear correspondence of the static values (0Hz or 1Hz) at a temperature of 27 °C. Also the tendency of the frequency dependency towards higher frequency is represented in a corresponding behaviour towards low temperatures in the DMTA measurement.

1.01 GPa

2.9 GPa

2.2 GPa

0.56 GPa

0.54 GPa

Figure 20a: Determine E-Modulus by correlation 4.2.

0.95 GPa

Figure 20b: DMTA measurement of E-modulus at 1Hz

dome tweeter frequency response calculation

Dome Tweeter loudspeaker for high frequency reproduction require a very exact modelling due to the fact that the sound pressure frequency response SPL shows high sensitivity concerning a variation of geometric data. Two procedures are possible either to use one frequency response and detect the frequency range where the specific parameter acts upon or to use multiple sample type results using the same materials (consistent set type of parameter). 4.1.3 Single frequency response topology sensitivity analysis of material parameters on the frequency response. Fig. 21a,b

Figure 11a: dome

Figure 21b: single frequency

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

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4.2.2 multi frequency response topology Sensitivity analysis of material parameter on a multitude of frequency responses of different sample types using the same materials. The changed geometry shapes of the dome membrane and the changed acoustic wave guide in front of the transducer create a completely different frequency response behaviour. The effects of the material parameters on those responses is also different.These different target response curves allow to identify the correct set of parameters Tweeter with wave guide 1. Fig. 22a,b Tweeter with wave guide 2. Fig. 23a,b Tweeter without wave guide. Fig. 24a,b

Figure 22a: Figure 22b:

Figure 23a: Figure 23b:

Figure 24a: Figure 24b: Conclusions The Problems arising with the design of Loudspeakers can be solved with Ansys FEM simulations. Also the specific formulations of frequency dependent material parameters can be implemented in Ansys to adapt to the requirements of loudspeaker Simulations. The definition and correlation of material parameters is the most time consuming part of the simulations. Once these parameters are defined the performance of the loudspeaker, the geometric design, the material choice, tolerance considerations, the weight and the related costs could be optimised (minimised). This is a much more time saving and cost optimised procedure of adoption to the required performance than to build a huge amount of samples. References [1] [2]

Becker G.W., Oberst H..: "Frequency dependent material parameter of Polyvinylchlorid”,Kolloid Zeitschrift 148, 1956, pp. 6 Cremer L., Heckel M.: "Körperschall", Springer Verlag, 1996, pp 224

20th CAD-FEM Users’ Meeting 2002 International Congress on FEM Technology

October 9-11, 2002 Kultur- und Congress Centrum Graf-Zeppelin-Haus, Friedrichshafen, Lake Constance, Germany

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