Factors Affecting the Accuracy of Numerical Simulation of Radiation from Loudspeaker Drive Units P.C.Macey PACSYS Limited Presented at ALMA European Symposium Frankfurt, Germany 4 th April 2009
Loudspeaker design Measurement Lumped parameter (Thiele, Small) Numerical simulations solving the underlying differential equations 1. Finite difference method 2. Finite element method 3. Boundary element method
Benefits of accurate simulation Less testing is necessary Radical design changes can be investigated more easily The correct design decisions are made more reliably The full set of results, displacement distribution, pressure field are known The underlying phenomena are understood better Extend to further simulation, e.g. waveguide design Publicity benefits
Sources of error Insufficient mesh refinement Ill-conditioning Inaccurate material properties Missing important structural details Approximate acoustic/structure coupling Radiation condition accuracy Viscothermal losses, enclosure venting
Reference Model Typical 165mm automotive unit, non-proprietary design. Axisymmetric configuration radiating into a half space. Detailed descriptions of junction
Full Model Generalised fluid-structure coupling Structural and acoustic mesh
Natural modes of structure Mode 1: 78 Hz. Main pistonic mode Mode 2: 569 Hz Spider mode
Natural modes of structure Mode 9: 2851 Hz. Cone breakup mode Mode 14: 4383 Hz. Cone breakup mode
Acoustic simulation Helmholtz equation 2 p k 2 p 0 (1) Continuity of normal velocity p i V n Sommerfeld radiation condition p i p lim r 0 r r c (2) (3) acoustic FE satisfy (1) and (2) acoustic BE satisfy (1), (2) and (3) Rayleigh Integral satisfies (1) and (3)
Rayleigh Integral The contribution from an element of vibrating area is added as though it were locally baffled and radiating into a half space. p x 2i Vg x, y da y D Exact for a flat, baffled diaphragm radiating into a half space
Simple test problem radiation from a hemispherical dome
Analytic solution Consider radiation into full space with radial velocity distribution. V cos for 0 2 V cos for 2 From the separability of the Helmholtz equation in spherical polar coordinates V r p a n 0 n P n 2 cos h kr n a n Solution for coefficients ' 2n 1 i cv V P cos sin d 2 r n 2h kr 0 n
Accuracy of Rayleigh Integral Assume dome radius 0.01m, Speed of sound 340 m/s Density of air 1.2 kg/m^3 Radiated pressure at 1m on axis 6.00E+00 5.00E+00 4.00E+00 3.00E+00 closed form solution Rayleigh integral 2.00E+00 1.00E+00 0.00E+00 0 5000 10000 15000 20000 25000 30000 35000 40000 Frequency (Hertz)
Dome radiating into half space Pressure contours at 10000Hz. Pressure contours at 40000Hz.
Generic loudspeaker accuracy of Rayleigh Integral
Model with tweeter mounted Note: Rayleigh Integral does not take diffraction effects into account!
Effect of tweeter mounting
Analysis with dustcap removed Pressure amplitude distribution at 5kHz The Rayleigh integral cannot analyse the case without the dustcap.
Acoustic/structure coupling Continuity of normal velocity p n i V n Continuity of stress p nn Uncoupled solution :- Solve for structure vibrating in vacuo then pure acoustic solution
Generic loudspeaker inaccuracy of approximate coupling
Solid v Shell elements Solid element mesh Shell element mesh Geometry modelled by shell elements
Solid v Shell elements Shell element models much quicker to create Shell elements are computationally less demanding Thin shell theory for cone, thick shell theory for surround Experience shows solid element models are generally more accurate Hybrid approaches are possible
Joint accuracy
Venting Cavity
Variation of Viscosity
Conclusions To obtain accurate results it is advisable to have Accurate material properties Fully coupled model Proper acoustic radiation model Solid elements to model details of components connections Model venting of rear acoustic cavities