## Physical and Computational Acoustics

Modal Analysis:

Modal analysis is a tool that allows to compare measurements and calculation in the low frequency region.

The modes are mainly calulated using the finite element method (FEM).

The measurement is recorded with an artifical excitation either by a shaker or an instrumented hammer. The admittances of the structure are derived using the power spectral density (PSD) of the excitation and the cross power spectral density of the response and the excitation.

The admittances of a discrete model as it is given in the FEM can be described by a quotient of two polynomials. The polynomial in the denominator is the same for all admittances of a structure.  The zeros of the polynomial in the denominator are the complex resonances of the structure.

A large number of methods exist for the esimation of the complex modes from the admittances. In the institute the Global Rational Fractional Polynomial (GRFP) method was implemented. An extension of the method allows to derivate directly the mode shape.

The numerical simulation of sound fields enhances the understanding of sound and noise propagation and supports for example the development of robust and efficient noise reduction methods. This is one of the reasons why we work on the development and application of computational methods to describe physical interactions between sounds and objects. Our research includes models of the dynamics of structures, sound fields, and vibrations with applications in environmental noise control, automotive audio, auditory perception, and speech production. Our research includes

• the development and application of efficient boundary element methods (BEM) for the Helmholtz equation to simulate the sound propagation of in 2D, 2.5D and 3D,
• the development of efficient methods for modeling systems with hysteretic behavior,
• modeling and (psychoacoustically) evaluating sound mitigation methods,
• the development of models and methods to simulate the human vocal tract.

Computers grow more powerful every year, however, the needs and the requirements of real life applications (e.g. the need for more detailed models, simulation of bigger objects with complex geometry, real-time applications, ...) grow even faster. Thus it is necessary to constantly develop new and robust methods and to introduce new ideas and concepts into the field of numerical and applied mathematics. By working in close cooperations with all other groups of the institute, the group is able to combine new mathematical concepts and developments (e.g. advances in Frame and wavelet theory) with their immediate application for real life problems (e.g. calculation of head related transfer functions for 3D virtual audio, evaluation of noise reduction measurements, ...).

### Lighthouse Topic: Detection and Simulation of Noise and its Propagation for Developing Efficient Countermeasures

Noise, i.e. unwanted sound, is a phenomenon that effects our every day lives, and apart from being annoying can result in health problems. Motivated by this reason the topic of noise and noise reduction is one of the lighthouse research topic at the ARI. Together with partners from science and industry the researchers at the ARI investigated and investigate the origin and propagation of noise as well as methods for noise mitigation in multiple projects (PAAB, Wiabahn, PASS, LARS, RELSKG, SysBahnLärm, ...). This includes

• Measurement and Simulation of Noise Sources,
• Simulation of Noise Propagation,
• Perception of Noise.

In connection with the international noise awareness day the ARI offers the interested public the opportunity to visit the institute and to experience the topics Acoustics' and "Noise"' with several interactive presentations and experiments.

Further links in connection with noise:

### Staff

A project for the future is the combination of Comutational Fluid Dynamics (CFD) especially the Large Eddy Simulation (LES) with the Fast Multipole Boundary Element Method (FMBEM). The flow acoustics of a panthograph can be simulatied by this combination.

A flutter instability in water was observed on rotor blades. The instability leads to high noise levels. A simple tool for the rapid simuation was developed. The panel method was combined with the finite element model of the blade to estimate the critical velocity of the fluid.

Display of the radiation pattern of a loudspeaker The Acoustic Holography is an altenrative to the beam forming method. This method is able to handle nearfield and farfield components. The nearfield components decay exponentially. Therefore the distance to the source has to be as small as possible. Using a regular grid in two dimensions the Fourier transfromation about time and the two directions in space is used. The knowledge of the wavelength in two dimensions, of the frequency and the wave speed in air allows the derivation of the wavelength and -type in the third dimension. The wavelength in the third dimension is used to project the coherent sound field from the grid plane into a plane infront of the grid as close as possible to the surface of the structure.

The project extends isotropic computation of vibrations in the soil to anisotropic material. Up to now mainly deterministic models have been used in practice.

To identify gear orders in a multiple motor component environment a specific method for the generation of order spectrograms has been implemented into S_TOOLS-STx. The method applies smoothing on the rpm-signal and uses re-sampling as well as the Discrete Fourier Transform (DFT) in combination with the anti aliasing filter to create order analysis spectrograms at reasonable computational cost.

Blind source seperation is based on PCA and ICA.

The principal component analysis (PCA) and independent component analysis (ICA) are methods to devide a mixture of sounds into uncorrelated or indendent components.

The PCA is based on the singular value decomposition (SVD) of a matrix or on the eigenvalue and eigenvector determination of a centered covarince or correlation matrix.

PCA is applied in the method Spatial Transfrom of sound fields (STSF) to receive uncorrelated components from a mixture. The components are assumed to be coherent and projected by the method acoustic holography.

The method PCA was applied to simultaneous measurements of vibrations on the structure and sound in the far field. The components from the PCA are seperated into near field and far field components using the reaction of the far field microphone.

The Doppler effect of moving sources in the far field was compensated either by correcting the transformation kernel from time to frequency domain or by re-sampling.

A better seperation of independent sources is possible, if ICA is applied. ICA in theory is based on the Kulback-Leibler divergency related to a Gaussian distribution to maximize a non-Gausianity. Approximations to this feature that are more stable and faster to optimize are used in the FastICA algorithms.

The FastICA code was applied to short pices of music and the notes were separated by this algorithm.

Beam forming of a 64 microphone array is applied when instationary or moving sound sources have to be processed. Numeric simulation corresponds with measurements of monopole sources in an anechoic room concerning dynamic relations and side lobe influence.

Measurements of noise of high speed trains (ICE-S from 200 km/h to 300 km/h) have been performed. Beam forming on horizontal as well as on vertical axis enables the identification of distinctive noise emissions generated from different sections of the trains passing by. Noise radiated from wheel-rail contact dominates at speeds up to 240km/h. At higher speeds flow noise originating from roof components and pantographs is clearly distinguishable.

The picture compares (left) the traditional Boundary Element Method (BEM) with the MLFMM (see larger picture).

The BEM is an important tool used in Acoustics. But the computational effort for traditional BEM, which is O(n2) (n is the number of unknowns), makes the solution of real life problems for high frequencies almost impossible. The combination with the Fast Multipole Method (FMM) reduces this effort to O(n·log2(n)), thus making it possible to also solve problems in the high frequency region.