Vibroacoustic Simulation. Alexander Peiffer

can be applied.

      1.7.2 Response of MIMO Systems to Random Load

      When the input can only be defined by a cross spectral matrix [Sff], the same is true for the response Gm(ω).

       (1.205)

      with

      and so

       (1.206)

      The system matrix H can be removed from the expected value operator

       (1.207)

      and finally

(1.208)

This expression determines the cross spectral response of a linear MIMO system excited by random load. Typical numerical system equations have millions of degrees of freedom. Therefore, in many cases the above equation cannot by used straightforwardly, because this would lead to triple multiplication of Hermitian-but-huge matrices. Thus, for random excitation we need different approaches for the response calculation. The conjugate version of Equation (1.208) is

       (1.209)

      Bibliography

      1 Julius S. Bendat and Allan G. Piersol. Engineering Applications of Correlation and Spectral Analysis. Wiley, New York, 1980. ISBN 978-0-471-05887-8.

      2 Cyril M. Harris and Charles E. Crede. Shock and Vibration Handbook. McGraw-Hill, New York, NY, U.S.A., second edition, 1976. ISBN 0-07-026799-5.

      3 P. A. Nelson and S. J. Elliott. Active Control of Sound. Academic, London, 1993. ISBN 0-12-515426-7.

      Notes

      1 1 In this book the convention ejωt for the complex harmonic function is used. Literature that deals with wave propagation often use e−jωt to have positive wavenumber for positive wave propagation. However, as in every textbook in acoustics I denote the used convention on the first page to avoid confusion.

2 Waves in Fluids

      2.1 Introduction

      The acoustic wave motion is described by the equations of aerodynamics that are linearized because of the small fluctuations that occur in acoustic waves compared to the static state variables. The fluid motion is described generally by three equations:

       Continuity equation – conservation of mass

       Newton’s law – conservation of momentum

       State law – pressure volume relationship.

      For lumped systems the velocity is simply the time derivative of the point mass position in space. The same approach can be used in fluid dynamics, but here the continuous fluid is subdivided into several cells and their movement is described by trajectories. This is called the Lagrange description of fluid dynamics. Even if the equation of motions are simpler in that formulation it is quite complicated to follow all coordinates of fluid volumes in a complex flow. Thus, the Euler description of fluid dynamics is used. In this description the conservation equations are performed for a control volume that is fixed in space and the flow passes through this volume.

      In this chapter, the three dimensional space is given by the Cartesian coordinates x={x,y,z}T and the velocity of the fluid v={ vx, vy, vz }T.

      2.2 Wave Equation for Fluids

      2.2.1 Conservation of Mass

For simplicity we consider first the flow in the x-direction as in Figure 2.1. The mass flow balance contains the following quantities:

      1 The elemental mass m=ρdV=ρA with A=dydz.

      2 Mass flow into the volume (ρvxA)x.

      3 Mass flow out of the volume (ρvxA)x+dx.

      4 Mass input from external sources m˙.

Figure 2.1 Mass flow in x-direction through control volume. Source: Alexander Peiffer.

      leading to equation

      

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