Vibroacoustic Simulation - Alexander Peiffer

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basic ideas, pitfalls, and results of some industrial examples from aerospace, automotive, and train industries. Special thanks goes to my former colleague Ulf Orrenius who wrote the train and motivation section of chapter 12. His great experience and knowledge strongly enriched the content of this chapter.

      This book is about simulation, but simulation is nothing without validation based on tests. In my view both – simulation and tests – are required to perform a good acoustic design and noise control engineering. Thus, chapter 13 briefly summarizes test and correlation methods together with an outlook to further topics of simulation and ongoing research in the field of acoustic simulation.

      In most cases the life of an acoustic engineer means solving the target conflict between the acoustic performance and costs, weight, and space requirements. This is the reason why design engineers are not always the best friends of acousticians during the design phase. The more important it is that you are able to calculate the effect of your decisions for efficient application of the sometimes rare acoustic resources. I hope that this book provides some support for this demanding task.

      Coming back to my initial motivation: If I would have to hold a lecture on vibroacoustic simulation now, I would sleep much better.

       Alexander Peiffer

      Planegg, Germany

      Special thanks goes to my former colleague Ulf Orrenius who wrote the train- and motivation section of chapter 12. His great experience and knowledge strongly enriched the content of this chapter.

      Simple systems with properties constructed by lumped elements as masses, springs and dampers are a good playground to understand and investigate the physics of dynamic systems. Many phenomena of vibration as resonance, forced vibration and even first means of vibration control can be explained and visualized by these lumped systems.

      In addition, a basic knowledge of signal and system analysis is required to put the principle of cause and effect in the right context. Every vibroacoustic system response depends on excitation by random, harmonic or specific signals in the time domain and we need a mathematical tool set to describe this.

      An excellent test case to demonstrate and define the principle effects of vibration is the harmonic oscillator. It consists of a point mass, a spring and a damper. The combination of many point masses connected via simple springs and dampers provides some further insight into dynamic systems.

      As those systems are described by components that have no dynamics in themselves they are called lumped systems. In principle all vibroacoustic systems can by modelled and approximated by this simplified approach.

      1.1 The Damped Harmonic Oscillator

      Figure 1.1 Damped harmonic oscillator with initial conditions a) and external force excitation b). Source: Alexander Peiffer.

      1.1.1 Homogeneous Solutions

      Without external excitation as shown in Figure 1.1 a) the motion depends on the initial conditions at time t = 0 with the displacement u(0)=u0 and velocity vx(0)=vx0. The damping is supposed to be viscous, thus proportional to the velocity Fxv=−cvu˙. The equation of motion

(1.1)

      

(1.2)

      with the two solutions

      Hence,

      with B1 and B2 depending on the initial conditions. The root in Equation (1.3) is zero when cv equals 4mks. This specific value is called the critical viscous damping damping

       c Subscript v c Baseline equals StartRoot 4 m k Subscript s Baseline EndRoot (1.5)

      We use the following definitions:

      ω0 is the natural angular frequency, ζ is ratio of the

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