Vibroacoustic Simulation. Alexander Peiffer

So the estimate would be:

       (1.167)

      In Bendat and Piersol (1980) it is shown that the relative error is given by:

       (1.168)

      So for a given time length T the more we average, the more precise is the spectral estimate, but we sacrifice spectral resolution against statistical precision.

      Figure 1.23 Sketch of a single-input–single-output system. Source: Alexander Peiffer.

      1.6 Systems

      Any car, building, air plane or machine represents a dynamic system. This dynamic system is excited by source function f(t), e.g. a force. The excitation is called the input of the system. This excitation leads to a specific response g(t) called the output of the system. The simplest cases are systems with one input and output, e.g. the forced harmonic oscillator. Those systems are called single-input–single-output systems (SISO). Real systems are usually driven by multiple inputs or sources and have continuous or multiple responses. Those are called multiple-input–multiple-output systems (MIMO). Every system is described by its transfer function: i.e. a functional expression H that relates the inputs to the outputs.

       (1.169)

      The practical advantage of such a formalism is that a complex realistic system is reduced to the quantities of interest. All intermediate steps of sound and vibration prediction are neglected. A practical example would be the force from the engine mount as input exciting the car system. A reasonable output could be the sound pressure at the drivers ear.

      We restrict our considerations to linear systems. Thus, the response of the system to the sum of two signals f1(t) and f2(t) is given by the sum of each single response

       (1.170)

      If the inputs are multiplied by constants a and b the responses scale linearly with the input

       (1.171)

      This is called the superposition principle.

      1.6.1 SISO-System Response in Frequency Domain

The Equation (1.28) for the displacement response of the damped harmonic oscillator

       (1.172)

      is an example for a system response in frequency domain. With u and F_s as complex amplitudes of harmonic signals we can express the system properties easily in the frequency domain

(1.173)

      Thus, H(ω) is the system transfer function in frequency domain. It is usually a complex function.

      1.6.2 System Response in Time Domain

      The product of Fourier transforms corresponds to a convolution in time domain (1.42), hence:

       (1.174)

      or alternatively, as

       (1.175)

      A system is called causal (and this is definitely the case for any mechanical system) when the impulse response is zero at times smaller than zero (it can only react when it has experienced an excitation), so

       (1.176)

      That means we can rearrange the integration limits as follows:

       (1.177)

      All those forms are convolution integrals according to (1.46) and the response can be written as

       (1.178)

      So, with this function we can express any response of a linear system using the above convolution integral. When considering a unit frequency excitation F(ω)=1 Equation (1.173) shows that the response equals the transfer function.

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