Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

So the estimate would be:

ModifyingAbove upper S With caret Subscript f g Baseline equals StartFraction 1 Over upper M EndFraction sigma-summation Underscript m equals 1 Overscript upper M Endscripts upper F Subscript m Superscript asterisk Baseline left-parenthesis omega right-parenthesis upper G Subscript m Baseline left-parenthesis omega right-parenthesis (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.

g left-parenthesis t right-parenthesis equals upper H left-bracket f left-parenthesis t right-parenthesis right-bracket (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

upper H left-bracket f 1 left-parenthesis t right-parenthesis plus f 2 left-parenthesis t right-parenthesis right-bracket equals upper H left-bracket f 1 left-parenthesis t right-parenthesis right-bracket plus upper H left-bracket f 2 left-parenthesis t right-parenthesis right-bracket (1.170)

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

upper H left-bracket a f 1 left-parenthesis t right-parenthesis plus b f 2 left-parenthesis t right-parenthesis right-bracket equals a upper H left-bracket f 1 left-parenthesis t right-parenthesis right-bracket plus b upper H left-bracket f 2 left-parenthesis t right-parenthesis right-bracket (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

bold-italic u left-parenthesis omega right-parenthesis equals StartFraction 1 Over k Subscript s Baseline minus m omega squared plus j c Subscript v Baseline omega EndFraction bold-italic upper F Subscript x Baseline left-parenthesis omega right-parenthesis (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:

g left-parenthesis t right-parenthesis equals integral Subscript negative normal infinity Superscript normal infinity Baseline h left-parenthesis t minus tau right-parenthesis f left-parenthesis tau right-parenthesis d tau (1.174)

or alternatively, as

g left-parenthesis t right-parenthesis equals integral Subscript negative normal infinity Superscript normal infinity Baseline h left-parenthesis tau right-parenthesis f left-parenthesis t minus tau right-parenthesis d tau period (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

h left-parenthesis t right-parenthesis equals 0 for t less-than 0 (1.176)

That means we can rearrange the integration limits as follows:

h left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline h left-parenthesis t minus tau right-parenthesis f left-parenthesis tau right-parenthesis d tau (1.177)

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

g left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis asterisk h left-parenthesis t right-parenthesis equals h left-parenthesis t right-parenthesis asterisk f left-parenthesis t right-parenthesis (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.

Скачать книгу