Vibroacoustic Simulation. Alexander Peiffer
Чтение книги онлайн.

Читать онлайн книгу Vibroacoustic Simulation - Alexander Peiffer страница 27

Название: Vibroacoustic Simulation

Автор: Alexander Peiffer

Издательство: John Wiley & Sons Limited

Жанр: Отраслевые издания

Серия:

isbn: 9781119849865

isbn:

СКАЧАТЬ alt="bold-italic upper S Subscript f g Baseline left-parenthesis omega right-parenthesis equals upper E left-bracket bold-italic upper F Superscript asterisk Baseline left-parenthesis omega right-parenthesis bold-italic upper G left-parenthesis omega right-parenthesis right-bracket"/> (1.166)

      Note that the cross correlation Rfg(τ) of time signals f(t) and g(t) can be reconstructed from the inverse Fourier transform.

      1.5.5 Estimation of Power and Cross Spectra

      As shown in section 1.5.4 and Equation (1.164) the expected value of the Fourier transform of an ensemble of signals of infinite duration is given by:

bold-italic upper S Subscript f g Baseline left-parenthesis omega right-parenthesis equals limit Underscript upper T right-arrow normal infinity Endscripts upper E left-bracket StartFraction 1 Over upper T EndFraction upper F Subscript k Superscript asterisk Baseline left-parenthesis omega right-parenthesis upper G Subscript k Baseline left-parenthesis omega right-parenthesis right-bracket

       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:

       StartFraction upper E left-bracket left-bracket ModifyingAbove bold-italic upper S With caret Subscript f g Baseline left-parenthesis omega right-parenthesis minus bold-italic upper S Subscript f g Baseline left-parenthesis omega right-parenthesis right-bracket squared right-bracket Over bold-italic upper S Subscript f g Superscript 2 Baseline left-parenthesis omega right-parenthesis EndFraction equals StartFraction upper T Subscript w Baseline Over upper T EndFraction equals StartFraction 1 Over upper M EndFraction (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

       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 Fs as complex amplitudes of harmonic signals we can express the system properties easily in the frequency domain

      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

       СКАЧАТЬ