Calculating Outputs From Inputs

Outputs can be calculated from Inputs & Transfer Functions in three different ways;

1. Output Time Waveforms or Fourier spectra from Input Time Waveforms or Fourier spectra & Transfer Functions

2. Cross spectra between Inputs & Outputs.

3. Output Auto spectra or PSDs from Auto spectra or PSDs & Transfer Functions

Output time Waveforms or Fourier Spectra

Output Fourier spectra are calculated from Transfer Functions and Input Fourier spectra using the formula;

{X(w)} = [H(w)] {F(w)}

where:

{F(w)} - Fourier spectra of Inputs (m - vector)

[H(w)] - Transfer Function matrix (n by m)

{X(w)} - Fourier spectra of Outputs (n - vector)

m - number of Inputs

n - number of Outputs

w - frequency variable

• If Input Time Waveforms are provided as a Source, they are transformed to Fourier spectra before using the above equation

• The calculated Output Fourier spectra are then transformed to Output Time Waveforms

The following block diagram depicts the Output calculation when the Inputs are provided as time waveforms, and either a Modal Model A Modal Model is a set of mode shapes that have been scaled so that they preserve the dynamic properties of a structure. Two types of modal models are used in ME'scope, Residue mode shapes and Unit Modal Mass (UMM) mode shapes. Residue mode shapes are obtained by curve fitting a set of calibrated FRFs. UMM mode shapes are obtained by re-scaling a set of Residue mode shapes. Modal models are used for SDM calculations, FEA Model Updating, FRF synthesis, and MIMO calculations. or Experimental FRFs are also provided for the Transfer Function matrix.

MIMO Output Calculation.

Cross Spectra

Cross spectra between Inputs & Outputs are calculated from Input Auto spectra & Transfer Functions using the formula;

[{X(w)}{F(w)}t] =  [H(w)] [{F(w)}{F(w)}t]

where:

[{F(w)}{F(w)}t] - Input Auto spectrum matrix (m by m)

[H(w)] - Transfer Function matrix (n by m)

[{X(w)}{F(w)}t] - Cross spectrum matrix between Inputs & Outputs (n by m)

 m - number of Inputs

 n - number of Outputs

 w - frequency variable (radians per second)

 t - denotes transposed conjugate

Output Auto Spectra

Output Auto spectra or PSDs are calculated from Input Auto spectra or PSDs & Transfer Functions using the formula;

[{X(w)}{X(w)}t] = [H(w)] [{F(w)}{F(w)}t] [H(w)]t

where:

[{F(w)}{F(w)}t] - Input Auto spectrum matrix (m by m)

[H(w)] - Transfer Function matrix (n by m)

[{X(w)}{X(w)}t] - Output Auto spectrum matrix (n by n)

 m - number of Inputs

 n - number of Outputs

 w - frequency variable (radians per second)

 t - denotes transposed conjugate

Only the diagonal elements of the Output Auto spectrum matrix are calculated.