What is a MIMO (Multi-Input Multi-Output) Model?

NOTE:  The commands in this tutorial are only available if the VES-3600 Advanced Signal Processing option is authorized by your ME'scope license. Check Help | About to verify authorization of this option.

Calculation of Transfer Functions, Outputs, and Inputs are all based upon use of a MIMO (Multi-Input Multi-Output) model of the dynamics of a structure. A MIMO model A MIMO model is a Multiple Input Multiple Output frequency domain matrix model containing a vector of Inputs, a matrix of Transfer functions and a vector of Outputs. The MIMO model is an equation where the Transfer function matrix is post-multiplied by the vector of Input DFTs to obtain the vector of Output DFTs. Inputs, Outputs & Transfer functions can be calculated using different forms of the MIMO model equation. is a frequency domain model where Fourier spectra of multiple Inputs are multiplied by elements of a Transfer Function A Transfer function is a cross-channel frequency domain function that defines the dynamic properties between an Output from and an Input to a structure or machine. A Transfer function is defined as the ratio (Output Fourier spectrum / Input Fourier spectrum). An FRF is a special case of a Transfer Function where the Input is an excitaton force, and the Output is the response caused by the excitation force. matrix to yield the Fourier Spectra of multiple Outputs.

The MIMO model is expressed with the equation:

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

where:

  {F(w)} = Input Fourier spectra (m - vector).

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

  {X(w)} = Output Fourier spectra (n - vector).

  m = number of Inputs.

  n = number of Outputs.

  w = frequency variable (radians per second).

NOTE:  Rows of the Transfer Function matrix correspond to Outputs, and columns correspond to Inputs.  Each Input and Output corresponds to a measurement DOF DOF is an acronym for degree-of-freedom. A DOF includes a Point number & direction. If each measurement (M#) in a Data Block, Shape Table or Acquisition window has a DOF defined for it, the DOF can be used to create Animation equations by assigning M#s to matching Points & directions on the model in the connected Structure window. Each Point number should correspond to a numbered Point on the model. Each DOF direction should correspond to a Measurement Axis direction at the Point on the model. Scalar data has no direction associated with it. (point & direction).  Each Transfer Function is a Cross-channel function between two DOFs, an Input DOF and an Output DOF.

Definitions

Transfer Function

• A Transfer Function is defined as the Fourier spectrum of an Output divided by the Fourier spectrum of an Input.

Frequency Response Function (FRF)

• An FRF is defined as the Fourier spectrum of a displacement, velocity, or acceleration response (Output) divided by the Fourier spectrum of the excitation force (Input) that caused the response.

Transmissibility

• A Transmissibility is defined as the Fourier spectrum of an Output divided by the Fourier spectrum of an Input with the same units.

NOTE:  An FRF and Transmissibility are special cases of a Transfer Function.

MIMO Calculations

Any one of the components of the MIMO model, (Inputs, Outputs, or Transfer Functions) can be calculated from the other two components using commands in the Transform | MIMO menu.

• Inputs and Outputs can be either time or frequency domain M#s

• Transfer Functions can be either experimentally derived or synthesized from modal parameters

The MIMO model is also used to calculate and display in animation a sinusoidal ODS due to multiple sinusoidal excitation forces at the same frequency.  (See the Transform | MIMO | Sinusoidal ODS command description for details.)