Theory of Fast RMS modal response computing

You can use random response analysis to compute the dynamic response of structures. Because a random excitation is non-deterministic, you can use a Power Spectral Density (PSD) function a driving excitation during the solution.

One difficulty with solving for RMS displacements or stresses using a PSD function is that the modal equations involve integration for every frequency, resulting in long solution times. The Fast RMS Method has been implemented to speed up the computation of RMS results.

The displacement of a structure may be written in modal coordinates:

Equation 28-6.

where:

• is the generalized mass frequency matrix

• is the modal damping matrix

• is the diagonalized natural frequency matrix

Then the complex response as a function of frequency is written as:

Equation 28-7.

where:

• is the modal coordinate matrix

• is the modal matrix

• is the applied loading vector

is defined as:

Equation 28-8.

The PSD of the modal coordinates is defined as:

Equation 28-9.

where:

• is the PSD matrix for random forces

• is a constant white noise input

The desired displacement or stress component matrix of the finite element model is . We can create a linear combination of the modal displacements written as:

Equation 28-10.

The PSD matrix of structural output  becomes:

Equation 28-11.

The root mean square (RMS) of , is written as:

Equation 28-12.

Solving Equation 7 will require long computation times due to integration at every node across every frequency.

Using linear time-invariant theory, the integration step can be replaced with a closed-form solution.

Considering the structural system being solved as a Linear Time Invariant system, the equations are written as:

Equation 28-13.

Equation 28-14.

where:

• is the state vector

• is the input vector

• is the output vector modified by matrices , and .

For a white noise random excitation , expressed as a PSD using the constant, symmetric, and positive semi-definite PSD matrix , the steady state variance state vector matrix is expressed as:

Equation 28-15.

where:

• is the expectation vector

• is a unique symmetric solution of the linear Lyapunov equation

Converting to:

Equation 28-16.

The steady state variance matrix of  is:

Equation 28-17.

Taking the root mean square of :

Equation 28-18.

Defining a matrix as the complex eigenvalues of using as the eigenvector matrix:

Equation 28-19.

Giving:

Equation 28-20.

Equation 15 may be reduced by pre-multiplying by and , which results in:

Equation 28-21.

In which:

Equation 28-22.

Equation 28-23.

The matrix is diagonal and can now directly be solved using Lyapunov's method.

Every (i,j) pair of the matrix can be written:

Equation 28-24.

Or in compact matrix form:

Equation 28-25.

in which is a square unit matrix and is the term-by-term division Hadamard operator.

The equations can be processed defining and as eigenvalues of matrix .

Unique solutions exist for all cases of which is true for engineering structural problems because all frequencies are positive. Therefore in engineering problems, all cases of interest exist only in the positive frequency domain ( to ).

Equation 28-26.

Therefore the equation for structural results , in matrix form, is written as:

Equation 28-27.

Equation 22 contains no integration, having been reduced to a matrix equation, therefore RMS results can be computed quickly. The Fast RMS Method uses this equation to rapidly compute RMS responses for a FEM model.

Similar computational techniques can be used for non-white noise excitation, or when solving for other structural results, such as von Mises stress.

For additional reading, see the references.

References:

1)  de la Fuente, E., 2008, “An Efficient Procedure to Obtain Exact Solutions in Random Vibration Analysis of Linear Structures,” Engineering Structures, 20, pp. 2981 – 2990

2)  de la Fuente, E., 2009, “Von Mises Stress in Random Vibrations of Linear Structures,” Computer and Structures, 87, pp. 1253-1262

3)  Rupp, C. and Antal, G., 2012, “Implementation of an Efficient RMS Algorithm for Random Analysis in Vibrata™,” ASME, IDET/CIE 2012

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