Random Response Analysis

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Random response analysis is used when a structure is subjected to a nondeterministic, continuous excitation. Cases likely to involve nondeterministic loads are those linked to conditions such as turbulence on an airplane structure, road surface imperfections on a car structure, noise loads on a given structure, etc.

The complex frequency response can be achieved by direct and modal frequency response. If $image\rra_1a.gif$ and $image\rra_1b.gif$ are the complex frequency responses (displacement, velocity or acceleration) of x th degree of freedom, due to load cases a and b respectively, the power spectral density of the response of x th degree of freedom $image\rra_2.gif$ is as follows:

$image\rra_3.gif$ .

$image\rra_4.gif$ is the power spectral density of the two sources, where the individual source a is the excited load case and b is the applied load case. If $image\rra_5.gif$ is the spectral density of the individual source (a th load case), the power spectral density of the response of x th degree of freedom due to a th load case will be:

$image\rra_6.gif$ .

The cross spectral density $image\rra_4.gif$ with two different sources $image\rra_7.gif$ could possibly be a complex number. The power spectral density of the response of x th degree of freedom due to a th and b th load cases will be:

$image\rra_3.gif$ .

The total power spectral density of the response will be the summation of the power spectral density of all individual load cases as well as all cross load cases.

The autocorrelation $image\rra_8.gif$ of a variable x(t) can be defined by the following equation:

$image\rra_9.gif$ .

The variance $image\rra_10.gif$ of the x(t) will be equal to $image\rra_11.gif$ . The variance $image\rra_10.gif$ can be expressed as a function of power spectral density $image\rra_11a.gif$ as follows:

$image\rra_12.gif$ .

The root mean square value of the response x(t) can also be written in the following equation:

$image\rra_13.gif$ .

The autocorrelation function and the power spectral density are Fourier transforms of each other. Therefore, the auto correlation can be described as follows:

$image\rra_14.gif$ .

There could be fatigue failure due to random vibration. The number of fatigue cycles of random vibration is evaluated by multiplying the vibration duration and another parameter called maximum number of positive zero crossing. The maximum number of positive zero crossing is defined in the following equation:

$image\rra_15.gif$ .

Whenever there is a request for XYPLOT, XYPEAK or XYPUNCH, OptiStruct will export the root mean square value and the maximum number of positive crossing to the *.peak file.

Setup for Random Response Analysis

Random response analysis is activated, for all subcases, through the inclusion of the RANDOM data selector in the Subcase Information section of the input. This selector identifies RANDPS and RANDT1 bulk data entries to be used for random response analysis. The input spectral density is described by the RANDPS bulk data entry. The RANDPS data refers to a TABRND1 bulk data entry, which contains the power spectral density of the loading versus frequency. The RANDT1 bulk data entry describes the time span for the auto-correlation. Loading for each frequency response subcase may be distinct, but all frequency response subcases must reference the same frequency data.

Output from Random Response Analysis

Three output requests may be used for random response analysis results. These output requests are placed in the I/O Options section of the input data. The three controllers are:

XYPEAK	Generates a .peak file containing a summary of the requested output.
XYPLOT	Generates a HyperGraph session file (_rand.mvw file) and related data file (.rand file) for the requested output. Also generates the .peak file.
XYPUNCH	Generates a .pch file for the requested output. Also generates the .peak file.

These output requests are different from most other OptiStruct output requests in that they may be combined on the same line.

The requests are formatted as follows:

Operation, Curve-type, Plot-type / Grid (Component) list

"Operation" can be any combination of XYPLOT, XYPUNCH, and XYPEAK.

"Curve-type" can be DISP, VELO, or ACCE to request displacement, velocity or acceleration, respectively.

"Plot-type" can be either PSDF or AUTO to request power spectral density function or autocorrelation, respectively.

"Grid (Component) list" must come after a slash "/" . Each entry in the list is comma separated. Each entry consists of a GRID or SPOINT ID followed by a component of motion (T1, T2, T3, R1, R2, or R3) in parentheses. For SPOINTs the component must be T1.

Example 1

Requesting random response results in a HyperGraph session file for the velocity PSDF for GRIDs 3 and 6 for component T2:

XYPLOT, VELO, PSDF / 3(T2), 6(T2)

Example 2

Requesting random response summary results to be written to the .peak file for the autocorrelation of displacement for GRID 223 for component R3:

XYPEAK, DISP, AUTO / 223(T3)

Example 3

Requesting random response results output, in all formats, for the acceleration PSDF for GRIDS 8 and 9 for components T1 and T2:

XYPEAK, XYPLOT, XYPUNCH, ACCE, PSDF / 8(T1), 9(T1), 8(T2), 9(T2)

Here the XYPEAK request is valid, but redundant as it is always created when XYPLOT or XYPUNCH is present.

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