Random vibration

This article explains the response evaluations for random analyses in Response Dynamics. It includes:

• Definitions of the tools the software uses to define a random process.

• Theory of random response evaluation.

In Response Dynamics, the solution to the random vibration analysis can be PSD response functions, RMS values, or the level crossing rate (LCR). The excitation is valid only when the input random PSD excitations are ergodic and stationary (1).

You can evaluate the responses using:

• Power spectral density (PSD) functions

• Correlation of multiple excitations

• PSD response functions

• PSD of Von Mises stress

• Response Root Mean Square (RMS)

• Level Crossing Rate (LCR)

Power spectral density (PSD) functions

The power spectral density (PSD) function, also called the auto-spectral density function, is the tool the software uses to define and evaluate a random process (excitations or responses).

The PSD function is a real number function, which describes random excitations in terms of the one-sided spectral density of its mean square value (1), which is defined as:

	Equation 1

in which ( )* is the transpose of the complex-conjugate

Correlation of multiple excitations

Use the Create Correlation command to correlate multiple random excitations. For more information, see PSD correlation.

PSD response functions

If the random excitations are incoherent (uncorrelated), the formula for the response PSD evaluation is:

	Equation 2

where:

i = the i-th random excitation

r = the response to the random excitation.

If the random excitations are coherent (correlative), then the formula for the response PSD evaluation is:

where:

	= the frequency response (transfer function) between response r and excitation i.
	= the CSD function between the excitations and i and j, as defined under the Cross Spectral Density section above.

PSD of Von Mises stress

The PSD of Von Mises stress is:

	Equation 3

where:

where ( )* represents the transpose of the complex conjugate, and:

Response Root Mean Square (RMS)

The root mean square (RMS) values calculated from a response PSD function designate the "zero-mean" standard deviation,, of the random process. Because the statistical distribution of the response may not be obvious, the RMS values cannot be used to predict the exact confidence level of the structure's behavior in response to the given random excitations.

For example, 3 * may not represent 99.7% confidence as a Gaussian process (normal distribution). However, it is commonly used as a good engineering measure for random analysis.

Calculating RMS

Let S(t) be the PSD function, while the RMS values are the square root of coordinates in the shaded area A under the function curve in the figure below, which can be calculated using the following formula:

For each pair of given frequencies (f 1, f 2), calculate:

(i) If C >0; S rising from f 1to f 2, where:

	Equation 4

(ii) If C< 0; S falling from f 1to f 2, then:

	Equation 5a
	Equation 5b

(iii) If C=0; S is flat from f 1to f 2, then:

	Equation 6

RMS von Mises Stress and RMS Strain are both computed using the method detailed by Segalman, et al [Reference (2)].

Level Crossing Rate (LCR)

Let:

	= the RMS value of a PSD function
	= the RMS value of its derivative

You can calculate the level crossing rate by:

	the crossing rate above a response level, a

such that:

	Equation 7

and then:

	zero crossing rate = apparent frequency

References

1. Bendat, Julius S., Piersol, Allan G., Random Data: Analysis and Measurement Procedures, Second Edition (New York: John Wiley & Sons, Inc., 1986).

2. Segalman, D.J., Fulcher, C.W.G., Reese, G.M., Field Jr., R.V. "An Efficient Method for Calculating RMS von Mises Stress in a Random Vibration Environment." Proceeding of the 16th International Modal Analysis Conference, Santa Barbara, CA. pp. 117-123.

3. Wirsching, Paul H., Paez, Thomas L., Ortiz, Heith. Random Vibrations: Theory & Practice. (New York: John Wiley & Sons, Inc., 1995), pp. 134-135.

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