Performing a Random Vibration Analysis

Sometimes, the loads on a structure are random in nature. An example is the wind load on a tower. In addition to the average wind, there are gusts caused by the turbulence in the flow. The gusts will give a time-dependent excitation, which can induce a dynamically amplified response in the structure. The frequency and amplitude of these gusts will be randomly distributed. If the structure is large, the peaks in the wind speed may not even occur synchronously at locations far from each other.

If the pressure is measured, it is possible to give a statistical representation of the wind load. The statistical representation of an input like a force to be used in a random vibration analysis is the power spectral density (PSD), which is a function of the frequency.

There are two main cases when a random vibration analysis is useful. In addition to the scenario outlined above, it is common that design standards (in particular for electronic components and devices) include requirements on random vibration testing. To simulate such a test, a random vibration analysis can be performed. In this case, the prescribed PSD of the excitation is a simplified envelope intended to cover a multitude of loading conditions.


	For a detailed theory, see Random Vibration Theory.

Correlation

When several loads act on a structure, they can have different degrees of interdependency. This is described by a statistical measure of their correlation. There are two important special cases of correlation, which significantly simplify the analysis.

If the loads are random in nature and completely independent of each other, they can be described as being uncorrelated. This would be the case when, for example, two separate drilling machines are used simultaneously at some distance from each other.

If several loads have the same source, they can be fully correlated. Vibration testing is a common example. In such case, several points on the structure are attached to the same shaker table. Thus, they will experience identical (but random) acceleration histories. Another example could be when a certain load is decomposed into components along different coordinate axes. Clearly, the X- and Y-components are just scaled versions of the total load, and thus they are fully correlated.

In a general case, the correlation between two loads is a function of both the distance between their positions and the frequency. The input to the analysis includes not only the PSD of each load but also their cross-correlation.

Spatial Variation of Random Loads

The PSDs and cross-correlation matrices are assumed to be functions of frequency only. It is not allowed to write expressions that are functions of the spatial coordinates. The expression for the load inside the physics interface, however, can have such dependencies as long as they appear as a pure multiplier to the spectrum.

In principle, however, the PSD and even the full cross-correlation functions could be considered to vary continuously over a structure, and thus be functions of the spatial coordinates. An example of this could be the wind pressure on a high tower.

If you want to model such situation, you need to split the loaded region into smaller parts, each with its own constant spectrum definition. Note that the number of off-diagonal cross-correlation functions increases quadratically with the subdivision.

Setting up a Random Vibration Analysis

Random vibration analysis is based on a modal representation of the structure. It represents a type of mode superposition and relies on the reduced-order model (ROM) concept.

In principle, you can perform random vibration analysis on any reduced-order model that is of the Frequency Domain, Modal type. This requires a number of settings and studies run in an appropriate order.

It is significantly more convenient to start by adding a study, either from the window or from the page in the Model Wizard.

The study is not a study in itself; rather, it adds a number of nodes to the Model Builder tree to facilitate a random vibration analysis. These nodes serve the purpose of setting up the ROM and providing input data to the random vibration analysis.

Two studies are added:

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One study with an study step for computing the eigenfrequencies and corresponding eigenmodes.

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One study with a study step, in which the ROM is created. It references the eigenfrequency study, as well as an embedded study step. The latter can mainly be considered as a placeholder which is a mandatory input to the model reduction. In case you have already computed the eigenmodes, you can change the setting of the in this node to point to the corresponding existing eigenfrequency study, and then delete the newly generated one.


	The Model Reduction study step is set up to be suitable for a random vibration analysis. In particular, it is required the Ensure reconstruction capability is selected.

Under , a node becomes available. It contains three subnodes:

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A node, in which you define the control parameters for the ROM. All loads that represents random excitations must have a value multiplied by one of such parameters.

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A node. This is a placeholder for the ROM to be created.

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A node. Here, you prescribe the PSD functions and the cross-correlation functions, if needed. They can be function of frequency only (the internal variable freq); it is not possible to prescribe a coordinate dependency.

Workflow

Here, you find recommended steps to perform a random vibration analysis.

Add a study.

In the node, add all required control parameters. The values that you assign to the parameters are not important. Control parameters are used in the same manner as ordinary parameters in expressions for the loads in the physics interface. You need as many control parameters as you have different PSD functions.

In the physics interface, create the loads that should have a random variation. Typically, the value is just the name of a control parameter, even though it is possible to also add a scalar multiplier to it.

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The reduced-order models on which the random vibration analysis is based are not unit aware. Always use the base units from the current unit system in the context of a random vibration analysis.

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Only ‘Neumann type’ loads like forces and pressures can be given a random excitation. It is not possible to use ‘Dirichlet type’ loads like prescribed displacements, velocities, or accelerations. This is similar to other mode superposition methods.

For the common case that a number of support points have the same acceleration PSD, the support acceleration can be replaced by a frame acceleration load, using the feature. This is how vibration testing is analyzed. See also Absolute Versus Relative Accelerations.

If you want any scalar outputs, define them as under in the component. You can use, for example, probes or functions like at3() to create scalar outputs. It is also possible to compute scalar results using, for example, during result presentation. Such evaluations will, however, require a larger computational effort than scalar outputs that are part of the ROM definition.

In case you want to reuse a previously computed eigenfrequency study, go to the study step. Change the settings for in the section to point to the correct eigenfrequency study.

Run the study containing the study step. It will automatically create a ROM that can be used for all further evaluations.

Define the PSD and cross-correlation functions required for representing the loading as functions under . These functions should depend on frequency only.


	It is common that an input PSD is provided in terms of straight lines in a log-log diagram of PSD value versus frequency, f. To mimic this behavior, you can use an interpolation function (say int1) where you enter pairs of log(f) and log(PSD) values. You then reference this function through an expression like exp(int1(log(freq[1/Hz]))). This will provide a linear interpolation in a log-log space.

Go to the node. Select the appropriate .

For each control parameter, enter the corresponding spectrum function. The argument to the function should be the built-in frequency variable freq.

If the correlation type is , enter also the off-diagonal part of the .

To make these settings available for result evaluation, run for the model reduction study.

Result Evaluation

For presentation of random vibration results, you can use the standard features under , with the provision that the quantity you are studying is a linear function of the displacements. You can use any linear expression, not only built-in variables.

In addition to linear variables, there is also a special case where the RMS of quadratic forms can be computed.

The evaluation of random vibration results is done through special operators, listed in Table 2-17, which are defined by the node (with a tag rvib1 in this example).

Table 2-17: Random Vibration Operators.
Operator	Syntax	Explanation
psd	rvib1.psd(expr)	Evaluates the PSD of an expression for the given frequency
cross	rvib1.cross(expr1, expr2)	Evaluates the cross-correlation between two expressions for a given frequency,
rms	rvib1.rms(expr, f1, f2, N)	Evaluates the RMS value of an expression within the frequency interval [f1, f2]. N points are used for integrating over the interval.
rms	rvib1.rms(expr)	Evaluates the RMS value of an expression. The frequency interval, as well as the integration settings are taken from settings in the corresponding Random Vibration node.
m	rvib1.m(K,expr, f1, f2, N)	Evaluates the K:th order moment of the PSD. Thus, rvib1.rms(expr, f1, f2, N) = sqrt(rvib1.m(0,expr, f1, f2, N)). The maximum supported moment order is K=4.
m	rvib1.m(K,expr)	Evaluates the K:th order moment of the PSD. The frequency interval, as well as the integration settings are taken from settings in the corresponding Random Vibration node. The maximum supported moment order is K=4.
q2	rvib1.q2(expr, f1, f2, N)	Evaluates the RMS of a quadratic form, for example the square of an effective displacement, or the square of the von Mises equivalent stress. The expression must not contain time derivatives like velocities or accelerations. Rather than rvib1.q2(utt^2,…), you can write rvib1.q2(u^2(2pi*freq)^4,…). The quadratic form itself must be real-valued. This means that evaluation using the von Mises stress can only be done if the material properties are real-valued. In particular, it should be noted that loss factor damping implies a complex-valued stiffness.
q2	rvib1.q2(expr)	Evaluates the RMS value of a quadratic form. The frequency interval, as well as the integration settings are taken from settings in the corresponding Random Vibration node.
q2sq	rvib1.q2sq(expr, f1, f2, N)	Evaluates the square root of the RMS of the quadratic form. The norm of the quadratic form is used as input. See q2 operator above for more information. The relation between these two operators is: rvib1.q2sq(sqrt(expr),…) = sqrt(rvib1.q2(expr,…))
q2sq	rvib1.q2sq(expr)	Evaluates the square root of the RMS of the square root of a quadratic form. The frequency interval, as well as the integration settings are taken from settings in the corresponding Random Vibration node.

The operator q2sq() is a convenience version of the q2() operator. Some important quantities have special variables, intended for use in the q2sq() operator, see Table 2-18.

Table 2-18: Predefined Variables for Use With the q2sq Operator.
Variable	Description
<phys>.mises_rv	von Mises equivalent stress
<phys>.disp_rv <phys>.vel_rv	Norm of displacement Norm of velocity
<phys>.utt_rv	Norm of acceleration
<phys>.rot_rv	Norm of rotation
<phys>.rotvel_rv	Norm of rotational velocity
<phys>.rotacc_rv	Norm of rotational acceleration


	You cannot use ordinary variables like <phys>.mises or <phys.disp> as arguments to the q2sq() operator. The reason is that the argument must never evaluate to zero.

It is convenient to plot the PSD of a result quantity computed, for example, at certain point in the structure for a given frequency range. Such evaluation sweeps over the frequency can be defined in a Global Evaluation Sweep node.

The RMS of a result quantity does not give direct information about its peak value. Formally, probability theory says that for a long enough process, the peak value can be arbitrarily large. In practice, it is common to assume that the peak value is three (or sometimes four) times larger than the RMS value. There are good reasons for this practice.

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The input signal is usually bounded for physical reasons. It is not a true Gaussian distribution.

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For testing, the shaker table accelerations are limited, and this is even expected in the requirements on the signal.

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The whole random vibration analysis assumes linearity. Nonlinear effects will typically limit the response.

Absolute Versus Relative Accelerations

It is common that the input to a random vibration analysis is a PSD for a base acceleration. Since it is not possible to directly prescribed nonzero displacements (or accelerations) at constrained points, the standard approach is to replace the base acceleration with an inertia load on the structure. This can be seen as a frame transformation, where the analysis is made in a coordinate systems that is connected to the fixture. This change of frame does not affect for example stress or strain results.

The computed accelerations, however, are measured relative to the support points, not relative to a room fixed frame. In most cases, you are interested in the absolute accelerations, that is what would be measured by an accelerometer. When the feature is used to describe the acceleration, special variables containing the absolute (room fixed) accelerations are also available.

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Bracket — Random Vibration Analysis: Application Library path

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Random Vibration Analysis of a Deep Beam: Application Library path