Performing a Random Vibration Analysis
Introduction
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.
Similarly, the computed results quantities are represented by their PSD, from which it is possible to derive a root-mean-square (RMS) value.
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.
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 Random Vibration (PSD) study, either from the Add Study window or from the Select Study page in the Model Wizard.
The Random Vibration (PSD) 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:
One study with an Eigenfrequency study step for computing the eigenfrequencies and corresponding eigenmodes.
One study with a Model Reduction study step, in which the ROM is created. It references the eigenfrequency study, as well as an embedded Frequency Domain 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 Training study 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 Global Definitions, a Reduced-Order Modeling node becomes available. It contains three subnodes:
A Global Reduced Model Inputs 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.
A Frequency Domain, Modal Reduced-Order Model node. This is a placeholder for the ROM to be created.
A Random Vibration 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.
1
Add a Random Vibration (PSD) study.
2
In the Global Reduced Model Inputs 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.
3
4
If you want any scalar outputs, define them as Variables under Definitions 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, Point Evaluation during result presentation. Such evaluations will, however, require a larger computational effort than scalar outputs that are part of the ROM definition.
5
In case you want to reuse a previously computed eigenfrequency study, go to the Model Reduction study step. Change the settings for Training study in the Model Reduction Settings section to point to the correct eigenfrequency study.
6
Run the study containing the Model Reduction study step. It will automatically create a ROM that can be used for all further evaluations.
7
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.
8
Go to the Random Vibration node. Select the appropriate Correlation type.
9
10
If the correlation type is Cross-correlated, enter also the off-diagonal part of the Correlation matrix.
11
To make these settings available for result evaluation, run Update Solution for the model reduction study.
Result Evaluation
For presentation of random vibration results, you can use the standard features under Results, 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 Random Vibration node (with a tag rvib1 in this example).
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.
 
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.
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 Base Excitation feature is used to describe the acceleration, special variables containing the absolute (room fixed) accelerations are also available.
Bracket — Random Vibration Analysis: Application Library path Structural_Mechanics_Module/Tutorials/bracket_random_vibration
Random Vibration Analysis of a Deep Beam: Application Library path Structural_Mechanics_Module/Verification_Examples/random_vibration_deep_beam