Bracket — Random Vibration Analysis

The various examples based on a bracket geometry form a suite of tutorials that summarizes the fundamentals of modeling structural mechanics problems in COMSOL Multiphysics and the Structural Mechanics Module.

This example shows how to perform a random vibration analysis for a structure using power spectral density (PSD). The computations are based on the modal reduced-order model (ROM).

Model Definition

The model geometry is represented in Figure 1.

Figure 1: Bracket geometry. Rigid connector and its center of rotation are shown using special physics symbols.

The bracket is made of structural steel. Four smaller holes in the bracket are fully constrained to represent a bolted connection.

A pin connects two larger holes in the bracket arms. The pin is assumed to be perfectly rigid and is modeled with a rigid connector. Thus, the pin is not represented in the model geometry.

The rigid connector has the translational and rotational degrees of freedom of a rigid body. Options for this feature include applying an external load to such rigid body. In this tutorial, you use two components of such load as random excitations applied to the structure. The input PSD for the load components are shown in Figure 2.

Figure 2: Input PSD for random load components. Log-log scale is used.

The two applied load components are assumed to be fully correlated. Physically, this means that it is the same excitation which causes the load in both directions.

You will compute the displacement response at a point with coordinates: X = 0, Y = 0, Z =-0.45, which is a representative location in the lower plate of the bracket.

Results and Discussion

Figure 3 shows the PSD for two displacement components calculated for the frequency range that contains the first three natural frequencies of the structure. Note that there are three peaks on the resulting PSD. The left and right ones correspond to two natural frequencies of the structure, while the middle peak corresponds to the maximum of the input excitation.

The cross-correlation between two displacement components is shown in Figure 4.

Figure 3: PSD for the horizontal (U) and vertical (W) displacement response at the selected location.

Figure 4: Cross-correlation between the horizontal and vertical displacement response at the selected location.

Figure 5 and Figure 6 show the PSD for the horizontal and vertical displacement, respectively, computed at a particular frequency.

Figure 5: PSD of u-displacement component (m2/Hz).

Figure 6: PSD of w-displacement component (m2/Hz).

Figure 7 and Figure 8 show the RMS for the horizontal and vertical displacement, respectively, computed using integration of their corresponding PSD over the frequency range from 150 to 800 Hz.

Figure 7: RMS of u-displacement component (m).

Figure 8: RMS of w-displacement component (m).

The interpretation of results from a random vibration analysis requires special attention. Usually, the RMS is the primary result. The RMS is defined as

(1)

for some long time T. In many cases, you are more interested in the maximum value than in the RMS. For many statistical distributions, for example the normal distribution, the tail of the distribution extends to infinity. Mathematically the response can then reach an arbitrarily high value with a probability that increases with time.

In practice, it is customary to assume that the maximum value does not exceed a certain factor times the RMS value. This factor is often taken as 3 or 4.

Notes About the COMSOL Implementation

The simplest way to set up a random vibration analysis in COMSOL Multiphysics is to add a predefined study called . This will automatically add three study nodes configured to act as a sequence.

The first added study is an eigenfrequency analysis that serves as a basis for the modal reduced-order model (ROM) required for the system response analysis. Note that any nodes should be manually disabled under this study node for all structural interfaces in the model. This is needed because the eigenmode extraction should be always done for undamped system.

The second added study is an ordinary frequency domain analysis that only sets up the type of the ROM that will be constructed using the last added study. There is no need to run this second study unless you intend to perform a non-random frequency response analysis of the structure.

It is enough to compute the last of the three studies. This will automatically compute the requested number of eigenfrequencies and then create a ROM based on the computed eigenmodes.

When you add a study, two nodes under will be also added automatically. Those are used for specifying the model control parameters (inputs) and to enter expressions for the input PSD to be used in the random vibration computations. Such computations are performed as postprocessing steps. However, if the PSD expressions are changed under , a solution update might be needed.

The evaluations are done using special operators that become available as part of the random vibration analysis.

The input PSD as functions of the frequency are often specified using linear interpolation in the log-log representation of the data. The model shows how to set up such functions.

Structural_Mechanics_Module/Tutorials/bracket_random_vibration

Modeling Instructions

Root

This file serves as starting point for various examples based on a bracket geometry. It sets up the model geometry, material, and mesh. It also adds a Solid Mechanics interface with a fixed constraint applied to the bolt holes.

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file bracket_basic.mph.

Add Study

In the toolbar, click