Random vibrations subject a component to non-deterministic motion which cannot be analyzed precisely. The mode shapes and natural frequencies remain the same for the component and randomness is an inherent characteristic of the input excitation. They present a significant challenge to engineers trying to meet design margins. Some examples for random vibrations are:

Automobile driving on a rough road, potholes, railroad tracks and other obstacles.

Motor vibration in vehicles.

Rocket motor ignition vibration during the first few seconds and during powered flight.

Hard disk drive motion.

Load on an airplane wing during flight.

Wave height in the ocean.

Motion of buildings, poles, bridges, and other structures during an earthquake.

In these examples, the motion varies randomly with time, and it is nonperiodic. Hence, the amplitude of vibration cannot be expressed as a deterministic mathematical function. Instead, we have to use the statistical nature of the input excitations, such as acceleration, force, velocity or displacements. Letâ€™s take a look at an example of a car travelling on a rough road. The figure below shows the time history of the vertical acceleration of the car.

Now, if we have to model the actual behavior of the car subjected to this load, we have to use an Explicit algorithm with a very small-time step. This method becomes computationally very expensive to solve for medium to large physical systems. An alternate method for evaluating these physical systems subjected to random loads is using a statistical or probabilistic approach. Most of the random excitations follow a Gaussian (normal) distribution as shown in the figure below. This shows that 68.26% of the random data corresponds to the 1Ïƒ interval and 99.7 % of the random data corresponds to the 3Ïƒ interval. Since the input excitation has statistical behavior, it is assumed that the output variables, such as displacements and stresses, have statistical nature as well.

In this method, the frequency data from the time history is acquired, along with the statistical data of the amplitude, and this is used as the load in a random vibration analysis. This spectrum is shown in the figure below and is termed as a Power Spectral Density (PSD).

## What Is A PSD Curve?

Now, letâ€™s understand what constitutes a PSD curve. The magnitude of the PSD curve is the mean square value of the input excitation. For the above acceleration vs time plot, the mean of the squared acceleration values is the power of the PSD. This power is distributed over a spectrum of frequencies (x axis) as it provides useful data when dealing with physical systems that have resonance. The power of the PSD is dependent on the frequency band width. Hence the change in frequency bands causes variation in the squared magnitudes. To overcome this, a consistent independent value of power, termed as density (y-axis), is calculated by dividing the squared magnitudes by the frequency bandwidth. Hence the units of PSD is G2/Hz.

## Random Vibration Analysis In FEA

The random vibration analysis in FEA is solved using mode superposition method. This is a linear analysis and requires an input of natural frequencies and eigenmode shapes of the physical system extracted from a linear modal analysis. The input PSD can be in terms of acceleration, velocity or displacement.

Non-zero displacements and rotations cannot be prescribed as boundary conditions in a random vibration analysis. The only loads that can be applied to the system are excitation loads (velocity, acceleration, displacement) applied through a PSD curve. Only one excitation direction is possible in each step. To compute the systemâ€™s response in multiple directions, different steps have to be used. The material density and elastic properties must be assigned to the region where dynamic response is required. Plasticity, thermal properties, rate dependent properties, electrical, diffusion and fluid flow properties cannot be included in a random response analysis since they are typically nonlinear, as are contact algorithms. While analyzing a multi-body system, components can either be tied together or connector elements can be used. The random response analysis cannot be used if contact plays a crucial role in determining the motion of the body. In that case, an alternative dynamic analysis method should be considered.

### Defining Frequency Range

Frequency range of interest for the random response analysis needs to be specified in the analysis. The response of the system will be calculated at multiple points between lowest frequency of interest and the first eigenfrequency in the range, between each eigenfrequency in the range and between the last eigenfrequency in the range and the highest frequency in the range as shown in the figure below.

### The Bias Parameter

The bias parameter is used to determine the spacing of result points in each of the frequency intervals as shown in the figure below. A bias parameter of 1 gives equally spaced result outputs in the frequency interval. However, most relevant information is usually clustered around the resonant frequencies of interest.

## Output from Random Vibration Analysis

The output of a random response analysis will be the PSD of stresses and displacements, and variance and root mean square values of these variables if required. Note that these are not the actual stresses of the physical system at any time point, but they are root mean square values of the stresses occurring in the system undergoing random vibrations. The figure below shows the RMS values of the von Mises stress for a steel structure subjected to random vibrations.

Since, the input excitations are assumed to have a normal distribution, the output variables will also have normal distributions. So, they can be extracted with different levels of confidence (68.26%, 95.44% or 99.72%). The computational cost of the simulation can be reduced by requesting the output only for selected element or node sets.

Once high stress regions are identified from the results, a spectral plot of stresses can be plotted. These plots can be used to identify the frequencies that contribute most to the RMS stress. Good insights into potential design changes can be obtained by reviewing frequency response displacements at the problematic frequencies determined from the stress spectrum.

## Final Thoughts

Random response analysis predicts the response of a physical system subjected to a non-periodic continuous excitation that is expressed in a statistical manner. This analysis is incorporated in the design phase by engineers to avoid issues in the physical system related to these dynamic effects. Hopefully, this article has provided some useful insights into the components of PSD curve as well as input and output parameters for a random response analysis.

Weâ€™re always here to help, so if you have questions about dynamic effects in your designs or models, or just FEA in general, donâ€™t hesitate to reach out!

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