# Summary

In this course, we discussed Random Vibration Analysis and Random Vibration Fatigue and some important aspects to consider when performing this analysis. Let’s summarize the key points from each lesson.

How to Perform Random Vibration Analysis

• Random Vibration Analysis is a linear analysis based on Mode Superposition Method. The input is in the form of PSD, which is statistical in nature, hence the Output is also statistical in nature.
• The Power Spectral Density (PSD) contains the loading (in terms of quantity squared) vs frequency. This is used as the load in Random Vibration Analysis.
• The main principle behind evaluating random vibrations is that for a given frequency, its average value tends to remain constant.
• The Random Excitation Input is assumed to follow Gaussian/Normal Distribution in which the 1-sigma value represents the occurrence 68.3% of the time, 2-sigma represents the occurrence 95.4%, and the 3-sigma value represents the occurrence 99.7% of the time.
• The square root of the area under the Response PSD is the RMS value. It is 1-sigma or 1 standard deviation value in statistical terms.
• It is worth noting that the number of modes that needs to be extracted from Modal analysis should range from 0 to 1.5 times the maximum input PSD frequency.

How to check the fidelity of the Input PSD curve

• The reason we pursue statistical results instead of time-history dynamic results is because we cannot easily characterize the loading, it is random, and solving the analysis using the time-history data is computationally expensive and may not always yield useful results.
• A good quality Input PSD is required to obtain reliable results, and the quality of the input is determined by Ansys Mechanical.
• For a good quality Input PSD, the PSD values between two consecutive points should not change more than an order of magnitude.
• If the PSD input curve is marked as green, it is considered as reliable and accurate.
• If the PSD input curve is yellow, it is a warning indicator. The results may not be reliable and accurate.
• If the PSD input curve is red, the results produced are not trustworthy. It is recommended to modify the input curve.
• In Ansys Mechanical, we can use the option “Improved Fit” to improve the Input PSD curve automatically. Intermediate data points are added to improve the Input PSD curve.

How to Interpret Random Vibration Results Correctly

• A random vibration analysis is a type of linear dynamic analysis that studies a structure's response to random vibrations. Both the input and output quantities are probabilistic in nature and the results have a Gaussian distribution.
• The directional results from the solver cannot be combined in the usual way due to their statistical nature. The X, Y, and Z displacements, for example, cannot be combined to determine the size of the overall displacement. The same holds true for other derived quantities such as Principal stresses and Equivalent stresses.
• A special algorithm called Segalman-Fulcher is used to compute Equivalent Stress. The 3-sigma rules of multiplying the RMS value by 3 gives us a conservative estimate on the upper bound of the equivalent stress.
• Displacements are always in the Nodal Solution coordinate system, while the stresses and strains are always in Elemental Solution coordinate system.
• Response PSD is plotted as the square of spectrum response over the excitation frequency range, thus provides information on how the power is distributed as a function of frequency. It can also give us valuable information about the model’s response at a certain frequency or frequency range.
• The centroid of the area under the Response PSD w.r.t frequency is referred to as the “Expected Frequency.” This also gives us an idea whether the output response is occurring at or near a dominant frequency.
• By increasing the “Clustering frequency points,” one can increase the accuracy since this feature will increase the number of frequency points generated on both sides of the natural frequencies.

How to Perform Random Vibration Fatigue Analysis

• Random vibration fatigue analysis helps to determine the life of structure when subjected to random vibration.
• The Fatigue life in Random Vibration Fatigue Analysis is calculated based on Stress-life approach.
• The S-N curve represented in Ansys Mechanical is linear or bilinear in nature. If the linear S-N curve is not directly defined, then Ansys Mechanical automatically considers a linear plot between the first and the last data points available.
• There are three main cycle counting methods – Narrow Band Formulation, Steinberg Formulation and Wirsching Formulation. Among these, the Steinberg Formulation is the most widely used method to calculate fatigue life and damage.
• The Steinberg Formulation assumes a Gaussian distribution with 68.27% cycles occurring at 1-sigma stress, 27.18% cycles occurring at 2-sigma stress and 4.28% of cycles occurring at 3-sigma stress.
• Miner's rule is used to sum up damage. One needs to turn on “Calculate Velocity” and “Calculate Acceleration” under the Analysis Settings which are vital to calculate fatigue life and damage.
• Fatigue life is reported in seconds, while damage is reported per duration exposure.