Random Vibration Analysis Introduction This analysis enables you to determine the response of structures to vibration loads that are random in nature. An example would be the response of a sensitive electronic component mounted in a car subjected to the vibration from the engine, pavement roughness, and acoustic pressure.

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• Thomson Theory Of Vibration Solution Manual

Loads such as the acceleration caused by the pavement roughness are not deterministic, that is, the time history of the load is unique every time the car runs over the same stretch of road. Hence it is not possible to predict precisely the value of the load at a point in its time history. Such load histories, however, can be characterized statistically (mean, root mean square, standard deviation). Also random loads are non-periodic and contain a multitude of frequencies. The frequency content of the time history (spectrum) is captured along with the statistics and used as the load in the random vibration analysis.

This spectrum, for historical reasons, is called Power Spectral Density or PSD. In a random vibration analysis since the input excitations are statistical in nature, so are the output responses such as displacements, stresses, and so on. Typical applications include aerospace and electronic packaging components subject to engine vibration, turbulence and acoustic pressures, tall buildings under wind load, structures subject to earthquakes, and ocean wave loading on offshore structures.

Points to Remember. The excitation is applied in the form of Power Spectral Density (PSD). The PSD is a table of spectral values vs. Frequency that captures the frequency content.

The PSD captures the frequency and mean square amplitude content of the load’s time history. The square root of the area under a PSD curve represents the root mean square (rms) value of the excitation. The unit of the spectral value of acceleration, for example, is G 2/Hertz.

The input excitation is expected to be stationary (the average mean square value does not change with time) with a zero mean. This analysis is based on the mode-superposition method. Hence a that extracts the natural frequencies and mode shapes is a prerequisite.

This feature covers one type of PSD excitation only- base excitation. The base excitation could be an acceleration PSD (either in acceleration 2 units or in G 2 units), velocity PSD or displacement PSD. The base excitation is applied in the specified direction to all entities that have a boundary condition.

Other support points in a structure such as Frictionless Surface are not excited by the PSD. Multiple uncorrelated PSDs can be applied. This is useful if different, simultaneous excitations occur in different directions. If stress/strain results are of interest from the random vibration analysis then you will need to request stress/strain calculations in the modal analysis itself. Only displacement results are available by default. Postprocessing.

The results output by the solver are one sigma or one standard deviation values (with zero mean value). These results follow a Gaussian distribution. The interpretation is that 68.3% of the time the response will be less than the standard deviation value. You can scale the result by 2 times to get the 2 sigma values. The response will be less than the 2 sigma values 95.45% of the time and 3 sigma values 99.73% of the time.

The Coordinate System setting for result objects is, by default, set to Solution Coordinate System and cannot be changed because the results only have meaning when viewed in the solution coordinate system. Since the directional results from the solver are statistical in nature they cannot be combined in the usual way. For example the X, Y, and Z displacements cannot be combined to get the magnitude of the total displacement. The same holds true for other derived quantities such as principal stresses. A special algorithm by Segalman-Fulcher is used to compute a meaningful value for equivalent stress. Preparing the Analysis Create Analysis System.

For this analysis type. Note: If you set the Mode Significance Level property to 0.0, the application considers all modes in mode superposition of random vibration responses. This can require significant computation time for large systems that use a large number of modes to obtain random vibration displacement responses. In this case, a Mode Significance Level setting that excludes insignificant modes from superimposing random vibration displacement responses is recommended. However, this performance improvement reduces solution accuracy. As a result, you need to use caution and carefully check your solution.

Set the Mode Significance Level to 1e-4 when you are concerned about solution processing time. During Random Vibration analyses, the velocity and acceleration responses are separate calculations, in addition to displacement responses. To further improve your solution time, do not request velocity and acceleration responses unless needed. The velocity and acceleration responses require approximately the same computation time. By default, Displacement is the only response calculated. To include velocity ( Calculate Velocity property) and/or acceleration ( Calculate Acceleration property) responses, set their respective Output Controls to Yes. By default, modal results are removed from result file to reduce its size.

To keep modal results, set the Keep Modal Results property to Yes. Note: Default settings can be modified using the Options dialog box. See the section of the Help under. Damping Controls enable you to specify damping for the structure in the Random Vibration analysis. Controls include: Constant Damping, Constant Damping Ratio, Stiffness Coefficient (beta damping), and a Mass Coefficient (alpha damping). They can also be applied as using the tab. A non-zero damping is required.

The Constant Damping Ratio has a default setting of 0.01. This value can be modified by setting the Constant Damping property to Manual. These settings enable you to save solution files from the Random Vibration analysis. The default behavior is to only keep the files required for postprocessing. You can use these controls to keep all files created during solution or to create and save a Mechanical APDL application database (db file). You must point to a modal analysis in the Initial Condition environment field. The must extract enough modes to cover the PSD frequency range.

A conservative rule of thumb is to extract enough modes to cover 1.5 times the maximum frequency in the PSD excitation. When a PSD analysis is linked to a modal analysis, additional solver files must be saved to achieve the PSD solution.

(See.) If the files were not saved, then the modal analysis has to be solved again and the files saved. Apply Loads and Supports. For this analysis type.

Any boundary condition must be defined in the prerequisite Modal Analysis. The only applicable load is a of spectral value vs. Remote displacement cannot coexist with other boundary condition types (for example, fixed support or displacement) on the same location for excitation. The remote displacement will be ignored due to conflict with other boundary conditions.

Four types of base excitation are supported: PSD Acceleration, PSD G Acceleration, PSD Velocity, and PSD Displacement. Each PSD base excitation should be given a direction in the nodal coordinate of the excitation points.

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Multiple PSD excitations (uncorrelated) can be applied. Typical usage is to apply 3 different PSDs in the X, Y, and Z directions. Correlation between PSD excitations is not supported. For this analysis type. Note: When using a random vibration system database from a version prior to the most current version of Mechanical, it is possible to encounter incompatibility of the file(s) file.mode, file.full, and/or file.esav, created by the modal system.

This incompatibility can cause the random vibration system’s solution to fail. In the event you experience this issue, use the Clear Generated Data feature and resolve the modal system. Please refer to the section of the MAPDL Structural Analysis Guide for more information. Review Results. For this analysis type.

Thomson Theory Of Vibration Solution Manual

If stress/strain results are of interest from the random vibration analysis then you will need to request stress/strain calculations in the modal analysis itself. You can use the Output Controls under Analysis Settings in the modal analysis for this purpose. Only displacement results are available by default. Linking a Random Vibration analysis system to a fully solved Modal analysis may result in zero equivalent stress. To evaluate correct equivalent stress in this situation, you need to re-solve the Modal analysis. Applicable results are Directional (X/Y/Z) Displacement/Velocity/Acceleration, normal and shear stresses/strains and equivalent stress. These results can be displayed as plots.

The displacement results are relative to the base of the structure (the fixed supports). The velocity and acceleration results include base motion effects (absolute). Since the directional results from the solver are statistical in nature they cannot be combined in the usual way. For example the X, Y, and Z displacements cannot be combined to get the magnitude of the total displacement. The same holds true for other derived quantities such as principal stresses. For directional acceleration results, an option is provided to displayed Transient Structural Analysis Using Linked in G (gravity) by selecting Yes in the Acceleration in G field. By default the 1 σ results are displayed.

You can apply a scale factor to review any multiples of σ such as 2 σ or 3 σ. The Details view as well as the legend for contour results also reflects the percentage (using Gaussian distribution) of time the response is expected to be below the displayed values. Meaningful equivalent stress is computed using a special algorithm by Segalman-Fulcher.

Note that the probability distribution for this equivalent stress is neither Gaussian nor is the mean value zero. However, the “3 σ” rule (multiplying the RMS value by 3) yields a conservative estimate on the upper bound of the equivalent stress. Force Reaction and Moment Reaction probes can be scoped to a Remote Displacement boundary condition to view Reactions Results.

The use of nodal averaging may not be appropriate in a random vibration analysis because the result values are not actual values but standard deviations. Moreover, the element coordinate system for the shell elements in a surface body may not all be aligned consistently when using the Default Coordinate System. Consider using unaveraged results for postprocessing instead. Using Command Objects within a Random Vibration Analysis In an effort to minimize disk space usage, only the results from the Random Vibration analysis are kept in the result file. The results from the Modal analysis are removed during the solution. If your command object contains commands which require this data, set the Keep Modal Results property in the to Yes.

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