11 (l)random vibrations methodology
- 2. 7/22/2019 2 Introduction What is random vibration analysis? • Random Vibrations is a spectrum analysis technique based on probability and statistics. • Meant for loads such as acceleration loads in a rocket launch that produce different time histories during every launch • A Random Vibration analysis computes the probability distribution of different results, such as displacement or stress, due to some random excitation • The analysis follows a modal analysis • An internal combination is done to compute the combined effect from each mode and their interactions.
- 3. 7/22/2019 3 Power Spectral Density • Power spectral density (PSD) is a statistical measure defined as the limiting mean-square value of a random variable. • It is used in random vibration analyses in which the instantaneous magnitudes of the response can be specified only by probability distribution functions that show the probability of the magnitude taking a particular value. • A PSD spectrum is a statistical measure of the response of a structure to random dynamic loading conditions. • It is a graph of the PSD value versus frequency, where the PSD may be a displacement PSD, velocity PSD, acceleration PSD, or force PSD. • The area under a PSD curve is the variance of the response (square of the standard deviation).
- 4. 7/22/2019 4 Inputs-Outputs •Input: –Natural frequencies and mode shapes from a modal analysis –Single or multiple PSD excitations applied to ground nodes •Output: –1sresults can be contoured like any other analysis. –Response PSD at one DOF (one point in one direction) 7/22/2019 4
- 5. 7/22/2019 5 Preparing for Analysis • Create Analysis System Because a random vibration analysis is based on modal responses, a modal analysis is a required prerequisite. The requirement then is for two analysis systems, a modal analysis system and a random vibration analysis system that share resources, geometry, and model data. From the Toolbox, drag a Modal template to the Project Schematic. Then, drag a Random Vibration template directly onto the Modal template
- 6. 7/22/2019 6 • Define Engineering Data Both Young's modulus (or stiffness in some form) and density (or mass in some form) must be defined in the modal analysis. Material properties must be linear but can be isotropic or orthotropic, and constant or temperature- dependent. Nonlinear properties, if any, are ignored. • Attach Geometry There are no specific considerations for a random vibration analysis. • Define Connections Only linear behavior is valid in a random vibration analysis. Nonlinear elements, if any, are treated as linear. If you include contact elements, for example, their stiffness's are calculated based on their initial status and are never changed. Only the stiffness of springs are taken into account in a random vibration analysis.
- 7. 7/22/2019 7 • Apply Mesh Controls/Preview Mesh There are no specific considerations for a random vibration analysis • Establish Analysis Settings You can specify the number of modes to use from the modal analysis. A conservative rule of thumb is to include modes that cover 1.5 times the maximum frequency in the PSD excitation table. You can also exclude insignificant modes by setting the Mode Significance Level property to between 0 (all modes selected) and 1 (no modes selected).
- 8. 7/22/2019 8 • Analysis Data Management These settings enable you to save solution files from the Random Vibration analysis. The default behavior is to only keep the files required for post processing. You can use these controls to keep all files created during solution or to create and save a Mechanical APDL application database (db file). • Note: The Inertia Relief option (under Analysis Settings) for a upstream Static Structural analysis is not supported in a Random Vibration analysis. • Define Initial Conditions You must point to a modal analysis in the Initial Condition environment field. The modal analysis 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 Analysis Data Management.) If the files were not saved, then the modal analysis has to be solved again and the files saved.
- 9. 7/22/2019 9 • Output Controls 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. • Damping Controls • 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 Material Damping using the Engineering Data 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.
- 10. 7/22/2019 10 • Apply Loads and Supports • Any Support Type boundary condition must be defined in the prerequisite Modal Analysis. • The only applicable load is a PSD Base Excitation of spectral value vs. frequency. • 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. • 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
- 11. 7/22/2019 11 Post-processing • Stress (normal, shear, equivalent) and Strain (normal, shear) results can also be reviewed. • Response PSD can be plotted at one DOF (one point in one direction, either absolute or relative to base excitation).
- 12. 7/22/2019 12 Post-Processing • 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
- 13. 7/22/2019 13 • Review Results • 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.
- 14. 7/22/2019 14 • These results can be displayed as contour 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.