The procedure for a PSD analysis consists of the following steps:
Obtain the spectrum solution.
Combine the modes.
Review the results.
Of these, the first two steps are the same as described for a single-point response spectrum analysis. The procedure for the remaining three steps is explained below.
In this step, the program uses mode shapes extracted by the modal solution to calculate the PSD response. The following requirements apply:
The mode shape file (Jobname.mode) must be available. The left mode shape file (Jobname.lmode) must also be available when the modal solution is obtained with the unsymmetric eigensolver.
The database must contain the same model from which the modal solution was obtained.
The Jobname.full, .esav, and .emat files must be available for the participation factors calculation.
The results file (Jobname.rst) must be available for writing the static solutions (base excitation) and other solution types.
The element modal load file (Jobname.mlv) must be available if load vectors were created (MODCONT,ON) and the element results were written on the Jobname.mode file (
MSUPkey= YES on the MXPAND command) during the modal analysis.
Note: If the PSD analysis is not performed in the same directory as the modal analysis, remote modal files usage must be activated (MODDIR). In this case, the PSD results file only contains PSD results. The modal analysis files in the modal analysis directory are not modified.
Enter SOLUTION (/SOLU).
Define the analysis type and analysis options:
For spectrum type (SPOPT), select Power Spectral Density (PSD).
If you are interested in element results and reaction forces, specify YES for
Elcalcon the SPOPT command. Element results and reaction forces caused by the spectrum are calculated only if they were also requested during the modal expansion pass. Note that you must have asked for element results during the modal analysis as well (MXPAND). For available element results, see Option: Number of Modes to Expand (MXPAND) in the Structural Analysis Guide.
Specify load step options. The following options are available for a random vibration analysis:
Spectrum Data
Type of PSD (PSDUNIT)
The PSD type can be displacement, velocity, force, pressure, or acceleration. Whether it is a base excitation or a nodal excitation is specified in Steps 4 and 5. If a pressure PSD is to be applied, the pressures should be applied in the modal analysis itself.
PSD-versus-frequency table
Define a piecewise-linear (in log-log scale) PSD versus frequency table. Since a curve-fitting polynomial is used for the closed-form integration of the curve, you should graph the input, which is overlaid with the fitted curve to ensure a good fit. If the fit is not good, you should add one or more intermediate points to the table until you obtain a good fit. For a good fit, the PSD values between consecutive points should not change by more than an order of magnitude.
PSDFRQ and PSDVAL are used to define the PSD-versus-frequency table. Step 6 describes how to apply additional PSD excitations (if any). The maximum number of tables is 200.
You can issue SPTOPT followed by STAT to list PSD tables, and issue PSDGRAPH to graph them.
Damping (Dynamics Options)
The following forms of damping are available: Alpha (Mass) Damping (ALPHAD), Beta (Stiffness) Damping (BETAD), and Frequency-Dependent Damping Ratio (MDAMP) result in a frequency-dependent damping ratio, whereas Damping Ratio (DMPRAT) specifies a damping ratio to be used at all frequencies. If you specify more than one form of damping, Mechanical APDL calculates an effective damping ratio at each frequency.
Note: Material-dependent damping ratio (MP,DMPR) is also available but only if specified in the modal analysis where an effective damping ratio is calculated based on the elements’ strain energies.
Residual Vector (RESVEC)
The residual vectors reduce the error caused when the higher modes are neglected in the analysis.
The remaining steps are specific to a random vibration analysis:
Apply the PSD excitation.
For base excitation, use the
UX,UY,UZlabels and theROTX,ROTY,ROTZlabels on the D (or DK, or DL, or DA) command. A value of 0.0 (or blank) can be used to remove a specification. Values other than 1.0 scale the participation factors.For uniform base motion using the SED command, specify
SEDX,SEDY, orSEDZ. A value of 0.0 (or blank) removes a specification.For nodal excitation, use the
FX,FY,FZlabels on the F (or FK) command. A value of 0.0 (or blank) can be used to remove a specification. Values other than 1.0 scale the participation factors.For pressure PSD excitation (where the pressure distribution was provided in the modal analysis), bring in the load vectors from the modal analysis (LVSCALE). You can use the scale factor to scale the participation factors.
Note: You can apply base excitations only at nodes that were constrained in the modal analysis. If you applied the constraints using solid model constraints (DK), you must use the same solid model commands in defining the PSD excitation. Any loads applied during the preceding modal analysis must be removed by deleting or zeroing them.
Begin participation factor calculations for the above PSD excitation (PFACT).
Use the
TBLNOfield to indicate which PSD table to use, andExcitto specify whether the calculations are for a base or nodal excitation.If you need to apply multiple PSD excitations on the same model, repeat steps 3, 4, and 5 for each additional PSD table. Then define, as necessary, the degree of correlation between the excitations, using any of the following commands: COVAL for cospectral values, QDVAL for quadspectral values, PSDSPL for a spatial relationship, PSDWAV for a wave propagation relationship, PSDGRAPH to graph the data overlaid with the fitted curve.
When you use the PSDSPL or PSDWAV command, you must use SPATIAL or WAVE, respectively, for
Parcoron the PFACT command. PSDSPL and PSDWAV relationships might be quite CPU intensive for multi-point base excitations. Nodal excitation and base excitation input must be consistent when using PSDWAV and PSDSPL (for example, FY cannot be applied to one node and FZ be applied to another). The PSDSPL and PSDWAV commands are not available for a pressure PSD analysis.Specify the output controls.
The only valid output control command for this analysis is PSDRES, which specifies the amount and form of output written to the results file. Up to three sets of solution quantities can be calculated: displacement solution, velocity solution, or acceleration solution. Each of these can be relative to the base or absolute.
Table 6.5: Solution Items Available in a PSD Analysis shows a summary of the possible solution sets. To limit the amount of data written to the results file, use OUTRES at the mode expansion step.
Table 6.5: Solution Items Available in a PSD Analysis
Solution Items Form Displacement Solution ( Lab= DISP on PSDRES)Displacements, stresses, strains, forces Relative, absolute, or neither Velocity Solution ( Lab= VELO on PSDRES)Velocities, stress velocities, force velocities, etc. Relative, absolute, or neither Acceleration Solution ( Lab= ACEL on PSDRES)Accelerations, stress accl's, force accl's, etc. Relative, absolute, or neither Start solution calculations (SOLVE).
Leave the SOLUTION processor (FINISH).
The modes can be combined in a separate solution phase. A maximum of 10000 modes can be combined. The procedure is as follows:
Enter Solution (/SOLU).
Define analysis type (ANTYPE,SPECTR).
Only the PSD mode combination method is valid in a random vibration analysis. This method triggers calculation of the one-sigma (1 σ, the standard deviation of the response, see Review the Results below) displacements, stresses, etc., in the structure. If you do not issue the PSDCOM command, the program does not calculate the one-sigma response of the structure. You can also specify the type of modal forces to be used in the combination.
ForceType=STATIC (default) combines the modal static forces (that is, stiffness multiplied by mode shape forces, both of which are stress-causing forces) whileForceType=TOTAL combines the summed modal static forces and inertia forces (that is, stiffness and mass forces, both of which forces are seen by the supports).The
SIGNIFandCOMODEfields on the PSD mode combination method (PSDCOM) offer options to reduce the number of modes to be combined (see the description of PSDCOM command). If you want to exercise these options, it is prudent to print the modal covariance matrices in Obtain the PSD Solution to first investigate the relative contributions of the modes toward the final solution.Start the solution (SOLVE).
After solution, leave the SOLUTION processor (FINISH).
Note: You can run multiple PSD analyses without performing the modal analysis
each time. To do so, you must activate
modeReuseKey on the SPOPT
command after the first PSD analysis and for each subsequent one so that the
database and necessary files are ready for the new analysis.
Results from a random vibration analysis are written to the structural results file, Jobname.rst. When modal remote file usage is not activated (MODDIR), they consist of the following quantities:
Expanded mode shapes from the modal analysis
Static solution for base excitation (PFACT,,BASE)
The following output, if mode combinations are requested (PSDCOM) and based on the PSDRES setting:
1 σ displacement solution (displacements, stresses, strains, and forces)
1 σ velocity solution (velocities, stress velocities, strain velocities, and force velocities)
1 σ acceleration solution (accelerations, stress accelerations, strain accelerations, and force accelerations)
1 σ is the standard deviation of the response; that is, for any output value the expectation is that this value will not be exceeded 68.3% of the time.
Note: Only component displacement, force, stress, and strain values are 1 σ values and follow a Gaussian or normal distribution. Combined values (e.g. USUM, SI, SEQV, S1, etc.), or component values transformed into another coordinate system are not statistically meaningful, and they should be avoided.
The only exception is the equivalent stress (SEQV), for which a specific method is used, such that multiplying it by 3 (the "3 σ" rule) yields a good approximation to its upper bound, see Equivalent Stress Mean Square Response in the Mechanical APDL Theory Reference. Note that you must not issue the AVPRIN command to obtain this result. SEQV is not calculated for beam and pipe elements.
You can review these results in POST1, the general postprocessor, and then calculate response PSDs in POST26, the time-history postprocessor.
Postprocessing operations read your data from the results file. Only the solution data you SAVE will be available if you resume the database after a SOLVE.
To review results in POST1, you first need to understand how the results data are organized on the results file. Table 6.6: Organization of Results Data from a PSD Analysis shows the organization.
Note: Load step 2 is left blank if you specify only nodal PSD excitation. Also, if you suppress the displacement, velocity, or acceleration solution using the PSDRES command, the corresponding load step is left blank. Also, the superelement displacement file (.dsub) is not written for load steps 3, 4, or 5 in a PSD analysis.
Table 6.6: Organization of Results Data from a PSD Analysis
| Load Step | Substep | Contents |
|---|---|---|
| 1 | 1 | Expanded modal solution for 1st mode |
| 2 | Expanded modal solution for 2nd mode | |
| 3 | Expanded modal solution for 3rd mode | |
| Etc. | Etc. | |
| 2 (Base excit. only) | 1 | Unit static solution for PSD table 1 |
| 2 | Unit static solution for PSD table 2 | |
| Etc. | Etc. | |
| 3 | 1 | 1 sigma displacement solution |
| 4 | 1 | 1 sigma velocity solution (if requested) |
| 5 | 1 | 1 sigma acceleration solution (if requested) |
For example, to read in the 1 σ displacement solution, issue the command:
SET,3,1
You may use Fact on the SET command to multiply the result values to obtain, for example, the 2 σ values using Fact=2 (the response will be less than these 2 σ values 95.4% of the time), or use Fact=3 for the 3 σ values (99.7% of the time).
Use the same options available for the SPRS analysis.
Note: Nodal averaging performed by the PLNSOL command may not be appropriate in a random vibration analysis because the result values are not actual values but standard deviations. Instead, consider using the PLESOL command to display unaveraged element results.
Note: Displacements, stresses, and strains are always in the solution nodal or element coordinate system (RSYS,SOLU).
If Elcalc = YES in MXPAND
and SPOPT commands, element results and reaction forces
of 1-σ solutions are calculated as explained in Variance of Element Results and Reaction
Forces in the Theory Reference.
To print the reaction forces variances at constrained nodes, issue the PRRSOL or PRRFOR command. If the PSD analysis is done using distributed-memory parallel processing, results from the PRRSOL command may be too conservative and PRRFOR is preferred.
You can calculate and display response PSDs for any results quantity available on the results file (displacements, velocities, and/or accelerations) if the Jobname.rst and Jobname.psd files are available. If you are postprocessing in a new session, the Jobname.db file corresponding to the PSD analysis solve must be available for resume.
The procedure to calculate the response PSD is as follows:
Enter POST26, the time-history postprocessor (/POST26).
Store the frequency vector by issuing STORE,PSD,
NPTS, whereNPTSis the number of frequency points to be added on either side of natural frequencies in order to "smooth" the frequency vector (defaults to 5). The frequency vector is stored as variable 1.Define the variables in which the result items of interest (displacements, stresses, reaction forces, etc.) are to be stored (NSOL, ESOL, and/or RFORCE).
Calculate the response PSD and store it in the desired variable (RPSD). The PLVAR command can then be used to plot the response PSD.
You can integrate the response PSD to obtain the variance and take its square root to obtain its 1 σ value. For example:
RPSD,4,3,,3,2 ! variable 4 is the relative accel RPSD of var 3 INT1,5,4,1 ! variable 5 is the integral of the RPSD *GET,VARIANCE,VARI,5,EXTREME,VLAST ! get the integral value STDDEV=SQRT(VARIANCE) ! convert to standard deviation (1-sigma)
Note: This value will correspond to the POST1 1 σ values. POST26 sums all the modes by default for the response PSD (see
SIGNIFon the RPSD command), whereas POST1 only sums the significant modes (SIGNIFequals 0.0001 by default for the PSDCOM command). You can use this comparison to verify that the significance factor on the PSDCOM command is small enough and that the curve-fitting for the input PSD curve was adequate.
Note: If the files of a PSD analysis performed with remote modal files usage (MODDIR command) have been transferred to a different machine, server, or disk, you can issue MODDIR again after entering POST26 to specify the directory path of the modal analysis files. Make sure to issue MODDIR before the STORE command.
The response PSD calculation of the equivalent stress (SEQV) is not supported.
You can calculate the covariance between two quantities available on the results file (displacements, velocities, and/or accelerations), if the Jobname.rst and Jobname.psd files are available.
The procedure to calculate the covariance between two quantities is as follows:
Enter POST26, the time-history postprocessor (/POST26).
Define the variables in which the result items of interest (displacements, stresses, reaction forces, etc.) are to be stored (NSOL, ESOL, and/or RFORCE).
Calculate the contributions of each response component (relative or absolute response) and store them in the desired variable (CVAR). The PLVAR command can then be used to plot the modal contributions (relative response) followed by the contributions of pseudo-static and mixed part responses to the total covariance.
Obtain the covariance (*GET,
Par,VARI,n,EXTREM,CVAR).
! Build the Model
/FILNAM, ! Jobname
/TITLE, ! Title
/PREP7 ! Enter PREP7
...
... ! Generate model
...
FINISH
!
! Obtain the Modal Solution
/SOLU ! Enter SOLUTION
ANTYPE,MODAL ! Modal analysis
MODOPT,LANB ! Block Lanczos method
MXPAND, ... ! Number of modes to expand, ...
D, ... ! Constraints
SAVE
SOLVE ! Initiates solution
FINISH
!
! Obtain the Spectrum Solution
/SOLU! Reenter SOLUTION
ANTYPE,SPECTR ! Spectrum analysis
SPOPT,PSD, ... ! Power Spectral Density; No. of modes;
! Stress calcs. on/off
PSDUNIT, ... ! Type of spectrum
PSDFRQ, ... ! Frequency pts. (for spectrum values vs.
! frequency tables)
PSDVAL, ... ! Spectrum values
DMPRAT, ... ! Damping ratio
D,0 ! Base excitation
PFACT, ... ! Calculate participation factors
PSDRES, ... ! Output controls
SAVE
SOLVE
FINISH
!
! Combine modes using PSD method
/SOLU ! Re-enter SOLUTION
ANTYPE,SPECTR ! Spectrum analysis
PSDCOM,SIGNIF,COMODE ! PSD mode combinations with significance factor and
! option for selecting a subset of modes for
! combination
SOLVE
FINISH
!
! Review the Results
/POST1 ! Enter POST1
SET, ... ! Read results from appropriate load step, substep
...! Postprocess as desired
...! (PLDISP; PLNSOL; NSORT; PRNSOL; etc.)
...
FINISH
!
! Calculate Response PSD
/POST26 ! Enter POST26
STORE,PSD ! Store frequency vector (variable 1)
NSOL,2,... ! Define variable 2 (nodal data)
RPSD,3,2,,... ! Calculate response PSD (variable 3)
PLVAR,3 ! Plot the response PSD
...
! Calculate Covariance
RESET ! Reset all POST26 specifications to initial defaults.
NSOL,2 ! Define variable 2 (nodal data).
NSOL,3 ! Define variable 3 (nodal data).
CVAR,4,2,3,1,1 ! Calculate covariance between displacement
! at nodes 2 and 3.
*GET,CVAR23U,VARI,4,EXREME,CVAR ! Obtain covariance.
FINISH