If viscous damping is present, the orthogonality property of the modes does not, in general, diagonalize the generalized damping matrix:
Equation 5-14.
Similarly, if structural damping is present, the orthogonality property does not, in general, diagonalize the generalized stiffness matrix:
Equation 5-15.
Consequently, in the presence of viscous damping matrix or structural damping, the equations of motion are coupled. For such cases, the modal frequency response method:
1. Solves the undamped system for the eigenvalues and corresponding eigenvectors.
2. Premultiplies the equation of motion of the damped system by the transpose of the eigenvector matrix, [f]T:
Equation 5-16.
The result,Equation 5-16, is expressed in modal coordinates and is coupled.
3. SolvesEquation 5-16using the direct frequency response approach described in the“Direct Frequency Response Analysis (SOL 108)”section.
Because only a subset of all the modes are typically required for an accurate representation of the response, the size ofEquation 5-16is typically much smaller than the size ofEquation 5-5in a direct frequency response solution. As a result, modal frequency response has computational benefits.
Modal damping occurs when damping is applied to each mode separately so that the equations of motion remain uncoupled. When modal damping is used, each mode has damping biwhere bi= 2miwizi. The equations of motion remain uncoupled and have the form:
Equation 5-17.
Each of the modal responses is computed from:
Equation 5-18.
The TABDMP1 bulk entry is used to define modal damping. On the TABDMP1 entry, you define frequency/damping pairs. The software uses this table to calculate the damping value to apply at a particular frequency. To activate a TABDMP1 table, include a SDAMPING case control command that references the TID of the TABDMP1 entry.
You can enter damping values on a TABDMP1 entry as either structural damping (default), Gi, damping factor, zi, or quality factor, Qi. The software converts the entered damping values to modal damping values usingEquation 5-19.
Equation 5-19.
where the i subscript refers to the ithmode, and not the ithexcitation frequency.
To find the damping at a specific modal frequency, the software linearly interpolates or linearly extrapolates the values provided on the TABDMP1 entry. For example, suppose damping factor/frequency data is entered on a TABDMP1 entry as follows:
1 2 3 4 5 6 7 8 9 10
TABDMP1 10 CRIT +TAB1
+TAB1 2.0 0.16 3.0 0.18 4.0 0.13 6.0 0.13 +TAB2
+TAB2 ENDT
Also suppose the structure being analyzed has modes at 1.0, 2.5, 3.6, and 5.5 Hz. The software will interpolate and extrapolate the TABDMP1 data to produce estimates of damping factor at the modal frequencies as indicated inTable 5-1.
Table 5-1. Results from Interpolating/Extrapolating TABDMP1 Data
Entered Computed
f zzz f zzz
2.0 0.16 1.0 0.14(1)
3.0 0.18 2.5 0.17
4.0 0.13 3.6 0.15
6.0 0.13 5.5 0.13
(1)The value is extrapolated from data at 2.0 Hz and 3.0 Hz.
By default, the damping values specified on the TABDMP1 entry are used to calculate bi in Equation 5-17. However, to process the damping values as complex stiffness,
specify PARAM,KDAMP,-1 in the bulk data section of the input file. When you specify PARAM,KDAMP,-1, Equation 5-17becomes:
Equation 5-20.
Also by default, NX Nastran computes modal damping only once from the original modal solution. This can lead to inaccurate results if the model contains elements with frequency dependent stiffness properties. For example, frequency dependent stiffness properties can be defined for certain elements using the PBUSHT and PELAST property entries.
You can improve accuracy of the results for such a model by specifying PARAM,SDAMPUP,YES in the bulk data section of the input file. When PARAM,SDAMPUP,YES is specified, the software uses an updated, frequency dependent modal stiffness to update the modal damping (including any frequency independent modal damping) for each frequency response calculation. The penalty for improving the accuracy of the results is that the runtime may increase, particularly if the number of modes is large.
Repeated eigenvalues and rigid body modes are not supported with this method. Rigid body modes may be removed either by adjusting F1 on the EIGRL bulk entry or by setting PARAM,LFREQ. If repeated eigenvalues are present, the method is deactivated automatically and the default method is used instead.
For example, the performance of four different solution and damping combinations was evaluated using a relatively small model of approximately 3000 DOF that contained one scalar spring element with frequency dependent stiffness specified using a PELAST entry. The frequency response up to 133 Hz was computed using modes with frequencies up to 350 Hz. The results from each solution and damping combination are depicted inFigure 5-3.
Figure 5-3. Frequency Response Predicted Using Various Combinations of Solutions and Damping.
The four solution and damping combinations were:
• Direct frequency response with damping specified using PARAM,G. (The “S108 param,G”
curve in Figure 5-3.)
• Modal frequency response with modal damping specified using the SDAMPING case control command only. (The “S111 SDAMP (default)” curve inFigure 5-3.)
• Modal frequency response with modal damping specified using the SDAMPING case control command and PARAM,SDAMPUP,YES. (The “S111 param,sdampup” curve inFigure 5-3.)
• Modal frequency response with damping specified using PARAM,G. (The “S111 param,G”
curve inFigure 5-3.) This solution is a direct solution to the reduced modal representation.
The results from all the solution and damping combinations were essentially identical except for those from the modal solution where PARAM,SDAMPUP,YES was not specified. Once PARAM,SDAMPUP,YES was specified, the modal damping solution was consistent with the others.
The models were run on an IBM Power4/1200MHz machine. By adding the
PARAM,SDAMPUP,YES specification to the modal solution, the runtime increased from 0:19 to 0:36. By comparison, the runtime for the direct frequency response solution was 0:58.