Dynamic reduction consists of using additional shape functions (Goq, in the DMAP) to improve the approximation used to reduce a superelement (over the approximation that exists if only a static reduction were used).
For reference, we will look at conventional dynamic analysis. For many problems, the mode shapes of a structure are used to transform the equations of motion into modal coordinates, thus simplifying the solution. If this approach and all modes of a structure are used, no approximation occurs in this transformation.
Dynamic reduction of superelements is similar to the transformation of a structure into modal coordinates. However, the transformation involves the use of both static and dynamic shapes.
The dynamic reduction method available when using superelements is Component Modal Synthesis.
As before, the static transformation matrix is generated. This operation is always performed.
In dynamics, a dynamic transformation can be performed in addition to the static transformation.
A series of dynamic shape-functions (Goq, in DMAP) is found and used to dynamically transform the superelement properties to the exterior DOFs.
When doing a dynamic reduction, the program calculates dynamic shapes based on the boundary conditions you have selected.
Note
You choose the shapes used to perform the dynamic reduction. The accuracy of the final solution depends on how many shape functions you use and how these shapes are calculated.
At this point, we must discuss the subsets of the A-set. The A-set in NX Nastran is the set containing all exterior DOFs for a superelement. When a superelement is processed, the component matrices are first generated for the G-set, then the matrices go through a series of processing steps until what remains is a set of reduced matrices that attach to the A-set DOFs. If the reduction is performed properly, these reduced matrices contain sufficient information to represent the dynamic behavior of the superelement.
The A-set is divided into several subsets for dynamic reduction. The following subsets of the A-set are used:
T Physical exterior DOF of the superelement Q Generalized coordinates in the A-set
Note
The A-set is divided into a number of subsets. These subsets determine whether an exterior DOF is held constrained during the calculation of the dynamic shape functions.
Note that this has no affect on whether the points are constrained once the superelement has been processed.
For purposes of dynamic reduction, the T-set is divided into three subsets:
B Physical exterior DOF that are constrained during the calculation of the dynamic transformation vectors.
C Physical exterior DOFs that are not constrained during the calculation of the dynamic transformation vectors.
R Reference DOFs—these are treated the same as the C-set during dynamic reduction.
The B-, C-, and R-sets allow you control over the way the dynamic transformation vectors are calculated. These sets define how the exterior DOFs are treated during the calculation of the dynamic transformation matrix for the superelement. The placement of a DOF in a B-, C-, or R-set does not determine how the point is treated in downstream superelements or whether the point is constrained in the final solution.
By default, all physical exterior DOFs for a superelement are placed in the B-set for that superelement. If you want the DOFs to be treated differently, you must place the DOFs in either the C- or R-set (see“Input and Output for Dynamic Reduction”).
Note
Placing an exterior DOF in the B- or C- or R-set for a superelement determines how that DOF is treated during the calculation of the dynamic shape functions for the superelement. These sets do not apply physical constraints to the model and have no impact on how the DOF is treated in downstream superelements.
Because the motion of a superelement is represented as a linear combination of the vectors used in the reduction, these vectors should resemble (as closely as possible) the actual deformations the superelement experiences in the final dynamic solution.
To illustrate, we will look at a simple problem—the vibration modes of a cantilever beam—using several different approaches. The model is shown inFigure 9-1.
Figure 9-1. Cantilever Beam Model for Dynamic Reduction
We divide the beam into two superelements using the following SESET entries (it will also be done using PART superelements):
SESET,1,7,THRU,11 SESET,2,2,THRU,5
Using this definition, we find the first four system modes, using one, two, and three component modes per superelement.
We look at this problem in several ways—first by placing all exterior DOF in the B-set (the default), then by placing all exterior DOFs in the C-set, and finally, by placing point 6 (which is exterior to superelement 2) in the C-set for superelement 2.
The input files are not shown here because we have not yet discussed the format; however, the results are shown.
The following table shows the first four natural frequencies (in-plane bending only) for this model. The first set of results represents a model with no superelements (the correct finite element solution); the second set of solutions represents the results found by using static reduction only; the third set represents those found by placing all exterior DOFs in the B-set (fixed boundary); the fourth set shows the results obtained by placing all exterior DOFs in the C-set (free-free); and the fifth set shows the results found by placing all exterior DOFs for superelement 1 in the B-set, while placing grid point 6 in the C-set for superelement 2. In the results for each set of boundary conditions, the first column represents the solution found by using only one mode for each superelement; the second column represents the results using two modes for each superelement, and so on.
Baseline Static
Reduction Fixed Boundary Free Boundary Mixed Boundary
one
Baseline Static
Reduction Fixed Boundary Free Boundary Mixed Boundary
349.73 359.07 349.73 349.73 349.07 359.07 355.68 349.76 349.73 349.73 349.73 2167.32 2934.10 2182.49 2168.12 2167.44 2872.97 2872.56 2407.86 2181.82 2168.04 2167.42 6007.8 6733.63 6011.36 6008.07 12377.89 11445.01 6042.50 6778.62 6011.01 6008.02 11650.2 18222 18514.4 11670.88 29918.47 13974.00 20474.5 11839.85 11669.57
How the exterior DOFs are handled during the calculation of the component modes can make a significant difference in the accuracy of the results, as shown by the results in the above table.
Using only static reduction is the same as if an A-set were defined for the model (without superelements), including only grid point 6. Surprisingly, this approach gives a reasonable estimate of the first mode, and a somewhat reasonable estimate of the second mode (within 35% of the correct value).
Note
For some models static reduction may be all that is needed.
When all exterior points are in the B-set, the component modes of superelement 1 are found with grid point 6 constrained, or cantilevered, and the component modes of superelement 2 are found with grid points 1 and 6 fixed, or in a fixed-fixed state. The results for this approach are accurate for this model. With only one mode for each superelement, the first two elastic modes of the model are predicted within 1%, and the third mode is predicted within 12%. When additional component modes are added, the results quickly converge to the correct solution. With three modes per superelement, the fourth system mode is predicted with an accuracy of better than 1%.
When all exterior points are in the C-set, the component modes of each superelement are found with no constraints applied on the exterior points, or (for this model) in the free-free state. The first system mode is predicted well (within 2%) using only one mode per superelement. Notice that when using only one mode per superelement, the model finds only three system modes, which is explained later in this section. For this model, predicting the second system mode using free-free CMS is difficult. Using three modes per superelement, the results for the second mode are only within 11% of the correct value, indicating how much impact the method used to handle the exterior points during CMS can have.
When the fourth approach is used, the component modes of superelement 1 are found with grid point 6 constrained (cantilevered), and the component modes of superelement 2 are found with grid point 1 fixed and no constraints on grid point 6 (cantilevered). These results are the best for all the methods shown.
Conclusion
We recommend dynamic reduction for almost all models using superelements in a dynamic solution. How you treat the exterior points during the dynamic reduction calculation is very important to the accuracy of your solution. We recommend calculating component modes using the boundary condition that best approximates how you expect the component to behave when combined with the rest of the structure.