1. Oliviero Giannini, Ute Gauger, Michael Hanss
Analysis of the uncertain dynamic behavior of an automotive control unit using the Component Mode Transformation Method.
Oliviero Giannini, Ute Gauger, Michael Hanss
Institut für Angewandte und Experimentelle Mechanik, Universität Stuttgart, Germany
Universität Stuttgart

2. Methodology: transformation method
The uncertain parameters are modeled by using fuzzy numbers
Defines a sampling strategy for the input parameters
The crisp model is evaluated several time
The crisp output of the model are postprocessed in order to obtain the fuzzy output.

3. Methodology: Transformation method
Advantages:
Extremely simple
Works with any FE program
Capable of any structural analysis
Complete ignorance on the model characteristics
Disadvantages:
High computational resources

4. Methodology: The Component Mode Transformation Method (CMTM)
Sub structuring case A (Substructures certain and connections uncertain)

5. Methodology: The Component Mode Transformation Method (CMTM)
Sub-structuring case B (Substructures and connection are uncertain)

6. Methodology: CMTM Advantages: Much faster than the standard method
Error generally negligible with respect to the size of uncertainties
Extremely appropriate to solve problem with complex geometry
Disadvantages:
Mainly suited for linear analysis (non-linearities must be introduced in the reduced model)
Needs knowledge about the original model (substructuring)
Cannot reduce the scale factor: 100 fuzzy parameters lead to an overwhelming number of models to evaluate (even if they are reduced)

7. Computational Gain
AsymptoticGain
Critical Number

8. Preliminary analysis of the computational cost
For the modal value of all the uncertain parameters:
Evaluate the full system model.
Create of the reduced models of the substructures.
Evaluate the Reduced Model.
Comparing the results of solution 1 and 3, it is possible to estimate the error introduced by the reduction process.
Comparing the time of solution 1 and 3, it is possible to estimate the Asymptotic Computational gain.
It is also possible to estimate if the the number of uncertain parameters is higher than the critical number

9. Automotive controll unit: Model

10. Automotive controll unit: Model

11. Uncertain parameters 0.06 0.04 0.08 General 0.12 0.14 0.16 108 107 109
Nominal value
Left-hand worst-case deviation
Right-hand worst-case deviation
Type
The x coordinates of the positioning of mass 1
0.06
0.04
0.08
General
The y coordinates of the positioning of mass 1
0.12
The x coordinates of the positioning of mass 2
0.14
0.16
The y coordinates of the positioning of mass 2
Constraint Stiffness kuz
108
107
109
Reduced

12. Substructuring strategies
Case 1: 1 substructure - the board with 4 uncertain parameters. The uncertain constraint is only considered in the reduced model.
Case 2: 2 substructures - the board is cut into 2 parts, each with 2 uncertain parameters. The uncertain clamping conditions are only considered in the reduced model.
Case 3: 2 substructures - the board is cut into 2 parts, each with 3 uncertain parameters. Two uncertain parameters are related to the position of the electronic component, the third to the constraints.

13. Preliminary analysis and comparison between the substructuring strategies

14. Automotive controll unit: eigenvalues

15. Automotive controll unit: eigenvalues

16. Automotive controll unit: eigenvalues

17. Automotive controll unit: FRF

18. Concluding remarks
The method couples the capabilities of the standard Transformation Method, with the computational advantages of the Component Mode Synthesis approach.
The method allows for a preliminary analysis, which provides reliable expectations on the computational gain as well as the errors.
The proposed application it is not suited for the application of the proposed method as highlighted by the preliminary analysis
However it is possible to reduce the computational time about a factor 3 with a negligible error on the results

19. Fuzzy Numbers to model the uncertainties

20. Input Parameter Hyperspace
general transformation method
reduced transformation method

21. Input Parameter Hyperspace
general transformation method
reduced transformation method

22. Computational Cost The actual advantage of the method depends on:
Number and type of uncertainties
How the uncertainty is modeled (general transformation or reduced transformation method)
The computational cost of the analysis and how it scales with the dimension of the model
The size of the reduced model depends on the number of used modes and not on the complexity of the geometry

23. Outline Methodlogy Application Concluding Remarks
The Component Mode Transformation Method
Analysis of the computational cost
Application
The test case
Comparison between different substructuring strategy
Results
Concluding Remarks

24. The CMTM procedure
Step 1: Definition and sampling of the parameter hyperspace.
The input parameter hyperspace is sampled according to the Transformation Method or, if possible, to the reduced Transformation Method.
Step 2: Substructuring of the model.
The uncertain model is sub-structured so that the minimum possible number of uncertain parameters characterizes a single substructure and the maximum number occurs in joints or boundary conditions.
Step 3: Application of the Transformation Method to the substructures
Since the model of each substructure is uncertain, the Transformation Method is applied. Instead of a single crisp reduced model of the substructure, a set of those is necessary to cover the portion of the parameter hyperspace inherent to that substructure.
Step 4: Assembling the crisp reduced model of the whole structure.
Each crisp reduced model of the overall structure is characterized by a well-defined set of crisp parameters (this is obtained at step 1 when the parameter hyperspace is sampled). Therefore, the models of the substructures to be used in this step must be chosen accordingly.
At this step, the uncertainties related to joints and boundary conditions are considered too.
Step 5: Solution and post-processing.
Each crisp reduced model is solved, and the fuzzy outputs are reconstructed from the data obtained from the analysis of each crisp model.

25. Uncertain parameters
The x- and y-coordinates of the positioning of mass 1 (x1,y1)
The x- and y-coordinates of the positioning of mass 2 (x2,y2)
x1 = (0.04) 0.06 (0.8) m (symmetric, triangular fuzzy number),
x2 = (0.12) 0.14 (0.16) m (symmetric, triangular fuzzy number),
y1 = y2 = (0.04) 0.08 (0.12) m (symmetric, triangular fuzzy number).
The uncertain clamping of the board is described by 1 uncertain parameter:
kuz = (107) 108 (109) N/m,
kroty = 0.001· kuz = (104) 105 (106) N/m.