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

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.

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

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

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

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)

Computational Gain AsymptoticGain Critical Number

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

Automotive controll unit: Model

Automotive controll unit: Model

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

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.

Preliminary analysis and comparison between the substructuring strategies

Automotive controll unit: eigenvalues

Automotive controll unit: eigenvalues

Automotive controll unit: eigenvalues

Automotive controll unit: FRF

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

Fuzzy Numbers to model the uncertainties

Input Parameter Hyperspace general transformation method reduced transformation method

Input Parameter Hyperspace general transformation method reduced transformation method

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

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

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.

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.