On the Modal and Non-Modal Model Reduction of Metallic Structures with Variable Boundary Conditions

Author(s)
Wolfgang Witteveen

ABSTRACT

Vibration mode based model reduction methods like Component Mode Synthesis (CMS) will be compared to methods coming from control engineering, namely moment matching (MM) and balanced truncation (BT). Conclusions based on the theory together with a numerical demonstration will be presented. The key issues on which the paper is focused are the reduction of metallic structures, the sensitivity of the reduced model to varying boundary conditions, full system response, accurate statics and the possibility to determine “a priori” the number of needed modes (trial vectors). These are important topics for the use of reduction methods in general and in particular for the implementation of FE models in multi body system dynamics where model reduction is widely used. The intention of this paper is to give insight into the methods nature and to clarify the strengths and limitations of the three methods. It turns out, that in the considered framework CMS delivers the best results together with a clear strategy for an “a priori” selection of the modes (trial vectors).

Vibration mode based model reduction methods like Component Mode Synthesis (CMS) will be compared to methods coming from control engineering, namely moment matching (MM) and balanced truncation (BT). Conclusions based on the theory together with a numerical demonstration will be presented. The key issues on which the paper is focused are the reduction of metallic structures, the sensitivity of the reduced model to varying boundary conditions, full system response, accurate statics and the possibility to determine “a priori” the number of needed modes (trial vectors). These are important topics for the use of reduction methods in general and in particular for the implementation of FE models in multi body system dynamics where model reduction is widely used. The intention of this paper is to give insight into the methods nature and to clarify the strengths and limitations of the three methods. It turns out, that in the considered framework CMS delivers the best results together with a clear strategy for an “a priori” selection of the modes (trial vectors).

KEYWORDS

Model Reduction; Vibration Modes; Component Mode Synthesis; Moment Matching; Balanced Truncation

Model Reduction; Vibration Modes; Component Mode Synthesis; Moment Matching; Balanced Truncation

Cite this paper

W. Witteveen, "On the Modal and Non-Modal Model Reduction of Metallic Structures with Variable Boundary Conditions,"*World Journal of Mechanics*, Vol. 2 No. 6, 2012, pp. 311-324. doi: 10.4236/wjm.2012.26037.

W. Witteveen, "On the Modal and Non-Modal Model Reduction of Metallic Structures with Variable Boundary Conditions,"

References

[1] A. K. Noor, “Recent Advances and Applications of Reduction Methods,” Applied Mechanics Reviews, Vol. 47, No. 5, 1994, pp. 125-146. doi:10.1115/1.3111075

[2] L. Meirovitch, “Computational Methods in Structural Dynamics,” Springer, Berlin, 1980.

[3] Q. Q. Zu, “Model Order Reduction Techniques,” Springer Verlag, London, 2004.

[4] R. J. Craig, “Coupling of Substructures for Dynamic Analyses—An overview,” The American Institute of Aeronautics and Astronautics, Vol. 6, No. 7, 2000, pp. 13131319.

[5] R. J. Craig, “A Review of Time-Domain and Frequency-Domain Component Mode Synthesis Methods,” International Journal of Analytical and Experimental Modal Analysis, Vol. 2, No. 2, 1987, pp. 59-72.

[6] D. de Clerk, “General Framework for Dynamic Substructuring: History, Review and Classification of Techniques,” The American Institute of Aeronautics and Astronautics, Vol. 46, No. 5, 2008, pp. 1169-1181. doi:10.2514/1.33274

[7] M. Lehner and P. Eberhard, “On the Use of Moment-Matching to Build Reduced Order Models in Flexible Multibody Dynamics,” Multibody System Dynamics, Vol. 16, No. 2, 2006, pp. 191-211. doi:10.1007/s11044-006-9018-2

[8] R. R. Craig Jr. and T.-J. Su, “Krylov Model Reduction Algorithm for Undamped Structural Dynamics Systems,” Journal of Guidance, Control, and Dynamics, Vol. 14, No. 6, 1991, pp. 1311-1313.

[9] M. Lehner, “Modellreduktion in Elastischen Systemen,” Doctoral Thesis, Shaker Verlag, Aachen, 2007.

[10] P. Benner, “Numerical Linear Algebra for Model Reduction in Control and Simulation,” 2012. http://www.tu-chemnitz.de/mathematik/preprint/2005/PREPRINT_18.pdf

[11] B. Lohmann and B. Salimbarhrami, “Ordnungsreduktion Mittels Krylov-Unterraummethoden,” Automatisierungstechnik, Vol. 52, No. 1, 2004, pp. 30-38. doi:10.1524/auto.52.1.30.25436

[12] D. G. Meyer and S. Srinivasan, “Balancing and Model Reduction for Second-Order Form Linear Systems,” IEEE Transactions on Automatic Control, Vol. 41, No. 11, 1996, pp. 1632-1644. doi:10.1109/9.544000

[13] T. Reis and T. Stykel, “Balanced Truncation Model Reduction of Second-Order Systems,” Mathematical and Computer Modelling of Dynamical Systems, Vol. 14, No. 5, 2008, pp. 391-406. doi:10.1080/13873950701844170

[14] P. Benner and J. Saak, “Efficient Balancing Based MOR for Large Scale Second Order Systems, Mathematical and Computer,” Modeling of Dynamical Systems, Vol. 17, No. 2, 2011, pp. 123-143. doi:10.1080/13873954.2010.540822

[15] B. Yan, S. X. D. Tan and B. McGaugy, “Second-Order Balanced Truncation for Passive-Order Reduction of RLCK Circuits,” IEEE Transactions on Circuits and Systems-II, Vol. 55, No. 9, 2008, pp. 942-946.

[16] J. Fehr and P. Eberhard, “Error-Controlled Model Reduction in Flexible Multibody Dynamics,” Journal of Computational and Nonlinear Dynamics, Vol. 5, No. 3, 2010, p. 31005.

[17] J. Fehr and P. Eberhard, “Simulation Process of Flexible Multibody Systems with Non-Modal Model Order Reduction Techniques,” Multibody System Dynamics, Vol. 25, No. 3, 2011, pp. 313-334. doi:10.1007/s11044-010-9238-3

[18] C. Nowakowski, J. Fehr and P. Eberhard, “Model Reduction for a Crankshaft Used in Coupled Simulations of Engines,” In: J. C. Samin and P. Fisette, Eds., ECCOMAS Thematic Conference on Multibody Dynamics, Brussels, 4-7 July 2011, Article ID: 2374-1604.

[19] T. Bonin, M. Z?h, A. Soppa, H. Fassbender, J. Saak and P. Benner, “Modale versus Moderne Ordnungsreduktionsverfahren,” Carl Hanser Verlag, München, 2009.

[20] K. Willox and J. Peraire, “Balanced Model Reduction via the Proper Orthogonal Decomposition,” AIAA Journal, Vol. 40, No. 11, 2002, pp. 2323-2330. doi:10.2514/2.1570

[21] H. Fa?bender and P. Benner, “Numerische Methoden zur Passivit?terhaltenden Modellreduktion,” Automatisierungstechnik, Vol. 54, No. 4, 2006, pp. 153-160. doi:10.1524/auto.2006.54.4.153

[22] C. Tobias and P. Eberhard, “Stress Recovery with Krylov-Subspaces in Reduced Elastic Multibody Systems,” Multibody System Dynamics, Vol. 25, No. 4, 2011, pp. 377-393. doi:10.1007/s11044-010-9239-2

[23] P. Fischer, W. Witteveen and M. Schabasser, “Integrated MBS-FE-Durability Analysis of Truck Frame Components by Modal Stresses,” Proceedings of the 15th European ADAMS Users’ Conference, Rome, 2000. http://www.mscsoftware.com/support/library/conf/adams/euro/2000/Steyr_Durability_Truck_Frame.pdf

[24] R. Schwertassek, S. V. Dombrowski and O. Wallrapp, “Modal Representation of Stress in Flexible Multibody Simulation,” Nonlinear Dynamics, Vol. 20, No. 4, 1999, pp. 381-399. doi:10.1023/A:1008322210909

[25] P. Koutsovasilis and M. Beitelschmidt, “Comparision of Model Reduction Techniques for Large Mechanical Systems,” Multibody System Dynamics, Vol. 20, No. 2, 2008, pp. 111-128. doi:10.1007/s11044-008-9116-4

[26] W. Witteveen, “Comparision of CMS, Krylov and Balanced Truncation Based Model Reduction from a Mechanical Application Engineer’s Perspective,” Conference Proceedings of Society for Experimental Mechanics series, Vol. 27, 2012, pp. 319-331.

[27] L. Gaul and A. Schmidt, “Experimental Determination and Modeling of Material Damping,” VDI-Berichte Nr. 2003, 2007, pp. 17-40. http://www.iam.uni-stuttgart.de/Mitarbeiter/Schmidt/papers/vdi-paper.pdf

[28] D. Ewings, “Modal Testing Theory, Practice and Application,” 2nd Edition, Research Studies Press, Baldock, 2000.

[29] S. Bogard, A. Schmidth and L. GauL, “Modeling of Damping in Bolted Structures,” VDI-Berichte Nr. 2003, 2007, pp. 97-110. http://www.iam.uni-stuttgart.de/Mitarbeiter/Schmidt/papers/vdi-paper_sergey.pdf

[30] L. Gaul and R. Nitsche, “The Role of Friction in Mechanical Joints,” Applied Mechanics Reviews, Vol. 54, No. 2, 2001, pp. 93-105. doi:10.1115/1.3097294

[31] R. R. Craig and M. C. C. Bampton, “Coupling of SubStructures for Dynamic Analysis”, AIAA Journal, Vol. 6, No. 7, 1968, pp. 1313-1319. doi:10.2514/3.4741

[32] R. J. Guyan, “Reduction of Stiffness and Mass Matrix,” AIAA Journal, Vol. 3, No. 2, 1965, p. 380.

[33] D. J. Rixon, “Dual Craig-Bampton with Enrichment to Avoid Spurious Modes,” Proceedings of the IMAC-XXVII, Orlando, 9-12 February 2009, Article ID: 170.

[34] B. H?gblad and L. Eriksson, “Model Reduction Methods for Dynamic Analysis of Large Structures,” Computers & Structures, Vol. 47, No. 4-5, 1993, pp. 735-749.

[35] M. Lehner and P. Eberhard, “A Two Step Approach for Modal Reduction in Flexible Multibody Dynamics,” Multibody System Dynamics, Vol. 17, No. 2-3, 2007, pp. 157176. doi:10.1007/s11044-007-9039-5

[36] P. Benner and A. Schneider, “Balanced Truncation Model Order Reduction for LTI Systems with Many Inputs or Outputs,” Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, Budapest, 5-9 July 2010, pp. 1971-1974.

[37] G. Kerschen, J. C. Golinval, A. F. Vakakis and L. A. Bergman, “The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview,” Nonlinear Dynamics, Vol. 41, No. 1-3, 2005, pp. 147-169. doi:10.1007/s11071-005-2803-2

[38] A. Chatterjee, “An Introduction to the Proper Orthogonal Decomposition,” Current Science, Vol. 78, No. 7, 2000, pp. 808-817.

[39] S. Volkwein, “Model Reduction Using Proper Orthogonal Decomposition,” 2012. http://www.uni-graz.at/imawww/volkwein/POD.pdf

[40] B. F. Feeny and R. Kappagantu, “On the Physical Interpretation of Proper Orthogonal Modes in Vibration,” Journal of Sound and Vibration, Vol. 211, No. 4, 1998, pp. 607-616. doi:10.1006/jsvi.1997.1386

[41] F. Ma and C. H. Ng, “On the Orthogonality of Natural Modes of Vibrations,” Mechanics Research Communications, Vol. 31, No. 3, 2004, pp. 295-299. doi:10.1016/j.mechrescom.2003.03.001

[42] T. Kim, “Frequency-Domain Karhunen-Loeve Method and Its Application to Linear Dynamic Systems,” AIAA Journal, Vol. 36, No. 11, 1998, pp. 2117-2123. doi:10.2514/2.315

[43] D. M. Tran, “Component Mode Synthesis Methods Using Interface Modes. Application to Structures with Cyclic Symmetry,” Computers and Structures, Vol. 79, No. 2, 2001, pp. 209-222. doi:10.1016/S0045-7949(00)00121-8

[44] E. Balmes, “Use of Generalized Interface Degrees of Freedom in Component Mode Synthesis,” Proceedings of IMAC XIV Conference, Milwaukee, 12-15 February 1996, pp. 204-210.

[45] W. Witteveen and H. Irschik, “Efficient Mode-Based Computational Approach for Jointed Structures: Joint Interface Modes,” AIAA Journal, Vol. 47, No. 1, 2009, pp. 252-263. doi:10.2514/1.38436

[46] MD R3 Nastran, 2008. www.mscsoftware.com

[47] Scilab, Version 5.3.0. www.scilab.org

[48] R. J. Allemang and D. L. Brown, “A Correlation Coefficient for Modal Vector Analysis,” Proceedings of the 1st International Modal Analysis Conference (IMAC I), Orlando, 8-10 November 1982, pp. 110-116.

[1] A. K. Noor, “Recent Advances and Applications of Reduction Methods,” Applied Mechanics Reviews, Vol. 47, No. 5, 1994, pp. 125-146. doi:10.1115/1.3111075

[2] L. Meirovitch, “Computational Methods in Structural Dynamics,” Springer, Berlin, 1980.

[3] Q. Q. Zu, “Model Order Reduction Techniques,” Springer Verlag, London, 2004.

[4] R. J. Craig, “Coupling of Substructures for Dynamic Analyses—An overview,” The American Institute of Aeronautics and Astronautics, Vol. 6, No. 7, 2000, pp. 13131319.

[5] R. J. Craig, “A Review of Time-Domain and Frequency-Domain Component Mode Synthesis Methods,” International Journal of Analytical and Experimental Modal Analysis, Vol. 2, No. 2, 1987, pp. 59-72.

[6] D. de Clerk, “General Framework for Dynamic Substructuring: History, Review and Classification of Techniques,” The American Institute of Aeronautics and Astronautics, Vol. 46, No. 5, 2008, pp. 1169-1181. doi:10.2514/1.33274

[7] M. Lehner and P. Eberhard, “On the Use of Moment-Matching to Build Reduced Order Models in Flexible Multibody Dynamics,” Multibody System Dynamics, Vol. 16, No. 2, 2006, pp. 191-211. doi:10.1007/s11044-006-9018-2

[8] R. R. Craig Jr. and T.-J. Su, “Krylov Model Reduction Algorithm for Undamped Structural Dynamics Systems,” Journal of Guidance, Control, and Dynamics, Vol. 14, No. 6, 1991, pp. 1311-1313.

[9] M. Lehner, “Modellreduktion in Elastischen Systemen,” Doctoral Thesis, Shaker Verlag, Aachen, 2007.

[10] P. Benner, “Numerical Linear Algebra for Model Reduction in Control and Simulation,” 2012. http://www.tu-chemnitz.de/mathematik/preprint/2005/PREPRINT_18.pdf

[11] B. Lohmann and B. Salimbarhrami, “Ordnungsreduktion Mittels Krylov-Unterraummethoden,” Automatisierungstechnik, Vol. 52, No. 1, 2004, pp. 30-38. doi:10.1524/auto.52.1.30.25436

[12] D. G. Meyer and S. Srinivasan, “Balancing and Model Reduction for Second-Order Form Linear Systems,” IEEE Transactions on Automatic Control, Vol. 41, No. 11, 1996, pp. 1632-1644. doi:10.1109/9.544000

[13] T. Reis and T. Stykel, “Balanced Truncation Model Reduction of Second-Order Systems,” Mathematical and Computer Modelling of Dynamical Systems, Vol. 14, No. 5, 2008, pp. 391-406. doi:10.1080/13873950701844170

[14] P. Benner and J. Saak, “Efficient Balancing Based MOR for Large Scale Second Order Systems, Mathematical and Computer,” Modeling of Dynamical Systems, Vol. 17, No. 2, 2011, pp. 123-143. doi:10.1080/13873954.2010.540822

[15] B. Yan, S. X. D. Tan and B. McGaugy, “Second-Order Balanced Truncation for Passive-Order Reduction of RLCK Circuits,” IEEE Transactions on Circuits and Systems-II, Vol. 55, No. 9, 2008, pp. 942-946.

[16] J. Fehr and P. Eberhard, “Error-Controlled Model Reduction in Flexible Multibody Dynamics,” Journal of Computational and Nonlinear Dynamics, Vol. 5, No. 3, 2010, p. 31005.

[17] J. Fehr and P. Eberhard, “Simulation Process of Flexible Multibody Systems with Non-Modal Model Order Reduction Techniques,” Multibody System Dynamics, Vol. 25, No. 3, 2011, pp. 313-334. doi:10.1007/s11044-010-9238-3

[18] C. Nowakowski, J. Fehr and P. Eberhard, “Model Reduction for a Crankshaft Used in Coupled Simulations of Engines,” In: J. C. Samin and P. Fisette, Eds., ECCOMAS Thematic Conference on Multibody Dynamics, Brussels, 4-7 July 2011, Article ID: 2374-1604.

[19] T. Bonin, M. Z?h, A. Soppa, H. Fassbender, J. Saak and P. Benner, “Modale versus Moderne Ordnungsreduktionsverfahren,” Carl Hanser Verlag, München, 2009.

[20] K. Willox and J. Peraire, “Balanced Model Reduction via the Proper Orthogonal Decomposition,” AIAA Journal, Vol. 40, No. 11, 2002, pp. 2323-2330. doi:10.2514/2.1570

[21] H. Fa?bender and P. Benner, “Numerische Methoden zur Passivit?terhaltenden Modellreduktion,” Automatisierungstechnik, Vol. 54, No. 4, 2006, pp. 153-160. doi:10.1524/auto.2006.54.4.153

[22] C. Tobias and P. Eberhard, “Stress Recovery with Krylov-Subspaces in Reduced Elastic Multibody Systems,” Multibody System Dynamics, Vol. 25, No. 4, 2011, pp. 377-393. doi:10.1007/s11044-010-9239-2

[23] P. Fischer, W. Witteveen and M. Schabasser, “Integrated MBS-FE-Durability Analysis of Truck Frame Components by Modal Stresses,” Proceedings of the 15th European ADAMS Users’ Conference, Rome, 2000. http://www.mscsoftware.com/support/library/conf/adams/euro/2000/Steyr_Durability_Truck_Frame.pdf

[24] R. Schwertassek, S. V. Dombrowski and O. Wallrapp, “Modal Representation of Stress in Flexible Multibody Simulation,” Nonlinear Dynamics, Vol. 20, No. 4, 1999, pp. 381-399. doi:10.1023/A:1008322210909

[25] P. Koutsovasilis and M. Beitelschmidt, “Comparision of Model Reduction Techniques for Large Mechanical Systems,” Multibody System Dynamics, Vol. 20, No. 2, 2008, pp. 111-128. doi:10.1007/s11044-008-9116-4

[26] W. Witteveen, “Comparision of CMS, Krylov and Balanced Truncation Based Model Reduction from a Mechanical Application Engineer’s Perspective,” Conference Proceedings of Society for Experimental Mechanics series, Vol. 27, 2012, pp. 319-331.

[27] L. Gaul and A. Schmidt, “Experimental Determination and Modeling of Material Damping,” VDI-Berichte Nr. 2003, 2007, pp. 17-40. http://www.iam.uni-stuttgart.de/Mitarbeiter/Schmidt/papers/vdi-paper.pdf

[28] D. Ewings, “Modal Testing Theory, Practice and Application,” 2nd Edition, Research Studies Press, Baldock, 2000.

[29] S. Bogard, A. Schmidth and L. GauL, “Modeling of Damping in Bolted Structures,” VDI-Berichte Nr. 2003, 2007, pp. 97-110. http://www.iam.uni-stuttgart.de/Mitarbeiter/Schmidt/papers/vdi-paper_sergey.pdf

[30] L. Gaul and R. Nitsche, “The Role of Friction in Mechanical Joints,” Applied Mechanics Reviews, Vol. 54, No. 2, 2001, pp. 93-105. doi:10.1115/1.3097294

[31] R. R. Craig and M. C. C. Bampton, “Coupling of SubStructures for Dynamic Analysis”, AIAA Journal, Vol. 6, No. 7, 1968, pp. 1313-1319. doi:10.2514/3.4741

[32] R. J. Guyan, “Reduction of Stiffness and Mass Matrix,” AIAA Journal, Vol. 3, No. 2, 1965, p. 380.

[33] D. J. Rixon, “Dual Craig-Bampton with Enrichment to Avoid Spurious Modes,” Proceedings of the IMAC-XXVII, Orlando, 9-12 February 2009, Article ID: 170.

[34] B. H?gblad and L. Eriksson, “Model Reduction Methods for Dynamic Analysis of Large Structures,” Computers & Structures, Vol. 47, No. 4-5, 1993, pp. 735-749.

[35] M. Lehner and P. Eberhard, “A Two Step Approach for Modal Reduction in Flexible Multibody Dynamics,” Multibody System Dynamics, Vol. 17, No. 2-3, 2007, pp. 157176. doi:10.1007/s11044-007-9039-5

[36] P. Benner and A. Schneider, “Balanced Truncation Model Order Reduction for LTI Systems with Many Inputs or Outputs,” Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems, Budapest, 5-9 July 2010, pp. 1971-1974.

[37] G. Kerschen, J. C. Golinval, A. F. Vakakis and L. A. Bergman, “The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview,” Nonlinear Dynamics, Vol. 41, No. 1-3, 2005, pp. 147-169. doi:10.1007/s11071-005-2803-2

[38] A. Chatterjee, “An Introduction to the Proper Orthogonal Decomposition,” Current Science, Vol. 78, No. 7, 2000, pp. 808-817.

[39] S. Volkwein, “Model Reduction Using Proper Orthogonal Decomposition,” 2012. http://www.uni-graz.at/imawww/volkwein/POD.pdf

[40] B. F. Feeny and R. Kappagantu, “On the Physical Interpretation of Proper Orthogonal Modes in Vibration,” Journal of Sound and Vibration, Vol. 211, No. 4, 1998, pp. 607-616. doi:10.1006/jsvi.1997.1386

[41] F. Ma and C. H. Ng, “On the Orthogonality of Natural Modes of Vibrations,” Mechanics Research Communications, Vol. 31, No. 3, 2004, pp. 295-299. doi:10.1016/j.mechrescom.2003.03.001

[42] T. Kim, “Frequency-Domain Karhunen-Loeve Method and Its Application to Linear Dynamic Systems,” AIAA Journal, Vol. 36, No. 11, 1998, pp. 2117-2123. doi:10.2514/2.315

[43] D. M. Tran, “Component Mode Synthesis Methods Using Interface Modes. Application to Structures with Cyclic Symmetry,” Computers and Structures, Vol. 79, No. 2, 2001, pp. 209-222. doi:10.1016/S0045-7949(00)00121-8

[44] E. Balmes, “Use of Generalized Interface Degrees of Freedom in Component Mode Synthesis,” Proceedings of IMAC XIV Conference, Milwaukee, 12-15 February 1996, pp. 204-210.

[45] W. Witteveen and H. Irschik, “Efficient Mode-Based Computational Approach for Jointed Structures: Joint Interface Modes,” AIAA Journal, Vol. 47, No. 1, 2009, pp. 252-263. doi:10.2514/1.38436

[46] MD R3 Nastran, 2008. www.mscsoftware.com

[47] Scilab, Version 5.3.0. www.scilab.org

[48] R. J. Allemang and D. L. Brown, “A Correlation Coefficient for Modal Vector Analysis,” Proceedings of the 1st International Modal Analysis Conference (IMAC I), Orlando, 8-10 November 1982, pp. 110-116.