Model reduction methods applied to a nonlinear mechanical system
DOI:
https://doi.org/10.31686/ijier.vol7.iss7.1611Keywords:
model reduction methods, Nonlinear mechanical system, finite element method, stability analysis, computational costAbstract
Modern structures of high flexibility are subject to physical or geometric nonlinearities, and reliable numerical modeling to predict their behavior is essential. The modeling of these systems can be given by the discretization of the problem using the Finite Element Method (FEM), however by using this methodology, it is a very robust model from the computational point of view, making the simulation process difficult. Using reduced models has been an excellent alternative to minimizing this problem. Most model reduction methods are restricted to linear problems, which
motivated us to maximize the efficiency of these methods considering nonlinear problems. For better accuracy, in this study, adaptations and improvements are suggested in reduction methods such as the Enriched Modal Base (EMB), the System Equivalent Reduction Expansion Process (SEREP), QUASI-SEREP and the Iterated Improved Reduced System (IIRS). The stability of a system is discussed according to the calculation of the Lyapunov exponents and phase space. Numerical simulations showed that the reduced models presented a good performance, according to the commitment of quality and speed of responses (or time saving).
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References
[1] Guedri M.; Bouhaddi N.; Majed R. Reduction of the Stochastic Finite Element
Models by using a Robust Dynamic Condensation Method. Journal of Sound and
Vibration, vol. 297, pp. 123–145, 2006.
[2] Gerges, Y. Méthods de Réduction de Modéles en Vibroacoustique Non-Linéaire,
PhD Thesis, University of Franche-Comté, Besançon, 2013.
[3] Borges, R. A.; Lima, A. M. G. ; Steffen Jr, V. Robust optimal design of a nonlinear
dynamic vibration absorber combining sensitivity analysis. Shock and Vibration, v.
17, p. 507-520, 2010.
[4] Lülf, F. A.; Tran, D. M.; Ohayon, R. Reduced bases for nonlinear structural
dynamic systems: A comparative study. Journal of Sound and Vibration, v. 332, n.
15, p. 3897– 3921, 2013.
[5] Lopes Jr. V, Steffen Jr. V., Savi M. A. Dynamics of Smarts Systems and structures:
Concepts and Applications, Springer, Switzerland, 2016.
[6] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A. Determining Lyapunov
exponents from a time series. Physica D: Nonlinear Phenomena, vol. 16, no. 3, pp.
285-317, 1985.
[7] Nayfeh, A.H. Problems in Perturbation, Wiley & Sons, New York, 1985.
[8] Paiva A., Savi M. A. An overview of constitute models for shape memory alloys.
Journal Mathematical Problems in Engineering, v. 2006, p. 1-30, 2006.
[9] Savi M. A., Pacheco P. M. C. L., Braga A. M. B. Chaos in a shape memory twobars
truss. International Journal of Non Linear Mechanics, v. 37, p. 1387-1395,
2002.
[10] Rabelo M. Silva, L., Borges, R.A., Gonçalves, R. Henrique, M., Computational
and Numerical Analysis of a Nonlinear Mechanical System with Bounded Delay,
International Journal of Nonlinear Mechanics, v. 91, 36-57, 2017.
[11] De Lima, A. M.G.; Bouhaddi, N.; Rade, D. A.; Belonsi, M. A Time-Domain
Finite Element Model Reduction Method for Viscoelastic Linear and Nonlinear
Systems. Latin American Journal of Solids and Structures, v. 12, no. 6, p. 1182-
1201, 2015.
[12] Moore, B. C. Principal component analysis in linear systems: Controllability,
observability, and model reduction. Automatic Control, IEEE Transactions on,
IEEE, v. 26, n. 1, p. 17-32, 1981.
[13] Qu, Z. Q. Model order reduction techniques with applications in finite element
analysis. [S.l.]: Springer Science & Business Media, 2004.
[14] Koutsovasilis, P.; Beitelschmidt, M. Model reduction comparison for the elastic
crankshaft mechanism. In: Proc. 2. Internacional Operational Modal Analysis
Conference - IOMAC, vol.1, pp. 95–106. Copenhagen, 2008.
[15] Gonçalves, D. F., Fonseca Jr., L. A., Rezende S. M. F., Borges, R. A., De Lima,
A. M. G. Reduction model methods applied of mechanical nonlinear system.
XXXVI Ibero-Latin American Congress in Computational Methods in Engineering,
2015.
[16] Guyan, R.J. Reduction of stiffness and mass matrices. AIAA Journal, v. 3, no. 2,
p. 380, 1965.
[17] Irons, B. Structural eigenvalues problems: elimination of unwanted variables.
AIAA Journal, v. 3, p. 961–962, 1965.
[18] Kim, J. G., Lee, P. S. An accurate error estimator for Guyan reduction. Computer
Methods Applied Mechanics and Engineering, v. 278, p. 1-19, 2014.
[19] O’Callahan, J.C. A procedure for an improved reduced system (IRS) model. In:
Proceedings of the Seventh International Modal Analysis Conference, Las Vegas,
pp. 17-21, 1989.
[20] Gordis, J.H. An analysis of the improved reduced system (IRS) model reduction
procedure. Modal Analysis: The International Journal of Analytical and
Experimental Modal Analysis, v. 9, n. 4, p. 269-285, 1994.
[21] Friswell, M.I.; Garvey, S.D.; Penny, J.E.T. Model reduction using dynamic and
iterated IRS techniques. Journal of Sound and Vibration, v. 186, n. 2, p. 311-323,
1995.
[22] Friswell, M.I.; Garvey, S.D.; Penny, J.E.T. The convergence of the iterated IRS
method. Journal of Sound and Vibration, v. 211, n. 1, p. 123-132, 1998.
[23] Xia, Y.; Hao, H. Improvement on the iterated IRS method for structural
eigensolutions. Journal of Sound and Vibration, v. 270, p. 713-727, 2004.
[24] Cho, M., Kim, H. Element-based node selection method for reduction of
eigenvalue problems. AIAA J, v. 42, n. 8, p. 1677-1684, 2004.
[25] Kim, H., Cho, M. Two-level scheme for selection of primary degrees of freedom
and semi-analytic sensitivity based on the reduced system. Computer Methods
Applied Mechanics and Engineering, v. 195, p. 4244-4268, 2006.
[26] Choi, D., Kim H., Cho, M. Improvement of substructuring reduction technique for
large eigenproblems using an efficient dynamic condensation method. Journal of
Mechanical Science and Technology, v. 22, p. 255-268, 2008.
[27] Hosseinzadeh, A. Z., Bagheri, A., Amiri, G. G., Koo, K. Y. A flexibility-based
method via the iterated improved reduction system and the cuckoo optimization
algorithm for damage quantification with limited sensors. Smart Materials and
Structures, v. 23, 2014.
[28] Boo, S. H., Lee, P. S. A dynamic condensation method using algebraic
substructuring. International Journal for Numerical Methods in Engineering, v. 109, n.
12, p. 1701-1720, 2017.
[29] O’Callahan, J.; Avitabile, P.; Riemer, R. System equivalent reduction expansion
process (SEREP). In: Proceedings of the Seventh International Modal Analysis
Conference, Las Vegas, p. 29-37, 1989.
[30] Kammer, D.C. Test-analysis-model development using exact model reduction.
The International Journal of Analytical and Experimental Modal Analysis, v. 2, n. 4,
p. 174-179, 1987.
[31] Friswell, M.I.; Inman, D.J. Reduced-order models of structures with viscoelastic
components. AIAA Journal, v. 37, n. 10, p. 1318-1325, 1999.
[32] Friswell, M.I., Penny, J.E.T., Garvey, S.D. Model reduction for structures with
damping and gyroscopic effects. In: Proceedings of ISMA-25 Leuven, Belgium, p.
1151-1158, 2000.
[33] Das, A. S.; Dutt, J. K. Reduced model of a rotor-shaft system using modified
SEREP. Mechanics Research Communications, v. 35, n. 6, p. 398-407, 2008.
[34] Burton, T. D.; Hemez, F. M.; Rhee, W. A Combined Model Reduction/SVD
Approach to Nonlinear Model Updating. IMAC-XVIII: A Conference on Structural
Dynamics, San Antonio, Texas, 2000.
[35] Burton, T. D.; Rhee, W. On the Reduction of Nonlinear Structural Dynamics
Models. Journal of Vibration and Control, v. 6, p. 531-556, 2000.
[36] Kim, J.; Burton, T. D. Reduction of Nonlinear Structural Models Having Non-
Smooth Nonlinearities. In Proc. IMAC XX, Los Angeles, CA, p. 324-330, 2002.
[37] Cho, M. Study on the system reduction under the condition of dynamic load.
Journal of Mechanical Science and Technology, n. 27, v. 1, p. 113-124, 2013.
[38] Masson, G.; Ait Brik, B.; Cogan, S.; Bouhaddi, N. Component Mode Synthesis
(CMS) based on an enriched Ritz approach for efficient structural optimization.
Journal of Sound and Vibration, v. 296, p. 845-860, 2006.
[39] Zienkiewicz, O.C.; Taylor, R. L; Zhu, J.Z. The Finite Element Method: Its Basis
and Fundamentals, Elsevier, Barcelona, 6ª edition, 2005.
Models by using a Robust Dynamic Condensation Method. Journal of Sound and
Vibration, vol. 297, pp. 123–145, 2006.
[2] Gerges, Y. Méthods de Réduction de Modéles en Vibroacoustique Non-Linéaire,
PhD Thesis, University of Franche-Comté, Besançon, 2013.
[3] Borges, R. A.; Lima, A. M. G. ; Steffen Jr, V. Robust optimal design of a nonlinear
dynamic vibration absorber combining sensitivity analysis. Shock and Vibration, v.
17, p. 507-520, 2010.
[4] Lülf, F. A.; Tran, D. M.; Ohayon, R. Reduced bases for nonlinear structural
dynamic systems: A comparative study. Journal of Sound and Vibration, v. 332, n.
15, p. 3897– 3921, 2013.
[5] Lopes Jr. V, Steffen Jr. V., Savi M. A. Dynamics of Smarts Systems and structures:
Concepts and Applications, Springer, Switzerland, 2016.
[6] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A. Determining Lyapunov
exponents from a time series. Physica D: Nonlinear Phenomena, vol. 16, no. 3, pp.
285-317, 1985.
[7] Nayfeh, A.H. Problems in Perturbation, Wiley & Sons, New York, 1985.
[8] Paiva A., Savi M. A. An overview of constitute models for shape memory alloys.
Journal Mathematical Problems in Engineering, v. 2006, p. 1-30, 2006.
[9] Savi M. A., Pacheco P. M. C. L., Braga A. M. B. Chaos in a shape memory twobars
truss. International Journal of Non Linear Mechanics, v. 37, p. 1387-1395,
2002.
[10] Rabelo M. Silva, L., Borges, R.A., Gonçalves, R. Henrique, M., Computational
and Numerical Analysis of a Nonlinear Mechanical System with Bounded Delay,
International Journal of Nonlinear Mechanics, v. 91, 36-57, 2017.
[11] De Lima, A. M.G.; Bouhaddi, N.; Rade, D. A.; Belonsi, M. A Time-Domain
Finite Element Model Reduction Method for Viscoelastic Linear and Nonlinear
Systems. Latin American Journal of Solids and Structures, v. 12, no. 6, p. 1182-
1201, 2015.
[12] Moore, B. C. Principal component analysis in linear systems: Controllability,
observability, and model reduction. Automatic Control, IEEE Transactions on,
IEEE, v. 26, n. 1, p. 17-32, 1981.
[13] Qu, Z. Q. Model order reduction techniques with applications in finite element
analysis. [S.l.]: Springer Science & Business Media, 2004.
[14] Koutsovasilis, P.; Beitelschmidt, M. Model reduction comparison for the elastic
crankshaft mechanism. In: Proc. 2. Internacional Operational Modal Analysis
Conference - IOMAC, vol.1, pp. 95–106. Copenhagen, 2008.
[15] Gonçalves, D. F., Fonseca Jr., L. A., Rezende S. M. F., Borges, R. A., De Lima,
A. M. G. Reduction model methods applied of mechanical nonlinear system.
XXXVI Ibero-Latin American Congress in Computational Methods in Engineering,
2015.
[16] Guyan, R.J. Reduction of stiffness and mass matrices. AIAA Journal, v. 3, no. 2,
p. 380, 1965.
[17] Irons, B. Structural eigenvalues problems: elimination of unwanted variables.
AIAA Journal, v. 3, p. 961–962, 1965.
[18] Kim, J. G., Lee, P. S. An accurate error estimator for Guyan reduction. Computer
Methods Applied Mechanics and Engineering, v. 278, p. 1-19, 2014.
[19] O’Callahan, J.C. A procedure for an improved reduced system (IRS) model. In:
Proceedings of the Seventh International Modal Analysis Conference, Las Vegas,
pp. 17-21, 1989.
[20] Gordis, J.H. An analysis of the improved reduced system (IRS) model reduction
procedure. Modal Analysis: The International Journal of Analytical and
Experimental Modal Analysis, v. 9, n. 4, p. 269-285, 1994.
[21] Friswell, M.I.; Garvey, S.D.; Penny, J.E.T. Model reduction using dynamic and
iterated IRS techniques. Journal of Sound and Vibration, v. 186, n. 2, p. 311-323,
1995.
[22] Friswell, M.I.; Garvey, S.D.; Penny, J.E.T. The convergence of the iterated IRS
method. Journal of Sound and Vibration, v. 211, n. 1, p. 123-132, 1998.
[23] Xia, Y.; Hao, H. Improvement on the iterated IRS method for structural
eigensolutions. Journal of Sound and Vibration, v. 270, p. 713-727, 2004.
[24] Cho, M., Kim, H. Element-based node selection method for reduction of
eigenvalue problems. AIAA J, v. 42, n. 8, p. 1677-1684, 2004.
[25] Kim, H., Cho, M. Two-level scheme for selection of primary degrees of freedom
and semi-analytic sensitivity based on the reduced system. Computer Methods
Applied Mechanics and Engineering, v. 195, p. 4244-4268, 2006.
[26] Choi, D., Kim H., Cho, M. Improvement of substructuring reduction technique for
large eigenproblems using an efficient dynamic condensation method. Journal of
Mechanical Science and Technology, v. 22, p. 255-268, 2008.
[27] Hosseinzadeh, A. Z., Bagheri, A., Amiri, G. G., Koo, K. Y. A flexibility-based
method via the iterated improved reduction system and the cuckoo optimization
algorithm for damage quantification with limited sensors. Smart Materials and
Structures, v. 23, 2014.
[28] Boo, S. H., Lee, P. S. A dynamic condensation method using algebraic
substructuring. International Journal for Numerical Methods in Engineering, v. 109, n.
12, p. 1701-1720, 2017.
[29] O’Callahan, J.; Avitabile, P.; Riemer, R. System equivalent reduction expansion
process (SEREP). In: Proceedings of the Seventh International Modal Analysis
Conference, Las Vegas, p. 29-37, 1989.
[30] Kammer, D.C. Test-analysis-model development using exact model reduction.
The International Journal of Analytical and Experimental Modal Analysis, v. 2, n. 4,
p. 174-179, 1987.
[31] Friswell, M.I.; Inman, D.J. Reduced-order models of structures with viscoelastic
components. AIAA Journal, v. 37, n. 10, p. 1318-1325, 1999.
[32] Friswell, M.I., Penny, J.E.T., Garvey, S.D. Model reduction for structures with
damping and gyroscopic effects. In: Proceedings of ISMA-25 Leuven, Belgium, p.
1151-1158, 2000.
[33] Das, A. S.; Dutt, J. K. Reduced model of a rotor-shaft system using modified
SEREP. Mechanics Research Communications, v. 35, n. 6, p. 398-407, 2008.
[34] Burton, T. D.; Hemez, F. M.; Rhee, W. A Combined Model Reduction/SVD
Approach to Nonlinear Model Updating. IMAC-XVIII: A Conference on Structural
Dynamics, San Antonio, Texas, 2000.
[35] Burton, T. D.; Rhee, W. On the Reduction of Nonlinear Structural Dynamics
Models. Journal of Vibration and Control, v. 6, p. 531-556, 2000.
[36] Kim, J.; Burton, T. D. Reduction of Nonlinear Structural Models Having Non-
Smooth Nonlinearities. In Proc. IMAC XX, Los Angeles, CA, p. 324-330, 2002.
[37] Cho, M. Study on the system reduction under the condition of dynamic load.
Journal of Mechanical Science and Technology, n. 27, v. 1, p. 113-124, 2013.
[38] Masson, G.; Ait Brik, B.; Cogan, S.; Bouhaddi, N. Component Mode Synthesis
(CMS) based on an enriched Ritz approach for efficient structural optimization.
Journal of Sound and Vibration, v. 296, p. 845-860, 2006.
[39] Zienkiewicz, O.C.; Taylor, R. L; Zhu, J.Z. The Finite Element Method: Its Basis
and Fundamentals, Elsevier, Barcelona, 6ª edition, 2005.
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How to Cite
Antonio Borges, R., Gonçalves, D. ., De lima, A. M., & Fonseca Júnior, L. (2019). Model reduction methods applied to a nonlinear mechanical system. International Journal for Innovation Education and Research, 7(7), 281-300. https://doi.org/10.31686/ijier.vol7.iss7.1611
