Robust Linearization and Eigenvalue Analysis of General Complex Constrained Multibody Systems

Abstract

The derivation of linearized equations and subsequent eigenvalue analysis is the basis for tasks such as frequency-domain response analysis, control design, and stability assessment for mechanical systems. However, for general multibody systems with redundant or nonholonomic constraints, practical challenges persist in achieving numerically stable linearization and reliable eigenvalue analysis. This study provides numerically reliable linearization and eigenvalue analysis algorithms of Lagrange's equations of the first kind for general multibody systems, and forms a systematic computational framework. First, a robust linearization methodology is developed to automatically generate linearized state-space equations. Coordinate partitioning is employed not only to eliminate redundant constraints and dependent coordinates, but also to significantly simplify the calculations. Second, a direct eigenvalue analysis on Redundant Coordinate Set Jacobians is performed instead of decomposing the state matrix, for its better numerical conditioning and sparsity. The equivalence of the eigenvalue problems is strictly proven theoretically. The methodology is validated by various benchmark problems and practical cases. The proposed approach has been implemented and released with a general-purpose rigid-flexible coupled multibody dynamics software INTESIM-FMBD (v7.0), and provides a reliable numerical tool for linearizing general mechanical systems.

Keywords

constrained multibody system / eigenvalue analysis / linearization

References

[1] V. D. Dwivedi and P. Wahi, “Influence of Gravity on the Vibration Characteristics of a Geometrically Nonlinear McPherson-Suspension Model,” Multibody System Dynamics 61, no. 4 (2024): 571–596, https://doi.org/10.1007/s11044-023-09946-0.

[2] G. Cassoni, A. Cocco, A. Tamer, A. Zanoni, and P. Masarati, “Rotorcraft Stability Analysis Using Lyapunov Characteristic Exponents Estimated From Multibody Dynamics,” CEAS Aeronautical Journal 15, no. 3 (2024): 703–719, https://doi.org/10.1007/s13272-024-00724-y.

[3] M. Hu, S. Cao, X. Cao, et al., “Dynamics and Stability Analysis of Unbalance Responses in Mid-Positioned Electrically Assisted Turbocharger Rotor,” International Journal of Mechanical System Dynamics 4, no. 4 (2024): 515–535, https://doi.org/10.1002/msd2.12129.

[4] L. Pyrhönen, S. Jaiswal, A. Garcia-Agundez, D. García Vallejo, and A. Mikkola, “Linearization-Based State-Transition Model for the Discrete Extended Kalman Filter Applied to Multibody Simulations,” Multibody System Dynamics 57, no. 1 (2023): 55–72, https://doi.org/10.1007/s11044-022-09861-w.

[5] A. Köthe and R. Luckner, “Applying Eigenstructure Assignment to Inner-Loop Flight Control Laws for a Multibody Aircraft,” CEAS Aeronautical Journal 13, no. 1 (2022): 33–43, https://doi.org/10.1007/s13272-021-00549-z.

[6] E. Kassarian, F. Sanfedino, D. Alazard, C. A. Chevrier, and J. Montel, “Linear Fractional Transformation Modeling of Multibody Dynamics Around Parameter-Dependent Equilibrium,” IEEE Transactions on Control Systems Technology 31, no. 1 (2023): 418–425, https://doi.org/10.1109/TCST.2022.3167610.

[7] P. Masarati and A. Zanoni, “Eigensolution Continuation by Projection Sensitivity of Minimal Coordinate Set Multibody Systems,” Multibody System Dynamics 66 (2025): 403–433, https://doi.org/10.1007/s11044-025-10072-2.

[8] M. Bos, S. Traversaro, D. Pucci, et al., “Efficient Geometric Linearization of Moving-Base Rigid Robot Dynamics,” Journal of Geometric Mechanics 14, no. 4 (2022): 507–543, https://doi.org/10.3934/jgm.2022009.

[9] J. L. Archut and B. Corves, “Real-Time Multibody Simulation of Vehicle Wheel Suspensions of Different Topologies With Elastokinematic Properties,” Multibody System Dynamics 63, no. 1 (2025): 255–271, https://doi.org/10.1007/s11044-024-09984-2.

[10] J. L. Archut, R. Cleven, M. Wahle, M. Hüsing, and B. Corves, “A Computationally Efficient Multibody Simulation Model of a Suspension System Including Elastokinematic Properties,” in Advances in Dynamics of Vehicles on Roads and Tracks II, eds. A. Orlova and D. Cole (Springer International Publishing, 2022), 606–616.

[11] L. Zhang, X. Rui, J. Zhang, J. Gu, and X. Zhang, “Study on the Direct Differential Method in Constraint Violations for the Reduced Multibody System Transfer Matrix Method,” in Proceedings of the 2nd International Conference on Mechanical System Dynamics, eds. X. Rui and C. Liu (Springer Nature, 2024), 2521–2534.

[12] G. Urkullu, I. F. De Bustos, V. García-Marina, and H. Uriarte, “Direct Integration of the Equations of Multibody Dynamics Using Central Differences and Linearization,” Mechanism and Machine Theory 133 (2019): 432–458, https://doi.org/10.1016/j.mechmachtheory.2018.11.024.

[13] F. González, P. Masarati, J. Cuadrado, and M. A. Naya, “Assessment of Linearization Approaches for Multibody Dynamics Formulations,” Journal of Computational and Nonlinear Dynamics 12, no. 4 (2017): 041009, https://doi.org/10.1115/1.4035410.

[14] F. González, P. Masarati, and J. Cuadrado, “On the Linearization of Multibody Dynamics Formulations,” in Volume 6: 12th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (American Society of Mechanical Engineers, 2016), V006T09A008, https://doi.org/10.1115/DETC2016-59227.

[15] L. Mariti, N. P. Belfiore, E. Pennestrì, and P. P. Valentini, “Comparison of Solution Strategies for Multibody Dynamics Equations,” International Journal for Numerical Methods in Engineering 88, no. 7 (2011): 637–656, https://doi.org/10.1002/nme.3190.

[16] A. Agúndez, D. García-Vallejo, E. Freire, and A. Mikkola, “The Dependent Coordinates in the Linearization of Constrained Multibody Systems: Handling and Elimination,” International Journal of Mechanical Sciences 268 (2024): 109036, https://doi.org/10.1016/j.ijmecsci.2024.109036.

[17] C. Yang, D. Cao, Z. Zhao, Z. Zhang, and G. Ren, “A Direct Eigenanalysis of Multibody System in Equilibrium,” Journal of Applied Mathematics 2012, no. 1 (2012): 638546, https://doi.org/10.1155/2012/638546.

[18] J. L. Ortiz and D. Negrut, “Exact Linearization of Multibody Systems Using User-defined Coordinates,” in SAE 2006 World Congress & Exhibition (2006), 2006-01-0587, https://doi.org/10.4271/2006-01-0587.

[19] D. Negrut and J. L. Ortiz, “A Practical Approach for the Linearization of the Constrained Multibody Dynamics Equations,” Journal of Computational and Nonlinear Dynamics 1, no. 3 (2006): 230–239, https://doi.org/10.1115/1.2198876.

[20] D. Negrut and J. L. Ortiz, “On an Approach for the Linearization of the Differential Algebraic Equations of Multibody Dynamics,” in Volume 4a: ASME/IEEE Conference on Mechatronic and Embedded Systems and Applications (ASME, 2005), 211–222, https://doi.org/10.1115/DETC2005-85109.

[21] A. G. Agúndez, D. García-Vallejo, E. Freire, L. Pyrhönen, and A. Mikkola, “An Efficient and Accurate Linearization Approach for Hydraulically Actuated Multibody Systems With Holonomic and Nonholonomic Constraints,” Nonlinear Dynamics 111, no. 11 (2023): 10331–10356, https://doi.org/10.1007/s11071-023-08345-6.

[22] J. Wang, X. Rui, X. Wang, J. Zhang, Q. Zhou, and J. Gu, “Eigenvalue Analysis of Planar Linear Multibody System Under Conservative Force Based on the Transfer Matrix Method,” International Journal of Mechanical System Dynamics 3, no. 1 (2023): 12–24, https://doi.org/10.1002/msd2.12048.

[23] C. M. Pappalardo, A. Lettieri, and D. Guida, “Stability Analysis of Rigid Multibody Mechanical Systems With Holonomic and Nonholonomic Constraints,” Archive of Applied Mechanics 90, no. 9 (2020): 1961–2005, https://doi.org/10.1007/s00419-020-01706-2.

[24] INTESIM Engineering Simulation Technology (Dalian) Co., Ltd., INTESIM-FMBD (2025), accessed 2026/02/15, http://www.intesim.com/42/23.

[25] P. Masarati, M. J. U. Quro, and A. Zanoni, “Projection Continuation for Minimal Coordinate Set Formulation and Singularity Detection of Redundantly Constrained System Dynamics,” Multibody System Dynamics 61, no. 4 (2024): 453–480, https://doi.org/10.1007/s11044-023-09930-8.

[26] P. Zhou, A. Zanoni, and P. Masarati, “A Projection Continuation Approach for Minimal Coordinate Set Constrained Dynamics,” Multibody System Dynamics 57, no. 3 (2023): 237–257, https://doi.org/10.1007/s11044-023-09882-z.

[27] C. W. Gear, B. Leimkuhler, and G. K. Gupta, “Automatic Integration of Euler–Lagrange Equations With Constraints,” Journal of Computational and Applied Mathematics 12–13 (1985): 77–90, https://doi.org/10.1016/0377-0427(85)90008-1.

[28] C. Führer and B. J. Leimkuhler, “Numerical Solution of Differential-Algebraic Equations for Constrained Mechanical Motion,” Numerische Mathematik 59, no. 1 (1991): 55–69, https://doi.org/10.1007/BF01385770.

[29] G. Guennebaud, B. Jacob, et al., Eigen (2010), https://libeigen.gitlab.io.

[30] Y. Qiu, Spectra—C++ Library for Large Scale Eigenvalue Problems, https://spectralib.org/.

[31] R. Masoudi, T. Uchida, D. Vilela, A. Luaces, J. Cuadrado, and J. McPhee, “A Library of Computational Benchmark Problems for the Multibody Dynamics Community,” in Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics 2013 (2013), 1153–1162, https://lim.ii.udc.es/docs/proceedings/2013_07_ECCOMAS_ALibrary.pdf.

[32] F. González and J. Cuadrado, Multibody Benchmark 3D Simple Pendulum (2015), https://www.iftomm-multibody.org/benchmark/problem/3D_simple_pendulum/.

[33] B. Minaker, Multibody Benchmark Wheel on Tipping Table (2015), https://www.iftomm-multibody.org/benchmark/problem/Wheel_on_tipping_table/.

[34] F. González, P. Masarati, and J. Cuadrado, Four-Bar Linkage With Diagonal Spring (2015), https://www.iftomm-multibody.org/benchmark/problem/Four-bar_linkage_with_diagonal_spring/.

[35] F. González, P. Masarati, and J. Cuadrado, Four-Bar Linkage With Diagonal Spring and Damper (2015), https://www.iftomm-multibody.org/benchmark/problem/Four-bar_linkage_with_diagonal_spring_and_damper/.

[36] B. Minaker, Multibody Benchmark Linearized Bicycle (2015), https://www.iftomm-multibody.org/benchmark/problem/Linearized_bicycle/.

[37] J. Meijaard, J. M. Papadopoulos, A. Ruina, and A. Schwab, “Linearized Dynamics Equations for the Balance and Steer of a Bicycle: A Benchmark and Review,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2084 (2007): 1955–1982, https://doi.org/10.1098/rspa.2007.1857.

[38] C. M. Pappalardo, A. Lettieri, and D. Guida, “A General Multibody Approach for the Linear and Nonlinear Stability Analysis of Bicycle Systems. Part I: Methods of Constrained Dynamics,” Journal of Applied and Computational Mechanics 7, no. 2 (2021): 655–670, https://doi.org/10.22055/jacm.2020.35438.2653.

[39] A. García-Agúndez, D. García-Vallejo, and E. Freire, “Linearization Approaches for General Multibody Systems Validated Through Stability Analysis of a Benchmark Bicycle Model,” Nonlinear Dynamics 103, no. 1 (2021): 557–580, https://doi.org/10.1007/s11071-020-06069-5.

[40] X. Cao, D. C. Bui, D. Takács, and G. Orosz, “Autonomous Unicycle: Modeling, Dynamics, and Control,” Multibody System Dynamics 61, no. 1 (2024): 43–76, https://doi.org/10.1007/s11044-023-09923-7.

[41] A. G. Agúndez, D. García-Vallejo, and E. Freire, “An Electric Kickscooter Multibody Model: Equations of Motion and Linear Stability Analysis,” Multibody System Dynamics 62, no. 4 (2024): 493–524, https://doi.org/10.1007/s11044-024-09974-4.

[42] A. G. Agúndez, D. García-Vallejo, E. Freire, and A. M. Mikkola, “Stability Analysis of a Waveboard Multibody Model With Toroidal Wheels,” Multibody System Dynamics 53, no. 2 (2021): 173–203, https://doi.org/10.1007/s11044-021-09780-2.

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