A General Multibody Approach for the Linear and Nonlinear Stability Analysis of Bicycle Systems. Part II: Application to the Whipple-Carvallo Bicycle Model

Document Type : Research Paper


1 Department of Industrial Engineering, University of Salerno, Via Giovanni Paolo II, 132, Fisciano, 84084, Salerno, Italy

2 Spin-Off MEID4 s.r.l., University of Salerno, Via Giovanni Paolo II, 132, Fisciano, 84084, Salerno, Italy


This paper represents the second contribution of a two-part research work presenting the application of the proposed multibody analysis approach to bicycle systems and the relative numerical results found. In this work, a nonlinear multibody model of a bicycle system is developed and implemented to perform a parametric analysis to understand the influence of the variation of the principal model parameters on the system stability under investigation. To demonstrate the effectiveness of the proposed approach, the case study considered in this paper is the dynamic analysis of the Whipple-Carvallo bicycle model. Considering the combined use of a robust numerical technique for nonlinear dynamical simulations with a specifically devised linearization procedure, the effects of the different geometric parameters and inertial properties on the bicycle stability are investigated. The numerical results obtained in this work using the proposed multibody techniques are useful to gain insight information about the dynamic behavior of the bicycle system in a straight motion. The proposed multibody methodology also demonstrated a high potential for analyzing complex multibody mechanical systems in virtue of the generality of the analytical and computational approaches adopted.


Main Subjects

[1] Shabana, A.A., Dynamics of multibody systems, Cambridge university press, New York, USA, 2003.
[2] Blundell, M., Harty, D., Multibody systems approach to vehicle dynamics, Elsevier, Oxford, UK, 2004.
[3] Shabana, A.A., Computational dynamics, John Wiley & Sons, London, UK, 2009.
[4] Meijaard, J.P., Papadopoulos, J.M., Ruina, A., Schwab, A.L., 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(2084), 2007, 1955–1982.
[5] Kooijman, J.D.G., Schwab, A.L., Meijaard, J.P., Experimental validation of a model of an uncontrolled bicycle, Multibody System Dynamics, 19(1-2), 2008, 115-132.
[6] Barbagallo, R., Sequenzia, G., Cammarata, A., Oliveri, S.M., Fatuzzo, G., Redesign and multibody simulation of a motorcycle rear suspension with eccentric mechanism, International Journal on Interactive Design and Manufacturing (IJIDeM), 12(2), 2018, 517-524.
[7] Palomba, I., Richiedei, D., Trevisani, A., Two-stage approach to state and force estimation in rigid-link multibody systems, Multibody System Dynamics, 39(1-2), 2017, 115-134.
[8] Carpinelli, M., Gubitosa, M., Mundo, D., Desmet, W., Automated independent coordinates’ switching for the solution of stiff DAEs with the linearly implicit Euler method, Multibody System Dynamics, 36(1), 2016, 67-85.
[9] Bruni, S., Meijaard, J.P., Rill, G., Schwab, A.L., State-of-the-art and challenges of railway and road vehicle dynamics with multibody dynamics approaches, Multibody System Dynamics, 49, 2020, 1-32.
[10] Tasora, A., Mangoni, D., Negrut, D., Serban, R., Jayakumar, P., Deformable soil with adaptive level of detail for tracked and wheeled vehicles, International Journal of Vehicle Performance, 5(1), 2019, 60-76.
[11] Soria, L., Peeters, B., Anthonis, J., Van Der Auweraer, H., Operational modal analysis and the performance assessment of vehicle suspension systems, Shock and Vibration, 19(5), 2012, 1099-1113.
[12] Popp, K., Schiehlen, W.O., Ground Vehicle Dynamics, Springer, Berlin, 2010.
[13] Wong, J.Y., Theory of Ground Vehicles, Wiley, Hoboken, 2008.
[14] Meli, E., Magheri, S., Malvezzi, M., Development and implementation of a differential elastic wheel–rail contact model for multibody applications, Vehicle system dynamics, 49(6), 2011, 969-1001.
[15] Cajic, M. S., Lazarevic, M. P., Determination of joint reactions in a rigid multibody system, two different approaches, FME Transactions, 44(2), 2016, 165-173.
[16] Vlasenko, D., Kasper, R., Generation of equations of motion in reference frame formulation for fem models, Engineering Letters, 16(4), 2008, 537-544.
[17] Geier, A., Kebbach, M., Soodmand, E., Woernle, C., Kluess, D., Bader, R., Neuro-musculoskeletal flexible multibody simulation yields a framework for efficient bone failure risk assessment, Scientific reports, 9(1), 2019, 1-15.
[18] Kebbach, M., Grawe, R., Geier, A., Winter, E., Bergschmidt, P., Kluess, D., D’Lima, D., Woernle, C., Bader, R., Effect of surgical parameters on the biomechanical behaviour of bicondylar total knee endoprostheses - A robot-assisted test method based on a musculoskeletal model, Scientific Reports, 9(1), 2019, 1-11.
[19] Nitschke, M., Dorschky, E., Heinrich, D., Schlarb, H., Eskofier, B.M., Koelewijn, A.D., Van Den Bogert, A.J., Efficient trajectory optimization for curved running using a 3D musculoskeletal model with implicit dynamics, Scientific Reports, 10(1), 2020, 1-12.
[20] Moore, J.K., Kooijman, J.D.G., Schwab, A.L., Hubbard, M., Rider motion identification during normal bicycling by means of principal component analysis, Multibody System Dynamics, 25(2), 2011, 225-244.
[21] Damsgaard, M., Rasmussen, J., Christensen, S.T., Surma, E., De Zee, M., Analysis of musculoskeletal systems in the AnyBody Modeling System, Simulation Modelling Practice and Theory, 14(8), 2006, 1100-1111.
[22] Whipple, F.J., The stability of the motion of a bicycle, Quarterly Journal of Pure and Applied Mathematics, 30(120), 1899, 312-348.
[23] Carvallo, M.E., Theorie de mouvement du monocycle et de la bycyclette, MPublishing, 1901.
[24] Cossalter, V., Motorcycle dynamics, Second Edition, Lulu. Com, 2006.
[25] Moore, J., Hubbard, M., Parametric study of bicycle stability (P207), The engineering of sport, 7, 2008, 311-318.
[26] Moore, J.K., Human control of a bicycle, Davis, CA, University of California, Davis, 2012.
[27] Franke, G., Suhr, W., and Rieß, F., An advanced model of bicycle dynamics, European Journal of Physics, 11(2), 1990, 116.
[28] Papadopoulos, J.M., Bicycle steering dynamics and self-stability: a summary report on work in progress, Cornell Bicycle Research Project, Cornell University, NY, 1987.
[29] Meijaard, J.P., Derivation of the Linearized Equations for an Uncontrolled Bicycle, Internal report, University of Nottingham, UK, 2004, 1-19.
[30] Schwab, A.L., Meijaard, J.P., Papadopoulos, J.M., A Multibody Dynamics Benchmark on the Equations of Motion of an Uncontrolled Bicycle, EUROMECH Nonlinear Dynamics Conference, August, 2005, 7-12.
[31] Kooijman, J.D.G., Schwab, A.L., A review on bicycle and motorcycle rider control with a perspective on handling qualities, Vehicle System Dynamics, 51(11), 2013, 1722-1764.
[32] Limebeer, D.J.N., Sharp, R.S., Single track vehicle modelling and stability, IEEE Control Systems Magazine, October, 2006, 34-61.
[33] Schwab, A.L., Meijaard, J.P., A review on bicycle dynamics and rider control, Vehicle System Dynamics, 51(7), 2013, 1059-1090.
[34] Xiong, J., Wang, N., Liu, C., Stability analysis for the Whipple bicycle dynamics, Multibody System Dynamics, 48(3), 2020, 311-335.
[35] Xiong, J., Wang, N., Liu, C., Bicycle dynamics and its circular solution on a revolution surface, Acta Mechanica Sinica, 36(1), 2019, 220-233.
[36] Basu-Mandal, P., Chatterjee, A., Papadopoulos, J.M., Hands-free circular motions of a benchmark bicycle, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2084), 2007, 1983-2003.
[37] Escalona, J.L., Recuero, A.M., A bicycle model for education in multibody dynamics and real-time interactive simulation, Multibody System Dynamics, 27(3), 2012, 383-402.
[38] Escalona, J.L., KÅ‚odowski, A., Munoz, S., Validation of multibody modeling and simulation using an instrumented bicycle: from the computer to the road, Multibody System Dynamics, 43(4), 2018, 297-319.
[39] Frosali, G., Ricci, F., Kinematics of a bicycle with toroidal wheels, Communications in Applied and Industrial Mathematics, 3(1), 2012, 424.
[40] Astrom, K.J., Klein, R.E., Lennartsson, A., Bicycle dynamics and control: adapted bicycles for education and research, IEEE Control Systems Magazine, 25(4), 2005, 26-47.
[41] Plochl, M., Edelmann, J., Angrosch, B., Ott, C., On the wobble mode of a bicycle, Vehicle system dynamics, 50(3), 2012, 415-429.
[42] Tomiati, N., Colombo, A., Magnani, G., A nonlinear model of bicycle shimmy, Vehicle system dynamics, 57(3), 2019, 315-335.
[43] Schwab, A.L., Meijaard, J.P., Kooijman, J.D.G., Lateral dynamics of a bicycle with a passive rider model: stability and controllability, Vehicle System Dynamics, 50(8), 2012, 1209-1224.
[44] Jansen, C., McPhee, J., Predictive dynamic simulation of Olympic track cycling standing start using direct collocation optimal control, Multibody System Dynamics, 49, 2020, 53-70.
[45] Cossalter, V., Lot, R., A motorcycle multi-body model for real time simulations based on the natural coordinates approach, Vehicle System Dynamics, 37(6), 2002, 423-447.
[46] Cossalter, V., Lot, R., Massaro, M., An advanced multibody code for handling and stability analysis of motorcycles, Meccanica, 46(5), 2011, 943-958.
[47] Sharp, R.S., Evangelou, S., Limebeer, D.J.N., Advances in the modelling of motorcycle dynamics, Multibody System Dynamics, 12(3), 2004, 251-283.
[48] Schwab, A.L., Meijaard, J.P., Papadopoulos, J.M., Benchmark results on the linearized equations of motion of an uncontrolled bicycle, Journal of mechanical science and technology, 19(1), 2005, 292-304.
[49] Wisse, M., Schwab, A.L., Skateboards, bicycles, and three-dimensional biped walking machines: Velocity-dependent stability by means of leanto-yaw coupling, The International Journal of Robotics Research, 24(6), 2005, 417-429.
[50] Zenkov, D.V., Bloch, A.M., Marsden, J.E., The energy-momentum method for the stability of non-holonomic systems, Dynamics and Stability of Systems, 13(2), 1998, 123-165.
[51] Ruina, A., Nonholonomic stability aspects of piecewise holonomic systems, Reports on mathematical physics, 42(1-2), 1998, 91-100.
[52] Borisov, A.V., Mamayev, I.S., The dynamics of a Chaplygin sleigh, Journal of Applied Mathematics and Mechanics, 73(2), 2009, 156-161.
[53] Neimark, J.I., Fufaev, N.A., Dynamics of nonholonomic systems (Vol. 33), American Mathematical Society, USA, 2004.
[54] Pollard, B., Fedonyuk, V., Tallapragada, P., Swimming on limit cycles with nonholonomic constraints, Nonlinear Dynamics, 97(4), 2019, 2453-2468.
[55] Zhang, C., Li, Y., Qi, G., Sheng, A., Distributed finite-time control for coordinated circumnavigation with multiple non-holonomic robots, Nonlinear Dynamics, 98(1), 2019, 573-588.
[56] Pappalardo, C. M., Guida, D., On the Dynamics and Control of Underactuated Nonholonomic Mechanical Systems and Applications to Mobile Robots, Archive of Applied Mechanics, 89(4), 2019, 669-698.
[57] Sharp, R.S., On the stability and control of the bicycle, Applied mechanics reviews, 61(6), 2008, 060803.
[58] Laulusa, A., Bauchau, O.A., Review of classical approaches for constraint enforcement in multibody systems, Journal of computational and nonlinear dynamics, 3(1), 2008, 011004.
[59] Bauchau, O.A., Laulusa, A., Review of contemporary approaches for constraint enforcement in multibody systems, Journal of Computational and Nonlinear Dynamics, 3(1), 2008, 011005.
[60] Bayo, E., Ledesma, R., Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics, Nonlinear Dynamics, 9(1-2), 1996, 113-130.
[61] Shabana, A.A., Sany, J.R., An augmented formulation for mechanical systems with non-generalized coordinates: application to rigid body contact problems, Nonlinear dynamics, 24(2), 2001, 183-204.
[62] Pappalardo, C.M., A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems, Nonlinear Dynamics, 81(4), 2015, 1841-1869.
[63] Pappalardo, C.M., Guida, D., A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems, Archive of Applied Mechanics, 88(12), 2018, 2153-2177.
[64] Pappalardo, C.M., Lettieri, A., Guida, D., Stability analysis of rigid multibody mechanical systems with holonomic and nonholonomic constraints, Archive of Applied Mechanics, 90, 2020, 1961-2005.
[65] Pappalardo, C.M., De Simone, M.C., Guida, D., Multibody Modeling and Nonlinear Control of the Pantograph/Catenary System, Archive of Applied Mechanics, 89(8), 2019, 1589-1626.
[66] Rivera, Z.B., De Simone, M.C., Guida, D., Unmanned Ground Vehicle Modelling in Gazebo/ROS-Based Environments, Machines, 7(2), 2019, 42.
[67] De Simone, M., Rivera, Z., Guida, D., Obstacle Avoidance System for Unmanned Ground Vehicles by using Ultrasonic Sensors, Machines, 6(2), 2018, 18.
[68] De Simone, M.C., Guida, D., Control Design for an Under-Actuated UAV Model, FME Transactions, 46(4), 2018, 443-452.
[69] De Simone, M.C., Guida, D., Identification and Control of a Unmanned Ground Vehicle by using Arduino, UPB Scientific Bulletin, Series D: Mechanical Engineering, 80(1), 2018, 141-154.
[70] De Simone, M.C., Rivera, Z.B., Guida, D., Finite Element Analysis on Squeal-Noise in Railway Applications, FME Transactions, 46(1), 2018, 93-100.