A symbolic approach for automatic generation of the equations of motion of multibody systems This paper describes a collection of methods and procedures for the automatic generation of the equations of motion of multibody systems using general-purpose Computer Algebra Software. A brief review of existing symbolic multibody systems is given, and advantages and disadvantages of symbolic approaches compared with numerical ones are discussed. Then, a set of methods for symbolic modeling of multibody systems is explained. The first step of the modeling procedure consists of the description of the multibody system, by defining objects (such as points, vectors, rigid bodies, forces and torques, special objects) and the relationships between them (kinematic chains, constraints). The second step is the derivation of the equations of motion, which can be performed in a quasiautomatic way. A further step is the linearization of the equations and the calculation of the system’s frequency response functions. By way of example, a dynamic model of the motorcycle is developed, obtaining the nonlinear equations of motion in a dependent coordinates’ formulation. Next, the equations of motion are linearized and reduced to an independent formulation, reobtaining the well known Sharp’s model of the straight running of the motorcycle. Root loci and frequency response functions are also calculated. This example demonstrates the power of the given symbolic procedures and shows how a model suitable for stability, handling and control analysis can be developed quickly and easily. The procedure described in this paper has been implemented in a Maple package called MBSymba, which is available on the web page www.dim.unipd.it/lot/mbsymba.html.

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  1. Gabiccini, M.; Bartali, L.; Guiggiani, M.: Analysis of driving styles of a GP2 car via minimum lap-time direct trajectory optimization (2021)
  2. Bruni, Stefano; Meijaard, J. P.; Rill, Georg; Schwab, A. L.: State-of-the-art and challenges of railway and road vehicle dynamics with multibody dynamics approaches (2020)
  3. Joubert, Nicolas; Boisvert, Maxime; Blanchette, Carl; St-Amant, Yves; Desrochers, Alain; Rancourt, Denis: Frame loads accuracy assessment of semianalytical multibody dynamic simulation methods of a recreational vehicle (2020)
  4. Shafei, A. M.; Shafei, H. R.: Dynamic modeling of tree-type robotic systems by combining $3\times 3$ rotation and $4\times 4$ transformation matrices (2018)
  5. Carpinelli, Mariano; Gubitosa, Marco; Mundo, Domenico; Desmet, Wim: Automated independent coordinates’ switching for the solution of stiff DAEs with the linearly implicit Euler method (2016)
  6. Shafei, A. M.; Shafei, H. R.: A systematic method for the hybrid dynamic modeling of open kinematic chains confined in a closed environment (2016)
  7. Pei, YongChen; Sun, YouHong; Wang, JiXin: Dynamics of rotating conveying mud drill string subjected to torque and longitudinal thrust (2013)
  8. Cossalter, V.; Doria, A.; Lot, R.; Massaro, M.: The effect of rider’s passive steering impedance on motorcycle stability: identification and analysis (2011)
  9. Cossalter, V.; Lot, R.; Massaro, M.: An advanced multibody code for handling and stability analysis of motorcycles (2011)
  10. Van Khang, Nguyen: Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems (2011)
  11. Keppler, Rainer; Seemann, Wolfgang: A reduction algorithm for open-loop rigid-body systems with revolute joints (2006)
  12. Lot, R.; Da Lio, M.: A symbolic approach for automatic generation of the equations of motion of multibody systems (2004)