Computational Methods for Analysis and Design

Both symbolic and object-oriented (numerical) implementations of multibody dynamic formulations have been developed. By solving the equations generated by these computer algorithms using appropriate numerical methods, kinematic and dynamic analyses of complex mechanical systems have been automated. The use of Maple to generate kinematic and dynamic equations in symbolic form gives several advantages over a conventional numeric-based approach:

• Physical insight into the structure of the kinematic and dynamic equations
• Symbolic equations are easily communicated between colleagues
• Speed of numerical solution is increased by identification of repeated sub-expressions and elimination of operations involving 1 or 0
• Symbolic identification of first integrals of motion (energy, momentum) is possible
• Design sensitivities can be partially computed symbolically, thereby facilitating subsequent optimization analyses

Our Maple algorithms, DynaFlex , are freely available for non-commercial research and development.

In addition, the overall design cycle has been semi-automated by using numerical methods of optimization. Once an initial system configuration has been established, and an objective function defined (e.g. minimum-time or minimum-power maneuver), the computer can be used to perform the iterative calculations needed to determine the "best" set of design or control parameters. We have developed a hybrid optimization approach that exploits the best features of genetic algorithms and gradient-based sequential quadratic programming methods.  A Beowulf cluster of 17 PCs has been built to perform these design optimization calculations in parallel.

Selected Publications:

• C. Schmitke, K. Morency, and J. McPhee, Using Graph Theory and Symbolic Computing to Generate Efficient Models for Multibody Vehicle Dynamics, to appear in J. Multibody Dyn., 2008.
• Y. He and J. McPhee, Application of Optimization Algorithms and Multibody Dynamics to Ground Vehicle Suspension Design, Int. J. Heavy Veh. Sys., v.14, 158-192, 2007.
• Y. He and J. McPhee, Application of SQP and Dynamic Mode Tracking to the Identification of the Critical Speed of Rail Vehicles, Int. J. Heavy Veh. Sys., v.14, 193-212, 2007.
• W. Zhou, D. Jeffery, G. Reid, C. Schmitke, and J. McPhee, Implicit Reduced Involutive Forms and Their Application to Engineering Multibody Systems, in Lecture Notes in Computer Science, v.3519, H. Li, P. Olver, and G. Sommer, eds., Springer-Verlag, 31-43, 2005.
• Y. He and J. McPhee, Mechatronic Vehicle Design via Multidisciplinary Optimization, CSME Forum, University of Western Ontario, London, June 2004, pp.504-513.
  
• W. Zhou, D. Jeffrey, G. Reid, C. Schmitke, and J. McPhee, Implicit Reduced Involutive Forms and Their Application to Engineering Multibody Systems, International Workshop on Geometric Invariance and Applications in Engineering, Xi’an, China, May 2004.
  
• Y. He and J. McPhee, Application of Optimization Algorithms and Multibody Dynamics to Ground Vehicle Suspension Design, submitted to Journal of Sound and Vibration, 2003.
  
• E. Zahariev and J. McPhee, Stabilization of Multiple Constraints in Multibody Dynamics Using Optimization and a Pseudo-inverse Matrix, Mathematical and Computer Modelling of Dynamical Systems, vol.9, no.4, 2003, pp.417-436.
  
• J. McPhee, P. Shi, and J.-C. Piedboeuf, Dynamics of Multibody Systems using Virtual Work and Symbolic Programming , Mathematical and Computer Modelling of Dynamical Systems , vol.8, no.2, 2002, pp.137-156.
  
• P. Shi and J. McPhee, Symbolic Programming of a Graph-Theoretic Approach to Flexible Multibody Dynamics, Mechanics of Structures and Machines, vol.30, no.1, pp.123-154, 2002.
  
• Y. He and J. McPhee, Comparative Study of Optimization Algorithms used in Ground Vehicle Suspension Design, submitted to Journal of Sound and Vibration, 2002.
  
• P. Shi, J. McPhee, and G. Heppler, Polynomial Shape Functions and Numerical Methods for Flexible Multibody Dynamics , Mechanics of Structures and Machines , vol.29, no.1, pp.43-64, 2000.
  
• C. Good and J. McPhee, Dynamics of Mountain Bicycles with Rear Suspensions: Design Optimization, Sports Engineering , vol.3, no.1, pp.49-56, 2000.
  
• A. Seth and J. McPhee, Prediction of Optimal Arm Motions using a Genetic Algorithm, Proceedings of 17th Canadian Congress of Applied Mechanics, Hamilton, Ontario, 30 May - 3 June, 1999.
  
• J. Heilig and J. McPhee, Determination of Minimum-Time Manoeuvres for a Robotic Manipulator using Numerical Optimization Methods, Mechanics of Structures and Machines , vol.27, no.2, pp.185-201, 1999.
  
• A.E. Baumal, J.J. McPhee, and P.H. Calamai, Genetic Algorithm Optimization for Active Vehicle Suspension Design , Mechanics in Design, edited by S.A. Meguid, Proceedings of C.S.M.E. Forum, pp.455-464, 1996. Winner of award for Best Paper Lead-authored by a Student.
  
• O.M. Oshinowo and J.J. McPhee, Object-Oriented Implementation of A Graph-Theoretic Formulation for Planar Multibody Dynamics , International Journal for Numerical Methods in Engineering , vol.40, pp.4097-4118, 1997.
  
• J.J. McPhee and C.E. Wells, Automated Symbolic Analysis of Mechanical System Dynamics, MapleTech , vol.3, no.1, pp.48-56, 1996.
  
• C.E. Wells and J. McPhee, A Symbolic Toolbox for Investigating the Structure of Multibody Dynamics Equations, Mechanics in Design, edited by S.A. Meguid, Proceedings of C.S.M.E. Forum, pp.75-84, 1996.
  
• A.E. Baumal and J.J. McPhee, Feller-Buncher Planar Motion Boom Numerical Optimization Project , Technical Report for Timberjack Inc., Woodstock, Ontario, 1996.