| The study of complex system dynamic behavior has become an essential aspect of many engineering disciplines. The ever increasing complexity of multibody dynamic systems has produced a profound need to treat the areas of dynamic formulation theory, computational science, and design optimization in a unified manner so to more effectively examine dynamic system behavior. The linking of these previously divided disciplines to form efficient yet robust unified methodologies is crucial to effectively extending design optimization to complex dynamic systems.; In an effort to facilitate multibody dynamic systems design optimization, a novel full-recursive, direct-differentiation-based sensitivity analysis method is developed. Based on an efficient state space forward dynamics algorithm, the presented algorithm differs from more traditional methods in a number of ways. First, the potential of recursive calculation of many quantities is fully exploited in the derived algorithm, which significantly reduces the differentiation operation complexity relative to that often encountered with more traditional methods. Second, the essential chain rule of differentiation is performed separately on the individual bodies to form essential derivative primitives which are then recursively assembled to reduce the overall differentiation cost. Third, the entire procedure relies on a three-sweeping recursive procedure allowing the decomposition and the solution of the state sensitivity equations as they are being form. This special feature eliminates the cumbersome procedures of explicitly forming the sensitivity equations and their associated cost of matrix factorization and multiplication. Finally, the use a of local representation of kinematic, and kinetic quantities, as well as their associated derivatives, results in a great simplification of many of the mathematical manipulations which are part of this sensitivity analysis process.; As a result, a constant formulation complexity operational cost) for each body, irrespective of its topological location, is achieved. Moreover, being fully analytic in nature, the algorithm is not susceptible to the accuracy and stability problems which plague finite differences sensitivity approximation methods. Most importantly, the overall computational expense associated with the determination of the sensitivity values through the use of this algorithm is bilinear in the number of design variables and the number of system dynamic degrees of freedom.; Procedure validity and practical performance is investigated through both analytic and empirical means. Based on a typical multi-degree-of-freedom chain system, the analytic operation count of the algorithm is examined and is found to be linear in the number of generalized coordinates n. The computational requirements between these analyses maintain a near constant ratio with respect to the increase in system generalized coordinates. Empirical simulations further indicate that this algorithm contributes to a significant reduction on CPU time with respect to conventional direct differentiation methods in producing state sensitivities. The provided numerical examples, include optimal trajectory of a slider-crank mechanism, vehicle suspension system performance study, and spacecraft appendage deployment, demonstrating the practical application range of this algorithm and its potential in multibody design synthesis. |
