Simulation of Engineering Systems Described by High-Index DAE and Discontinuous ODE Using Single Step Methods

<p> This dissertation presents numerical methods for solving two classes of or-dinary diferential equations (ODE) based on single-step integration meth-ods. The &filig;rst class of equations addressed describes the mechanical dynamics of constrained multibody systems. These equations are ordinary di&fflig;erential equations (ODE) subject to algebraic constraints. Accordinly they are called di&fflig;erential-algebraic equations (DAE).</p><p> Speci&filig;c contributions made in this area include an explicit transforma-tion between the Hessenberg index-3 form for constrained mechanical systems to a canonical state-space form used in the nonlinear control communities. A hybrid solution method was developed that incorporates both sliding-mode control (SMC) from the controls literature and post-stabilization from the DAE related literature. The process of developing the hybrid method produced insights into both areas in a way that allowed both areas to bene&filig;t from the other&rsquo;s strengths. First, the hybrid method produced an accurate and e&ffilig;cient method for simulating sliding-mode control systems. A technique called post-stabilization provides a more e&ffilig;cient method for simulating SMC systems than conventional methods using the discontinuous control term. Sec-ond, use of SMC mathematical framework allows the hybrid method to handle arbitrary, or inconsistent initial conditions.</p><p> The second class of equations addressed here are discontinuous ODE. Speci&filig;c contributions made in solving DODE include further classi&filig;cation of discontinnuities into parametric or structural discontinuities as well as unilateral or bilateral events. Consistent event location and discontinuity sticking from Park and Barton[56] originally addressed bilateral events only and were implemented in a single-step environment and then extended to address uni-lateral events as well. An e&fflig;ective detection scheme was developed using low-order interpolants for detecting most events in the correct order. For rare cases when the detection scheme fails, a try-catch model was implemented to deal with two possible failure scenarios. The detection and location methods successfully handled all events in the correct order for the benchmark problems solved. Lastly, a region of concurrency was developed that can provide large e&ffilig;ciency gains for some systems containing multiple closely spaced events.</p>