This thesis explores the dynamics of a three-dimensional four-bar mechanical linkage subject to random external forcing. The Lagrangian... Show moreThis thesis explores the dynamics of a three-dimensional four-bar mechanical linkage subject to random external forcing. The Lagrangian formulation of the equations of motion are index-3 stochastic di erential-algebraic equations (SDAE) that describe the time evolution of the sample paths of the generalized coordinates, velocities, and Lagrange multipliers as stochastic processes. We solve the SDAEs using two di erent approaches: inverse dynamics, Case Study 1, via independent, successive solution of the nonlinear equations for each kinematic variable, where the time evolution of one generalized coordinate is prescribed; and direct dynamics, Case Study 2, via direct solution of the SDAEs in the index-1 formulation, using fourth-order stochastic backward di erentiation formula (BDF) with modi ed Newton iteration and position and velocity stabilization (Ascher and Petzold [2]), where the (deterministic) input driving torque is prescribed. For the particular application of a three-dimensional swing gate security system, we conduct numerical experiments for both approaches. In Case Study 1, we simulate the random external forcing as a Gaussian wind speed process that applies stochastic wind drag onto the gate. The kinematic variables are deterministic, while the required input driving torque is a stochastic process. In Case Study 2, we apply the external forcing as a resistive torque with additive Gaussian noise modeling the wind drag; the kinematic variables are stochastic processes. For both cases, we apply four mean wind speeds: 0 mph (deterministic only), 10 mph, 20 mph, and 30 mph, from which we compute the deterministic solution and three stochastic sample paths for each stochastic process. The overall conclusions are that direct solution is possible for inverse dynamics, that the solution of index-1 SDAEs in multibody dynamics is tractable since the mass matrix is symmetric and positive de nite, and that the deterministic solution is the expectation of the sample paths. M.S. in Applied Mathematics, December 2011 Show less