Linearized equations of motion are important in engineering applications, especially with respect to stability analysis and control design. Traditionally, the full, nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. Here, this development is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems, where a hybrid system is described with both discrete and continuous generalized coordinates. The results presented require only velocity level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest. This study shows that the entire nonlinear equations of motion do not have to be generated in order to construct the linearized equations of motion. Several examples are presented to illustrate application of these results to both continuous and hybrid system problems.

This thesis studies the dynamics and stability of some bicycles. The dynamics of idealized bicycles is of interest due to complexities associated with the behaviour of this seemingly simple machine. It is also useful as it can be a starting point for analysis of more complicated systems, such as motorcycles with suspensions, frame flexibility and thick tyres. Finally, accurate and reliable analyses of bicycles can provide benchmarks for checking the correctness of general multibody dynamics codes. The first part of the thesis deals with the derivation of fully nonlinear differential equations of motion for a bicycle. Lagrange’s equations are derived along with the constraint equations in an algorithmic way using computer algebra.Then equivalent equations are obtained numerically using a Newton-Euler formulation. The Newton-Euler formulation is less straightforward than the Lagrangian one and it requires the solution of a bigger system of linear equations in the unknowns. However, it is computationally faster because it has been implemented numerically, unlike Lagrange’s equations which involve long analytical expressions that need to be transferred to a numerical computing environment before being integrated. The two sets of equations are validated against each other using consistent initial conditions. The match obtained is, expectedly, very accurate. The second part of the thesis discusses the linearization of the full nonlinear equations of motion. Lagrange’s equations have been used.The equations are linearized and the corresponding eigenvalue problem studied. The eigenvalues are plotted as functions of the forward speed ν of the bicycle. Several eigenmodes, like weave, capsize, and a stable mode called caster, have been identified along with the speed intervals where they are dominant. The results obtained, for certain parameter values, are in complete numerical agreement with those obtained by other independent researchers, and further validate the equations of motion. The bicycle with these parameters is called the benchmark bicycle. The third part of the thesis makes a detailed and comprehensive study of hands-free circular motions of the benchmark bicycle. Various one-parameter families of circular motions have been identified. Three distinct families exist: (1)A handlebar-forward family, starting from capsize bifurcation off straight-line motion, and ending in an unstable static equilibrium with the frame perfectly upright, and the front wheel almost perpendicular. (2) A handlebar-reversed family, starting again from capsize bifurcation, but ending with the front wheel again steered straight, the bicycle spinning infinitely fast in small circles while lying flat in the ground plane. (3) Lastly, a family joining a similar flat spinning motion (with handlebar forward), to a handlebar-reversed limit, circling in dynamic balance at infinite speed, with the frame near upright and the front wheel almost perpendicular; the transition between handlebar forward and reversed is through moderate-speed circular pivoting with the rear wheel not rotating, and the bicycle virtually upright. In the fourth part of this thesis, some of the parameters (both geometrical and inertial) for the benchmark bicycle have been changed and the resulting different bicycles and their circular motions studied showing other families of circular motions. Finally, some of the circular motions have been examined, numerically and analytically, for stability.

The main objective of this work is to demonstrate some new computational methods for estimation, optimization and modeling of dynamical systems that use automatic differentiation. Particular focus will be upon dynamical systems arising in Aerospace Engineering. Automatic differentiation is a recursive computational algorithm, which enables computation of analytically rigorous partial derivatives of any user-specified function. All associated computations occur, in the background without user intervention, as the name implies. The computational methods of this dissertation are enabled by a new automatic differentiation tool, OCEA (Object oriented Coordinate Embedding Method). OCEA has been recently developed and makes possible efficient computation and evaluation of partial derivatives with minimal user coding. The key results in this dissertation details the use of OCEA through a number of computational studies in estimation and dynamical modeling. Several prototype problems are studied in order to evaluate judicious ways to use OCEA. Additionally, new solution methods are introduced in order to ascertain the extended capability of this new computational tool. Computational tradeoffs are studied in detail by looking at a number of different applications in the areas of estimation, dynamical system modeling, and validation of solution accuracy for complex dynamical systems. The results of these computational studies provide new insights and indicate the future potential of OCEA in its further development.