Ordinary Differential Equations and Applied Dynamics Commons^™

Rigid Body Constrained Motion Optimization And Control On Lie Groups And Their Tangent Bundles, Brennan S. Mccann

Doctoral Dissertations and Master's Theses

Rigid body motion requires formulations where rotational and translational motion are accounted for appropriately. Two Lie groups, the special orthogonal group SO(3) and the space of quaternions H, are commonly used to represent attitude. When considering rigid body pose, that is spacecraft position and attitude, the special Euclidean group SE(3) and the space of dual quaternions DH are frequently utilized. All these groups are Lie groups and Riemannian manifolds, and these identifications have profound implications for dynamics and controls. The trajectory optimization and optimal control problem on Riemannian manifolds presents significant opportunities for theoretical development. Riemannian optimization is an attractive …

2n-Dimensional Canonical Systems And Applications, Andrei Ludu, Keshav Baj Acharya

Publications

We study the 2N-dimensional canonical systems and discuss some properties of its fundamental solution. We then discuss the Floquet theory of periodic canonical systems and observe the asymptotic behavior of its solution. Some important physical applications of the systems are also discussed: linear stability of periodic Hamiltonian systems, position-dependent effective mass, pseudo-periodic nonlinear water waves, and Dirac systems.

Nonlocal Symmetries For Time-Dependent Order Differential Equations, Andrei Ludu

Publications

A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.

Differential Equations Of Dynamical Order, Andrei Ludu, Harihar Khanal

Publications

No abstract provided.

Generalized Thomas-Fermi Equations As The Lampariello Class Of Emden-Fowler Equations, Haret C. Rosu, S.C. Mancas

Publications

A one-parameter family of Emden-Fowler equations defined by Lampariello’s parameter p which, upon using Thomas-Fermi boundary conditions, turns into a set of generalized Thomas-Fermi equations comprising the standard Thomas-Fermi equation for p = 1 is studied in this paper. The entire family is shown to be non integrable by reduction to the corresponding Abel equations whose invariants do not satisfy a known integrability condition. We also discuss the equivalent dynamical system of equations for the standard Thomas-Fermi equation and perform its phase-plane analysis. The results of the latter analysis are similar for the whole class.

Improving Airplane Touchdown Control By Utilizing The Adverse Elevator Effect, Nihad E. Daidzic Ph.D., Sc.D.

International Journal of Aviation, Aeronautics, and Aerospace

The main objective of this original research article is to understand the short-term dynamic behavior of the transport-category airplane during landing flare elevator control application. Increasing the pitch angle to arrest the sink rate, the elevator will have to produce negative lift to rotate the airplane’s nose upward. This has an immediate adverse effect of initially accelerating airplane downward. A mathematical model of landing flare based on the flat-Earth longitudinal dynamics of rigid airplane was developed which is realistic only on very short time-scales as pitch stiffness and damping were neglected. Pilot control scenarios using impulse and step elevator pull-up …

Computational Models For Nanosecond Laser Ablation, Harihar Khanal, David Autrique, Vasilios Alexiades

Publications

Laser ablation in an ambient environment is becoming increasingly important in science and technology. It is used in applications ranging from chemical analysis via mass spectroscopy, to pulsed laser deposition and nanoparticle manufacturing. We describe numerical schemes for a multiphase hydrodynamic model of nanosecond laser ablation expressing energy, momentum, and mass conservation in the target material, as well as in the expanding plasma plume, along with collisional and radiative processes for laser-induced breakdown (plasma formation). Numerical simulations for copper in a helium background gas are presented and the efficiency of various ODE integrators is compared.

A Non-Autonomous Second Order Boundary Value Problem On The Half-Line, Gregory S. Spradlin

Greg S. Spradlin Ph.D.

By variational arguments, the existence of a solution to a nonautonomous second-order boundary problem on the half-line is proven. The corresponding autonomous problem has no solution, revealing significant differences between the autonomous and the non-autonomous case.

Models Of Phototransduction In Rod Photoreceptors, Harihar Khanal, Vasilios Alexiades

Publications

Phototransduction is the process by which photons of light generate an electrical response in retinal rod and cone photoreceptors, thereby initiating vision. We compare the electrical response in salamander rods from increasingly more (spacialy) detailed models of phototransduction: 0-dimensional (bulk), 1-dimensional (longitudinal), 2-dimensional (axisymmetric), and 3-dimensional (with incisures). We discuss issues of finding physical parameters for simulation and validation of models, and also present some computational experiments for rods with geometry of mouse and human photoreceptors.

Scattered Homoclinics To A Class Of Time-Recurrent Hamiltonian Systems, Gregory S. Spradlin

Greg S. Spradlin Ph.D.

A second-order Hamiltonian system with time recurrence is studied. The recurrence condition is weaker than almost periodicity. The existence is proven of an infinite family of solutions homoclinic to zero whose support is spread out over the real line.

Laplace Transform Of Spherical Bessel Functions, Andrei Ludu

Andrei Ludu

No abstract provided.

Simulation Of Engineering Systems Described By High-Index Dae And Discontinuous Ode Using Single Step Methods, Marc Compere

Publications

This dissertation presents numerical methods for solving two classes of or-dinary diferential equations (ODE) based on single-step integration meth-ods. The first class of equations addressed describes the mechanical dynamics of constrained multibody systems. These equations are ordinary differential equations (ODE) subject to algebraic constraints. Accordinly they are called differential-algebraic equations (DAE).

Specific 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 …

Interfering Solutions Of A Nonhomogeneous Hamiltonian System, Gregory S. Spradlin

Greg S. Spradlin Ph.D.

A Hamiltonian system is studied which has a term approaching a constant at an exponential rate at infinity. A minimax argument is used to show that the equation has a positive homoclinic solution. The proof employs the interaction between translated solutions of the corresponding homogeneous equation. What distinguishes this result from its few predecessors is that the equation has a nonhomogeneous nonlinearity.