Tarek A. Elgohary

Generalized Frequency Domain Modeling and Analysis For A Flexible Rotating Spacecraft

A flexible rotating spacecraft is modeled as a three body hybrid system consisting of a rigid hub a flexible appendage following the Euler-Bernoulli beam assumptions and a tip mass and inertia. Hamilton’s extended principle is used to derive the equations of motion and the boundary conditions of the system. This work compares the frequency domain accuracy provided by series approximation methods versus analytical models. Applying the Laplace transform to the integro-partial derivation equations of motion model, leads to a generalized state space model for the frequency domain representation of the system. Both approximate and exact transfer function models are developed and compared. Eigen decomposition is used to solve the flexible appendage sub-problem and then to find the solution for the full system of equations. The analytic frequency domain model is manipulated by introducing a spatial domain state space, where a standard convolution integral representation is used to invoke the boundary conditions that act at the tip mass for the free end of the beam. Closed-form solutions are obtained for the convolution integral forcing terms. The closed form solution is used to generate transfer functions for both the rigid and the flexible modes of the system in terms of the input torque. A numerical example is presented to compare the frequency response of the closed form solution transfer function to the numerical assumed modes solution. The difference resulting from in the natural frequencies resulting from the series truncation is highlighted and discussed. The closed form solution proves to be more accurate with no truncation errors and is suitable for control design iterations.

Nonlinear analysis of a 2-DOF piecewise linear aeroelastic system

We study the dynamics of a two- degree-of-freedom (pitch and plunge) aeroelastic system where the aerodynamic forces are modeled as a piecewise linear function of the effective angle of attack. Stability and bifurcations of equilibria are analyzed. We find border collision and rapid bifurcations. Bifurcation diagrams of the system were calculated utilizing MATCONT and Mathematica. Chaotic behavior with intermittent switches about the two nontrivial equilibria was also observed.

Dynamics and Controls of a Generalized Frequency Domain Model Flexible Rotating Spacecraft

Modeling a flexible rotating spacecraft as a distributed parameters system of a rigid hub attached to a flexible appendage is very common. When considering large angle maneuvers the same model applies to flexible robotic manipulators by adding a tip mass at the end of the flexible appendage to account for the payload. Following Euler-Bernoulli beam theory the dynamics for both no tip mass and tip mass models are derived. A Generalized State Space (GSS) system is constructed in the frequency domain to completely solve for the input-output transfer functions of the models. The analytical solution of the GSS is obtained and compared against the classical assumed modes method. The frequency response of the system is then used in a classical control problem where a Lyapunov stable controller is derived and tested for gain selection. The assumed modes method is used to obtain the time response of the system to verify the gain selections and draw connections between the frequency and the time domains. The GSS approach provides a powerful tool to test various control schemes in the frequency domain and a validation platform for existing numerical methods utilized to solve distributed parameters models.

Generalized Frequency Domain Solution for a Hybrid Rigid Hub Timoshenko Beam Rotating Aerospace Structure

A hybrid system consisting of a rotating rigid hub and a flexible appendage following the Timoshenko beam assumptions where shear deformations are taken into account is introduced. Generalization of Lagrange’s equations utilizing Hamilton’s extended principle is used to derive the equations of motion and the boundary conditions of the system. Applying the Laplace transform to the integro-partial equations of motion leads to a generalized state space model for the frequency domain representation of the system. The beam sub-problem is then solved and utilized for insights for the solution of the full system. Boundary conditions at the beam free end are imposed to obtain the full solution for the state space model. The solution is used to generate transfer functions for both the rigid and the flexible modes of the system in terms of the input torque at the rigid rotating hub. No modal truncation errors are introduced into the transfer function calculations. Numerical results are presented for transfer functions frequency response using the generalized state space solution methodology.

A New Method for Space Objects Probability of Collision

The state of a dynamical system and its uncertainty, as defined by its probability density function (PDF), are valuable for numerous fields in science and engineering. There have been numerous methods proposed to estimate and quantify this uncertainty. In astrodynamics, space situational awareness (SSA) is a major area that relies on uncertainty quantification to estimate a space object’s state and its associated uncertainty. This data is invaluable for making informed decisions regarding probability of collision, tracking, and catalog maintenance. A new method for uncertainty quantification based on orthogonal polynomials and the application of Liouville’s theorem is developed. The method identifies the region of extreme probability at the time of interest and populates that region with structured points. The associated PDF is computed based on the a-priori PDF of the initial conditions and/or the nominal values of the system parameters (e.g. drag coefficient). High dimension orthogonal polynomials are used to approximate the PDF at the target time. Having an analytical expression for the propagated PDF enables rigorous probabilistic analysis. The present method is applied to several problems to compute the probability of collision between two objects. Numerical experiments show orders of magnitude improvement in computational cost versus classical Monte Carlo Methods. The new approach is easy to implement, extensible to higher dimensions, computationally efficient and provides a rigorous approach to address probability of collision problems in SSA.

A Simple Perturbation Algorithm for Inverting the Cartesian to Geodetic Transformation

A singularity-free perturbation solution is presented for inverting the Cartesian to Geodetic transformation. Geocentric latitude is used to model the satellite ground track position vector. A natural geometric perturbation variable is identified as the ratio of the major and minor Earth ellipse radii minus one. A rapidly converging perturbation solution is developed by expanding the satellite height above the Earth and the geocentric latitude as a perturbation power series in the geometric perturbation variable. The solution avoids the classical problem encountered of having to deal with highly nonlinear solutions for quartic equations. Simulation results are presented that compare the solution accuracy and algorithm performance for applications spanning the LEO-to-GEO range of missions.

Nonlinear Analysis of a 2-DOF Piecewise Linear Aeroelastic System

Aerodynamic forces for a 2-DOF aeroelastic system oscillating in pitch and plunge are modeled as a piecewise linear function. Equilibria of the piecewise linear model are obtained and their stability/bifurcations analyzed. Two of the main bifurcations are border collision and rapid/Hopf bifurcations. Continuation is used to generate the bifurcation diagrams of the system. Chaotic behavior following the intermittent route is also observed. To better understand the grazing phenomenon sets of initial conditions associated with the system behavior are defined and analyzed.Copyright