Journal Papers by Tarek A. Elgohary
Elgohary, Tarek A., Dong, Leiting, Junkins, John L. and Atluri, Satya N. "A Simple, Fast, and Accurate Time-Integrator for Nonlinear Dynamical Systems.", CMES: Computer Modeling in Engineering & Sciences, vol. 100, no. 3, pp. 249 - 275, 2014. In this study, we consider Initial Value Problems (IVPs) for strongly nonlinear dynamical systems... more In this study, we consider Initial Value Problems (IVPs) for strongly nonlinear dynamical systems, and study numerical methods to analyze short as well as long-term responses. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables of positions as well as velocities. For each discrete-time interval Radial Basis Functions (RBFs) are as- sumed as trial functions for the mixed variables in the time domain. A simple col- location method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. Three numerical exam- ples are provided to compare the present algorithm with explicit as well implicit methods in terms of accuracy, required size of time-interval (or step) and compu- tational cost. The present algorithm is compared against, the second order central difference method, the classical Runge-Kutta method, the adaptive Runge-Kutta- Fehlberg method, the Newmark-b and the Hilber-Hughes-Taylor methods. First the highly nonlinear Duffing oscillator is analyzed and the solutions obtained from all algorithms are compared against the analytical solution for free oscillation at long times. A Duffing oscillator with impact forcing function is next solved. So- lutions are compared against numerical solutions from state of the art ODE45 nu- merical integrator for long times. Finally, a nonlinear 3-DOF system is presented and results from all algorithms are compared against ODE45. It is shown that the present RBF-Coll algorithm is very simple, efficient and very accurate in obtaining the solution for the nonlinear IVP. Since other presented methods require a much smaller step size and higher computational cost, the proposed algorithm is advanta- geous and has promising applications in solving nonlinear dynamical systems. The extension of the present algorithm to orbit propagation problems with perturba- tions, will be pursued in our future studies. Issues of numerical stability for various time-integrators will also be explored in future studies.
Elgohary, Tarek A., Dong, Leiting, Junkins, John L. and Atluri, Satya N. (2014), "Time Domain Inverse Problems in Nonlinear Systems Using Collocation and Radial Basis Functions ", CMES: Computer Modeling in Engineering & Sciences, Vol. 100, No. 1, pp. 59-84 In this study, we consider ill-posed time-domain inverse problems for dynamical systems with vari... more In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. The duffing optimal control problem with various prescribed initial and final conditions, as well as the orbital transfer Lambert's problem are solved by the proposed RBF collocation method as examples. It is shown that this method is very simple, efficient and very accurate in obtaining the solutions, with an arbitrary solution as the initial guess. Since methods such as the Shooting Method and the Pseudo-spectral Method can be unstable and require an accurate initial guess, the proposed method is advantageous and has promising applications in optimal control and celestial mechanics. The extension of the present study to other optimal control problems, and other orbital transfer problems with perturbations, will be pursued in our future studies.
Elgohary, Tarek A., Dong, Leiting, Junkins, John L. and Atluri, Satya N. (2014), "Solution of Post-Buckling & Limit Load Problems, Without Inverting the Tangent Stiffness Matrix & Without Using Arc-Length Methods", CMES: Computer Modeling in Engineering & Sciences, Vol. 98, No. 6, pp. 543-563. In this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit... more In this study, the Scalar Homotopy Methods are applied to the solution of post-buckling and limit load problems of solids and structures, as exemplified by simple plane elastic frames, considering only geometrical nonlinearities. Explicitly derived tangent stiffness matrices and nodal forces of large-deformation planar beam elements, with two translational and one rotational degrees of freedom at each node, are adopted following the work of ]. By using the Scalar Homotopy Methods, the displacements of the equilibrium state are iteratively solved for, without inverting the Jacobian (tangent stiffness) matrix. It is well-known that, the simple Newton's method (and the Newton-Raphson iteration method that is widely used in nonlinear structural mechanics), which necessitates the inversion of the Jacobian matrix, fails to pass the limit load as the Jacobian matrix becomes singular. Although the so called arc-length method can resolve this problem by limiting both the incremental displacements and forces, it is quite complex for implementation. Moreover, inverting the Jacobian matrix generally consumes the majority of the computational burden especially for large-scale problems. On the contrary, by using the presently developed Scalar Homotopy Methods, convergence near limit loads, and in the post-buckling region, can be easily achieved, without inverting the tangent stiffness matrix and without using complex arc-length methods. The present paper thus opens a promising path for conducting post-buckling and limit-load analyses of nonlinear structures. While the simple Williams' toggle is considered as an illustrative example in this paper, extension 1 Department of Aerospace Engineering, Texas A&M University, College Station, TX. Student Fellow, Texas A&M Institute for Advanced Study.
The mathematical model for a flexible spacecraft that is rotating about a single axis rotation is... more The mathematical model for a flexible spacecraft that is rotating about a single axis rotation is described by coupled rigid and flexible body degrees-of-freedom, where the equations of motion are modeled by integro-partial differential equations. Beam-like structures are often useful for analyzing boom-like flexible appendages. The equations of motion are analyzed by introducing generalized Fourier series that transform the governing equations into a system of ordinary differential equations. Though technically straightforward, two problems arise with this approach: (1) the model is frequency-truncated because a finite number of series terms are retained in the model, and (2) computationally intense matrix-valued transfer function calculations are required for understanding the frequency domain behavior of the system. Both of these problems are resolved by: (1) computing the Laplace transform of the governing integro-partial differential equation of motion; and (2) introducing a generalized state space (consisting of the deformational coordinate and three spatial partial derivatives, as well as single and double spatial integrals of the deformational coordinate). The resulting math model is cast in the form of a linear state-space differential equation that is solved in terms of a matrix exponential and convolution integral. The structural boundary conditions defined by Hamilton's principle are enforced on the closed-form solution for the generalized state space. The generalized state space model is then manipulated to provide analytic scalar transfer function models for original integro-partial differential system dynamics. Symbolic methods are used to obtain closed-form eigen decomposition-based solutions for the matrix exponential/convolution integral algorithm. Numerical results are presented that compare the classical series based approach with the generalized state space approach for computing representative spacecraft transfer function models.
A singularity-free perturbation solution is presented for inverting the Cartesian to Geodetic tra... more 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.
In Book by Tarek A. Elgohary
Turner James D., Elgohary Tarek A., Majji, Manoranjan and Junkins, John L. (2012), "High Accuracy Trajectory and Uncertainty Propagation Algorithm for Long-Term Asteroid Motion Prediction", Adventures on the Interface of Mechanics and Control, K. T. Alfriend, M. Akella, J. E. Hurtado, J. Juang an...
Conference Papers by Tarek A. Elgohary
Several analytical and numerical methods exist to solve the orbit propagation of the two-body pro... more Several analytical and numerical methods exist to solve the orbit propagation of the two-body problem. Analytic solutions are mainly implemented for the un- perturbed/classical two-body problem. Numerical methods can handle both the unperturbed and the perturbed two-body problem. The literature is rich with nu- merical methods addressing orbit propagation problems such as, Gauss-Jackson, Higher order adaptive Runge-Kutta and Talyor series based methods. More re- cently, iterative methods have been introduced for orbit propagation based on the Chebyshev-Picard methods. In this work, Radial Basis Functions, RBFs, are used with time collocation to introduce a fast, accurate integrator that can readily handle orbit propagation problems. Optimizing the shape parameter of the RBFs is also introduced for more accurate results. The algorithm is also applied to Lmabert’s problem. Two types of orbits for the unperturbed two-body problem are presented; (1) a Low Earth Orbit (LEO) and (2) a High Eccentricity Orbit (HEO). The initial conditions for each orbit are numerically integrated for 5, 10 and 20 full orbits and the results are compared against the Lagrange/Gibbs F&G analytic solution, Mat- lab ode45 and the higher order rkn12(10). An Lambert’s orbit transfer numerical example is also introduced and the results are compared against the F&G solution. The algorithm is shown to be capable of taking large time steps while maintaining high accuracy which is very significant in long-term orbit propagation problems.
A hybrid system consisting of a rotating rigid hub and a flexible appendage following the Timoshe... more 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.
Modeling a flexible rotating spacecraft as a distributed parameters system of a rigid hub attache... more 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.
Several methods exist for integrating the Keplerian Motion of two gravitationally interacting bod... more Several methods exist for integrating the Keplerian Motion of two gravitationally interacting bodies, even when gravitational perturbation terms are included. The challenge is that the equations of motion become very stiff when the perturbation terms are included, which forces the use of small time steps, higher-order methods, or extended precision calculations. Recently, Turner and Elgohary have shown that by introducing two scalar Lagrange-like invariants that it is possible to integrate the two-body and two-body plus J2 perturbation term using a recursive formulation for developing an analytic continuation-based power series that overcomes the limitations of standard integration methods. Numerical comparisons with RK12(10), and other state of the art integration methods indicate performance improvements of~70X, while main-taining~mm accuracy for the orbit predictions. Extensions for J3 through J6 are currently under development. With accurate trajectories available, the next important theoretical development becomes extending the series-based solution for the state transition matrices (STM) for both the two-body and two-body plus J2 perturbation. STMs are useful for many celestial mechanics optimization calculations. Second and third order STM models are developed to support uncertainty propagation investigations. The application of scalar Lagrange-like invariants generates highly efficient state trajectory, STM, and higher-order STMs models. The proposed mathematical models are expected to be broadly useful for celestial mechanic applications for optimization, uncertainty propagation, and nonlinear estimation theory.
A natural geometric perturbation variable is identified as the ratio of the major and minor Earth... more A natural geometric perturbation variable is identified as the ratio of the major and minor Earth ellipse radii minus one. A singularity-free perturbation solution is presented for inverting the Cartesian to Geodetic transformation, which yields millimeter accuracy throughout the LEO through GEO range of satellite applications. Geocentric latitude is used to model the satellite ground track position vector. Rapidly converging perturbation solutions are developed for the satellite height above the Earth and the geocentric latitude as a perturbation power series in the geometric perturbation variable. Very compact series coefficients are recovered for the fourth order series approximations. The perturbation solution algorithm presented in this work provide three significant benefits over existing approaches for the problem: (1) No highly sensitive quartic polynomial solution algorithms are required; (2) A non-iterative algorithm inverts the transformation without requiring special starting guesses for the power series solution; and (3) Uniform solution accuracy is obtained for the Equator and the Polar regions. Simulation results are presented that compare the solution accuracy and algorithm performance for applications spanning the LEOto-GEO range of missions.
A flexible rotating spacecraft is modeled as a three body hybrid system consisting of a rigid hub... more 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 di↵erence 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.
Aerodynamic forces for a 2-DOF aeroelastic system oscillating in pitch and plunge are modeled as ... more 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. ρ Air density L Aerodynamic lift M Aerodynamic moment U Freestream velocity 1
The mathematical model for a flexible spacecraft that is rotating about a single axis rotation is... more The mathematical model for a flexible spacecraft that is rotating about a single axis rotation is described by coupled rigid and flexible body degrees-of-freedom, where the equations of motion are modeled by integro-partial differential equations. Beam-like structures are often useful for analyzing boom-like flexible appendages. The equations of motion are analyzed by introducing generalized Fourier series for the deformational coordinate that transforms the governing equations into a system of ordinary differential equations. Though technically straightforward, two problems arise with this approach: (1) the model is frequency-truncated because a finite number of series terms are retained in the model, and (2) computationally intense matrix-valued transfer function calculations are required for understanding the frequency domain behavior of the system. Both of these problems are resolved by: (1) computing the Laplace transform of the governing integro-partial differential equation of motion; and (2) introducing a generalized state space (consisting of the deformational coordinate and three spatial partial derivatives, as well as single and double spatial integrals of the deformational coordinate). The Laplace domain equation of motion is cast in the form of a linear state-space differential equation that is solved in terms of a matrix exponential and convolution integral. The structural boundary conditions defined by Hamilton's principle are enforced on the closed-form solution for the generalized state space. The generalized state space model is then manipulated to provide analytic scalar transfer function models for the original integro-partial differential system dynamics. Symbolic methods are used to obtain closed-form eigen decomposition-based solutions for the matrix exponential/convolution integral algorithm. Numerical results are presented that compare the classical series based approach with the generalized state space approach for computing representative spacecraft transfer function models. * Research Professor,
Master Thesis by Tarek A. Elgohary
The nonlinear dynamic analysis of aeroelastic systems is a topic that has been covered extensivel... more The nonlinear dynamic analysis of aeroelastic systems is a topic that has been covered extensively in the literature. The two main sources of nonlinearities in such systems, structural and aerodynamic nonlinearities, have analyzed numerically, analytically and experimentally. In this research project, the aerodynamic nonlinearity arising from the stall behavior of an airfoil is analyzed. Experimental data was used to fit a piecewise linear curve to describe the lift versus angle of attack behavior for a NACA 0012 2 DOF airfoil. The piecewise linear system equilibrium points are found and their stability analyzed. Bifurcations of the equilibrium points are analyzed and applying continuation software the bifurcation diagrams of the system are shown. Border collision and rapid/Hopf bifurcations are the two main bifurcations of the system equilibrium points. Chaotic behavior represented in the intermittent route to chaos was also observed and shown as part of the system dynamic analysis. Finally, sets of initial conditions associated with the system behavior are defined. Numerical simulations are used to show those sets, their subsets and their behavior with respect to the system dynamics. Poincaré sections are produced for both the periodic and the chaotic solutions of the system. The proposed piecewise linear model introduced some interesting dynamics for
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Journal Papers by Tarek A. Elgohary
In Book by Tarek A. Elgohary
Conference Papers by Tarek A. Elgohary
Master Thesis by Tarek A. Elgohary