scholarly journals Optimization of Low-Thrust Limited-Power Trajectories in a Noncentral Gravity Field—Transfers between Orbits with Small Eccentricities

2009 ◽  
Vol 2009 ◽  
pp. 1-35 ◽  
Author(s):  
Sandro da Silva Fernandes
Keyword(s):  
Gravity Field ◽  
Perturbation Technique ◽  
Canonical System ◽  
State Variables ◽  
Low Thrust ◽  
Lie Series ◽  
Zonal Harmonic ◽  
First Order ◽  
Mayer Problem ◽  
Limited Power

Numerical and first-order analytical results are presented for optimal low-thrust limited-power trajectories in a gravity field that includes the second zonal harmonicJ2in the gravitational potential. Only transfers between orbits with small eccentricities are considered. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin Maximum Principle, successive canonical transformations are performed and a suitable set of orbital elements is introduced. Hori method—a perturbation technique based on Lie series—is applied in solving the canonical system of differential equations that governs the optimal trajectories. First-order analytical solutions are presented for transfers between close orbits, and a numerical solution is obtained for transfers between arbitrary orbits by solving the two-point boundary value problem described by averaged maximum Hamiltonian, expressed in nonsingular elements, through a shooting method. A comparison between analytical and numerical results is presented for some maneuvers.

scholarly journals A First-Order Analytical Theory for Optimal Low-Thrust Limited-Power Transfers between Arbitrary Elliptical Coplanar Orbits

2008 ◽  
Vol 2008 ◽  
pp. 1-30 ◽  
Author(s):  
Sandro da Silva Fernandes ◽  
Francisco das Chagas Carvalho
Keyword(s):  
Canonical Transformation ◽  
Separation Of Variables ◽  
Canonical System ◽  
State Variables ◽  
Low Thrust ◽  
Transformation Theory ◽  
First Order ◽  
Mayer Problem ◽  
Limited Power ◽  
Hori Method

A complete first-order analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory. The optimization problem is formulated as a Mayer problem of optimal control theory with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin maximum principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a Mathieu transformation. The short periodic terms are then eliminated from the maximum Hamiltonian through an infinitesimal canonical transformation built through Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton-Jacobi equation through separation of variables technique. For transfers between close orbits a simplified solution is straightforwardly derived by linearizing the new Hamiltonian and the generating function obtained through Hori method.

System Properties of One-Dimensional Distributed Systems

1977 ◽  
Vol 99 (2) ◽  
pp. 85-90 ◽  
Author(s):  
L. S. Bonderson
Keyword(s):  
Distributed Systems ◽  
Sufficient Conditions ◽  
State Variables ◽  
One Dimensional ◽  
First Order ◽  
Lumped Element ◽  
Order Form ◽  
Power Product ◽  
System Properties ◽  
Necessary And Sufficient

The system properties of passivity, losslessness, and reciprocity are defined and their necessary and sufficient conditions are derived for a class of linear one-dimensional multipower distributed systems. The utilization of power product pairs as state variables and the representation of the dynamics in first-order form allows results completely analogous to those for lumped-element systems.

Weakly Nonlinear Double-Diffusive Oscillatory Magneto-Convection Under Gravity Modulation

2020 ◽  
Vol 18 (9) ◽  
pp. 725-738
Author(s):  
Palle Kiran ◽  
S. H. Manjula
Keyword(s):  
Mass Transfer ◽  
Prandtl Number ◽  
Gravity Field ◽  
Heat And Mass Transfer ◽  
Landau Equation ◽  
Perturbation Technique ◽  
Critical Rayleigh Number ◽  
Oscillatory Mode ◽  
Stationary Mode ◽  
Double Diffusive

An imposed time-periodic gravity field effect on double-diffusive magneto-convection for oscillatory mode has been investigated. The gravity field consisting of steady and periodic modes. A layer is confined with an electrically conducting fluid with Boussines q approximation and heated from below cooled from above. While using the perturbation technique we study nonlinear double-diffusive convection just above the critical state of the onset convection. The growth rate of the disturbances is confined with a critical Rayleigh number to investigate oscillatory convection. Analysis of finite- amplitude convection has been derived through the complex Ginzburg-Landau equation (CGLE). The convective heat and mass transfer obtained through CGLE at third-order under solvability conditions. This convective amplitude is required to estimate heat and mass transfer in terms of the Nusselt and Sherwood numbers. It is found that increasing the frequency of modulation causes diminishing heat and mass transfer. The effect of Prandtl number Pr, magnetic Prandtl number Pm, and amplitude δ enhances heat/mass transfer. It is found that an oscillatory mode of convection enhances the heat and mass transfer than the stationary mode. Further, streamlines, isotherms, and isohalines have their usual nature on double-diffusive magnetoconvection.

Electro-elastic analysis of functionally graded piezoelectric variable thickness cylindrical shells using a first-order electric potential theory and perturbation technique

2020 ◽  
Vol 31 (17) ◽  
pp. 2044-2068
Author(s):  
Mohammad Parhizkar Yaghoobi ◽  
Mehdi Ghannad
Keyword(s):  
Potential Theory ◽  
Electric Potential ◽  
Perturbation Technique ◽  
Functionally Graded ◽  
Homogeneous Distribution ◽  
Matched Asymptotic Expansion ◽  
First Order ◽  
Electromechanical Behavior ◽  
Present Solution ◽  
Functionally Graded Piezoelectric

In this research, an analytical solution is presented for the functionally graded piezoelectric cylindrical variable wall thickness that is subjected to mechanical and electrical loading. The non-homogeneous distribution of materials is considered as a power function. The first-order electric potential theory, first-order shear deformation theory, and the energy method are used for extracting the system of governing equations. The solution is accomplished using the matched asymptotic expansion method of the perturbation technique. The effects of non-homogeneous properties on the electromechanical are discussed. Since the intensity of variations in the distribution of properties in functionally graded piezoelectric cylinders can be changed using non-homogeneity constant, the electromechanical behavior of the cylinder can be changed by non-homogeneity constant. By reducing the electric or displacement field in functionally graded piezoelectric cylinders, de-polarization or loss of piezoelectric properties may be averted. Results indicate that non-homogeneity constant has a significant effect on the electromechanical behavior. However, in some cases, the effects of non-homogeneity constant may be neglected. Comparing these results with those predicted by the plane elasticity theory and finite element method shows good agreement. In fact, the present solution can be considered as an objective function to optimize the properties and behavior.

A New Perturbation Method for Oscillatory Systems

2003 ◽  
Author(s):  
Ronald E. Mickens
Keyword(s):  
Harmonic Oscillator ◽  
Numerical Solutions ◽  
Perturbation Technique ◽  
Equivalent Linearization ◽  
Oscillatory Solutions ◽  
Research Grants ◽  
First Order ◽  
Oscillatory Systems ◽  
Analytical Approximations ◽  
New Perturbation

We present preliminary results on a new perturbation technique that combines the fundamental elements of the methods of equivalent linearization and first-order averaging. The importance of this procedure is derived from the fact that it can be applied to ODE’s having oscillatory solutions, but for which no harmonic oscillator limiting equation exists. The procedure is illustrated by showing how it can be used to calculate analytical approximations to two nonlinear ODE’s. Comparison with the corresponding accurate numerical solutions is also given. The work presented here was supported in part by research grants from DOE and the MBRS-SCORE Program at Clark Atlanta University.

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