Non-linear solution to the maximum height orbit transfer guidance problem

2012 ◽  
Vol 226 (6) ◽  
pp. 620-630 ◽  
Author(s):  
H H Afshari ◽  
E Taheri
Keyword(s):  
Linear Problem ◽  
Perturbation Technique ◽  
Variable Mass ◽  
Pseudospectral Method ◽  
Performance Measure ◽  
Maximum Height ◽  
Orbit Transfer ◽  
Linear Solution ◽  
Two Body Problem ◽  
Non Linear

An optimal control solution to the highly non-linear problem of orbit transfer mission is achieved by using a newly proposed analytical perturbation technique. The problem is classified as a two-point boundary value problem in order to optimize a performance measure in a given time. Assuming a constant thrust operating in a given length of time, it is sought to find the thrust direction history of a transfer from a given initial orbit to the largest possible orbit. The system dynamical model is stated by regarding a variable mass spacecraft moving in the variable gravitational field of the Earth, based on the two-body problem. To assess the perturbation solution fidelity, a numerical solution based on the Gauss pseudospectral method has been employed. The main novelty of this work is in applying a new analytical solution strategy that is a combination of perturbation technique and backward integration to a highly non-linear problem in the calculus of variations approach.

scholarly journals Low order model for the dynamics of bi-stable composite plates

2011 ◽  
Vol 22 (17) ◽  
pp. 2025-2043 ◽  
Author(s):  
A.F. Arrieta ◽  
G. Spelsberg-Korspeter ◽  
P. Hagedorn ◽  
S.A. Neild ◽  
D.J. Wagg
Keyword(s):  
Dynamic Response ◽  
Linear Problem ◽  
Ritz Method ◽  
Stable State ◽  
Composite Plates ◽  
Mode Shapes ◽  
Order Model ◽  
Linear Solution ◽  
Non Linear ◽  
Low Order

This article presents the derivation and validation of a low order model for the non-linear dynamics of cross-ply bi-stable composite plates focusing on the response of one stable state. The Rayleigh–Ritz method is used to solve the associated linear problem to obtain valuable theoretical insight into how to formulate an approximate non-linear dynamic model. This allows us to follow a Galerkin approach projecting the solution of the non-linear problem onto the mode shapes of the linear problem. The order of the non-linear model is reduced using theoretical results from the linear solution yielding a low order model. The dynamic response of a bi-stable plate specimen is studied to simplify the model further by only keeping the non-linear terms leading to observed oscillations. Simulations for the dynamic response using the derived model are presented showing excellent agreement with the experimentally observed behaviour. Additionally, deflection shapes are measured and compared with the calculated mode shapes, showing good agreement.

Non-linear wave propagation in a relaxing gas

1969 ◽  
Vol 39 (2) ◽  
pp. 329-345 ◽  
Author(s):  
H. Ockendon ◽  
D. A. Spence
Keyword(s):  
Numerical Solutions ◽  
Perturbation Technique ◽  
Finite Amplitude ◽  
Linear Wave ◽  
Singular Perturbation Technique ◽  
Linear Solution ◽  
Non Linear ◽  
Linear Wave Propagation ◽  
Two Temperatures ◽  
Propagation Of Waves

We consider the propagation of waves of small finite amplitude ε in a gas whose internal energy is characterized by two temperatures T (translational) and Ti (internal) in the form e = CvfT + CvfTi, and Ti is governed by a rate equation dTi/dt = (T − Ti)/τ. By means of approximations appropriate for a wave advancing into an undisturbed region x > 0, we show that to order εδ, the equation satisfied by velocity takes the non-linear form \[ \bigg(\tau\frac{\partial}{\partial t}+1\bigg)\bigg\{\frac{\partial u}{\partial t}+\bigg(a_1+\frac{\gamma + 1}{2}u\bigg)\frac{\partial u}{\partial x}-{\textstyle\frac{1}{2}}\lambda\frac{\partial^2u}{\partial x^2}\bigg\}=(a_1-a_0)\frac{\partial u}{\partial x}, \] where a1, a0 are the frozen and equilibrium speeds of sound in the undisturbed region, δ = ½(1 − (a20/a21)), and λ is the diffusivity of sound due to viscosity and heat conduction (λ may be neglected except when discussing the fine structure of a discontinuity). Some numerical solutions of this model equation are given.When ε is small compared with δ, it is also possible to construct a solution for the flow produced by a piston moving with a constant velocity by means of a sequence of matched asymptotic expansions. The limit reached for large times for either compressive or expansive pistons is the expected non-linear solution of the exact equations. For a certain range of advancing piston speeds, this is a fully dispersed wave with velocity U in the range a0 < U < a1. If U > a1 the solution is discontinuous, and indeterminate in the absence of viscosity; a singular perturbation technique based on λ is then used to determine the structure of the wave head.

Deformation of Rectangular Slender Webplates with Boundary Members Flexible in the Webplate Plane

1966 ◽  
Vol 17 (4) ◽  
pp. 371-394 ◽  
Author(s):  
J. Djubek
Keyword(s):  
Linear Problem ◽  
Numerical Results ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Non Linear

SummaryThe paper presents a solution of the non-linear problem of the deformation of slender rectangular plates which are stiffened along their edges by elastically compressible stiffeners flexible in the plane of the plate. The webplate is assumed to be simply-supported along its contour. Numerical results showing the effect of flexural and normal rigidity of stiffeners are given for a square webplate loaded by shear and compression.

scholarly journals Thermodynamics of irreversible plant cell growth

2011 ◽  
Vol 75 (3) ◽  
pp. 183-190 ◽  
Author(s):  
Mariusz Pietruszka ◽  
Sylwia Lewicka ◽  
Krystyna Pazurkiewicz-Kocot
Keyword(s):  
Cell Wall ◽  
Cell Growth ◽  
Water Absorption ◽  
Abiotic Factors ◽  
Plant Cells ◽  
Wall Stress ◽  
Growth Equation ◽  
Experimental Conditions ◽  
Linear Solution ◽  
Non Linear

The time-irreversible cell enlargement of plant cells at a constant temperature results from two independent physical processes, e.g. water absorption and cell wall yielding. In such a model cell growth starts with reduction in wall stress because of irreversible extension of the wall. The water absorption and physical expansion are spontaneous consequences of this initial modification of the cell wall (the juvenile cell vacuolate, takes up water and expands). In this model the irreversible aspect of growth arises from the extension of the cell wall. Such theory expressed quantitatively by time-dependent growth equation was elaborated by Lockhart in the 60's.The growth equation omit however a very important factor, namely the environmental temperature at which the plant cells grow. In this paper we put forward a simple phenomenological model which introduces into the growth equation the notion of temperature. Moreover, we introduce into the modified growth equation the possible influence of external growth stimulator or inhibitor (phytohormones or abiotic factors). In the presence of such external perturbations two possible theoretical solutions have been found: the linear reaction to the application of growth hormones/abiotic factors and the non-linear one. Both solutions reflect and predict two different experimental conditions, respectively (growth at constant or increasing concentration of stimulator/inhibitor). The non-linear solution reflects a common situation interesting from an environmental pollution point of view e.g. the influence of increasing (with time) concentration of toxins on plant growth. Having obtained temperature modified growth equations we can draw further qualitative and, especially, quantitative conclusions about the mechanical properties of the cell wall itself. This also concerns a new and interesting result obtained in our model: We have calculated the magnitude of the cell wall yielding coefficient (T) [m<sup>3</sup> J<sup>-1</sup>•s<sup>-1</sup>] in function of temperature which has acquired reasonable numerical value throughout.

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