AA 4362 Astrodynamics InPlane Orbital Maneuvering with Continuous Thrust

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AA 4362 AstrodynamicsIn-Plane Orbital
Maneuvering with Continuous Thrust
Vallado Section 6.7
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Orbital Energy
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Keplers Laws
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Orbital Dynamics
Must resort to Newtons laws to describe
these orbits
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Perifocal Coordinate System
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Velocity Vector
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Acceleration Vector
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Acceleration Vector (contd)
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Newtons Second Law
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Newtons Second Law
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Resolution of Forces
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Resolution of Forces
? 0
Ignore effects of Lift/Drag
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Gravitational Forces
Ignore J2 effect
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Dynamics Equations
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Dynamics Equations (contd)
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Vehicle Mass
Initial mass of vehicle
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Collected State Equations
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Generalized Equations of Motion
J2 not included
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Vector Form of State Equations
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Integrated Equations of Motion
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Numerical Approximation ofthe Integral
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Numerical Approximation ofthe Integral (contd)
Trapezoidal rule
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Or we can use Finite Differences
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Numerical Approximation ofthe Integral (contd)
Trapezoidal rule
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Numerical Approximation ofthe Integral (contd)
trapezoidal rule
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Predictor/Corrector Algorithm
trapezoidal rule
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Predictor/Corrector AlgorithmModified for
variable Thrust
trapezoidal rule
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Higher Order Integrators
Simple Second Order predictor/corrector works
well for Small-to-moderate step sizes but at
larger step sizes can be come unstable Good
to have a higher order integration scheme in our
bag of tools 4th Order Runge-Kutta method is
one most commonly used Lots of arcane
derivations and Mystery with regard to This
method lets clear this up!!!
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4th Order Runge-Kutta Method
Lets add two more points To the curve before
summing
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4th Order Runge-Kutta Method(contd)
The basic Differential equation is
Approximate the first derivative by finite
difference
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4th Order Runge-Kutta Method(contd)
Now correct this derivative estimate with what
we have learned This is almost
equivalent to what we have already done
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4th Order Runge-Kutta Method(contd)
Repeat this process twice more to give us 4
points on the curve
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4th Order Runge-Kutta Method(contd)
Finally take a weighted average of the
results
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4th Order Runge-Kutta Method(contd)
What happens if the Input (Thrust) is not
Constant? Simply split the difference
between Fthrust k and Fthrust k1
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4th Order Runge-Kutta Method(contd)
split the difference between Fthrust k and
Fthrust k1
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Initial Conditions
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Initial Conditions (contd)
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Comparison of Runge-Kutta Integrator to
Trapezoidal Rule
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Comparison of Runge-Kutta Integrator to
Trapezoidal Rule
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Comparison of Runge-Kutta Integrator to
Trapezoidal Rule
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Comparison of Runge-Kutta Integrator to
Trapezoidal Rule
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Comparison of Runge-Kutta Integrator to
Trapezoidal Rule
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Comparison of Runge-Kutta Integrator to
Trapezoidal Rule
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Solving for the Instantaneous Orbit
Or orbit once thrust has been terminated
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Orbit Eccentricity
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Eccentricity Example
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Eccentricity Example (contd)
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Eccentricity Example (contd)
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Eccentricity Example (contd)
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Eccentricity Example (contd)
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Eccentricity Example (contd)
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Eccentricity Example (contd)
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Worked Example
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Worked Example (contd)
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Worked Example (contd)
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Worked Example (contd)
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Worked Example (contd)
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Worked Example (contd)
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Worked Example (contd)
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Worked Example (contd)
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Worked Example (contd)
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Worked Example (contd)
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Compare to Hohmann transfer using Conventional
Propulsion
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EP, in the Right Circumstances
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EP, in the Right Circumstances
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Homework 15
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Homework 15 (contd)
. Or if you are really adventurous modulate
you thrust level so that you can insert
directly Into a final circular orbit without an
apogee kick . Implement both Trapezoidal and
Runge-Kutta Integration schemes compare
algorithm performance as Time interval DT becomes
progressively larger