jkirk5
commented
3 months ago Need investigation on if we would like to pursue this implementation (either picking a single version of the EOM to use, or switching between current and new implementation as needed by phase setup)
from @robfalck:
Desired capability or behavior.
Summary of Issue
The initial implementation of the Solved ODE assumes TAS as the control input. For constant EAS and constant Mach phases of flight, it would be better if it allowed Mach or EAS to be the input.
Description
Equations for Mach as the Input
With
ManddM_dras the inputs, we can recover the time derivative ofMachas$$\begin{align} \frac{dM}{dt} &= \frac{dM}{dr} V \cos \gamma & & & (1) \end{align}$$
Since the definition of Mach is
$$\begin{align} M &= \frac{V}{a} & & & (2) \end{align}$$
Then its time derivative is
$$\begin{align} \dot{M} &= \frac{\dot{V}a - \dot{a}{V}}{a^2} & & & (3) \end{align}$$
where $a$ is the local speed of sound. The time derivative of the speed of sound is:
$$\begin{align} \dot{a} &= \frac{da}{dh} \frac{dh}{dt} = \frac{da}{dh} V \sin \gamma & & & (4) \end{align}$$
Note that $\frac{da}{dh}$ comes from the atmosphere model and will be available in the next version of dymos.
Solving equation (3) for $\dot{V}$ gives the approximate rate of change of true airspeed based on the Mach interpolation polynomials:
$$\begin{align} \frac{dV}{dt}_{approx} &= \dot{M} a + \frac{\dot{a}V}{a} & & & (5) \end{align}$$
Equations for equivalent airspeed as the Input
True airspeed $V$ and equivalent airspeed $V_{eq}$ are related through the equation
$$\begin{align} V &= \frac{V_{eq}}{\sqrt{\frac{\rho}{\rho_sl}}} & & & (6) \end{align}$$
After some simplification, the approximate rate of change of true airspeed based on the EAS interpolating polynomials is
$$\begin{align} \dot{V} &= \dot{V{eq}} \left( \frac{\rho{SL}}{\rho} \right) ^{\frac{3}{2}} - \frac{V{eq} \dot{\rho}}{2} \left( \frac{\rho{SL}^{\frac{3}{2}}}{\rho^{\frac{5}{2}}} \right) & & & (7) \end{align}$$
where
$$\begin{align} \dot{\rho} &= \frac{d\rho}{dh} V \sin \gamma & & & (8) \end{align}$$
$\frac{d \rho}{dh}$ is available as an output from the 1976 standard atmosphere model in dymos.
Is your feature request related to a problem? Please describe.
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Associated Bug Report
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