Add ability to control the solved ODE with Mach or EAS

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 M and dM_dr as the inputs, we can recover the time derivative of Mach as

$$\begin{align}
\frac{dM}{dt} &= \frac{dM}{dr} V \cos \gamma & & & (1)
\end{align}$$

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.

No response

Associated Bug Report

No response

Start by locating the Solved ODE implementation and the 1976 standard atmosphere model in Aviary, then trace how TAS and its derivatives are currently supplied. Compare those entry points with the Mach and EAS equations in the issue; done means the solved ODE can use Mach or EAS as the control input, with coverage for the new behavior.