diff --git a/contributing/contributing.html b/contributing/contributing.html index 0b4509911..ffc6920a1 100644 --- a/contributing/contributing.html +++ b/contributing/contributing.html @@ -1055,21 +1055,21 @@

Notebook Creation#

Both sets of equations of motion have common aerodynamic models.

First, the lift coefficient is computed using a model which assumes linearity in lift wrt the angle of attack.

-
-(21)#\[\begin{align} +
+(21)#\[\begin{align} C_L &= C_{L0} + \frac{\alpha}{\alpha_{max}} \left(C_{L-max} - C_{L0}\right) \end{align}\]

Next, the drag-due-to-lift factor is computed (Equations 12.48 and 12.61 in Raymer[@raymer2012aircraft]).

-
-(22)#\[\begin{align} +
+(22)#\[\begin{align} K_{nom} &= \frac{1}{ \pi AR e} \\ b &= \frac{span}{2} \\ K &= 33 K_{nom} \frac{ \left(\frac{h + h_w}{b} \right) ^{\frac{3}{2}}}{1.0 + 33 \left( \frac{h + h_w}{b}\right) ^{\frac{3}{2}}} @@ -956,8 +956,8 @@

The Aerodynamic Model

Finally, the lift and drag are computed after computing the dynamic pressure.

-
-(23)#\[\begin{align} +
+(23)#\[\begin{align} q &= 0.5 \rho v^2 \\ L &= q S C_L \\ D &= q S \left( C_{D0} + K C_{L}^2 \right) @@ -968,8 +968,8 @@

Stall Speed\(\frac{v}{v_{stall}}\)). This constraint is used to trigger the beginning of rotation and is used as a boundary constraint at the end of the initial climb. Stall speed is given by Equation 5.6 in Raymer[@raymer2012aircraft].

-
-(24)#\[\begin{align} +
+(24)#\[\begin{align} W &= m g \\ v_{stall} &= \sqrt{\frac{2 * W}{\rho S C_{L-max}}} \end{align}\]
@@ -977,8 +977,8 @@

Stall Speed

Runway Equations of Motion#

The runway equations of motion are used to integrate range and speed as the vehicle rolls along the runway.

-
-(25)#\[\begin{align} +
+(25)#\[\begin{align} F_r &= mg - L \cos \alpha - T \sin \alpha \\ \dot{v} &= \frac{T \cos \alpha - D - F_r \mu_r}{m} \\ \dot{r} &= v @@ -1005,8 +1005,8 @@

Runway Equations of Motion

Flight Equations of Motion#

The flight equations of motion include two additional state variables: the flight-path angle (\(\gamma\)) and altitude (\(h\)).

-
-(26)#\[\begin{align} +
+(26)#\[\begin{align} \dot{v} &= \frac{T}{m} \cos \alpha - \frac{D}{m} - g \sin \gamma \\ \dot{\gamma} &= \frac{T}{m v} \sin \alpha + \frac{L}{m v} - \frac{g \cos \gamma}{v} \\ \dot{h} &= v \sin \gamma \\ @@ -1531,7 +1531,7 @@

Building and running the problem -
<dymos.phase.phase.Phase at 0x7f30432bf290>
+
<dymos.phase.phase.Phase at 0x7f27407db490>
@@ -1733,8 +1733,8 @@

Building and running the problem
Full total jacobian for problem 'problem' was computed 3 times, taking 0.9877428739999914 seconds.
+
Full total jacobian for problem 'problem' was computed 3 times, taking 1.0517292810000072 seconds.
 Total jacobian shape: (178, 166) 
 
 
@@ -1792,10 +1792,10 @@ 
Building and running the problem
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/core/total_jac.py:1646: DerivativesWarning:Constraints or objectives [('traj.phases.climb->path_constraint->gam', inds=[(0, 0)])] cannot be impacted by the design variables of the problem.
@@ -1868,11 +1868,11 @@ 
Building and running the problem
Full total jacobian for problem 'problem' was computed 3 times, taking 3.3772143690000007 seconds.
+
Full total jacobian for problem 'problem' was computed 3 times, taking 3.264605084999971 seconds.
 Total jacobian shape: (178, 166) 
 
 
@@ -1333,10 +1333,10 @@ 
Building and running the problem
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/core/total_jac.py:1646: DerivativesWarning:Constraints or objectives [('traj.phases.climb->path_constraint->gam', inds=[(0, 0)])] cannot be impacted by the design variables of the problem.
diff --git a/examples/brachistochrone/brachistochrone.html b/examples/brachistochrone/brachistochrone.html
index 3f083d9f4..c3ea4a131 100644
--- a/examples/brachistochrone/brachistochrone.html
+++ b/examples/brachistochrone/brachistochrone.html
@@ -866,8 +866,8 @@ 
State variables
 
System dynamics#

 
From the free-body diagram above, the evolution of the state variables is given by the following ordinary differential equations (ODE).

-

-(13)#\[\begin{align}
+

+(13)#\[\begin{align}
     \frac{d x}{d t} &= v \sin(\theta) \\
     \frac{d y}{d t} &= -v \cos(\theta) \\
     \frac{d v}{d t} &= g \cos(\theta)
@@ -885,8 +885,8 @@ 
The initial and final conditions
-(14)#\[\begin{align}
+

+(14)#\[\begin{align}
     x_0 &= 0 \\
     y_0 &= 10 \\
     v_0 &= 0
@@ -894,8 +894,8 @@ 
The initial and final conditions
-(15)#\[\begin{align}
+
 
Optimization terminated successfully    (Exit mode 0)
@@ -1394,7 +1394,7 @@ 
Solving the problem with Legendre-Gauss-Lobatto collocation in Dymos
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/recorders/sqlite_recorder.py:226: UserWarning:The existing case recorder file, dymos_solution.db, is being overwritten.
 

 

-
Full total jacobian for problem 'problem5' was computed 3 times, taking 0.014200435000020661 seconds.
+
Full total jacobian for problem 'problem5' was computed 3 times, taking 0.013965556000016477 seconds.
 Total jacobian shape: (12, 51) 
 
 No coloring was computed successfully.
diff --git a/examples/brachistochrone/brachistochrone_tandem_phases.html b/examples/brachistochrone/brachistochrone_tandem_phases.html
index 155f89bc3..d75404819 100644
--- a/examples/brachistochrone/brachistochrone_tandem_phases.html
+++ b/examples/brachistochrone/brachistochrone_tandem_phases.html
@@ -997,7 +997,7 @@ 
Setup and run
Model viewer data has already been recorded for Driver.
 

 

-
Full total jacobian for problem 'problem' was computed 3 times, taking 0.16316950800000996 seconds.
+
Full total jacobian for problem 'problem' was computed 3 times, taking 0.16410940899999105 seconds.
 Total jacobian shape: (223, 287) 
 

 

@@ -1006,10 +1006,10 @@ 
Setup and run
/usr/share/miniconda/envs/test/lib/python3.11/site-packages/openmdao/visualization/opt_report/opt_report.py:625: UserWarning: Attempting to set identical low and high ylims makes transformation singular; automatically expanding.
@@ -1842,10 +1842,10 @@ 
Plots#
     
   
   
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