State Space Model Example¶

The preceding theory can be tested using data taken from the Appendix of ‘Introduction to Aircraft Flight Dynamics’ [Sch98] which, helpfully, provides the stability derivatives both in reduced dimensional form, and in non-dimensional form. This will allow verification of the conversion between the two forms.

MD DC-8 Jet Transport Aircraft Data¶

The following data are reproduced from Appendix B.4[Sch98].

Four conditions are given:

Power Approach
Holding
Cruise
$V_{NE}$

Constants¶

\[\begin{split}\begin{align} S &=2600\text{ft}^2\\ b &= 142.3\text{ft}\\ \bar{c} &= 23\text{ft} \end{align}\end{split}\]

Condition and Geometric Data¶

Condition	1	2	3	4
$h$, ft	0	15,000	33,000	33,000
$M$	0.218	0.443	0.84	0.88
$V_\infty$\text{ft s}^{-1}$	243	468	825	868
$q_\infty$ \text{lb ft}^2$	70.4	164	271	300
$W$ lb	190,000	190,000	230,000	230,000
$I_{xx}$ slug-ft$^2$	3.09E+06	3.11E+06	3.77E+06	3.77E+06
$I_{yy}$ slug-ft$^2$	2.94E+06	2.94E+06	3.56E+06	3.56E+06
$I_{zz}$ slug-ft$^2$	5.58E+06	5.88E+06	7.13E+06	7.13E+06
$I_{xz}$ slug-ft$^2$	28,000	-64500	45,000	53,700
$\delta_{flap}$ deg	35	0	0	0

Longitudinal Nomalised Derivatives¶

Condition	1	2	3	4
$h$, ft	0	15,000	33,000	33,000
$M$	0.218	0.443	0.84	0.88
$X_u$	-0.0291	-0.0071	-0.014	-0.0463
$X_\alpha$	15.32	15.03	3.544	-22.36
$Z_u$	-0.251	-0.133	-0.074	0.062
$Z_\alpha$	-152.8	-354	-664.3	-746.9
$M_u$	0	0	-0.0008	-0.0025
$M_\alpha$	-2.118	-5.01	-9	-12.003
$M_\dot{\alpha}$	-0.26	-0.337	-0.42	-0.449
$M_q$	-0.792	-0.991	-0.924	-1.008
$X_{\delta_e}$	0	0	0	0
$Z_{\delta_e}$	-10.17	-23.7	-34.69	-39.06
$M_{\delta_e}$	-1.351	-3.241	-4.589	-5.12

Lateral-Directional Normalised Derivatives¶

Condition	1	2	3	4
$h$, ft	0	15,000	33,000	33,000
$M$	0.218	0.443	0.84	0.88
$Y_\beta$	-27.05	-47.19	-71.73	-81.28
$L_\beta$	-1.334	-2.684	-4.449	-5.111
$L_p$	-0.949	-1.234	-1.184	-1.299
$L_r$	0.611	0.391	0.337	0.352
$N_\beta$	0.762	1.272	2.175	2.497
$N_p$	-0.119	-0.048	-0.013	-0.008
$N_r$	-0.268	-0.253	-0.231	-0.254
$Y_{\delta_r}$	5.782	13.47	18.38	20.35
$L_{\delta_r}$	0.185	0.375	0.561	0.639
$N_{\delta_r}$	-0.389	-0.861	-1.173	-1.298
$Y_{\delta_a}$	0	0	0	0
$L_{\delta_a}$	0.725	1.622	2.12	2.329
$N_{\delta_a}$	0.05	0.036	0.052	0.062
$L_\beta^*$	-1.327	-2.711	-4.424	-5.076
$L_p^*$	-0.95	-1.233	-1.184	-1.299
$L_r^*$	0.609	0.397	0.335	0.349
$N_\beta^*$	0.756	1.302	2.148	2.459
$N_p^*$	-0.124	-0.035	-0.021	-0.017
$N_r^*$	-0.264	-0.257	-0.228	-0.251
$L_{\delta_{r}}^*$	0.181	0.393	0.547	0.621
$N_{\delta_{r}}^*$	-0.388	-0.866	-1.169	-1.294
$L_{\delta_{a}}^*$	0.726	1.622	2.12	2.33
$N_{\delta_{a}}^*$	0.053	0.018	0.065	0.08

Lateral-Directional Dimensionless Derivatives¶

Condition	1	2	3	4
$h$, ft	0	15000	33000	33000
$M$	0.218	0.443	0.84	0.88
$C_{y_\beta}$	-0.8727	-0.6532	-0.7277	-0.7449
$C_{\ell_\beta}$	-0.1582	-0.1375	-0.1673	-0.1736
$C_{\ell_\hat{p}}$	-0.385	-0.416	-0.516	-0.538
$C_{\ell_\hat{r}}$	0.248	0.132	0.147	0.146
$C_{n_\beta}$	0.1633	0.1232	0.1547	0.1604
$C_{n_\hat{p}}$	-0.087	-0.0331	-0.011	-0.006
$C_{n_\hat{r}}$	-0.196	-0.161	-0.019	-0.199
$C_{y_{\delta_r}}$	0.1865	0.1865	0.1865	0.1865
$C_{\ell_{\delta_r}}$	0.0219	0.0192	0.0211	0.0217
$C_{n_{\delta_r}}$	-0.0834	-0.0834	-0.0834	-0.0834
$C_{y_{\delta_a}}$	0	0	0	0
$C_{\ell_{\delta_a}}$	0.086	0.0831	0.0797	0.0791
$C_{n_{\delta_a}}$	0.0106	0.035	0.0037	0.004

Longitudinal Dimensionless Derivatives¶

Condition	1	2	3	4
$h$, ft	0	15000	33000	33000
$M$	0.218	0.443	0.84	0.88
$C_D$	0.1095	0.0224	0.0188	0.0276
$C_{D_\alpha}$	0.487	0.212	0.272	0.486
$C_{D_M}$	0.0202	0.0021	0.1005	0.365
$C_{L}$	1.038	0.445	0.326	0.295
$C_{L_\alpha}$	4.81	4.876	6.744	6.899
$C_{L_M}$	0.02	0.048	0	-1.2
$C_{m_\alpha}$	-1.478	-1.501	-2.017	-2.413
$C_{m_M}$	-0.006	-0.02	-0.17	-0.5
$C_{m_\dot{\alpha}}$	-3.84	-4.1	-6.62	-6.83
$C_{m_\hat{q}}$	-11.7	-12.05	-14.6	-15.2
$C_{D_{\delta_e}}$	0	0	0	0
$C_{L_{\delta_e}}$	0.328	0.328	0.352	0.358
$C_{m_{\delta_e}}$	-0.943	-0.971	-1.008	-1.016

These data are all saved in an excel file, which is available for download. They can be pulled out of the excel file using the following scripts:

## Import Pandas
import pandas as pd

## Import the Nomdimensional Longitudinal
DC8Data = 'Data/DC8Data.xlsx'
Nondim_lon_data = pd.read_excel(DC8Data, sheet_name="Longitudinal Dimensionless")
other_data = pd.read_excel(DC8Data, sheet_name="Geometric")

# These are now imported as a pandas dataframe - which measn they're in slightly an awkward format to work with
# But you *could* just hard code them all manually. This is just how I'd like to import them.

# This extracts the names of all the longitudinal nondimensional derivatives
lon_derivs_names = Nondim_lon_data["Condition"].tolist()[2:]

# Which condition? Let's look at the first condition
cond = 3

# Make an empty dictionary
DC8_lon_nondim = {}

# Now we can iterate through the list of deriatives and store each value in the dictionary
for deriv in lon_derivs_names:
    # Get the row corresponding to this derivative
    this_row = Nondim_lon_data[Nondim_lon_data["Condition"] == deriv]
    
    # Get the derivative corresponding to the condition
    this_derivative = this_row[cond]
    
    # Add it to the dictionary
    DC8_lon_nondim[deriv] = this_derivative.item()
    
# Print the dictionary to show what cool work we've done
print("The longitudinal non-dimenional deriviatives, taken from the Appendix are:", DC8_lon_nondim)

# Do the same with the dimensional deribatives
## Import the Nomdimensional Longitudinal
DC8Data = 'Data/DC8Data.xlsx'
dim_lon_data = pd.read_excel(DC8Data, sheet_name="Longitudinal")


# This extracts the names of all the longitudinal nondimensional derivatives
lon_derivs_names = dim_lon_data["Condition"].tolist()[2:]



# Make an empty dictionary
DC8_lon_dim_data = {}

# Now we can iterate through the list of deriatives and store each value in the dictionary
for deriv in lon_derivs_names:
    # Get the row corresponding to this derivative
    this_row = dim_lon_data[dim_lon_data["Condition"] == deriv]
    
    # Get the derivative corresponding to the condition
    this_derivative = this_row[cond]
    
    # Add it to the dictionary
    DC8_lon_dim_data[deriv] = this_derivative.item()
    
print("From the table, the data are:", DC8_lon_dim_data)

The longitudinal non-dimenional deriviatives, taken from the Appendix are: {'Cd': 0.0188, 'Cda': 0.272, 'Cdm': 0.1005, 'Cl': 0.326, 'Cla': 6.744, 'Clm': 0.0, 'Cma': -2.017, 'Cmm': -0.17, 'Cmda': -6.62, 'Cmq': -14.6, 'Cde': 0.0, 'Cle': 0.352, 'Cme': -1.008}
From the table, the data are: {'Xu': -0.014, 'Xalpha': 3.544, 'Zu': -0.074, 'Zalpha': -664.3, 'Mu': -0.0008, 'Malpha': -9.149, 'Malphadot': -0.42, 'Mq': -0.924, 'Xde': 0.0, 'Zde': -34.69, 'Mde': -4.589}

# Now we can convert them to the dimensional form by going through the equations in the table we produced
# We need to get out some of the constants from the table, which have some awkward names
DC8_lon_dim = {}
q = other_data[other_data["Condition"] == "Q lb/ft^2"][cond].item()
m = other_data[other_data["Condition"] == "W lb"][cond].item()/32.17 # Data is as W to need to divide by slugs
U0 = other_data[other_data["Condition"] == "V f/s"][cond].item()
M = other_data[other_data["Condition"] == "M"][cond].item()
S = 2600 # Wing area in ft^2
b = 142.3 # span in ft
c = 23 # MAC in ft

# We'll work in US customary units to confirm that the conversion between nondimensional and dimensional works
# as expected - if it *does*, then we can convert to good-old SI units

# Xu
DC8_lon_dim['Xu'] = -q * S / m / U0 * (2 * DC8_lon_nondim['Cd'] + M * DC8_lon_nondim['Cdm'])
print(f"From our conversion, Xu is {DC8_lon_dim['Xu']:1.4f}, from the table it is {DC8_lon_dim_data['Xu']:1.4f}")

# Xw
DC8_lon_dim['Xw'] = q * S / m / U0 * (DC8_lon_nondim['Cl'] - DC8_lon_nondim['Cda'])

print(DC8_lon_dim['Xw']*U0)

From our conversion, Xu is -0.0146, from the table it is -0.0140
5.3218131652173915

\[\newcommand{\od}[2]{\frac{\text{d}#1}{\text{d}#2}}\]

\[\newcommand{\pd}[2]{\frac{\partial#1}{\partial#2}}\]

\[\newcommand{\ppd}[2]{\frac{\partial^2#1}{\partial#2^2}}\]

\[\newcommand{\pppd}[2]{\frac{\partial^3#1}{\partial#2^3}}\]

Here we reach an issue - we have calculate $X_u$, but the table has presented $X_\alpha$. We can use the relationship

\[X_w\triangleq\frac{1}{m}\pd{X}{\alpha}\pd{\alpha}{w}=X_\alpha\pd{\alpha}{w}=X_\alpha\frac{1}{U_0}\]

print(f"From our conversion, Xw is {DC8_lon_dim['Xw']:1.4f}, from the table it is {DC8_lon_dim_data['Xalpha']/U0:1.4f}")

# Xq = 0 as it doesn't exist in the table
DC8_lon_dim['Xq'] = 0

# Xde
DC8_lon_dim['Xde'] = -q * S / m * DC8_lon_nondim['Cde']
print(f"From our conversion, Xde is {DC8_lon_dim['Xde']:1.4f}, from the table it is {DC8_lon_dim_data['Xde']:1.4f}")

# Zu
DC8_lon_dim['Zu'] = -q * S / m / U0 * (2*DC8_lon_nondim['Cl'] + M * DC8_lon_nondim['Clm'])
print(f"From our conversion, Zu is {DC8_lon_dim['Zu']:1.4f}, from the table it is {DC8_lon_dim_data['Zu']:1.4f}")

From our conversion, Xw is 0.0065, from the table it is 0.0043
From our conversion, Xde is -0.0000, from the table it is 0.0000
From our conversion, Zu is -0.0779, from the table it is -0.0740

# Try the corvair
S = 2000
b = 120
c = 18.94
W = 126000
q = 61
m = W / 32.17
# W = 155000
V = 227
M = 0.203

Cd = 0.154
Cdm = 0
Xu = -q*S/m/V * (2 * Cd + M * Cdm)
print(Xu)

Cl = 1.029
Cda = 0.43
Xa = q * S / m * (Cl - Cda)
print(Xa)

Clm = 0
Zu = -q * S / m / V * (2*Cl + M * Clm)
print(Zu)

Clda = 4.66
Zda = -q*S/m * (c/2/V) * Clda

-0.042263475281448845
18.658089365079366
-0.2823968575624082

Condition	1	2	3	4
\(h\), ft	0	15,000	33,000	33,000
\(M\)	0.218	0.443	0.84	0.88
\(V_\infty\)\text{ft s}^{-1}$	243	468	825	868
\(q_\infty\) \text{lb ft}^2$	70.4	164	271	300
\(W\) lb	190,000	190,000	230,000	230,000
\(I_{xx}\) slug-ft\(^2\)	3.09E+06	3.11E+06	3.77E+06	3.77E+06
\(I_{yy}\) slug-ft\(^2\)	2.94E+06	2.94E+06	3.56E+06	3.56E+06
\(I_{zz}\) slug-ft\(^2\)	5.58E+06	5.88E+06	7.13E+06	7.13E+06
\(I_{xz}\) slug-ft\(^2\)	28,000	-64500	45,000	53,700
\(\delta_{flap}\) deg	35	0	0	0

Aircraft Flight Mechanics by Harry Smith, PhD