Linearisationdraft

Worked Example

Data for several aircraft are presented in [Heffley:1972wb] and, honestly, this is a good exercise in showing you what a tremendous amount of being an engineer is - extracting data from old reports and doing stuff with it.

Data for Boeing 747

Data are given on Page 217 of the aforementioned report for the B-747 in power approach configuration.

Configuration Information

h=SL

V_\infty=165\text{KTAS}

S=5500\text{ft}^2

b=195.68\text{ft}

\bar{c}=27.31\text{ft}

\theta_0=0

note that the trim theta is not actually given, but assumed to be zero

Nondimensional Stability Derivatives

Longitudinal	Lateral-Directional
$C_L=1.11$	$C_{y_\beta}=-0.96\text{rad}^{-1}$
$C_D=0.102$	$C_{\ell_\beta}=-0.221\text{rad}^{-1}$
$C_{L_\alpha}=5.70\text{rad}^{-1}$	$C_{n_\beta}=0.150\text{rad}^{-1}$
$C_{D_\alpha}=0.66\text{rad}^{-1}$	$C_{\ell_{\hat{p}}}=-0.45$
$C_{m_\alpha}=-1.26\text{rad}^{-1}$	$C_{n_{\hat{p}}}=-0.121$
$C_{L_{\dot{\alpha}}}=-6.7\text{rad}^{-1}$	$C_{\ell_{\hat{r}}}=0.101$
$C_{m_{\dot{\alpha}}}=-3.2\text{rad}^{-1}$	$C_{n_{\hat{r}}}=-0.30$
$C_{L_{\hat{q}}}=5.40$	$C_{\ell_{\delta_a}}=0.0461\text{rad}^{-1}$
$C_{m_{\hat{q}}}=-20.8$	$C_{n_{\delta_a}}=0.0064\text{rad}^{-1}$
$C_{L_\text{M}}=-0.81$	$C_{y_{\delta_r}}=0.175\text{rad}^{-1}$
$C_{m_\text{M}}=0.27$	$C_{\ell_{\delta_r}}=0.007\text{rad}^{-1}$
$C_{L_{\delta_e}}=0.338\text{rad}^{-1}$	$C_{n_{\delta_r}}=-0.109\text{rad}^{-1}$
$C_{m_{\delta_e}}=-1.34\text{rad}^{-1}$

Note that in [Heffley:1972wb], the derivatives with respect to angular rates are erroneously given units of /rad, when they clearly are non-dimensional rates. The derivatives with respect to aerodynamic angles correctly are given units of /rad.

Some units have dimensions of $[\text{T}^{-1}]$ and hence have the same value in either unit system. Expand the below to see this. Other derivatives do not.

Now the system matrix can be constructed:

\begin{aligned} \begin{bmatrix} \dot{u}\\\dot{w}\\\dot{q}\\\dot{\theta}\end{bmatrix} &= \begin{bmatrix} X_u & X_w & 0 & -g\cdot\cos\Theta_e\\ Z_u & Z_w & U_0 & -g\cdot\sin\Theta_e\\ M_u^* & M_w^* & M_q^* & M_\theta^*\\ 0 & 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} {u}\\{w}\\{q}\\{\theta} \end{bmatrix} + \begin{bmatrix} 0\\Z_{\delta_e}\\M_{\delta_e}^*\\0 \end{bmatrix}\left[\delta_e\right]\end{aligned}

well…almost - you’ll see two things that need to be performed:

The starred terms need to be made from the products of the stability derivatives.
We have a $Z_q$ term from conversion of the stability derivatives, but no corresponding place to put it.

The first problem is easy. From Equations (7)

Now we’ve got all the terms, we don’t know where to put $Z_q$ . The answer should be obvious - it goes where the $U_0$ term is - it doesn’t replace it, but is added to it. So:

The same can be performed for the lateral-directional matrix:

And put into the system matrix for lateral-directional motion

\boldsymbol{A}=\begin{bmatrix}Y_v & 0 & -U_0 & g\cdot\cos\theta_0 & 0\\L_v^* & L_p^* & L_r^* & 0 & 0\\N_v^* & N_p^* & N_r^* & 0 & 0\\0 & 1 & \tan\theta_0 & 0 & 0 \\ 0 & 0 & \sec\theta_0 & 0 & 0\end{bmatrix}

Transient Longitudinal Response

Note that we’ve not made either of the control matrices yet - and there’s a good reason for that. If we consider the linearised model of the aircraft created, the stability of the aircraft (i.e., its response to a perturbation) will be governed by the system matrix.

In the first instance, it is desired to understand the stick fixed stability and understand what will happen to the aircraft if no inputs are made.

Consider what the two matrices developed actually show us; if the aircraft is disturbed from a given condition, then if the disturbance is a symmetric state variable ( $u, w, q, \theta$ ), then the output is a time rate of change of all other symmetric variables.

The control matrices tell us how the aircraft responds due to control input. Therefore, if we’re interested in aircraft response in the absense of pilot or control-system input, then we’re only interested in the A matrices themselves. The A matrices give the open loop response of the aircraft.

Since these are ODEs, we could write a means to time-march through the solution - but there are some handy inbuilt tools in Python to look at the responses of state-space systems to different inputs.

You will probably need to install these to work on your system - look here.

But if we create the $B$ matrix, it can be used to excite the aircraft with unity impulse input. For this reason, it makes sense to make the two control matrices with derivatives that have been scaled by a factor of $\frac{\pi}{180}$ to allow a unity degree control input.

Interactive demo

As much as a pain as the non-dimensional to dimensional procedure is, it is useful in that you can take fully non-dimensional data and then use whatever unit system you prefer, and you’ll get results in those units.

Note that there is a flag to convert between US Customary (ugh) and SI units in the code above. You can see that there’s some very annoying fiddly bits that have to be performed with the US Customary units to ensure consistency - that is, the mass moments of inertia are given in slugs-ft^2, but the masses etc. are presented in lbs. This can cause some dimensional moments of inertia to be about thirty-two times too large, which has a large effect on the end result.

My strong advice is if you’re required to do anything in US Customary units, to convert to SI at the start, and then convert to US Customary at the end. There’s just too many pitfalls in the US Customary system.

Response characteristics

We’ll spend the next module understanding the actual characteristics of the aircraft in terms of dynamic response, but we can start to intuit some of this now:

There are two phases to the aircraft response - a short-term response that is heavily damped, affecting mainly heave (i.e., angle of attack) and pitch rate, and a long-term response that is lightly-damped, affecting mainly speed and pitch attitude, but also showing in the other state variables.

Non-state variables can be explored - we could go back and change the state variable $w$ for $\alpha$ , for example, and then explore the change in AoA. This would require calculating a new set of stability derivatives (many sources use $\alpha$ in place of $u$ ). Instead, perturbational angle of attack can be determined from the linear relationship

You will see that the angle of attack is affected much more by the short-term response than the long-term response.

Transient Lateral Response

A similar exercise can be performed for the lateral response

Getting eigenvalues from the graph data

This bit wont make much sense until you watch the corresponding lecture…

Other Condition Data

The data as presented in the original report are only tabulated for a couple of aircraft conditions for the Boeing 747, with the remainder of the data presented in graphical form. Thankfully, someone has already done the hard work of extracting those data from the graph and presenting in tabular form (it wasn’t me, the following is taken from Caughey’s online course notes, borrowed with thanks).

The condition number confers to the Mach/SL diagram on Page 212 of the original report.

Derivative	Condition 2	Condition 5	Condition 7	Condition 9	Condition 10
$h$ /ft	0	20,000	20,000	40,000	40,000
$M$	0.25	0.50	0.80	0.80	0.90
$\alpha$ / $^\circ$	5.70	6.80	0.0	4.60	2.40
$W$ /lb	564,032	636,636	636,636	636,636	636,636
$I_{xx}$ /(slug-ft $^2$ )	14.3 x $10^6$	18.4 x $10^6$	18.2 x $10^6$	18.2 x $10^6$	18.2 x $10^6$
$I_{yy}$ /(slug-ft $^2$ )	32.3 x $10^6$	33.1 x $10^6$	33.1 x $10^6$	33.1 x $10^6$	33.1 x $10^6$
$I_{zz}$ /(slug-ft $^2$ )	45.3 x $10^6$	49.5 x $10^6$	49.7 x $10^6$	49.7 x $10^6$	49.7 x $10^6$
$I_{xz}$ /(slug-ft $^2$ )	-2.23 x $10^6$	-2.76 x $10^6$	0.97 x $10^6$	-1.56 x $10^6$	-0.35 x $10^6$
$C_L$	1.11	0.68	0.266	0.66	0.521
$C_D$	0.102	0.0393	0.0174	0.0415	0.0415
$C_{L_\alpha}$	5.7	4.67	4.24	4.92	5.57
$C_{D_\alpha}$	0.66	0.366	0.084	0.425	0.527
$C_{m_\alpha}$	-1.26	-1.146	-629	-1.033	-1.613
$C_{L_{\dot{\alpha}}}$	6.7	6.53	5.99	5.91	5.53
$C_{m_{\dot{\alpha}}}$	-3.2	-3.35	-5.4	-6.41	-8.82
$C_{L_q}$	5.4	5.13	5.01	6	6.94
$C_{m_q}$	-20.8	-20.7	-20.5	-24	-25.1
$C_{L_M}$	0	-0.0875	0.105	0.205	-0.278
$C_{D_M}$	0	0	0.008	0.0275	0.242
$C_{m_M}$	0	0.121	-0.116	0.166	-0.114
$C_{L_{\delta_e}}$	0.338	0.356	0.27	0.367	0.3
$C_{m_{\delta_e}}$	-1.34	-1.43	-1.06	-1.45	-1.2
$C_{y_\beta}$	-0.96	-0.9	-0.81	-0.88	-0.92
$C_{l_\beta}$	-0.221	-193	-0.164	-0.277	-0.095
$C_{n_\beta}$	0.15	0.147	0.179	0.195	0.207
$C_{l_p}$	-0.45	-0.323	-0.315	-0.334	-0.296
$C_{n_p}$	-0.121	-0.069	0.0028	-0.042	0.023
$C_{l_r}$	0.101	0.212	0.0979	0.3	0.193
$C_{n_r}$	-0.3	-0.278	-0.265	-0.327	-333
$C_{l_{\delta_a}}$	0.0461	0.0129	0.012	0.0137	0.0139
$C_{n_{\delta_a}}$	0.0064	0.0015	0.0008	0.0002	-0.003
$C_{y_{\delta_r}}$	0.175	0.1448	0.0841	0.1157	0.062
$C_{l_{\delta_r}}$	0.007	0.0039	0.009	0.007	0.0052
$C_{n_{\delta_r}}$	-0.109	-0.1081	-0.099	-0.1256	-0.091