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"# Stability Derivatives\n",
"\n",
"To complete the linearisation process, linear expressions for the **aerodynamic** and **propulsive** force and moment perturbations are required. The end result of this process will be the **stability derivatives** that we incepted in the static stability module, and we'll find that there are a _lot_ of them. Rather than the approach taken in the previous modules, which constrained motion to single degrees of freedom, an approach is adopted that allows representation of motion in all axes including all cross-couplings.\n",
"\n",
"It is assumed that the external forces ($X,Y,Z$) and moments ($L,M,N$) are functions of the instantaneous values of the disturbance velocities (translational and angular), control angles, and the derivatives of both. That is:\n",
"\n",
"$$\\begin{aligned}\n",
" X &= f_1\\left(u, \\dot{u}, v, \\dot{v}, w, \\dot{w}, p, \\dot{p}, q, \\dot{q}, r, \\dot{r},\\delta_a, \\dot{\\delta}_a, \\delta_e, \\dot{\\delta}_e,\\delta_r, \\dot{\\delta}_r\\right)\\\\\n",
" Y &= f_2\\left(u, \\dot{u}, v, \\dot{v}, w, \\dot{w}, p, \\dot{p}, q, \\dot{q}, r, \\dot{r},\\delta_a, \\dot{\\delta}_a, \\delta_e, \\dot{\\delta}_e,\\delta_r, \\dot{\\delta}_r\\right)\\\\\n",
" Z &= f_3\\left(u, \\dot{u}, v, \\dot{v}, w, \\dot{w}, p, \\dot{p}, q, \\dot{q}, r, \\dot{r},\\delta_a, \\dot{\\delta}_a, \\delta_e, \\dot{\\delta}_e,\\delta_r, \\dot{\\delta}_r\\right)\\\\\n",
" L &= f_4\\left(u, \\dot{u}, v, \\dot{v}, w, \\dot{w}, p, \\dot{p}, q, \\dot{q}, r, \\dot{r},\\delta_a, \\dot{\\delta}_a, \\delta_e, \\dot{\\delta}_e,\\delta_r, \\dot{\\delta}_r\\right)\\\\\n",
" M &= f_5\\left(u, \\dot{u}, v, \\dot{v}, w, \\dot{w}, p, \\dot{p}, q, \\dot{q}, r, \\dot{r},\\delta_a, \\dot{\\delta}_a, \\delta_e, \\dot{\\delta}_e,\\delta_r, \\dot{\\delta}_r\\right)\\\\\n",
" N &= f_6\\left(u, \\dot{u}, u, \\dot{w}, w, \\dot{w}, p, \\dot{p}, u, \\dot{r}, r, \\dot{r},\\delta_e, \\dot{\\delta}_e, \\delta_r, \\dot{\\delta}_r,\\delta_a, \\dot{\\delta}_a\\right)\\end{aligned}$$\n",
"\n",
"Simple expressions for $f_1$ through $f_6$ can be adopted for simple cases (I have an example of this in the previous notes but haven't converted to online yet). There are clearly more complex ways to model aircraft, so the representation above is used as a generality. The method used to linearise the external forces and moments is to represent them by a Taylor series expansion:\n",
"\n",
"```{admonition} *Hopefully* revision - Taylor Series:\n",
"\n",
"Let $f(x)$ be a function having derivatives of all orders in the interval $(a-\\delta)first, third, and fifth equations being **symmetric** (flight constrained to X-Z plane) and the second, fourth, and sixth equations being **asymmetric**."
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