### Flight Mechanics

* General Equations
* Linearised Equations
* Longitudinal Motion
* Lateral Motion
* Aerodynamic Derivatives

### Linearised Equations of Aircraft Motion

The generalised equations of motion for a symmetric aircraft as described in the previous section provide a complete and accurate description of the motion. However due to their nonlinear and differential nature, along with the requirement to include complex vector rotations, they are largely limited to time domain numerical simulation analyses. They are frequently implemented in flight simulators and computer based mathematical models of flight trajectories.

It is often more useful to further simplify these equations to give forms that offer direct assessment of the stability of the aircraft and its response to control inputs. Specifically if the general equations are linearised about a trimmed flight condition then this produces a set of equations that can be directly solved using frequency domain methods.

### Trim Conditions

A trimmed flight condition is one in which the nett forces and moments acting on the aircraft (body axes reference frame) are zero and independent of time. This is valid for only a fixed length of time as it assumes no change in aircraft mass and thus neglects fuel burn. In this situation the rates of change of the fundamental state variables are zero. The rates of change of linear velocities and rotational velocities are zero.

Typical examples of trimmed flight conditions are:

• Symmetric motion
• Anti-symmetric motion

Note: The linearised version of the equations are valid only for small perturbations of the state variables about the trim values. This is due to the nonlinearities in the general equations which would become dominant if the perturbations are too large. The difference between the linearised solution and the full, correct solution for the aircraft will potentially diverge significantly over time as well as being inaccurate inbetween two significantly separated trim conditions. However at a trimmed condition, the general nonlinear and the linearised representations do produce identical solutions for the dynamics of the vehicle.

### Linearisation

The general equations of motion are linearised by considering small displacements relative to a trimmed flight condition. This is a "small perturbation model". The linear equations are derived by a substitution of the small perturbation quantities into the general equations and then removing higher order rate terms which can be considered negligible. Assuming trim state variables (1), then variables can be written as a trim value plus a small disturbance. For example consider the gravity component along the zb axis

$$g_z(θ,φ)=g_z\cos(θ)\cos(φ)$$

Substituting for the angle variables as trim values plus small disturbances, ie. $θ=θ_1+Δθ$ and $φ=φ_1+Δφ$ then

\cl"ma-join-align"{\table g_z(θ,φ), = g\cos(θ_1+Δθ)\cos(φ_1+Δφ); , = g(\cos(θ_1)\cos(Δθ)-\sin(θ_1)\sin(Δθ))(\cos(φ_1)\cos(Δφ)-\sin(φ_1)\sin(Δφ))}

As the perturbation angles are small, it can be assumed that $\cos(Δθ)≈1$, $\cos(Δφ)≈1$, $\sin(Δθ)≈Δθ$ and $\sin(Δφ)≈Δφ$, so that

\cl"ma-join-align"{\table g_z(θ,φ), = g(\cos(θ_1)-Δθ\sin(θ_1))(\cos(φ_1)-Δφ\sin(φ_1)); , = g\cos(θ_1)\cos(φ_1)-g\sin(θ_1)\cos(φ_1)Δθ-g\cos(θ_1)\sin(φ_1)Δφ+g\sin(θ_1)\sin(φ_1)ΔθΔφ ; , = g_z(θ_1,φ_1)-g\sin(θ_1)\cos(φ_1)Δθ-g\cos(θ_1)\sin(φ_1)Δφ}

If the small perturbation substitutions are made to the complete z-axis linear motion equation

$$m(w↖{.}-uq+vp)=mg\cos(θ)\cos(φ)+F_{Az}+F_{Tz}$$

becomes

$$\table , ,m[(w↖{.}_1+Δw↖{.})-(u_1+Δu)(q_1+Δq)+(v_1+Δv)(p_1+Δp)]; ,=,mg\cos(θ_1)\cos(φ_1)-mg\sin(θ_1)\cos(φ_1)Δθ-mg\cos(θ_1)\sin(φ_1)Δφ+F_{Az}+ΔF_{Az}+F_{Tz}+ΔF_{Tz}$$

expanding and neglecting small higher order terms gives,

\cl"ma-join-align"{\table , [m(w↖{.}_1-u_1q_1+v_1p_1)-mg\cos(θ_1)\cos(φ_1)-F_{Az}-F_{Tz}]; ,+ m(Δw↖{.}-u_1Δq-q_1Δu+v_1Δp+p_1Δv); ,= - mg\sin(θ_1)\cos(φ_1)Δθ-mg\cos(θ_1)\sin(φ_1)Δφ+ΔF_{Az}+ΔF_{Tz}}(85)

By definition, at the trim state (1), an equilibrium exists, the forces are in balance and the z-direction acceleration is zero, $w↖{.}_1=0$. Therefore

$$[m(w↖{.}_1-u_1q_1+v_1p_1)-mg\cos(θ_1)\cos(φ_1)-F_{Az}-F_{Tz}]=0$$

This is called the "Trim Equation". There will be one trim equation for each of the six degrees of freedom. Equ. 85 thus reduces to

\cl"ma-join-align"{\table , m(Δw↖{.}-u_1Δq-q_1Δu+v_1Δp+p_1Δv); ,= -mg\sin(θ_1)\cos(φ_1)Δθ-mg\cos(θ_1)\sin(φ_1)Δφ+ΔF_{Az}+ΔF_{Tz}}

In practice the Δ symbol is dropped from the velocity, angle and rate perturbation variables. So it must be remembered that in the following sections, any unsubscripted variables represent small disturbances to a trimmed state variable. Thus the z-axis motion equation for small perturbations from a trimmed condition becomes,

$$m(w↖{.}-u_1q-q_1u+v_1p+p_1v)= -mgθ\sin(θ_1)\cos(φ_1)-mgφ\cos(θ_1)\sin(φ_1)+F_{Az}+F_{Tz}$$

### Summary of Linearised Equations of Motion

The previous linearisation process can be applied to all the general equations to obtain the following,

$$m(u↖{.}-v_1r-r_1v+w_1q+q_1w)= -mgθ\cos(θ_1)+F_{Ax}+F_{Tx}$$
$$m(v↖{.}+u_1r+r_1u-w_1p-p_1w)= -mgθ\sin(θ_1)\sin(φ_1)+mgφ\cos(θ_1)\cos(φ_1)+F_{Ay}+F_{Ty}$$
$$m(w↖{.}-u_1q-q_1u+v_1p+p_1v)= -mgθ\sin(θ_1)\cos(φ_1)-mgφ\cos(θ_1)\sin(φ_1)+F_{Az}+F_{Tz}$$
$$I_{xx}p↖{.}-I_{xz}r↖{.}-I_{xz}(p_1q+q_1p)+(I_{zz}-I_{yy})(r_1q+q_1r)=M_{Ax}+M_{Tx}$$
$$I_{yy}q↖{.}+(I_{xx}-I_{zz})(p_1r+r_1p)+2I_{xz}(p_1p-r_1r)=M_{Ay}+M_{Ty}$$
$$I_{zz}r↖{.}-I_{xz}p↖{.}+(I_{yy}-I_{xx})(p_1q+q_1p)+I_{xz}(q_1r+r_1q)=M_{Az}+M_{Tz}$$

and the equations for orientation, angular rates, become,

\cl"ma-join-align"{\table φ↖{.}, =p+φ(q_1\cos(φ_1)-r_1\sin(φ_1))\tan(θ_1)+θ(q_1\sin(φ_1)+r_1\cos(φ_1))\sec^2(θ_1)+q\sin(φ_1)\tan(θ_1)+r\cos(φ_1)\tan(θ_1); θ↖{.}, =-φ(q_1\sin(φ_1)+r_1\cos(φ_1))+q\cos(φ_1)-r\sin(φ_1); ψ↖{.}, =φ(q_1\cos(φ_1)-r_1\sin(φ_1))\sec(θ_1)+θ(q_1\sin(φ_1)+r_1\cos(φ_1))\sec(θ_1)\tan(θ_1)+q\sin(φ_1)\sec(θ_1)+r\cos(φ_1)\sec(θ_1)}

Note: The trim equations are significant in determining the initial flight conditions about which the perturbations occur. This means that two trim conditions havinf different values for the initial states will produce significanlty different flight dynamics for the vehicle. Eg. Two level flight trim states could have different airspeeds (V) and angles of attack ($α≈w/V$), so there may be significant differences in the dynamic behaviour of the aircraft between the two conditions.

The force and moment terms in the above equations are also just the perturbations from the trimmed state. These will need to be defined more accurately in further sections, but it should be noted that thrust variations (force perturbations for propeller driven aircraft can be quite large. Whereas force perturbations for jet powered aircraft are usually negligible.

The trim conditions for steady level flight are as follows,

$$p_1=q_1=r_1=0$$
$$ψ↖{.}_1=θ↖{.}_1=φ↖{.}_1=0$$
$$φ_1=0$$
$$v_1=0$$
$$θ_1=α$$

For this trim condition the linearised equations reduce to

$$m(u↖{.}+w_1q)= -mgθ\cos(θ_1)+F_{Ax}+F_{Tx}$$
$$m(v↖{.}+u_1r-w_1p)= mgφ\cos(θ_1)+F_{Ay}+F_{Ty}$$
$$m(w↖{.}-u_1q)= -mgθ\sin(θ_1)+F_{Az}+F_{Tz}$$
$$I_{xx}p↖{.}-I_{xz}r↖{.}=M_{Ax}+M_{Tx}$$
$$I_{yy}q↖{.}=M_{Ay}+M_{Ty}$$
$$I_{zz}r↖{.}-I_{xz}p↖{.}=M_{Az}+M_{Tz}$$
$$φ↖{.} =p+r\tan(θ_1)$$
$$θ↖{.} =q$$
$$ψ↖{.} =r\sec(θ_1)$$

At this trim condition, perturbation variables u,w,q and θ only occur in the 1st,3rd,5th and 8th equation. Similarly perturbation variables v,r,p,φ and ψ only occur in the 2nd, 4th, 6th,7th and 9th equations. Thus if the aerodynamic and propulsive forcing functions are also assumed to follow a similar distribution, the full set of nine equations can be solved as 2 independent sets of equations. One for longitudinal motion in u,w,q and θ and another for lateral motion in v,r,p,φ and ψ

### Linearised Aerodynamic Forces

To solve the linearised perturbation equations of motion, expressions for the aerodynamic and propulsive forces are required. These forcing functions are dependent on the aircraft state variables, any pilot control inputs and random fluctuations such as air turbulence. To ensure that they are included in a linear form, an approximation of the forcing functions is used by taking a series expansion about the trim condition. As a first approximation, the small higher order rate terms are neglected, although in some situations the second order rates of some state variables may be significant. The linearised aerodynamic force and moment variables will generally be functions of the wind axes angles. Perturbation angles can be related to body axes velocity components as , $α≈w/u_1$, $α↖{.}≈w↖{.}/u_1$, $β≈v/u_1$ and $β↖{.}≈v↖{.}/u_1$.

Assuming aerodynamic forcing functions in x and z are only produced by longitudinal perturbations, then the first order expansions are

$$F_{Ax}= {∂F_{Ax}}/{∂u}u+{∂F_{Ax}}/{∂w}w + {∂F_{Ax}}/{∂w↖{.}}w↖{.}+{∂F_{Ax}}/{∂q}q+{∂F_{Ax}}/{∂δ_e}δ_e$$
$$F_{Az}= {∂F_{Az}}/{∂u}u+{∂F_{Az}}/{∂w}w + {∂F_{Az}}/{∂w↖{.}}w↖{.}+{∂F_{Az}}/{∂q}q+{∂F_{Az}}/{∂δ_e}δ_e$$
$$M_{Ay}= {∂M_{Ay}}/{∂u}u+{∂M_{Ay}}/{∂w}w + {∂M_{Ay}}/{∂w↖{.}}w↖{.}+{∂M_{Ay}}/{∂q}q+{∂M_{Ay}}/{∂δ_e}δ_e$$

and the lateral-directional perturbation forcing functions,

$$F_{Ay}= {∂F_{Ay}}/{∂v}v + {∂F_{Ay}}/{∂p}p+{∂F_{Ay}}/{∂r}r+{∂F_{Ay}}/{∂δ_a}δ_a+{∂F_{Ay}}/{∂δ_r}δ_r$$
$$M_{Ax}= {∂M_{Ax}}/{∂v}v + {∂M_{Ax}}/{∂p}p+{∂M_{Ax}}/{∂r}r+{∂M_{Ax}}/{∂δ_a}δ_a+{∂M_{Ax}}/{∂δ_r}δ_r$$
$$M_{Az}= {∂M_{Az}}/{∂v}v + {∂M_{Az}}/{∂p}p+{∂M_{Az}}/{∂r}r+{∂M_{Az}}/{∂δ_a}δ_a+{∂M_{Az}}/{∂δ_r}δ_r$$

Note:The affects of control inputs have been included by the addition of first order terms for δe, a control input producing pitch disturbances, δa, a control input producing roll disturbances and δr, a control input producing yaw disturbances. Also to simplify the notation, any propulsive force perturbations will be assumed to be included in the above derivatives and moment functions will be simplified using the notation,

$$M_{Ax}+M_{Tx}=L\text" , "M_{Ay}+M_{Ty}=M\text" , "M_{Az}+M_{Tz}=N$$

### Longitudinal Equations of motion for Steady Level Flight

In summary the longitudinal small perturbation equations of motion for the steady level flight trimmed condition can written as shown below. The translation eqations can be divided by the aircraft mass, m, and the aerodynamic derivatives can be replaced by mass or inertia specific dimensional derivatives using simplifying notation, ie. ${{∂F_{Ax}}/{∂u}}/m ≡ X_u$. The moment equation can be divided by the mass moment of inertia about the y-axis Iyy, ie. ${{∂M_{Ay}}/{∂u}}/{I_{yy}}={{∂M}/{∂u}}/{I_{yy}}≡M_u$. The terms $X_u$, $X_w$,....$M_u$, $M_w$, etc. are call the dimensional aerodynamic derivatives for longitudinal motion. The assumption is made that the steady level flight angle of attack α is small so that $\cos(θ_1)≈1$ then the small perturbation longitudinal motion equations become,

$$u↖{.}+w_1θ↖{.}=-gθ+X_u u+X_w w+X_{w↖{.}}w↖{.}+X_q θ↖{.}+X_{δ_e} δ_e$$
$$w↖{.}-u_1θ↖{.}=-gθ\sin(θ_1)+Z_u u+Z_w w+Z_{w↖{.}}w↖{.}+Z_q θ↖{.}+Z_{δ_e} δ_e$$
$$θ↖{..}=M_u u+M_w w+M_{w↖{.}}w↖{.}+M_q θ↖{.}+M_{δ_e} δ_e$$

### Lateral Equations of motion for Steady Level Flight

In summary the lateral small perturbation equations of motion for the steady level flight trimmed condition can written as shown below. The translation eqation can be divided by the aircraft mass, m, and the aerodynamic derivatives can be replaced by mass or inertial specific dimensional derivatives using simplifying notation, ie. ${{∂F_{Ay}}/{∂p}}/m ≡ Y_p$. The moment equations can be divided by the mass moment of inertia about the appropriate axis Ixx or Izz respectively, ie. ${{∂M_{Ax}}/{∂p}}/{I_{xx}}={{∂L}/{∂p}}/{I_{xx}}≡L_p$ and ${{∂M_{Az}}/{∂p}}/{I_{zz}}={{∂N}/{∂p}}/{I_{zz}}≡N_p$. Similar to the above definitions, the terms $Y_p$,$L_p$,$N_p$, etc. are called the dimensional aerodynamic derivatives for lateral motion. The assumption is made that the steady level flight angle of attack α is small so that $\cos(θ_1)≈1$ then the small perturbation lateral motion equations become,

$$v↖{.}+u_1r-w_1p=-gφ+Y_v v+Y_p p + Y_r r + Y_{δ_a} δ_a + Y_{δ_r} δ_r$$
$$p↖{.}-{I_{xz}}/{I_{xx}} r↖{.}=L_v v + L_p p+L_r r+L_{δ_a} δ_a + L_{δ_r} δ_r$$
$$r↖{.}-{I_{xz}}/{I_{zz}} p↖{.}=N_v v + N_p p+N_r r+N_{δ_a} δ_a + N_{δ_r} δ_r$$