Flight Mechanics

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
  
  • Steady level flight
  • Steady climb or descent
• Anti-symmetric motion
  
  • Steady turn
  • Steady yaw or sideslip

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 $z_b$ axis

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

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

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

becomes

expanding and neglecting small higher order terms gives,

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

This is called the "Trim Equation". There will be one trim equation for each of the six degrees of freedom. Equ. 44 thus reduces to

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,

Summary of Linearised Equations of Motion

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

and the equations for orientation, angular rates, become,

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.

Steady Level Flight

For this trim condition the linearised equations reduce to

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

and the lateral-directional perturbation forcing functions,

Note: The affects of control inputs have been included by the addition of first order terms for $δ_e$, a control input producing pitch disturbances, $δ_a$, a control input producing roll disturbances and $δ_r$, 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,

Longitudinal Equations of motion for Steady Level Flight

Lateral Equations of motion for Steady Level Flight