Stability derivatives

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Stability derivatives are a means of linearising the equations of motion of an atmospheric flight vehicle so that conventional control engineering methods may be applied to assess their stability.

The dynamics of atmospheric flight vehicles is potentially very difficult to analyse, because the forces and moments on the vehicle are seldom simple linear functions of its states. In order to address this problem, and render the analysis of stability and the design of autopilots tractable, it is necessary to deal with linear approximations to the equations of motion. The analysis is then applied to a range of flight conditions.

A stability derivative is an incremental change in a force or moment acting on the vehicle corresponding to an incremental change in one of the states.

1 Nomenclature
2 Equations of Motion
3 Stability Derivative Contributions
4 Response
5 Comments
6 Control Derivatives
7 See also
8 References

[edit] Nomenclature

In aeronautics, the following naming convention is commonly used: The stability axis set differs between aircraft and missiles.

Aircraft analysis uses wind axes:

The x axis points along the velocity vector

The z axis lies in the plane of incidence (i.e. the plane containing the velocity vector and the longitudinal axis), perpendicular to the x axis, such that in level flight, the z axis points downwards.

The y axis completes a right-handed orthogonal set, so that in level flight, it points in the direction of the starboard wing.

Missiles and rockets use body axes:

The x axis points along the missile body

The y axis points to starboard

The z axis completes a right-handed orthogonal axis set, i.e. points downwards relative to the xy plane

The following notation is commonly used for the physical quantities involved:

Body Forces: X,Y,Z - for longitudinal, lateral and normal forces.

Moments: L,M,N - for roll,pitch and yawing moments respectively

Velocities: u,v,w - for longitudinal, lateral and normal components.

It is helpful to think of these as projections of the velocity vector on to the three body axes, rather than in terms of the translational motion of the vehicle. As the body rotates relative to the velocity vector, these components change, even when there is no net change in speed. In this context the difference between speed and velocity is central to understanding, and not mere pedantry. In general, the body longitudinal axis does not coincide with the velocity vector.

Angular velocities: p,q,r - for roll, pitch and yaw rates

[edit] Equations of Motion

The use of stability derivatives is most conveniently demonstrated with missile or rocket configurations, because these exhibit greater symmetry than aeroplanes, and the equations of motion are correspondingly simpler. If it is assumed that the vehicle is roll-controlled, the pitch and yaw motions may be treated in isolation. It is common practice to consider the yaw plane, so that only 2D motion need be considered. Furthermore, it is assumed that thrust equals drag, and the longitudinal equation of motion may be ignored.

The body is oriented at angle $ψ$ (psi) with respect to inertial axes. The body is oriented at an angle $β$ (beta) with respect to the velocity vector, so that the components of velocity in body axes are:

u = U c o s β

v = U s i n β

where U is the speed. Resolving into fixed (inertial) axes:

u f = U c o s (β) c o s (ψ) - U s i n (β) s i n (ψ) = U c o s (β + ψ)

v f = U s i n (β) c o s (ψ) + U c o s (β) s i n (ψ) = U s i n (β + ψ)

The acceleration with respect to inertial axes is:

$\frac {du_f}{dt}=\frac {dU} {dt} cos(\beta+\psi)-U\frac {d(\beta+\psi)} {dt} sin(\beta+\psi)$

$\frac{dv_f}{dt}=\frac{dU}{dt}sin(\beta+\psi)+U\frac{d(\beta+\psi)}{dt}cos(\beta+\psi)$

From Newton's Second Law, this is equal to the force acting divided by the mass. Now forces arise from the pressure distribution over the body, and hence are generated in body axes, and not in inertial axes, so the body forces must be resolved to inertial axes, as Newton's Second Law does not apply in its simplest form to an accelerating frame of reference.

Resolving the body forces:

X f = X c o s (ψ) - Y s i n (ψ)

Y f = Y c o s (ψ) + X s i n (ψ)

Newton's Second Law, assuming constant mass:

$X_f=m\frac{du_f}{dt}$

$Y_f=m\frac{dv_f}{dt}$

where m is the mass. Equating the inertial values of acceleration and force yields the equations of motion:

$X=m\frac{dU}{dt}cos(\beta)-mU\frac{d(\beta+\psi)}{dt}sin(\beta)$

$Y=m\frac{dU}{dt}sin(\beta)+mU\frac{d(\beta+\psi)}{dt}cos(\beta)$

Treating $β$ and the time derivatives as small quantities, the small perturbation equations of motion become:

$X=m\frac{dU}{dt}$

$Y=mU\frac{d(\beta+\psi)}{dt}$

The first resembles the usual expression of Newton's Second Law, whilst the second is essentially the centrifugal acceleration. The equation of motion governing the rotation of the body is derived from the time derivative of angular momentum:

$N=C\frac{d^2\psi}{dt^2}$

where C is the moment of inertia about the yaw axis. Assuming constant speed, there are only two state variables; $β$ and $\frac{d\psi}{dt}$ , which will be written more compactly as the yaw rate r. There is one force and one moment, which for a given flight condition will each be functions of $β$ , r and their time derivatives. For typical missile configurations the forces and moments depend, in the short term, on $β$ and r. The forces may be expressed in the form:

$Y=Y_0 + \frac {\partial Y}{\partial \beta} \beta +\frac {\partial Y}{\partial r}r$

where $Y 0$ is the force corresponding to the equilibrium condition (usually called the trim) whose stability is being investigated. It is common practice to employ a shorthand:

$\frac{\partial Y}{\partial \beta}=Y_\beta$

The partial derivative $\frac{\partial Y}{\partial \beta}$ and all similar terms characterising the increments in forces and moments due to increments in the state variables are called stability derivatives. Typically, $\frac{\partial Y}{\partial r}$ is insignificant for missile configurations, so the equations of motion reduce to:

$\frac{d\beta}{dt}=\frac{Y_\beta}{mU}\beta-r$

$\frac{dr}{dt}=\frac{N_\beta}{C}\beta+\frac{N_r}{C}r$

[edit] Stability Derivative Contributions

Each stability derivative is determined by the position, size, shape and orientation of the missile components. In aircraft, the directional stability determines such features as dihedral of the main planes, size of fin and area of tailplane, but the large number of important stability derivatives involved precludes a detailed discussion within this article. The missile is characterised by only three stability derivatives, and hence provides a useful introduction to the more complex aeroplane dynamics.

Consider first $Y β$ , a body at an angle of attack $β$ generates a lift force in the opposite direction to the motion of the body. For this reason $Y β$ is always negative.

At low angles of attack, the lift is generated primarily by the wings, fins and the nose region of the body. The total lift acts at a distance $x c p$ ahead of the centre of gravity (it has a negative value in the figure), this, in missile parlance, is the centre of pressure . If the lift acts ahead of the centre of gravity, the yawing moment will be negative, and will tend to increase the angle of attack, increasing both the lift and the moment further. It follows that the centre of pressure must lie aft of the centre of gravity for static stability. $x c p$ is the static margin and must be negative for static stability. Alternatively, positive angle of attack must generate positive yawing moment on a statically stable missile, i.e. $N β$ must be positive. It is common practice to design manoeuvrable missiles with near zero static margin (neutral static stability).

The need for positive $N β$ explains why arrows and darts have flights and unguided rockets have fins.

The effect of angular velocity is mainly to decrease the nose lift and increase the tail lift, both of which act in a sense to oppose the rotation. $N r$ is therefore always negative. There is a contribution from the wing, but since missiles tend to have small static margins (typically less than a calibre), this is usually small. Also the fin contribution is greater than that of the nose, so there is a net force $Y r$ , but this is usually insignificant compared with $Y β$ and is usually ignored.

[edit] Response

Manipulation of the equations of motion yields a second order homogenous linear differential equation in the angle of attack $β$ :

$\frac{d^2\beta}{dt^2}-(\frac{Y_\beta}{mU}+\frac{N_r}{C})\frac{d\beta}{dt}+(\frac{N_\beta}{C}+\frac{Y_\beta}{mU}\frac{N_r}{C})\beta=0$

The qualitative behaviour of this equation is considered in the article on directional stability. Since $Y β$ and $N r$ are both negative, the damping is positive. The stiffness does not only depend on the static stability term $N β$ , it also contains a term which effectively determines the angle of attack due to the body rotation. The distance of the centre of lift, including this term, ahead of the centre of gravity is called the manoeuvre margin. It must be negative for stability.

This damped oscillation in angle of attack and yaw rate, following a disturbance, is called the 'weathercock' mode, after the tendency of a weathercock to point into wind.

[edit] Comments

We have chosen the state variables to be the angle of attack $β$ and the yaw rate r, and have omitted the speed perturbation u, together with the associated derivatives e.g. $Y u$ . This may appear arbitrary. However, we know that the timescale of the speed variation is much greater than that of the variation in angle of attack, so that its effects are negligible as far as the directional stability of the vehicle is concerned. Similarly, the effect of roll on yawing motion is also ignored, because missiles generally have low aspect ratio configurations and the roll inertia is much less than the yaw inertia, consequently the roll loop is expected to be much faster than the yaw response, and is consequently ignored. These simplifications of the problem based on a priori knowledge, represent an engineer's approach. Mathematicians prefer to keep the problem as general as possible and only simplify it at the end of the analysis, if at all.

Aircraft dynamics is much more complex than missile dynamics, mainly because the simplifications, such as separation of fast and slow modes, and the similarity between pitch and yaw motions, are not obvious from the equations of motion, and are consequently deferred until a late stage of the analysis. Subsonic transport aircraft have high aspect ratio configurations, so that yaw and roll cannot be treated as decoupled. However, this is merely a matter of degree; the basic ideas needed to understand aircraft dynamics are covered in this simpler analysis of missile motion.

[edit] Control Derivatives

Deflection of control surfaces modifies the pressure distribution over the vehicle, and these are dealt with by including perturbations in forces and moments due to control deflection. The fin deflection is normally denoted $ζ$ (zeta). Including these terms, the equations of motion become:

$\frac{d\beta}{dt}=\frac{Y_\beta}{mU}\beta-r+\frac{Y_\zeta}{mU}\zeta$

$\frac{dr}{dt}=\frac{N_\beta}{C}\beta+\frac{N_r}{C}r+\frac{N_\zeta}{C}\zeta$

Including the control derivatives enables the response of the vehicle to be studied, and the equations of motion used to design the autopilot.

[edit] See also

[edit] References

Babister A W: Aircraft Dynamic Stability and Response. Elsever 1980, ISBN 0-08-024768-7
Friedland B: Control System Design. McGraw-Hill Book Company 1987. ISBN 0-07-100420-3

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Categories: Aeronautics | Aerodynamics