Components of acceleration

It is necessary to define the resultant acceleration at any point within the rigid aircraft in terms of the general motion relative to the body axis system. The aircraft, and therefore the body axes are assumed to be moving relative to the initial reference axes system. for example the earth axis system see Chapter 1, Section An arbitrary reference point within the body has co-ordinates y, relative to the centre of gravity, 0. The local components of the velocity and the acceleration of the reference point are v, and w and and respectively relative to the three axis directions.

Note that is used here rather than to distinguish it from the normal acceleration factor referred to in Chapter 3, Section 3.2.3, which is of opposite sign. As the body is rigid

loading a n d structural layout

there is no motion of the reference point relative to the origin, The resultant absolute velocities of the reference point as a consequence of the angular motion are:

Each term on the right-hand side represents the linear velocity at the reference point due to its displacement from the axis system. The absolute velocities of the reference point are obtained by adding the overall velocities of the aircraft:

The acceleration components of the reference point each consist of three terms. The first is the linear acceleration along a given reference direction. The other two terms a consequence of the centrifugal acceleration components of the angular velocities at the reference point. Thus. for example, in the plane the angular velocity about they axis is q and at the reference point it has a component in the 0: direction. The corresponding centrifugal acceleration is in the direction. From in the plane when is zero, w' is also equal to Substituting to eliminate from gives acceleration component due to parallel to the axis Ox as

In general:

o' - pw'

+

w -

Substitution of Eqns into Eqns gives:

Differentiation of Eqns gives, with respect to time:

airframe dynamics

Using Eqns and (4.42):

Note that if the disturbance from the initial condition is considered for a point on the Ox axis, so that y and are zero, the acceleration components reduce to:

Am -

+

w +

^-

Thus for a point on the Ox axis the lateral acceleration due to the angular velocities is a function of

+

pq) while the normal acceleration is a function of - pr). These terms are the effective yawing and pitching accelerations referred to in Chapter Figs 3.6 and 3.4, respectively. When the motion is solely in the vertical plane the pitching acceleration is simply and when it is solely in the horizontal plane the yawing acceleration is simply

+

Generalized

force

and moment equations Forces

Since the origin used to define the acceleration components given in ( 4 . 4 3 ) is the centre of gravity of the aircraft, the components of force acting on the aircraft as a whole may be written as:

where is the mass of the aircraft.

Aircrafl loading a n d structural layout

Moments

The moments actingon the aircraft are balanced by the summation of the products of the local masses and accelerations as follows:

About the axis rolling moment L is balanced by:

-

About t h e y axis the pitching moment M is balanced by:

About the axis the yawing N is balanced by:

-

Am is a local item of mass located at and subjected to accelerations m. and Only angular effects are present and, from Eqn. (4.43). for the pitching moment equation:

Expanding and collecting like terms:

By definition the moments and products of inertia are:

Moment of about axis: I ,

+

Moment of inertia about axis: I,

+

Moment of inertia about axis:

Product of inertia in the plane:

Product of inertia in the plane: I, Product of inertia in the plane:

Using these definitions, Eqn. ( 4 . 4 7 ) becomes:

airframe dynamics

Similarly:

The products of inertia and are theoretically zero for an aircraft having conventional symmetry about the plane. Hence the moment equations reduce to:

Initial steady trimmed conditions

In the initial steady trimmed condition the velocity components along the body axes O x and are and respectively, while that along the axis, is zero. The body O x axis is inclined to the earth reference at a pitch angle of but the initial roll and yaw angles and both zero.

The force and moment balance in the trimmed case is in Section

Disturbed forces and moments 4.4.5.1 Introduction

When the motion of the aircraft is disturbed from the trimmed condition the velocity components along the O x , O y and axes, respectively, are:

=

+

W =

+

^w

and the total angles, denoted by the suffix relative to the same axes are:

The rates of change of the velocity components with time are:

loading and structural layout

Substituting Eqns (4.51) into the force equations (4.45) yields:

Disturbance forces and moments

applied, or disturbance, forces and moments may arise from or more of five causes:

(a) Gravity forces due to the movement of the hody axes relative to the initial reference. earth, axes and are denoted by the suffix There are no gravity disturbance moments when the origin of the axis system is the centre of gravity of the aircraft.

(h) Aerodynamic forces and moments on the airframe due to the disturbance of the aircraft from the initial trimmed condition, denoted by the suffix a.

Pilot, autopilot, inputs to the control denoted by the suffix, c . Atmospheric disturbances in the form of gusting or continuous turhulcnce, denoted by the suffix ad.

Powerplant effects which arise as a consequence changes in engine thrust as occurs. for example, when an engine fails. denoted by the suffix pp.

Thus for example:

4.4.5.3 Gravitational effects

The gravitational effects are sometimes included with the inertial terms on the right- hand side of Eqns (4.45). Since in the trimmed condition the axis Ox is inclined at an angle to the earth reference axis

When the aircraft is disturbed through the angles and roll, pitch, and yaw, respectively, the use of the axis transformation equation [Chapter gives:

Rigid airframe dynamics

Airframe aerodynamic effects

The aerodynamic force and moment contributions to the disturbed motion equations are a consequence of the linear and angular velocities and accelerations imposed upon the airframe. Thus, in addition to a possible constant tenn, there are potentially twelve contributions to each of the force and moment equations although in practice certain of the effects are likely to be insignificant. In addition to the twelve contributions due to the perturbed motion there may also be the constant term derived from the initial steady trimmed conditions. Thus, for example:

where the suffixes indicate the cause of the effect.

Comparableequationsapply to forcesand to themoments. as theconstant refer to the trimmed flight condition, the only terms actually are due to the components of the lift, relative to the earth axes, which initially balance the weight. Thus:

It is commonpractice to express eachof the individual effects as apower series, for instance:

and similar for the other parameters.

4.4.5.5 Control effects

The forces and moments consequential the operation of the controls are similar to those of the aerodynamic terms described in the previous paragraph. They arise from the three control inputs, namely and roll, pitch, and yaw, respectively. Thus:

and similarly for the other terms.

4.4.5.6 Atmospheric disturbances

The terms in the force and moments equations which arise as a result of disturbances in the atmosphere may be represented in a similar way to those of the airframe aerodynamic effects. They are normally neglected in control and stability analysis

loading and structural layout

but are of considerable importance in loading calculations. These effects are further discussed in Chapter 6.

4.4.5.7 Powerplant effects

The contributions to the equations of motion which are a consequence of the changes in output relative to the trimmed case can also be expressed in a similar way to the airframe aerodynamic They are further considered in Sections 4.6.2

In document Aircraft Loading and Structural Layout (Page 120-127)