In this section, several basic aerospace engineering concepts, definitions, and nomenclature are introduced. Some of these concepts may be familiar from basic physics, but with a new focus on aerospace applications. Others concepts are new and specific to aerospace engineering or flight test. We start by defining several aerospace vehicle axis systems and the associated conditions that
k
k k
Introductory Concepts 115
Y P(x, y, z)
Z
V
X
w O
u→
→v
→
→
Figure 2.9 Velocity vector and velocity components in Cartesian coordinates.
define the vehicle’s orientation and motion. An idealized, point mass model of an aerospace vehicle is discussed, which is useful in analyzing the vehicle state by applying Newton’s laws of motion.
The speed of sound and Mach number are introduced and used to discuss the different regimes of flight. Finally, several aerospace concepts are introduced that are captured by a defining diagram or chart, these being the flight envelope, the aircraft load factor versus airspeed plot, and the aircraft weight and balance plot.
2.3.1 Aircraft Body Axes
There are several different axis systems that may be used to define the orientation or attitude of an aerospace vehicle. The selection of a particular system usually depends on the type of problem that is being analyzed. In this text, we usually deal with a three-dimensional coordinate system that is rigidly attached to the aircraft, called the body axis system.
The origin of the body axis system is located at the aircraft center of gravity, commonly referred to as the “CG”, as shown in Figure 2.10. The xb-axis points out through the aircraft nose along a defined reference line, which may be a line through the fuselage or wing (usually the wing chord, to be defined in Chapter 3). The yb-axis points out of the aircraft’s right wing and is positive in that direction. Using the right-hand rule, the zb-axis points through the bottom of the aircraft and is positive in that direction. The xb-zbplane is a symmetry plane that “cuts” the aircraft into two symmetrical halves. The xb, yb, and zb axes are also referred to as the longitudinal, lateral, and vertical axes, respectively.
The body axes are “bolted” to the aircraft and do not change their orientation, relative to the aircraft, as the aircraft translates and rotates in three-dimensional space. The xb- and yb-axes always points out of the aircraft nose and right wing, respectively, regardless of the aircraft orientation.
Aircraft moments of inertia and products of inertia are referenced to the body axis system, since they then remain constant, regardless of changes in the aircraft orientation. The body axis system is often the frame of reference for the pilot, since the pilot is attached to this coordinate system as the aircraft translates and rotates.
The motion of the aircraft can be described about the three body axes. In general, an aircraft has six degrees of freedom, three linear translations and three angular rotations. An aircraft can translate
k k
Roll
Yaw Pitch
CG
xb
yb Lateral axis zb
Vertical axis
Longitudinal axis
Figure 2.10 Aircraft body axis system.
forward (and aft, if it is a helicopter or airship) along the longitudinal axis, move to its right or left along the lateral axis, and move up or down along the vertical axis. As shown in Figure 2.10, the aircraft can rotate about each of the three body axes. Rotation about the longitudinal axis is called roll, hence this axis is called the roll axis. Rotation about the lateral axis is called pitch, hence this axis is called the pitch axis. Rotation about the vertical axis is called yaw, hence this axis is called the yaw axis.
2.3.2 Angle-of-Attack and Angle-of-Sideslip
Consider an aircraft that is flying at a velocity, V∞, as shown in Figure 2.11, where the nose of the aircraft may not be pointing in the direction of the velocity vector. The orientation of the aircraft can be defined with respect to the velocity vector in terms of two angles, the angle-of-attack,𝛼, and the angle-of-sideslip,𝛽, as shown.
xb
yb zb
V∞
α β
Figure 2.11 Orientation of an aircraft with respect to the velocity vector, V∞.
k
k k
Introductory Concepts 117
v
yb xb
w u
Projection of V∞
in xb - zb plane
zb
β V∞
α
Figure 2.12 Definition of angle-of-attack,𝛼, and angle-of-sideslip, 𝛽. (Source: Adapted from Dynamics of Flight: Stability and Control, B. Etkin and L.D. Reid, Fig. 1.7, p. 16, (1996), [7], with permission from John Wiley & Sons, Inc.)
The aircraft angle-of-attack,𝛼, measured in the xb-zbplane, is defend as the angle between the aircraft longitudinal axis (the xb-axis) and the projection of the velocity vector in the xb-zbplane, as shown in Figure 2.12. The projection of the velocity vector is defined by its components, u and w, along the xb- and zb-axes, respectively. Positive angle-of-attack is measured up from the velocity vector to the aircraft reference line. Later, in Chapter 3, we define an angle-of-attack specific to the airfoil section of a wing.
The aircraft angle-of-sideslip𝛽, is the angle between the aircraft xb-zbsymmetry plane and the velocity vector. The sideslip angle is not measured in the xb-ybplane, since the velocity vector is not necessarily in this plane. If the angle-of-sideslip is zero, then the angle-of-attack is simply the angle between the aircraft longitudinal axis and the total velocity vector. Positive angle-of-sideslip is with the aircraft nose pointing to the left with respect to the velocity vector. Positive sideslip angle is also referred to as “wind in the right ear” as this is what the pilot would feel in an open cockpit airplane with the nose pointing left relative to the velocity vector.
Let us define the velocity vector, ⃗V∞, as
⃗V∞= û𝚤 + v ̂𝚥 + w ̂k (2.18)
where u, v, and w are the components of the velocity in the xb, yb, and zbaxis directions, respectively, and̂𝚤, ̂𝚥, and ̂k are the unit vectors along these axes, respectively. The magnitude of the velocity, V, is given by
V∞=√
u2+ v2+ w2 (2.19)
Using these definitions of the velocity, the angle-of-attack is defined as 𝛼 = tan−1w
u (2.20)
and the angle-of-sideslip is defined as
𝛽 = sin−1 v
V∞ (2.21)
The angle-of-attack and angle-of-sideslip are two important parameters that are frequently used in describing the aircraft orientation, especially in the areas of aerodynamics and stability and control.
k k Example 2.4 Calculation of Angle-of-attack and Angle-of-sideslip The components of velocity
of an aircraft, in the body axis system, are u = 173.8kt (nautical mile per hour), v = 1.27kt, and w = 13.2kt. Calculate the magnitude of the velocity, the angle-of-attack, and the angle-of-sideslip.