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Unformatted text preview: AA 311 Lecture 19: Introduction to Static Stability To discuss stability of a steady motion, we must first introduce some terminology to describe the motion. Suppose we fix a reference frame to some point in the aircraft, as shown in Figure 1. We denote by x B the unit vector pointing through the nose of the aircraft. This axis is often referred to as the longitudinal axis . We let z B represent the unit vector pointing through the belly of the aircraft; this is often called the directional axis . Finally, we define the lateral axis in terms of the unit vector y B = z B x B . Viewing the aircraft from behind, y B points to the right. x I y I z I x B y B z B V ! Figure 1: Inertial and body-fixed reference frames. To describe the orientation of the aircraft, we define an inertial reference frame, which is denoted by the fixed unit vectors x I , y I , and z I . The reason we choose to describe the aircrafts orientation with respect to an inertial frame is that Newtons laws of motion only hold in an inertially fixed frame. We will typically consider an earth-fixed frame to be an inertial frame. Although the resulting equations of motion will technically be incorrect, the error due to the earths rotation, its revolution about the sun, etc. will be small over the time periods of interest in studying stability and control. As the aircraft is assumed to be rigid, the location of any point in the airplane is uniquely determined by the position and orientation of the body-fixed reference frame. Therefore, we will often represent the aircraft simply by its body-fixed reference frame. Suppose that the aircraft (i.e., the body frame) translates at some velocity with respect to the inertial frame. We let V = u v w denote the translational velocity of the body with respect to the inertial frame, but expressed in the body frame . 1 Also, suppose that the aircraft rotates at some angular velocity with respect to the the inertial frame. We let = p q r denote the angular velocity of the body with respect to the inertial frame, but expressed in the body frame. 1 Note the distinction, here! While any given vector can be expressed in any given reference frame, derivatives are always taken with respect to a specific frame . 1 Axis Linear Aerodynamic Angular Angular Aerodynamic Velocity Force Displacement Velocity Moment Longitudinal ( x B ) u X p L Lateral ( y B ) v Y q M Directional ( z B ) w Z r N The angular displacement variables , , and do not generally represent angles about the body-fixed axes. These angles, referred to as the Euler angles , define a series of three rotations which transform vectors from the inertial frame to the body frame, and vice versa. The parameterization of vehicle attitude will be discussed in detail in later courses. Until then, we will only consider simple motions in which, for example, the pitch angle is truly a rotation about the lateral ( y B ) axis....
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- Fall '09