Introduction to Aeronautics: A Design Perspective Chapter 6: STABILITY AND CONTROL “The balancing of a gliding or flying machine is very simple in theory. It merely consists in causing the center of pressure to coincide with the center of gravity. But in actual practice there seems to be an almost boundless incompatibility of temper which prevents their remaining peaceably together for a single instant, so that the operator, who in this case acts as peacemaker, often suffers injury to himself while attempting to bring them together.”
6.1 DESIGN MOTIVATION
Simply stated, stability and control is the science behind keeping the aircraft pointed in a desired direction. Whereas performance analysis sums the forces on an aircraft, stability and control analysis requires summing the moments acting on it due to surface pressure and shear stress distributions, engine thrust, etc., and ensuring those moments sum to zero when the aircraft is oriented as desired. Stability analysis also deals with the changes in moments on the aircraft when it is disturbed from equilibrium, the condition when all forces and moments on it sum to zero. An aircraft which tends to drift away from its desired equilibrium condition, or which oscillates wildly about the equilibrium condition, is said to lack sufficient stability. The Wright brothers intentionally built their aircraft to be unstable because this made them more maneuverable. As the quotation from Wilbur Wright above suggests, such an aircraft can be very difficult and dangerous to fly.
Control analysis determines how the aircraft should be designed so that sufficient control authority (sufficiently large moments generated when controls are used) is available to allow the aircraft to fly all maneuvers and at all speeds required by the design specifications. Good stability and control characteristics are as essential to the success of an aircraft as are good lift, drag, and propulsion characteristics. Anyone who has flown a toy glider which is out of balance or which has lost its tail surfaces, or who has shot an arrow or thrown a dart with missing tail feathers, knows how disastrous poor stability can be to flying. Understanding stability and control and knowing how to design good stability characteristics into an aircraft are essential skills for an aircraft designer.
6.2 THE LANGUAGE The science of stability and control is complex, and only an orderly, step-by-step approach to the problem will yield sufficient understanding and acceptable results. This process must begin by defining quite a number of axes, angles, forces, moments, displacements, and rotations. As much as possible, these definitions will be consistent with those used in aerodynamic and performance analysis, but occasionally the complexity and unique requirements of stability and control problems dictate that less intuitive definitions and reference points be used.
One of the least intuitive elements of stability and control analysis is the coordinate system as shown in Figure 6.1. Note that the vertical (z) axis is defined as positive downward! The reason for this choice is a desire to have consistent and convenient definitions for positive moments. Positive moment directions are defined consistent with the right hand rule used in vector mathematics, physics, and mechanics. This rule states that if the thumb of a person’s right hand is placed parallel to an axis of a coordinate system, then the fingers of that hand will point in the positive direction of the moment about that axis. Since the moment about the aerodynamic center of an airfoil or wing was defined in Chapter 3 as being positive in a nose-up direction, the right-hand rule requires that the lateral (spanwise) axis of the aircraft coordinate system be positive in the direction from the right wing root to the right wing tip. A natural starting point for the coordinate system is the aircraft’s center of gravity, since it will rotate about this point as it moves through the air. The aircraft’s longitudinal axis (down its centerline) is chosen parallel to and usually coincident with its aircraft reference line (defined in Chapter 4), but positive toward the aircraft’s nose so that a moment tending to raise the left wing and lower the right wing is positive. This axis is chosen as the x axis to be consistent with performance analysis. Making x positive toward the front allows the aircraft’s thrust and velocity to be taken as positive quantities. Since a rotation about the longitudinal axis to the right or clockwise is positive, for consistency it is desired that a moment or rotation about the aircraft’s vertical axis such that the nose moves to the right be considered positive. This requires that the vertical axis be positive downward so that the right-hand rule is satisfied.
The only choice which remains is whether the lateral or vertical axis should be the y axis. The y axis is generally taken as vertical in performance analysis, but an x,y,z coordinate system must satisfy another right-hand rule in order to be consistent with conventional vector mathematics. The right-hand rule for 3-dimensional orthogonal (each axis perpendicular to the others) coordinate systems requires that if the thumb of a person’s right hand is placed along the coordinate system’s x axis, the fingers point in the shortest direction from the system’s y axis to its z axis (try this on Figure 6.1). To satisfy this right-hand rule as well as all the previous choices for positive directions, the coordinate system’s y axis must be the aircraft’s lateral axis (positive out the right wing), and the z axis must be the vertical axis (positive down). A coordinate system such as this which has its origin at the aircraft center of gravity and is aligned with the aircraft reference line and lateral axis is referred to as a body axis system.
Figure 6.1 Aircraft Body Axes and Positive Moment Directions
For consistency with aerodynamic analysis, the nose-up moment is labeled m. Since m is the moment about the y axis, the moment about the xaxis is labeled and the moment about the z axis is labeled n, to make them easier to remember. Note that the symbol is used instead of l to avoid confusion with airfoil lift and the number 1, and lower case is used for and n to avoid confusion with the symbols for wing lift and normal force. Unfortunately, there is no consistent way to avoid confusion between the pitching moment on an airfoil and the whole-aircraft pitching moment just described, since both have been given the symbol m. To partly alleviate this problem, the symbol M will be used for finite wing and whole-aircraft pitching moments when they are not used in conjunction withand n. Forces on the aircraft may be broken into components along the x, y and z axes. These force components are labeled X, Y, and Z respectively.