 # PART 3: Equations of Motion

The equations of motion constitute the core of the mathematical model of flight dynamics. These equations relate the forces acting on the aircraft to its position, velocity, acceleration and orientation in space. Their derivation is more than an intellectual exercise. The correct interpretation of these equations depends to a very great extent on knowledge of how they were obtained, the reference frames to which they apply, and the underlying assumptions made along the way.

## Simplifying Assumptions

In this model, the aircraft is treated as a rigid body with six degrees of freedom. This is, of course, an idealization of actual flight dynamics, but avoids the complexities that a consideration of elastic forces and the movement of aircraft parts, such as engine rotors, ailerons, and the like would introduce. Other simplifying assumptions regarding the dynamics of flight have been made to reduce computational complexity. The assumptions that earth rate is constant and zero and that Coriolis accelerations can be neglected simplify the equations of motion for the aircraft. Control of the hypothetical aircraft is accomplished by adjusting values of engine thrust, aerodynamic lift, and bank angle. This models an aircraft which uses ailerons, elevators, and engines alone to control speed, heading and altitude. Aircraft yaw in all flight maneuvers is assumed to be constant and zero. Also ignored are the complexities introduced by considering the effect aircraft shape and motion have on the external forces which operate on the aircraft. In this model, the aircraft is capable of both translational and rotational motion.

## Derivation From First Principles, the Force Equation

The aircraft moves, translationally, under the influence of gravity and of the aerodynamic forces, lift, drag, and thrust. These forces are assumed to act on the aircrafts center of mass. With respect to the Inertial frame of reference, the total force, , acting on the aircraft is, (3-1)

where is the vector sum, (3-2)

The relationship between force and acceleration, embodied in Equation 3-1, is not expressed in a form well-suited to our purpose. It needs to be expressed relative to a frame of reference in which the forces acting on the aircraft and the state variables of interest (i.e., velocity, roll, and pitch) assume a simple form. The North-Oriented, Local-Level frame provides just such a frame.

The time rate of change of a vector, , as observed in a stationary frame is related to its time rate of change as observed in a rotating frame by the equation, (3-3)

Where is the angular velocity of the rotating frame relative to the stationary frame. This relationship may be used to rewrite the acceleration term appearing on the left side of Equation 3-1. Thus, we are able to express the aircrafts acceleration in the Inertial frame in terms of its acceleration as observed in the ECEF frame as, (3-4)

where is the earths spin velocity given by Equation 2-1 (Part 2 LINK).

Reapplying Equation 3-3 to the leftmost term of Equation 3-4, the equation (3-5)

is obtained. The earths spin velocity is constant. Thus, (3-6)

The aircrafts acceleration relative to the Inertial frame may now be expressed in the ECEF frame as, (3-7)

Substituting the right hand side of Equation 3-7 for the term, in Equation 3-1, and defining , Equation 3-1 becomes, (3-8)

In this equation the subscripts indicating the frame relative to which the derivatives are taken have been dropped. But it is important to remember that this equation is valid only in the ECEF frame of reference. We are still short of our objective, which is to relate velocity and force in the NOLL frame of reference. This is accomplished by finding an expression for each term of Equation 3-8 valid in the NOLL frame.

In the VO frame of reference, the aircraft velocity is . Applying the transformations of Equations 2-2 and 2-3 (See Part 2), the velocity relative to the NOLL frame is seen to be, (3-9)

This is equivalent to, (3-10)

Using Equations 2-5 and 3-10, an expression for , the Coriolis acceleration term, is obtained. That is,  (3-11)

The position vector, relative to the NOLL frame, which locates the aircraft is, simply, . Using this expression for and Equation 2-5, (3-12)

and, (3-13)

The resolution of forces (i.e., weight, lift, drag, and thrust) onto coordinate axes is most easily accomplished in the VO frame of reference, where the aircrafts velocity vector is parallel to the positive axis. The drag force is opposite in direction to , while lift is normal to the velocity vector. The aircrafts thrust vector, in general, is displaced from aircraft velocity by the thrust angle of attack, , and lies in the plane formed by the lift and drag force vectors. The aircrafts weight, , acts vertically and downward along the negative axis. The diagrams in Figures 3-1 (a), (b), and (c) illustrate the relationships between forces and coordinate axes. It should be remembered that the thrust vector lies in the lift-drag plane. It does not necessarily lie in the plane, as might be inferred by the diagrams of Figure 3-1.

The vector sum of all the forces acting on the aircraft are given by, (3-14)

The components of force relative to the NOLL frame are found by applying the inverse of the transformation appearing in Equation 2-9 to Equation 3-14. That is, (3-15)

where, (3-16)

is the component of total force normal to the aircrafts velocity vector, and (3-17)

is the component parallel to . The total force in the NOLL frame is,  (3-18) Again applying the relationship expressed in Equation 3-3 and equation, (3-19)

for the aircrafts position vector expressed in the NOLL frame, the equation, (3-20)

relating aircraft velocities as observed in the ECEF and the NOLL frames is obtained. The value for given by Equation 2-7 when substituted in Equation 3-20 yields, (3-21) Equating components of the unit vectors in Equations 3-21 and 3-10, the kinematic relations, (3-22) (3-23) (3-24)

are obtained. Applying Equations 2-7 and 3-3 once again, this time to Equation 3-21, an expression for , , in terms of time derivatives as observed in the NOLL frame of reference is obtained. This equation,  (3-25)

after evaluation using the kinematic relations of Equations 3-22, 3-23, and 3-24, yields,     (3-26)

All terms of Equation 3-26 are expressed as components along NOLL frame axes. Substitution of Equations 3-11, 3-13, 3-18, and 3-26 into Equation 3-8 yields in a system of simultaneous differential equations. Solving these equations for , ,and , the three differential equations,  (3-27) (3-28)  (3-29)

are obtained. These are the equations of motion for the aircraft. Considerable simplification of these equations is possible if the effects of the earths rotation on flight dynamics is neglected. The effects of earth spin are reflected in those terms of Equations 3-27 through 3-29 which include the earth rate, . For a non-rotating earth, the equations of motion become, (3-30) (3-31) (3-32)

Ignoring terms involving results in further simplification, without significant loss of precision in most applications. The relatively small contribution of these terms to aircraft state and the simplification of the control laws derived using the equations of motion, justify their removal. The simplified equations are, (3-33) (3-34) (3-35)