Solution
2.3.5 The Free-Body Diagram and the Four Forces
When we consider the flight of aircraft, it is sometimes convenient to think about the aircraft as a point mass, that is, it is assumed that all of the vehicle’s mass is concentrated at a single point, called the center of mass or center of gravity (CG). This point mass assumption allow us to analyze the aircraft motion as a free-body problem, that is, the motion of a single point mass that is free of its surroundings, acted upon by distinct forces. The free-body diagram depicts this point mass representation of the vehicle, with vectors drawn to show the magnitude and direction of the forces acting on it.
In many instances, we apply Newton’s first and second laws of motion to a free-body diagram in order to analyze the vehicle state or motion. Newton’s first law states that a body remains in an equilibrium state, either at rest (zero velocity) or in motion at constant velocity. Newton’s second law deals with the non-equilibrium state, where the sum of the net force acting on a body is equal to the time rate of change of the body’s momentum, m ⃗V. For many of the situations of interest to us, we assume that the mass of the body is constant, so that Newton’s second law becomes
∑⃗F = d
dt(m ⃗V) = md ⃗V
dt = m⃗a (2.22)
where⃗a is the time rate of change of the velocity, or the acceleration.
In the text, we consider two cases for the motion of an aircraft, unaccelerated motion with a straight-line flight path and accelerated motion with a curved flight path. Unaccelerated flight is associated with the climb, cruise, and descent flight conditions and accelerated flight is associated with takeoff, landing, and turning flight. In Chapter 5, we analyze the vehicle performance by applying Newton’s laws to the vehicle translational motion. In Chapter 6, we analyze the vehicle stability and control by applying Newton’s laws to the vehicle’s curved or rotational motion.
2.3.5.1 Wings-Level, Unaccelerated Flight
For the case of unaccelerated flight, the acceleration is zero, the velocity is constant, and
Equation (2.22) is simply ∑
⃗F = 0 (2.23)
k
k k
Introductory Concepts 121
Horizon Flight path
V∞
T D
L
W
Figure 2.15 The four forces acting on an aircraft in level, unaccelerated flight.
Let us now draw a free-body diagram for an aircraft flying in level, unaccelerated flight at a constant altitude and constant airspeed. The aircraft’s flight path is horizontal to the surface of the earth as is its velocity, V∞. There are four distinct forces acting on the aircraft, lift, L, drag, D, thrust, T, and weight, W, as shown in Figure 2.15. The lift is perpendicular to the velocity vector and the drag is parallel to the velocity vector. The thrust acts along a vector defined by the propulsion system, defined by a thrust vector angle,𝛼T, relative to the velocity vector. For simplicity, it is often assumed that the thrust vector angle is zero so that the thrust is parallel to the velocity vector, as shown in Figure 2.15. The weight acts in a direction towards the center of the earth, along the gravity vector. Ignoring the curvature of the earth, the weight acts downward.
If we assume that the aircraft in Figure 2.15 is in steady level flight, that is, flying at constant altitude and constant airspeed, we can apply Equation (2.23) to the forces in the directions perpen-dicular and parallel to the velocity vector to obtain
∑F⟂= L − W = 0 (2.24)
∑F∥= T − D = 0 (2.25)
where F⟂ and F∥ are the components of force perpendicular to and parallel to the flight path, respectively. This simply gives us the obvious result that for steady, constant velocity flight, the lift equals the weight and the thrust equals the drag.
L = W (2.26)
D = T (2.27)
Despite its simplicity, this equilibrium case will be useful in future analyses. If we divide Equation (2.27) by (2.26), we obtain an expression relating the thrust-to-weight ratio, (T∕W), to the lift-to-drag ratio, (L∕D), for an aircraft in steady level flight.
T
W = 1
L∕D (2.28)
These two non-dimensional ratios are important parameters in many aspects of aircraft aerodynam-ics, performance, and design. Broadly speaking, the thrust-to-weight ratio is a propulsion related
k k parameter and the lift-to-drag ratio is an aerodynamics related parameter. The thrust is dependent
on the propulsion system and the weight is a function of the aircraft structure, payload, and fuel.
The thrust can be changed during a flight, for example by the pilot selecting a different throttle set-ting. The aircraft total weight changes during a flight, usually decreasing, due to fuel consumption.
Therefore, the thrust-to-weight ratio varies continuously during a flight, having different values at different phases of flight, such as takeoff, cruise, or landing. The thrust-to-drag ratio is a measure of the propulsion system’s capability to accelerate the aircraft mass. We can see this by apply-ing Newton’s second law to the propulsion system thrust force (T = ma) and the aircraft weight (W = mg), as follows
T W = ma
mg = a
g (2.29)
where m is the aircraft mass, g is the acceleration due to gravity, and a is the aircraft acceleration.
Equation (2.29) shows that the thrust-to-weight ratio is directly proportional to the aircraft’s accel-eration. A higher thrust-to-weight ratio indicates that an aircraft has a higher acceleration or climb capability. If the thrust-to-weight ratio is greater than one, then the vehicle is capable of acceler-ating in a vertical climb. High performance fighter aircraft may have this capability, while it is a requirement for a vertical takeoff rocket vehicle.
Thrust-to-weight ratios for various types of vehicles are shown in Table 2.4. Aircraft thrust-to-weight ratios are usually specified for the maximum static thrust that is produced at sea level divided by the aircraft maximum takeoff weight. Thrust-to-weight ratios are also quoted for a jet or rocket engine alone, as a measure of the engine’s acceleration capability without an airframe.
The lift-to-drag ratio is a measure of the aerodynamic efficiency of an aircraft. The more aero-dynamic lift that an aircraft can produce in relation to the aeroaero-dynamic drag, the more aerodynam-ically efficient it is. The lift and drag are strongly influenced by the size and design of the aircraft wing. The lift and drag, and hence the lift-to-drag ratio, vary with the airspeed. We are often inter-ested in the maximum value of the lift-to-drag ratio, denoted as (L∕D)max. Values of the lift-to-drag ratio are given for various types of vehicles in Table 2.4. Examine these values closely, as it is worthwhile to obtain a “feel” for the L∕D of different types of vehicles.
2.3.5.2 Climbing, Unaccelerated Flight
Consider now the case of climbing, unaccelerated flight, where the aircraft is in a constant airspeed climb, as shown in Figure 2.16. The flight path angle,𝛾, is defined as the angle between the aircraft’s
Table 2.4 Lift-to-drag and thrust-to-weight ratios for various types of aerospace vehicles.
Type of aerospace vehicle Lift-to-drag ratio, L/D Thrust-to-weight ratio, T/W
Wright Flyer I 8.3 —
General aviation airplane 7–15 —
High performance glider 40–60 —
Commercial airliner 15–25 0.25–0.4
Military fighter airplane 4–15 0.6–1.1
Helicopter 4–5 —
Space capsule (Apollo capsule) ∼0.35 (reentry) —
Lifting space plane (Space Shuttle) 4–5 (subsonic glide) 1.5 (lift-off) Values are for cruise flight conditions unless otherwise noted.
k
k k
Introductory Concepts 123
W sin γ Horizon
Flight path
γ
W cos γ γ
W
D T L
V∞
Figure 2.16 The four forces acting on an aircraft in climbing, unaccelerated flight.
velocity vector and the horizon. The same four forces of lift, drag, weight, and thrust act on the aircraft at its center of gravity. Again, it is assumed that the thrust vector angle is zero, aligning the thrust with the velocity vector. As shown in Figure 2.16, the weight vector points vertically downward, making an angle𝛾 with respect to the direction perpendicular to the flight path.
Even though the aircraft is climbing, it is not accelerating or decelerating, hence, Equation (2.23) is still valid. Summing the forces perpendicular and parallel to the flight path, we have
∑F⟂= L − W cos𝛾 = 0 (2.30)
∑F∥= T − D − W sin𝛾 = 0 (2.31)
Solving for the lift and drag, we have
L = W cos𝛾 (2.32)
D = T − W sin𝛾 (2.33)
Solving Equation (2.33) for thrust, we have
T = D + W sin𝛾 (2.34)
Equation (2.32) states that, for a constant airspeed climb, the lift must equal the component of weight perpendicular to the flight path. Equation (2.34) states that, for a steady climb, the thrust must equal the drag plus a component of the weight in the direction opposite to the flight path.
Comparing Equation (2.34) with Equation (2.27) for level flight, we see that, as expected, more thrust is required for a constant airspeed climb than for level flight at constant airspeed, by an additional amount equal to W sin𝛾. Note the case of level, unaccelerated flight simply corresponds to the case of zero flight path angle.
2.3.5.3 Descending, Unaccelerated Flight
Consider now the case of steady, unaccelerated flight, where the aircraft is in a constant airspeed descent, as shown in Figure 2.16. The flight path angle is now negative, −𝛾, since the flight path is below the horizon. The angle,𝜃, is defined as the magnitude of the negative flight path angle. The
k k
W sin γ Horizon
Flight path
W cos γ θ θ
W
D L
T V∞
Figure 2.17 The four forces acting on an aircraft in descending, unaccelerated flight.
same four forces of lift, drag, weight, and thrust act on the aircraft at its center of gravity. Again, it is assumed that the thrust vector angle is zero, aligning the thrust with the velocity vector. As shown in Figure 2.17, the weight vector points vertically downward, making an angle,𝜃 with respect to the direction perpendicular to the flight path.
Even though the aircraft is descending, it is not accelerating or decelerating, and Equation (2.23) is valid. Summing the forces perpendicular and parallel to the flight path, we have
∑F⟂= L − W cos𝜃 = 0 (2.35)
∑F∥= T − D + W sin𝜃 = 0 (2.36)
Solving for the lift and drag, we have
L = W cos𝜃 (2.37)
D = T + W sin𝜃 (2.38)
Solving Equation (2.38) for thrust, we have
T = D − W sin𝜃 (2.39)
Equation (2.37) states that, for a constant airspeed descent, the lift must equal the component of weight perpendicular to the flight path. Equation (2.39) states that, for a steady descent, the thrust equals the drag minus a component of the weight in the direction of the flight path. Thus, we see that less thrust is required for descending flight at constant airspeed than for constant airspeed, level, or climbing flight, since there is the component of weight, W sin𝜃, acting in the direction of the thrust. Note again, that the case of level, unaccelerated flight simply corresponds to that of zero flight path angle.
In summary, by using the point mass assumption for the vehicle, drawing a free-body diagram, and applying Newton’s second law of motion, we can develop the equations relating the forces to the state of the vehicle. We apply this procedure to many vehicle motion problems in the future, especially in analyzing vehicle aerodynamics, performance, and stability and control.
k
k k
Introductory Concepts 125