Flight Theory and Aerodynamics. Joseph R. Badick
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Average time to liftoff:
Total takeoff distance:
ROTATIONAL MOTION
Without derivation, some of the relationships among tangential (tip) velocity, Vt; radius of rotation, r; revolutions per minute, rpm; centripetal forces, CF; weight of rotating parts, W; and acceleration of gravity, g, are shown below. A more detailed discussion regarding rotorcraft can be found in Chapter 15 of this textbook.
(1.7)
(1.8)
(1.9)
For our discussion, the units of work will be measured in ft‐lb.
ENERGY AND WORK
Energy is the ability to do work. In physics, work has a meaning different from the popular definition. You can push against a solid wall until you are exhausted but, unless the wall moves, you are not doing any work. Work requires that a force must move an object (displacement) in the direction of the force. Another way of saying this is that only the component of the force in the direction of movement does any work:
There are many kinds of energy: solar, chemical, heat, nuclear, and others. The type of energy that is of interest to us in aviation is mechanical energy.
There are two kinds of mechanical energy: The first is called potential energy of position, or more simply potential energy, PE. No movement is involved in calculating PE. A good example of this kind of energy is water stored behind a dam. If released, the water would be able to do work, such as running a generator. As a fighter aircraft zooms to a zenith point, it builds PE; once it starts to accelerate downward, it converts PE to KE. PE equals the weight, W, of an object multiplied by the height, h, of the object above some base plane:
(1.10)
The second kind of mechanical energy is called kinetic energy, KE. As the name implies, kinetic energy requires movement of an object. It is a function of the mass, m, of the object and its velocity, V:
(1.11)
The total mechanical energy, TE, of an object is the sum of its PE and KE:
(1.12)
The law of conservation of energy states that the total energy (of a closed system) remains constant. Both potential and kinetic energy can change in value, but the total energy must remain the same. For example, when a ball is thrown upward, if the height of the thrower is the reference plane, its energy is all kinetic when it leaves the thrower’s hand. As it rises, PE is continually increasing, but KE is always decreasing by the same amount, so the sum remains constant. At the top of its travel, PE is at its maximum (the same amount as the KE it had when it left the thrower’s hand) and KE is zero. Energy cannot be created or destroyed, but can change in form.
EXAMPLE
An aircraft that weighs 15 000 lb is flying at 10 000 ft altitude at an airspeed of 210 kts. Calculate the potential energy, kinetic energy, and the total energy.
PE: PE = Wh → PE = 15000 lb × 10000 ft → PE = 1.5 × 108
KE:
Total Energy: TE = PE + KE → TE = 1.79 × 108
Application 1.2
Consider a general aviation airplane that weighs 3000 lb with a designated approach speed over the runway threshold of 65 kts., calculate the KE. Now, consider if that same airplane approaches the runway with an extra 10 kts. of speed due to poor planning, calculate the new KE.
Why does only a 10 kts. change in approach speed result in such a wide margin of KE? What are the consequences of this “extra” energy?
POWER
In our discussion of work and energy, we have not mentioned time. Power is defined as “the rate of doing work” or work/time. We know:
and
James Watt defined the term horsepower (HP) as 550 ft‐lb/s:
If the speed is measured in knots, Vk, and the force is the thrust, T, of a jet engine, then
EXAMPLE
An aircraft’s turbojet engine produces 8000 lb of thrust at 180 kts., what is the equivalent horsepower that engine is producing?
Equation 1.13 is very useful in comparing thrust‐producing aircraft (turbojets) with power‐producing aircraft (propeller aircraft and helicopters); a more detailed discussion will follow in future chapters.
Application 1.3
Consider the example calculation provided to solve for horsepower (HP).
Would the horsepower remain the same if the thrust remained 8000 lb but the aircraft slowed to a speed of 160 kts.? Why or why not? How can the equation be altered to solve for thrust (T) if an aircraft was maintaining a constant speed with a known HP?
FRICTION
If two surfaces are in contact with each other, then a force develops between them when an attempt is made to move them relative to each other. This force is called friction. Generally, we think of friction as something to be avoided because it wastes energy and causes parts to wear. In our discussion on drag, we will discuss the parasite drag on an airplane in flight and the thrust or power to overcome that force. Friction is not always our enemy; however, without it there would be no traction between an aircraft’s tires and the runway. Once an aircraft lands, lift is reduced and a portion of the weight contributes to frictional force. Depending on the aircraft type, aerodynamic braking, thrust reversers, and spoilers will be used to assist the brakes and shorten the landing, or rejected takeoff distance.