Mathematics of helicopters
The mathematics of helicopters involves the principles and calculations that underpin their unique flight mechanics and design features. Helicopters achieve vertical flight through spinning blades that generate lift, a concept that dates back to early innovations by figures like Leonardo da Vinci and furthered by engineers such as Thomas Edison in the quest for efficient propeller designs. Modern helicopters are equipped with main rotors and tail rotors to manage torque effects, allowing for stable flight and maneuverability.
Key mathematical aspects include modeling the forces at play during various flight conditions, such as hovering, climbing, or descending, which depend on the thrust generated relative to the helicopter's weight. Factors such as pressure differential around the rotors and stability equations related to inertial velocities are crucial for understanding helicopter dynamics. Additionally, phenomena like transverse flow effect and ground resonance highlight the complex interactions between rotor blades and airflow, influencing both performance and safety. As helicopters continue to evolve, mathematical modeling plays a vital role in enhancing their design and capabilities, making it an integral area of study within aviation.
Mathematics of helicopters
Summary:Helicopters apply vertical thrust to overcome their weight.
A helicopter is a type of aircraft that overcomes gravitational force by employing spinning blades to generate vertical thrust. The ideas of vertical flight can be traced back to the Chinese and to Leonardo da Vinci. Thomas Edison studied several different propeller designs and concluded that a feasible helicopter needed a lightweight engine that could produce a large amount of power. Mathematicians such as Theodore Karmen and George de Bothezat also worked on helicopter design in the early twentieth century. In modern helicopters, downward force is supplied by an engine driver rotor. A helicopter has many advantages over a fixed-wing aircraft, such as the ability to take off and land vertically, to hover, and to fly backwards and laterally in the air. As the main rotor spins, it generates a torque that could set the helicopter into a fatal spin. To compensate for this, helicopters have a smaller rotor and blades on their tails.
Flight Controls
A helicopter has four main flight control inputs that enable it to perform various aerial maneuvers: the cyclic control, the collective pitch control, the anti-torque pedals, and the throttle. The cyclic control changes the pitch of the rotor blades cyclically, enabling the helicopter to move in the desired direction. The collective pitch control controls the altitude of the rotorcraft. The anti-torque pedals change the pitch of the tail, altering the amount of thrust.
Mathematically Modeling Helicopter Flight
Helicopters fly by sucking air from above their rotors and forcing it downwards with a thrust equal to (if hovering), greater than (if climbing), or less than (if descending) their weight. The pressures at various points around a helicopter are given by
here P0 is the rest pressure of the air far above the rotors, P + ∆P is the pressure below the rotors, vin is the velocity of the air as it is sucked in, and vout is the velocity of the air as it is forced down.
There are also equations governing the stability and flight of a helicopter. These take into account the inertial velocities in the moving axes system, the Euler rotations defining the orientation of the fuselage axes with respect to Earth, and the aircraft mass. In the early twenty-first century, mathematicians model areas of helicopter flight and performance, such as aerobatic maneuvers that push the limits of the system and that help inform improvements and future designs of new helicopters.
Transverse Flow and Ground Resonance Effects
In forward flight, because the air is being accelerated for a longer period of time as it travels to the rear of the rotor system, air passing through the rear portion of the rotors has a greater downwash angle than the air passing through the forward portion. This pressure difference causes a decrease in the angle of attack, resulting in less lift in the rear of the rotorcraft, increased angle of attack, and more lift in the front. This is called the “transverse flow effect” and it causes easily recognizable vibrations.
When a helicopter is resting on the ground with its rotor spinning, a destructive harmonic vibration called “ground resonance effect” can develop and is caused by a reaction of the rotor blades to the lateral motion of the helicopter. Ground resonance effect develops when the rotor blades move out of phase with each other and cause the rotor disc to become unbalanced.
Bibliography
Ganiev, R. F., and I. G. Pavolov. “The Theory of Ground Resonance of Helicopters.” International Applied Mechanics 9, no. 4 (1973).
Leishman, J. G. Principles of Helicopter Aerodynamics. New York: Cambridge University Press, 2006.
Padfield, Gareth D. Helicopter Flight Dynamics. Oxford, England: Blackwell, 1996.
Wagtendonk, W. J. Principles of Helicopter Flight. Newcastle, WA: Aviation Supplies & Academics, 2006.