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AE 2610 Introduction to Aerospace Engineering
pg. 1/1
Homework #10
1. Consider two rods with identical length and diameter of 10 ft and 1 inch, respectively. One rod is AM-350
stainless steel, and the other is 2024 aluminum. Both rods are under the same tension load of 15,000 lb.
Which rod will elongate the most, and by what amount compared to the other?
2. Consider a rod made from 2024 aluminum with a diameter of 0.5 inch. What is the maximum load in
tension that can be carried by this rod before it yields?
3. Consider an airplane with a tricycle landing gear stationary on the ground. The weight of the airplane is
5158 lb. Assume the nose wheel strut and the two main gear struts are perpendicular to the ground. The
diameter of the nose wheel strut is 1 inch; that of each main gear strut is 3 inches. The nose wheel strut is
located 5.62 ft ahead of the center of gravity of the airplane. The two main gear struts are located 1.12 ft
behind the center of gravity. Calculate the compressive stresses in the nose wheel strut and the main gear
struts.
4. Consider a kite flying in a wind, supported and restrained by a cord from the ground. The lift on the kite
is 20 lb, and the cord makes an angle of 60 degrees with the ground. The cord is taut and strait. The
diameter of the cord is 0.1 inch. Calculate the tensile stress in the cord. Ignore the weight of the cord.
5. A 1 ft square plate made of 2024 aluminum is subjected to a shear load along its edges. The displacement
of one side relative to the opposite side is 0.1 inch. Calculate the shear stress acting on the edges.
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to Aerospace Engineering
Lecture 11 : The Standard Atmosphere (TSA)
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Definition of Altitude : Geometric vs. Absolute
Ã¯ÂÂ­ Altitude in general represents the distance above the ground
Ã¯ÂÂ´ Geometric Altitude (hG): Geometric height above sea level
Ã¯ÂÂ´ Absolute Altitude (hA): Geometric Altitude + Radius of Earth (ha+rE)
2
Ã¯Æ’Â¦ rE Ã¯Æ’Â¶
Ã¯Æ’Â¦ rE Ã¯Æ’Â¶
g Ã¯â‚¬Â½ g0 Ã¯Æ’Â§ Ã¯Æ’Â· Ã¯â‚¬Â½ g0 Ã¯Æ’Â§
Ã¯Æ’Â·
h
r
h
Ã¯â‚¬Â«
G Ã¯Æ’Â¸
Ã¯Æ’Â¨ AÃ¯Æ’Â¸
Ã¯Æ’Â¨ E
Western Michigan University
2
2
Department of Mechanical and Aerospace Engineering
The Standard Atmosphere (TSA)
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
The Standard Atmosphere (TSA)
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
The Standard Atmosphere (TSA)
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
The Standard Atmosphere (TSA)
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
Earth Atmosphere Model
Western Michigan University
7
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 12: Propulsion (1)
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Definitions, Principles & Terminologies
Ã¯ÂÂ­ Propulsion [two Latin words, Pro (=before, forward) + Pellere(=to drive)]
Ã¯ÂÂ´ Propulsion means to push forward or drive an object forward.
Ã¯ÂÂ´ A propulsion system is a mechanism that produces THRUST to push an
object forward
Ã¯ÂÂ´ THRUST is the force generated by engine to move an aircraft through the air
Ã¯ÂÂ­ NewtonÃ¢â‚¬â„¢s 3rd Law
Ã¯ÂÂ´ Thrust is a mechanical force generated often through the reaction from
accelerating the mass of gas.
Ã¯ÂÂ´ The engine does work on the gas as the gas is accelerated to the rear, the
engine is accelerated in the opposite direction. (Action & Reaction)
Ã¯ÂÂ´ The acceleration of the air mass produces a force on the aircraft
Ã¯ÂÂ­ How does gas (air) accelerated?
Ã¯ÂÂ´ Energy is needed to be expanded, which must be generated as heat by the
combustion of some fuel
Ã¯ÂÂ´ The thrust equation describes how the acceleration of the gas produces a
force
Ã¯ÂÂ´ Depending upon how the aircraft propulsion system generates this energy,
the major aircraft propulsion systems can be categorized as
PROPELLER/PROP, JET/TURBINE, RAMJET, and ROCKET.
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
Thrust Equation
Ã¯ÂÂ­ NewtonÃ¢â‚¬â„¢s 2nd Law
Ã¯ÂÂ´ Force = Mass x Acceleration Conservation of Momentum
Ã¯â€šÂ· Momentum = Mass x Velocity
Ã¯â€šÂ· The momentum of a moving object is conserved if no force is applied to it.
Ã¯â€šÂ· The time rate of change of momentum of a moving object is equal to the force
applied to the object.
Ã¯ÂÂ­ Mass Flow Rate
Ã¯ÂÂ´ Mass Flow Rate = Density x Velocity x Area
Ã¯ÂÂ­ Thrust Equations
F Ã¯â‚¬Â½ Ã¯â‚¬Â¨ mÃ¯â‚¬Â¦ eVe Ã¯â‚¬Â­ mÃ¯â‚¬Â¦ oVo Ã¯â‚¬Â© Ã¯â‚¬Â« Ã¯â‚¬Â¨ pe Ã¯â‚¬Â­ po Ã¯â‚¬Â© Ae
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
Thrust Equation: Observations
F Ã¯â‚¬Â½ Ã¯â‚¬Â¨ mÃ¯â‚¬Â¦ eVe Ã¯â‚¬Â­ mÃ¯â‚¬Â¦ oVo Ã¯â‚¬Â© Ã¯â‚¬Â« Ã¯â‚¬Â¨ pe Ã¯â‚¬Â­ po Ã¯â‚¬Â© Ae
Ã¯ÂÂ­ Usually, the magnitude of the Pressure-Area term is small relative to
mass flow rate-velocity term.
Ã¯ÂÂ­ Two possible ways to produce high thrust
Ã¯ÂÂ´ Make Engine Airflow Rate as high as possible as long as Ve>Vi
Ã¯â€šÂ· Propeller & High-bypass turbofan engine: Large amount of air is processed each
second, but the air velocity is not changed very much.
Ã¯ÂÂ´ Make the Exit Velocity much greater than the incoming velocity
Ã¯â€šÂ· Turbojets & Turbojets w/After Bunner
Ã¯â€šÂ· A moderate amount of airflow is accelerated to a high velocity
Ã¯ÂÂ­ After some minor modifications on Thrust Equation, we can study
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Propeller propulsion system
Turbo Jet propulsion system
Ramjet propulsion system
Rocket propulsion system
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
Power & Efficiency
Ã¯ÂÂ­ Power = (Thrust) x (AC Speed)
Ã¯ÂÂ´ Power is the rate of using energy, or doing work
Ã¯â€šÂ· Required Power: the power required to fly (accelerate, climb, turning, etc.)
Ã¯â€šÂ· Engine Power: the power that is actually produced by the engine & delivered to the
propeller
Ã¯â€šÂ· Propulsive Power: the power actually used to produce thrust
Ã¯â€šÂ· Waste Power = Engine Power Ã¢â‚¬â€œ Propulsive Power (Propulsive Power is always
smaller than Engine Power due to inefficiencies)
Ã¯ÂÂ­ Efficiency:
Ã¯ÂÂ´ Zero Waste Power = Best Propulsive System Efficiency
Ã¯â€šÂ· Engine Efficiency: relates to the fuel burning into engine power
Ã¯â€šÂ· Propulsive Efficiency = Propulsive Power / Engine Power
Ã¯â€šÂ· Total Efficiency = Engine Efficiency x Propulsive Efficiency : Measure of how much
power the system develops for a certain quantity of fuel burned.
Ã¯ÂÂ´ Waste Power includes
Ã¯â€šÂ· the energy supply to power the other aircraft system (Engine Efficiency)
Ã¯â€šÂ· poor Propeller or Turbine Blade Design (Propulsive Efficiency) in terms of size,
rotation speed, blade pitch angle, etc.
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Propeller Propulsion System : Introduction
Ã¯ÂÂ­ Propeller based Propulsion System Components
Ã¯ÂÂ´ Internal Combustion Reciprocating (piston) Engine or Turbine engine burns
air/fuel mixture to turn a propeller
Ã¯ÂÂ´ Turning propeller produces THRUST
Ã¯ÂÂ­ Propellers
Ã¯ÂÂ´ Propeller is simply a rotating wing
Ã¯â€šÂ· Propellers usually have between 2 and 6 blades.
Ã¯â€šÂ· The blades are usually long and thin with airfoil shape cross sections
Ã¯â€šÂ· Airfoil produces LIFT perpendicular to free stream velocity
Ã¯ÂÂ´ Since the propeller blades rotate, the tip velocity is higher than the root velocity
Ã¯â€šÂ· To make LIFT constant along the blade span, blades must be twisted
Ã¯â€šÂ· Tip velocity high, small AOA & Root velocity low, high AOA
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
Propeller Propulsion System : Propeller Characteristics (1)
Ã¯ÂÂ­ Propellers
Ã¯ÂÂ´ The total area of the propeller blades determines the ability to convert the
engineÃ¢â‚¬â„¢s power into thrust
Ã¯ÂÂ´ Blade Size & Rotation Speed
Ã¯â€šÂ· ground clearance
Ã¯â€šÂ· tip speed must be below the speed of sound
Ã¯â€šÂ· too small blade will cause the engine to Ã¢â‚¬Å“over-revÃ¢â‚¬Â at high RPM, while too large
blade will cause the engine to operate at less than the optimum
Ã¯â€šÂ· Modern AC piston engines operate at between 2200-2600RPM with Prop diameter
72-76 inches.
Ã¯ÂÂ´ Number of Blades
Ã¯â€šÂ· Increasing number of blades increases the total area, and in turn increases the
power
Ã¯â€šÂ· If the total area is the same for 2 bladed vs. 3,4,5 or even 6-bladed, the efficiency
will be close to the same
Ã¯â€šÂ· 2 bladed propeller is ideal for low speed AC, more bladed propeller is needed for
high power requirements such as faster climb & higher speeds
Western Michigan University
7
Department of Mechanical and Aerospace Engineering
Propeller Propulsion System: Propeller Characteristics (2)
Ã¯ÂÂ­ Propeller Pitch
Ã¯ÂÂ´ With the size & rotation speed fixed, the only way to increase to thrust
(thus efficiency) is from the propeller pitch angle
Ã¯â€šÂ· The efficiency of fixed propeller depends on the rotation speeds: Narrow range
of AC speed for the optimum efficiency
Ã¯â€šÅ¸ Variable Pitch Propeller (pilot controls pitch)
Ã¯â€šÅ¸ Constant Speed Propeller (pilot controls both rotation speed & pitch)
Ã¯â€šÅ¾
Ã¯â€šÅ¾
Throttle to control the power output
RPM control to set the rotation speed thus the speed of the engine
Western Michigan University
8
Department of Mechanical and Aerospace Engineering
Internal Combustion Reciprocating (Piston) Engine
Ã¯ÂÂ­ Piston Engine
Ã¯ÂÂ´ converts chemical energy (Fuel + Air) to mechanical energy (Propeller
rotation)
Ã¯ÂÂ´ Air is less dense as altitude gets high, so air pump is needed
Ã¯â€šÂ· Turbo-charging: energy expelled in the exhaust to run a small pump in the air
intakes
Ã¯â€šÂ· Supercharger: powered mechanically (or electric motor) through a belt on the
engine shaft to pump additional air into the cylinders
Ã¯ÂÂ´ Piston engine power is function of altitude, not of the speed of the
airplane. (Note that propeller thrust decreases with speed
Western Michigan University
9
Department of Mechanical and Aerospace Engineering
Gas Turbine Engine: Operation Principle
Ã¯ÂÂ­ How it operates?
Air comes through INLET where it is diffused, then
Go into COMPRESSOR where it is compressed, then
Go into BURNER where it is mixed with fuel and burned, then
Some of combustion energy turns the TURBINE to power
COMPRESSOR, and
Ã¯ÂÂ´ The hot air gas pass through NOZZLE at the exhaust end of turbine to
give THRUST
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Western Michigan University
10
Department of Mechanical and Aerospace Engineering
Gas Turbine vs. Piston-Propeller
Ã¯ÂÂ­ Energy Conversion
Ã¯ÂÂ´ Gas Turbine converts the combustion (chemical) energy into the exhaust in
terms of high speed hot air gas
Ã¯ÂÂ´ Piston/Prop converts the combustion (chemical energy into mechanical
energy to turn a propeller
Ã¯ÂÂ­ Power, Thrust & Efficiency
Ã¯ÂÂ´ Turbine Engine: available power increases with speed and the thrust is
independent of speed.
Ã¯ÂÂ´ Piston/Prop: available power is constant with speed and the thrust decrease
with speed
Western Michigan University
11
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 13: Propulsion (2)
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Introduction to Gas Turbine Engine
Ã¯ÂÂ­ Turbine Engine Components
Ã¯ÂÂ´ Inlet (Diffuser)
Ã¯ÂÂ´ Engine Core
Ã¯â€šÂ· Compressor
Ã¯â€šÂ· Burner
Ã¯â€šÂ· Turbine
Ã¯ÂÂ´ Nozzle
Ã¯ÂÂ­ Gas Turbine Engine
Classification
Ã¯ÂÂ´ By Compressor Type
Ã¯â€šÂ· Centrifugal Flow
Ã¯â€šÂ· Axial Flow
Ã¯â€šÂ· Centrifugal-Axial Flow
Ã¯ÂÂ´ By Power Usage produced by
Engine
Ã¯â€šÂ·
Ã¯â€šÂ·
Ã¯â€šÂ·
Ã¯â€šÂ·
Turbojet Engines
Turbofan Engines
Turboshaft Engines
Afterburner
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
Inlet (Diffuser)
Ã¯ÂÂ­ Inlet duct is normally considered airframe part made by aircraft
manufacturer, not by engine manufacturer!
Ã¯ÂÂ­ Main Functions
Ã¯ÂÂ´ Engine Functions:
Ã¯â€šÂ· Recover as much of the total pressure of the free air stream as possible and
deliver this pressure to the front of the engine compressor
Ã¯â€šÂ· Deliver air to compressor under all flight conditions with a little turbulence
Ã¯ÂÂ´ Airframe Function
Ã¯â€šÂ· Must hold to a minimum drag
Ã¯ÂÂ­ Types
Ã¯ÂÂ´ Subsonic Duct (Subsonic Diffuser)
Ã¯ÂÂ´ Supersonic Duct (Supersonic to Subsonic Diffuser)
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
Compressor – Centrifugal
Ã¯ÂÂ­ Main Functions
Ã¯ÂÂ´ Prevents the combustion gases from blowing out the front of engine
Ã¯ÂÂ´ Increase the pressure & density of the air (thus the oxygen) to burn fuel
Ã¯ÂÂ­ Types
Ã¯ÂÂ´ Centrifugal Compressor (or Impellers)
Ã¯â€šÂ· It compresses the air by accelerating air outward perpendicular (i.e. radially) to the
longitudinal axis of the machine
Ã¯â€šÂ· Principle Advantage: Light weight, simplicity & low cost (mostly for smaller jets)
Ã¯â€šÂ· Disadvantage: Significant energy loss due to air direction change
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
Compressor – Axial
Ã¯ÂÂ­ Types continued
Ã¯ÂÂ´ Axial Flow Compressor
Ã¯â€šÂ· A series of rotating blades pushes the air back along the longitudinal axis of
machine and thus energy to it. The blade has airfoil shape cross-section, producing
force and thereby increasing the pressure (since it is ducted)
Ã¯â€šÂ· Rows of blade makes up the axial compressor (each low has 30 to 40 blades, and
is called ROTOR). The ROTORÃ¢â‚¬â„¢s function is to increase energy by adding
pressure into the air
Ã¯â€šÂ· Between rotating blades, there is a stationary set of blades called STATOR. The
STATORÃ¢â‚¬â„¢s function is to further increase the pressure by slowing it down from the
speed leaving from the rotor
Ã¯â€šÂ· By having multiple stage ROTOR-STATOR (10-12), it produce fairly high
compressions with high efficiency
Ã¯â€šÂ· Advantage: handle large volumes of airflow and high pressure ratio
Ã¯â€šÂ· Disadvantage: susceptable to foreign object damage, expensive, very heavy
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Compressor Ã¢â‚¬â€œ Multiple Spool Axial & Axial-Centrifual
Ã¯ÂÂ­ Types continued
Ã¯ÂÂ´ Multiple Spool Axial Compressor
Ã¯â€šÂ· Spool is defined as a group of compressor stages rotating at the same speed
Ã¯â€šÅ¸ Single Spool
Ã¯â€šÅ¸ Two Spool: two rotors operate independently by two concentric shaft (Low
Pressure Compressor for Turbine)
Ã¯ÂÂ´ Axial-Centrifugal
Ã¯â€šÂ· Combination of axial-centrifugal compressor combination (mostly for small turboprop & turbo-shaft engine)
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
Burner
Ã¯ÂÂ­ Combustion Difference between Jet and Piston Engine
Ã¯ÂÂ´ Jet Engine: Compression of Air is continuous
Ã¯ÂÂ´ Piston Engine: Compression is cyclic
Ã¯ÂÂ­ Burner in Jet Engine
Ã¯ÂÂ´ Best combustion efficiency is to keep the post-combustion temperature
as high as possible (typically 2500F or 1500C)
Ã¯ÂÂ´ Needs to be cooled by air bleed
Ã¯ÂÂ´ Types of Burner: Can, Annular & Can-Annular type
Western Michigan University
7
Department of Mechanical and Aerospace Engineering
Turbines
Ã¯ÂÂ­ Main Function
Ã¯ÂÂ´ It extract kinetic energy from the expanding gases coming from burner, and
converting it to shaft horsepower to drive the compressor and the engine
accessory
Ã¯ÂÂ´ Nearly Ã‚Â¾ of all combustion energy is needed to drive the compressor
Ã¯ÂÂ­ Important Characteristics
Ã¯ÂÂ´ Turbine Vane: non-rotating, first set of blade right after burner which turns
exhaust gas into turbine blades.
Ã¯ÂÂ´ Turbine Blades: rotating blade connected to compressor by a shaft.
Ã¯ÂÂ´ Turbine is the reverse process of the compressor. The air expands and cools
through each stage, removing energy from the air.
Ã¯ÂÂ´ Multiple Spool Compressor need multiple turbine sections
Western Michigan University
8
Department of Mechanical and Aerospace Engineering
Nozzle (Exhaust Duct), After Burner & Thrust Vectoring
Ã¯ÂÂ­ Nozzle Function
Ã¯ÂÂ´ To collect and straighten the gas flow as it comes from the turbine
Ã¯ÂÂ´ To increase the velocity of the gases before discharging from the end
of the exhaust nozzle
Ã¯ÂÂ­ Thrust Reversers
Ã¯ÂÂ´ Partially turns the exhaust forward to produce negative thrust
Ã¯ÂÂ­ Thrust Vectoring
Ã¯ÂÂ´ Changing the nozzle direction in a particular direction
Ã¯ÂÂ­ After Burner or Thrust Augmentation
Ã¯ÂÂ´ Increases thrust by adding fuel to the exhaust gases after passing
turbine section for uncombined oxygen
Ã¯ÂÂ´ Approximately 25% of air passing through engine is consumed by
combustion, and 75% is available for additional combustion if more
fuel is added.
Ã¯ÂÂ´ Most after burners will produce 50% more thrust
Ã¯ÂÂ´ They are used only for take-off, climb, and maximum burst speed
Western Michigan University
9
Department of Mechanical and Aerospace Engineering
Turbine Engine Type by Power Usage: Turbojet & Turbofan
Ã¯ÂÂ­ Turbojet
Ã¯ÂÂ´ All thrust force is derived from the exhaust
gas
Ã¯ÂÂ´ Requires high accelerating of air mass into
the engine
Ã¯ÂÂ´ Minimum extraction of power by turbine
Ã¯ÂÂ´ Drawbacks
Ã¯â€šÂ· Inefficiency from very high exhaust velocity
Ã¯â€šÂ· High Noise due to high exhaust velocity
Ã¯ÂÂ­ Turbofan or Fanjet
Ã¯ÂÂ´ Ducted enclosed fan at the front driven
mechanically by turbine
Ã¯ÂÂ´ Fan acts like a prop, but 30-40 blades
Ã¯ÂÂ´ Large fan gives more thrust
Ã¯ÂÂ´ Approximately 75% of Thrust comes from
fan, and less than 25% comes from the
exhaust gas
Ã¯ÂÂ´ Bypass Ratio of 8:1 for thrust of 90% to 10%
Ã¯ÂÂ´ Less noise
Western Michigan University
10
Department of Mechanical and Aerospace Engineering
Turbine Engine Type by Power Usage: Turboprop & Turboshaft
Ã¯ÂÂ­ Turboprop
Ã¯ÂÂ´ Propeller runs to faster than the turbine, so
gearbox is needed
Ã¯ÂÂ´ Good at lower speed, but less efficient at
higher speed
Ã¯ÂÂ´ Small commuter aircraft, GA, corporate jets
Ã¯ÂÂ´ Great power at less noise
Ã¯ÂÂ´ Lighter than piston engine, but expensive
Ã¯ÂÂ­ Turboshaft
Ã¯ÂÂ´ Similar to Turboprop, but the shaft is one
which the turbine is mounted drives
something other than an aircraft propeller
Ã¯ÂÂ´ Helicopter Engine to turn main rotors
through reduction gear box
Western Michigan University
11
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 14 : Aircraft Performance (1)
Introduction
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Introduction to Aircraft Performance
Ã¯ÂÂ­ The subjects of aircraft performance includes
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
How FAST it will fly : Maximum Velocity
How FAST it will climb or descent: Rate of Climb (R/C) & Rate of Descent (R/D)
How FAST it will take-off or land: Take-off Distance & Landing Distance
How FAR it will fly: Range
How HIGH it will fly: Ceiling
Haw LONG it will fly: Endurance
Ã¯ÂÂ­ Aircraft Performance requires information about
Ã¯ÂÂ´ Four forces on Aircraft: LIFT, DRAG, THRUST, WEIGHT
Ã¯ÂÂ´ The Lift to Drag ratio (L/D) represents the AERODYNAMIC characteristics of a given
airplane
Ã¯ÂÂ´ High L/D means Good Aerodynamic Efficiency
Ã¯ÂÂ´ Thrust to Weight ratio (T/W)
Ã¯ÂÂ´ Wing Loading (W/S)
Ã¯ÂÂ­ Aeronautical Engineers vs. Pilot on L/D
Ã¯ÂÂ´ To aeronautical Engineers, L/D is the most important design parameters
Ã¯ÂÂ´ To pilots, the maximum L/D means minimum drag
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
Performance & Airspeed
Ã¯ÂÂ­ Each aircraft has specific Airspeeds for the best performance
Ã¯ÂÂ´ Engineers need to design / Pilot must know, at what speed his/her plane has
the best performance
Ã¯ÂÂ­ Airspeed Measurement
Ã¯ÂÂ´ PITOT-STATIC System based on BernoulliÃ¢â‚¬â„¢s Equation
Ã¯ÂÂ­ Types of Airspeed (see pg. 7.8 of Pilot’s Handbook ofof Aeronautical
Knowledge)
Ã¯ÂÂ´ Indicated Airspeed (IAS)
Ã¯â€šÂ· Value on the ASI (Airspeed Indicator) in the cockpit
Ã¯ÂÂ´ Equivalent Airspeed (EAS)
Ã¯â€šÂ· EAS for Incompressible Air (M0.3) is the Ã¢â‚¬Å“Compressibility-corrected value of CASÃ¢â‚¬Â
Ã¯ÂÂ´ True Airspeed (TAS): Actual aircraft speed defined from BernoulliÃ¢â‚¬â„¢s Equation
Ã¯â€šÂ· TAS is Ã¢â‚¬Å“Altitude-corrected valueÃ¢â‚¬Â of EAS
Ã¯ÂÂ´ Ground Speed
Ã¯â€šÂ· Aircraft Speed observed by a ground observer (Air Traffic Tower)
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
Level Flight : L=W, T=D
Ã¯ÂÂ­ For airplane flying at level with a given speed, the propulsion system must
produce Thrust equal to the amount of Drag.
Ã¯ÂÂ´ Thrust Required (TR) is the Drag of airplane at a given speed
Ã¯ÂÂ´ Drag Polar Curve = Thrust Required (TR) Curve
Ã¯ÂÂ­ Thrust (Drag) vs. Power
Ã¯ÂÂ´ It is useful to deal with Power rather than Thrust (Drag) which are forces
Ã¯â€šÂ· The concept of POWER is used to characterize Piston-Prop engines
Ã¯â€šÂ· The concept of POWER is useful to deal with efficiencies
Ã¯â€šÂ· Power is defined as Force x Velocity
Ã¯ÂÂ­ Power Required (PR) Curve
Ã¯ÂÂ´ Power required is just Drag x Velocity, and has nothing to do with engine power
Ã¯ÂÂ­ Power Available (PA) Curve from a Piston Engine
Ã¯ÂÂ´ Break Horse Power (BHP): The horse power delivered at the shaft by piston
engine
Ã¯ÂÂ´ Shaft Horse Power (SHP): The horse power delivered to the propeller
Ã¯ÂÂ´ Thrust Horse Power (THP): The amount of horse power that gets converted to
thrust
Ã¯ÂÂ´ THP = (Propulsive Efficiency) x BHP
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
TR, PR, TA, PA & Maximum Speed : Prop Engine Aircraft
Cessna 182 Power Required
Cessna 182 Thrust Required
1000
800
900
700
800
600
600
PR (hp)
TR (LBS)
700
500
400
300
500
400
300
200
200
100
100
0
0
0
50
100
150
200
250
300
0
50
100
TAS (KTS)
150
200
250
300
200
250
300
TAS (KTS)
Cessna 182 TR vs. TA
Cessna 182 PR vs. PA
3000
800
700
2500
Power (hp)
Thrust (LBS)
600
2000
1500
1000
500
400
300
200
500
100
0
0
0
50
100
150
200
250
300
0
50
TAS (KTS)
100
150
TAS (KTS)
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
TR, PR, TA, PA & Maximum Speed : Jet Engine Aircraft
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
Definition of Aircraft Angles in Longitudinal Plane
a: Angle of Attack
g : Flight Path Angle (Angle of Climb)
q : Pitch Angle
X-body axis
(zero-lift-line)
VÃ¯â€šÂ¥
g
Western Michigan University
7
Department of Mechanical and Aerospace Engineering
Definitions of Aircraft Angles in Lateral/Directional Plane
ÃŽÂ¨: Heading Angle
ÃÂ¤: Bank Angle
Western Michigan University
8
Department of Mechanical and Aerospace Engineering
Aircraft Aerodynamic Angles
a: Angle of Attack
b : Side-slip Angle
Western Michigan University
9
AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
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AE 2610 Introduction to Aerospace Engineering: Lecture 15- Supplementary
Dept. of Mechanical & Aerospace Engineering
Western Michigan University
pg. 11/11
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to Aerospace Engineering
Lecture 01: Introduction
Western Michigan University Ã‚Â© Dr. Kapseong Ro
1
Department of Mechanical and Aerospace Engineering
Course Goal & Objectives (1)
Ã¯ÂÂ­ Course Goal
Learn broad conceptual understanding of the major branches of
aerospace engineering discipline leading to aerospace vehicle design
Ã¯ÂÂ­ Course Objectives
Ã¯ÂÂ´ Apply mathematics and physics concept, so far learned, to the solution of
introductory level problems in aerospace engineering
Ã¯ÂÂ´ Introduce fundamental aerospace principals through lectures and
experiments for subsequent courses in aerospace engineering
Ã¯ÂÂ´ Expose to some hands-on skills on experimental techniques in aerospace
engineering
Ã¯ÂÂ´ Motivate the engineering students to pursue education and subsequent
professional career in aerospace engineering
Western Michigan University Ã‚Â© Dr. Kapseong Ro
2
Department of Mechanical and Aerospace Engineering
Course Goal & Objectives (2)
Ã¯ÂÂ­ Laboratory Session
Ã¯ÂÂ´ Expose to various experimental techniques in aeronautical engineering.
Ã¯ÂÂ´ To get hands-on experience and learning by means of Small Wind
Tunnel (SWT), Flight Simulator, Propeller Test Bench, and various
computer software for aeronautical engineering.
Ã¯ÂÂ´ To learn about working as a team member through group experiments,
group report preparation, and group presentation.
Western Michigan University Ã‚Â© Dr. Kapseong Ro
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Department of Mechanical and Aerospace Engineering
Pre/Co-Requisites
Ã¯ÂÂ­ Any pre- & co- requisites?
Ã¯ÂÂ´ Calculus I (MATH 1220 or MATH1700) required
Ã¯ÂÂ´ Physics (PHYS 2050 & PHYS 2060 Mechanics & Heat, Lab) must be
taken concurrently
Ã¯ÂÂ­ Who needs to take this course?
Ã¯ÂÂ´ Anyone who is interested in flying and aircraft.
Ã¯ÂÂ´ All aerospace engineering major students (required)
Ã¯ÂÂ´ Elective for some non aerospace engineering majors
Ã¯ÂÂ´ Pre-requisite for AE 3610.
Western Michigan University Ã‚Â© Dr. Kapseong Ro
4
Department of Mechanical and Aerospace Engineering
What is Aeronautics/Aeronautical Engineering?
Ã¯ÂÂ­ Aeronautics/Aeronautical Engineering is
Ã¯ÂÂ´ the study of the science of flight
Ã¯ÂÂ´ the method of designing an airplane or other flying machine
Ã¯ÂÂ­ There are four basic branches of aeronautics/aeronautical
engineering
Ã¯ÂÂ´ Aerodynamics (AE3610, AE3710)
Ã¯â€šÂ· Study of how air flows around an object such as Airplane
Ã¯ÂÂ´ Propulsion (AE4660 & AE4670)
Ã¯â€šÂ· Study of operating and design principles of an engine that produces the thrust
for an aircraft to take-off and flying through air
Ã¯ÂÂ´ Materials & Structure (AE4630)
Ã¯â€šÂ· Study of materials used in airplane construction, and how those materials
make the plane strong enough to fly effectively
Ã¯ÂÂ´ Performance, Stability & Control (AE3800 & AE4600)
Ã¯â€šÂ· Study of airplane motion such as speed, distance, height, time aloft, turning,
stability and control
Western Michigan University Ã‚Â© Dr. Kapseong Ro
5
Department of Mechanical and Aerospace Engineering
What does an Aeronautical Engineer do?
Ã¯ÂÂ­ Aeronautical engineers must understand the four basic branches of
aeronautical engineering
Ã¯ÂÂ´ To design aircraft (AE 4690 Aircraft Design)
Ã¯ÂÂ­ What is Aerospace Engineering?
Ã¯ÂÂ´ Aerospace Engineering includes
Ã¯â€šÂ· Study of Aeronautics and Astronautics
Ã¯ÂÂ´ Astronautics is
Ã¯â€šÂ· Study of Science and Applied/Engineering Science of the space outside of the
Earth Atmosphere
Ã¯â€šÂ· Aeronautics is common fundamental study for Aerospace Engineering
Ã¯â€šÂ· Astronautics is a specialization of Aerospace Engineering
Western Michigan University Ã‚Â© Dr. Kapseong Ro
6
Department of Mechanical and Aerospace Engineering
Spacecraft, Space Shuttle, Satellite
Western Michigan University Ã‚Â© Dr. Kapseong Ro
7
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to Aerospace Engineering
Lecture 02: History of Aviation
(Ref.: Chapter 1 of Intro. to Flight by J.D. Anderson)
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
History of Aviation: Very Early Development
Ã¯ÂÂ­ Early Thinking: Imitation of a Bird
Ã¯ÂÂ´ Greek Myth: Daedalus & Icarus
Ã¯ÂÂ´ Strapping a pair of wings to
arms, and leaping from towers
or roofs while flapping
vigorously; Always Disastrous!
Ã¯ÂÂ´ Ornithopters (by Leonardo da
Vinci, 1486-1490): wings
flapping up & down powered by
human arm, leg, or body through
mechanism: flapping wing was
always disastrous!
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
History of Aviation: Very Early Development
Ã¯ÂÂ­ The 1st aerial voyage in history:
11/21/1783
Ã¯ÂÂ´ The MontgolfierÃ¢â‚¬â„¢s hot-air balloon
lifts from the ground near Paris,
France
Ã¯ÂÂ´ 25 minutes airborne time
Ã¯ÂÂ´ Lasted for almost 100 years for
human flight
Ã¯ÂÂ´ Aerostatic Flight, No contribution
to human heavier-than-air flight
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
History of Aviation: The True Airplane Inventor
Ã¯ÂÂ­ The True Airplane Inventor, Sir George
Cayley,1773-1857)
Ã¯ÂÂ´ The origin of the modern airplane
Ã¯â€šÂ· Fixed-Wing for LIFT
Ã¯â€šÂ· Separate mechanism for THRUST (by
paddles)
Ã¯â€šÂ· Cruciform Horizontal & Vertical tail for
STABILITY
Ã¯ÂÂ´ Separation of LIFT & THRUST in
contrast to combined LIFT & THRUST
concept of flapping wing ornithopter
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
History of Aviation: The True Airplane Inventor
Ã¯ÂÂ­ Sir George Cayley
Ã¯ÂÂ´ Whirling-arm apparatus for testing
airfoils
Ã¯ÂÂ´ The 1st modern configuration
airplane of history: a glider with
fixed wing and an adjustable h. &
v. tail
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
History of Aviation: The True Airplane Inventor
Ã¯ÂÂ­ Sir George Cayley
Ã¯ÂÂ´ Triplane, the boy carrier
Ã¯â€šÂ· Three wings mounted top of one another
Ã¯â€šÂ· Strictly speaking, it is Ã¢â‚¬Å“poweredÃ¢â‚¬Â airplane; that is equipped with propulsive
flappers
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
History of Aviation: The True Airplane Inventor
Ã¯ÂÂ­ Sir George Cayley
Ã¯ÂÂ´ Human-carrying glider, from
MechanicsÃ¢â‚¬â„¢ Magazine, 1852
Ã¯â€šÂ· A main wing at an angle of incidence for
LIFT, with a dihedral for lateral stability
Ã¯â€šÂ· An adjustable cruciform tail for
longitudinal and directional stability
Ã¯â€šÂ· A pilot-operated elevator and rudder
Ã¯â€šÂ· A fuselage in the form of a car, with a
pilotÃ¢â‚¬â„¢s seat and three-wheel
undercarriage
Ã¯â€šÂ· A tubular beam and box beam
construction
Ã¯ÂÂ´ Not discovered by later aviation
enthusiasts
Western Michigan University
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Department of Mechanical and Aerospace Engineering
History of Aviation: The interregnum
Ã¯ÂÂ­ No major advances in next 50 years
of CayleyÃ¢â‚¬â„¢s success (1853-1891)
Ã¯ÂÂ­ Some considerable activities though
uncoordinated directions
Ã¯ÂÂ´ William Samuel Henson
(1812~1888): Aerial steam carriage
Ã¯â€šÂ· Engine inside a closed fuselage
Ã¯â€šÂ· Driving two propellers
Ã¯â€šÂ· Tricycle landing gear
Ã¯â€šÂ· A single rectangular wing of relatively
high aspect ratio
Ã¯â€šÂ· Keeps CayleyÃ¢â‚¬â„¢s concept. Never built
but well-known publicity for wide
publication, so that one of the most
influential airplane in history
Western Michigan University
8
Department of Mechanical and Aerospace Engineering
History of Aviation: The interregnum
Ã¯ÂÂ­ No major advances in next 50
years of CayleyÃ¢â‚¬â„¢s success
Ã¯ÂÂ­ Some considerable activities
though uncoordinated directions
Ã¯ÂÂ´ John Stringfellow:
Ã¯â€šÂ· Friend of Samuel Henson
Ã¯â€šÂ· Small Steam Engine Builder
Ã¯â€šÂ· Steam powered triplane, a model
exhibited at the 1st aeronautical
exhibition in London, 1868.
Ã¯â€šÂ· Made strong influence upon
Octave Chanute
Western Michigan University
9
Department of Mechanical and Aerospace Engineering
History of Aviation: The interregnum
Ã¯ÂÂ­ Some considerable activities
though uncoordinated
Ã¯ÂÂ´ Felix Du Temple:
Ã¯â€šÂ· First successful powered model
airplane in history using a
monoplane, swept-forward
wings powered by clockwork in
1858
Ã¯â€šÂ· The worldÃ¢â‚¬â„¢s first powered takeoff
by a piloted, full-size airplane
Ã¯â€šÂ· No sustained flight
Ã¯ÂÂ´ Alexander F. Mozhaiski
Ã¯â€šÂ· Powered takeoff using steam
engine
Ã¯â€šÂ· No sustained flight
Western Michigan University
10
Department of Mechanical and Aerospace Engineering
History of Aviation: Otto Lilienthal, Glider Man
Ã¯ÂÂ­ Otto Lilienthal (1848-1896): The Glider Man
Ã¯ÂÂ´ Born & educated in Germany
Ã¯ÂÂ´ Designed & flew the 1st successful controlled gliders in history
Ã¯ÂÂ´ Published a book called Ã¢â‚¬Å“Bird Flight as the Basis of AviationÃ¢â‚¬Â, later read by the
WrightÃ¢â‚¬â„¢s
Ã¯ÂÂ´ Made philosophical conclusion: To learn practical aerodynamics, one needs to get up
in the air Ã¯Æ’Â  lead to design of his own glider in 1889 & 1890, both unsuccessful
Ã¯ÂÂ´ In 1891, the 1st successful glider flew.
Western Michigan University
11
Department of Mechanical and Aerospace Engineering
History of Aviation: Otto Lilienthal, The Glider Man
Ã¯ÂÂ­ Otto Lilienthal (1848-1896): The Glider Man
Ã¯ÂÂ´ Made over 2000 successful glider flights
Ã¯ÂÂ´ Aerodynamic Data obtained were published, with the rise of photography and printing
industry
Ã¯ÂÂ´ The WrightÃ¢â‚¬â„¢s got crystallized idea of flight by LilienthalÃ¢â‚¬â„¢s paper
Ã¯ÂÂ´ Killed in monoplane glider flight subject to a temporary gust on a fine summer day
Western Michigan University
12
Department of Mechanical and Aerospace Engineering
History of Aviation: Percy Pilcher
Ã¯ÂÂ­ Percy Pilcher (1867-1899): Extending the Glider Tradition
Ã¯ÂÂ´ British engineer interested in gliders, and guided by Lilienthal
Ã¯ÂÂ´ Built the Ã¢â‚¬Å“HawkÃ¢â‚¬Â, a hang glider
Ã¯ÂÂ´ Had firm sights set on powered flight
Ã¯ÂÂ´ Calculated an engine specs (40lb weight, 4hp, 5 ft dia. Propeller)
for his Hawk, and bench tested
Ã¯ÂÂ´ Killed in flight of Hawk
Western Michigan University
13
Department of Mechanical and Aerospace Engineering
Aeronautics comes to America
Ã¯ÂÂ­ Octave Chanute (1832-1910)
Ã¯ÂÂ´ French born, naturalized US citizen in Chicago
Ã¯ÂÂ´ Published an aeronautics classic, Ã¢â‚¬Å“Progress in Flying Machines.Ã¢â‚¬Â
Ã¯ÂÂ´ A close relationship and interchange of ideas with the WrightÃ¢â‚¬â„¢s
Ã¯ÂÂ´ Built a successful biplane glider for major contribution to aviation, directly influenced
the Wright Flyer of 1903
Western Michigan University
14
Department of Mechanical and Aerospace Engineering
Aeronautics comes to America
Ã¯ÂÂ­ Samuel p. Langley (1834-1906)
Ã¯ÂÂ´ Secretary of the Smithsonian
Institution
Ã¯ÂÂ´ Designed & built a series of
powered aircraft, two attempted
powered flights just weeks before
the WrightsÃ¢â‚¬â„¢
Ã¯ÂÂ´ Built so called Ã¢â‚¬Å“AerodromesÃ¢â‚¬Â, a
powered (steam engine) flight
model
Western Michigan University
15
Department of Mechanical and Aerospace Engineering
Aeronautics comes to America
Ã¯ÂÂ­ Samuel p. Langley (1834-1906)
Ã¯ÂÂ´ Built a small gasoline engine for
aerodrome power plant with
Charles Manly
Ã¯ÂÂ´ 3.2hp gasoline engine for Ã‚Â¾
scale Aerodrome flight model,
successfully flown
Ã¯ÂÂ´ 52.4hp, 208lb gasoline engine
built for full scale Aerodrome
Ã¯ÂÂ´ Two trials at Potomac river with
Manly onboard, but both were
unsuccessful
Western Michigan University
16
Department of Mechanical and Aerospace Engineering
Wright Brothers: Inventors of the 1st Practical Airplane
Ã¯ÂÂ­ Wilbur (1867-1912) & Orville (1871-1948) Wright
Ã¯ÂÂ´ One of the most important developments in aviation history: Observed that birds Ã¢â‚¬Å“regain their
lateral balance when partly overturned by a gust of wind, by a torsion of the tips of the wingsÃ¢â‚¬Â Ã¯Æ’Â
The use of wing twist to control airplanes in lateral (rolling) motion, and Chanute named the
term Ã¢â‚¬Å“Wing warpingÃ¢â‚¬Â
Ã¯ÂÂ´ In Aug. of 1899, built their first aircraft, a biplane kite with a wingspan of 5 ft with four controlling
strings from the ground to test Wing warping
Ã¯ÂÂ´ In Sept. of 1900, built a full size biplane, a 17ft wingspan and a horizontal elevator in front of the
wings
Western Michigan University
17
Department of Mechanical and Aerospace Engineering
Wright Brothers : Inventors of the 1st Practical Airplane
Ã¯ÂÂ­ Wilbur (1867-1912) & Orville (1871-1948) Wright
Ã¯ÂÂ´ July ~ Aug. 1901, The Wright brotherÃ¢â‚¬â„¢s #2 glider (22ft wing span, biplane) was built
and flight tested at Kill Devil Hills, NC: Almost manned flight
Ã¯ÂÂ´ Discovered that the existing aeronautical data by Lilienthal & Langley were not
accurate & enough Ã¯Æ’Â  built their wind tunnel testing facilities and tested more than
200 different airfoils
Western Michigan University
18
Department of Mechanical and Aerospace Engineering
Western Michigan University
19
Department of Mechanical and Aerospace Engineering
Wright Brothers: Inventors of the 1st Practical Airplane
Ã¯ÂÂ­ Wilbur (1867-1912) & Orville (1871-1948) Wright
Ã¯ÂÂ´ Sept. 1902, The Wright brotherÃ¢â‚¬â„¢s #3 glider (32ft wing span, biplane) was built,
added a movable, vertical rudder behind the wings.
Ã¯ÂÂ´ When this movable rudder was connected with wing warping, the #3 glider made a
smooth, banked turn.
Ã¯ÂÂ´ This combined use of rudder with wing warping was another major contribution of
the WrightÃ¢â‚¬â„¢s to flight control in particular, and aeronautics in general.
Ã¯ÂÂ­ 1000 perfect flights
Ã¯ÂÂ­ 622.5 ft, 26sec
Western Michigan University
20
Department of Mechanical and Aerospace Engineering
Wright Brothers: Inventors of the 1st Practical Airplane
Ã¯ÂÂ­ Wilbur (1867-1912) & Orville
(1871-1948) Wright
Ã¯ÂÂ´ Built their own engine 12hp,
200lb, and designed their own
propeller through their own
research.
Ã¯ÂÂ´ Finally built the Wright Flyer I: 40
ft wingspan, double rudder,
double elevator, gasoline-fueled
Wright engine driving two pusher
props through bicycle chains
Ã¯ÂÂ´ 12/17/1903, Orville became the
history of manned powered flight
of 59seconds, 852ft.
Ã¯ÂÂ´ Not publicized
Western Michigan University
21
Department of Mechanical and Aerospace Engineering
Wright Brothers: Inventors of the 1st Practical Airplane
Ã¯ÂÂ­ Wilbur (1867-1912) & Orville
(1871-1948) Wright
Ã¯ÂÂ´ During 1904, built the Wright
Flyer II near Dayton, OH : 80
brief flight, it has smaller wing
camber, and more powerful
engine, 5min 4sec duration 2.75
mi.
Ã¯ÂÂ´ During 1905, built the Wright
Flyer III: increased airfoil camber,
the 1st practical airplane, 38 min
3sec, 24 mi. performed figure 8
flight
Ã¯ÂÂ´ Evolved to the Wright type A in
1908 with 40hp engine.
Western Michigan University
22
Department of Mechanical and Aerospace Engineering
Technological Race by America vs. Europe after the Wrights
Ã¯ÂÂ­ The WrightÃ¢â‚¬â„¢s were widely accepted as the 1st successful manned,
powered flight, thenÃ¢â‚¬Â¦.
Ã¯ÂÂ´ Technological race between America and Europe begins
Ã¯ÂÂ´ The Wrights machine is statically unstable from the beginning: meaning it
would not fly Ã¢â‚¬Å“by themselvesÃ¢â‚¬Â, rather they had to be constantly controlled by
a pilot.
Ã¯ÂÂ´ Europeans believed in inherently stable aircraft, and French & British built a
controllable but stable airplanes, which is safer & easier to fly.
Ã¯ÂÂ´ Concept of Ã¢â‚¬Å“Static StabilityÃ¢â‚¬Â has carried over to virtually all airplane designs
to the present.
Western Michigan University
23
Department of Mechanical and Aerospace Engineering
The Aeronautical Triangle: Langley, The Wrights & Curtiss
Ã¯ÂÂ­ A relationship that dominated the early development of aeronautics from
1886 ~ 1916.
Alexander Graham Bell
Western Michigan University
24
Department of Mechanical and Aerospace Engineering
Curtiss vs. the Wrights
Western Michigan University
25
Department of Mechanical and Aerospace Engineering
AE 2610 Introduction to Aerospace Engineering
Lecture 05: Basic Aerodynamics – Pressure and Airspeed Measurement
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Static, Dynamic & Total (Stagnation) Pressure
Ã¯ÂÂ­ Static Pressure
Ã¯ÂÂ´ Measure of the purely random motion of molecules in the gas (or fluid)
Ã¯ÂÂ´ The atmospheric pressure in our room is close to 1 ATM (sea level
pressure), and it is STATIC pressure
Ã¯ÂÂ´ In case of moving flow, it is the pressure you feel when you ride along with
the gas (or fluid) at the local flow velocity.
Ã¯ÂÂ­ Dynamic Pressure
Ã¯ÂÂ´ Dynamic pressure is a defined quantity for a moving flow of gas (or fluid)
Ã¯ÂÂ´ Dynamic Pressure, q = (1/2)*ÃÂ*V2
Ã¯ÂÂ­ Total Pressure (= Stagnation Pressure)
Ã¯ÂÂ´ Static Pressure + Dynamic Pressure
Ã¯ÂÂ´ PT = P + q = P + (1/2)*ÃÂ*V2
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
BernoulliÃ¢â‚¬â„¢s Equation Application: Pressure Measurement (I)
Ã¯ÂÂ­ Measurement of Static Pressure
Ã¯ÂÂ´ Manometer: an instrument for comparing pressures; typically a glass U-tube
containing mercury, in which pressure is indicated by the difference in levels
in the two arms of the tube
Ã¯ÂÂ´ Barometer: an instrument for measuring atmospheric pressure, usually to
determine altitude or weather changes (similar to manometer with zero
pressure in sealed end)
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
BernoulliÃ¢â‚¬â„¢s Equation Application: Pressure Measurement (II)
Ã¯ÂÂ­ Measurement of Total Pressure
Ã¯ÂÂ´ Pitot Tube: an open end facing directly in to
the flow (perpendicular to flow direction,
and the other end connected to pressure
gage)
Ã¯â€šÂ· the gas is trapped inside the pressure gage
chamber, and the gas velocity inside the
tube will go to zero (stagnation)
Ã¯ÂÂ­ Measurement of Dynamic Pressure
Ã¯ÂÂ´ Static Pressure Tab: pressure at a small
hole made on a parallel boundary (such as
wall) of flow. Because the flow moves over
the opening, the pressure measured
indicates only the random motion of the
molecules (i.e. static pressure)
Ã¯ÂÂ´ Dynamic Pressure = Total – Static
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
Measurement of Flow Speed (Airspeed)
Ã¯ÂÂ­ Dynamic Pressure & Flow Speed
Ã¯ÂÂ´ Dynamic Pressure is the difference
between Total and Static Pressure
q1 = pT -p1
2 Ã¯â‚¬Â¨ pT -p1 Ã¯â‚¬Â©
V1 =
ÃÂ
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Airspeed Indicator & Other Pitot-static Tube based Instruments
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
Other Application of BernoulliÃ¢â‚¬â„¢s Equation
Ã¯ÂÂ­ Venturi
Ã¯ÂÂ´ A tube with a constriction used to reduce or
control fluid flow, as one in the air inlet of a
carburetor
Western Michigan University
7
Department of Mechanical and Aerospace Engineering
AE 2610 Introduction to Aerospace Engineering
Lecture 05: Basic Aerodynamics – Pressure and Airspeed Measurement
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Static, Dynamic & Total (Stagnation) Pressure
Ã¯ÂÂ­ Static Pressure
Ã¯ÂÂ´ Measure of the purely random motion of molecules in the gas (or fluid)
Ã¯ÂÂ´ The atmospheric pressure in our room is close to 1 ATM (sea level
pressure), and it is STATIC pressure
Ã¯ÂÂ´ In case of moving flow, it is the pressure you feel when you ride along with
the gas (or fluid) at the local flow velocity.
Ã¯ÂÂ­ Dynamic Pressure
Ã¯ÂÂ´ Dynamic pressure is a defined quantity for a moving flow of gas (or fluid)
Ã¯ÂÂ´ Dynamic Pressure, q = (1/2)*ÃÂ*V2
Ã¯ÂÂ­ Total Pressure (= Stagnation Pressure)
Ã¯ÂÂ´ Static Pressure + Dynamic Pressure
Ã¯ÂÂ´ PT = P + q = P + (1/2)*ÃÂ*V2
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
BernoulliÃ¢â‚¬â„¢s Equation Application: Pressure Measurement (I)
Ã¯ÂÂ­ Measurement of Static Pressure
Ã¯ÂÂ´ Manometer: an instrument for comparing pressures; typically a glass U-tube
containing mercury, in which pressure is indicated by the difference in levels
in the two arms of the tube
Ã¯ÂÂ´ Barometer: an instrument for measuring atmospheric pressure, usually to
determine altitude or weather changes (similar to manometer with zero
pressure in sealed end)
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
BernoulliÃ¢â‚¬â„¢s Equation Application: Pressure Measurement (II)
Ã¯ÂÂ­ Measurement of Total Pressure
Ã¯ÂÂ´ Pitot Tube: an open end facing directly in to
the flow (perpendicular to flow direction,
and the other end connected to pressure
gage)
Ã¯â€šÂ· the gas is trapped inside the pressure gage
chamber, and the gas velocity inside the
tube will go to zero (stagnation)
Ã¯ÂÂ­ Measurement of Dynamic Pressure
Ã¯ÂÂ´ Static Pressure Tab: pressure at a small
hole made on a parallel boundary (such as
wall) of flow. Because the flow moves over
the opening, the pressure measured
indicates only the random motion of the
molecules (i.e. static pressure)
Ã¯ÂÂ´ Dynamic Pressure = Total – Static
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
Measurement of Flow Speed (Airspeed)
Ã¯ÂÂ­ Dynamic Pressure & Flow Speed
Ã¯ÂÂ´ Dynamic Pressure is the difference
between Total and Static Pressure
q1 = pT -p1
2 Ã¯â‚¬Â¨ pT -p1 Ã¯â‚¬Â©
V1 =
ÃÂ
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Airspeed Indicator & Other Pitot-static Tube based Instruments
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
Other Application of BernoulliÃ¢â‚¬â„¢s Equation
Ã¯ÂÂ­ Venturi
Ã¯ÂÂ´ A tube with a constriction used to reduce or
control fluid flow, as one in the air inlet of a
carburetor
Western Michigan University
7
Department of Mechanical and Aerospace Engineering
AE 2610 Introduction to Aerospace Engineering
Lecture 06: Basic Aerodynamics – Lift Production (1)
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Description of Aerodynamic Force
Ã¯ÂÂ­ What is Aerodynamic Force?
Ã¯ÂÂ´ Aerodynamic force is produced by the motion of fluid (air) past an object
Ã¯ÂÂ´ It is caused by the net pressure imbalance along the surface of the body
Ã¯ÂÂ´ Lift and Drag are components of aerodynamic force vector that are defined to be
perpendicular and parallel to the flow direction, respectively.
Ã¯ÂÂ­ How is LIFT generated?
Ã¯ÂÂ´ Lift occurs when a flow of gas or liquid is turned (thus its direction is deflected) by a
solid object.
Ã¯ÂÂ´ The flow is turned in one direction, and the lift is generated in the opposite direction by
NewtonÃ¢â‚¬â„¢s 3rd Law (Action & Reaction)
Ã¯ÂÂ´ Because air is gas and its molecules are free to move, any solid surface can deflect a
flow
Ã¯ÂÂ´ For an AIRFOIL, both upper & lower surface contribute the flow turning.
Ã¯ÂÂ­ Incorrect Airfoil Lift Theory
Ã¯ÂÂ´ Neglecting the upper surfaceÃ¢â‚¬â„¢s part in turning the flow leads to an INCORRECT LIFT
THEORY
Ã¯ÂÂ­ NO Fluid, NO Lift & NO Motion, NO Lift
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
Incorrect Lift Theory (I)
Ã¯ÂÂ­ Longer Path or Equal Transit Theory
Ã¯ÂÂ´ Top of airfoil is shaped to provide longer path than bottom, so that air
molecules have farther to go over to the top
Ã¯ÂÂ´ Air molecules must move faster over the top to meet molecules at the
trailing edge that have gone underneath
Ã¯ÂÂ´ From BernoulliÃ¢â‚¬â„¢s equation, higher velocity produce lower pressure on the
top
Ã¯ÂÂ´ Difference in pressure produce lift
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
Incorrect Lift Theory (II)
Ã¯ÂÂ­ Skipping Stone Theory
Ã¯ÂÂ´ Lift is the result of simple ACTION & REACTION as air molecules strike
bottom of the air foil imparting momentum to the foil
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
Incorrect Lift Theory (III)
Ã¯ÂÂ­ Venturi Theory
Ã¯ÂÂ´ Upper surface of airfoil behaves like a Ventrui nozzle constricting the flow
Ã¯ÂÂ´ Through the constriction, flow speeds up (velocity x area = constant)
Ã¯ÂÂ´ From BernoulliÃ¢â‚¬â„¢s equation, high velocity gives low pressure
Ã¯ÂÂ´ Decreased pressure on upper surface produces lift
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Mathematical & Physical Description of Lift
Ã¯ÂÂ­ Mathematical Description : Circulation Theory
Ã¯ÂÂ­ Physical Description
Ã¯ÂÂ´ NewtonÃ¢â‚¬â„¢s Laws of Motion
Ã¯â€šÂ· The 1st Law : Law of Inertia
Ã¯â€šÂ· The 2nd Law : Conservation of Momentum (Force = Mass x Acceleration)
Ã¯â€šÂ· The 3rd Law : Action & Reaction
Ã¯ÂÂ´ Coanda Effect
Ã¯â€šÂ· If a stream of water (air) is flowing along a solid surface which is curved slightly
from the stream, the water will tend to follow the surface.
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
Airfoil & Lift
Ã¯ÂÂ­ Airfoil Lift
Ã¯ÂÂ´ An airfoil speeds up the air flowing over it proportional to the amount of
area obstructed by the upper and lower portions of the airfoil.
Ã¯ÂÂ´ The area of the stream is reduced more over the top surface.
Ã¯ÂÂ´ A greater lowering of pressure on the top surface than on the bottom.
Ã¯ÂÂ´ Pressure force imbalance between the top & the bottom, causing LIFT
Western Michigan University
7
Department of Mechanical and Aerospace Engineering
Airfoil Terminology
Western Michigan University
8
Department of Mechanical and Aerospace Engineering
Angle of Attack (ÃŽÂ±) & Forces (L, D / N, A) on Airfoil
Geometric Relations
L = N cosÃŽÂ± Ã¢â‚¬â€œ A sin ÃŽÂ±
D = N sinÃŽÂ± + A cosÃŽÂ±
L: Lift
D: Drag
A: Axial Force
N: Normal Force
Western Michigan University
9
Department of Mechanical and Aerospace Engineering
Lift on Cambered and Symmetrical Airfoils (I)
Ã¯ÂÂ­ Positive Cambered Airfoil
Ã¯ÂÂ´ At a positive angle of attack, the flow no
longer divides at the tip of leading edge
Ã¯ÂÂ´ The flow divides at a point farther down on
the nose of the airfoil
Ã¯ÂÂ´ The dividing point is called stagnation
point, because flow velocity at this point is
zero. (streamline completely stops)
Ã¯ÂÂ´ Increased effective upper area due to the
change in location of the stagnation point
Ã¯ÂÂ´ The area of the airflow over the top
surface got reduced
Ã¯ÂÂ´ By Continuity & BernoulliÃ¢â‚¬â„¢s Equations:
Lower Area of Airflow Ã¢â€ â€™ Higher Velocity Ã¢â€ â€™
Lower Pressure
Western Michigan University
10
Department of Mechanical and Aerospace Engineering
Lift on Cambered and Symmetrical Airfoils (II)
Ã¯ÂÂ­ Symmetrical Airfoil
Ã¯ÂÂ´ At zero angle of attack the same amount
of flow area is obstructed, resulting in
same pressure loss both on the top & the
bottom
Ã¯ÂÂ´ No pressure difference, no lift
Ã¯ÂÂ´ At a positive angle of attack, the same
effects occur as with the cambered airfoil
(except that it needs more angle of
attack)
Ã¯ÂÂ­ Negative Cambered Airfoil
Ã¯ÂÂ´ Equivalent to a positive cambered airfoil
at inverted flight
Ã¯ÂÂ´ The inverted angle must be great
enough, though, that the effective lower
area of the airfoil (in fact, the upper)
must be greater than the upper (in fact,
the lower).
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11
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 07: Basic Aerodynamics Ã¢â‚¬â€œ Lift Production (2)
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1
Department of Mechanical and Aerospace Engineering
Pressure Distribution on Airfoil
Ã¯ÂÂ­ Pressure varies along the surfaces
Ã¯ÂÂ´ Arrows indicate the amount and the
direction of pressure forces at various
points along the airfoil surface
Ã¯ÂÂ´ Arrows pointing away from the surface
indicate lower pressure than the
surrounding air (Negative Pressure)
Ã¯ÂÂ´ Arrows pointing toward the surface
indicate higher pressure, meaning
surrounding air exert a force on the
surface (Positive Pressure)
Ã¯ÂÂ´ Force = Pressure x Area
Zero Lift
Cruise Lift
Aerodynamic Force =
Ftop.surface Ã¢â‚¬â€œ Fbottom.surface
Lift near Stall
Ã¯ÂÂ´ Component of Aerodynamic Force
perpendicular to the incoming flow
direction is called LIFT
Ã¯ÂÂ´ At a high alpha, pressure at top surface
got much lowered while pressure at
bottom got positive
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2
Department of Mechanical and Aerospace Engineering
Center of Pressure & Pitching Moment
Ã¯ÂÂ­ Center of Pressure
Ã¯ÂÂ´ Pressure is distributed force, and can be replaced
by a single resultant force at a point called
Ã¢â‚¬Å“Center of PressureÃ¢â‚¬Â
Ã¯ÂÂ´ Center of Pressure varies with angle of attack,
and is also different for top & bottom surfaces
(most cases, LSCP is forward of that of USCP)
Ã¯ÂÂ´ This difference causes a Moment (Torque), and
is called Ã¢â‚¬Å“Pitching MomentÃ¢â‚¬Â
Ã¯ÂÂ´ With changes in angle of attack, both the Lift &
Center of Pressure changes
Ã¯ÂÂ´ However, there is a point on the airfoil where the
pitching moment remain constant with all angles
of attack. This point is called Ã¢â‚¬Å“Aerodynamic
CenterÃ¢â‚¬Â
Ã¯ÂÂ´ Most airfoils, Aerodynamic Center is a 25%
(quarter chord) of chord length aft of the leading
edge
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Department of Mechanical and Aerospace Engineering
The Stall
Ã¯ÂÂ­ Lift vs. Angle of Attack
Ã¯ÂÂ´ Lift increases as Angle of Attack increases, but
there is a limit called Ã¢â‚¬Å“Stalling PointÃ¢â‚¬Â
Ã¯ÂÂ´ Stall results from a separated flow due to
Ã¯â€šÂ· Friction force: fluid including air has Ã¢â‚¬Å“stickinessÃ¢â‚¬Â
Ã¯â€šÂ· Pressure gradient
Ã¯ÂÂ´ Separated flow causes turbulent air on the top
surface, and there is no longer pressure lowering
mechanism over the surface. Lift decreases
drastically.
Ã¯ÂÂ´ Lift vs. Angle of Attack curve : linear up to stall
Ã¯ÂÂ´ To reduce the stall chances
Ã¯â€šÂ· Round shape leading edge
Ã¯â€šÂ· Warn pilot by detecting it before it occurs
Ã¯ÂÂ´ Vane-type stall warning: takes advantage of the
relation between stall angle of attack & stagnation
point
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
Early Airfoil Development
Ã¯ÂÂ­ Flat plate can produce LIFT
Ã¯ÂÂ´ Not efficient airfoil (large drag), and sharp LE
cause stall at small AOA
Ã¯ÂÂ´ A flat plate with the nose bent downward
enable the airfoil to achieve much high AOA
without stalling, as well as sharp LE still
induces the separation.
Ã¯ÂÂ´ Sharp LE may be replaced by rounded nose
Ã¢â‚¬â€œ early glider
Ã¯ÂÂ­ Lots of experiments were performed by
early guys
Ã¯ÂÂ´ The Wrights tested 200 different airfoil
shapes
Ã¯ÂÂ­ Design & Test led Airfoil Designation
Ã¯ÂÂ´ Chronological: RAF 6 series, Clark Y series
Ã¯ÂÂ´ Characterized: NACA series
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5
Department of Mechanical and Aerospace Engineering
NACA Airfoil Designation
Ã¯ÂÂ­
Ã¯ÂÂ­
Ã¯ÂÂ­
Ã¯ÂÂ­
NACA 4 digit series airfoil family: e.g. NACA 2412, NACA 0012
NACA 5 digit series airfoil family: e.g. NACA 23012 (Beech Bonanza)
NACA 6 series (Cherokees, Mooneys, P-51 Mustang
NASA Supercritical Airfoil (Richard Whitcomb)
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6
Department of Mechanical and Aerospace Engineering
Lift & Lift Coefficient
Ã¯ÂÂ­ Lift is a force component of the resultant aerodynamic force
Ã¯ÂÂ­ Lift depends on
Flow Speed, VÃ¢Ë†Å¾ (Free-stream velocity)
Flow Density, Ã¯ÂÂ²Ã¢Ë†Å¾ (Free-stream density)
Surface Area, Sref (Reference area)
Incidence angle relative to the free-stream flow direction (Angle
of attack, ÃŽÂ± )
Ã¯ÂÂ´ Flow Viscosity (We will discuss this later when we study Ã¢â‚¬Å“DragÃ¢â‚¬Â
the other component of the resultant aerodynamic force)
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ´
Ã¯ÂÂ­ Lift can be expressed in terms of
Ã¯ÂÂ´ Free-stream Dynamic Pressure : qÃ¢Ë†Å¾ = (1/2)*Ã¯ÂÂ²Ã¢Ë†Å¾*(VÃ¢Ë†Å¾)^2
Ã¯Æ’Â¦1
Ã¯Æ’Â¶
L Ã¯â‚¬Â½ Ã¯Æ’Â§ Ã¯ÂÂ² Ã¯â€šÂ¥VÃ¯â€šÂ¥2 Ã¯Æ’Â· Ã¯Æ’â€” Sref Ã¯Æ’â€” CL Ã¯â‚¬Â¨Ã¯ÂÂ¡ , Re Ã¯â‚¬Â© Ã¯â‚¬Â½ qÃ¯â€šÂ¥ Sref CL Ã¯â‚¬Â¨Ã¯ÂÂ¡ , Re Ã¯â‚¬Â©
Ã¯Æ’Â¨2
Ã¯Æ’Â¸
where Re Ã¯â‚¬Â½ Reynolds Number
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7
Department of Mechanical and Aerospace Engineering
High Lift Devices (Flaps)
Ã¯ÂÂ­ Camber & Lift
Ã¯ÂÂ´ More Camber, More Lift
Ã¯ÂÂ´ More Camber, More Drag
Ã¯ÂÂ´ At landing & take-off, High Lift &
High Drag can be tolerated, but at
other times Drag must be kept at
minimum
Ã¯ÂÂ´ Needs to alter lift during flight
Ã¯ÂÂ­ How to alter Camber during flight?
Ã¯ÂÂ´ Make wing section (airfoil) flexible
=> wing warping
Ã¯ÂÂ´ Make some portion of trailing edge
to be movable, and this movable
surface called FLAP
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8
Department of Mechanical and Aerospace Engineering
High Lift Devices (Flaps)
Western Michigan University
9
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to Aerospace Engineering
Lecture 08: Basic Aerodynamics Ã¢â‚¬â€œ Drag Production
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1
Department of Mechanical and Aerospace Engineering
Pressure Drag, Viscous Drag & Parasite Drag
Ã¯ÂÂ­ Drag
Ã¯ÂÂ´ Drag is the TERM used to denote Resistance to Airflow.
Ã¯ÂÂ´ Any object subject to airflow has associated Drag.
Ã¯ÂÂ­ Pressure, Viscosity & Parasite Drag
Ã¯ÂÂ´ Pressure Drag: force felt by your hand when it is perpendicular to the
direction of flow
Ã¯ÂÂ´ The airflow along the surface of a body creates a frictional force on the
body, due to some viscosity (stickiness).
Ã¯ÂÂ´ This frictional force is called skin friction drag or viscous drag
Ã¯ÂÂ´ Parasite Drag = Pressure Drag + Skin Friction Drag
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2
Department of Mechanical and Aerospace Engineering
Boundary Layers & Reynolds Number (I)
Ã¯ÂÂ­ Boundary Layers
Ã¯ÂÂ´ The viscosity of air causes it to stick at the surface of an object, so the
velocity on the surface is zero for any airflow velocity
Ã¯ÂÂ´ Proceeding away from the surface, the velocity gradually builds up until it
reaches to the free stream velocity.
Ã¯ÂÂ´ The distance is not very great but does have some definite thickness.
Ã¯ÂÂ´ This thickness area is called Boundary Layer
Ã¯ÂÂ­ Laminar & Turbulent Boundary Layers
Ã¯ÂÂ´ Laminar Flow: Smooth, layered fashion in which the streamlines all remain in
the same relative position
Ã¯ÂÂ´ Turbulent Flow: Stream line break up and become all intermingled, moving in
a random, irregular pattern
Ã¯ÂÂ´ Transition: The area where a boundary layer changes from Laminar to
Turbulent flow
Ã¯ÂÂ­ Reynolds Number
Ã¯ÂÂ´ A measure of the viscous qualities of fluid (non-dimensional number)
Ã¯ÂÂ´ Re = (density ÃªÅ¾Â velocity ÃªÅ¾Â length)/(viscosity)
ÃÂVc Vc
Re=
Western Michigan University
ÃŽÂ¼
=
ÃŽÂ½
3
Department of Mechanical and Aerospace Engineering
Viscosity, Boundary Layer, Laminar & Turbulent Flow
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4
Department of Mechanical and Aerospace Engineering
Boundary Layers & Reynolds Number (II)
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5
Department of Mechanical and Aerospace Engineering
Wakes & Pressure Drag (I)
Ã¯ÂÂ­ The relative wake behind
A. curved shape pointed into the wind
B. A flat plate
C. A curved shape pointed downwind
D. An aerodynamic shape
Ã¯ÂÂ­ Flow around a Circular Cylinder
A.
B.
C.
D.
E.
F.
G.
Stagnation Point
Laminar Boundary Layer
Transition Point
Turbulent Broundary Layer
Separation Point
Separated Flow
Wake
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6
Department of Mechanical and Aerospace Engineering
Wake & Pressure Drag (II)
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7
Department of Mechanical and Aerospace Engineering
Wakes & Pressure Drag (III)
Western Michigan University
8
Department of Mechanical and Aerospace Engineering
Drag Coefficient & Drag Quantity (I)
Dynamic Pressure
Ã¯Æ’Â¦1
Ã¯Æ’Â¶
q = Ã¯Æ’Â§ ÃÂV 2 Ã¯Æ’Â·
Ã¯Æ’Â¨2
Ã¯Æ’Â¸
Drag Coefficient
D
CD =
, D = qS Ã¯Æ’â€” C D
qS
Ã¯ÂÂ­ Reference Area, S, used for DRAG
Coefficient calculation
Ã¯ÂÂ´ Drag is generated by 3D body, yet is
proportional to 2D area of body as
defined.
Ã¯ÂÂ´ Customary: projected frontal area for
bluff body (such as fuselage)
Ã¯ÂÂ´ Wetted Area: total surface of body
Ã¯ÂÂ´ Although same projected area
(reference area), Drag coefficient will be
quite different for a bluff body and an
aerodynamically shaped body.
For a 2D Airfoil
D’
cd =
, D’ = qc Ã¯Æ’â€” c d
qc
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9
Department of Mechanical and Aerospace Engineering
Flow over an Airfoil & Drag
Western Michigan University
10
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 09: Basic Aerodynamics Finite Wing Theory 1
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1
Department of Mechanical and Aerospace Engineering
Finite Wing Span Effects (I)
Ã¯ÂÂ­ Airfoil (2-D Flow) vs. Finite Wing (3-D Flow)
Ã¯ÂÂ´ Airfoil: Wing section profile (chord) Ã¢â€ â€™ 2D Flow Analysis
Ã¯ÂÂ´ Finite Wing: Wing planform (chord & finite wingspan) Ã¢â€ â€™ 3D Flow Analysis
Ã¯ÂÂ­ Difference between Flow over an Airfoil vs. a Finite Wing
Ã¯ÂÂ´ Airfoil: Low Pressure on top & High Pressure on bottom
Ã¯ÂÂ´ Finite Wing:
Ã¯â€šÂ· Low Pressure on top & High Pressure on bottom, while Wingtip Vortex at each
wing tip.
Ã¯â€šÂ· As the wing moves forward, this wing tip vortices trails on behind the wing
(sometimes called Trailing Vortices)
Ã¯â€šÂ· Strength of vortices is proportional to aircraft weight (requires more Lift, in turn
great pressure difference) Ã¢â€ â€™ High Pressure Differential cause High Strength
Vortices
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2
Department of Mechanical and Aerospace Engineering
Finite Wing Span Effects (II)
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3
Department of Mechanical and Aerospace Engineering
Finite Wing Span Effects (III)
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4
Department of Mechanical and Aerospace Engineering
Finite Wing Span Effects (V)
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5
Department of Mechanical and Aerospace Engineering
Wingtip Vortices
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6
Department of Mechanical and Aerospace Engineering
Wingtip Vortices
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7
Department of Mechanical and Aerospace Engineering
Finite Wing Downwash Effects
Ã¯ÂÂ­ Downwash
Ã¯ÂÂ´ The tip vortices tend to exert a downward motion to the air leaving the
trailing edge, and this downward push to air is called downwash.
Ã¯ÂÂ´ Downwash pushing downward on airstream causes rearward tilted lift vector
to result.
Ã¯ÂÂ´ With a tilted lift vector, not all of the lift is acting perpendicular to the
incoming airstream (drag is induced, Induced Drag), therefore a little more
angle of attack is needed to make up for this loss of lift.
Ã¯ÂÂ´ This additional angle of attack is called the Induced Angle of Attack
Western Michigan University
8
Department of Mechanical and Aerospace Engineering
Wing Planform Geometry Parameters
Ã¯ÂÂ­ Wing Area
Ã¯ÂÂ­ Average Chord
Ã¯ÂÂ­ Aspect Ratio
Ã¯ÂÂ­ Root Chord & Tip Chord
Ã¯ÂÂ­ Taper Ratio (TR)
Ã¯ÂÂ­ Sweep Angle
Ã¯ÂÂ­ Mean Aerodynamic
Chord
Ã¯ÂÂ­ Dihedral Angle
Ã¯ÂÂ­ Wing Twist Angle
Western Michigan University
9
Department of Mechanical and Aerospace Engineering
Planform vs. Spanwise Lift Distribution
Ã¯ÂÂ­ Spanwise Lift Distribution
Ã¯ÂÂ´ Wingtip vortices cause a reduction in lift to extend inboard, and thus gradually
reduces lift as spanwise direction increase toward the tip.
Ã¯ÂÂ´ Lift at each tip is equal to zero since the top & bottom surface pressures are
equal
Ã¯ÂÂ´ Aerodynamic theory shows that elliptical lift distribution in spanwise direction
has the minimum induced drag
Ã¯ÂÂ´ Elliptical lift distribution requires elliptical wing planform shape
Ã¯ÂÂ´ Manufacturing costs are high for elliptical wing planform
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10
Department of Mechanical and Aerospace Engineering
Planform Shapes vs. Lift Distribution
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11
Department of Mechanical and Aerospace Engineering
Lift Coefficient & Lift Quantity
Force = Pressure Ã¯â€šÂ´ Area
Ã¯Æ’Â¦1 2Ã¯Æ’Â¶
Dynamic Pressure, q= Ã¯Æ’Â§ ÃÂV Ã¯Æ’Â·
Ã¯Æ’Â¨2
Ã¯Æ’Â¸
Wing Area, S
L
Lift Coefficient, CL =
qS
LÃ¯â€šÂ¢
For a 2D Airfoil, cl =
qc
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12
Department of Mechanical and Aerospace Engineering
Lift Controlling Devices (Flaps, Slots & Slats, Spoilers)
Western Michigan University
13
Department of Mechanical and Aerospace Engineering
Lift Controlling Devices (Flaps, Slots & Slats, Spoilers)
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14
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to Aerospace Engineering
Lecture 10: Basic Aerodynamics
(Finite Wing Theory 2)
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Induced Drag
Ã¯ÂÂ­ Finite Wingspan (3-D Wing) causes
Ã¯ÂÂ´ Wingtip Vortices
Ã¯ÂÂ´ Downwash
Ã¯ÂÂ´ Lift vector, being perpendicular to the actual flow, gets tilted backward,
therefore needs more LIFT to balance out the aircraft WEIGHT.
Ã¯ÂÂ´ Lift vector tilted backward resulting in a component of lift in the streamwise
direction, opposite to the flight path.
Ã¯ÂÂ´ This component of force is called INDUCED DRAG, which is a direct result
of Lift production of a finite wingspan aircraft
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
FormulaÃ¢â‚¬â„¢s for Finite Wing Induced Drag
In d u c e d D ra g C oe ffic ie n t
C Di =
Di
, D i = qS Ã¯Æ’â€” C Di
qS
C 2L
C Di =
Ã¯â‚¬Â½ K C 2L ,
Ã¯ÂÂ° eA R
1
w h e re K Ã¯â‚¬Â½
,
Ã¯ÂÂ° eA R
a n d e is c a lle d
“O sw a ld E ffic ie n c y F a c tor”
0 < e Ã¯â€šÂ£ 1 .0 Ã¯ÂÂ­ How do we reduce Induced Drag? Ã¯ÂÂ´ Increase wing efficiency: e=1 for elliptical wing (minimum induced drag) Ã¯ÂÂ´ Increase Aspect Ratio: High Aspect Ratio (Long & Skinny Wing, Low Wingspan) Ã¯ÂÂ´ Reduce Weight Ã¯ÂÂ´ Increase Flight Speed Ã¯ÂÂ´ Fly at low altitude (since the air is more dense at low altitude) A t le ve l fligh t, W e igh t= L ift W W = L = qSC L Ã¯Æ’Å¾ C L Ã¯â‚¬Â½ qS D i = qSC Di Ã¯â‚¬Â½ qSK C 2 L Ã¯â‚¬Â½ qSKW 2 Ã¯â‚¬Â¨q S Ã¯â‚¬Â© 2 KW 2 KW 2 Di Ã¯â‚¬Â½ Ã¯â‚¬Â½ qS Ã¯Æ’Â¦1 2 Ã¯Æ’Â¶ Ã¯Æ’Â§ ÃÂV Ã¯Æ’Â· S Ã¯Æ’Â¨2 Ã¯Æ’Â¸ Western Michigan University 3 Department of Mechanical and Aerospace Engineering Total Airplane Drag Total Airplane Drag DTotal = Dp + Di = (qSÃ¯Æ’â€” CDp ) + (qSÃ¯Æ’â€” CDi ) KW2 Ã¯Æ’Â¦1 2Ã¯Æ’Â¶ DTotal Ã¯â‚¬Â½ Ã¯Æ’Â§ ÃÂV Ã¯Æ’Â·SÃ¯Æ’â€” CDp Ã¯â‚¬Â« Ã¯Æ’Â¦1 2Ã¯Æ’Â¶ Ã¯Æ’Â¨2 Ã¯Æ’Â¸ Ã¯Æ’Â§ ÃÂV Ã¯Æ’Â·S Ã¯Æ’Â¨2 Ã¯Æ’Â¸ K2 2 DTotal Ã¯â‚¬Â½ K1V Ã¯â‚¬Â« 2 V Ã¯ÂÂ­ A total airplane has Ã¯ÂÂ´ Fuselage, External tank, L/G, etc. results in Parasite Drag Ã¯ÂÂ´ Wings (main wing, h-tail, v-tail) results in both Parasite and Induced Drag Ã¯ÂÂ´ Total Airplane Drag = Parasite Drag + Induced Drag Ã¯ÂÂ´ Parasite Drag grows with Velocity Ã¯ÂÂ´ Induced Drag drops off inversely with Velocity Ã¯ÂÂ´ What velocity is the total airplane drag minimum? Western Michigan University 4 Department of Mechanical and Aerospace Engineering Total Airplane Drag Western Michigan University 5 Department of Mechanical and Aerospace Engineering Special Types of Parasite Drag Ã¯ÂÂ­ All airplane drags can be classified as either Parasite or Induced Ã¯ÂÂ´ Parasite Drags include Pressure Drag, Skin Friction Drag Ã¯ÂÂ´ For some special case, some parasite drags may not be explained as Pressure Gradient or Skin Friction effects Ã¯ÂÂ­ Interference Drag Ã¯ÂÂ´ Junction of two different bodies (wing + fuselage + nacelle + pylon + cowling) where small vortices & boundary layer interactions are formed Ã¯ÂÂ­ Cooling Drag Ã¯ÂÂ´ Related with engine cooling, which requires continuous flow of air over the engine parts. Ã¯ÂÂ´ Considerable drag is generated due to pressure and frictional effects from flow around engine parts as needed for engine cooling Ã¯ÂÂ­ Profile Drag Ã¯ÂÂ´ Parasite Drag for a wing only, mostly due to skin friction by large surface area. Ã¯ÂÂ´ At high angle of attack, large amount of pressure drag is generated. Western Michigan University 6 Department of Mechanical and Aerospace Engineering Ground Effect Ã¯ÂÂ­ When flying close to the ground Ã¯ÂÂ´ Wingtip vortices break up some portion by the ground plane Ã¯ÂÂ´ Reduced strength wingtip vortices reduces wing downwash Ã¯ÂÂ´ Reduced wing downwash reduces induced drag Ã¯ÂÂ´ Less Angle of Attack is required Ã¯ÂÂ´ Mostly apparent for Low-Wing configuration aircraft, and when flying flat surface (such as sea surface) Western Michigan University 7 Department of Mechanical and Aerospace Engineering Ground Effect - WIG Western Michigan University 8 Department of Mechanical and Aerospace Engineering Wing Locations relative to Fuselage Ã¯ÂÂ­ Low Wings Ã¯ÂÂ´ Easy landing gear placement & storage Ã¯ÂÂ´ Wing Structure can typically carry through lower fuselage Ã¯ÂÂ´ Engines need long landing gear such that fuselage doesnÃ¢â‚¬â„¢t require upsweep Ã¯ÂÂ­ High Wings Ã¯ÂÂ´ Better Stability Ã¯ÂÂ´ Fuselage closer to the ground (good for military transports) Ã¯ÂÂ´ External struts interfere with lift less Ã¯ÂÂ´ More rooms for big flaps Ã¯ÂÂ­ Mid Wings Ã¯ÂÂ´ Lowest drag configuration Ã¯ÂÂ´ Allows for stores underneath while maintaining visibility above Ã¯ÂÂ´ Best for aerobatics since itÃ¢â‚¬â„¢s symmetric Ã¯ÂÂ­ Biplanes Ã¯ÂÂ´ More structurally efficient Ã¯ÂÂ´ Less span required Ã¯ÂÂ´ Good for aerobatics due to faster roll rates Western Michigan University 9 Department of Mechanical and Aerospace Engineering AE 2610 Introduction to Aerospace Engineering Lecture 15 : Aircraft Performance (2) Climb Performance Analysis Western Michigan University 1 Department of Mechanical and Aerospace Engineering Climb : Excess Power Ã¯ÂÂ­ Airplanes ability to climb depends on power curves (PR & PA) Ã¯ÂÂ´ When A/C climbs, Wing Lift is less than Weight, and thus additional Lift should come from Thrust. Ã¯ÂÂ´ A/C is climbing on Excess Power (or Excess Thrust, or Power Differential) of the engine Western Michigan University 2 Department of Mechanical and Aerospace Engineering Climb : Force Diagram a: Angle of Attack g : Flight Path Angle (Angle of Climb) q : Pitch Angle X-body axis (zero-lift-line) VÃ¯â€šÂ¥ g Western Michigan University 3 Department of Mechanical and Aerospace Engineering Important Climb Performance Speed : Vx vs. Vy Ã¯ÂÂ­ Two Climb Scenarios Ã¯ÂÂ´ The Best Angle of Climb: The steepest climb Ã¯â€šÂ· Pilot wants to gain altitude at the shortest distance Ã¯â€šÂ· Vx : Speed at the maximum climb angle, occurs at maximum Excess Thrust Ã¯ÂÂ´ The Best Rate of Climb (R/C): The fastest climb Ã¯â€šÂ· R/C = Excess Power / Weight Ã¯â€šÂ· Vy : Speed at the maximum R/C, occurs at just above minimum Drag Western Michigan University 4 Department of Mechanical and Aerospace Engineering Climb : Hodograph Ã¯ÂÂ­ Two Climb Scenarios Ã¯ÂÂ´ The Best Angle of Climb: The steepest climb Ã¯ÂÂ´ The Best Rate of Climb (R/C): The fastest climb Ã¯â€šÂ· R/C = EP/ W Ã¯â€šÂ· Vy : Speed at the maximum R/C, occurs at just above minimum Drag 40 35 30 VV (FPS) Ã¯â€šÂ· Pilot wants to gain altitude at the shortest distance Ã¯â€šÂ· Vx : Speed at the maximum climb angle, occurs at maximum Excess Thrust Cessna 182 Hodograph 25 20 15 10 5 0 0 50 100 150 200 250 300 VH (FPS) Western Michigan University 5 Department of Mechanical and Aerospace Engineering AE 2610 Introduction to Aerospace Engineering Lecture 18: Aerospace Materials Western Michigan University 1 Department of Mechanical and Aerospace Engineering Aircraft Structural Materials: Brief History Ã¯ÂÂ­ Pre-1930Ã¢â‚¬â„¢s, Airplane Construction Material Ã¯ÂÂ´ Pre-1930Ã¢â‚¬â„¢s, most airplanes were made with WOOD and FABRIC Ã¯ÂÂ´ Wood Ã¯â€šÂ· Strong, Easy to work, Plentiful Ã¯â€šÂ· It had to have very straight grain Ã¯â€šÂ· It rots, and its properties varied (no manufacturing control) Ã¯ÂÂ´ Fabric Ã¯â€šÂ· Cotton covered the airframe and carried light wing loads Ã¯ÂÂ­ During 1920Ã¢â‚¬â„¢s Ã¯ÂÂ´ Steel tubing replaced wood on many aircraft Ã¯ÂÂ´ Metal alloys were just making progress Ã¯ÂÂ´ Aluminum wasnÃ¢â‚¬â„¢t strong enough Ã¯ÂÂ­ During 1930Ã¢â‚¬â„¢s Ã¯ÂÂ´ Duralumin (invented in Germany 1909) started to be used Ã¯ÂÂ´ More knowledge of materials allowed engineers to create stronger yet nonbrittle metals Western Michigan University 2 Department of Mechanical and Aerospace Engineering Historical Development of Structures & Materials Western Michigan University 3 Department of Mechanical and Aerospace Engineering Aircraft Structural Materials: Traditional Ã¯ÂÂ­ Tube & Fabric Ã¯ÂÂ´ Same construction techniques as with wood Ã¯ÂÂ´ Welded joints are critical and must be Ã¢â‚¬Å“heat treatedÃ¢â‚¬Â Ã¯ÂÂ´ Ã¢â‚¬Å“Tube-and-fabricÃ¢â‚¬Â construction used until the early 50Ã¢â‚¬â„¢s on small planes Ã¯ÂÂ­ Alloys Ã¯ÂÂ´ Alloys are basic metals (iron, aluminum) with added elements Ã¯ÂÂ´ Duralumin : added a little copper and magnesium to aluminum Ã¯ÂÂ´ Steel is an iron alloy (with varying amounts of carbon) Ã¯ÂÂ­ Heat Treating Ã¯ÂÂ´ Crystals of the metal depend on rate of cooling Ã¯ÂÂ´ Quenching (rapid cooling) results in brittle metals Ã¯ÂÂ´ Annealing is the process of heating a metal and slowly cooling it Ã¯ÂÂ­ Aircraft Metals Ã¯ÂÂ´ Steel Alloys: too heavy but used for tubing Ã¯ÂÂ´ Aluminum Alloys: most widely used for modern aircraft Ã¯ÂÂ´ Titanium Alloys: good for high temperature (SR71) but very expensive Western Michigan University 4 Department of Mechanical and Aerospace Engineering Metals Manufacturing Ã¯ÂÂ­ Primary Manufacturing Process Ã¯ÂÂ´ Melting: Liquid casting Ã¯ÂÂ´ Plastic deformation at room temperature Ã¢â‚¬â€œ deforming by high forces Ã¯â€šÂ· Shearing Ã¯â€šÂ· Forming Ã¯â€šÅ¸ Bulk material (hot): Extrusion, Drawing, Forging Ã¯â€šÅ¸ Sheet/plate material (room temperature): Bending, Drawing, Rolling Western Michigan University 5 Department of Mechanical and Aerospace Engineering Aircraft Structural Materials: Composite Ã¯ÂÂ­ Composite Materials Ã¯ÂÂ´ A material system composed of two or more distinct constituents that are mechanically combined to possess unique and desired properties (in general, high strength but low weight) Ã¯ÂÂ´ Example: Fiberglass (Glass fibers & Polyester), Plywood (wood fibers & glue), Carbon Epoxy, Kevlar Ã¯ÂÂ­ Pros & Cons of Composites Ã¯ÂÂ´ Pros: Ã¯â€šÂ· Potential weight savings Ã¯â€šÂ· Potential to be tailored to a specific application Ã¢â‚¬â€œ strength, stiffness, thermal expansion, stealth Ã¯â€šÂ· Complex shapes can be manufactured Ã¯â€šÂ· Reduced part count in a structure, and therefore reduced cost Ã¯ÂÂ´ Cons Ã¯â€šÂ· Difficult to predict strength Ã¯â€šÂ· Less knowledge of the material forces Ã¢â‚¬Å“overbuildingÃ¢â‚¬Â, which negates the weight advantage Ã¯â€šÂ· Engineers are still learning about the fatigue properties of composite Ã¯â€šÂ· Environmental Sensitivity (Temperature, UV, Lightning) Western Michigan University 6 Department of Mechanical and Aerospace Engineering Fiber Reinforced Composite & Aerospace Application Ã¯ÂÂ­ Fiber reinforced plastic (FRP) composite material Ã¯ÂÂ´ Fiber reinforced composite materials consist of Fibers & Matrix Ã¯â€šÂ· Fibers give strength & stiffness Ã¯â€šÂ· Matrix (glue or resin) provides load transfer among fibers Ã¯ÂÂ´ Why thin fibers? Ã¯â€šÂ· The smaller the diameter, the fewer number of internal flaws Ã¯â€šÂ· More bonding surface area Ã¯â€šÂ· More flexibility Ã¯â€šÂ· Molecular alignment Ã¯ÂÂ­ Aerospace Composite Materials Ã¯ÂÂ´ Fibers: Aramid (Kevlar), Glasses, Carbon (Graphite) Ã¯ÂÂ´ Matrix: Polyester, Epoxy Ã¯ÂÂ­ Composites in Airplane Ã¯ÂÂ´ Plywood Ã¢â‚¬â€œ World War II Ã¯ÂÂ´ Polymer matrix composite Ã¯â€šÂ· Graphite/Epoxy, Aramid/Epoxy, Fiberglass Ã¯ÂÂ´ Metal matrix composites Ã¯ÂÂ´ Ceramic matrix composites Western Michigan University 7 Department of Mechanical and Aerospace Engineering Composite Application Trends & Advantage Western Michigan University 8 Department of Mechanical and Aerospace Engineering Composites in Airplane: Civil Transport Western Michigan University 9 Department of Mechanical and Aerospace Engineering Polymers & Ceramics Ã¯ÂÂ­ Types of Polymers Ã¯ÂÂ´ Elastomers: rubbers Ã¯ÂÂ´ Plastics: thermoplastic, thermoset Ã¯ÂÂ´ Fibers: natural fibers, synthetic fibers, nylon Ã¯ÂÂ­ Properties of Polymers Ã¯ÂÂ´ Low temperature: elastic & brittle behavior Ã¯ÂÂ´ Medium temperature: rubbery behavior & glass transition temperature Ã¯ÂÂ´ High temperature: viscous (liquid) Ã¯ÂÂ­ Types of Ceramics Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Glass: window panes, lenses, fibers, etc. Clay: cement, lime, etc. Cements: cement, lime Other Ã¯â€šÂ· Ã¯â€šÂ· Ã¯â€šÂ· Cutting tools & abrasive materials Armor reinforcement Heat resistant (1600-1700Ã¯â€šÂ°C) materials for engines and space shuttle heat protection system Western Michigan University 10 Department of Mechanical and Aerospace Engineering AE 2610 Introduction to Aerospace Engineering Lecture 19 : Aircraft Design Western Michigan University 1 Department of Mechanical and Aerospace Engineering Aircraft Design Process Western Michigan University 2 Department of Mechanical and Aerospace Engineering Design Specifications Ã¯ÂÂ­ Specifications Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Payload (passengers, cargo, or weapons) Range or Endurance Cruise Speed or Top Speed Take-off & Landing Distance (Field Length) Ceiling Economic Requirement: Cost, Fuel Consumption, Maintainability, Reliability Ã¯ÂÂ­ Airworthiness Requirements Ã¯ÂÂ´ FAA Part 23: Light Airplane (12,500 lb MTOW) Design Requirements Ã¯â€šÂ· Airplane Categories: Normal, Utility, Acrobatic Ã¯ÂÂ´ FAA Part 25: Airplane with MTOW > 12,500lb
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Design Phases & Process
Ã¯ÂÂ­ Design Phases
Ã¯ÂÂ´ Conceptual Design (CD)
Ã¯â€šÂ· General Concepts of how the plane will look like
Ã¯â€šÂ· Prop or Jet?; Sing or Multi-engine?; High or Low wing?; fixed or retractable L/G?,
etc.
Ã¯ÂÂ´ Preliminary Design (PD)
Ã¯â€šÂ· Sometimes considered as Aerodynamic Design
Ã¯â€šÂ· Largely, the consideration of aerodynamics in arriving at an overall configuration Ã¢â‚¬â€œ
the external shape of the airplane
Ã¯ÂÂ´ Detail Design (DD)
Ã¯â€šÂ· Sometimes considered as Structural Design
Ã¯â€šÂ· Often, PD is modified during DD
Ã¯ÂÂ­ Design Process
Ã¯ÂÂ´ Aircraft Design needs Trade-Off
Ã¯â€šÂ· It can not achieve the best for all performance, and should be the missionoriented
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Initial Conception
Ã¯ÂÂ­ Study Specifications or Requirements
Ã¯ÂÂ´ Identify Requirements & Corresponding Performance Characteristics
Ã¯ÂÂ´ In general,
Performance Item
Payload capacity
Speed
Range
Take-off Distance
Landing Distance
Climb & Ceiling
Desirable Characteristics
Large size, high power, low W
Low D (small component size), high power
Low W, low D, low SFC, large fuel cap.
High power, large S, high lift airfoil, low D, low W
Large S, high lift airfoil, low W
High power, low D, low W
Ã¯ÂÂ´ Note that some performance items are conflicting requirements
Ã¯ÂÂ´ Design Parameters
Ã¯â€šÂ·
Ã¯â€šÂ·
Ã¯â€šÂ·
Ã¯â€šÂ·
(W/S) : Wing Loading
(P/W) : Power Loading or (T/W): Thrust to Weight Ratio
(L/D) : Glide Ratio (mostly for glide performance)
(L/W) : Load Factor (for turn, pull-up, pull-down maneuver)
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Fuselage Design
Ã¯ÂÂ­ Fuselage is the first part to design & develop
Ã¯ÂÂ´ Fuselage shape is optimized for low drag: Fineness Ratio (l/d) < 3 Ã¯ÂÂ´ Practical fuselage design is usually cylindrical shape (modified from the optimum) Western Michigan University 6 Department of Mechanical and Aerospace Engineering Wing Design Ã¯ÂÂ­ High vs. Low Wing Ã¯ÂÂ´ High Wing has Ã¯â€šÂ· Ã¯â€šÂ· Ã¯â€šÂ· Ã¯â€šÂ· Batter (L/D) ratio (clean upper surface) Better Lateral stability: requires little or no dihedral Shorter Landing Distance (due to the ground effect) Better crash fire protection Ã¯ÂÂ´ Low Wing has Ã¯â€šÂ· Ã¯â€šÂ· Ã¯â€šÂ· Ã¯â€šÂ· Better landing gear support Better roll maneuverability (dihedral) Easier refueling Short takeoff distance (due to the ground effect) Ã¯ÂÂ­ Planform Selection Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Best Aerodynamic Shape: Elliptical (expensive to manufacture) Rectangular Shape is easy to make, but more drag & heavy In between, tapered wing is the design of choice Rectangular up to some spanwise location, and tapered wing to the tip (Cessna & Piper single engine aircraft) Western Michigan University 7 Department of Mechanical and Aerospace Engineering Wing Design Ã¯ÂÂ­ Optimum Airfoil Selection Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Low drag coefficient (Cdo) Minimum drag at design lift coefficient Maximum lift coefficient as high as possible Pitching moment coefficient as near zero as possible Sufficient thickness for spar, fuel, and landing gear Ã¯ÂÂ­ Two important properties Ã¯ÂÂ´ Increasing Thickness will Ã¯â€šÂ· Increase maximum lift coefficient (up to approximately 18%) Ã¯â€šÂ· Increase drag coefficient Ã¯â€šÂ· Provides greater space for structure & fuel Ã¯ÂÂ´ Increasing Camber will Ã¯â€šÂ· Increase design lift coefficient Ã¯â€šÂ· Increase (in negative value) pitching moment Ã¯â€šÂ· Increase lift coefficient at a given angle of attack Western Michigan University 8 Department of Mechanical and Aerospace Engineering Power Plant Selection Ã¯ÂÂ­ For slow-speed light airplane, Piston-Prop is the standard Ã¯ÂÂ´ Mostly Air Cooled, Horizontally-opposed cylinder design Ã¯ÂÂ´ Choose HP based on PR curve Ã¯ÂÂ­ Important Factors in Power Plant Selection Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ If the same power rating, choose the highest P/W ratio SFC (mostly the same for modern engines, 0.42-0.5 lb/(h*BHP) Some other minor factors: Turbocharging, Gearing, Fuel Injection, Gearing Propeller Selection: Engine manufacturer gives the best prop selection Ã¯â€šÂ· Number of blades & diameters Ã¯â€šÂ· Controllable pitch or Fixed Pitch Ã¯â€šÂ· FAA regulation on Propeller (at least 7 inches for Tricycle-geared A/C, 9 inches with tailwheel types) Western Michigan University 9 Department of Mechanical and Aerospace Engineering Wing & Power Plant Type Western Michigan University 10 Department of Mechanical and Aerospace Engineering Wing & Power Plant Type Western Michigan University 11 Department of Mechanical and Aerospace Engineering Landing Gear Configuration Ã¯ÂÂ­ Tail Wheel Type Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Ã¯ÂÂ´ Old Fashioned for easy to make & install Add very little weight Main L/G is usually ahead of CG, and easy to attach to the wing or fuselage Disadvantages Ã¯â€šÂ· CG behind Main L/G -> premature landing impact ->A/C to rotate to high AOA ->
lift increase -> bounce back into air
Ã¯â€šÂ· Yawing tendency (breaking or crosswind) due to CG behind main L/G
Ã¯â€šÂ· Low pilot visibility
Ã¯ÂÂ­ Tricycle Gear
Ã¯ÂÂ´ CG ahead of main L/G, cures the disadvantages of tail wheel type
Ã¯ÂÂ´ Disadvantages
Ã¯â€šÂ· Nose gear carries high load, resulting in very heavy weight contribution
Ã¯â€šÂ· Causing High Drag
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Tail Design
Ã¯ÂÂ­ Horizontal & Vertical Tail
Ã¯ÂÂ´ Stability & Control Issue (Trim, Balance)
Ã¯ÂÂ´ Parameter to design: Horizontal Tail Volume (VH), Vertical Tail Volume (VV)
Ã¯ÂÂ­ Tail Configuration
Western Michigan University
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Spring 2021
pg. 1/9
AE 2610 INTRODUCTION TO AEROSPACE ENGINEERING
Hours &
Location:
MW 3:30PM Ã¢â‚¬â€œ 4:20PM (Lecture) @ https://wmich.webex.com/meet/kapseong.ro
MW 5:30PM Ã¢â‚¬â€œ 7:50PM (Lab) @CEAS F209 & Applied Aerodynamics Lab (Kilgore Rd.)
Instructor:
Office Hours:
Dr. Kapseong Ro (Email: kapseong.ro@wmich.edu) (Tel: 276 Ã¢â‚¬â€œ 3412)
One-to-one or Group-wise Webex meeting upon request
Text:
Reference:
Ã¢â‚¬Å“Introduction to Flight,Ã¢â‚¬Â 8th Edition, John D. Anderson, Jr., McGraw Hill, 2012
1. http://history.nasa.gov/SP-367/cover367.htm
2. FAA-H-8083-25B
Grading:
2 Midterm Exams
Final Exam (Comprehensive)
10 HW Sets
Lab (Group Work)
40%
25%
15%
20%
Course Goal
The objective of this course is to give students a broad conceptual understanding of major branches
of aerospace engineering discipline leading to aerospace vehicle design.
Course Objectives
Ã¯â€šÂ§ To apply mathematics and physics courses to the solution of introductory level problems in aerospace
engineering.
Ã¯â€šÂ§ To introduce the principals of aeronautics for use in subsequent course in aerospace curriculum.
Ã¯â€šÂ§ To expose the student to the principals of aircraft design
Ã¯â€šÂ§ To motivate the engineering students to pursue education and subsequent professional career in
aerospace engineering.
Tentative Topics to be covered:
1. History of Aviation
2. Basic Aerodynamics
3. Propulsion
4. Airplane Performance
5. Stability & Control
6. Structures & Materials
7. High-speed Flight
8. Airplane Design (through DBF)
9. Aerospace Testing (through Lab
Exercises)
Pre-requisite: MATH 1220 or MATH
1700 & PHYS1060 Concurrently
Students are responsible for making themselves aware of and
understanding the University policies and procedures that pertain to
Academic Honesty. These policies include cheating, fabrication,
falsification and forgery, multiple submission, plagiarism, complicity
and computer misuse. The academic policies addressing Student
Rights and Responsibilities can be found in the Undergraduate
Catalog at http://catalog.wmich.edu/index.php?catoid=32 and the
Graduate Catalog at http://catalog.wmich.edu/index.php?catoid=33.
If there is reason to believe you have been involved in academic
dishonesty, you will be referred to the Office of Student Conduct. You
will be given the opportunity to review the charge(s) and if you
believe you are not responsible, you will have the opportunity for a
hearing. You should consult with your instructor if you are uncertain
about an issue of academic honesty prior to the submission of an
assignment or test.
Dept. of Mechanical & Aerospace Engineering, Western Michigan University
Ã‚Â© Dr. Kapseong Ro
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 03
Basic Aerodynamics – Fundamental Theorem (1)
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Airplane Flying in the Air
Ã¯ÂÂ­ What is air?
Ã¯ÂÂ´ Air is a gaseous substance that has weight (Molecular Weight = 29)
Ã¯ÂÂ´ It has molecules that are constantly moving around, creating air pressure.
Ã¯ÂÂ´ Moving air around an object will create force, e.g. LIFT & DRAG
Ã¯ÂÂ­ How does a wing LIFT airplane?
Ã¯ÂÂ´ Airfoil, the section of airplane wings, is curved on the top such that air moves
faster on the top of a wing, slower underneath the wing. The slow air pushes up
from below while the faster air pushes from the top. This forces wing to lift up
into the air
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
Solid vs. Fluids
Ã¯ÂÂ­ Phases of Matter
Ã¯ÂÂ´ In general, the phases of matter are distinguished by the pressure &
temperature transforming into other phases as conditions change to favor
existence of the other form.
Ã¯ÂÂ´ An example is melting and its complement freezing
Ã¯ÂÂ­ Solid vs. Fluids
Ã¯ÂÂ´ Solid
Ã¯â€šÂ· holds a rigid shape without a container
Ã¯ÂÂ´ Fluid
Ã¯â€šÂ· Liquid is incompressible fluid
Ã¯â€šÅ¸ Is able to conform to the shape of its container but retaining a (nearly)
constant volume independent of pressure.
Ã¯â€šÂ· Gas is compressible fluid
Ã¯â€šÅ¸ Not only will a gas conform to the shape of its container but it will expand to
fill the container
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
Solid vs. Fluids
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
Fundamental Quantities
Ã¯ÂÂ­ Aerodynamics is a science that deals with the flow of air
Ã¯ÂÂ´ Aerodynamicist: A person who practices this science
Ã¯ÂÂ´ Hydro-dynamics & Hydro-dynamicist
Ã¯ÂÂ´ Aerostatics vs. Aerodynamics, Hydrostatics vs. Hydrodynamics
Ã¯ÂÂ­ Four Fundamental Physical Properties of Air
Ã¯ÂÂ´ Pressure (P): the normal force per unit area exerted on a surface due to the
momentum change of air (or any gas) molecules impacting on that surface.
Ã¯ÂÂ´ Density (ÃÂ): the mass of given substance (air) per unit volume.
Ã¯ÂÂ´ Temperature (T): a measure of the average kinetic energy of the particles in
the air (or any gas). (Kinetic Energy, KE=(3/2)*k*T, k: Boltzman Constant)
Ã¯ÂÂ´ Velocity (V): the velocity at any point in a flowing air is the velocity of an
infinitesimally small fluid element as it sweeps through that point.
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Understanding Unit System
Ã¯ÂÂ­ Standard International (SI) units vs. Customary (English/Imperial) units
Ã¯ÂÂ­ Consistent Unit System
Ã¯ÂÂ­ Unit Conversion
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 04
Basic Aerodynamics – Fundamental Theorem (2)
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Some Laws from Physics (I)
Ã¯ÂÂ­ Equations of State of Perfect Gas
Ã¯ÂÂ´ A gas (including air) is a collection of particles (molecules : atoms, electrons,
etc.), and each particle is subject to intermolecular forces.
Ã¯ÂÂ´ A perfect gas is one in which intermolecular forces are negligible (each
particle is separated for enough distance)
Ã¯ÂÂ´ A perfect gas has relationships between Pressure, Density & Temperature,
given by P = ÃÂRT, where R is the Universal Gas Constant (=287 joule/kgÃªÅ¾ÂK)
Ã¯ÂÂ´ Air at standard condition is regarded as a perfect gas
Ã¯ÂÂ­ Laws of Conservation of Mass, Momentum & Energy
Ã¯ÂÂ´ Mass is a physical quality expressing the amount of matter in a body.
Ã¯ÂÂ´ Momentum of a body is defined as Mass x Velocity
Ã¯ÂÂ´ Energy is the capacity of a body or system to do work.
Western Michigan University
2
Department of Mechanical and Aerospace Engineering
Some Laws from Physics (II) : Continuity Equation
Ã¯ÂÂ­ Continuity Equation
Ã¯ÂÂ´ Law of Conservation of Mass applied to a moving fluid or gas (airflow =
moving air)
Ã¯ÂÂ´ The net mass flow rate INWARD across any closed surface is equal to the
rate of increase of the mass within the closed surface
Ã¯ÂÂ´ mass flow rate = density x velocity x area, Ã¯ÂÂ²VA
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
Some Laws from Physics (III) : BernoulliÃ¢â‚¬â„¢s Equation
Ã¯ÂÂ­ BernoulliÃ¢â‚¬â„¢s Equation
Ã¯ÂÂ´ Law of Conservation of Momentum and Conservation of Energy applied to a
moving fluid or gas (airflow = moving air)
Ã¯ÂÂ´ Logical Derivation
Ã¯â€šÂ· When an incompressible fluid flows along a horizontal flow tube of varying cross
section, its velocity must change. (Continuity Equation)
Ã¯â€šÂ· A force is required to produce this acceleration, and for this force to be caused by
the fluid surrounding a particular fluid element,
Ã¯â€šÅ¸ The pressure must be different in different regions
Ã¯â€šÅ¸ Same pressure everywhere = no net force on any fluid element
Ã¯ÂÂ´ BernoulliÃ¢â‚¬â„¢s equation is a general expression that relates the pressure
difference between two points in a flow (can be internal flow or external
flow)
Ã¯ÂÂ´ Mathematical Expression: P + (1/2)*ÃÂ*V2 + ÃÂ*g*h = constant
1
1
2
P1+ ÃÂV1 +ÃÂgh1=P2 + ÃÂV22 +ÃÂgh2
2
2
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
BernoulliÃ¢â‚¬â„¢s Equation
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Flow Visualization: Streamlines
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 16: Stability & Control
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Definitions & Terminologies
Ã¯ÂÂ­ Stability & Control
Ã¯ÂÂ´ Stability is the tendency of an airplane to fly a prescribed flight condition
Ã¯ÂÂ´ Control, whether an airplane is stable or unstable, is the ability of a pilot to
change the airplane’s flight conditions
Ã¯ÂÂ­ Equilibrium Flight
Ã¯ÂÂ´ For an airplane to be in equilibrium for a particular flight condition, the sum
of all the forces and moments on it must be zero.
Ã¯â€šÂ· Example: Straight level flight (Lift = Weight, Thrust = Drag)
Ã¯ÂÂ´ Trim or Balance means that No Net Moment exists that tends to move the
airplane out of the current condition. (Center of Gravity = Center of Lift)
Ã¯ÂÂ­ Static Stability vs. Dynamic Stability
Ã¯ÂÂ´ Static Stability deals with the airplaneÃ¢â‚¬â„¢s tendency to return to its original
flight condition when disturbed
Ã¯ÂÂ´ Dynamic Stability deals with how the motion caused by a disturbance
changes with time
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Static Stability
Western Michigan University
3
Department of Mechanical and Aerospace Engineering
Dynamic Stability
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Longitudinal & Lateral/Directional Stability & Control
Ã¯ÂÂ­ Longitudinal Stability & Control (S&C)
Ã¯ÂÂ´ Longitudinal S&C is concerned with an airplaneÃ¢â‚¬â„¢s Pitching motion
Ã¯ÂÂ´ Elevator & Thrust are the means of control
Ã¯ÂÂ­ Lateral/Directional Stability & Control
Ã¯ÂÂ´ Lateral S&C relates to an airplaneÃ¢â‚¬â„¢s Rolling motion
Ã¯ÂÂ´ Directional S&C relations to an airplaneÃ¢â‚¬â„¢s Yawing motion
Ã¯ÂÂ´ These two are closely interrelated, therefore simply referred to as
Lateral/Direction S&C
Ã¯ÂÂ´ Aileron & Rudder are the means of control
Ã¯ÂÂ­ Longitudinal & Lateral/Directional S&C can be considered Independent
Ã¯ÂÂ´ Pitching motion is decoupled from Rolling & Yawing for most aircraft
configurations (XZ-plane of symmetry) and flight conditions
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Center of Gravity, Center of Pressure, Aerodynamic Center
Ã¯ÂÂ­ Definitions
Ã¯ÂÂ´ Center of Gravity: the point through which the resultant of the gravitational
forces (=Weight) on a body always acts.
Ã¯ÂÂ´ Center of Pressure: the point through which the resultant force of pressure
distribution (=Lift & Drag) on airfoil surface applies.
Ã¯â€šÂ· Center of Pressure changes as the angle of attack changes
Ã¯ÂÂ´ Aerodynamic Center: the point on the airfoil that the aerodynamic moment is
independent of angle of attack. (Constant moment no matter what the angle
of attack is)
Ã¯â€šÂ· Aerodynamic Center is very close to the 0.25c (quarter chord)
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Longitudinal Stability : (Flying Wing Ã¢â‚¬â€œ analogous to Airfoil)
Ã¯ÂÂ­ Stability of Airfoil
Ã¯ÂÂ´ Symmetric Airfoil
Ã¯ÂÂ´ Cambered Airfoil
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Longitudinal Stability : Wing & H-Tail Configuration
Ã¯ÂÂ­ Horizontal Stabilizer (or Horizontal Tail)
Ã¯ÂÂ´ One desire is to have an aircraft that is Balanced & Stable, but if the only
balanced wing is neutrally stable, Balance & Stability can not be achieved at
the same time.
Ã¯ÂÂ´ Adding Horizontal Tail allows the adjustment of the Center of Pressure
Ã¯â€šÂ· UP-Lifting HT will make CP moves AFT
Ã¯â€šÂ· DOWN-lifting HT will make CP moves FWD
Ã¯ÂÂ­ Neutral Points of Wing-Horizontal Tail Configuration
Ã¯ÂÂ´ NP is a point of wing-tail configuration where the aerodynamic moment is
independent of angle of attack.
Ã¯ÂÂ´ CG & NP position determines the stability of Wing-Horizontal Tail
Configuration
Ã¯â€šÂ· Stable Wing-Tail Configuration: CG is ahead of NP
Ã¯â€šÂ· Neutrally Stable Wing-Tail Configuration: CG & NP coincides
Ã¯â€šÂ· Unstable Wing-Tail Configuration: CG is behind of NP
Ã¯ÂÂ´ Horizontal TailÃ¢â‚¬â„¢s function is to maintain stability (Not to control pitch, elevator
is for pitch control)
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Longitudinal Stability : Wing-Horizontal Tail Configuration
Ã¯ÂÂ­ STABLE Wing-Horizontal Tail Configuration
1. CG is ahead of NP
2. Balance (Trim)
a. Wing lift produces a torque that tries to push the aircraft nose down
b. H.S. down-lift produces aircraft nose up torque to counter-balance (Not much H.S.
down-lift required due to long moment arm)
c. Due to H.S. down-lift, CP moves FWD to coincide with CG, and aircraft gets
balanced
3. When disturbance increases AOA, Wing produces more lift from increased
AOA
4. H.S. down-lift reduced due to decrease in its negative AOA
5. Due to reduced H.S. down-lift, CP moves AFT, and aircraft is no longer
balanced
6. But it is stable, because there is restoring torque
Western Michigan University
9
Department of Mechanical and Aerospace Engineering
Longitudinal Stability : Wing-Horizontal Tail Configuration
Ã¯ÂÂ­ Neutrally Stable Wing-Horizontal Tail Configuration
1. CG is at NP
2. Balance
a. Wing lift is a little bit ahead of CG
b. H.S. up-lift needs to produce for balance
c. At this situation, the CP is independent of the angle of attack.
3. When disturbance increases AOA, both wing lift & tail lift increase to balance
such that CG remains stationary.
4. There is no restoring moment, but remain balanced.
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Longitudinal Stability : Wing-Horizontal Tail Configuration
Ã¯ÂÂ­ Unstable Wing-Horizontal Tail Configuration
Ã¯ÂÂ´ CG is ahead of NP
Ã¯ÂÂ´ Balance (Trim)
Ã¯â€šÂ·
Ã¯â€šÂ·
Ã¯â€šÂ·
Wing lift produces a torque that tries to pull the A/C nose up
HS up-lift produces aircraft nose down torque to counter-balance
Due to HS up-lift, the aircraft gets balanced
Ã¯ÂÂ´ When disturbance increases AOA, Wing produces more lift from increased
AOA
Ã¯ÂÂ´ H.S. up-lift can not catch up the growing wing lift that causes nose-up moment
to counter-balance.
Ã¯ÂÂ´ The aircraft gets farther apart from the initial flight condition
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Longitudinal Stability : Wing-Horizontal Tail Configuration
Ã¯ÂÂ­ Why the Wing Lift grows faster than Tail lift?
Ã¯ÂÂ´ Tail is aerodynamically less efficient since it is Low AR (short & stubby)
Ã¯ÂÂ´ Wing Downwash, less AOA for tail (e.g. 10 degree Wing AOA increase but only
6 degree Tail AOA increase)
Ã¯ÂÂ­ Aircraft Center of Gravity must be maintained
Ã¯ÂÂ´ To maintain static stability (CG must be ahead of NP)
Ã¯ÂÂ´ Unstable aircraft possible, but requires continuous control (pilot work-load)
Ã¯ÂÂ´ aircraft CG must be maintained for all flight conditions, so called the CG
envelope.
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Computation of Aircraft CG
Work out Examples
Western Michigan University
13
Department of Mechanical and Aerospace Engineering
AE 2610
Introduction to
Aerospace Engineering
Lecture 17: Aircraft Structures
Western Michigan University
1
Department of Mechanical and Aerospace Engineering
Solid vs. Fluid & Applied Forces
Ã¯ÂÂ­ Solid vs. Fluid (Liquid or Gas)
Ã¯ÂÂ´ The molecules of solid is closely compact together and forming a distinctive
boundary, and the intermolecular force resists to the external force to
maintain the equilibrium.
Ã¯â€šÂ· Finite Deformation under the externally applied force
Ã¯ÂÂ´ The molecules of fluid is very loose and cannot maintain its boundary, and
has no resistance to the external force
Ã¯â€šÂ· Continuous Deformation under the externally applied force
Ã¯ÂÂ­ Types of Externally Applied Force
Ã¯ÂÂ´ Normal Force: Forces applied perpendicular to the surface
Ã¯â€šÂ· Compression or Compressive Force
Ã¯â€šÂ· Tension or Tensile Force
Ã¯ÂÂ´ Tangential Force: Forces applied parallel to and on the surface
Ã¯â€šÂ· Shear Force
Ã¯â€šÂ· Friction Force
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Stress & Strain
Ã¯ÂÂ­ Stress
Ã¯ÂÂ´ The intermolecular force that resists to the externally force is called the Internal
Force
Ã¯ÂÂ´ The internal force per unit area is called STRESS, and denoted by Ã¯ÂÂ³ (normal
stress) and Ã¯ÂÂ´ (shear stress), Unit: N/m2, lb/ft2
Ã¯ÂÂ­ Strain
Ã¯ÂÂ´ The change in length per unit length is called STRAIN, and denoted by Ã¯ÂÂ¥
(normal strain) and Ã¯ÂÂ± (shear strain). Unit: No dimension or in radian for shear
strain
Ã¯ÂÂ­ HookeÃ¢â‚¬â„¢s Law
Ã¯ÂÂ´ The stress is directly proportional to the strain up to a certain limiting value of
the stress, called the yield stress,
Ã¯â€šÂ· i.e. (Ã¯ÂÂ³ = EÃŽâ€¡Ã¯ÂÂ¥) or (Ã¯ÂÂ´ = G ÃŽâ€¡Ã¯ÂÂ±)
Ã¯ÂÂ´ Proportionality Constant
Ã¯â€šÂ· YoungÃ¢â‚¬â„¢ Modulus or Modulus of Elasticity (E)
Ã¯â€šÂ· Shear Modulus or Modulus of Rigidity (G)
Ã¯ÂÂ­ Stress-Strain Diagram
Western Michigan University
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Department of Mechanical and Aerospace Engineering
Types of Structure: Beams, Trusses, Plates & Shells
Ã¯ÂÂ­ Beams
Ã¯ÂÂ´ In bending: an axial bar with
transverse load
Ã¯ÂÂ´ In torsion: an axial bar with
torque
Ã¯ÂÂ´ Example: Wing in Flight is
analogous to beam in bending
& torsion
Ã¯ÂÂ­ Trusses
Ã¯ÂÂ´ A truss is a collection of beams
Ã¯ÂÂ´ Example: Bridges, Fuselage
and wings of older airplanes
Ã¯ÂÂ­ Plates & Shells
Ã¯ÂÂ´ They carry shear forces, and it
can be more efficient than
rectangular beams
Ã¯ÂÂ´ Example: fuselages & modern
wings
Western Michigan University
4
Department of Mechanical and Aerospace Engineering
Typical Aircraft Structural Elements (I)
Western Michigan University
5
Department of Mechanical and Aerospace Engineering
Aircraft Main Structural Elements (II)
Ã¯ÂÂ­
Ã¯ÂÂ­
Ã¯ÂÂ­
Ã¯ÂÂ­
Wing Box & Carry-through
Main Spars
Bulkheads
Longerons
Western Michigan University
6
Department of Mechanical and Aerospace Engineering
Aircraft Structure Ã¢â‚¬â€œ Typical Wing Structure (I)
Western Michigan University
7
Department of Mechanical and Aerospace Engineering
Aircraft Structure Ã¢â‚¬â€œ Typical Wing Structure (II)
Western Michigan University
8
Department of Mechanical and Aerospace Engineering
Aircraft Structure Ã¢â‚¬â€œ Typical Fuselage Structure (I)
Western Michigan University
9
Department of Mechanical and Aerospace Engineering
Aircraft Structure Ã¢â‚¬â€œ Typical Fuselage Structure (II)
Western Michigan University
10
VII Power Series Methods
VII
Power Series Methods
1/8
7.2. Power Series Solutions
Example
Write the first four nonzero terms for each of the power series
solution of y 00 Ã¢Ë†â€™ 2xy 0 + 3y = 0, and determine the interval of
convergence of each series.
P
n
We write the solution as yP
(x) = Ã¢Ë†Å¾
Solution:
n=0 cn x . Then
P
Ã¢Ë†Å¾
Ã¢Ë†Å¾
00
0
nÃ¢Ë†â€™1
and y (x) = n=2 n(n Ã¢Ë†â€™ 1)cn x nÃ¢Ë†â€™2 .
y (x) = n=1 ncn x
Substituting,
P
PÃ¢Ë†Å¾
PÃ¢Ë†Å¾
Ã¢Ë†Å¾
nÃ¢Ë†â€™2
nÃ¢Ë†â€™1
n
n(n
Ã¢Ë†â€™
1)c
x
Ã¢Ë†â€™
2x
nc
x
+
3
n
n
n=0 cn x = 0. Or,
Pn=2
PÃ¢Ë†Å¾ n=1 n PÃ¢Ë†Å¾
Ã¢Ë†Å¾
nÃ¢Ë†â€™2
n
Ã¢Ë†â€™ n=1 2ncn x + n=0 3cn x = 0. Shifting
n=2 n(n Ã¢Ë†â€™ 1)cn x
th…
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