FLIGHT MECHANICS BDA 20603 DR. ZAMRI BIN OMAR D1-206 zamri@uthm.edu.my 4537618 Books: Roskam, Anderson, Nelson, etc.

LECTURE OUTLINE Flight Mechanics is composed of: Airplane Performance Airplane Stability and Control What is maximum speed of airplane? How fast can it climb to a given altitude? How far can it fly on a given tank of fuel (Range)? How long can it stay in the air (Endurance)? Can the airplane land in a suitable distance? Is airplane stable (wind, weight, etc.)? Can airplane be controlled?

FLIGHT MECHANICS: LECTURE OUTLINE Behavior of entire airplane How fast can this airplane fly? How far can this airplane fly on a single tank of fuel? How long can this airplane stay in the air on a single tank of fuel? How fast and how high can it climb? How does it perform?

EXAMPLE: CONCORDE Supersonic Jetliner : The one and only. Wing Span: 25.6m Wing Area: 358.3 m2 Length: 61.66 m Height: 12.2 m Useful load: 111130 kg Empty Weight: 78700 kg Cruise speed: Mach 2.02 Range : 7250 km Mach 2.02=2,472 km/h Crew: 3 (pilot, co-pilot, and flight engineer) Capacity: 92–120 passengers (128 in high-density layout)[223] Length: 202 ft 4 in (61.66 m) Wingspan: 84 ft 0 in (25.6 m) Height: 40 ft 0 in (12.2 m) Fuselage internal length: 129 ft 0 in (39.32 m) Fuselage width: maximum of 9 ft 5 in (2.87 m) external 8 ft 7 in (2.62 m) internal Fuselage height: maximum of 10 ft 10 in (3.30 m) external 6 ft 5 in (1.96 m) internal) Wing area: 3,856 ft2 (358.25 m2) Empty weight: 173,500 lb (78,700 kg) Useful load: 245,000 lb (111,130 kg) Powerplant: 4 × Rolls-Royce/SNECMA Olympus 593 Mk 610 afterburning turbojets Dry thrust: 32,000 lbf (140 kN) each Thrust with afterburner: 38,050 lbf (169 kN) each Maximum fuel load: 210,940 lb (95,680 kg) Maximum taxiing weight: 412,000 lb (187,000 kg) Performance Maximum speed: Mach 2.04[224] (≈1,350 mph, 2,172 km/h) at cruise altitude Cruise speed: Mach 2.02[224] (≈1,320 mph, 2,124 km/h) at cruise altitude Range: 3,900 nmi (4,500 mi, 7,250 km) Service ceiling: 60,000 ft (18,300 m) Rate of climb: 5,000 ft/min (25.41 m/s) lift-to-drag: Low speed– 3.94, Approach– 4.35, 250 kn, 10,000 ft– 9.27, Mach 0.94– 11.47, Mach 2.04– 7.14 Fuel consumption: 46.85 lb/mi (13.2 kg

EXAMPLE: AIRBUS A380 / BOEING 747 COMPARISON Wingspan: 79.8 m AR: 7.53 GTOW: 560 T Wing Loading: GTOW/b2: 87.94 Wingspan: 68.5 m AR: 7.98 GTOW: 440 T Wing Loading: GTOW/b2: 93.77

OVERALL AIRPLANE DRAG Wing or airfoil Entire Airplane No longer concerned with aerodynamic details Drag for complete airplane, not just wing Wing or airfoil Entire Airplane Tail Surfaces Engine Nacelles Landing Gear

DRAG POLAR CD,0 is parasite drag coefficient at zero lift (aL=0) CD,i drag coefficient due to lift (induced drag) Oswald efficiency factor, e, includes all effects from airplane CD,0 and e are known aerodynamics quantities of airplane Example of Drag Polar for complete airplane

4 FORCES ACTING ON AIRPLANE Model airplane as rigid body with four natural forces acting on it Lift, L Acts perpendicular to flight path (always perpendicular to relative wind) Drag, D Acts parallel to flight path direction (parallel to incoming relative wind) Propulsive Thrust, T For most airplanes propulsive thrust acts in flight path direction May be inclined with respect to flight path angle, aT, usually small angle Weight, W Always acts vertically toward center of earth Inclined at angle, q, with respect to lift direction Apply Newton’s Second Law (F=ma) to curvilinear flight path Force balance in direction parallel to flight path Force balance in direction perpendicular to flight path

GENERAL EQUATIONS OF MOTION (6.2) Free Body Diagram Parallel to flight path:

GENERAL EQUATIONS OF MOTION (6.2) Free Body Diagram Parallel to flight path: Perpendicular to flight path:

STATIC VS. DYNAMIC ANALYSES Examine two forms of these equations: Static Performance: Zero Accelerations (dV/dt = 0, V2/rc = 0) Maximum velocity Maximum rate of climb Maximum range Maximum endurance Dynamic Performance: Accelerating Flight Take-off and landing characteristics Turning flight Accelerated flight and rate of climb

LEVEL, UNACCELERATED FLIGHT Equations of motion reduce to very simple expressions Aerodynamic drag is balanced by thrust of engine Aerodynamic lift is balanced by weight of airplane Level means horizontal, so q = 0° Unaccelerated flight means all accelerations = 0 For most conventional airplanes aT is small enough such that cos(aT) ~ 1

FORCES ON AN AIRPLANE

ACTUAL FORCES ON AN AIRPLANE

LEVEL, UNACCELERATED FLIGHT TR is thrust required to fly at a given velocity in level, unaccelerated flight Notice that minimum TR is when airplane is at maximum L/D L/D is an important quantity Airplane’s power plant must produce a net thrust which is equal to drag

THRUST REQUIREMENT (6.3) TR for airplane at given altitude varies with velocity Thrust required curve: TR vs. V∞

PROCEDURE: THRUST REQUIREMENT Select a flight speed, V∞ Calculate CL Minimum TR when airplane flying at (L/D)max Calculate CD Calculate CL/CD Calculate TR This is how much thrust engine must produce to fly at selected V∞

THRUST REQUIREMENT (6.3) Usually around 2º-8º Thrust required, TR, varies inversely with L/D Minimum TR when airplane is flying at L/D is a maximum L/D is a measure of aerodynamic efficiency of an airplane Maximum aerodynamic efficiency → Minimum TR Recall Homework Problem #5.6, find (L/D)max for NACA 2412 airfoil Usually around 2º-8º

THRUST REQUIREMENT (6.3) Different points on TR curve correspond to different angles of attack At b: Small q∞ Large CL (or CL2) and a to support W D large At a: Large q∞ Small CL and a D large

THRUST REQUIRED VS. FLIGHT VELOCITY Zero-Lift TR (Parasitic Drag) Lift-Induced TR (Induced Drag) Zero-Lift TR ~ V2 (Parasitic Drag) Lift-Induced TR ~ 1/V2 (Induced Drag)

THRUST REQUIRED VS. FLIGHT VELOCITY At point of minimum TR, dTR/dV∞=0 (or dTR/dq∞=0) CD,0 = CD,i at minimum TR and maximum L/D Zero-Lift Drag = Induced Drag at minimum TR and maximum L/D

MAXIMUM VELOCITY Intersection of TR curve and maximum Maximum flight speed occurs when thrust available, TA=TR Reduced throttle settings, TR < TA Cannot physically achieve more thrust than TA which engine can provide Intersection of TR curve and maximum TA defined maximum flight speed of airplane

AIRPLANE POWER PLANTS Consider two types of engines common in aviation today Reciprocating piston engine with propeller Average light-weight, general aviation aircraft Rated in terms of POWER Jet (Turbojet, turbofan) engine Large commercial transports and military aircraft Rated in terms of THRUST

THRUST VS. POWER Jets Engines (turbojets, turbofans for military and commercial applications) are usually rate in Thrust Thrust is a Force with units (N = kg m/s2) For example, the PW4000-112 is rated at 98,000 lb of thrust Piston-Driven Engines are usually rated in terms of Power Power is a precise term and can be expressed as: Energy / time with units (kg m2/s2) / s = kg m2/s3 = Watts Note that Energy is expressed in Joules = kg m2/s2 Force * Velocity with units (kg m/s2) * (m/s) = kg m2/s3 = Watts Usually rated in terms of horsepower (1 hp = 550 ft lb/s = 746 W) Example: Airplane is level, unaccelerated flight at a given altitude with speed V∞ Power Required, PR=TR*V∞ [W] = [N] * [m/s]

AIRPLANE POWER PLANTS

Propeller Drive Engine POWER AVAILABLE (6.6) Propeller Drive Engine Jet Engine

POWER REQUIRED (6.5) PR varies inversely as CL3/2/CD Recall: TR varies inversely as CL/CD

POWER REQUIRED (6.5) PR vs. V∞ qualitatively (Resembles TR vs. V∞)

POWER REQUIRED (6.5) Zero-Lift PR Lift-Induced PR Zero-Lift PR ~ V3 Lift-Induced PR ~ 1/V

POWER REQUIRED At point of minimum PR, dPR/dV∞=0

POWER REQUIRED V∞ for minimum PR is less than V∞ for minimum TR

WHY DO WE CARE ABOUT THIS? We will show that for a piston-engine propeller combination To fly longest distance (maximum range) we fly airplane at speed corresponding to maximum L/D To stay aloft longest (maximum endurance) we fly the airplane at minimum PR or fly at a velocity where CL3/2/CD is a maximum Power will also provide information on maximum rate of climb and altitude

POWER AVAILABLE AND MAXIMUM VELOCITY (6.6) Propeller Drive Engine PA PR 1 hp = 550 ft lb/s = 746 W

POWER AVAILABLE AND MAXIMUM VELOCITY (6.6) Jet Engine PA = TAV∞ PR

ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7) Recall PR = f(r∞) Subscript ‘0’ denotes seal-level conditions

Vmax,ALT < Vmax,sea-level ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7) Propeller-Driven Airplane Vmax,ALT < Vmax,sea-level

RATE OF CLIMB (6.8) Boeing 777: Lift-Off Speed ~ 180 MPH How fast can it climb to a cruising altitude of 30,000 ft?

RATE OF CLIMB (6.8) Governing Equations:

RATE OF CLIMB (6.8) Rate of Climb: TV∞ is power available Vertical velocity Rate of Climb: TV∞ is power available DV∞ is level-flight power required (for small q neglect W) TV∞- DV∞ is excess power

Propeller Drive Engine RATE OF CLIMB (6.8) Propeller Drive Engine Jet Engine Maximum R/C Occurs when Maximum Excess Power

EXAMPLE: F-15 K Weapon launched from an F-15 fighter by a small two stage rocket, carries a heat-seeking Miniature Homing Vehicle (MHV) which destroys target by direct impact at high speed (kinetic energy weapon) F-15 can bring ALMV under the ground track of its target, as opposed to a ground-based system, which must wait for a target satellite to overfly its launch site.

GLIDING FLIGHT (6.9) To maximize range, smallest q occurs at (L/D)max

EXAMPLE: HIGH ASPECT RATIO GLIDER q To maximize range, smallest q occurs at (L/D)max A modern sailplane may have a glide ratio as high as 60:1 So q = tan-1(1/60) ~ 1°