Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 1 Aircraft Equations of Motion -Objectives 1 • • Axis systems Coordinate transformation Aircraft Equations of Motion 2 • 3 6 dof forces and moments Eqs. • EOM into longitudinal and lateral motions AE-426 Flight Dynamics --- 201 4 • Kinematic Eqs. And its need 2 Aircraft Equations of Motion -Introduction •What is Flight Dynamics? •Is the study of aircraft motion and its characteristics due to internally or externally generated disturbances. Aircraft Equations of Motion AE-426 Flight Dynamics -- 201 3 Aircraft Equations of Motion -Introduction Dynamics Studying the motion with acceleration of particles and rigid bodies (Kinetics) Kinematics Studying the motion including the effects of forces on motion Describing motion i.e. position, velocity, and acceleration, without reference to forces Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 4 Aircraft Equations of Motion -Introduction Six degrees of freedom (6-DOF) motions Experiences motions in three dimensions - 3 translational degrees describes the trajectory - 3 rotational degrees describes the orientation Rigid Aircraft Body Derive the general equations of motion (EOM) Aircraft Equations of Motion Define the axis system, or coordinate systems, or reference frame in which these equations are derived AE-426 Flight Dynamics --- 201 5 Aircraft Equations of Motion -Aircraft Axis Systems Why we need axis systems? • To derive aircraft EOM. • To define a particular vector (aerodynamic forces, weight, thrust, …) Characteristics of these axis systems • Right-hand Orthogonal Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 6 Aircraft Equations of Motion -Aircraft Axis Systems (Cont.) Earth Axis System (Non-Rotating): XE, YE, ZE • Inertial. Why? • Rotational velocity of the earth is neglected • Acceptable for supersonic airplanes but not hypersonic • Fixed on Earth, XE to North, YE to East, ZE to center of Earth. Body Axis System (Rotating): Xb, Yb, Zb • Non-inertial. • Fixed on aircraft c.g., Xb out to nose, Yb out to right wing, Zb down of aircraft. • Easy to find moments and products of inertia. cg α Yb Ys Xb Zs Stability Axis System: Xs, Ys, Zs • Obtained by rotating body axis by angle of attack α. • Xs in direction of relative wind V∞, Ys out to right wing, Zs down of aircraft. Zb Xs O XE V∞ AE-426 Flight Dynamics --- 201 7 YE Aircraft Equations of Motion ZE Aircraft Equations of Motion -Coordinate Transformations Weight Vector •πΉ =πΉ =πΉ 0 = 0 π 0 = 0 ππ Aerodynamic Forces •πΉ −π· =πΉ = πΉ −πΏ Thrust Vector •πΉ Aircraft Equations of Motion π cos π 0 =πΉ = −π sin π AE-426 Flight Dynamics --- 201 8 Aircraft Equations of Motion -Coordinate Transformations (Cont) cg O XE Yb YE ZE Zb Aircraft Equations of Motion Xb AE-426 Flight Dynamics --- 201 9 Aircraft Equations of Motion -Coordinate Transformations (Cont) Earth to Body Axis Transformation • Three consecutive Euler rotations through Euler angles in order. β 1st rotation through YAW (Heading) angle Ψ. β 2nd rotation through PITCH angle Θ. β 3rd rotation through ROLL angle Φ. Earth Axis System 0 • Angles limits: −90β −180β Aircraft Equations of Motion Ψ, Θ, Φ Body Axis System ≤ Ψ ≤ 360β ≤ Θ ≤ 90β ≤ Φ ≤ 180β AE-426 Flight Dynamics --- 201 10 Aircraft Equations of Motion -Coordinate Transformations (Cont) STEP (1): Rotate πΉβ = π π€Μ + π π₯Μ + π π in Earth axis by Ψ about z-axis to get: STEP (2): Rotate πΉβ = π π€Μ + π π₯Μ + π π in intermediate axis by Θ about y-axis to get: • Intermediate axis π€Μ , π₯Μ , π with πΉβ = π π€Μ + π π₯Μ + π π • π = +π cos Ψ + π sin Ψ • π = −π sin Ψ + π cos Ψ • π = +π cos Ψ sin Ψ 0 π π • π = − sin Ψ cos Ψ 0 π 0 0 1 π π • Another intermediate axis π€Μ π π₯Μ + π π • π = π cos Θ − π sin Θ •π =π • π = π sin Θ + π cos Θ cos Θ 0 − sin Θ π • π = 0 1 0 sin Θ 0 cos Θ π • πΉβ = π Ψ πΉβ Aircraft Equations of Motion , π₯Μ , π with πΉβ = π π€Μ + π π π • πΉβ = π Θ πΉβ = π Θ π Ψ πΉβ AE-426 Flight Dynamics --- 201 STEP (3): Rotate πΉβ = π π€Μ + π π₯Μ + π π in intermediate axis by Φ about x-axis to get: • Body axis π€Μ, π₯Μ, π with πΉβ = π • π = +π • π = +π cos Φ + π sin Φ • π = −π sin Φ + π cos Φ π 1 0 0 • π = 0 cos Φ sin Φ π 0 − sin Φ cos Φ π€Μ + π π₯Μ + π π π π π • πΉβ = π Φ πΉβ = π Φ π Θ π Ψ πΉβ 11 Aircraft Equations of Motion -Coordinate Transformations (Cont) Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 12 Aircraft Equations of Motion -Coordinate Transformations (Cont) Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 13 Aircraft Equations of Motion -Coordinate Transformations (Cont) Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 14 Aircraft Equations of Motion -Coordinate Transformations (Cont) Stability to Body Axis Transformation • Required to transform aerodynamic forces. • Rotate stability axis system about y-axis through positive angle of attack Body Axis System πΌ about y Stability Axis System • Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 15 Aircraft Equations of Motion -Coordinate Transformations (Cont) Summary of Axes Transformation • Earth to body axis transformation: πΉβ = π Φ π Θ π Ψ πΉβ • Body to Earth axis transformation: πΉβ = π −Ψ π −Θ π −Φ πΉβ πΉβ = π Ψ π Θ π Φ πΉβ • Stability to body axis transformation: πΉβ = π πΌ πΉβ • Overall; Earth Axis System Aircraft Equations of Motion Ψ, Θ, Φ Body Axis System AE-426 Flight Dynamics --- 201 πΌ Stability Axis System 16 Aircraft Equations of Motion -Aircraft Force Equations • Developed from Newton’s 2nd Law; π ππ ππ‘ Inertial ( = πΉ ββ ) . . • Newton’s 2nd Law is valid only in an inertial reference frame. • Iner al frame → ο¬xed in space with no rela ve mo on. • Assumptions to develop EOM: – Aircraft is a rigid body. – Aircraft mass is constant. – Earth axis system is assumed to be inertial. • Three translational EOM. Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 17 Aircraft Equations of Motion -Aircraft Force Equations (Cont) • Since Newton’s 2nd Law applies only in an inertial (non-rotating) axis system BUT ALL THE ACTION happen on a non-inertial (rotating) axis system, then, we need to write aircraft EOM in its body axis system. • So, we need a vector transformation relationship; ππ΄β ππ‘ ππ΄β = ππ‘ + π × π΄β Where : represents any vector which is to be transformed. : is the angular rotation vector of rotating system relative to non-rotating system. Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 18 Aircraft Equations of Motion -Aircraft Force Equations (Cont.) Steps to Develop Aircraft Force EOM STEP β : Starting with Newton’s 2nd Law; π ππ ππ‘ Inertial π π π ππ‘ Inertial π πβInertial = πΉβ = πΉβ = πΉβ STEP β‘ : Expressing πβInertial in body axis system and accounting for the rotation of body axis on the aircraft; Aircraft Equations of Motion Μ πβInertial Body = πBody + πBody × πBody πΜ π€Μ Μ πβInertial Body = π + π Μ π Body π πΜ + ππ − π π π ⇒ πΜ + π π − ππ π π Body πΜ + ππ − ππ Body AE-426 Flight Dynamics --- 201 π₯Μ π π 19 Aircraft Equations of Motion -Aircraft Force Equations (Cont) STEP β’ : Newton’s 2nd Law becomes; πβInertial Body πΜ + ππ − π π π πΜ + π π − ππ πΜ + ππ − ππ Body = πΉβBody = πΉ πΉ πΉ Body STEP β£ : Expanding RHS of πΉβBody ; πΉ πΉ πΉβBody = πΉ = πΉ πΉ Body πΉ Body πΉ + πΉ πΉ Body πΉ + πΉ πΉ Body STEP β€ : Combine and separate Eq. πβInertial Body = πΉβBody ; π πΜ + ππ − π π π πΜ + π π − ππ = −ππ sin Θ + = ππ sin Φ cos Θ + π πΜ + ππ − ππ = ππ cos Φ cos Θ + Aircraft Equations of Motion −π· cos πΌ + πΏ sin πΌ πΉ −π· sin πΌ − πΏ cos πΌ AE-426 Flight Dynamics --- 201 + π cos Φ + 0 + −π sin Φ 20 Aircraft Equations of Motion -Aircraft Moment Equations • Developed from Newton’s 2nd Law; • Angular momentum of aircraft; • Three rotational EOM. Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 21 Aircraft Equations of Motion -Aircraft Moment Equation (Cont) Steps to Develop Aircraft Moment EOM STEP β : Starting with a small elemental mass dm of aircraft located at dome distance from c.g. and rotating with positive angular rates about c.g.; πβdm = π₯π€Μ + π¦π₯Μ + π§π STEP β‘ : Expressing the velocity of dm and accounting for its rotation around c.g.; Aircraft Equations of Motion πdm = ππβdm + πBody × πβdm ππ‘ Body πdm = πBody × πβdm πdm = ππ§ − π π¦ π€Μ + π π₯ − ππ§ π₯Μ + ππ¦ − ππ₯ π AE-426 Flight Dynamics --- 201 22 Aircraft Equations of Motion -Aircraft Moment Equation (Cont) Steps to Develop Aircraft Moment EOM STEP β : Starting with a small elemental mass dπ of aircraft located at some distance from c.g. and rotating with positive angular rates about c.g.; πβdm = π₯π€Μ + π¦π₯Μ + π§π STEP β‘ : Expressing the velocity of dπ and accounting for its rotation around c.g.; π = ππβ + πBody × πβ ππ‘ Body π = πBody × πβ π = ππ§ − π π¦ π€Μ + π π₯ − ππ§ π₯Μ + ππ¦ − ππ₯ π STEP β’ : Linear momentum of dπ and accounting for its rotation around c.g.; Linear Momentum = = Aircraft Equations of Motion dππ dπ ππ§ − π π¦ π€Μ + π π₯ − ππ§ π₯Μ + ππ¦ − ππ₯ π AE-426 Flight Dynamics --- 201 23 Aircraft Equations of Motion -Aircraft Moment Equation (Cont) STEP β£ : Angular momentum of dπ; dπ» = πβ dπ» dπ» dπ» = π€Μ π₯ dπ ππ§ − π π¦ dπ» = π π¦ + π§ dπ» = π π₯ + π§ dπ» = π π₯ + π¦ × dππ π₯Μ π¦ dπ π π₯ − ππ§ π π§ dπ ππ¦ − ππ₯ dπ − ππ₯π¦ dπ − π π₯π§ dπ dπ − π π¦π§ dπ − ππ₯π¦ dπ dπ − ππ₯π§ dπ − ππ¦π§ dπ STEP β€ : Integrating; dπ» = π π¦ +π§ dπ − π π₯π¦ dπ − π π₯π§ dπ dπ» = π π₯ +π§ dπ − π π¦π§ dπ − π π₯π¦ dπ dπ» = π Aircraft Equations of Motion π₯ +π¦ dπ − π π₯π§ dπ − π AE-426 Flight Dynamics --- 201 π¦π§ dπ Moments of Inertia πΌ πΌ πΌ Products of Inertia πΌ πΌ πΌ 24 Aircraft Equations of Motion -Aircraft Moment Equation (Cont) Integrating; π» = ππΌ − ππΌ − π πΌ π» = ππΌ − π πΌ − ππΌ π» = π πΌ − ππΌ − ππΌ The same can be obtained applying basic physics using, π» = πΌβπ. If aircraft has an xz plane of symmetry, then π» = ππΌ − π πΌ π» = ππΌ π» = π πΌ − ππΌ π» = ππΌ Aircraft Equations of Motion − π πΌ π€Μ + ππΌ π₯Μ + π πΌ − ππΌ AE-426 Flight Dynamics --- 201 π 25 Aircraft Equations of Motion -Aircraft Moment Equation (Cont) STEP β₯ : Taking the time rate of angular momentum vector π» = ππΌ − π πΌ ππ» ππ‘ = ππ» ππ‘ +π π€Μ + ππΌ π₯Μ + π πΌ − ππΌ π; ×π» Where, πΜπΌ − π Μ πΌ πΜ πΌ = π ΜπΌ − πΜπΌ ππ» ππ‘ + ππΌ Μ − π πΌ Μ + ππΌ Μ + π πΌ Μ − ππΌ Μ Since aircraft mass distribution is constant, ππ» ππ‘ πΜπΌ − π Μ πΌ πΜ πΌ = π Μ πΌ − πΜπΌ Combining all, ππ» ππ‘ πΜπΌ − π Μ πΌ πΜ πΌ = π ΜπΌ − πΜπΌ Aircraft Equations of Motion + ππΌ π€Μ π − π πΌ π₯Μ π ππΌ π π π πΌ − ππΌ AE-426 Flight Dynamics --- 201 πΜπΌ + ππ πΌ − πΌ βΉ πΜ πΌ − ππ πΌ − πΌ π Μ πΌ + ππ πΌ − πΌ − π Μ + ππ πΌ + π −π πΌ + ππ − πΜ πΌ 26 Aircraft Equations of Motion -Aircraft Moment Equation (Cont) STEP β¦ : The three moment equations in the body axis are found from: ππ» ππ‘ =π Where, π πΏ = π π πΏ π π = rolling moment = pitching moment yawing moment πΏ + π π Since aircraft mass distribution is constant, πΜπΌ πΜ πΌ + ππ πΌ − πΌ − π Μ + ππ πΌ = πΏ − ππ πΌ − πΌ + = π π Μ πΌ + ππ πΌ + π −π πΌ ππ − πΜ πΌ = π ππππ’πππ πππππππππ‘πππ π‘ππππ Aircraft Equations of Motion −πΌ ππ¦ππ ππππππ π πππ π‘ππππ πππ’πππππ π‘ππππ AE-426 Flight Dynamics --- 201 27 Aircraft Equations of Motion -Aircraft EOM’s • Six differential EOM; • • • • Dependent variables: Independent variable: Nonlinear ODE. No closed-form analytical solution, only numerical methods. Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 28 Aircraft Equations of Motion -Aircraft EOM’s (Cont) Longitudinal EOM EOM can be decoupled π πΜ + ππ − π π = −ππ sin Θ + −π· cos πΌ + πΏ sin πΌ + π cos Φ πΜπΌ − ππ πΌ − πΌ + π − π πΌ = π π πΜ + ππ − ππ = ππ cos Φ cos Θ + −π· sin πΌ − πΏ cos πΌ − π sin Φ πΜ πΌ Lateral EOM + ππ πΌ − πΌ − π Μ + ππ πΌ =πΏ +πΏ π πΜ + π π − ππ = ππ sin Φ cos Θ + πΉ π Μ πΌ + ππ πΌ −πΌ + ππ − πΜ πΌ +πΉ =π +π • BENEFIT: easier to solve three EOM simultaneously for many flight conditions. Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 29 Aircraft Equations of Motion -Aircraft Kinematic EOM’s • Additional Eqs are required with the 6 EOM to completely solve aircraft motion, Why? ο Only 6 Eqs, with ο More than 6 unknowns: • Are found by relating body axis system angular velocity rates ( to time rate of change of Euler angle “Euler rates” ( ), How? ) ο Since the magnitude of the three body rate must equal the magnitude of the three Euler rates; ο the following equality must satisfy: Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 30 Aircraft Equations of Motion -Aircraft Kinematic EOM’s (Cont) • Note this: • For the above to be valid, both sides must be on the same axis system. • Euler rates vectors are found from the Euler rotation sequences (Recall slide 9); Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 31 Aircraft Equations of Motion -Aircraft Kinematic EOM’s (Cont) • Rewrite in body axis system: • Hence; • Carrying out the mathematics; or Aircraft Equations of Motion AE-426 Flight Dynamics --- 201 32 examples EX 4.1: given the following Euler angles and rates EX 4.2: given the following Euler angles and rates Ψ = 0 πππ, ΨΜ = 10 πππ Θ = 0 πππ, ΘΜ = 0 π πππ Φ = 90 πππ, ΦΜ = 0 π Ψ = 0 πππ, ΨΜ = 0 The roll, pitch, and yaw rates are The roll, pitch, and yaw rates are π=0 , π = 10 πππ π πππ Φ = 90 πππ, ΦΜ = 0 π Θ = 0 πππ, ΘΜ = 20 ,π =0 π=0 a 10 deg/s Euler yaw rate ΨΜ is felt by the pilot as 10 deg/s pitch rate Q Aircraft Equations of Motion ,π = 0 , π = −20 a 20 deg/s Euler pitch rate ΘΜ is felt by the pilot as -20 deg/s yaw rate R AE-426 Flight Dynamics --- 201 33