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3. Translational and Rotational Dynamics MAE 345 2017.Pptx Translational and Rotational Dynamics! Robert Stengel! Robotics and Intelligent Systems MAE 345, Princeton University, 2017 Copyright 2017 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE345.html 1 Reference Frame •! Newtonian (Inertial) Frame of Reference –! Unaccelerated Cartesian frame •! Origin referenced to inertial (non-moving) frame –! Right-hand rule –! Origin can translate at constant linear velocity –! Frame cannot rotate with respect to inertial origin ! x $ # & •! Position: 3 dimensions r = # y & –! What is a non-moving frame? "# z %& •! Translation = Linear motion 2 Velocity and Momentum of a Particle •! Velocity of a particle ! $ ! x! $ vx dr # & # & v = = r! = y! = # vy & dt # & # ! & # & " z % vz "# %& •! Linear momentum of a particle ! v $ # x & p = mv = m # vy & # & "# vz %& 3 Newtons Laws of Motion: ! Dynamics of a Particle First Law . If no force acts on a particle, it remains at rest or continues to move in straight line at constant velocity, . Inertial reference frame . Momentum is conserved d ()mv = 0 ; mv = mv dt t1 t2 4 Newtons Laws of Motion: ! Dynamics of a Particle Second Law •! Particle acted upon by force •! Acceleration proportional to and in direction of force •! Inertial reference frame •! Ratio of force to acceleration is particle mass d dv dv 1 1 ()mv = m = ma = Force ! = Force = I Force dt dt dt m m 3 ! $ " % fx " % fx # & 1/ m 0 0 $ ' = $ ' f Force = # fy & = force vector $ 0 1/ m 0 '$ y ' $ ' # & #$ 0 0 1/ m &' f fz $ z ' "# %& # & 5 Newtons Laws of Motion: ! Dynamics of a Particle Third Law For every action, there is an equal and opposite reaction Force on rocket motor = –Force on exhaust gas FR = !FE 6 One-Degree-of-Freedom Example of Newtons Second Law 2nd-order, linear, time-invariant ordinary differential equation 2 d x(t) fx (t) ! "Defined as" 2 ! ""x(t) = v"x (t) = dt m Corresponding set of 1st-order equations (State-Space Model) dx ()t 1 ! x" (t) ! x (t) ! v (t) x (t) ! x(t), Displacement dt 1 2 x 1 dx()t dx2 ()t fx (t) x2 (t) ! , Rate ! ""x1(t) = x"2 (t) = v"x (t) = dt dt m 7 State-Space Model is a Set of 1st- Order Ordinary Differential Equations State, control, and output vectors for the example ! $ ! $ x1(t) x1(t) x(t) = # & ; u(t) = u() t = fx (t); y(t) = # & "# x2 (t) %& "# x2 (t) %& Stability and control-effect matrices ! 0 1 $ ! 0 $ F= # & ; G= # & " 0 0 % " 1 / m % Dynamic equation x!(t) = Fx(t) + Gu(t) 8 State-Space Model of the 1-DOF Example Output equation y(t) = Hxx(t) + Huu(t) Output coefficient matrices ! 1 0 $ ! 0 $ Hx = # & ; Hu = # & " 0 1 % " 0 % 9 Dynamic System Dynamic Process: Current state Observation Process: Measurement may depend on prior state may contain error or be incomplete x : state dim = (n x 1) y : output (error-free) u : input dim = (m x 1) dim = (r x 1) w : disturbance dim = (s x 1) n : measurement error p : parameter dim = (l x 1) dim = (r x 1) z : measurement t : time (independent variable, 1 x 1) dim = (r x 1) 10 State-Space Model of Three- Degree-of-Freedom Dynamics ! $ ! x! $ vx # & # & ! x $ ! # & # y & # vy & y # & # & ! $ # z! & ! $ # & r!(t) # vz & r(t) z x! ()t " # & = # & = = Fx()t + Gu()t x()t ! # & = # & v!x # & vx # v!(t) & # & fx / m "# v(t) %& # & " % # & # & # ! & vy vy f m # & # y / & # & "# vz %& # v!z & # & " % fz / m " % ! x $ ! 0 0 0 1 0 0 $# & ! 0 0 0 $ # & y # & 0 0 0 0 1 0 # & 0 0 0 ! $ # &# & # & fx # 0 0 0 0 0 1 & z # 0 0 0 &# & = # & + # fy & # 0 0 0 0 0 0 &# vx & # 1/ m 0 0 & # & # &# & # & fz 0 0 0 0 0 0 vy 0 1/ m 0 "# %& # &# & # & "# 0 0 0 0 0 0 %&# & "# 0 0 1/ m %& " vz % 11 What Use are the Dynamic Equations? Compute time response Compute trajectories Determine stability Identify modes of motion 12 Forces! 13 External Forces: Aerodynamic/Hydrodynamic ! $ ! C $ X # X & # & 1 2 faero = Y = # CY & 'V A # & 2 # Z & # & " % " CZ % ! = air density, functionof height ! C $ "#h # X & = !sealevele dimensionless # CY & = V = airspeed # & aerodynamic coefficients CZ 1/2 1/2 " % $ 2 2 2 & $ T & =%vx + vy + vz ' = %v v' Inertial frame or body frame? A = referencearea 14 External Forces: Friction 15 External Forces: Gravity •! Flat-earth approximation –! g is gravitational acceleration –! mg is gravitational force –! Independent of position –! z measured up ! 0 $ # & = = 2 mg flat m # 0 &; go 9.807 m / s "# –go %& 16 External Forces: 1/2 Gravity r = ()x2 + y2 + z2 L = Latitude ! = Longitude •! Spherical earth, inertial ! g $ # x & frame ground = # gy & = ggravity # & –! ”Inverse-square gravitation "# gz %& –! Non-linear function of position ! x $ ! x / r $ ! cos L cos( $ µ # & µ # & µ 14 # & µ = ' 3 y = ' 2 y / r = ' 2 cos Lsin( –! = 3.986 x 10 m/s r # & r # & r # & "# z %& "# z / r %& "# sin L %& •! Spherical earth, rotating frame gr = ggravity + grotation –! ”Inverse-square gravitation " x % " x % µ $ ' $ ' –! Centripetal acceleration = ! + (2 3 $ y ' $ y ' –5 r ! ! = 7.29 x 10 rad/s #$ z &' #$ 0 &' 17 State-Space Model with Round-Earth Gravity Model (Non-Rotating Frame) ! x! $ ! x $ ! 0 0 0 $ # & ! 0 0 0 1 0 0 $# & # & y! # & y 0 0 0 # & 0 0 0 0 1 0 # & # &! $ # ! & # &# & # 0 0 0 & x z # 0 0 0 0 0 1 & z # & # & = # & ' 3 ! # µ / r 0 0 &# y & # vx & # 0 0 0 0 0 0 &# vx & # & # & 3 "# z %& # v!y & 0 0 0 0 0 0 # vy & # 0 µ / r 0 & # & # &# & # & # 0 0 0 0 0 0 & 3 # v!z & " %# vz & # 0 0 µ / r & " % " % " % " 0 0 0 1 0 0 %" x % $ '$ ' $ 0 0 0 0 1 0 '$ y ' $ 0 0 0 0 0 1 '$ z ' = 3 $ ' $ !µ / r 0 0 0 0 0 ' $ '$ vx ' 3 $ 0 !µ / r 0 0 0 0 '$ vy ' $ '$ ' !µ 3 #$ 0 0 / r 0 0 0 &'#$ vz &' Inverse-square gravity model introduces nonlinearity 18 Vector-Matrix Form of Round-Earth Dynamic Model ! $ 0 I3 ! r! $ # & ! r $ # & = # 'µ & # & v! v " % # 3 I3 0 & " % " r % What other forces might be considered, and where would they appear in the model? 19 Point-Mass Motions of Spacecraft •! For short distance and low speed, flat-Earth frame of reference and gravity are sufficient •! For long distance and high speed, round-Earth frame and inverse-square gravity are needed 20 Mass of an Object xmax ymax zmax m = ! dm = ! ! ! "(x, y,z)dx dydz Body xmin ymin zmin "(x, y,z) = density of the body Density of object may vary with (x,y,z) 21 More Forces! 22 External Forces: Linear Springs Scalar, linear spring f = !k() x ! xo = !k"x; k = springconstant Uncoupled, linear vector spring " % " % kx ()x ! xo " % $ ' kx 0 0 (x $ ' $ ' $ ' $ ' fS = ! ky ()y ! yo = ! 0 ky 0 $ (y ' $ ' $ ' $ ( ' $ k z ! z ' $ 0 0 kz ' # z & # z ()o & # & ()xo, yo,zo : Equilibrium (reference) position 23 External Forces: Viscous Dampers Scalar, linear damper f = !d() v ! vo = !d"v; d = dampingconstant Uncoupled, linear vector damper " % dx vx ! vx " % " % $ ()o ' dx 0 0 vx $ ' $ ' $ ' = ! ! (! $ 0 dy 0 ' $ vy ' fD $ dy ()vy vyo ' $ ' $ ' $ ' 0 0 dz vz $ dz vz ! vz ' #$ &' #$ &' # ()o & ()vxo ,vyo ,vzo : Equilibrium (reference) velocity [usually = 0] 24 State-Space Model with Linear Spring and Damping ! 0 0 0 0 0 0 $ # &! x ' x $ ! x! $ ! x $ # 0 0 0 0 0 0 &# ()o & # & ! 0 0 0 1 0 0 $# & ! # 0 0 0 0 0 0 &# ' & # y & # &# y & ()y yo # 0 0 0 0 1 0 & # 'k 'd &# & # z! & # z & # x 0 0 x 0 0 &# ' & # & # 0 0 0 0 0 1 &# & ()z zo ! = + # m m &# & # vx & # 0 0 0 0 0 0 &# vx & # & # 'k 'd &# vx & # ! & # & y y vy # 0 0 0 0 0 0 & vy # 0 0 0 0 &# & # & # & m m vy # 0 0 0 0 0 0 & # &# & # v!z & " %# vz & " % " % # 'kz 'dz &# v & # 0 0 0 0 &" z % " m m % Spring Damping Effects Effects 25 State-Space Model with Linear Spring and Damping Stability Effects Reference Effects ! 0 0 0 1 0 0 $ ! 0 0 0 1 0 0 $ # & # & ! x! $ # 0 0 0 0 1 0 &! x $ # 0 0 0 0 1 0 &! $ # & # & xo y! # 0 0 0 0 0 1 & y # 0 0 0 0 0 1 &# & # & # & # y & # 'k 'd & # k d & o # z! & # x 0 0 x 0 0 &# z & # x 0 0 x 0 0 &# & # & # & zo ! = # m m & + # m m &# & # vx & # vx & # 'k 'd & # k d &# 0 & # ! & y y # & y y # & vy # 0 0 0 0 & vy # 0 0 0 0 & 0 # & # m m &# & # m m &# & # v!z & # vz & # 0 & " % # 'kz 'dz &" % # kz dz &" % # 0 0 0 0 & # 0 0 0 0 & m m m m " % " % 26 Vector-Matrix Form ! $ ! $ ! $ r! 0 I3 ()r ' ro # & = # & # & " v! % "# 'K / m 'D / m %& # v & " % ! $ ! $ ! $ ! $ ! r! $ 0 I3 r 0 'I3 ro # & = # & # & + # & # & " v! % "# 'K / m 'D / m %& " v % "# K / m D / m %& "# 0 %& 27 Rotational Motion! 28 Assignment # 2 Document the physical characteristics and flight behavior of a Syma X11 quadcopter! https://www.youtube.com/watch?v=EwO6U7DbqSo https://www.youtube.com/watch?v=kyiuy2CHzj0 29 Center of Mass ! $ xcm # & 1 ! = rcm # ycm & ' r dm # & m Body " zcm % x y z ! x $ max max max # & = ' ' ' # y & ((x, y,z)dx dydz x y z min min min "# z %& Reference point for rotational motion 30 Particle in Inverse- Square Field Angular Momentum of a Particle •! Moment of linear momentum of differential particles that make up the body –! Differential mass of a particle times –! Component of velocity perpendicular to moment arm from center of rotation to particle dh = ()r ! dmv = ()r ! v dm 31 Cross Product of Two Vectors i j k r ! v = x y z = ()yvz " zvy i + ()zvx " xvz j + ()xvy " yvx k vx vy vz ()i, j, k : Unit vectors along ()x, y,z This is equivalent to " % yv ! zv " % $ ()z y ' " 0 !z y % vx $ ' $ ' = $ ! ' = ! v = ! $ ()zvx xvz ' $ z 0 x ' $ y ' rv $ ! ' $ ' $ ' y x 0 vz xvy ! yvx # & #$ &' #$ () &' 32 Cross-Product-Equivalent Matrix r ! v = rv! Cross-product equivalent of radius vector Velocity vector # & ! $ 0 "z y vx % ( # & r ! ! r" = % z 0 "x ( v = # vy & % " ( # & $ y x 0 ' "# vz %& 33 Angular Momentum of an Object Integrate moment of linear momentum of differential particles over the body xmax ymax zmax h ! " ()r ! v dm = " " " ()r ! v #(x, y,z)dx dydz Body xmin
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