Trebuchets, SuperBall Missles, and Related Multi-Frame Mechanics

Transcription

1 Trebuchets, SuperBall Missles, and Related Multi-Frame Mechanics A millenial embarrassment (and redemption) for Physics Computer Aided Development of Principles, Concepts, and Connections 1. Theoretical Analysis F k =!! q k + Γ k mn q! m q! n. Numerical Synthesis A Post-Modern Scientific Method 3. Laboratory Observation Bill Harter and Dave Wall University of Arkansas and HARTER-Soft Elegant Educational Tools Since 001

2 Multistage Throwing Devices Am. J. Phys. 39, 656 (1971) (A class project ) Superball Missles ( ) BANG! (Bigger BANG!) (Still Bigger BANG!) The Trebuchet (~10 3 BC-150?) What Galileo might have tried to solve What Galileo did solve (simple harmonic pendulum)

3 Elementary Trebuchet Model - Multiple Rotating and Translating Frames φ θ R Siege of Kenilworth 115 (Re-enacted on NOVA) x (Translational recoil possible, too) Pre-Launch Coordinate Manifold φ = 135 Y 135 θ =0 Post-Launch Coordinate Manifold θ = - 45 Y φ = 180 θ -45 X X

4 Early Human Agriculture and Infrastructure Building Activity Throwing Slinging Splitting Chopping Cultivating and Digging Reaping What Trebuchet mechanics is really good for... Bull whip cracking Fly-fishing Hammering Ring-The-Bell (at the Fair) Later Human Recreational Activity Lacrosse Hammer throwing Tennis rallying Tennis serving Baseball & Football Batting Golfing Cultivating and Digging Water skiing Space Probe Planetary Slingshot

5 Trebuchet analogy with racquet swing - What we learn Driving Force: Gravity Large force F nearly parallel to velocity v so v increases rapidly. m L v F Force F nearly perpendicular to velocity v so v increases very little. Early on (Gain the energy/momentum) L F v m Most velocity v gained earlier here. Later on (Steer or guide) F mostly serves to steer m here. L Rotation of body provides most of energy of arm-racquet lever L. Energy Input Most of speed gained early by arm-racquet system L. Preparation Center-of-mass for semi-rigid arm-racquet system L is cocked. L L Follow-Through Arm-racquet system L flies nearly freely. Small applied forces mostly for steering. Ball hit occurs.

6 An Opposite to Trebuchet Mechanics- The Flinger Early on (Not much happening) Driving Force : Gravity Not much increase in velocity v F Later on (Last-minute cram for energy) Maximum increase in velocity v just before m slides off end F v Anti-analogy can be useful pedagogy Trebuchet-like experiment Flinger experiment α skateboard wheel slides on pool que-stick skateboard wheel swings

7 Trebuchet model in rotating beam Assume: Constant beam ω ω - R 9 o clock R 6 o clock 1 mv beam (trebuchet) = 1 mω + Final 3 o clock Initial ( 6 o'clock position) Final Trebuchet KE Final ( 3 o'clock position) ( ) 1 mω ( + ) = 1 mω ( ) Initial 6 o clock R = + 6 o clock Initial 9 o clock 1 mv =V ( r 0 ) V ( r f )= 1 mω r 1 f mω r 0 beam v beam R = + -rb 9 o clock beam = 1 mω ( 4 ) Flinger model in rotating beam Assume: Constant beam ω ω Initial Final 1 mv beam(flinger) = 1 mω + Final Initial 3 o clock 3 o clock ( ) 1 mω = 1 Final Flinger KE v beam beam ( ) mω + Flinger KE is mω more than 6 o clock trebuchet but misdirected mω Flinger KE is ( r b ) less than 9 o clock trebuchet and misdirected

8 Trebuchet model in lab Assume: Constant beam ω ω - R 9 o clock R 6 o clock ( ω + )= v rotation v beam v lab Flinger model in lab ω Initial ( ω r ) b + = v rotation Final v lab v beam ( ) v ω half -cocked 6 o clock beam (trebuchet) = ω ( 4 ) fully-cocked 9 o clock v lab ( trebuchet )= ω( + + ) half -cocked 6 o clock ω( + + ) fully-cocked 9 o clock = 5.00ω = 5.16ω = 5.00ω 5.8ω 6.00ω 5.8ω ( =, =1), ( = 1.5, =1.5), ( = 1, = ) beam v lab (flinger) ( ) v = ω + ( flinger)= ( ) ( ) = ω ( + ) r b = ω (compare) = 3.74ω = 3.96ω = 4.1ω ( =, =1), ( = 1.5, =1.5), = 1, = ( )

9 Many Approaches to Mechanics (Trebuchet Equations) Each has advantages and disadvantages (Trebuchet exposes them) U.S. Approach Quick n dirty Newton F=Ma Equations Cartesian coordinates French Approach Tres elegant Lagrange Equations in Generalized Coordinates F = d T dt!q T q German Approach Pride and Precision Riemann Christoffel Equations in Differential Manifolds F k =!! q k + Γ k mn q! m q! n Anglo-Irish Appproach Powerfully Creative numerics graphics. Numerical Synthesis Hamilton s Equations Phase Space!p j = H,!q k = H. q j p k Unified Approach 1. Theoretical Analysis A Post-Modern Scientific Method F k =!! q k + Γ k mn q! m q! n 3. Laboratory Observation All approaches have one thing in common: The Art of Approximation Physics lives and dies by the art of approximate models and analogs.

10 Another thing in common: sum Equations Require Kinetic Energy T= 1 in terms of coordinates and derivitives. γ µν!qµ!q ν T = 1 ( MR +mr ) θ! 1 mr θ! φ! cos(θ φ) 1 mr! φ! θ cos(θ φ) + 1 m! φ sum It helps to use Covariant Metric γ µν matrix: = 1 (!θ φ!) γ γ θ,θ θ, φ!θ γ φ, θ γ φ,φ!φ The γ µν give Covariant Momentum p µ = γ µν!q ν (a.k.a. canonical momentum) p θ p φ = γ γ θ,θ θ, φ!θ γ φ, θ γ φ,φ!φ The inverse γ µν give Contravariant Momentum (a.k.a. generalized velocity)!θ!φ = γθ, θ γ θ,φ γ φ,θ γ φ, φ p θ p φ sum!q ν p ν = γ νµ p µ

11 Trebuchet equations nonlinear and Lagrange-Hamilton methods are a bit messy.. Lagrangian L=T-V d dt d dt L!θ = L!φ = +F θ +F φ Lagrange quations need rearrangement to solve numerically MgR sin θ + mgr sin θ = MR + mr Riemann Christofffel Equations give less mess.. L θ L φ ( )!!...they are immediately computer integrable. (..and help with qualitative analysis..)!p θ -!p φ - mg sin φ = m!! φ mr!! θ cos(θ φ) + mr! θ sin(θ φ) F k =!! q k 1 + Γ k mn q! m q! n where : Γ mn; L =F θ θ = MgR sin θ + mgr sin θ L =F φ = mg sin φ φ θ mr φ!! cos(θ φ) mr φ! sin(θ φ) T = γ mn q! m! γ n q m + γ m q n γ mn θ!! φ!! = 1 m mr cos(θ φ) mr!φ sin(θ φ) + ( mr MR)g sin θ µ mr cos(θ φ) MR + mr mr!θ sin(θ φ) mg sin φ where: µ = m MR + mr sin (θ φ) q n q

12 Super-elastic Bounce m Analogous Superball Models -Bang Model m Space Plot (x versus y) m m 1 BANG! m 1 Velocity Plot (V y1 versus V y ) Class of W. G. Harter, Velocity Amplification in Collision Experiments Involving Superballs, Am. J. Phys. 39, 656 (1971) (A class project ) Optimal Throw m 1 :m = 3:1 100% Energy Transfer FINAL V is times INITIAL V END m m Bang! m m 1 1 m 1 Bang1! END <- m1 turn - around -> END m1:m = 7:1 m 1 first hits ground (Bang 1) (Bang) Fastest Throw 0% Energy Transfer START m 1 -m collision FINAL V is 3 times INITIAL V m 1 :m = :1 Graphic Solution

13 Lab View θ B θ B θ B!θ FINAL = 1 4mr MR 1 + 4mr MR Beam-Relative View START φ B -π/ (9 o'clock) r R FINAL Beam angular velocity for r= φ B MID φ B = 0 (6 o'clock) r R FINAL φ B π/ (3 o'clock) R r!θ INITIAL φ B ( )!θ FINAL = 0 (!θ FINAL =! ) θ INITIAL Hamiltonian Model Approximation conserves total energy and total angular momentum (Assumes internal forces large compared to gravity which is then ignored after initial impulse) FINAL beam-relative lever angular velocity for r=!φ FINAL =!θ FINAL +!θ INITIAL =!θ INITIAL 3 θ! INITIAL Optimal Throw Quickest Throw

14 M R!θ FINAL = = r FINAL Beam angular velocity for r= 1 4mr MR 1 + 4mr MR 0!θ INITIAL m!θ INITIAL Optimal Throw Quickest Throw r!φ FINAL =!θ FINAL +!θ INITIAL =!θ INITIAL 3 θ! INITIAL FINAL beam-relative lever angular velocity for r= Optimal Throw Quickest Throw FINAL Bottom line lab velocity for r= KE mass m 1 FINAL = mr (!φ FINAL +!θ FINAL ) = 1 ( θ! mr INITIAL ) (!θ FINAL = 0) ( 4 θ! INITIAL )!θ FINAL = θ! INITIAL ( ) Consistent with fully-cocked 9 o clock velocity ( ) ω + +

15 Coupled Rotation and Translation (Throwing) Early non-human (or in-human) machines: trebuchets, whips.. (3000 BC-154 AD) X-stimulated pendulum: (Quasi-Linear Resonance) Y φ A x (t) X For small φ (cos φ ~1 ) : Forced Harmonic Resonance d φ g A x (t) dt + φ = A Newtonian F=Ma equation Lorentz equation (with Γ=0) Y-stimulated pendulum: (Non-Linear Resonance) Y φ X Parametric Resonance d φ g A y (t) A y (t) For small φ (sin φ ~φ ) : + ( _ + ) φ = 0 dt A Schrodinger-like equation (Time t replaces coord. x) General φ : ( AD) General case: A Nasty equation! d φ g+a y (t) A x (t) dt + sin φ + cos φ = 0

16 The Arkansas Whirler (a) (b) (picture of Hog) Chaotic motion from both linear and non-linear resonance (a) Trebuchet, (b) Whirler. Positioned for linear resonance Positioned for nonlinear resonance

17 Schrodinger Equation Parametric Resonance Related to Jerked-Pendulum Trebuchet Dynamics Schrodinger Wave Equation d φ + ( E V(x) )φ = 0 dx With periodic potential V(x) = V 0 cos(nx) Mathieu Equation d φ + ( E +V dx 0 cos(nx) )φ = 0 E = N ω y g N ω y dx = dt (Let N= to get edge modes) Jerked Pendulum Equation () d φ + g dt + A y t φ = 0 On periodic roller coaster: y=-a y cos w y t A y ()= t ω Ay y cos(ω y t) Nx =ω y t d φ + g dt + ω dx = dt y d φ + N g dx ω + ω y y Connection Relations N QM Energy E-to-ω y Jerk frequency Connection ω y Ay Ay cos(ω y t) φ = 0 cos(nx) φ = 0 V 0 = N A y QM Potential V 0 -A y Amplitude Connection

18 E = N ω y g E QM Energy E-Related toω y Jerk frequency V=.0 Bands 3 - B 1 B Stable Hanging Band() ω y = N g E QM Potential V 0 -A y Amplitude Connection A + 1 Unstable Resonance A - Gap () Stable Hanging Band(1) B 1+ Unstable Resonance Gap (1) V 0 = N A y B A 1 1 Stable Inverted Band(0) V0

19 ω y (A 1 )= A 1 Mode - A Mode A y =0.5 ω y (A )= A y =0.5 sinφ(t) sinφ(t) V=.0 Bands B 1 B Stable Hanging Band() ω y (B )= A y =0.5 B Mode sinφ(t) Equivalent V-well bottoms Equivalent V-well tops A + 1 Unstable Resonance A - Gap () Stable Hanging Band(1) B 1+ Unstable Resonance Gap (1) 1 - B 1 Mode A y =0.5 sinφ(t) Equivalent V-well tops B 1 + A 1 Stable Inverted Band(0) ω y (B 1 )= Unstable Resonance Gap (1) 0 + A 1 Mode A y =0.5 sinφ(t) Y-acceleration:A(t)=-A y ω y cosωy t ω y (A 1 )=.9646 Equivalent V-well bottoms

20 Supernova Superballs (Bigger BANG!) (Still Bigger BANG!) Class of W. G. Harter, Velocity Amplification in Collision Experiments Involving Superballs, Am. J. Phys. 39, 656 (1971) (A class project ) Coming Next to Theaters Near You??!! Super Trebuchet? (Multi-) Supersonic? Most important: Quantum multi trebuchets...they re already inside you! (Proteins RNA)