Tennis racket (Dzhanibekov) effect: torque-free rotational motion
by Christian Peterson and William Schwalm
by Christian Peterson and William Schwalm
Introduction
Introduction
This notebook and animation accompany the paper titled "Euler’s rigid rotators, Jacobi elliptic functions, and the Dzhanibekov or tennis racket effect" that is published in the American Journal of Physics. In the paper, the torque-free rotational motion of a general rigid body is developed analytically and is applied to the flipping motion of a T-handle spinning in zero gravity that can be seen in videos on the internet (https://www.youtube.com/watch?v=1n-HMSCDYtM).
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Euler's equations for rigid body motion are first solved using Jacobi elliptic functions in the body frame, then are transformed into the inertial frame using the Euler rotation matrix. This notebook is provided to anyone who might want access to calculation used in the paper or to use it as a starting point for a project with rigid body motion.
Dzhanibekov effect Animation
Dzhanibekov effect Animation
The principle moments of inertia of the T-handle are calculated approximatelyusing the moments of inertia of thick rods in conjunction with Steiner’s parallel axis theorem. The figures above show consecutive times during one half cycle of a flip in the
motion described by a particular solution with initial conditions that seem to be appropriate. Although the behavior of our solution agrees qualitatively with the motion in the video of the T-handle, we have made no attempt to adjust the model to fit the period (τ) of the flipping motion because of the unknown moments of inertia of the T-handle and the extreme sensitivity of the solution to the initial conditions of θ, of which we have no way of determining from the video.
motion described by a particular solution with initial conditions that seem to be appropriate. Although the behavior of our solution agrees qualitatively with the motion in the video of the T-handle, we have made no attempt to adjust the model to fit the period (τ) of the flipping motion because of the unknown moments of inertia of the T-handle and the extreme sensitivity of the solution to the initial conditions of θ, of which we have no way of determining from the video.
Code
Code
Estimated Moments of Inertia
ρ=8050;(*DensityofSteel*)a1=0.004;(*RadiusofSmallRod(m)*)c1=.05;(*LengthofSmallRod(m)*)a2=.0075;(*RadiusofLargeRod(m)*)c2=.1;(*LengthofLargeRod(m)*)m1=ρπc1a1^2;(*Massofthesmallrod(kg)*)m2=(.1)ρπc2a2^2;(*MassofthelargeRod(kg);Theobjectisapproximatedashollowbymakinglessdensethanthesmallrod*)μ=(a2^2c2)/(a1^2c1);(*CenterofMassoftheT-Handle*)Irod1={{1/12m1(3a1^2+4c1^2),0,0},{0,1/12m1(3a1^2+4c1^2),0},{0,0,1/2m1a1^2}};Irod2={{1/12m2(3a2^2+4c2^2),0,0},{0,1/2m2a2^2,0},{0,0,1/12m2(3a2^2+4c2^2)}};h1={{(μ(c1+2a2)/(2(1+μ)))^2,0,0},{0,(μ(c1+2a2)/(2(1+μ)))^2,0},{0,0,0}};h2={{((c1+2a2)/(2(1+μ)))^2,0,0},{0,((c1+2a2)/(2(1+μ)))^2,0},{0,0,0}};InertiaMatrix=MatrixForm[(Irod1+m1h1)+(Irod2+m2h2)](*InertiaMatrixofT-handle*)
Rigid Body Parameters
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I1=InertiaMatrix[[1,1,1]];I2=InertiaMatrix[[1,2,2]];I3=InertiaMatrix[[1,3,3]];
Initial Conditions
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θ0=π/180;L=.0024;ω20=LSin[θ0]/I2;ω30=LCos[θ0]/I3;(*~50Rad/s,estimatedfromtheT-handlevideo*)T=1/2(I2ω20^2+I3ω30^2);
Constants
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k=ω30Sqrt[(I3(I1-I3))/(I2(I1-I2)ω20^2+I3(I1-I3)ω30^2)];A=ω30Sqrt[(I3(I3-I2))/(I2(I1-I2))];b=Sqrt[((I3-I2)(I2(I1-I2)ω20^2+I3(I1-I3)ω30^2))/(I1I2I3)];
Body Frame Solutions
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ω1[t_]:=AJacobiCN[bt+EllipticK[k],k];ω2[t_]:=(ω20/Sqrt[1-k^2])JacobiDN[bt+EllipticK[k],k];ω3[t_]:=ω30JacobiSN[bt+EllipticK[k],k];
Period of Elliptic Functions
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τ=4EllipticK[k]/b;
Time Dependant Rotation Angles
θ[t_]:=ArcCos[I3ω3[t]/L]//N;ψ[t_]:=ArcTan[I1ω1[t],I2ω2[t]]//N;ϕ[t_]:=Lt/I3+((L^2-2I3T)EllipticPi[(ω30^2I3^2)/L^2,JacobiAmplitude[bt,k],k])/(bLI3)//N;
Numerical Tabulation of Angles
θψϕ=Table[{θ[t],ψ[t],ϕ[t]},{t,0,7.3,.007}];RRfromθψϕ[{θ_,ψ_,ϕ_}]:=Transpose[{{Cos@ψCos@ϕ-Cos@θSin@ϕSin@ψ,-Sin@ψCos@ϕ-Cos@θSin@ϕCos@ψ,Sin@θSin@ϕ},{Sin@ϕCos@ψ+Cos@θSin@ψCos@ϕ,-Sin@ψSin@ϕ+Cos@θCos@ϕCos@ψ,-Sin@θCos@ϕ},{Sin@θSin@ψ,Sin@θCos@ψ,Cos@θ}}];RR=RRfromθψϕ/@θψϕ;
Simultaion
Simultaion
Simulation=Animate[Graphics3D[{LightGray,Specularity[White,20],Cylinder[{-(.6)*RR〚n,1〛,(.6)*RR〚n,1〛},.20],LightGray,Specularity[White,20],Cylinder[{{0,0,0},(.65)*RR〚n,3〛},.10],Red,Arrow@Tube@{{0,0,0},RR〚n,1〛},(*Line[RR〚1;;n,1〛],*)Green,Arrow@Tube@{{0,0,0},RR〚n,2〛},(*Line[RR〚1;;n,2〛],*)Blue,Arrow@Tube@{{0,0,0},RR〚n,3〛},Line[RR〚1;;n,3〛]},PlotRange1],{n,1,1040,1},AnimationRunningFalse,AnimationRate3,SaveDefinitionsTrue]
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