Gyroscope with variable R

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Loren Barker
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1 369(1a).wxm 1 / 11 Gyroscope with variable R (%i1) (%o0) kill(all); one 1 Coorinates (%i1) (%o1) t],!() epens([x,y,theta,phi,psi,r,v,t,u,l,phi,omega],t); t,y() t,ò() t,'() t,é() t,r() t,v() t,t() t,u() t,l() t,phi() t [x() Potential energy (%i) (U) U: m*g*(h*cos(theta)+r); gm() hcos() Ò+R 3 Lagrange function (%i3) (T_rot) T_rot: T: 1/*I[1]*(iff(phi,t)^*sin(theta)^+iff(theta,t)^) +1/*I[3]*(iff(phi,t)*cos(theta)+iff(psi,t))^; I 1 t Ò + sin() Ò t ' + I 3 Ò+ t ' cos() t É (%i4) (T_trans) (%i5) (L) T_trans: 1/*m*iff(R+h*cos(theta),t)^; m Ò R h sin() t t Ò L: T_rot+T_trans-U; m Ò R h sin() t I 3 Ò+ t ' cos() t É t Ò + gm() hcos() Ò+R 4 Lagrange equations 4.1 theta equation I 1 + sin() Ò t Ò t ' +

2 369(1a).wxm / 11 (%i6) D1: iff(l, iff(theta,t)); (D1) I 1 t Ò hmsin() R hsin() t Ò Ò t Ò (%i7) E1: expan(ratsimp(iff(d1,t) - iff(l,theta) = 0)); Ò (E1) h msin() t Ò+I 1 mcos() Òsin() t Ò+h Ò t Ò I 1 t ' cos() Òsin() Ò+I 3 t ' cos() Òsin() Ò+I 3 t ' t É sin() Ò g hmsin() Ò tr hmsin() Ò=0 (%i8) E1a: solve(e1, iff(theta,t,)); (E1a) mcos() Òsin() t Ò= (h Ò () I 3 I 1 t ' cos() Ò+I 3 [ t Ò + h msin() Ò +I 1 )] 4. phi equation t ' (%i9) D: iff(l, iff(phi,t)); (D) I 1 t ' sin() Ò +I 3 cos() Ò t É+ g tr hm sin() Ò)/( t ' cos() Ò+ t É (%i10) E: expan(ratsimp)(iff(d,t) - iff(l,phi) = 0); (E)() I 1 I 3 t ' cos() Ò I 3 t É sin() Ò t Ò+I 1 t' sin() Ò +I 3 t' cos() Ò +I 3 té cos() Ò=0 (%i11) Ea: solve(e, iff(phi,t,)); (Ea) [ t '= () I 1 I 3 Ò I 3 t ' cos() t É sin() t Ò+I 3 cos() té Ò ] I 1 sin() Ò +I 3 cos() Ò 4.3 psi equation Ò

3 369(1a).wxm 3 / 11 (%i1) D3: iff(l, iff(psi,t)); (D3) I 3 Ò+ t ' cos() t É (%i13) E3: expan(ratsimp(iff(d3,t) - iff(l,psi) = 0)); (E3) I 3 t ' sin() Ò ' cos() Ò+I 3 t Ò+I 3 (%i14) E3a: solve(e3, iff(psi,t,)); (E3a) [ t É= t ' sin() Ò t Ò t' cos() Ò] 4.4 R equation (%i15) D4: iff(l, iff(r,t)); (D4) m Ò t R hsin() t Ò t (%i16) E4: expan(ratsimp(iff(d4,t) - iff(l,r) = 0)); (E4) hmsin() Ò tò hmcos() (%i17) E4a: solve(e4, iff(r,t,)); (E4a) [ t R=hsin() Ò Ò t Ò+hcos() t Ò +gm+ Ò t Ò g] t É=0 tr m=0 5 Lagrange equations with constants of motion (%i18) kill(l); (%o18) one (%i19) theta_otot: 1/I[1]*sin(theta)*(phi_ot^*cos(theta)*(I[1]-I[3] I 1 I 3 phi_ot cos() Ò I 3 phi_ot psi_ot+ghm sin() Ò (theta_otot) () (%i0) psi_ot: 1/I[3]*(L[psi]-I[3]*phi_ot*cos(theta)); (psi_ot) L É I 3 phi_ot cos() Ò I 3 I 1 (%i1) phi_ot: (L[phi]-L[psi]*cos(theta))/(I[1]*sin(theta)^); (phi_ot) L ' L É cos() Ò I 1 sin() Ò

4 369(1a).wxm 4 / 11 (%i) theta_ot: eta; (theta_ot) Ñ (%i3) R_otot: -g; (R_otot) g (%i4) R_ot: v; (R_ot) v 5.1 right-han sies of Lagrange equations, Hamilton form (%i30) Eq5: theta_otot /*=eta_ot*/; Eq5a: eta /*= theta_ot*/; Eq6: phi_ot; Eq7: psi_ot; Eq8: R_otot; Eq9: v; () I 1 I 3 phi_ot cos() Ò I 3 phi_ot psi_ot+ghm sin() Ò (Eq5) (Eq5a)Ñ L ' L É cos() Ò (Eq6) I 1 sin() Ò L É I 3 phi_ot cos() Ò (Eq7) (Eq8) (Eq9) g v I 3 I 1 (%i31) str: [I[1]=0.5, I[3]=3., L[phi]=.8, L[psi]=, m=10, g=9.81, h=0. (str) [I 1 =0.5,I 3 =3,L ' =0.8,L É =,m=10,g=9.81,h=0.]

5 369(1a).wxm 5 / 11 (%i37) Eq5aA: ev(eq5a, str, eval); Eq5A: ev(eq5, str, eval); Eq6A: ev(eq6, str, eval); Eq7A: ev(eq7, str, eval); Eq8A: ev(eq8, str, eval); Eq9A: ev(eq9, str, eval); (Eq5aA)Ñ (Eq5A).0(.0() 0.8 cos() Ò 10.0() 0.8 cos() Ò cos() Ò (Eq6A) (Eq7A) (Eq8A) 9.81 (Eq9A) v sin() Ò 4.0() 0.8 cos() Ò 6.0() 0.8 cos() Ò cos() Ò sin() Ò sin() Ò +19.6)sin() Ò sin() Ò 6.0() 0.8 cos() Ò cos() Ò sin() Ò 3 (%i38) s: rk([eq5a, Eq5aA, Eq6A, Eq7A, Eq8A, Eq9A], [eta, theta, phi, psi,v,r], [0, %pi/4, %pi/4, 0, 0, 0], [t,0,8,0.01])$ 5. Graphics (%i44) c1: makelist([p[1],p[]],p,s)$ c: makelist([p[1],p[3]],p,s)$ c3: makelist([p[1],p[4]],p,s)$ c4: makelist([p[1],p[5]],p,s)$ c5: makelist([p[1],p[6]],p,s)$ c6: makelist([p[1],p[7]],p,s)$ 5.3 Plot theta, phi, psi

6 369(1a).wxm 6 / 11 (%i45) wxplot([[iscrete, c1], [iscrete, c]], [xlabel, "time"], [legen, "theta_{ot}", "theta"])$ (%t45) (%i46) wxplot([[iscrete, c3], [iscrete, c4]], [xlabel, "time"], [legen, "phi", "psi"])$ (%t46)

7 369(1a).wxm 7 / 11 (%i47) wxplot([[iscrete, c5], [iscrete, c6]], [xlabel, "time"], [legen, "v", "R"])$ (%t47) 5.4 Plot angular velocities (%i48) omega[3]: ratsimp(ev(phi_ot*cos(theta)+psi_ot)); (%o48) L É I 3 (%i49) omega[1]: phi_ot*sin(theta)*sin(psi)+theta_ot*cos(psi); (%o49) sin() É() L ' L É cos() Ò I 1 sin() Ò +Ñcos() É (%i50) omega[]: phi_ot*sin(theta)*cos(psi)-theta_ot*sin(psi); (%o50) cos() É() L ' L É cos() Ò I 1 sin() Ò Ñsin() É

8 369(1a).wxm 8 / 11 (%i53) omeg1: ev(omega[1], str); omeg: ev(omega[], str); omeg3: ev(omega[3], str);.0 sin() É 0.8 cos() Ò (omeg1) (omeg) (omeg3) 3 sin() Ò.0 cos() 0.8 cos() Ò É () sin() Ò +Ñcos() É Ñsin() É (%i54) om1: om: om3: om4: []; (om1) [] (%i55) for i:1 thru length(c1) o ( block([t], str1: [eta=c1[i][], theta=c[i][], phi=c3[i][], psi=c4[i][ t: c1[i][1], om1: appen(om1, [[t, ev(omeg1, str1)]]), om: appen(om, [[t, ev(omeg, str1)]]), om3: appen(om3, [[t, ev(omeg3, str1)]]), om4: appen(om4, [[t, ev(sqrt(omeg1^+omeg^+omeg3^), str1)] /*print(t,om1[i],om[i],om3[i]) */ )); (%o55) one (%i56) kill(y); (%o56) one

9 369(1a).wxm 9 / 11 (%i57) wxplot([[iscrete, om1], [iscrete, om], [iscrete, om3], [iscrete, om4]], [xlabel, "time"], [legen, "omega[1]", "omega[]", "omega[3]", " omega "])$ (%t57) 5.5 Plot space curve of centre of mass (%i58) cl1: makelist([p[1],10,p[3],p[4]],p,s)$ (%i59) cl1[1]; (%o59) [0.11,10, , ]

10 369(1a).wxm 10 / 11 (%i60) transf(cl1) := ( block([i,t,r,theta,phi], cl: [], for i:1 thru length(cl1) o ( [t,r,theta,phi]: cl1[i], x: r*sin(theta)*cos(phi), y: r*sin(theta)*sin(phi), z: r*cos(theta), /*print(t,x,y,z),*/ cl: appen(cl, [[x,y,z]]) ), return(cl))); (%o60) transf() cl1 :=block([i,t,r,ò,'],cl:[],for i thru length() cl1 o ([t,r,ò,']:cl1 i,x:r sin() Òcos() ',y:rsin() Òsin() ', z:rcos() Ò,cl:appen() cl,[[x,y,z]] ),return() cl ) (%i61) cl: transf(cl1); << Ausruck länger, als im Konfigurationsialog erlaubt >> (%i6) cr: points(cl)$ (%i63) wxraw3(line_with=,color=blue, cr)$ 0 errors, 0 warnings (%t63) 5.6 Plot space curve of omega vector

11 369(1a).wxm 11 / 11 (%i64) cl1: makelist([p[1],10,p[3],p[4]],p,s)$ (%i65) cl1[1]; (%o65) [0.11,10, , ] (%i67) cl: []; for i:1 thru length(om1) o ( [x,y,z]: [om1[i][],om[i][],om3[i][]], /*print(t,x,y,z),*/ cl: appen(cl, [[x,y,z]]) ); (cl) [] (%o67) one (%i68) cl[1]; (%o68) [0.0, , 3 ] (%i69) cr: points(cl)$ (%i70) wxraw3(line_with=,color=blue, cr)$ (%t70)

Gyroscope free falling with R

369(1b.wxm 1 / 13 Gyroscope free falling with R (%i1 kill(all; (%o0 one 1 Coorinates (%i1 epens([x,y,theta,phi,psi,r,v,t,u,l,phi,omega],t; (%o1 [x( t,y( t,( t,'( t,( t,r( t,v( t,t( t,u( t,l( t,phi( t,