Gyroscope free falling with R
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1 369(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, t ]! Potential energy (%i U: m*g*(h*cos(theta+r; h cos( +R (U g m 3 Lagrange function (%i3 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 + t ' I 3 t ' cos( + t (T_rot + (%i4 T_trans: 1/*m*iff(R+h*cos(theta,t^; m R h t t (T_trans (%i5 L: T_rot+T_trans-U; m R h t t I 1 (L + I 3 t ' cos( + t g m ( h cos( +R 4 Lagrange equations 4.1 theta equation t + t ' +
2 369(1b.wxm / 13 (%i6 D1: iff(l, iff(theta,t; (D1 I 1 h m t R h t t (%i7 E1: expan(ratsimp(iff(d1,t - iff(l,theta = 0; (E1 h m +I t 1 t +h m cos t I 1 t ' cos sin( +I 3 t ' cos sin( +I 3 t ' t g h m R h m sin( =0 t (%i8 E1a: solve(e1, iff(theta,t,; (E1a t = (h m cos ( I 3 I 1 t ' cos( +I 3 [ h m +I 1 ] 4. phi equation (%i9 D: iff(l, iff(phi,t; (D I 1 t ' +I 3 cos t + t ' t + g t ' cos t R h m sin( /( + t (%i10 E: expan(ratsimp(iff(d,t - iff(l,phi = 0; (E ( I 1 I 3 t ' cos I 3 t t +I 1 ' +I t 3 ' cos +I t 3 cos( =0 t (%i11 Ea: solve(e, iff(phi,t,; (Ea [ t '= ( I 1 I 3 t ' cos I 3 t t +I 3 t cos ] I 1 +I 3 cos 4.3 psi equation
3 369(1b.wxm 3 / 13 (%i1 D3: iff(l, iff(psi,t; (D3 I 3 + t ' cos t (%i13 E3: expan(ratsimp(iff(d3,t - iff(l,psi = 0; (E3 I 3 t ' t +I 3 ' cos( +I t 3 =0 t (%i14 E3a: solve(e3, iff(psi,t,; (E3a [ = t t ' t ' cos( ] t 4.4 R equation (%i15 D4: iff(l, iff(r,t; (D4 m R h t t (%i16 E4: expan(ratsimp(iff(d4,t - iff(l,r = 0; (E4 h m h m cos( t t +g m+ t R m=0 (%i17 E4a: solve(e4, iff(r,t,; (E4a [ R=h sin( +h cos( t t t g] 5 Lagrange equations with constants of motion (%i18 kill(l; (%o18 one 5.1 Rewriting the theta_otot equation (%i19 theta_a: ratsubst(theta_ot, iff(theta,t, rhs(first(e1a; (theta_a (sin( (cos t + g + I 1 I 3 t ' h m theta_ot I 3 t ' t R h m/(h m +I 1
4 369(1b.wxm 4 / 13 (%i0 theta_b: ratsubst(phi_ot, iff(phi,t, theta_a; (theta_b (h m cos theta_ot + I 3 phi_ot I 1 phi_ot cos( +I 3 phi_ot sin( /(h m +I 1 t + g (%i1 theta_c: ratsubst(psi_ot, iff(psi,t, theta_b; (theta_c (h m cos theta_ot + I 3 I 1 phi_ot cos( +I 3 phi_ot psi_ot + h m +I 1 t R h m g t R h m sin( /( (%i theta_: ratsubst(r_otot, iff(r,t,, theta_c; (theta_ (h m cos theta_ot + I 3 I 1 phi_ot cos h m +I 1 +I 3 phi_ot psi_ot + 5. Rewriting the R_otot equation (%i3 R_a: rhs(first(e4a; (R_a h +h cos( t t g (%i4 R_b: ratsubst(theta_ot, iff(theta,t, R_a; (R_b h cos theta_ot +h t g g R otot h m sin( /( (%i5 R_c: ratsubst(theta_otot, iff(theta,t,, R_b; (R_c h theta_otot+h cos theta_ot g 5.3 Solving for theta_otot an R_otot (%i6 GA: theta_otot = theta_; (GA theta_otot= (h m cos theta_ot + I 3 I 1 phi_ot cos g R otot h m sin( /( h m +I 1 (%i7 GB: R_otot = R_c; +I 3 phi_ot psi_ot + (GB R otot =h theta_otot+h cos theta_ot g
5 369(1b.wxm 5 / 13 (%i8 GC: solve([ga,gb], [theta_otot, R_otot]; ( I 1 I 3 phi_ot cos I 3 phi_ot psi_ot (GC [[theta_otot=, I 1 R otot =(I 1 h cos theta_ot + I 1 I 3 h phi_ot cos I 3 h phi_ot psi_ot I 1 g/ I 1 ]] 5.4 Preparing RHS (%i9 theta_otot: rhs(first(first(gc; I 1 I 3 phi_ot cos I 3 phi_ot psi_ot (theta_otot I 1 (%i30 R_otot: rhs(secon(first(gc; (R_otot (I 1 h cos theta_ot + I 1 I 3 h phi_ot cos I 3 h phi_ot psi_ot I 1 g/ I 1 (%i31 psi_ot: 1/I[3]*(L[psi]-I[3]*phi_ot*cos(theta; L I 3 phi_ot cos (psi_ot I 3 (%i3 phi_ot: (L[phi]-L[psi]*cos(theta/(I[1]*sin(theta^; L ' L cos (phi_ot I 1 (%i33 theta_ot: eta; (theta_ot (%i34 R_ot: v; (R_ot v 5.5 right-han sies of Lagrange equations, Hamilton form
6 369(1b.wxm 6 / 13 (%i40 Eq5: theta_otot /*= eta_ot*/; Eq5a: eta /*= theta_ot*/; Eq6: phi_ot; Eq7: psi_ot; Eq8: R_otot; /*= v_ot*/ Eq9: v; /*= R_ot*/ ( I 1 I 3 phi_ot cos I 3 phi_ot psi_ot (Eq5 (Eq5a (Eq6 L ' L cos I 1 (Eq7 L I 3 phi_ot cos I 3 I 1 (Eq8 (I 1 h cos theta_ot + I 1 I 3 h phi_ot cos I 3 h phi_ot psi_ot I 1 g/ I 1 (Eq9 v (%i41 str: [I[1]=0.5, I[3]=3., L[phi]=.8, L[psi]=, m=30, g=9.81, h=0.]; (str [I 1 =0.5,I 3 =3,L ' =0.8,L =,m=30,g =9.81,h=0.] (%i4 str: [I[1]=0.5, I[3]=4, L[phi]=1., L[psi]=3.8, m=10, g=9.81, h=0.] (str [I 1 =0.5,I 3 =4,L ' =1,L =3.8,m=10,g =9.81,h=0.]
7 369(1b.wxm 7 / 13 (%i48 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 0(Eq5A.0 ( cos ( cos( cos B cos A C cos (Eq6A (Eq7A.0 ( cos ( cos( cos (Eq8A.0 ( 1 ( cos ( cos( cos B A C.8 ( cos( cos cos (Eq9A v (%i49 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]$ 5.6 Graphics (%i55 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$ 5.7 Plot theta, phi, psi
8 369(1b.wxm 8 / 13 (%i56 wxplot([[iscrete, c1], [iscrete, c]], [xlabel, "time"], [legen, "theta_{ot}", "theta"]$ (%t56 (%i57 wxplot([[iscrete, c3], [iscrete, c4]], [xlabel, "time"], [legen, "phi", "psi"]$ (%t57
9 369(1b.wxm 9 / 13 (%i58 wxplot([[iscrete, c5], [iscrete, c6]], [xlabel, "time"], [legen, "v", "R"]$ (%t Plot space curve of centre of mass (%i59 cl1: makelist([p[1],p[7],p[3],p[4]],p,s$ (%i61 cl1[1];cl1[length(cl1]; (%o60 [ 0.0, 0.0, , ] (%o61 [ 8.0, , , ] (%i6 transf(cl1 := ( block([i,t,h,r,theta,phi], cl: [], h: 0., for i:1 thru length(cl1 o ( [t,r,theta,phi]: cl1[i], x: h*sin(theta*cos(phi, y: h*sin(theta*sin(phi, z: h*cos(theta + R, /*print(t,x,y,z,*/ cl: appen(cl, [[x,y,z]], return(cl; (%o6 transf( cl1 :=block([i,t,h,r,,'],cl:[],h:0.,for i thru length( cl1 o ([t,r,,']:cl1 i,x:h cos( ',y:h sin( ',z :h cos( +R,cl:appen( cl,[[x,y,z]],return( cl
10 369(1b.wxm 10 / 13 (%i64 cl: transf(cl1$ cr: points(cl$ (%i65 wxraw3(line_with=,color=blue, view=[76, 8], cr$ 0 errors, 0 warnings (%t Plot angular velocities (%i66 omega[3]: ratsimp(ev(phi_ot*cos(theta+psi_ot; (%o66 L I 3 (%i67 omega[1]: phi_ot*sin(theta*sin(psi+theta_ot*cos(psi; (%o67 ( L ' L cos I 1 + cos (%i68 omega[]: phi_ot*sin(theta*cos(psi-theta_ot*sin(psi; (%o68 cos ( L ' L cos I 1
11 369(1b.wxm 11 / 13 (%i71 omeg1: ev(omega[1], str; omeg: ev(omega[], str; omeg3: ev(omega[3], str; cos (omeg1.0 cos cos (omeg (omeg (%i7 om1: om: om3: om4: []; (om1 [] + cos (%i73 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] */ ; (%o73 one (%i74 kill(y; (%o74 one
12 369(1b.wxm 1 / 13 (%i75 wxplot([[iscrete, om1], [iscrete, om], [iscrete, om3], [iscrete, om4]], [xlabel, "time"], [legen, "omega[1]", "omega[]", "omega[3]", " omega "]$ (%t Plot space curve of omega vector (%i76 cl1: makelist([p[1],10,p[3],p[4]],p,s$ (%i78 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 [] (%o78 one (%i79 cr: points(cl$
13 369(1b.wxm 13 / 13 (%i80 wxraw3(line_with=,color=blue, cr$ (%t80 ;
Gyroscope with variable R
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()
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