3 00097
.... 3..3 4..4 5..6 6 0 7 3.4.4
) ) Lagrange 3) ) 4) 5)
3 r r A B C θ ( α ) r r 3
4 Lagrange B C x = r cosθ () y = r sinθ () x = r cosθ + r cos( θ + θ ) (3) y = r sinθ + r sin( θ + θ ) (4) x = rθ sinθ (5) y = rθ cosθ (6) (7) (8) B C m m T a T b Ta = m( x + y ) = mrθ (9) Tb = m( x + y ) (0) U U a b U = mr sinθ gcosφ () a U = m { r sinθ + r sin( θ + θ )} gcosφ () b Lagrange L( = T + T U U ) a b a b 4
(3) (4) (5) θ (6) θ (7) (8) (9) θ (0) 5
5 r r θ θ θ θ α 0 θ = α () θ = 0 () θ (3) (4) 6
(5) (6) θ (6) θ θ (6) C r r B r A r r, r r, r r = r sinα r = r + r cosα (6) θ 0 (7) A J 7
J = m {( r + r cos α) + ( r sin α) } + mr (8) (9) (7) (9) θ = {( T J) ( r + r cosα r sin α )} = ( Tr Jr ) (30) / / 0 (3) µ ( = m m) γ ( = r r) (3) 3 γ =..4 α π 6 3π 4 δ 4 θ 0 5 α = π π γ.0.5 δ 4 θ 0 r r α α r r 8
γ 3.5.5 0.5 0.5 0.75.5.5.75.5 α 3 γ α δ 3.5.5 0.5 0.5 0.75.5.5.75.5 α 4 γ α θ 0 γ 0.8 0.7 0.6 0.5 0.4 0.3 0. 0....3.4.5 r 5 α γ δ 4 0 8 6 4...3.4.5 r 6 α γ 0 θ 9
6 Mathematica η α µ γ (30) (3) η θ η = π 3 α = π µ = 0.3 γ =.4 7 8 θ 3π 0 θ θ 0 7 θ θ 8 θ 7 ( η = π 3, α = π, µ = 0.3, γ =.4 ) 8 ( η = π 3, α = π, µ = 0.3, γ =.4 ) 0
8 γ =. 9 0 9 7 0 θ θ 0 9 ( η = π 3, α = π, µ = 0.3, γ =.) 0 ( η = π 3, α = π, µ = 0.3, γ =.)
η η = π α = π µ = 0. γ =.4 η ( η π =, α = π, µ = 0., γ =.4 ) ( η π =, α = π, µ = 0., γ =.4 )
7 Lagrange 3
(000) 4
m =.0 m = 0.3 r =.0 r =. gdt = 0 q= 3.459ê80 s0 = 0 q alp = 90 q s0 = 360 q alp g= 9.8 fai = 30 q gcf = gcos@faid gdt tr0 = tr@td = tr0 rr = Hr+ r Cos@alpDL + Hr Sin@alpDL jj = m rr+ m r a = tr0êjj v = Sqrt@Ha Hr+ r Cos@alpDL + gcf Cos@s0 alpdl ê Hr Sin@alpDLD unc = v êaê uncêq ank = s0+ unc ankêq NDSolveA9gcf HHm+ ml r Cos@s3@tDD + m r Cos@s3@tD +s@tddl mrrsin@s@tdd s3@td s@td m r r Sin@s@tDD s@td + m r s3'@td + m r s3'@td + m r s3'@td + mrrcos@s@tdd s3'@td + m r s'@td + m r r Cos@s@tDD s'@td tr@td,s3@td == s3'@td, m r Igcf Cos@s3@tD + s@tdd + r Sin@s@tDD s3@td +Hr + r Cos@s@tDDL s3'@td + r s'@tdm 0, s@td == s'@td,s3@0d == v, s3@0d == ank, s@0d == 0, s@0d == alp=, 8s@tD, s@td, s3@td, s3@td<, 8t, 0, 3<E gy3= Plot@Evaluate@s3@tD s0 ê.out@d@@ddd, 8t, 0, 3<, Frame > TrueD gz3= Plot@Evaluate@s3@tD ê. Out@ D@@DDD, 8t, 0, 3<, Frame > TrueD gy= Plot@Evaluate@s@tDê.Out@ 3D@@DDD, 8t, 0, 3<, Frame > True, PlotStyle > 8Dashing@80.0, 0.0<D<D gz= Plot@Evaluate@s@tD ê. Out@ 4D@@DDD, 8t, 0, 3<, Frame > True, PlotStyle > 8Dashing@80.0, 0.0<D<D gzz= Show@gz, gz3, PlotRange > 80, <, FrameLabel > 8" t "," H θ ê tl,h θ ê tl "<D gyy= Show@gy, gy3, PlotRange > 8, 3<, FrameLabel > 8" t "," θ,θ "<D 5
. 0.3.. 0 0.074533.09439.5708 4.739 9.8 0.53599 0..663 0.6033 0.8399 0.55 3.57.64439 5.53 88s@tD InterpolatingFunction@880., 3.<<, <>D@tD, s@td InterpolatingFunction@880., 3.<<, <>D@tD, s3@td InterpolatingFunction@880., 3.<<, <>D@tD, s3@td InterpolatingFunction@880., 3.<<, <>D@tD<<.5 0.5 0-0.5 - -.5-0 0.5.5.5 3 6
.5.4.3.. 0.9 0.8 0 0.5.5.5 3 0-0.5 - -.5 0 0.5.5.5 3.5 0.5 0-0.5 - -.5 0 0.5.5.5 3 H θê tl,h θê tl.75.5.5 0.75 0.5 0.5 0 0.5.5.5 3 t 7
3 θ,θ 0-0 0.5.5.5 3 t 8
r =.0 r =. aln = 30 alx = 35 q= 3.459ê80 fad = 30 fai = fad q gdt = 0 gcf = gdt 9.8 Cos@faiD as = xn = aln q xx = alx q f@x_d = Hr+ r Cos@xD + gcfê aslêhr Sin@xDL gx= Plot@f@xDê, 8x, xn, xx<, Frame > TrueD fv@x_d = Sqrt@f@xDD gv= Plot@fv@xD, 8x, xn, xx<, Frame > TrueD r =.4 f@x_d = Hr+ r Cos@xD + gcfê aslêhr Sin@xDL gx= Plot@f@xDê, 8x, xn, xx<, Frame > True, PlotStyle > 8Dashing@80.0, 0.0<D<D fv@x_d = Sqrt@f@xDD gv= Plot@fv@xD, 8x, xn, xx<, Frame > True, PlotStyle > 8Dashing@80.0, 0.0<D<D grr= Show@gx, gx, PlotRange > 80, 3.<, FrameLabel > 8" α ", " γ "<D grr= Show@gv, gv, PlotRange > 80, 3.<, FrameLabel > 8" α ", " "<D.. 30 35 0.074533 30 0.53599 0 0 0.53599 9
.3569. H.+. Cos@xDL Csc@xD.5 0.5 0 0.5 0.75.5.5.75.5. è H.+. Cos@xDL Csc@xD.8.6.4. 0.8 0.5 0.75.5.5.75.5.4. H.4+. Cos@xDL Csc@xD.5.75.5.5 0.75 0.5 0.5 0.75.5.5.75.5. è H.4+. Cos@xDL Csc@xD 0
.8.6.4. 0.5 0.75.5.5.75.5 γ 3.5.5 0.5 0.5 0.75.5.5.75.5 α 3.5.5 0.5 0.5 0.75.5.5.75.5 α
r =.0 q= 3.459ê80 ald = 90 alp = ald q fad = 30 fai = fad q gdt = 0 gcf = gdt 9.8 Cos@faiD as = xn =.0 xx =.5 f@x_d = Hx+ r Cos@alpD + gcfê aslêhr Sin@alpDL gx= Plot@f@xDê, 8x, xn, xx<, Frame > TrueD fv@x_d = Sqrt@f@xD asd gv= Plot@fv@xD, 8x, xn, xx<, Frame > TrueD ald = 0 alp = ald q f@x_d = Hx+ r Cos@alpD + gcfê aslêhr Sin@alpDL gx= Plot@f@xDê, 8x, xn, xx<, Frame > True, PlotStyle > 8Dashing@80.0, 0.0<D<D fv@x_d = Sqrt@f@xD asd gv= Plot@fv@xD, 8x, xn, xx<, Frame > True, PlotStyle > 8Dashing@80.0, 0.0<D<D grr= Show@gx, gx, PlotRange > 80, 0.8<, FrameLabel > 8" r "," γ "<D grr= Show@gv, gv, PlotRange > 80, 5<, FrameLabel > 8" r "," "<D. 0.074533 90.5708 30 0.53599 0 0
..5. H3.6795 0 7 + xl 0.75 0.7 0.65 0.6 0.55 0.5...3.4.5. " 3.6795 0 7 + x 3.5 3.5.5...3.4.5 0.09439.547 H 0.5 + xl 0.55 0.5 0.45 0.4 0.35 0.3...3.4.5.803 è 0.5 + x 3
.5 0.5 0 9.5 9 8.5...3.4.5 γ 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 4 0 8 6 4...3.4.5 r...3.4.5 r 4