44 4 4.1 d P = d dv M v = F M = F 4.1 M v P = M v F 4.1 d H = M 4.2 H M Z K I z R R J x k i P r! j Y y - XY Z I, J, K -xyz i, j, k P R = R + r 4.3 X Fig. 4.1 Fig. 4.1 ω P [ ] d d = + ω 4.4 [ ]
4 45 4.3 [ ṙ ] = v = Ṙ = Ṙ + ω r v + ω r 4.5 P dm P dp = vdm M = P = dp = v dm = v + ω r dm = v dm + ω r dm = M v 4.6 dm 4.1 H H = r dp = r vdm = r v + ω r dm = v r dm + r ω r dm = r ω r dm 4.7 ω r ω = p i + q j + r k, r = x i + y j + z k 4.8 ω r = qz ryi + rx pzj + py qxk 4.9 r ω r = { ypy qx zrx pz } i + { zqz ry xpy qx } j 4.7 I xx = H x H y H z y 2 + z 2 dm, I yy = + { xrx pz yqz ry } k 4.1 H = H x i + H y j + H z k 4.11 I xx I xy I xz p = I yx I yy I yz q I zx I zy I zz r 4.12 z 2 + x 2 dm, I zz = x 2 + y 2 dm 4.13 x, y, z moment of inertia 4.12 I xy = I yx = xy dm, I yz = I zy = yz dm, I zx = I xz = zx dm 4.14 product of inertia xz y I xy, I yz 4.12 { } [ ] { } H = I ω [ I ] inertia tensor 4.15
46 4.2 4.13 I xx = M κ 2 xx, I yy = M κ 2 yy, I zz = M κ 2 zz 4.16 y 2 + z 2 dm z 2 + x 2 dm x 2 + y 2 dm κ 2 xx =, κ 2 yy =, κ 2 zz = 4.17 M M M κ xx, κ yy, κ zz x, y, z gyrational radius z z B θ r y L D ' y x x Fig. 4.2 Fig. 4.3 Fig. 4.2 x = r sin θ cos φ y = r sin θ sin φ z = r cos θ d = r 2 sin θ dr dθ dφ ρ R I xx = y 2 + z 2 dm = ρ y 2 + z 2 d π = ρ R r 4 dr 2π π sin 2 θ sin 2 φ + cos 2 θ sin θ dθ dφ sin 3 θ dθ = 4 π 3, cos 2 θ sin θ dθ = 2 2π 3, sin 2 φ dφ = π, I xx = ρ R5 5 κ xx = R r 4 dr = R5 5 8π 3 = ρ 4 2 3 πr3 5 R2 M κ 2 xx 4.18 2 R.632R =.316 2R 4.19 5 R.632 I xx = I yy = I zz
4 47 L B D Fig. 4.3 x, y, z ρ I xx = y 2 + z 2 ρ d = ρld B3 D3 + ρlb 12 12 = M B2 + D 2 M κ 2 xx 4.2 12 Iyy = M L2 + D 2 M κ 2 yy, I zz = M L2 + B 2 M κ 2 zz 4.21 12 12 M = ρlbd D/B = 1, B/L = 1/5 4.21 κ xx.48b, κ yy = κ zz.294l 4.22 4.22 κ xx.35b κ yy κ zz.25l 4.1 L = 2a B = 2b D = 2c 3 x a 2 + y b 2 + z c 2 = 1 I xx, I yy, I zz c/b = 1, b/a = 1/5 x, y, z x/a = r cos θ, y/b = r sin θ cos φ, z/c = r sin θ sin φ r 1, θ π, φ 2π κ xx.316b, κ yy = κ zz.228l Fig. 4.4 -xyz r x, y, z xyz -x y z z z x x' r r z' P r' y' y Fig. 4.4 I zz = x 2 + y 2 dm = x + x 2 dm + y + y 2 dm = x 2 + y 2 dm + x 2 + y 2 dm + 2x x dm + 2y y dm = x 2 + y 2 M + I zz
48 I xx = I xx + M y 2 + z2, Ixy = I xy M x y I yy = I yy + M z 2 + x2, Iyz = I yz M y z I zz = I zz + M x 2 + y2, Izx = I zx M z x 4.23 4.3 Fig. 4.5 compound pendulum θ h Mg Fig. 4.5 = h θ M I 4.2 I θ = M gh sin θ 4.24 κ I = M κ 2 4.24 M κ2 h θ = M g sin θ 4.25 l ml θ = mg sin θ l = κ2 h 4.26 l T = 2π g = 2π κ I = 2π gh M gh 4.27 κ = T 2π gh 4.28 h T κ 4.23 I = I M h 2 = M κ 2 h 2 M κ 2 4.29 κ = κ 2 h 2 4.4 -xyz P 4.5 v = v + ω r v 2 = v v = v 2 + ω r ω r + 2v ω r
4 49 T = 1 v 2 dm 2 = 1 2 M v2 + 1 ω r ω r dm + v ω 2 r dm = ω r ω r = ω r ω r r dm 4.3 4.31 4.3 T = 1 2 M v2 + 1 2 ω r ω r dm = 1 2 M v2 + 1 2 ω H = 1 2 M v2 + 1 2 { ω } T [ I ]{ ω } 4.32 H 4.7 4.15 { ω } T T 2 4.32 T U E = T + U = 4.33 v = h θ 4.29 T = 1 2 M v2 + 1 2 I θ 2 = 1 2 M h2 θ2 + 1 I M h 2 2 θ2 = 1 2 I θ 2 U = M gh 1 cos θ 4.33 4.34 1 2 I θ 2 + M gh 1 cos θ = C 1 2 I θ 2 M gh cos θ = C 4.35 4.24 4.24 θ I θ θ = M gh θ sin θ I 1 2 θ 2 = M gh cos θ + C 4.35 4.5 -xyz xy -XY Z X Y 2 1 F = F X I+F Y J R = X I+Y J a = ẌI + ŸJ 4.36
5 Y v x y u Y v j i R J I X X Fig. 4.6 z H x =, H y =, H z = I zz r = I zz 4.37 z Z k = K H = I zz k = I zz K 4.38 z M = M z k = M z K 4.1 4.2 M Ẍ = F X, M Ÿ = F Y, I zz = Mz 4.39 F X, F Y Y X 2 v = ui + vj i, j a = v = ui + u di dj + vj + v i, j 4.4 ω = rk = k di = ω i = k i = j dj = ω j = k j = i 4.4 4.41 a = u v i + v + u j 4.42 F = F x i + F y j H = I zz k, M = M z k M u v = F x M v + u = F y I zz = Mz 4.43
4 51 u, v, Ẋ + i Ẏ = u + i v e i X = X + Y = Y + t t u cos v sin u sin + v cos 4.44 4.43 F m = F xm, F ym,, M m =,, M zm v = u m, v m,, ω =,, r m x,, F xm = F x, F ym = F y, M zm = M z + F y x u m = u, v m = v x r, r m = r = } 4.45 4.43 m M u v x 2 = F x M v + u + x = Fy Izz + M x 2 + M x v + u = M z 4.46 z I zz + M x 2 4.23 4.6 3 6 4.1 4.2 [ I ] -xyz 4.1, 4.2 -XY Z v F v = ui + vj + wk, F = F x i + F y j + F z k 4.47 M 4.1 [ ] dv M + ω v = F 4.48 ω 4.8 ω = pi + qj + rk 4.48 M u + wq vr = F x M v + ur wp = F y M ẇ + vp uq = F z 4.49
52 4.43 M = M x i + M y j + M z k 4.2 [ dh ] + ω H = M 4.5 4.12 4.8 I xx ṗ + I zz I yy qr ṙ +I xy q rp + Ixz + pq + Iyz q 2 r 2 = M x I yy q + I xx I zz rp ṙ ṗ +I yz pq + Iyx + qr + Izx r 2 p 2 = M y I zz ṙ + I yy I xx pq ṗ +I zx qr + Izy q + rp + Ixy p 2 q 2 = M z 4.51 I xy = I yz = I xz = 4.51 I xx ṗ + I zz I yy qr = Mx I yy q + I xx I zz rp = My I zz ṙ + I yy I xx pq = Mz 4.52 X x 1 Y y 1 x 1 x 2 y 1 y 2 x 1 x 2 x 3 θ Z z 1 x 2 x 3, x y 1 y, θ y 1 1 y 2 x 1 y, 1 y 2 θ φ y φ z 1, z 2 θ φ z θ z 3 z 1, z z 3 2 z 1, z 2 Fig. 4.7
4 53 1 -XY Z -x 1 y 1 z 1 2 z 1 -x 2 y 2 z 2 3 y 2 θ -x 3 y 3 z 3 4 x 3 ϕ -xyz 1 2 4 u 1 v 1 w 1 u 2 v 2 w 2 u 3 v 3 w 3 = = = cos sin u 2 sin cos v 2 1 w 2 cos θ sin θ u 3 1 v 3 sin θ cos θ w 3 1 u cos ϕ sin ϕ v sin ϕ cos ϕ w 4.53 4.54 4.55 3 Ẋ u E = Ẏ Ż = E, θ, ϕ cos θ cos sin ϕ sin θ cos cos ϕ sin cos ϕ sin θ cos + sin ϕ sin cos θ sin sin ϕ sin θ sin + cos ϕ cos cos ϕ sin θ sin sin ϕ cos v w sin θ sin ϕ cos θ cos ϕ cos θ 4.56 4.57, θ, ϕ Eulerian angles u, v, w Ẋ, Ẏ, Ż X = t Ẋ, Y = t Ẏ, Z = t Ż 4.58 4.49, 4.52 u, v, w, p, q, r 4.57 z 1 = z 2 θ z 3 x 3 -x 3 y 3 z 3 sin θ,, cos θ θ y 2 = y 3 ϕ x 3 -x 3 y 3 z 3 ϕ sin θ, θ, cos θ -xyz p, q, r p q r = 1 cos ϕ sin ϕ sin ϕ cos ϕ ϕ sin θ θ cos θ
54 = 1 sin θ cos ϕ cos θ sin ϕ sin ϕ cos θ cos ϕ ϕ θ 4.59 ϕ 1 sin ϕ tan θ cos ϕ tan θ θ = cos ϕ sin ϕ sin ϕ sec θ cos ϕ sec θ p q r 4.6 ϕ, θ, p, q, r ϕ, θ, ϕ = t ϕ, θ = t θ, = t 4.61 4.56 4.58 ϕ, θ, 3 ϕ roll heel θ pitch trim yaw