215 11 13

1 2 1.1....................... 2 1.2.................... 2 1.3..................... 2 1.4...................... 3 1.5............... 3 1.6........................... 4 1.7.................. 4 1.8..................... 5 2 6 2.1....................... 6 2.2.................. 6 2.3.................... 6 3 9 3.1................ 9 3.2.......... 9 3.3.......... 1 3.4.......... 12 4 13 4.1............ 13 4.2................. 14 4.3..................... 14 4.4...................... 15 4.5.............. 15 5 18 5.1....................... 18 5.2..................... 18 5.3.................. 18 5.4......................... 19 i

A 24 A.1.................... 24 A.2......................... 24 A.3............................ 25 A.4...................... 25 A.5....................... 26 B 3 B.1...... 3 B.2.......................... 3 C 31 C.1....................... 31 C.2......................... 32 C.3..................... 32 C.4.............. 32 D 33 D.1....................... 33 D.2....................... 33 D.3........... 34 ii

natural science Newtonequation of 3 motion 5 universe Schrödinger quantum mechanics classical mechanics theory of elasticity 1

1 1.1 point mass rigid bodyelastic body fluid dynamics statics 1.2 force f 5 f 4 f 1 f 3 f 2 f 2 f 3 f 1 f 4 f 5 1.3 moment of force torque N N N O AOA N O r A f f 1 f 2 OA O O 2

N = r f (1.1) r f r f sin θ θ r f r f 1.4 F 1 F 2 F 1 + F 2 = F 1 = F 2 (1.2) F h1 F h2 F v1 F v2 F h1 + F h2 = F v1 + F v2 = 1.5 truss rahmen beam theory 3

1.6 frictional force tribology F s N F s = µ s N (1.3) µ s static coefficient of friction F d F d = µ d N (1.4) N F µ d dymanic coefficient of friction 1.7 1 1 m 4

1.8 f 1,f 2,f 3,,, δx 1, δx 2, δx 3,,, δx i f i δx i = (1.5) principle of virtual work 5

2 2.1 mass (inertial x f x p x mass) r = y, f = f y, p = p y,,, m f z f z p z r ṙ = dr dt, r = d2 r dt 2 (2.1) f = m r, (2.2) momentum p = mṙ f = ṗ (2.3) ẋ ṙ = ẏ, ḟ = f ẋ, ṗ = f y p y ż f z p z p ẋ,,, 2.2 t f(t) t = p(t + t) p(t) (2.4) f t impulse (2.4) 2.3 (2.3) t t + t t+ t fdt = p(t + t) p(t) t t F (2.4) t = t 1 x = x 1, 6

ẋ = v 1 t = t 2 x = x 2, ẋ = v 2 x 2 f dx = 1 x 1 2 mv 2 2 1 2 mv 1 2 (2.5) E k = 1 2 mv2 (2.6) kinetic energy (2.5) (2.5) x 2 x 1 f dx = t 2 t 1 f dx dt dt = t 2 t 1 f vdt = t 2 t 1 mẍ vdt = t 2 t 1 m v vdt = t 2 t 1 mv vdt = t 2 t 1 mv dv dt dt = v 2 v 1 mvdv = [ 1 2 mv2 ] v2 v 1 (2.5) m g mg ẍ = m ÿ (2.7) z x x = v x t (2.8) y = (2.9) z = v z t gt 2 /2 (2.1) (2.8) (2.1) z x t = r = r v = v r = r + v t 1 2 t 2 g (2.8) (2.1) t z = g 2v x 2 + vz x 2 v x x pendulum m T L sin θ r = L L cos θ, f = T sin θ T cos θ mg, (2.11) L z O m x Lsin T mg T θ f = m r g sin θ = L θ (2.12) 7

θ sin θ θ gθ = L θ (2.13) ( ) g θ = A sin L t + α (2.14) A α kr = m r (2.14) 8

3 3.1 r r ξ a ξ r = η a =,,, ζ a η a ζ 3.2 ξ, η, ζ x, y, z r = r + r (3.1) (2.2) f = f v = r f m v = m r (3.2) (3.2) x z O r y r r' (3.1) x x ξ y = + η y z z ζ O' 9

v v 3.3 r r R r = Rr (3.3) z O R R 1 = R T ṘRT. ṘRT a ṘR T a = ω a (3.4) ω ṙ = R(ω r + ṙ ) (3.5) r = R{ω (ω r ) + ω r + 2ω ṙ + r } (3.6) x A 11 A 12 A 13 A = A 21 A 22 A 23, A 31 A 32 A 33 R xξ R xη R xζ R = R yξ R yη R yζ,,, R zξ R zη R zζ y f mω (ω r ) m ω r 2mω ṙ = m r (3.7) centrifugal force Coriolis force (3.4) R ω ω (3.7) cos α sin α α β sin α cos β sin β cos α cos β R = sin α sin β cos β cos α sin β (3.8) z O α β R ω (3.4) α sin β sin β ω = α cos β = β + ( α) cos β (3.9) β 1 x y 1

ω β cos α cos α ω = α = β + ( α) 1 (3.1) β sin α sin α (3.9) (3.1) z η ω ω R O z z x y α(rad) ζ ξ ω α ω R ω ω cos α ω = ω = ω (3.11) ω sin α (3.7) R ṘR T = β α cos β β α sin β α cos β α sin β (3.4) ω R ω x t sin α cos ωt sin ωt cos α cos ωt R = sin α sin ωt cos ωt cos α sin ωt z O cos α sin α y m r =, f = (3.12) ζ mg R E ζ R E g (3.11) sin α mω (ω r ) = m(ζ cos α)ω 2 (3.13) cos α 11

2mω ṙ = 2mω ζ cos α (3.14) ζ < Foucault 2m 3.4 r = r + Rr (3.15) 3.2 (3.15) f = Rf, v = Rv, ω = Rω, 3.3 R ω f mω (ω r ) m ω r 2mω ṙ x O ṙ = R(ω r + ṙ + v ) m(ω v r = R{ω (ω r ) + ω r + 2ω ṙ + (ω + v ) = m r (3.16) v + v ) + r } ( ) (3.7) ( ) f = m r m(ω v + v ) (3.16) (3.2) z r r r' O' y 12

4 4.1 O ω ω ω r dm df = rdm (4.1) r r df = V V r rdm = V d {r (ω r)}dm (4.2) dt N N = d ( ) r (ω r)dm = d (Iω) (4.3) dt V dt (y 2 + z 2 ) xy zx I = xy (z 2 + x 2 ) yz V zx yz (x 2 + y 2 ) dm (4.4) I inertia tensor angular momentum (4.3) L = Iω (4.5) N = L (4.6) (4.6) f = ṗ P Q v dm r O ρ dv dm = ρdv dm v = ṙ = ω r d dt {r (ω r)} = d dt {r ṙ} = r r a 11 a 12 a 13 V a 21 a 22 a 23 dm a 31 a 32 a 33 V a 11dm V a 12dm V a 13dm = V a 21dm V a 22dm V a 23dm V a 31dm V a 32dm V a 33dm r (ω r) = r 2 ω (r ω)r x ω x r = y ω = z ω z (4.3) (4.4) ω y 13

p = mv m L = Iω I 4.5 4.2 (4.6) t N(t) t = L(t + t) L(t) (4.7) N(t) t (4.7) 4.3 de = 1 2 dm v 2 1 E = de = 2 ω r 2 dm = 1 2 (Iω) ω = 1 2 L ω (4.8) V V ω r 2 = ω T (y 2 + z 2 ) xy zx xy (z 2 + x 2 ) yz zx yz (x 2 + y 2 ) ω (2.6) 1 2 m v 2 (2.6) (4.8) J Ns Nms 14

4.4 (4.6) (4.4) I = I x I xy I zx I xy I y I yz I zx I yz I z (4.9) I x, I y, I z (moment of inertia)i xy, I yz, I zx r r G r r = r G + r (4.1) M = V dm (4.11) (yg 2 + z2 G ) x Gy G z G x G I = M x G y G (zg 2 + x2 G ) y Gz G z G x G y G z G (x 2 G + y2 G ) (η 2 + ζ 2 ) ξη ζξ + ξη (ζ 2 + ξ 2 ) ηζ dm (4.12) V ζξ ηζ (ξ 2 + η 2 ) (4.12) M 3.2 x x G ξ y = + η y G z z G ζ (4.12) V r r Gdm = V r dm = V ξdm = V ηdm = ζdm = V I z = V (x2 + y 2 )dm = (x G 2 + y G 2 )M + V (ξ2 + η 2 )dm. 4.5 (4.6) (4.5) N 15

ω N N N ω I Iµ = λµ (4.13) µ (4.13) µ λ I eigenvectoreigenvalue E(I λe)µ = µ( ) det(i λe) = 1 µ = µ i, λ = I pi, (i = 1 3) (4.14) E = 1 1 µ i (4.13) µ ix µ i µ i = µ iy N pi ω pi (4.6) N pi = I pi ω pi (4.15) µ iz i 1 3 µ i I pi (4.15) µ 1x µ 2x µ 3x I p1 P = µ 1y µ 2y µ 3y, I p = I p2 (4.16) µ 1z µ 2z µ 3z I p3 P I = P I p P T (4.17) (4.17) I p P I I P T I p P 123- xyz- Iµ 1 = I p1 µ 1, Iµ 2 = I p2 µ 2, Iµ 3 = I p3 µ 3 IP = P I p (4.17) 16

I p (= P T IP ) µ 1 µ 2 µ 3 1 2 3 P T P T µ 1 = 1, P T µ 2 = 1, P T µ 3 = 1 (4.18) I P T N L N p (= P T N) L p (= P T L) N = L N p = d dt (L p) = d dt (I pω p ) = İpω p + I p ω p (4.19) ω µ i I p I p1 I p3 ω p i ω pi I p i N p N ( = L ) N p = P T N = P T L = d dt P T L ( ) ( ) P T Iω P T IP P T ω = d dt = d dt = d dt (I pω p ) i İpω p = i = 3 N p = I p ω p ω p = N pi µ i = I pi ω pi µ i (4.2) (4.15) ω p3 3 θ I p I p1 cos 2 θ + I p2 sin 2 θ (I p1 I p2 ) cos θ sin θ (I p1 I p2 ) cos θ sin θ I p1 sin 2 θ + I p2 cos 2 θ I p3 I p İp 1 1 1 2 2 1 2 2 θ 2.3 3 3 İpω p = N = I θ N = mgl sin θi = ml 2 g sin θ = L θ 2.3 3 I p ω p ω p 17

5 5.1 N = L 3.3 L L R r = Rr 3.3 L = R(µ L + L ) (5.1) µ a ṘRT a = µ a µ R (5.1) N = µ L + L (5.2) L = d dt (RL ) = ṘL + R L = RR T ṘL + R L = R(µ L ) + R L = R(µ L + L ) (5.2) µ (3.7) ω ω µ L = Iω ω 5.2 E = 1 2 L ω = 1 2 L ω (5.3) 1 2 L ω = 1 2 (RL ) (Rω ) = 1 2 (RL ) T Rω = 1 2 L T R T Rω = 1 2 L T ω = 1 2 L ω 5.3 gyro 18

ω µ 3.3 α = π/2 z ζ ξ x y µt ξ ω (5.2) L L = L µ N = µ L (5.4) x N g = N = µ L (5.5) µ ζ L ξ η 5.4 N g = µ N g z t O I ξ ω = µi ξ ω y (precession) O ζ < α < π/2 I λ I = I ξ I ξ I ζ, λ = λ, (5.6) x z O y Mg z H O I ξ I ζ λ ζξ x β µ α β µ = µ α + µ β µ α = α, µ β = β, µ = β cos α α β sin α (5.7) 3.3 3.3 α β 19

H g N = HMg cos α ω = λ + µ ω = λ + µ µ L = L = I ζ αλ (I ζ I ξ ) α β sin α I ζ βλ cos α + (Iζ I ξ ) β 2 sin α cos α I ξ ( β cos α + α β sin α) I ξ α I ζ ( λ + β sin α + α β cos α) (5.8) (5.9) (5.1) (5.2) I ζ αλ + (I ζ 2I ξ ) α β sin α + I ξ β cos α = (5.11) HMg cos α = I ζ βλ cos α + (Iζ I ξ ) β 2 sin α cos α I ξ α (5.12) λ + β sin α + α β cos α = (5.13) α. (3.7) (5.2) λα, β ω µ (5.11) (5.13) t = α = α β = β λ = λ α = α β = β λ = λ α = α = const. µ µ α β = β = const.λ = λ = const. (5.14).pdf (I ζ λ) 2 4HMg(I ξ I ζ ) sin α β = I ζ λ 1 1 4HMg(I ξ I ζ ) sin α/(i ζ λ) 2 2(I ξ I ζ ) sin α (5.15) α = α = const. λ (5.15) 2

(5.15) α β HMg λi ζ (5.16) λ β H > λ β A 2T H I ζ = ρπt A 4, M = 2ρπT A 2 β 2Hg A 2 λ (5.17) G A 2T H O λ = α = β = (5.11) (5.13) t λ = α β{(i ζ 2I ξ ) sin 2 α I ξ cos 2 α} = (5.18) {} α β = β = λ = λ = const., β = β = const., HMg cos α = I ξ α (5.19) α 2.3 θ α sin cos 21

F F F (Friction) F λ λ > F = F (5.8) N = HMg cos α F (5.2) (5.8) (5.2) α α = αλ (5.22) I ζ αλ + I ξ β cos α = (5.21) HMg = I ζ βλ + (Iζ I ξ ) β 2 sin α (5.22) F = I ζ ( λ + β sin α) (5.23) β = I ζ λ 1 1 4HMg(I ξ I ζ ) sin α/(i ζ λ) 2 2(I ξ I ζ ) sin α (5.24) β (5.23) (5.23) λ = λ F t (5.25) I ζ λ I ζ F (5.24) λ β β β > (5.21) β (5.24) α β HMg λi ζ α = I ξ I ζ λ β cos α (5.26) α < (5.25) 22

(5.24) (5.26) (5.26) λi ξ /I ζ 23

A A.1 raar raar r = x y, a = a x a y z a z A 11 A 12 A 13 A = A 21 A 22 A 23, r =, R = ξ η, a = a ξ a η ζ a ζ R xξ R xη R xζ R yξ R yη R yζ,,,, A 31 A 32 A 33 R zξ R zη R zζ ṙ = ẋ ẏ ż V Adm =, ȧ = ȧ x ȧ y ȧ z, ṙ = ξ η ζ,,, V A 11dm V A 12dm V A 13dm V A 21dm V A 22dm V A 23dm V A 31dm V A 32dm V A 33dm,,, A.2 T r T = ( x y z ), a T = ( a x a y a z ), A T = 1 A 11 A 21 A 31 A 12 A 22 A 32 A 13 A 23 A 33 24

1 A 1 A = AA 1 = E = 1 E 1 A.3 A 11 A 12 A 13 x Ar = A 21 A 22 A 23 y A 31 A 32 A 33 z = a b = a x b x + a y b y + a z b z =a T b A 11 x + A 12 y + A 13 z A 21 x + A 22 y + A 23 z A 31 x + A 32 y + A 33 z e x e y e z a y b z a z b y a b = a x a y a z = a z b x a x b z b x b y b z a x b y a y b x A.4 ABC ab R d d (Aa) = Ȧa + Aȧ, dt d d (a b) = ȧ b + a ḃ, dt A(BC) = (AB)C = ABC (AB) T = B T A T R 1 = R T R(a b) = (Ra) (Rb) a (b a) = a 2 b (a b)a (AB) = ȦB + AḂ dt (A.6) (a b) = ȧ b + a ḃ dt (A.7) (A.1) (A.2) (A.3) (A.4) (A.5) A a r = Rr, r = R T r, (A.8) 25

A = RA R T, A = R T AR (A.9) A.5 R R 1 = R T ab a = Ra b = Rb a b a b = a b (Ra ) (Rb ) = a b (Ra ) T (Rb ) = a b (a T R T )(Rb ) = a b a T R T Rb = a b a (R T Rb ) = a b a (R T Rb b ) = E a (R T R E)b = ab R T R = ER T = R 1 R ṘR T RR T = E ṘR T + RṘT = { ( } ṘR T = RṘT = RṘT) T T { (ṘT ) T T = R T} 26

T = {ṘR } T ṘR T A anti s. r A anti s. r = ω r ω A anti s. A 12 A 13 A anti s. = A 12 A 23 A 13 A 23 ω x x ω = r = y ω y ω z A anti s. r = (A 13 )z ( A 12 )y ( A 12 )x ( A 23 )z ( A 23 )y (A 13 )x ω x y ω y x A anti s. ω ω = A 23 A 13 A 12 z ω r = ω y z ω z y ω z x ω x z r ω R r ṘR T r = ω r R T Ṙr = ω r r = Rr ṘRT r = ω r R T R T ṘR T r = R T (ω r) R T Ṙr = ω r r r R T Ṙr = ω r 27

I λ µ λ λ µ µ Iµ = λµ (1) µ Iµ µ = λµ µ (Iµ) T µ = λµ µ µ T I T µ = λµ µ (2) (1) I Iµ = λ µ z z µ z 1 z 2 (z 1 z 2 ) = z1z 2 µ Iµ = λ µ µ z 1 = x 1 + iy 1 z 2 = x 2 + iy 2 µ T Iµ = λ µ µ (3) (z 1 z 2 ) = (x 1 x 2 y 1 y 2 ) i(x 1 y 2 + x 2 y 1 ) (1) I (2) (3) z1z 2 = (x 1 iy 1 )(x 2 iy 2 ) = (x 1 x 2 y 1 y 2 ) i(x 1 y 2 + x 2 y 1 ) (2) λ = λ λ (1) (2) (z 1 z 2 ) = z1z 2 µ 1 µ 2 λ 1 λ 2 Iµ 1 = λ 1 µ 1 (1) Iµ 2 = λ 2 µ 2 (2) (1) (2) µ λ 1 λ 2 (1) µ 2 µ 2 (Iµ 1 ) = λ 1 µ 2 µ 1 µ T 2 Iµ 1 = λ 1 µ 2 µ 1 (1 ) (2) µ 1 µ 1 (Iµ 2 ) = λ 2 µ 1 µ 2 µ T 1 Iµ 2 = λ 2 µ 1 µ 2 28

(µ T 1 Iµ 2 ) T = λ 2 µ 1 µ 2 µ T 2 I T µ 1 = λ 2 µ 1 µ 2 (2 ) I I = I T (1 ) (2 ) (λ 1 λ 2 )µ 1 µ 2 = λ 1 λ 2 µ 1 µ 2 = µ 1 µ 2 = 29

B B.1 abc x t aẍ + bẋ + cx = (B.1) aλ 2 + bλ + c = λ = λ 1, λ 2 x(t) = C 1 e λ1t + C 2 e λ 2t (B.2) λ x(t) = C 1 e λt + C 2 te λt (B.3) C 1 C 2 λ e iθ = cos θ + i sin θ (B.4) C 1 C 2 B.2 ẍ + cx = (B.5) λ = ±i c (B.6) (B.8) x(t) = C 1 e +i ct + C 2 e i ct (B.6) = B 1 cos( ct) + B 2 sin( ct) (B.7) = A sin( ct + α) (B.8) 3

C C.1 A r r 2 r = m r (C.1) A N = L z xy z = xy r = r cos θ r sin θ (C.2) A = mr 2 ( r r θ 2 ) (C.3) = 2ṙ θ + r θ (C.4) (C.4) L I z = mr 2 ω z = θ L = mr 2 θ L θ = L mr 2 (C.5) (C.5) (C.3) r = L 2 /(ma) 1 B cos(θ + β) (C.6) B β cos θ sin θ r r = sin θ cos θ 1 cos θ sin θ r r θ 2 r = sin θ cos θ 2ṙ θ + r θ 1 u = 1/r du/dθ r θ (C.5) r = L d 2 u mr 2 dθ (C.3) 2 d 2 u dθ L 2 m u = A L u = C 1 cos(θ + C 2 ) ma L 2 31

C.2 xy r(θ) = ±l 1 ± ϵ cos θ (C.7) (C.7) ± θ + l ϵ ϵ ϵ = < ϵ < 1 ϵ = 1 1 < ϵ 2 1-1 2. 1..5. y/l x/l C.3-2 -1 1 2 ϵ =.,.5, 1,, 2. r(θ) = l 1±ϵ cos θ C.4 32

D D.1 z R E z ω α (rad) ξη ζ d ζ d ξ d φ O ζ = R E m L P T L R E g x t O y D.2 T sin θ cos φ f = T sin θ sin φ T cos θ mg (D.1) O d P L m R E O Lsin d z cos α ω =, ω = ω, ω sin α (D.2) (3.7) T sin θ cos φ ξ sin 2 α + ζ sin α cos α T sin θ sin φ + mω 2 η T cos θ mg 2mω ξ sin α cos α + ζ cos 2 α ξ = m η ζ η sin α ξ sin α + ζ cos α η cos α (D.3) 33

r = ξ η ζ = L sin θ cos φ L sin θ sin φ R E + L L cos θ (D.4) (D.3) ξ, η, ζ (D.4) θ, φ, T D.3 x ẋ ẍ r ξ = L( θ cos θ cos φ φ sin θ sin φ) η = L( θ cos θ sin φ + φ sin θ cos φ) ζ = L θ sin θ ξ = L( θ cos θ cos φ θ 2 sin θ cos φ θ φ cos θ sin φ) L( φ sin θ sin φ + φ 2 sin θ cos φ + θ φ cos θ sin φ) η = L( θ cos θ sin φ θ 2 sin θ sin φ + θ φ cos θ cos φ) +L( φ sin θ cos φ φ 2 sin θ sin φ + θ φ cos θ cos φ) ζ = L( θ sin θ + θ 2 cos θ) (D.3) 1.pdf (D.3) ζ T ξ η L R E R E T θ φ ω φ 2 sin θ = θcos θ = 1 3 θθ 2, φ 2 θ, ω φθ, ω φθ 2 (gθ + L θ) cos φ L(2ω θ sin α + 2 θ φ + φθ) sin φ = (gθ + L θ) sin φ L(2ω θ sin α + 2 θ φ + φθ) cos φ = (gθ + L θ) 2 + L 2 (2ω θ sin α + 2 θ φ + φθ) 2 = () 4 φθ φ φ φ φ 34

L θ + gθ = φ = ω sin α (D.5) (D.6) (D.5) (D.6) φ = ω φ = θ θ c 215 TAKAHASHI Kunio, 35