転がり軸受 理論・実践ガイドブック 5.定格荷重と寿命
|
|
|
- ゆりか やまのかみしゃ
- 2 years ago
- Views:
Transcription
1 C r C a ISO 8 JIS B Lundberg Palmgren ISO C r b m f c(icosα) 0.7 Z D w.8 b m f c(ilwe cosα) 7 9 Z 4 D we 9 7 α = 90 b m f c Z D w.8 b m f c L we 7 9 Z 4 D we 9 7 C a α 90 b m f c(cosα) 0.7 tanαz D w.8 b m f c(lwe cosα) 7 9 tanαz 4 D we 9 7 b m f c i α ( ) Z D w (mm) L we (mm) D we (mm) D w 5.4 mm D.8 w.647d.4 w 5. b m b m f c D wcosα/dpw D pw (mm) 5.4 f c D wecosα/dpw
2 5.. Lundberg Palmgren Lundberg- Palmgren L-P Hertz ISO τ 0 Z 0 V N S Lundberg ln S τ 0 c N e V Z 0 h (5.) S τ 0 (MPa) Z 0 (mm) N V V az 0 πd a (mm) d (mm) c e h u L 0 6 N = ul τ 0 c u e L e V ln h Z 0 S = C τ 0 c u e L e V h (5.) Z 0 C (5.) S u τ 0 Z 0 V L F u τ 0 Z 0 V F L 90 % 90 % (5.) S = 0.9 F L L = F C τ 0 Z 0 V u 6 Z 0 d a V 5. τ 0 Z 0 V τ 0 Z 0 V τ 0 = Tσ max Z 0 = ζ b T ζ b/a 5. V V = az 0 πd Hertz σ max = Q πab a = μ Q E 0 ρ b = v Q E 0 ρ Q (N) σ max (MPa) a (mm) b (mm) E 0 m = E 0 E E m μ ν ρ (5.) Z 0 τ 0 V ln S T c u e L e πdd w h ζ h μ c v c+h E 0 D w ρ (c+h )/ Q D w D w π (c h+)/ (5.) Q c S = 0.9 ln(/s) (5.) Q D w (c h+)/ L e T c u e πdd w h ζ h μ c v c+h E 0 D w ρ (c+h )/ Lundberg e = 0/9 c = / h = 7/ (5.4) C E 0/ π A ϕ QL = A ϕd w.8 (5.5) ϕ = T T. ζ 0.4 μ.8 v.5 u D w ζ (D w ρ). d T ζ b/a = 0. (5.6) d 0 6 Q c L = (5.5) Q c = A ϕd w.8 (5.7) 7
3 ϕ Σρ d Σρ ρ ρ 4 = D w D w γ = D w cosα D pw γ γ fd w D pw α r f = r i r e D w D w ρ γ = γ f Hertz cosτ cosτ = D w γ γ ρ + fd w D w ρ cosτ = γ γ + f (5.8) + (5.9) (5.8) (5.9) ( + cosτ) D w ρ γ = γ ( cosτ) D w ρ = f (5.0) D w ρ = d d = D pw D w cosα 4 ( + cosτ)( γ) (5.8) (5.9) (5.0) (5.) (5.) 4 ϕ ζ T (5.) (5.0) Ω ϕ Ω Ω = cosτ + cosτ = f ( γ) (5.4) f Ω = ( + cosτ). T T. ζ ζ Lundberg Ω Ω Ω =.Ω 0.4 (5.4) (5.5) (5.6) (5.) ϕ = f f 0.4 ( γ).9 D w D pw 0.4 μ.8 v.5 (5.5) 0. u 5 u u = Z (d m D w cosα) D pw = Z D w cosα D pw u = 0.5Z ( γ) ϕ = f f (5.7) Q c = 0.089A f f 0.4 ( γ).9 ( γ) γ cosα 0.4 ( γ).9 ( γ) γ cosα (5.6) 0. Z (5.7) 0. Z D w A = 98. = D pw D w cosα D pw Q c = 98. f f 0.4 ( γ).9 ( γ) γ cosα 0. Z D w.8 (5.8) = D pw ( γ) Σρ d (5.6) ϕ = T T. ζ ζ 0.4 μ.8 v.5 (+cosτ). ( γ). 4. D w D pw 0. γ 0. u (5.) 8 9
4 6 Q m L (5.5) (5.7) Q m L = Q c L = Q c Q m (5.9) i e 7 C r (.9) C r = ZQ max J r (ε) cosα ε J r (ε) Q C Q max Q mi Q mi = = Z j = Z Q i j = π π 0 Q ψ dψ 0 6 (L = ) (5.9) Q m Q C Q ci = Q mi = π π 0 Q ψ dψ Q ψ = Q max ε ( cosψ).5 (5.0) (5.) ΔS n S e S e = ΔS ΔS ΔS ΔSn ln S e = ln ΔS + ln ΔS + ln (5.) ln S e = C Q (c h+)/ L e πd C = T c u e ζ h μ c v c+h ΔS + + ln E 0 ρ (c+h )/ ΔS n (5.) (5.4) l Δl ΔS n Q (ψ) ln ΔS n = C Q (ψ) (c h+)/ L e Δl Δ (πd) = Δl (5.) ln S e = C L e + ψ0 ψ 0 Q (ψ) 0 dl c = / h = 7/ dl dψ dl = (d/) dψ ln S e = C L e d + ψ0 ψ 0 Q (ψ) 0 dψ (5.5) (5.4) (5.5) C Q 0 L e πd = C L e d + ψ0 ψ 0 Q (ψ) 0 dψ 0 6 Q ce Q ci = Q max π + ψ0 ψ 0 ε ( cosψ) 4.5 dψ / = Q max J (ε) (5.) Q ce = π + ψ0 ψ 0 Q (ψ) 0 dψ 0 J (ε) = π + ψ0 ψ 0 ε ( cosψ) 4.5 dψ / ψ 0. 0
5 Q ψ = Q max ε ( cosψ).5 Q ce = Q max = Q max J (ε) J (ε) = π + ψ0 ψ 0 ε ( cosψ) 5 dψ 0 (5.6) π + ψ0 ψ 0 ε ( cosψ) 5 dψ 0 J J ε ε J J ε J J J J Q max C r C r = ZQ max J r (ε) cosα (5.) (5.6) Q C C r C ri = ZQ ci J r (ε) J (ε) cosα C re = ZQ ce J r (ε) J (ε) cosα ε = 0.5 J r(0.5) = 0.88 J (0.5) = J (0.5) = (5.8) C ri = 9.9 C re = 8. f f f f 0.4 ( γ).9 ( + γ) γ 0. (cosα) 0.7 Z D w ( + γ).9 L = C r P Li = C ri P ( γ) γ 0. (cosα) 0.7 Z D w.8 Le = 5..4 L0 9 = L i L e 0 9 C re P C r = (C ri 0 + C re 0 ) 0. = Cri + C re C ri 0 0. (5.7) (5.8) (5.9) (5.0) (5.)
6 i P C i C i L L L = C P L = C P i i L e = L e + L e + + L e i C P e = C P i e + C P i e + + C e = (ic ) e i e = 0/9 C = i 0.7 C 4 C P i e = i (5.7) (5.8) (5.) (5.) C r = f c (i cosα) 0.7 Z D w.8 f c = 9.9λ +.04 γ + γ.7 f i f e f e f i ic P γ0. ( γ).9 ( + γ) f i f i 0.4 e (5.) (5.) (5.4) f c ISO f i f e λ Lundberg Palmgren ISO 99 b m b m % L 0 = C P L 0 = C 0 P L C (N) C r C a P (N) P r P a L 0h = 06 60n L 0 L 0h h n (min ) (5.5) (5.6) (5.7) 90 % L na = a a a L 0 L a a a 5 (5.8)
7 a 90 % 90 % % 5.6 % L n a 90 L 0 95 L L L L 0.7 L na (5.8) a a ISO a 007 ISO 8 aiso 0 JIS B 58 aiso L nm (5.9) 99 L L L nm = a aiso L 0 (5.9) a a a 5.7 a a a a L mm /s 0 mm /s D pw n D pw (mm) n (min ) NTN a TS TS TS aiso aiso ISO 8:007 (5.40) aiso = f e C C u κ (5.40) P C u NTN e C e C 5.8 D pw (d+d)/ κ κ ν ν 5.4 κ = ν ν (5.4) ν n (D pw) 5. (5.4) (5.4) 6 7
8 5.8 e C D pw 00 mm e C D pw 00 mm aiso 0. κ 0.4 aiso = aiso = κ aiso = aiso = κ κ κ κ κ e C C u P e C C u P 0.8 e C C u P e C C u P n 000 min ν = n D pw n 000 min ν = 4 500n D pw (5.4) (5.4) aiso P P = P r P = P a P =.5P a ν, mm /s D pw, mm n,min aiso = aiso = κ κ e C C u N κ P N P = P r P = P a P =.5P a 0.8 e C C u P e C C u P ν 8 9
9 aiso C u P e C κ aiso aiso 50 κ 4 κ = 4 κ NTN κ = aiso κ = F(L) = e αl β F(L) L α β (5.44) β β = 0/9 β = 9/8 lnln (5.44) = β ln L + ln α (5.45) F(L) F(0.) 90 % L 0 F(L' ) L' lnln 0. = β ln L 0 + ln α (5.46) e CC u/p 5. aiso aiso κ = e CC u/p e CC u/p 5.4 aiso aiso κ = e CC u/p lnln F(L' ) = β ln L ' + ln α (5.47) (5.47) (5.46) (L 0) L ' L 0 = ln F(L' ) ln 0.9 β = a % β = / ISO (5.48) 破損確率 % σ 範囲 理論寿命 β=9/8 または 0/9 β=/ 寿命比 ( 破損確率 50 % のときを.0) 5.5 aiso aiso %
10 a = L ' L 0 = ln F(L' ) ln (5.48) 90.0 荷重 n n (min ) t n t + t + t + + t n = n m n m = t n + t n + + t n n n n t n t n n t n 回転総数 n m 破損確率 (%) 過去の ISO および JIS による線図 最新の ISO および JIS による線図 信頼度 (%) 5.9 j n = ϕ j L m L j L j (5.49) L j j 0 6 L m 0 6 ϕ j j 信頼度係数 a ϕ j = n j t j n m ϕ j = t j (5.50) 5.8 a 5.. Miner Miner Rule Miner Miner ϕ L m L + ϕ L m L + ϕ L m L + + ϕ n L m L n = L m Miner P L 0 6 L /L L L/L = 00 % L m = ϕ L + ϕ L + ϕ L + + ϕ n L n (5.5) n L L n
11 5..4 S(L) = F(L) = e αl β F(L) S(L) L α β (5.5) S C L D L L 5. (5.5) ln ln ln S C = α D L D β S C = α L β S C = α L β (5.55) 残存確率 S C 装置 軸受 軸受 (5.5) ln S(L) = αlβ (5.5) 5.0 残存確率 装置 軸受 軸受 (5.54) (5.55) α ln ln ln S D = S = S = L β ln L D S C L β ln L S C L β ln L S C L D L L 寿命 5. (5.56) 寿命 S D = S S L S D S S 5. (5.5) 残存確率 5.0 装置 ln S D = ln S + ln S (5.56) (5.57) (5.57) ln ln ln S D = α D L β S = α L β (5.54) S = α L β S D S S 軸受 軸受 L D β = L β + L β n L D e = L e + L e + + L n e (5.58) S(L) D S(L) S(L) S D S S L 寿命 L D L L L n e 0/9 9/
12 5..5 ISO JIS ε = 0.5 (Δ r) 0 Δ r 0 (.) ε = ( cosψ ) = Δ r δ r ' (4.7) (4.8) (5.6) 5. δ r = cosα Q D w (4.7) δ ψ ψ δmax ψ δmax δ r ' di + Dw de d e (mm) d i (mm) d i + D w (mm) D w (mm) δ r ' (mm) δ ψ ψ (mm) δ max (mm) ψ ( ) ψ ( ) Δ r (mm) δ r = cosα (.6) Q max = F r ZJ r (ε)cosα Q 0.9 L we 0.8 ε ε J r = = F r Δ r D w Z cos 5 α D w Z cos 5 α = 0.448K F r r = Δ r f (ε) F r Δ r (4.8) (.6) (5.6) 5. ψ δ ψ = ψ / ε ε J r F = r Δ r L 0.8 we Z 0.9 cos.9 α = 0.5K F r 0.9 r = Δ r f (ε) (5.64) = δ r ' cosψ Δ r (5.59) ψ = 0 δ max = δ r ' Δ r (5.60) ψ 0 (5.6) (5.64) K r ε J r ε. J r δ ψ = δ r ' cosψ Δ r = 0 cosψ = Δ r δ r ' (5.6) 46 47
13 5.0 K r ( 0 5 ) K r ( 0 6 )
14 ISO L = C P p P (N) Lundberg-Palmgren L = Q c Q m p L 0 6 Q c (N) Q m = (N) (5.65) F r ε ε = 0.5 (.6) L ε L = J r(ε) J (0.5) J r(0.5) J (ε) J r. J 5. p (5.70) J r J (5.7) (5.7) f (ε) 5. ε ε J r F r = Δ r D w Z cos 5 α = 0.448K F r r = Δ r f (ε) ε ε J r F r = Δ r l 0.8 eff Z 0.9 cos.9 α (5.7) F Q r max(ε) = ZJ r(ε)cosα (5.66) = 0.5K F r 0.9 r = Δ r f (ε) (5.7) Q max(0.5) = F r ZJ r(0.5)cosα (5.67) Δ r J ε ε = 0.5 J (ε) = Q m(ε) Q J m(0.5) (0.5) = Q max(ε) Q max(0.5) (5.66) (5.67) (5.68) F r = Q m(ε) Z J r(ε) J (ε) cosα F r = Q m(0.5) Z J r(0.5) J (0.5) cosα Q m(0.5) Q m(ε) = J r(ε) J (0.5) J r(0.5) J (ε) (5.65) (5.69) L ε L = Q p c Q m(ε) Q p = c Q m(0.5) Q m(0.5) Q m(ε) p (5.68) (5.69) 5. J ε ε f (ε) L e L f (ε) L e L
15 5..6 Lundberg- Palmgren 5.4 ψ β β 0 4 L OSC = ΩL ROT β β 0 ψ Ⅱ β0 β 0 Ⅰ Ⅱ 5.4 (5.7) 5.4 Ω Ω Ω = A S Y S + A O Y O + Y R X e + AX i + BX r (5.74) L OSC L ROT A A S A O X i X e X r Y O Y S Y R B i e r O S R Ω D w/d pw Z α ε A i /A e A i /A e D w/d pw D w/d pw α = 0 ε = 0.5 Ω Z D w/d pw Ω 5.4 揺動寿命係数 Ω 00 0 深溝 6 深溝 5 円ころ 揺動半角 β ( ) 深溝 0 円ころ 0 円ころ Ω 揺動寿命係数 Ω 玉軸受 ころ軸受 揺動半角 β ( ) β
16 Ω Z = 6 Z = 0 0 % 5.6 (60 /Z) 5..7 ISO L i = Q ci Q mi p Le = L = (L i e + L e e ) ¹ e Q ce Q me p β c = 60 D pw Z D pw D w cosα = 60 Z D w cosα D pw D pw (mm) D w (mm) Z α ( ) (5.75) Q ci Q ce (N) Q mi Q me (N) L i L e L j D w/d pw D w D d = 0.6 D pw D + d = 0.5 D d D + d d (mm) D (mm) A j δ a+θr icosψ j δ rcosψ j 5 4 α ij V A j α V α ej S 5.7 j f i f e S S = ( f i + f e )D w δ a δ r ψ j j α α ij α ej R i δ ij δ ej 54 55
17 5.7 tanα ij = V V tanα ej = A j V A j V V + V {( f i 0.5) D w + δ ij } = 0 (A j V ) + (A j V ) {( f e 0.5) D w + δ ej } = 0 Q ij = K i δ ij.5 Q ej = K e δ ej.5 K i K e 5.8 λ ej M Gj D w α ej (5.76) (5.77) (5.78) (5.79) Q ej Q c 5. (5.8) Q c = 98.λ f f 0.4 ( γ).9 ( γ) γ cosα 0. Z D w.8 f f i f e λ λ = 0.95 α ( ) Z D w (mm) D w 5.4 mm D w.8.647d w.4 γ = D w cosα D pw D pw (mm) (5.8) JIS λ (5.8) Q i (N) Q e (N) M G 6.. F C (N) λ λ i = 0 λ e = j j Q ij F Cj α ij λ ij M Gj D w M Gj 5.8 Q c = 550.8λv ( γ)9 7 ( γ) 4 γ cosα λν λν = 0.8 D we (mm) L we (mm) λν 9 Z 4 D we 9 7 L we 7 9 (5.84) Q ij sinα ij Q ej sinα ej M Gj D w (λ ij sinα ij λ ej sinα ej ) = 0 (5.80) Q ij cosα ij Q ej cosα ej + M Gj D w (λ ij cosα ij λ ej cosα ej ) + F Cj = 0 (5.8) (5.76) (5.8) j = Z Z Q ij Q ej Q mi = Z Z wi Q ij j = wi Qme = Z Z we we Q ej j = (5.8) wi we wi we 0/ 0/ Q ij Q ej 56 57
18 4 Q c Q m L c L ci = Q ci Q mi p Lce = L c = (L ci e + L ce e ) ¹ e Q ce Q me p 0/ e 0/9 9/8 p L 0 ISO L 0 L = L c L 0 L 寿命 h 遠心力を考慮する 606 F r = 500 N ラジアル内部すきま 0 遠心力を考慮しない 内輪回転速度 min F r = 000 N n = 500 min 0.0 mm 606 C r = 600 N L 0 = K r = (5.7) 0 6 = 996 (h) Δ r f (ε) = 0.448K r F r 0.0 = = L ε/l 0 = L ε L ε = = 500 (h) 5- NU08 F r = 0 kn β = / min NU08 d = 40 mm D = 80 mm Z = C r = 64.5 kn L ROT = ( ) = (h) 5.6 Ω =.9 L OSC = = (h) β C β c 60 =. ( ) 0.5(80 40)
19 X Y X Y X Y X Y X Y JIS α f 0 F r /C 0r e F a F e F / a r F e / r F a F e F / a r F e / r P r P r = XF r + YF a (5.85) F r (N) F a (N) X Y JIS X Y X Y X Y F a / F r e F a / F r e α α' X Y 0 X 0.4ξ η Y 0.4 η tanα' 0.65 ξ tanα' ξ η 0.65 η tanα' cotα cotα e ξ tan α'.5tanα 5.6 ξ η α 0.67cotα α = 90 α 90 P a sinα.5 cosα α = 5 α 5 α 5.5 sinα.5 sinα.75 f = cosα cosα = 0.974cosα 0 F a cosα C 0r sinα P a = XF r + YF a F r (N) F a (N) X Y (5.86) 60 6
20 JIS X Y 5.8 P a =.F r + F a (5.87) 5.. X Y X Y JIS 5.0 F r /F a 0.55 X = tan50. α = 50 F a /F r e F a /F r.5tan50 F r /F a / X Y JIS F a / F r e X Y α 0 tanα sinα.5tanα tanα sinα F r P r A B C cota F acota Y P r 5.0 AC C i /C e = 0 C i /C e = 0.75 C i C e ABC Lundberg-Palmgren ABC X Y F a / F r e X.5tanα sinα.5tanα sinα tanα tanα Y e.5tanα.5tanα (5.88) (5.89) ε 5.0 F r P r = C i + C e F a cotα' P r = pe J r(0.5) J (0.5) C pe i + C e J (ε) J r(ε) J (ε) J (0.5) pe + pe + C e + C i C pe e + C i pe J r(0.5) J (0.5) J (ε) J (0.5) pe J (ε) J r(ε) ¹ pe pe ¹ pe J a(ε) J r(0.5) (5.88) (5.89) 6 6
21 (Y ) F r = ε = J ( ) = J ( ) = J a( ) = (5.89) F a cotα' P r = J (0.5) C pe i + ¹ pe J (0.5) C e C pe i + C e J (0.5) C pe i + ¹ pe J (0.5) C e Y = C pe i + C e J (0.5) J r(0.5) J r(0.5) J (0.5) cotα' (5.90) (5.9) P r = Y F a Y C i C e 0 (5.9) J (0.5)/J (0.5) J (0.5) J (0.5) Y ξ tanα' = e F a F r e OB AB P r = F r (5.86) X = Y = 0 F a F r > e OB C (a,0) BC a F a cotα' P r P r P r = a ξ a + a ξ a F r + cotα' a F r P r = F a (5.86) Y = J (0.5) J (0.5) J r(0.5) J (0.5) cotα' = J r(0.5) cotα' = 0.4 cotα' J (0.5) J (0.5) (5.9) X = a ξ a Y = cotα' a (5.96) η Y = Y η η = sinα.5 (5.9) (5.94) Y Y F r = 0 F a cotα' P r (5.86) P r = Y F a cotα' (5.9) (5.9) Y = 0.4cotα' η (5.96) (5.97) a = η 0.4 (5.98) (5.96) X = 0.4ξ η (5.97) (5.98) (5.99) (5.97) (5.99) 5.5 X Y F a cotα' P r = cotα' Y B (ξ,) OB F r P r = ξ F a cotα' P r F a F r = ξ tanα' (5.95) 64 65
22 BrgⅠ F rⅠ Fa BrgⅡ F rⅡ 0.5F r Y 0.5F r + F a Y F a = 0.5F r + F a Y - F ac (ε = 0.5) F a + F ac (ε = 0.5) F ac (ε = 0.5) F ac (ε 0.5) F ac (ε = 0.5) F ac (ε 0.5) F ac (ε 0.5) = F a + F ac (ε 0.5) F acⅠ (ε > 0.5) 軸受 Ⅰ F rⅠ F a 軸受 Ⅱ F rⅡ F acⅡ (ε < 0.5) 5. (5.0) BrgⅡ BrgⅠ Fa F rⅡ F rⅠ BrgⅠ BrgⅡ Fa F rⅠ F rⅡ BrgⅡ BrgⅠ Fa F rⅡ F rⅠ 0.5F r Y 0.5F r Y 0.5F r Y 0.5F r + F a Y 0.5F r Y - F a = 0.5F r F a Y + F a - 0.5F F a = r + F a Y F a = 0.5F r + F a Y 0.5F r F a Y - F ac (ε 0.5) F ac (ε = 0.5) (5.0) F ac (ε 0.5) = F a + F ac (ε = 0.5) P r P r = X F r + Y F ac (ε 0.5) (5.00) (5.0) (5.0) (5.0) X Y F r F r F a F ac F ac 0 ε = 0.5 F ac (ε = 0.5) = F r Y (5.00) 5. F acⅠ 66 軸受 Ⅰ F rⅠ F a 軸受 Ⅱ F rⅡ 5. F acⅡ P r = X F r + Y F a + F r Y P r P r = X F r + Y F ac (ε 0.5) (5.00) (5.0) P r = X F r + Y F r Y = X F r + F r X X (5.05) P r = F r X (5.04) (5.05) (5.06) 右に移動
23 F ac (ε = 0.5) F a + F ac (ε = 0.5) F ac (ε 0.5) F ac (ε = 0.5) F ac (ε 0.5) = F ac (ε = 0.5) F a (5.07) (5.08) 5.7 NTN F a /F r. F a F r 0.8 (e = 0.80) X = 0.6 Y =.4 P r P r = X F r + Y F ac (ε 0.5) P r = X F r + F r = F r (5.09) P r P r = X F r + Y F r Y F a (5.0) ) ) NTN F r tanα F a 0.6 ε ε = 0 (5.) NTN ε F r tan0 F a ( ε >.7 ε = 0) F a F r. (5.) P r =0.6F r +.4F a 5.0 ε ε F r (N) F a (N) α ( ) J r J a (5.) ε ε F rtanα F a J r (ε) Ja (ε) JIS C r C a C a = C r Y Y Y (5.4) 68 69
24 C a = C r' Y' = C r i 0.7 Y' (5.5) C r ' (N) C a (N) Y' Y (5.4) (5.5) C r ' = YC a 0.7 Y Y (F r = 0) L = C a F a P r ' = 0.6F r +.4F a C r ' =0.76C a 0.7 F a /F r. L = C r ' P r ' (5.6) F r = 500 N F a = 500 N n = 000 min 606 C r = 600 N C 0r = 00 N f 0 =.6 f 0 F a = = 0.60 C 0r e = 0.5 F a 500 = = = ey =.8 F r 500 P r = = 755 (N) L 0 = = (h) F r =.8 kn F a = kn n = 000 min F r = = 8.4 (kn) F r = = 4.4 (kn) (+) F ac = =.65 (kn) F ac = =.75 (kn) 軸受 軸受 F r F a = kn F r =.8 kn F r F a = (kn) F ac F a + F ac 70 7
25 08 e = X = 0.4 Y = 0.4.5/0.7.6 P r = ( +.75) = 8.76 (kn) P r = 4.4 (kn) 06 C r = 60.5 (kn) 08 C r = 88 (kn) L 0 = L 0 = = 00 (h) = (h) QJ0 F r = kn F a = 5 kn n = 000 min QJ0 C a = 57.5 (kn) F a = 5 =.5 >. F r P r ' = = 7.46 (kn) C r ' = = 7 (kn) L = = (h) ISO 76 JIS B f 0 5. f m D w/ m.04 m.06 m.08 m C 0r f 0 i ZD w cosα f 0 i ZL we D wecosα C 0a f 0 ZD w sinα f 0 ZL we D wesinα σ max 4 00 MPa MPa MPa f 0 Z D w (mm) D we (mm) L we (mm) i α ( ) 7 7
26 5. f σ max σ max σ max E(κ) κ γ γ m π 4 ( + γ) E(κ) κ γ γ m f 0 σ max 5. E(κ) γ = D wcosα/d pw D wecosα/d pw m D w/ D pw (mm) κ a/b a b 44 0 σ max σ max ( γ) ( γ) 5. f 0 γ
27 5.4. Hertz (5.0) (5.) Q max = χ D w E γ γ f σ max C 0r (.9) C 0r = 0.ZQ max cosα (5.) (5.) σ max = Q max πab Q max = πab σ max (5.7) σ max Q max a b (5.) (5.) C 0r =.07 σ max E χ γ γ f ZD w cosα a = b = χ E π E πχ Q ρ Q ρ v E v E χ χ = a/b E Σρ ν E (5.7) (5.9) Q max = π χ v E E ρ σmax E = MPa ν = 0. (5.8) (5.9) i f 0 C 0r = f 0 izd w cosα f 0 =.07 σ max E χ γ γ f σ max = Q max πl we b Q max = πl we b σ max σ max Q max L we b (5.4) (5.5) (5.6) Q max = χ E ρ σmax Σρ (5.0) b = 8Q πl we ρ v E (5.7) ρ = ρ + ρ + ρ + ρ = D w γ γ f D w γ = D wcosα/d pw f f i = r i /D w f e = r e /D w r i r e (5.) Σρ ν E (5.6) (5.7) Q max = π v E L we ρ σ max 76 77
28 E = MPa ν = 0. Q max = L we ρ σ max Σρ ρ = D we γ D we γ = D wecosα/d pw (5.8) (5.9) (5.8) (5.9) 5.4. ISO 76 JIS B 59 P 0r = X 0 F r + Y 0 F a P 0r = F r (5.) (5.4) Q max = ( γ)l we D we σ max (5.0) X 0 Y (5.) (5.0) C 0r = 44 σ max ( γ)zlwe D we cosα 5.4 X 0 Y 0 X 0 Y 0 X 0 Y 0 ISO i f 0 C 0r = f 0 izd we cosα f 0 = 44 σ max (5.) ( γ) (5.) α = α = α = α = α = α = α = α = cotα 0.44cotα cotα 0.44cotα 78 79
29 α 90 P 0a = X 0 F r + Y 0 F a α = 90 P 0a = F a X 0 Y (5.5) (5.6) F r (.9) Q max = 5F r Zcosα (5.40) P 0r (5.9) (5.40) 5.5 X 0 Y 0 X 0 Y X 0 Y 0 X 0 Y 0 ISO Stellrecht α F a F r 5. α' Z F a /Z Q a = F a Zsinα' (5.7) 5F r /Z Q r =.tanα.7 F r/fa F r Z =.5F r cosα' Zcosα' F r/fa 0.44cotα 0.44cotα F r/fa 0.67cotα α' F a Z F a Z Q a Q r Q a 5F r Z Q r 5. (5.8) 5P 0r Zcosα =.5 Zcosα' P 0r = F r + F a Zsinα' cosα F r + cosα F a cosα' 5sinα' Y 0 η 0 = 0.sinα (5.4) P 0r = cosα cosα F r + cosα' 5sinα' ( 0.sinα) F a (5.) (5.4) X 0 Y 0 cosα X 0 = cosα' cosα Y 0 = 5sinα' ( 0.sinα) (5.4) (5.4) (5.4) (5.44) X 0 Y 0 ISO Palmgren C 0r (F a = C 0r) X 0 Y 0 α = 5 α' = 6.6 (5.4) (5.44) X 0 Y 0 5. X 0 = Y 0 = cos5 = cos6.6 cos5 = sin6.6 ( 0.sin5 ) X 0 Y 0 Q max = Q r + Q a =.5 F r + F a Zcosα' Zsinα' (5.9) 80 8
30 C 0r / (F a = C 0r /) X 0 Y 0 (5.6) (5.7) α' X 0 Y / / X 0 Y α α' X 0 Y α (5.4) P 0r = 0.5F r + 0.cotα F a (5.45) ISO Y 0 0 % X 0 Y 0 (5.4) 5.4 X 0 = 0.5 Y 0 = 0.cotα EHL EHL EHL 5.5. EHL h 0 = 0.66 (η 0 U) R p h 0 R U U R 5.5 h h η 0 η 0 = νρ 0 9 (N s/mm ) U (mm/s) U = U + U U U R (mm) R = (R R ) p (MPa) h 0 (mm) ω (rad/s) h (mm) ν (mm /s) ρ (kg/mm 0 6 ) ω R z U y (5.46) 8 8
31 EHL Hertz 5.6 Dowson-Higginson Hertz 油膜圧力分布 油膜厚さ分布 5.7 ω 0 Z 圧力スパイク h 0 ヘルツ面圧分布 5.6 y 5.7 Martin Grubin Dowson-Higginson Dowson-Higginson H.S.Cheng Dowson Archard-Cowking Hamrock-Dowson h 0 R = 4.9 U W h 0 R =.95 (GU ) 0.77 W 0.09 h 0 R =.6 G U W 0. h 0 R =.65 G U W 0. h 0 R =.65 h 0 R =.8 U W h 0 η 0 U R R =.04ϕ0.74 (GU ) 0.74 P max E 0.74 W h 0 R =.6 G U ( e W K ) 0. h 0 R G U W K K = a/b a b 84 85
32 Ertel-Grubin h 0 = 0.8 = K A B C D D we ( γ).09 η0 α ncosθ γ γ 0.77 E' L we Q max η 0 : 絶対粘度,α: 圧力粘度係数動粘度は運転時の粘度を使用 K K = 0.8E' 0.09 E' = 8.5 (GPa) A A = D we ( γ).09 γ cosθ γ θ 0.77 L 0.09 D we (mm) γ = D we cosθ D pw (mm) D pw L we (mm) B B = (η 0 α) η 0 (N s/mm ) α C C = n 0.77 n (min ) D D = Q max 0.09 Q max (kn) (η0α) パラフィン系鉱油 ナフテン系鉱油 合成油 ν 0 ( 動粘度 SUS) (η0α)
33 Archard-Cowking h 0 = 0.0ϕ 0.74 = K A B C D D w ( γ).48 η0 α ncosθ γ γ K K = 0.0E' E' = 8.5 (GPa) A A = ϕ 0.74 D w ( γ).48 γ cosθ γ ϕ θ D w (mm) γ = D w cosθ D pw B B = (η 0 α) C C = n 0.74 n (min ) D D = Q max 0.74 E' Q max Q max Q max (kn) Q max = 5 F r Z F Q a max = Zsinθ A 0 NUE 0 NUE () 0 - NUE NUE %
34 5.9 A () % Λ Λ = h 0 R + R h 0 R R (rms) (rms) (R a) R rms =.R a 油膜形成率 % 潤滑係数 F( または α) 表面痛みの起こる領域 油膜パラメータ Λ ASME 推奨 AB の平均曲線 ひどい滑り運動を伴う軸受で表面痛みの起こる可能性のある領域寿命増加領域 A B Harris 5.8 Λ 4 00 % L 0 Λ = % 5.9 Tallian Skurka ASME ISO a ISO 0.5 A 曲線 :J.C.Skurka B 曲線 :T.E.Tallian 油膜パラメータ Λ
35 d m n = D pw n ϕ T h 0 ' = ϕ h 0 h 0 ' ϕ ϕ = ϕ T ( L 0.6 ) (0.645/ϕ (0.5/( L)) T) ϕt 鉱油 5 mm /s ϕ T = G Q L L 0.57 e 0.875S G = αe' α (mm /N) α = { log (νρ)} 0 ν ρ (0.88) E' Q L = Q max L we E' R y R y = D we ( γ) γ = D we cosθ D pw d m n 0 4 Λ' = ϕ T Λ ϕ T Λ Wolveridge Hamrock-Dowson 5. (x 0 x B) 5.0 b x 0 x B 5. Goksem b 0. Q max (kn) L we (mm) D we (mm) S = (u u ) (u + u ) u = D pw ( γ)(ω i ω c) u = D we ω r u (mm/s) u (mm/s) ω i (rad/s) ω c (rad/s) ω r (rad/s) D pw (mm) u = u S = 0 L = βη T (u + u ) T β T = ln 4k b T T v v η = ν ρ k b = 0.85 (W/mK) ρ (g/cm ) ν (mm /s) ν (98.9 ) (mm /s) T (K) T (98.9 = 7. K) 9 9
36 5-6 N09E Grubin F r = N n = min ν = 00 SUS Z = 5 R i = 0.5 μm R r = 0.5 μm h 0 = 0.8 D we ( γ).09 η0 α ncosθ γ γ = K A B C D K = = 0.78 A = B = C = = 558 D = = Q max = =.65 (kn) E' L Q max 0.09 h 0 = = (mm) Λ = (. 0.5) + (. 0.5) =. 700C Archard-Cowking F a = N n = min ν = 0 mm /s Z = R i = 0. μm R r = 0. μm h 0 = 0.0ϕ 0.74 = K A B C D D w ( γ).48 η 0 α ncosθ γ γ 0.74 E' Q max K = = 0.44 A = B = mm /sec = 60 SUS NTN C = = 77 D = Q max = sin5 =. (kn) h 0 = = (mm) Λ = (. 0.) + (. 0.) = N09E min 80 SUS N09E d = 45 mm D = 85 mm B = C = 809 h 0 = = (mm) Λ = d m n d m n = 0.85 (. 0.5) + (. 0.5) = = SUS = 5.8 mm /s = cp.9 cp 5.0 ϕ T = 0.75 h 0 ' = = 0.64 μm Λ' = =
37 N09E mm /s mm /s 5.8 mm /s 80 SUS 5-8 h 0 = (mm) Λ =.6 G = αe α(mm /N) α = { log( )} 0 = ν (mm /s) ρ (0.88) E' E' = (N/mm ) G = = 89 Q L = Q max L we E' R y D we = mml we = 9 mm D pw = 65.5 mm L = ( 6.66) = η = = 0.09 (Nsec/m ) β = ln = 75 (K) k b = 0.85 W/mK ρ0.88 g/cm ν 5.8 mm /s ν (98.9 ) 7.5 mm /s T. K T (98.9 = 7. K) ϕ T = = 0.75 ϕ = 0.75 ( ) (0.645/0.75)(0.5/( )) = % h 0 ' = = 0.50 μm Λ' = =. R y = 65.5 = (mm) Q max = 4.6F r Z Q L = = = S ω i = πn π = = 047 (rad/s) ω c = 65.5 ω r = u = 65.5 u = = 45.6 (rad/s) = 65 (N) 047 = 09 (rad/s) ( ) = mm/s = 6.66 (m/s) = mm/s = 6.66 (m/s) u = u S =
38 5.6 Palmgren F(L) = e αlβ F(L) L α β (5.47) 5.6. L 0 L 50 (5.47) ln F(L) = αl β 残存確率 F(t)% 寿命 5. t lng = k logg k =.0 log F(L) = α L β (5.48) α = α k (5.48) loglog F(L) (5.49) Y = loglog X = logl A = logα F(L) = β logl + logα (5.49) n = 8 5. L 0 L Y = βx-a Y X 5. β 98 99
39 No 5.0 n 残存確率 F(t)% 寿命 (h) 00 t (h) (h) DATA DATA No No
40 0 5.4 g n j (λ) = n C j [ jλ j ( λ) n j ] n j (5.50) % 95 % λ 5 λ % G n j (λ 5 ) = 0.05 G n j (λ 95 ) = G n j λ 0.5 λ λ m 5.0 G n j (λ m ) = λm 0 g n j (λ) dλ = 0.5 λ m λ m = ¹ n + j n ( ¹ n) (5.5) メディアンランク 5 % 90 % 90 % 信頼幅 % 累積分布関数 : G n j (λ) による繰返し計算 順位率 順位 番の分布 n = 0 の場合 5.5 n = 0 個数 順位 番の中央値 順位 番の中央値 順位 番の分布 順位 メディアンランク 順位率 累積破損確率 (%) 50 0 n = λ m 95 % ランク 5 % ランク 寿命線図 寿命
41 5.6. = n n + + (5.5) (5.5) (5.5) 5.7 n n = (h) (h) (8+ )/(+5) (8+ )/(+5) (8+ )/(+5) (8+5.50)/(+) 残存確率 F(t)% L0 L t = = 5 = 5.50 = 04 05
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
untitled
9118 154 B-1 B-3 B- 5cm 3cm 5cm 3m18m5.4m.5m.66m1.3m 1.13m 1.134m 1.35m.665m 5 , 4 13 7 56 M 1586.1.18 7.77.9 599.5.8 7 1596.9.5 7.57.75 684.11.9 8.5 165..3 7.9 87.8.11 6.57. 166.6.16 7.57.6 856 6.6.5
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4
35-8585 7 8 1 I I 1 1.1 6kg 1m P σ σ P 1 l l λ λ l 1.m 1 6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m
K E N Z U 2012 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.2................................... 4 1.2.1..................................... 4 1.2.2.................................... 5................................
[Ver. 0.2] 1 2 3 4 5 6 7 1 1.1 1.2 1.3 1.4 1.5 1 1.1 1 1.2 1. (elasticity) 2. (plasticity) 3. (strength) 4. 5. (toughness) 6. 1 1.2 1. (elasticity) } 1 1.2 2. (plasticity), 1 1.2 3. (strength) a < b F
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
all.dvi
38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
s s U s L e A = P A l l + dl dε = dl l l
P (ε) A o B s= P A s B o Y l o s Y l e = l l 0.% o 0. s e s B 1 s (e) s Y s s U s L e A = P A l l + dl dε = dl l l ε = dε = l dl o + l lo l = log l o + l =log(1+ e) l o Β F Α E YA C Ο D ε YF B YA A YA
液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
N/m f x x L dl U 1 du = T ds pdv + fdl (2.1)
23 2 2.1 10 5 6 N/m 2 2.1.1 f x x L dl U 1 du = T ds pdv + fdl (2.1) 24 2 dv = 0 dl ( ) U f = T L p,t ( ) S L p,t (2.2) 2 ( ) ( ) S f = L T p,t p,l (2.3) ( ) U f = L p,t + T ( ) f T p,l (2.4) 1 f e ( U/
TOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................
(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
Untitled
II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
![II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re](/thumbs/94/118770263.jpg "II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re")
II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
4 Mindlin -Reissner 4 δ T T T εσdω= δ ubdω+ δ utd Γ Ω Ω Γ T εσ (1.1) ε σ u b t 3 σ ε. u T T T = = = { σx σ y σ z τxy τ yz τzx} { εx εy εz γ xy γ yz γ
Mindlin -Rissnr δ εσd δ ubd+ δ utd Γ Γ εσ (.) ε σ u b t σ ε. u { σ σ σ z τ τ z τz} { ε ε εz γ γ z γ z} { u u uz} { b b bz} b t { t t tz}. ε u u u u z u u u z u u z ε + + + (.) z z z (.) u u NU (.) N U
I
I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
dvipsj.8449.dvi
9 1 9 9.1 9 2 (1) 9.1 9.2 σ a = σ Y FS σ a : σ Y : σ b = M I c = M W FS : M : I : c : = σ b
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =
72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
( ) ± = 2018
30 ( 3 ) ( ) 2018 ( ) ± = 2018 (PDF ), PDF PDF. PDF, ( ), ( ),,,,., PDF,,. , 7., 14 (SSH).,,,.,,,.,., 1.. 2.,,. 3.,,. 4...,, 14 16, 17 21, 22 26, 27( ), 28 32 SSH,,,, ( 7 9 ), ( 14 16 SSH ), ( 17 21, 22
数学の基礎訓練I
I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h)
1 16 10 5 1 2 2.1 a a a 1 1 1 2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h) 4 2 3 4 2 5 2.4 x y (x,y) l a x = l cot h cos a, (3) y = l cot h sin a (4) h a
e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,
K E N Z U 01 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.................................... 4 1..1..................................... 4 1...................................... 5................................
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [
![H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [](/thumbs/99/141438442.jpg "H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [")
3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
keisoku01.dvi
2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.
1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π
![1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π](/thumbs/101/149329485.jpg "1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π")
. 4cm 6 cm 4cm cm 8 cm λ()=a [kg/m] A 4cm A 4cm cm h h Y a G.38h a b () y = h.38h G b h X () S() = π() a,b, h,π V = ρ M = ρv G = M h S() 3 d a,b, h 4 G = 5 h a b a b = 6 ω() s v m θ() m v () θ() ω() dθ()
: , 2.0, 3.0, 2.0, (%) ( 2.
2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................
8 300 mm 2.50 m/s L/s ( ) 1.13 kg/m MPa 240 C 5.00mm 120 kpa ( ) kg/s c p = 1.02kJ/kgK, R = 287J/kgK kPa, 17.0 C 118 C 870m 3 R = 287J
26 1 22 10 1 2 3 4 5 6 30.0 cm 1.59 kg 110kPa, 42.1 C, 18.0m/s 107kPa c p =1.02kJ/kgK 278J/kgK 30.0 C, 250kPa (c p = 1.02kJ/kgK, R = 287J/kgK) 18.0 C m/s 16.9 C 320kPa 270 m/s C c p = 1.02kJ/kgK, R = 292J/kgK
/02/18
3 09/0/8 i III,,,, III,?,,,,,,,,,,,,,,,,,,,,?,?,,,,,,,,,,,,,,!!!,? 3,,,, ii,,,!,,,, OK! :!,,,, :!,,,,,, 3:!,, 4:!,,,, 5:!,,! 7:!,,,,, 8:!,! 9:!,,,,,,,,, ( ),, :, ( ), ( ), 6:!,,, :... : 3 ( )... iii,,
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
NETES No.CG V
1 2006 6 NETES No.CG-050001-V 2007 5 2 1 2 1 19 5 1 2 19 8 2 i 1 1 1.1 1 1.2 2 1.3 2 2 3 2.1 3 2.2 8 3 9 3.1 9 3.2 10 3.3 13 3.3.1 13 3.3.2 14 3.3.3 14 3.3.4 16 3.3.5 17 3.3.6 18 3.3.7 21 3.3.8 22 3.4
N cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
201711grade1ouyou.pdf
2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
Microsoft Word - 11問題表紙（選択）.docx
A B A.70g/cm 3 B.74g/cm 3 B C 70at% %A C B at% 80at% %B 350 C γ δ y=00 x-y ρ l S ρ C p k C p ρ C p T ρ l t l S S ξ S t = ( k T ) ξ ( ) S = ( k T) ( ) t y ξ S ξ / t S v T T / t = v T / y 00 x v S dy dx
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k
63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5
ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
untitled
( 9:: 3:6: (k 3 45 k F m tan 45 k 45 k F m tan S S F m tan( 6.8k tan k F m ( + k tan 373 S S + Σ Σ 3 + Σ os( sin( + Σ sin( os( + sin( os( p z ( γ z + K pzdz γ + K γ K + γ + 9 ( 9 (+ sin( sin { 9 ( } 4
さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a
... A a a a 3 a n {a n } a a n n 3 n n n 0 a n = n n n O 3 4 5 6 n {a n } n a n α {a n } α {a n } α α {a n } a n n a n α a n = α n n 0 n = 0 3 4. ()..0.00 + (0.) n () 0. 0.0 0.00 ( 0.) n 0 0 c c c c c
支持力計算法.PDF
. (a) P P P P P P () P P P P (0) P P Hotω H P P δ ω H δ P P (a) ( ) () H P P n0(k P 4.7) (a)0 0 H n(k P 4.76) P P n0(k P 5.08) n0(k P.4) () 0 0 (0 ) n(k P 7.56) H P P n0(k P.7) n(k P.7) H P P n(k P 5.4)
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j
![D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j](/thumbs/103/158001152.jpg "D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j")
6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..
The Physics of Atmospheres CAPTER :
The Physics of Atmospheres CAPTER 4 1 4 2 41 : 2 42 14 43 17 44 25 45 27 46 3 47 31 48 32 49 34 41 35 411 36 maintex 23/11/28 The Physics of Atmospheres CAPTER 4 2 4 41 : 2 1 σ 2 (21) (22) k I = I exp(
5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6 cos π 6.7 MP 4 P P N i i i i N i j F j ii N i i ii F j i i N ii li i F j i ij li i i i
i j ij i j ii,, i j ij ij ij (, P P P P θ N θ P P cosθ N F N P cosθ F Psinθ P P F P P θ N P cos θ cos θ cosθ F P sinθ cosθ sinθ cosθ sinθ 5 c P 5 kn n t π (.5 P 7 MP π (.5 n t n cos π. MP 6 4 t sin π 6
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i
http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
c 2009 i
I 2009 c 2009 i 0 1 0.0................................... 1 0.1.............................. 3 0.2.............................. 5 1 7 1.1................................. 7 1.2..............................
( ) 2002 1 1 1 1.1....................................... 1 1.1.1................................. 1 1.1.2................................. 1 1.1.3................... 3 1.1.4......................................
8 i, III,,,, III,, :!,,,, :!,,,,, 4:!,,,,,,!,,,, OK! 5:!,,,,,,,,,, OK 6:!, 0, 3:!,,,,! 7:!,,,,,, ii,,,,,, ( ),, :, ( ), ( ), :... : 3 ( )...,, () : ( )..., :,,, ( ), (,,, ),, (ϵ δ ), ( ), (ˆ ˆ;),,,,,,!,,,,.,,
4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t
![4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t](/thumbs/70/62705042.jpg "4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t")
1 1.1 sin 2π [rad] 3 ft 3 sin 2t π 4 3.1 2 1.1: sin θ 2.2 sin θ ft t t [sec] t sin 2t π 4 [rad] sin 3.1 3 sin θ θ t θ 2t π 4 3.2 3.1 3.4 3.4: 2.2: sin θ θ θ [rad] 2.3 0 [rad] 4 sin θ sin 2t π 4 sin 1 1
untitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
Z: Q: R: C: 3. Green Cauchy
7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
プランマブロック
CAT. NO. B00- SD 300TSTAC 46 4 6 8 0 SN 500 0 SN 600 4 SN 3300 8 SN 3400 0 SSN 500TSTAC V 500 48 V 600 5 SSN 600TSTAC 6 SSN 00B 30 SSN 300B 34 SD 500 38 SD 600 40 V 00 56 V 300 60 SD 3300 4 SD 3400 44
19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional
19 σ = P/A o σ B Maximum tensile strength σ 0. 0.% 0.% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional limit ε p = 0.% ε e = σ 0. /E plastic strain ε = ε e
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)
gr09.dvi
.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
TOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 6 3 6.1................................ 3 6.2.............................. 4 6.3................................ 5 6.4.......................... 6 6.5......................
untitled
- k k k = y. k = ky. y du dx = ε ux ( ) ux ( ) = ax+ b x u() = ; u( ) = AE u() = b= u () = a= ; a= d x du ε x = = = dx dx N = σ da = E ε da = EA ε A x A x x - σ x σ x = Eε x N = EAε x = EA = N = EA k =
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
Z: Q: R: C:
0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x
untitled
(a) (b) (c) (d) Wunderlich 2.5.1 = = =90 2 1 (hkl) {hkl} [hkl] L tan 2θ = r L nλ = 2dsinθ dhkl ( ) = 1 2 2 2 h k l + + a b c c l=2 l=1 l=0 Polanyi nλ = I sinφ I: B A a 110 B c 110 b b 110 µ a 110
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
Z: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
![6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2](/thumbs/92/109118076.jpg "6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2")
1 6 6.1 (??) (P = ρ rad /3) ρ rad T 4 d(ρv ) + PdV = 0 (6.1) dρ rad ρ rad + 4 da a = 0 (6.2) dt T + da a = 0 T 1 a (6.3) ( ) n ρ m = n (m + 12 ) m v2 = n (m + 32 ) T, P = nt (6.4) (6.1) d [(nm + 32 ] )a
) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
Gmech08.dvi
63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
SO(2)
TOP URL http://amonphys.web.fc2.com/ 1 12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6
LLG-R8.Nisus.pdf
d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =
, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,
6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,
70 : 20 : A B (20 ) (30 ) 50 1
70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................
Gmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2