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1 11 2 5
2 Lax Friedrichs i
3 Lax Friedrichs ii
4 ( ) ( ) [km/h] 9[m] 100[km/h] 10[km/h] 3) Lax Friedrichs 7 Lax Friedrichs t x x 0 (t) dx 0 (t)/dt d 2 x 0 (t)/dt 2 x i (t)(i = 1, 2, 3, ) 2 v i v i = dx i (t)/dt N N v i (t)(i = 1, 2, 3,, N) 1 1 v(x, t) 1
5 x x x x Fig. 2.1 x 1 t 1 v(x 1, t 1 ) x 2 v(x 2, t 1 ) x 1 t 2 v(x 1, t 2 ) x i t j x i (t j ) v(x i, t j ) v i (t j ) v(x i (t j ), t j ) = v i (t j ) (2.1) v(x, t) x, t q q q(x, t) 1 ρ L d ρ = 1 L + d (2.2) L d Fig
6 2.3 3 ( ) v 0 ρ 0 τ v 0 τ v 0 τ ρ 0 v 0 τ τ ρ 0 v 0 τ q q = ρ 0 v 0 (2.3) ( ) = ( ) ( ) x, t q(x, t) = ρ(x, t)v(x, t) (2.4) 2.4 a b Fig. 2.3 Fig.2.3 x = a, x = b N N = b a ρ(x, t)dx (2.5) x = a, x = b ( q(a, t), q(b, t)) dn/dt x = a x = b q(x, t) 3
7 dn dt = q(a, t) q(b, t) (2.6) (2.5), (2.6) d b ρ(x, t)dx = q(a, t) q(b, t) (2.7) dt a x = a, x = b (2.7) x = a, x = b x y b b x t x a ρ(y, t)dy = q(a, t) q(x, t) (2.8) x ρ(x, t) = q(x, t) (2.9) t x (2.4) ρ t + (ρv) = 0 (2.10) x 2.5 Lighthill Whitham v = v(ρ) (2.11) 1) (2.11) (2.10) ρ t + (ρv(ρ)) = 0 (2.12) x (2.10) ( ) v max v ρ = v (ρ) 0 (2.13) ρ max v(ρ max ) = 0 (2.14) 4
8 q = ρv(ρ) (2.15) (ρ = 0) 2. (v = 0 ρ = ρ max ) Fig.2.4 q 0 max Fig (2.10) 2.5 (2.2) L d 3) 5
9 ( ) + ( ) = ( ) Table 3.1 3) [km/h] [m] [m] [m] Table Fig.3.1 v = Aρ + v max (3.1) A = v max ρ max (3.2) (3.1) (3.2) (2.2) v = v max ρ max 1 L + d + v max (3.3) ρ = ρ max d = 0 (3.3) 6
10 v v max 0 max Fig. 3.1 v = v maxl d + L + v max (3.4) (3.4) Table 3.1 Fig. 3.2 d data L=0.004 L=0.005 L=0.006 L= v Fig. 3.2 L ( ) + ( ) [km] v max = 100 [km/h] 7
11 3.3 1 f(x) = f 0 + x 1 x 0 x x 0 (f 1 f 0 ) (3.5) (0 v 20) (20 < v 30) (30 < v 40) (90 < v 100) (3.6) d = v (0 v 20) d = v (20 < v 30) d = v (30 < v 40) d = v (40 < v 50) d = v 8 (50 < v 60) d = v 0 (60 < v 70) d = v (70 < v 80) d = v (80 < v 90) d = v (90 < v 100) (3.6) (3.6) (2.2) Fig L=0.004 L=0.005 L=0.006 L= v Fig
12 3.4 1 Table. 3.1 Fig d 0.12 data v Fig v = k ρ p + q (3.7) k = 3830, p = 30, q = 17 (3.7),(2.2) d d = v q k + p(v q) L (3.8) L = 0.005[km] k, p, q d = v (3.9) (v + 17) Table [km/h], 40[km/h], 50[km/h] 20[km/h], 60[km/h], 70[km/h] 80[km/h] 9
![2) d d = v q k + p(v q) L (3.8) L = 0.005[km] k, p, q d = v + 17 0.005 (3.](/docs-images/41/22678902/images/page_12.jpg "9) 3830 30(v + 17) Table 3.")
13 [km/h] [km] [km] [km] Table ) t 0 = 1.0[s] Table 4.1 µ [km/h] Table 4.1 µ = 0.53 Table 3.1 t 0 v d v 2 v 2 0 = 2ad (v 0 : a : ) (4.1) d = v2 2a ( ) d = t 0 v v2 2a (4.2) (4.3) 10
![0925-0.0934 0.1676 100 0.112 0.1115-0.1124 0.3606 Table 3.2 4 4) t 0 = 1.0[s] Table 4.](/docs-images/41/22678902/images/page_13.jpg "1 µ [km/h] 20 30 40 50 60 70 80 90 100 0.60 0.59 0.58 0.55 0.53 0.50 0.47 0.47 0.47 Table 4.1 µ = 0.53 Table 3.")
14 ma = µmg (4.4) a a = µg (4.5) (2.2),(4.3),(4.5) 1 2µg v2 + t 0 v 1 ρ + L = 0 (4.6) 2 ( v = t 0 + t ) µg ( 1 ρ + L) / 1 µg = µ 2 g 2 t µg( 1 ρ L) µgt 0 (v > 0) (4.7) t 0 = 1.0(s), µ = 0.53, g = 9.8(m/s 2 ), L = 5.0(m) (3.6) (4.7) Fig.4.1 (4.7) 100[km/h] [m/s] 27.8[m/s] 2 (4.7) v [m/s] 30 data keisan Fig
15 5 (Fig. 5.1) N v mgsin F mgcos mg Fig. 5.1 ma = F mg sin θ (5.1) F = µn = µmg cos θ (5.2) a a = g(µ cos θ + sin θ) (5.3) N F mgsin mgcos v mg Fig. 5.2 (Fig. 5.2) ma = F + mg sin θ F = µn = µmg cos θ a = g(µ cos θ sin θ) (5.4) 12
16 (5.3) θ x θ(x) ( θ(x) > 0 θ(x) < 0) (4.5) a = g(µ cos θ(x) + sin θ(x)) (5.5) (4.7) µ µ cos θ(x) + sin θ(x) (5.6) v(ρ, x) = (µ cos θ(x) + sin θ(x)) 2 g 2 t g(µ cos θ(x) + sin θ(x))( 1 ρ L) (2.10) (µ cos θ(x) + sin θ(x))gt 0 (5.7) ρ t + q(ρ, x) x = 0 (5.8) q(ρ, x) = ρv(ρ, x) (5.9) Lax Friedrichs t x t+ P4 P P x- x P x+ t Fig. 6.1 Lax Friedrichs 13
17 ρ t + q(ρ, x) x = 0 (6.1) { ρ(x, 0) = ρ0 (x) (0 < x < M) ρ x (0, t) = ρ x (M, t) = 0 (t > 0) (6.2) Lax Friedrichs (6.1) 2 q ρ q(ρ(x, t), x) = x ρ x + q x x x = q ρ ρ x + q x (6.3) (6.1) ρ t + q ρ ρ x + q x = 0 (6.4) Lax Friedrichs 1 ρ t ρ(p 4 ) ρ(p 1 ) t ( ρ(p 1 ) = ρ(p ) 3) + ρ(p 2 ) 2 (Fig. 6.1) q(ρ) x (6.4) ρ(p 4 ) (ρ(p 2 ) + ρ(p 3 ))/2 t ρ(p 4 ) = ρ(p 3) + ρ(p 2 ) 2 + q(ρ(p 3), x + x) q(ρ(p 2 ), x x) 2 x (6.5) = 0 (6.6) t 2 x {q(ρ(p 3), x + x) q(ρ(p 2 ), x x)} (6.7) M t x x x P P P P t P-1 P1 P3 P x 5 P7 Fig
18 2 Lax Friedrichs 1 0 x = 0 x P 1 P 1 P 0 P 1 P 1 M x = M x P 7 P 5 P 6 P 5 P Courant Friedrichs Lewy (CFL) CFL q ρ (ρ, x) t x 1 (6.8) q ρ v ρ (ρ, x) = (ρv(ρ, x)) ρ = v(ρ, x) + ρv ρ (ρ, x) µg (6.9) = ρ 2 µ 2 g 2 t µg(1/ρ L) q ρ = µ 2 g 2 t µg( 1 ρ L) µgt µg 0 ρ µ 2 g 2 t µg(1/ρ L) (6.10) x M 1000[m] x 1000 ( x = 1) ρ(x, 0) ρ(x, 0) = [ /m] (7.1) 50[km/h] 60[s] Fig
19 x t (a) 3 (b) 2 Fig (x = 500) 5 (8 % ) ( 10 % 3) 4) ) 5 x=0 x=500 x=1000 Fig. 7.2 Fig. 7.3 (x = 500) 5 (8 % ) 16
20 x t (a) 3 (b) 2 Fig x=0 x=500 x=1000 Fig. 7.4 Fig ρ(x, 0) ρ(x, 0) = 0.01 sin θ + [ /m] (7.2) (2.2) Table [km/h] ρ 60 [km/h] ρ [km/h] 17
![7.4 Fig. 7.5 7.3 ρ(x, 0) ρ(x, 0) = 0.01 sin θ + [ /m] (7.2) (2.](/docs-images/41/22678902/images/page_20.jpg "2) Table 3.1 40 [km/h] ρ 60 [km/h] ρ 40 60 [km/h] 17")
21 x t (a) 3 (b) 2 Fig. 7.5 Fig x t (a) 3 (b) 2 Fig. 7.6 Fig. 7.7, 7.8 Fig
22 x t (a) 3 (b) 2 Fig (a) (b) (c) 40 (d) 60 Fig
23 Fig Fig. 7.10, Fig x t (a) 3 (b) 2 Fig
24 (a) (b) (c) 40 (d) 60 Fig
25 5 5 5 Fig Lax Friedrichs ( ) Fig. 8.1 A B 22
26 B A Fig
27 [1] R.Harberman : MATHMATICAL MODEL : TRAFFIC FLOW (,, 1981) [2], :, (, 1969), pp63-65 [3] :, pp98-99 [4] :, pp
24.15章.微分方程式
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DT-870/5100 &DT-5042RFB ...3 1-1...3 1-1...6 1-3...16 2....17...21 3-1...21 3-2...21 3-2...22 3-3...23 3-4...24...25 4-1....25 4-2...27 4-3...28 4-4...33 4-5...36...37 5-1....39 5-2...40 5-3...43...49
Gmech08.dvi
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n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ
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9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =
5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =
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85 4 86 Copright c 005 Kumanekosha 4.1 ( ) ( t ) t, t 4.1.1 t Step! (Step 1) (, 0) (Step ) ±V t (, t) I Check! P P V t π 54 t = 0 + V (, t) π θ : = θ : π ) θ = π ± sin ± cos t = 0 (, 0) = sin π V + t +V
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f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
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467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 B =(1+R ) B +G τ C C G τ R B C = a R +a W W ρ W =(1+R ) B +(1+R +δ ) (1 ρ) L B L δ B = λ B + μ (W C λ B )
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WINET情報
WINETCONTENTS 1 2 3 Q&A Q A 4 Q A 5 Q A 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 No. 10. 1. 11. 2. 12. 3. 4. 13. 5. 14. 6. 7. 15. 8. 9. 16. 33 17. 33. 34. 35. 18. 19. 36.
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sec13.dvi
13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13:
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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
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(1) / 130 g 25 g 520% 170 g 30 g 560% 70 mg 600 mg 11.6% 0 10.5 mg 0% (1) (2) / 50100 g 25 g 200400% 50100 g 30 g 167333% 5001000 mg 600 mg 83167% 1020 mg 10.5 mg 95190% (2) / (1) 45.6 g 30 g 152% (2)
z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z
Tips KENZOU 28 6 29 sin 2 x + cos 2 x = cos 2 z + sin 2 z = OK... z < z z < R w = f(z) z z w w f(z) w lim z z f(z) = w x x 2 2 f(x) x = a lim f(x) = lim f(x) x a+ x a z z x = y = /x lim y = + x + lim y
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x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v
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