mt_4.dvi
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1 ( )
2 PI
3 i
4 PI URG USB ii
5 2.1 L B RSR 9N RSR 12N RSR 15N M PI ( ) PI PI PI ( PWM ) ( ) ( ) ( ) ( ) URG USB QV-700N ( ) ( ) URG iii
6 iv
7 1 1.1 [1][2][3] [4][5][6]
8 [7]
![[a] [b]](/docs-images/102/154259660/images/9-0.jpg "147cm")
9 [a] [b] 147cm 137cm 2.1: L B 15mm 20mm 130cm 2.2: 3
10 88cm 2.3: 4
11 3 3.1 M cm 3.3 ( 2.3) 88cm ( 2.2) 130cm ( 2.1) cm 150cm mm 322mm 420mm 5
12 gf 500gf 3.1: gf gf 300gf mm 20mm 5mm 3.7 [8] [9][10][11] 6
13 End-effector Arm 3.1:
14 cm 110cm 40cm [a] 4.1 [b] [c] [d] ( L 1 ) 2 n l l = L 1 +3n (4.1) 8
![Belt-holder grasping Timing-belt on First stage Belt-holder grasping Timing-belt on Third stage [d] Belt-holder grasping Timing-belt on Third stage [a] [b]](/docs-images/102/154259660/images/15-0.jpg "motor [c] Belt-holder grasping Timing-belt on Second stage (1) (2) (3) 4.1: 4.3 1 3 1-2 2-3 3-4 4 1kgf 4.1 4.1: 1-2 2-3 3-4 17.9Nm 11.0Nm 5.4Nm 4.2 (THK ) 4.")
15 Belt-holder grasping Timing-belt on First stage Belt-holder grasping Timing-belt on Third stage [d] Belt-holder grasping Timing-belt on Third stage [a] [b] motor [c] Belt-holder grasping Timing-belt on Second stage (1) (2) (3) 4.1: kgf : Nm 11.0Nm 5.4Nm 4.2 (THK ) 4.2 9
16 4.2: RSR 15N RSR 12N RSR 9N 63.1Nm 28.9Nm 18.4Nm 350mm 370mm 375mm 417g 270g 147g 4.2: RSR 9N RSR 12N RSR 15N 4.4 ( ) 4 2.0mm ( ) τ 1,τ l ng (n =1, 2) 2 0.5kg τ n (n =1, 2) τ g 10
17 l1g l1 l l2 l2g Robot τ1 τ2 M1 Mj2 M2 4.3: 4.3: mm(l 1 ) 1.5kg(M 1 ) 2-0.3kg(M j2 ) 2+ (0.5kg) 100mm(l 2 ) 0.7kg(M 2 ) τ w τ n = τ ng + τ nw (n =1, 2) (4.2) τ ng (n =1, 2) τ 1g = {M 1 l 1g + M j2 l 1 + M 2 (l 1 + l 2g )} g (4.3) τ 2g = M 2 l 2g g (4.4) τ nw (n =1, 2) τ nw = I n ω n (n =1, 2) (4.5) I n (n =1, 2) I l1 M l1 1 I 1 = x 2 +l 2 M 2 dx + x 2 dx + M j2 l1 2 (4.6) 0 l 1 l 1 l 2 l2 M 2 I 2 = x 2 dx (4.7) l 2 0 τ 1 =21.9Nm, τ 2 =1.96Nm 0.5m/sec kg 0.37Nm 11
18 Maxon 4.4: 1 24V20W Nm 2 24V2.5W Nm 24V20W Nm M M-1 160cm 12
19 4.4: M-1 4.5: 388mm 1218mm 225deg 270deg DC 3 25cm/sec 500g 2.8kg 13
20 4.5:
21 4.6: 15
22 PWM x : x y : y z : z x v : x y v : y L 0 : L h : L 1 : L 2 : θ a : θ b : θ 1 : θ 2 : θ v : 16
23 5.1: Z Y θ 1 L 0 (x v, y v ) L h L 1 θ b θ 2 L 2 (x,y,z) θ a, θ v X 5.2: 17
24 5.3 (x, y, z) (θ a,θ b ) (θ 1,θ 2 ) (L 1 ) (x v,y v,θ v ) x = x v + {L 0 L 1 cos θ 1 + L 2 cos (θ 1 + θ 2 )} cos θ v (5.1) y = y v + {L 0 L 1 cos θ 1 + L 2 cos (θ 1 + θ 2 )} sin θ v (5.2) z = L h + L 1 sin θ 1 + L 2 sin (θ 1 + θ 2 ) (5.3) θ a = θ v (5.4) θ b = θ 1 + θ 2 π (5.5) 5.4 (x, z, θ b ) (θ 1,θ 2 ) (L 1 ) θ 1 = arctan x + L 2 cos θ b + π (5.6) z L 2 sin θ b 2 θ 2 = θ b arctan x + L 2 cos θ b + π (5.7) z L 2 sin θ b 2 L 1 = (x + L 2 cos θ b ) 2 +(z L 2 sin θ b ) 2 (5.8) 5.5 PI PI [12] PI ) ( ) τ ref = K P (θ ref θ + K I θ ref θ dt (5.9) τ ref : [Nm] θ ref : [deg] 18
25 θ : [deg] K P : K I : PWM 0 PWM PWM δ = 1 V 0 ( R τ ref K τ + K e ω ) (5.10) δ : PWM /255 V 0 : [V ] R : [Ω] τ ref : [Nm] K τ : [Nm/A] K e : [V/rpm] ω : [rpm] PWM PI mm 400mm 200mm 0mm
26 200 Reference angle Actual angle 150 angle [deg] time [sec] Reference length Actual length length [mm] time [sec] 200 Reference angle Actual angle 150 angle [deg] time [sec] 5.3: PI ( ) 20
27 5.4 θ 1 l 1 τ ref θ 1 = K Pθ1 ( θ ref 1 θ 1 ) + K Iθ1 ( θ ref 1 θ 1 ) dt + mg l 1 2 cos θ 1 (5.11) τ ref θ 1 : [Nm] θ ref 1 : [deg] θ 1 : [deg] K Pθ1 : K Iθ1 : m : [kg] g : [m/sec 2 ] l 1 : [m] θ 1 τ ref l 1 = K Pl1 ( l ref 1 l 1 ) + K Il1 ( l ref 1 l 1 ) dt + mg sin θ 1 (5.12) τ ref l 1 : [Nm] l ref 1 : [mm] l 1 : [mm] K Pl1 : K Il1 : PI 5.5 PWM 21
28 5.4: PI PI I PI 5.5 PWM (x, z, θ b ) PI (θ b ) (x, z, θ b ) 60mm 22
29 l 1 [mm] time [sec] θ 1 [deg] without gravity compensation with gravity compensation time [sec] PWM duty [y/255] 40 without gravity compensation with gravity compensation time [sec] 5.5: PI PI ( PWM ) 23
30 (x, z, θ b ) (θ 1,θ 2,l) W t [Nm] v t 5.13 W a [Nm] v a 5.14 v t = W tl 3 3EI v a = W al 4 8EI (5.13) (5.14) l : [m] E : [Pa] I : 10mm θ 1 l 1 x y x 10cm f(x, y) =e x2 +y (a,b,c,d) f(θ 1,l 1 )=ae b((θ 1 c) 2 +(l 1 d) 2 ) (5.15) 24
31 a*exp(b*((x-c)*(x-c)+(y-d)*(y-d))) gaps caused by deflection gap [mm] θ 1 [deg] l 1 [mm] : 5.6 a =70, b = , c = 173, d = 800 (x, z, θ b ) (x, z, θ b ) (θ 1,l 1,θ 2 ) 2. θ 1 l 1 ( x, z, θ b ) 3. (x, z, θ b ) ( x, z, θ b ) 4. (x + x, z + z,θ + θ b ) 25
32 (θ 1,l 1,θ 2 ) 5. θ 1 l 1 ( x, z, θ b ) 6. (x + x, z + z,θ + θ b ) ( x, z, θ b ) (x, z, θ b ) 1mm (θ 1,l 1,θ 2 ) 1mm (x + x, z + z,θ + θ b ) 1mm mm PWM 26
33 No Yes 5.7: 27
34 5.8: ( ) ( ) 28
35 ( ) ( ) 0( ) PWM 5.8 PWM PC PI PI 29
36 PWM duty [y/255], Angular Velocity [deg/sec] 60 Load Value Time [sec] PWM duty [y/255], Angular Velocity [deg/sec] 60 Load Value Time [sec] 5.9: ( ) ( ) 30
37 5.10: 31
38 PC PC PC 32
39 : 33
40 15 20cm mm PC PC LxSystem[13] URG 6.3 URG-X002S( URG ) URG 6.1 URG 34
41 6.2: 35
![6.3: URG [rad] [mm] (x, y) θ URG (x p,y p ) x p = r cos(φ + θ)+x + L cos θ (6.](/docs-images/102/154259660/images/42-0.jpg "1) y p = r sin(φ + θ)+y + L sin θ (6.")
42 6.3: URG [rad] [mm] (x, y) θ URG (x p,y p ) x p = r cos(φ + θ)+x + L cos θ (6.1) y p = r sin(φ + θ)+y + L sin θ (6.2) x p : x y p : y r : φ : x : x y : y θ : L : URG
43 6.1: URG URG-X002S mm, mm 10mm, mm 1% ms/scan 170g 50mm 50mm 70mm URG n (x i,y i ) y = ax + b a,b a = b = n n n n x i y i x i y i i=1 i=1 i=1 ( n n ) n x 2 i 2 (6.3) x i i=1 i=1 n n n x 2 n i y i x i y i x i i=1 i=1 i=1 i=1 ( n n ) n x 2 2 (6.4) i x i i=1 i=1 (x r,y r ) d d = (ax r y r + b) 2 1+a 2 (6.5) URG USB cm 37
![6.4: USB QV-700N 6.4 Logicool USB QV-700N PC 6.4.2 Camera Calibration Toolbox for Matlab[15] 6.](/docs-images/102/154259660/images/44-0.jpg "4.3 USB RGB 5 5 6.5 38")
44 6.4: USB QV-700N 6.4 Logicool USB QV-700N PC Camera Calibration Toolbox for Matlab[15] USB RGB
45 6.5: ( ) ( ) (u, v) ( u, v) U V 1 θu θv URG L (x obj,y obj,z obj ) x obj = L (6.6) y obj = L tan( θ v v) (6.7) z obj = L tan( θ u u) (6.8) L
46 Y Z Projected Surface Object Projected Surface Object v θv* v Yobj u θu* u Zobj X X Xobj Xobj Top View Side View 6.6: 1. L 3 2. URG 3. URG L URG
47 図 6.7: エレベータによるフロア移動実験の様子 41
![y [mm] 4000 2000](/docs-images/102/154259660/images/48-0.jpg "0-2000 -4000-6000")
![Floor [mm] 42](/docs-images/102/154259660/images/48-4.jpg "6.8: URG")
48 y [mm] L rd Floor Elevator x [mm] y [mm] L402 4th Floor Elevator x [mm] : URG
49
50 図 6.9: 引き出し操作実験の様子 44
51 7 PI 3mm PC 45
52 46
53 [1] NEC PaPeRo, [2] ApriAlpha, press/ /pr j2001.htm [3] PFU MARON-1, [4], 13. [5] T.Tomizawa, A.Ohya, S.Yuta : Book Browsing System using an Autonomous Mobile Robot Teleoperated via the Internet, Proceedings of IROS 02. [6], 15. [7],,, 03, 2A1-1F-B6 ( ) [8], 9. [9], 11. [10], 12. [11], 14. [12] PID, (1992) [13] 15 47
54 [14] H.Kawata, W.Santosh, T.Mori, A.Ohya, S.Yuta : Development of Ultra-Small Lightweight Optical Range Sensor System, Proceedings of IROS 05. [15] Camera Calibration Toolbox for Matlab : doc/index.html 48
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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- 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 =
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
(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
Chap10.dvi
=0. f = 2 +3 { 2 +3 0 2 f = 1 =0 { sin 0 3 f = 1 =0 2 sin 1 0 4 f = 0 =0 { 1 0 5 f = 0 =0 f 3 2 lim = lim 0 0 0 =0 =0. f 0 = 0. 2 =0. 3 4 f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0.
() 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 ()
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
4 5 ( 5 3 9 4 0 5 ( 4 6 7 7 ( 0 8 3 9 ( 8 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 S θ > 0 θ < 0 ( P S(, 0 θ > 0 ( 60 θ
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
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
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..................................
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 =
mugensho.dvi
1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
1 [ 1] (1) MKS? (2) MKS? [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0 10 ( 1 velocity [/s] 8 4 O
![1 [ 1] (1) MKS? (2) MKS? [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0 10 ( 1 velocity [/s] 8 4 O](/thumbs/93/114442636.jpg "1 [ 1] (1) MKS? (2) MKS? [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0 10 ( 1 velocity [/s] 8 4 O")
: 2014 4 10 1 2 2 3 2.1...................................... 3 2.2....................................... 4 2.3....................................... 4 2.4................................ 5 2.5 Free-Body
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 ),
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
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..............
18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
1 1.1 [ 1] velocity [/s] 8 4 (1) MKS? (2) MKS? 1.2 [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0
![1 1.1 [ 1] velocity [/s] 8 4 (1) MKS? (2) MKS? 1.2 [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0](/thumbs/93/114340663.jpg "1 1.1 [ 1] velocity [/s] 8 4 (1) MKS? (2) MKS? 1.2 [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0")
: 2016 4 1 1 2 1.1......................................... 2 1.2................................... 2 2 2 2.1........................................ 2 2.2......................................... 3 2.3.........................................
A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6
1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
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 Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
![II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R](/thumbs/95/125540821.jpg "II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R")
II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =
Gmech08.dvi
51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y
1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
( ) 2002 1 1 1 1.1....................................... 1 1.1.1................................. 1 1.1.2................................. 1 1.1.3................... 3 1.1.4......................................
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
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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
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[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
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)
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D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y
5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x
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Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e
7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z
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( : December 27, 2015) CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x) f (x) y = f(x) x ϕ(r) (gradient) ϕ(r) (gradϕ(r) ) ( ) ϕ(r)
( : December 27, 215 CONTENTS I. 1 II. 2 III. 2 IV. 3 V. 5 VI. 6 VII. 7 VIII. 9 I. 1 f(x f (x y f(x x ϕ(r (gradient ϕ(r (gradϕ(r ( ϕ(r r ϕ r xi + yj + zk ϕ(r ϕ(r x i + ϕ(r y j + ϕ(r z k (1.1 ϕ(r ϕ(r i
. sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y = e x, y = e x 6 sinhx) coshx) 4 y-axis x-axis : y = cosh x, y = s
. 00 3 9 [] sinh x = ex e x, cosh x = ex + e x ) sinh cosh 4 hyperbolic) hyperbola) = 3 cosh x cosh x) = e x + e x = cosh x ) . sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y =
6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P
6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P
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
x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
熊本県数学問題正解
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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)
arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = =
arctan arctan arctan arctan 2 2000 π = 3 + 8 = 3.25 ( ) 2 8 650 π = 4 = 3.6049 9 550 π = 3 3 30 π = 3.622 264 π = 3.459 3 + 0 7 = 3.4085 < π < 3 + 7 = 3.4286 380 π = 3 + 77 250 = 3.46 5 3.45926 < π < 3.45927
QMI_09.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 3.1.2 σ τ 2 2 ux, t) = ux, t) 3.1) 2 x2 ux, t) σ τ 2 u/ 2 m p E E = p2 3.2) E ν ω E = hν = hω. 3.3) k p k = p h. 3.4) 26 3 hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.5) h
QMI_10.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx
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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,
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B. 41 II: ;; 4 B [] S 1 S S 1 S.1 O S 1 S 1.13 P 3 P 5 7 P.1:.13: 4 4.14 C d A B x l l d C B 1 l.14: AB A 1 B 0 AB 0 O OP = x P l AP BP AB AP BP 1 (.4)(.5) x l x sin = p l + x x l (.4)(.5) m d A x P O