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1 8 2 F (s) =L f(t) F (s) =L f(t) := Z 0 f()e ;s d (2.2) s s = + j! f(t) (f(0)=0 f(0) _ = 0 d n; f(0)=dt n; =0) L dn f(t) = s n F (s) (2.3) dt n Z t L 0 f()d = F (s) (2.4) s s =s f(t) L _ f(t) Z Z ;s L f(t) _ df() = 0 d ;s 0 e;s d = f()e ; f() de;s d 0 d 0 Z = f()e + s f()e ;s d = ;f(0) + sf (s) (2.5) 0 f(t) n L d n f(t) = ;s n; f(0) ; s n;2 f(0) _ dn; f(0) ;; dt n dt n; + s n F (s) (2.6) (f(0) = 0 0) (2.6) (2.3) f(t) g(t) = _g(t) Z t 0 _ f(0) = 0 d n; f(0)=dt n; = f()d G(s)
2 2.3 v R (t) =Ri R (t) (2.5) Z v C (t) = t i C ( )d (2.6) C 0 v L (t) =L d dt i L(t) (2.7) (a) (b) (c) 2.7 R []C [F]L [H]v R (t)v C (t) v L (t) [V]i R (t)i C (t)i L (t) [A] 2. RL 2.8 RL u(t) vin(t) y(t) i(t) (a) 2.8 u(t) = vin(t)y(t) = i(t) ( ) 2.8 RL L _y(t) +Ry(t) =u(t) (2.8) (b) (2.8) u(s) y(s) P (s)
![7 R []C [F]L [H]v R (t)v C (t) v L (t) [V]i R (t)i C (t)i L (t) [A] 2. RL 2.](/docs-images/47/20489871/images/page_2.jpg "8 RL u(t) vin(t) y(t) i(t) (a) 2.8 u(t) = vin(t)y(t) = i(t) ( ) 2.")
3 2.4 () 5 () 2.3 f r (t) ( r (t)) () c f r (t) =;c _z(t) (2.34) r (t) =;c (t) _ (2.35) (a) (b) 2.3 () 2.3 (a) 2.3 (a) M z(t) =f (t) ; c _z(t) (2.36) u(t) =f (t) y(t) =z(t) M y(t) +c _y(t) =u(t) (2.37) (b) (2.37) P (s) P (s) = Ms 2 (2.38) + cs (b) u(t) = (t)y(t) =
4 3. 37 (p i ) k i =(s ; p i )y(s) (3.30) s=p i bs + b0 y(s) = (s ; p)(s ; k p2) = s ; k2 p + s ; p2 (3.3) s ; p (s ; p)y(s) =k + k2(s ; p) s ; p2 (3.32) s = p k =(s ; p)y(s) s=p i k k2 (3.3) (3.32) (3.33) (3.30) k = sy(s) s=0 = s + s=0 = k2 =(s +)y(s) s=; = s s=; = ; (b) p i (i = 2 n) p = p 2 = = p` (3.27) y(s) = k ` (s ; p )` + + k 2 (s ; p ) 2 + k s ; p + k`+ s ; p`+ + k`+2 s ; p` k n s ; p n (3.34) y(s) y(t) =L ; y(s) = k `t` + + k 2 t + k e pt `! +k`+ e p`+t + k`+2 e p`+2t + + k n e pnt (3.35)
5 38 3 k i 8 >< >: (p i ) d`;i f(s ; p )`y(s)g k i = (` ; i)! ds`;i k j =(s ; p j )y(s) s=pj s=pi (i = 2 `) (j = ` + `+2 n) b2s2 + bs + b0 y(s) = (s ; p) 2 k2 (s ; p3) = (s ; p) 2 k k3 + s ; p + s ; p3 (3.37) (s ; p) 2 (s ; p) 2 k3(s ; p)2 y(s) =k2 + k(s ; p) + s ; p3 (3.38) s = p k2 =(s ; p) 2 y(s) s=p k2 (3.38) s df(s ; p) 2 y(s)g ds (3.36) (3.37) (3.38) (3.39) = k + k3f2(s ; p)(s ; p3) ; (s ; p)2 g (s ; p3) 2 (3.40) (3.40) s = p k = df(s ; p)2 y(s)g ds s=p (3.4) k k3 pi (3.36)
6 44 3 (a) T> 0 (b) T< T T > 0 T < 0 T > 0 T T > 0 t = T y(t) y = K 63.2 % 3.6 y(t) = K T e;t=t (t 0) (3.52) RL S ON () (2) (3) u(t) e(t) y(t) i(t) RL T S ON i(t) i R
7 (a) 0 < (b) ; <<0 (c) 3.4 y(t) =KL ; s ; 2 (d) ;! n (s +! n ) 2 ; s +! n = K ; e ;!nt (! n t +) (3.57) (3.57) = y = K ( 3.4 (c)) = ; ( 3.4 (d)) (iii) jj > P (s) (2 ) p = ;( + p 2 ; )! n p 2 = ;( ;p 2 ; )! n t 0
8 48 3 ; y(t) =KL s + p2 p ; p ; p 2 s ; p s ; p 2 ; + e! nt ; ; ; e ;! nt # = K " ; e;!nt 2 (3.58) =p 2 ; jj >>0 (3.58) > y = K ( 3.4 (c)) <; ( 3.4 (d)) 2 (a) Tp (3.54)(3.57)(3.58) 0 << 0 << (3.54) y(t) =K ( ; e ;!nt p ; 2 sin! dt + cos! d t!) y(t) _y(t) = K! n p ; 2 e;!nt sin! d t (3.59) T p! d t = (sin! d t =0 t>0) T p T p = = (3.60)! d! np ; 2 (b) Amax y max y max = y(t p )=K ; +e ;!ntp (3.6) y = K A max
9 A max = y max ; y = Ke ;!ntp = Kexp ; p ; 2! (3.62) (c) (3.59) i y i! d t i =(2i ; ) (i = 2 ) t i y(t) i A i = A i+ =A i = A i+ A i y i = y(t i )=K ; +e ;!nti (3.63) A i = y i ; y = Ke ;!nti (3.64) = e;!nti+ e ;!nti = exp ; 2 p ; 2 0 =log e 2! (3.65) 0 = ; (3.66) p ; () () <0 () =0 () 0 << A max 0 <<!! A max = Kexp ; 2 = exp ; p ; p 2 ; 2 y = K
10 6. 09 u(t) =Asin!t (A >0!>0) y(t) u(t) =Asin!t u(s) =A!=(s 2 +! 2 ) y(t) y(s) =P (s)u(s) = k = A! +! s + A! s 2 +! 2 = A(j ;!) k2 = 2 2( +! 2 ) k s + + k2 s + j! + k3 = ; A(j +!) 2( +! 2 ) k3 s ; j! y(t) y(t) =L ; [y(s)] = k e ;t + k 2e ;j!t + k 3e j!t (6.2) = A! +! 2 e;t + A (sin!t ;! cos!t) (t 0) (6.3) +! 2 (6.3) e ;t = 0 y(t) u(t) = A sin!t y(t) = A (sin!t ;! cos!t) +! 2 = B(!) sin(!t + G p(!)) (= y app(t)) (6.4) A B(!) = p +! 2 Gp(!) =tan; (;!) =;tan ;! (a) u(t) = sin0:t 6.2 (b) P (s) ==(s +) u(t) = sin0t 6.2 A =! =0: 0 u(t)y(t) y(t) y app(t) 6.2 u(t) y(t)
11 6. 3 P (j!)= N (j!)n2(j!) N m (j!) D(j!)D2(j!) D n (j!) = jn (j!)je j\n(j!) jn2(j!)je j\n2(j!) jn m (j!)je j\nm(j!) jd(j!)je j\d(j!) jd2(j!)je j\d2(j!) jd n (j!)je j\dn(j!) jn (j!)jjn2(j!)jjn m (j!)j = jd(j!)jjd2(j!)jjd n (j!)j exp ( j mx nx \N i (j!) ; \D i (j!) i= i=!) (6.20) (6.9) jp (j!)j \P (j!) jp jn (j!)jjn2(j!)jjn m (j!)j (j!)j = jd(j!)jjd2(j!)jjd n (j!)j mx (6.2) \P (j!)= \N i (j!) ; \D i (j!) (6.22) i= i= P (s) ==(s +)(s +2) P (s) = D(s) =s + D2(s) =s +2 D (s)d 2(s) p jd (j!)j = +! 2 \D (j!) = tan ;! jd 2(j!)j = p +4! 2 \D 2(j!) = tan ; 2! P (s) jp (j!)j = jd = ; (j!)jjd 2(j!)j p( (6.23) +! 2 )( + 4! 2 ) \P (j!) =; \D (j!)+\d 2(j!) = ; tan ;! + tan ; 2! (6.24) (6.23) (6.5) = tan ;! 2 = tan ; 2! nx
12 8 6 M p jp (0)j =! p u(t) y(t) u(t) () P (s) = +Ts jp (j!)j \P (j!) (6.27) jp (j!)j = p +(!T ) 2 \P (j!)=;tan;!t (6.28) (6.28) ( 20log!T << 0 jp (j!)j = 20log 0 = 0 [db] \P (j!) = ;tan ; 0=0[deg] 8 <!T = : 20log 0jP (j!)j =20log 0 p = ;3:0 [db] 2 \P (j!)=;tan ; =;45 [deg] 8 <!T >> : 20log 0jP (j!)j = 20log 0!T = ;20log 0!T [db] \P (j!) = ;tan ; = ;90 [deg] ! ==T 0[dB] ;20 [db/dec] 0 <! =5T 0 [deg]! 5=T ;90 [deg] (20 )
![T = : 20log 0jP (j!)j =20log 0 p = ;3:0 [db] 2 \P (j!)=;tan ; =;45 [deg] 8 <!T >> : 20log 0jP (j!)j = 20log 0!](/docs-images/47/20489871/images/page_12.jpg "T = ;20log 0!T [db] \P (j!) = ;tan ; = ;90 [deg] 6.76.8 6.9! ==T 0[dB] ;20 [db/dec] 0 <! =5T 0 [deg]!")
13 20 6! b jp (j!)j () T (!<< =T 0 [db] 0 [rad]!>> =T ) () ()! c ==T () (6.27) (=2 0) =2 6.6 P (s) ==( + 0s) (2 )
14 7. 37 (P (s)c(s) ) P (s)c(s) P (s)c(s) (; 0) (a) (b) 7.5 P (s)c(s) 7.2 P P (s) = +Ts T >0 C(s) =kp kp > 0 7. L(s) :=P (s)c(s) =kp =( + Ts) jl(j!)j \L(j!) jl(j!)j = kp p +(!T) 2 \L(j!) =;tan;!t! =0 jl(j!)j = kp \L(j!) = 0 [deg]! = jl(j!)j =0 \L(j!) =;90 [deg] L(j!) kp L(j!) = +j!t = kp ( ; j!t) +(!T) 2 L(j!) Im L(j!) =0! =0 P (j!)c(j!) 7.6 kp > 0 P (j!)c(j!) (; 0) kp > 0
![T) 2 \L(j!) =;tan;!t! =0 jl(j!)j = kp \L(j!) = 0 [deg]! = jl(j!)j =0 \L(j!) =;90 [deg] L(j!](/docs-images/47/20489871/images/page_14.jpg ") kp L(j!) = +j!t = kp ( ; j!t) +(!T) 2 L(j!) Im L(j!) =0! =0 P (j!)c(j!) 7.6 kp > 0 P (j!")
15 P (s)c(s) P (s)c(s) P P (s) = (s +) 3 (7.) C(s) =k P k P > 0 (7.2) 7. k P k P > 0 L(s) :=P (s)c(s) jl(j!)j \L(j!) jl(j!)j = k P j +j!j 3 = k P ( +! 2 ) 3=2 \L(j!) =;3tan;! (7.3)! =0 jl(j!)j = k P \L(j!) = 0 [deg]! = jl(j!)j =0 \L(j!) =;270 [deg] L(j!) L(j!) = kp D(j!) D(j!) =(; 3!2 )+j!(3 ;! 2 ) D(j!) Im D(j!) =0 0 <!<! = p 3 \L(j!) =;80 [deg] p! =! pc! pc = 3 7.! =! pc jl(j! pc)j = k P ( +! 2 pc) 3=2 = kp 8 < (7.4) 0 <k P < 8 7. k P =8
![)j = k P \L(j!) = 0 [deg]! = jl(j!)j =0 \L(j!) =;270 [deg] L(j!) L(j!) = kp D(j!) D(j!) =(; 3!2 )+j!(3 ;! 2 ) D(j!](/docs-images/47/20489871/images/page_15.jpg ") Im D(j!) =0 0 <!<! = p 3 \L(j!) =;80 [deg] p! =! pc! pc = 3 7.! =! pc jl(j! pc)j = k P ( +! 2 pc) 3=2 = kp 8 < (7.")
16 D =0 2 Ann Bn Cn D u(t) y(t) ( ) 3 (2.) n>m() D = (2.39) (t) =0 J (t) =;c _ (t) ; M`g(t) + (t) (8.3) x(t) u(t) y(t) " # " # x (t) (t) x(t) = = x 2(t) _(t) u(t) = (t) y(t) =(t) _x (t) = (t) _ =x 2(t) _x 2(t) = (t) = ;;c (t) _ ; M`g(t) + (t) J = ; M`g J x(t) ; c J x2(t) + J u(t) y(t) =(t) =x (t) (8.3) A = 4 ; M`g ; c 5 B = 4 5 C = 0 D =0 (8.4) J J J 2 MATLAB 8.5. (73 ) 3
17 D = P{D P{D u(t) =k P e(t) ; k D _y(t) e(t) =r ; y(t) (8.28) u(t) = (t)y(t) =(t) x (t) =(t) x 2(t) = _ (t) (8.28) P{D u(t) =k P ; r ; x (t) ; k Dx 2(t) = ;k P " # x (t) ;k D + k P r x 2(t) = Kx(t) +Hr K = ;k P ;k D H = kp (8.29) (8.27) r =0(8.27) H =0 u(t) =Kx(t) (8.30) (8.30) x(t) (8.30) (8.) ; _x(t) = A + BK x(t) (8.3) A + BK K t! x(t)! 0 (8.30) K
18 y(t) =x (t) =(t) r (8.45)(8.46) K y(t) r = 8.6 Q = diag q q > 0 0 R = (r = x(0) = 0) Z t u(t) =Kx(t) +k I w(t) w(t) := e( )d e(t) =r ; y(t) (8.47) 0 x(t) = y(t) (5.0) (8.47) _y(t)t I{PD x e (t) = x(t) T w(t)t 8.7
19 () P (s) = s ;2 +2 2s + (2) P (s) = ;p 2j ; 3s 2 +2s C 2.2 P (s) = CLs 2 + RCs + Cs 2.3 () P (s) = RCs + (2) P (s) = RCs P (s) = Js 2 + cs 2.5 P (s) = Ms + c 2.6 P (s) = Ms 2 +(c + c 2 )s + k T (t) = 2 M _z(t)2 U (t) =0D(t) = 2 c _z(t)2 q(t) =z(t)(t) =f (t) (2.46) (2.36) r T = M c K = 2.9! n = p = R C K = c CL 2 L 2.0 ;:043 0:5935j0:4793 0:752j;:0000 :442j P (s) = s 2 ; :2j s () ; s s s ; (2) s s s 4 (3) s s s 2 +4 (4) ; s +3 s s 2 +6s +0 (5) 6 s +5 (s +2) 3 (6) s 2 +4s () f (t) =2+3e;2t (2) f (t) = t2 2 e ;t (3) f (t) =cos5t + 5 sin5t (4) f (t) =e ;t 2cos2t + 32 sin2t 3.4 () y(t) = 3 2 ( ; e ;2t ) (2) y(t) =2; 3e;t + e;2t cos2t + 2 sin2t (3) y(t) = 2( ; te;t ; e;t ) (4) y(t) =; e;t 3.5 () y = 2 (2) y =2
20 203 K 3.6 y(t) =L; y(t) = K +Ts T e ;t=t y(0) = K T 3.7 () P (s) = Ls + R T = L R (2) y(t) =K( ; e;t=t )= R ( ; e ;Rt=L )(t 0)y = R (3) R T L T R = =50[]L = RT =0:2 [H] y 3.9 R 2r L r C k 3.0 ()! n = M = c K = 2p km k (2) K = y =0:04! d = A max = 6:28 = ; log e = 2:77 T p T p K (3)! n = q! d = 6:87 = = 0:404 k = = 25! n K M = k =! n 2 0:530c =2! nm = 2:94 3. () (2) (3) (4) () s = ; p 3j 2 (2) s = ; p , 4.4 () k P > ; 2 5 (2) k P > ; <k I < 2 5 (2 + 5k P ) () C(s) =2e = 2 C(s) =5e = () C(s) =2y = 5 5 C(s) =5y = G yr(s) = 0s +6:25 s 3 +2s 2 +2s +6:25, G yd(s) = (2) e =0 (2) y =0 5s s 3 +2s 2 +2s +6:25, s 3 +2s 2 +2s G er(s) = s 3 +2s 2 +2s +6:25, G ;5s ed(s) = s 3 +2s 2 +2s +6:25 G yr(s)gyd (s)
21 () y(t) = 2 et ; 2 (cost +sint) t! et! y(t) y(t) (6.7) 6.2 () jp (j!)j = p 25 +! 2, \P! (j!)=;tan; 5 (2) jp (j!)j = p 25 +! 2, \P (j!) =! tan; 5 (3) 2 jp (j!)j = p 4+! 4, \P 2! (j!)=;tan; 2 ;! 2 (4) jp (j!)j =! 2s +! 2 (9 +! 2 )(6 +! 2 )(25 +! 2 ), \P (j!) = 90 + tan ;! ; tan ;! ;! 3 tan; ;! 4 tan; 5 (5) jp (j!)j = ( +! 2 ) 5, \P (j!)=;0tan;! 6.3 y(t) = sin(t + ), = tan ; 2 ; tan; B(! ) A = 0, B(! 2) A =, B(! 3) A = 0, B(! 4) A = 6.5 P (j!) ==( + j!t) =x + jy jp (j!)j ==p +(!T ) 2 = px 2 + y 2 \P (j!)=;tan ;!T =tan ; y=x y=x = ;!T p +(!T ) 2 = p px +(;y=x) 2 = 2 + y 2 00 (x ; =2) 2 + y 2 =(=2) () P (j!) = e ;j!l = cos!l ; jsin!l jp (j!)j =\P (j!) = ;!L 6.8 (a) (2) jp (j!)j ==p +(!T ) 2 \P (j!) =;(!L +tan ;!T ) 6.8 (b) =0) =0) 7.2 () 0 <k P < 8 (! =! pc = p 5 Im P (j! pc)c(j! pc) (2) 0 <k P < 6(! =! pc = p 2 Im P (j! pc)c(j! pc)
22 7.3 () 8>< >: 8>< >:!gc =!pc = 0 [rad/s] k G M =80; P 20log 0 4 qk [db] =2 ; P 02 [rad/s] (k P > 0 4 ) P M = 80 ; 4tan ; q (2) k P = k =2 ; P [deg] (k P > 0 4 ) P k P =8:5 P M =49:98 [deg]pi k P =5:8T I =0:85 P M =50:23 [deg]pid k P =8:5T I =0:85T D =0:04 P M =50:30 [deg] S(s)T (s) 8 8. () (2.36) x(t) = T z(t) _z(t) Z t _x(t) =" 0 0 ;c=m y(t) = 0 x(t) # x(t) +" 0 =M # u(t) P (s) ==(Ms 2 + cs) (2) RCL (2.20)(2.2) T x(t) = i( )d i(t) 0 _x(t) =" 0 ;=CL ;R=L y(t) = =C 0 x(t) 8.2 e At = e ;2t" # ;3t" 3 ;2 # ; + e # x(t) +" 0 =L # u(t) P (s) ==(CLs 2 + RCs +) ;6 ;2 6 3
213 March 25, 213, Rev.1.5 4........................ 4........................ 6 1 8 1.1............................... 8 1.2....................... 9 2 14 2.1..................... 14 2.2............................
214 March 31, 214, Rev.2.1 4........................ 4........................ 5............................. 7............................... 7 1 8 1.1............................... 8 1.2.......................
2012 September 21, 2012, Rev.2.2
212 September 21, 212, Rev.2.2 4................. 4 1 6 1.1.................. 6 1.2.................... 7 1.3 s................... 8 1.4....................... 9 1.5..................... 11 2 12 2.1.........................
( ) 5. VSS (VIM ) 10. ( ) 11. (ANN ) ( )
1.... ( ) 5. VSS.. 8. 9. (VIM ) 1. ( ) 11. (ANN ) 1. 1. ( ) 1 Lagrange 1..1 Lagrange q, Lagrange D(q)q + C(q; _q)_q + G(q) = (1.1) D(q)q C(q; _q)_q G(q) ( ) D(q) D(q) m ; M < M m (D(q)) (1.) (D(q)) M
[1.1] r 1 =10e j(ωt+π/4), r 2 =5e j(ωt+π/3), r 3 =3e j(ωt+π/6) ~r = ~r 1 + ~r 2 + ~r 3 = re j(ωt+φ) =(10e π 4 j +5e π 3 j +3e π 6 j )e jωt
![[1.1] r 1 =10e j(ωt+π/4), r 2 =5e j(ωt+π/3), r 3 =3e j(ωt+π/6) ~r = ~r 1 + ~r 2 + ~r 3 = re j(ωt+φ) =(10e π 4 j +5e π 3 j +3e π 6 j )e jωt](/thumbs/101/152334931.jpg "[1.1] r 1 =10e j(ωt+π/4), r 2 =5e j(ωt+π/3), r 3 =3e j(ωt+π/6) ~r = ~r 1 + ~r 2 + ~r 3 = re j(ωt+φ) =(10e π 4 j +5e π 3 j +3e π 6 j )e jωt")
3.4.7 [.] =e j(t+/4), =5e j(t+/3), 3 =3e j(t+/6) ~ = ~ + ~ + ~ 3 = e j(t+φ) =(e 4 j +5e 3 j +3e 6 j )e jt = e jφ e jt cos φ =cos 4 +5cos 3 +3cos 6 =.69 sin φ =sin 4 +5sin 3 +3sin 6 =.9 =.69 +.9 =7.74 [.]
DE-resume
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名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
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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
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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 =
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β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
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