II. T = 1 m!! U = mg!(1 cos!) E = T + U! E U = T E U! m
U,E mg! U = mg!(1! cos)! < E < mg! mg! < E! L = T!U = 1 m!! mg!(1! cos) d L! L = L = L m!, =!mg!sin m! + mg!sin = d =! g! sin & g! d =! sin
! = v v = sin! (!, v) dv d! = v! = sin! v v dv d! = sin! 1 d(v ) d! = d sin! = d! cos! v cos! = A A A =!! < A < v =,! = n (n =!1,,1,!) A = < A v!!!
U = mg!(1! cos) cos =1! 1! + 1 4! 4! <<1 U U * = k + a ' & 4 ) k * mg!, a =! 1 ( 6 L = T!U T!U * = 1 m!! k + a ' & 4 ) ( d +L + = m! +L, + =!k(1 + a ), m! + mg!(1 + a ) = + - (1 + a ) = - * g!! = f (!) = K! + K'! 3 d! = sin! d! 1 d! ' & ) = ( cos! + E' E'* E 1 d! ' & ) ( k * = (1 cos!) + E E! ' 1 cos! = sin & ) ( d! ' & ) ( = 4 k sin! ' & ) (
E <! (k <1) U =! (1 cos) U max =! E <! = U max d!! = k = sin!! ksin! = sin d = k cos! 1 k sin! d! d!! ' = 4( & k ) sin ' & * d! = (! k ) sin = ( k 1) sin + = ( k cos+, ( = d! k cos+ = k cos+ k cos+ 1) k sin + d+ = d+ 1) k sin + * ( = + d+ - = F(+,k) 1) k sin + <! <! < < /! T 4 = / d = K(k) 1 k sin k = sin! K(k) = ( ) * T 4 T = ' * ' /!! k! d k ' / ( d = ' 1& k sin '
E >! (k >1) d! & ' sin! = sin* +! = 4( k ) sin & ' = 4( k 1 ) 1 k sin! & ' d! cos! = d* cos* d! = ( k 1 ) 1 k sin! = ( k 1 ) 1 k sin * ( k = 1 1 ) 1 k sin * d* cos* 1 cos! d! cos! 1 <! < < < cos = d cos = 1 sin = 1 sin! = cos! cos = cos! kt =! kt & = K 1 ) ' ( k * + d! 1 1 k sin! E =! (k =1)! t = d 1 + sin cos = log 1 sin sin = tanh! t (t & ±') = ±(
! x f (u) u = v! x v m x =!x + f (v! x ) v >> x f (v! x ) f (v )! x f '(v ) = x! f (v ) m =! + f (v ) ( ' * + f (v )! f '(v ) & ) + m + f '(v ) + = f (u) f '(u)! = ± =! 1, x = Ae! 1 t + Be! t v u = v! x
Van der Pol!Rayleigh Load Rayleigh (John William Strutt 184-1919) m x + kx = a x! b x 3 y = x x = y y = 1 m (!kx + ay! by 3 ) Rayleigh m d x dx + m! + x = x = e t m + m! + = D =! 4 m D =! 4 m <! = 1 ± i 4 m = 1 ± i = 4 m & x = ae 1 t cos(t + ') > (
! = A + B dx & ( ' A >, B > dx! < )! > Rayleigh m d x + ( dx + dx )!A + B ', * & - +.x = Rayleigh (Varistor, Surge Protector) v R = R i + R i 3 R >, R < v L = L di v C = q C i = dq, v L + v R + v C = V R & '!()!(!)!' ) ( ()!&!! i i d q + 1 ' ) L R! dq + ) dq ( + R &, *) -) +. q =. / 1 LC V L V R V C Rayleigh
u J! d! = F(v), v = u + d J d! + d! + F u + d! & ( + mgasin! = ' U F u + d! & ( = F(u) + F'(u) d! ' + 1 d! & F''(u) ( + 1 3 d! & F'''(u) ( +! ' 6 ' F''(u) = u sin!! J d! + d! d! + F(u) + F'(u) + 1 3 d! & F'''(u) ( + mga! = 6 ' mga) * mga! + F(u) J d ) d) + ( + F'(u)) + 1 6 d) & F'''(u) ( ' 3! d! + mga) = Rayleigh a!van der Pol Rayleigh Van der Pol m d x + ( dx + dx )!A + B ', * & - +.x = / = t dx =. m, y = 3B A A 3B y, dx d x = A dy 3B Rayleigh m A 3B dy + (!A + Ay ) A 3B y + x = m d y + A(!a + y ) dy + y = d y d + (!1 + y ) dy d + y =, A m Van der Pol
Van der Pol Rayleigh d z +!(z 1) dz + z = z = 3 y 3 y +!(3 y 1) 3 y + 3 y = y +!(3 y 1) y + y = y = y + C, y = y + C 1 t 3 y y = d ( y 3 ) = y 3 + C 3 t y +!( y 3 y ) + y = C t x = y C x!(1 x ) x + x = t Van der Pol Rayleigh Lienard Rayleigh d x +! dx ' + x = & dx = v dv = ( ) x +! dx, * '- + &. dv dx = ( x +!(v) v
!Van der Pol i =!v + v 3!i = v R + C dv + 1 L v M i L C R v d v! d * C! 1 & ( v! ) - +, RC' C v3. / + 1 LC v = t! = LC, x = v d x d! d + LC d! C 1 ' & ) * LC., - RC( C x / x + x =! LC C 1 ( & ) ' RC*,! 3+ d x d,!(1 x ) dx d, + x = L C Van der Pol
!Van der Pol Limit Cycle x!(1! x ) x + x = >!(1! x ) : Limit Cycle Limit Cycle Limit Cycle x = asint a LimitCycle!(1 x ) x t +T!(1 x ) x dx =, x = acost, T = t t +T!(1 x ) x dx =!(1 a sin t)a cos t = t & ) & )!a ( cos t a 4 sin t+ =!a ( 1 a + = ' * ' 4 * a = & ' (!!!&! &!!!!! =.
!Van der Pol! <<1 d x!(1! x ) dx + x = t, t 1, t,! t =1 t 1 = t ~ t, t 1 ~ t, t ~ t,! t n ~ n t,! n =1, =, =,! x = x +!x 1 +! x +! x = x (t, t 1, t,!) x 1 = x 1 (t, t 1, t,!) dx = dx + dx + = dx = dx +! dx dx +! +! dx +! +! dx 1 dx +! +! ' +! dx +! dx 1 dx +! dx +! ' +! +! dx 1 dx +! +! ' +! & & & = dx +! dx 1 + dx ' +! dx + dx 1 + dx ' +! & & t d x = d x +! d x 1 + d x d x ' +! & + d x + d x 1 + d x ' +! 1 &!(1! x ) dx =!(1! x ) dx )! (1! x ) dx 1 + dx & dx, * (! x x 1 - + '. + O(3 )
Van der Pol d x +! d x 1 + d x ' (!(1( x ) dx + x +!x 1 = & d x! : + x =! 1 : d x 1 + x 1 = ( d x + (1( x ) dx x = acos(t +!) t a,! t 1 x (t,t 1 ) = a(t 1 )cos[ t +!(t 1 )]! x = a(t 1 )sin[ t + (t 1 )] t x = da sin(t + ) a d cos(t + ) t 1 t x 1 t + x 1 = da sin(t + ) + a d cos(t + ) a 1 a cos (t + ) [ ]sin(t + ) sin 3 x =! 1 4 sin 3x + 3 sin x 4 [ 1! a cos (t + )]sin(t + ) = 1! 1 & 4 a ( sin(t ' + )! 1 4 a sin(3t + 3) ) x 1 )t + x * 1 = da! a 1! 1 &- + 4 a (., '/ sin(t + ) + a d cos(t + ) + 1 4 a3 sin(3t + 3) a = a(t 1),! =!(t 1 )
x 1 cost sint Van der Pol cos(t +!) sin(t +!) da! a 1! 1 4 a ' = & d( =!(t 1 ) = const da 1 = = 1 8 a(a )(a + ) 4a + 1 1 8 a + 1 & ( a + ' t 1 = ln a + ln(a 4) + const 4 a = 1 Ce t 1 x = 1! Ce!t 1 cos(t + (t 1 )) t 1 = a(t 1 ) = A C =1! 4 A x = 1 + 4 cos(t + )) A!1 ' e!(t & A < 4! cos(t + ) A > 4! cos(t + )
!Van der Pol! Relaxation Oscillation &!!!&!& &!!! &!!&!& &!!! & &!& & ' (!!!! & '!!!! =.1! =.35