动
处 运动 动 (-khz) 说 颜 乐 风 帜
简谐 动! (t) =!t + x(t) =x cos(!t + ) x(t) x 频! = 频 T = f F = kx ) mẍ = kx ) ẍ + k m x = 悬 弹 频 x(t) =x cos(! t + ) with! = r k m
弹 谐 动 E k = 1 mv = 1 mẋ = 1 mx! sin (! t + ) 势 E p = 1 kx = 1 kx cos (! t + ) 总 E = E k + E p = 1 kx = 1 mx! F = kx ) mẍ = kx ) ẍ + k m x = 悬 弹 频 x(t) =x cos(! t + ) with! = r k m
简谐 动! (t) =!t + x(t) =x cos(!t + ) x(t) x 频! = 频 T = f ma = ml = mg sin mg g = l 频 (t) = cos (! t + ) with! = r g l
轴 i z (t) =!t + x(t) =x cos(!t + ) 实轴 z(t) =x e i!t+ x(t) =Re[z(t)] z = cos + i sin = e i ż = i!z and z =! z i i =? z + k m z = 阶齐 线
变换 为 T! n = n T x(t) =x + X n [a n cos(! n t)+b n sin(! n t)] = x + X n c n cos(! n t + n ) 备 x = 1 T a n = T b n = T Z T/ T/ Z T/ T/ Z T/ T/ x(t)dt x(t) cos(! n t)dt x(t)sin(! n t)dt c n = p a n + b n n = arctan b n a n
H O N CO 动 1 k1 k k x1 x 动 动
k k k 动 x1 x x 1, = A 1, e i!t m 1 ẍ 1 = k 1 x 1 k(x 1 x ) m ẍ = k x k(x x 1 ) k + k1 m 1! k A1 k k + k m! A = (k + k 1 m 1! )(k + k m! ) k = k 1 = k = k and m 1 = m )! 1 = k m and! = 3k m
k k k 动 x1 x x 1, = A 1, e i!t k m! k A1 k k m! A =! 1 =! = r k r 3k m ) A 1 A =1 ) m ) A 1 A = 1 ) x1 (t) x (t) x1 (t) x (t) = Ae i! 1t = Be i! t 1 1 1 1
k k x1 x m 1 ẍ 1 = k 1 x 1 k(x 1 x ) m ẍ = k x k(x x 1 ) k! 1 = k m : x1 (t) x (t)! = 3k m : x1 (t) x (t) x1 (t) x (t) = Ae i! 1t 1 1 = Ae i! 1t = Be i! t + Be i! t 1 1 1 1 1 1 x1 () x () ẋ1 () ẋ () = = x1 A + B =Re x A B!1 A +! =Re i B! 1 A! B ) A = x 1 + x and B = x 1 x
k k k x1 (t) x (t) = Ae i! 1t 1 1 + Be i! t 1 1 x1 x A = x 1 + x and B = x 1 x 1. k 1 = k = k x 1 = x 1. k 1 = k = k x 1 = x 1. k 1 = k = 1k x 1 6= x =.5.5.5 4 6 8 1 1 4 6 8 1 1 4 6 8 -.5 -.5 -.5-1. -1. coupled pendulum: http://v.youku.com/v_show/id_xmzc5njqznja4.html, http://v.youku.com/v_show/id_xmzc3odayodqw.html, http://v.youku.com/v_playlist/f1734636o1p1.html -1.
弹 质 l m = l m x(t) =x cos! t 1 kx + 1 mv = 1 kx )! = r k m 设弹 长 缩 dm = du and v(u) = u l v E ks = Z l m! m + m 3 1 v(u) dm = 1 )! = m v l 3 s Z l k m + m /3 u du = 1 m 3 v! V s l
运动 f m v f 1 弹 f / v ) mẍ = f 1 + f f 1 = kx f = v = ẋ 阶线 f / v ) 阶 线 阶齐 ẍ + ẋ +! x = with = m and! = r k m z(t) =Ae i(!t+ ) with x(t) =Re(z(t)) q! +i! +! = )! = ±! + i
运动 f v 弹 mẍ = f 1 + f f 1 = kx f 1 f = v = ẋ m <! : x(t) =Ae t cos q! t + >! : x(t) =Ae ( p! )t + Be ( +p! )t z(t) =Ae i(!t+ ) with x(t) =Re(z(t)) q! +i! +! = )! = ±! + i
<! : x(t) =Ae t cos q! t + e t T x() = x and ẋ() = ) A = x and = T = p! >! = T! = q! 质 测 =! r1 Q =! 1 4Q
>! : x(t) =Ae ( p! )t + Be ( +p! )t x() = x and ẋ() = ) x = A + B q = (A + B)+! (A B)
临 运动 ẍ + ẋ +! x = with = m and! = r k m =! : x(t) =(A + Bt)e t x() = x and ẋ() = ) A = x and B A = 为 临 过 运动 http://v.youku.com/v_show/id_xmzc5odezmzqw.html
动 f m f(t) f 1 v 弹 驱动 mẍ = f 1 + f + f(t) f 1 = kx f = v = ẋ f(t) =f cos(!t) 阶 齐 线 = 齐 + 齐 ẍ + ẋ +! x = f m cos(!t) z + ż +! z = f m ei!t 齐 x(t) =Be t cos q! t + z(t) =Ae i(!t ) 历险记 http://v.youku.com/v_show/id_xmzgxntgotq4.html
动 阶 齐 线 ẍ + ẋ +! x = f m cos(!t) z + ż +! z = f m ei!t z(t) =Ae i(!t )! +i! +! A = f m ei = f m A(!! )= f m cos A! = f m sin A = (cos + i sin ) f /m p (!! ) +4! = arctan!!!
动. 1.5 1..5 -.5-1. -1.5 1 3 4 阶 齐 线 ẍ + ẋ +! x = f m cos(!t) x(t) =A cos(!t ) 1..5 A = f /m p (!! ) +4! -.5-1. 1 3 4 = arctan!!! 1..5 -.5 1 3 4 齐 x(t) =Be t cos q! t + -1.
动 - Af m! 1 阶 齐 线 8 6 4 /! =.1 /! =.5 /! = /! =. 质 Q =!..5 1. 1.5..5 3. ẍ + ẋ +! x = f m cos(!t) x(t) =A cos(!t ) A = f /m p (!! ) +4! = arctan!!! /! =1! max =!/! q! and A max = f m p! http://v.youku.com/v_show/id_xmzgxntg5nji4.html
动 -. -. -.4 /! =.1 /! =.1 /! =. 阶 齐 线 ẍ + ẋ +! x = f m cos(!t) x(t) =A cos(!t ) -.6 /! =.5 /! =1 A = f /m p (!! ) +4! -.8-1...5 1. 1.5..5 3. = arctan!!!!/! 动 http://v.youku.com/v_show/id_xmzc5mtuzodq.html
动 统 x(t) =A cos(!t ) A = f /m p (!! ) +4! v(t) =ẋ(t) =!A(!)sin(!t ) = B(!) cos(!t + /) = arctan!!! f = v P = f v = v =! A sin (!t ) ) P = 1 T Z T P (t)dt = 1! A f = f cos(!t) P = fv =!Af cos(!t)sin(!t ) = 1!Af [sin(!t )+sin ] ) P = 1 T Z T = 1! A P (t)dt = 1!Af sin
动 统 1-1 4 6 8 - f = v P = f v = v =! A sin (!t ) ) P = 1 T Z T P (t)dt = 1! A f = f cos(!t) P = fv =!Af cos(!t)sin(!t ) = 1!Af [sin(!t )+sin ] ) P = 1 T Z T = 1! A P (t)dt = 1!Af sin
统 质 - Q 值 统 储 Q = E 质 = E d 1 ka!a = k! = m!! '! f = v P = f v = v =! A sin (!t ) ) P = 1 T Z T P (t)dt = 1! A f = f cos(!t) P = fv =!Af cos(!t)sin(!t ) = 1!Af [sin(!t )+sin ] ) P = 1 T Z T = 1! A P (t)dt = 1!Af sin
动 mẍ = f 1 + f + f(t) 弹 驱动 f 1 = kx f = v = ẋ f(t) =f cos(!t) 驱动 驱动 f(t) =f Tacoma 桥 http://v.youku.com/v_show/id_xmtinda1ntmy.html http://v.youku.com/v_show/id_xmzc5njmyntuy.html
线 dt = ml + mg sin = ) + g l sin + g l = r g ) (t) = cos l t + = g l sin ) s l g ) ) Z t Z t Z t dt = d = d = s d l p ) T =4 cos cos g g l g l g l Z t Z t Z t sin dt sin d sin d ) (t) () = (t) = g l Z (cos cos ) d p cos cos
线 线 运动 ẍ + ẋ +! x = f m cos(!t) 线 弹 驱动 线 运动 mẍ + k ẋ + k 1 x 3 = k 3 cos t 线 弹 驱动 摆 http://v.youku.com/v_show/id_xmzgxnzq5ntqw.html
维 频 动 轴 A x 1 (t) =A 1 cos(!t + 1 ) A x (t) =A cos(!t + ) 1 A 1 实轴 x(t) =x 1 (t)+x (t) =A cos(!t + ) A = A 1 + A +A 1 A cos( 1 ) 1.5 1. A max = A 1 + A A min = A 1 A.5 -.5-1. -1.5 1 3 4 tan = A 1 sin 1 + A sin A 1 cos 1 + A cos
维 频 动 - x 1 (t) =A 1 cos(! 1 t + 1 ) and x (t) =A cos(! t + ), x(t) =x 1 (t)+x (t) A 1 = A and 1 = A 1 A 变换! 1! A 1 6= A and 1 = A 1 A 变换! 1!
维 频 动 y y(t) x(t) x x(t) =A 1 cos(!t + 1 ) y(t) =A cos(!t + ) = 1
维 频 动 - 萨 图 y y(t) x(t) x x(t) =A 1 cos(! 1 t + 1 ) y(t) =A cos(! t + ) = 1
维 频 动 - 萨 图 y y(t) x(t) x!! 1 = p x(t) =A 1 cos(! 1 t + 1 ) y(t) =A cos(! t + ) = 1
维 频 动 - 萨 结 y y(t) z x(t) x x(t) =A 1 cos(n x t + x ) y(t) =A cos(n y t + y ) z(t) =A cos(n z t + z )
业 1. 计 单摆摆动 变为 1/e 给 这 N 单摆 Q 值. 计 测 单摆 Q 值 单摆 给 单摆 摆长 摆 质 长 质 钥 单摆 习题 8.14 5 习题课