动 Prepared by Jiang Xiao

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1 动

2 处运动动 (-khz) 说颜乐风帜

3 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

4 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

5 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

6 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

7 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

8 H O N CO 动 1 k1 k k x1 x 动动

9 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

10 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

11 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 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

12 k k k x1 (t) x (t) = Ae i! 1t 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 = coupled pendulum:

13 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

14 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

15 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

16 <! : 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

17 >! : x(t) =Ae ( p! )t + Be ( +p! )t x() = x and ẋ() = ) x = A + B q = (A + B)+! (A B)

18 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

19 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

20 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!!!

21 动阶齐线 ẍ + ẋ +! x = f m cos(!t) x(t) =A cos(!t ) 1..5 A = f /m p (!! ) +4! = arctan!!! 齐 x(t) =Be t cos q! t + -1.

22 动 - Af m! 1 阶齐线 /! =.1 /! =.5 /! = /! =. 质 Q =! 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!

23 动 /! =.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! = arctan!!!!/! 动

24 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

25 动统 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

26 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

t) 驱动驱动 f(t) =f Tacoma 桥 http://v.youku.

27 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

28 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

30 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

31 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 ) A max = A 1 + A A min = A 1 A tan = A 1 sin 1 + A sin A 1 cos 1 + A cos

32 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!

33 维频动 y y(t) x(t) x x(t) =A 1 cos(!t + 1 ) y(t) =A cos(!t + ) = 1

34 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

35 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

36 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 )

37 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习题课

运动 Prepared by Jiang Xiao

顿运动 Prepared by Jiang Xiao 运动 Prepared by Jiang Xiao 运动运动运动 Aristotle 384BC - 322BC 变运动维运动 Galileo 1564-1642 Newton 1642-1727 线运动态变这态顿 - 惯顿线运动态变这态满惯满满转 0.034 m/s 2 转 0.006 m/s 2 银转动