第八章：振动

Prepared by Jiang Xiao

无处不在的周期运动和振动

心跳

乐器

听(20-20kHz)、说

风中的旗帜 水波

看、颜色

Prepared by Jiang Xiao

简谐振动

x(t) x0

�

!�(t) = !t+ �0

x(t) = x0 cos(!t+ �0)

相位：

初相位

! =2⇡

T= 2⇡f角频率：

频率

F = �kx ) mx = �kx

) x+k

m

x = 0

垂直悬挂的弹簧呢？

本征频率

x(t) = x0 cos(!0t+ �0) with !0 =

rk

m

Prepared by Jiang Xiao

弹性谐振子的能量

F = �kx ) mx = �kx

) x+k

m

x = 0

垂直悬挂的弹簧呢？

本征频率

x(t) = x0 cos(!0t+ �0) with !0 =

rk

m

Ek =

1

2

mv

2=

1

2

mx

2=

1

2

mx

20!

20 sin

2(!0t+ �0)

Ep =

1

2

kx

2=

1

2

kx

20 cos

2(!0t+ �0)

E = Ek + Ep =

1

2

kx

20 =

1

2

mx

20!

20

动能：

势能：

总机械能：

Prepared by Jiang Xiao

简谐振动

x(t) x0

�

�(t) = !t+ �0

x(t) = x0 cos(!t+ �0)

!

! =2⇡

T= 2⇡f角频率：

相位：

初相位

频率

ma = ml¨✓ = mg sin ✓ ⇡ mg✓

0 =

¨✓ � g

l✓

✓(t) = ✓0 cos (!0t+ �0) with !0 =

rg

l

本征频率

Prepared by Jiang Xiao

复平面

�

实轴

虚轴zi

z = cos�+ i sin� = ei�

�(t) = !t+ �0

x(t) = x0 cos(!t+ �0)

z(t) = x0ei!t+�0

x(t) = Re[z(t)]

ii =?

z = i!z and z = �!2z

二阶齐次线性微分方程

z +k

mz = 0

Prepared by Jiang Xiao

傅立叶变换

x0 =

1

T

Z T/2

�T/2x(t)dt

an =

2

T

Z T/2

�T/2x(t) cos(!nt)dt

bn =

2

T

Z T/2

�T/2x(t) sin(!nt)dt

cn =

pa

2n + b

2n

�n = � arctan

bn

an

任何周期为T的函数：

!n =2n⇡

T

x(t) = x0 +

X

n

[an cos(!nt) + bn sin(!nt)]

= x0 +

X

n

cn cos(!nt+ �n)

完备正交函数

Prepared by Jiang Xiao

双振子和耦合双振子

H2、O2、N2、CO双振子： 振动自由度：1

振动自由度：2

耦合双振子：

振动自由度：2

k1 k2k

x1 x2

Prepared by Jiang Xiao

耦合双振子

m1x1 = �k1x1 � k(x1 � x2)

m2x2 = �k2x2 � k(x2 � x1)

✓k + k1 �m1!2 �k

�k k + k2 �m2!2

◆✓A1

A2

◆= 0

(k + k1 �m1!2)(k + k2 �m2!

2)� k2 = 0

k1 = k2 = k and m1 = m2 ) !1 =k

mand !2 =

3k

m

振动自由度：2

x1,2 = A1,2ei!t

k kk

x1 x2

Prepared by Jiang Xiao

耦合双振子

振动自由度：2

x1,2 = A1,2ei!t

✓2k �m!2 �k

�k 2k �m!2

◆✓A1

A2

◆= 0

k kk

x1 x2

!1 =

rk

m

) A1

A2= 1 )

✓x1(t)x2(t)

◆= Ae

i!1t

✓11

◆

!2 =

r3k

m

) A1

A2= �1 )

✓x1(t)x2(t)

◆= Be

i!2t

✓1�1

◆

Prepared by Jiang Xiao

耦合双振子

k kk

x1 x2m1x1 = �k1x1 � k(x1 � x2)

m2x2 = �k2x2 � k(x2 � x1)

✓x1(t)x2(t)

◆= Ae

i!1t

✓11

◆+Be

i!2t

✓1�1

◆

!1 =k

m

:

✓x1(t)x2(t)

◆= Ae

i!1t

✓11

◆

!2 =3k

m

:

✓x1(t)x2(t)

◆= Be

i!2t

✓1�1

◆

✓x1(0)x2(0)

◆=

✓x10

x20

◆= Re

✓✓A+B

A�B

◆◆

✓x1(0)x2(0)

◆=

✓00

◆= Re

✓i

✓!1A+ !2B

!1A� !2B

◆◆ ) A =x10 + x20

2and B =

x10 � x20

2

Prepared by Jiang Xiao

耦合双振子

k kk

x1 x2

✓x1(t)x2(t)

◆= Ae

i!1t

✓11

◆+Be

i!2t

✓1�1

◆

A =x10 + x20

2and B =

x10 � x20

2

x10 6= x20 = 0

20 40 60 80

-1.0

-0.5

0.5

1.0

k1 = k2 = 10k

2 4 6 8 10 12

-1.0

-0.5

0.5

1.0

x10 = �x20

k1 = k2 = k

2 4 6 8 10 12

-1.0

-0.5

0.5

1.0

x10 = x20

k1 = k2 = k

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/f17324636o1p10.html

Prepared by Jiang Xiao

弹簧的有效质量

x(t) = x0 cos!0t

1

2

kx

2+

1

2

mv

2=

1

2

kx

20

) !0 =

rk

m

l0

m

假设弹簧上各点同步伸长或缩短： dm = ⌘du and v(u) =u

l0v

Eks =

Z l0

0

1

2v(u)2dm =

1

2

m0v2

l30

Z l0

0u2du =

1

2

m0

3v2

m ! m+m0

3) !0 =

sk

m+m0/3

m0 = ⌘l0

!0 ⌧ Vs

l0

Prepared by Jiang Xiao

阻尼运动

vf2

f1m

mx = f1 + f2

f1 = �kx

f2 = ��v = ��x

弹性力：阻尼力：

f2 / v )f2 / v2 )

二阶线性微分方程二阶非线性微分方程

z(t) = Ae

i(!t+�0) with x(t) = Re (z(t))

�!2 + 2i�! + !20 = 0 ) ! = ±

q!20 � �2 + i�

x+ 2�x+ !

20x = 0 with � =

�

2mand !0 =

rk

m

二阶齐次微分方程：

Prepared by Jiang Xiao

阻尼运动

vf2

f1m

mx = f1 + f2

f1 = �kx

f2 = ��v = ��x

弹性力：阻尼力：

� < !0 : x(t) = Ae

��tcos

✓q!

20 � �

2t+ �0

◆

� > !0 : x(t) = Ae

�(��p

�2�!20)t

+Be

�(�+p

�2�!20)t

弱阻尼：

强阻尼：

z(t) = Ae

i(!t+�0) with x(t) = Re (z(t))

�!2 + 2i�! + !20 = 0 ) ! = ±

q!20 � �2 + i�

Prepared by Jiang Xiao

弱阻尼

� < !0 : x(t) = Ae

��tcos

✓q!

20 � �

2t+ �0

◆

T

T =2⇡p

!20 � �2

>2⇡

!0= T0

x(0) = x0 and x(0) = 0

) A = x0 and �0 = 0

初始条件：e��t

!0 =q!20 � �2 = !0

r1� 1

4Q2

Q =!0

2�品质因子：

如何测量？

Prepared by Jiang Xiao

强阻尼

� > !0 : x(t) = Ae

�(��p

�2�!20)t +Be

�(�+p

�2�!20)t

x(0) = x0 and x(0) = 0

) x0 = A+B

0 = ��(A+B) +q

�

2 � !

20(A�B)

初始条件：

Prepared by Jiang Xiao

临界阻尼运动

x+ 2�x+ !

20x = 0 with � =

�

2mand !0 =

rk

m

� = !0 : x(t) = (A+Bt)e��t

x(0) = x0 and x(0) = 0

) A = x0 and B �A� = 0

初始条件：

为什么临界阻尼比过阻尼下降的快？

阻尼运动：http://v.youku.com/v_show/id_XMzc5ODEzMzQw.html

Prepared by Jiang Xiao

受迫振动

v

f2

f1m

f(t)弹性力：

阻尼力：

mx = f1 + f2 + f(t)

f1 = �kx

f2 = ��v = ��x

f(t) = f0 cos(!t)驱动力：

x+ 2�x+ !

20x =

f0

m

cos(!t)

二阶非齐次线性微分方程：

z + 2�z + !20z =

f0m

ei!t通解 = 齐次方程通解 + 非齐次方程特解

特解： z(t) = Aei(!t��)齐次方程通解：

x(t) = Be

��tcos

✓q!

20 � �

2t+ �0

◆

丁丁历险记：http://v.youku.com/v_show/id_XMzgxNTg0OTQ4.html

Prepared by Jiang Xiao

受迫振动

A(!20 � !2

) =

f0m

cos �

2A�! =

f0m

sin �

z + 2�z + !20z =

f0m

ei!t

��!2

+ 2i�! + !20

�A =

f0m

ei� =

f0m

(cos � + i sin �)

特解： z(t) = Aei(!t��)

x+ 2�x+ !

20x =

f0

m

cos(!t)

二阶非齐次线性微分方程：

A =f0/mp

(!20 � !2)2 + 4�2!2

� = arctan2�!

!20 � !2

Prepared by Jiang Xiao

受迫振动

齐次方程通解：

x(t) = Be

��tcos

✓q!

20 � �

2t+ �0

◆

x+ 2�x+ !

20x =

f0

m

cos(!t)

二阶非齐次线性微分方程：

10 20 30 40

-1.0

-0.5

0.5

1.0

10 20 30 40

-1.0

-0.5

0.5

1.0

10 20 30 40

-1.5

-1.0

-0.5

0.5

1.0

1.5

2.0

特解： x(t) = A cos(!t� �)

A =f0/mp

(!20 � !2)2 + 4�2!2

� = arctan2�!

!20 � !2

Prepared by Jiang Xiao

受迫振动 - 共振

特解： x(t) = A cos(!t� �)

A =f0/mp

(!20 � !2)2 + 4�2!2

� = arctan2�!

!20 � !2

!/!0

Af0m!20

0.0 0.5 1.0 1.5 2.0 2.5 3.00

2

4

6

8

10

!max

=q!2

0

� 2�2 and Amax

=f0

2m�p

!2

0

� �2

�/!0 = 0

�/!0 = 0.1

�/!0 = 0.2�/!0 = 0.5

�/!0 = 1

声音震碎玻璃：http://v.youku.com/v_show/id_XMzgxNTg5NjI4.html

x+ 2�x+ !

20x =

f0

m

cos(!t)

二阶非齐次线性微分方程：

Q =!0

2�

品质因子：

Prepared by Jiang Xiao

受迫振动 - 相位

特解： x(t) = A cos(!t� �)

A =f0/mp

(!20 � !2)2 + 4�2!2

� = arctan2�!

!20 � !2

!/!0

0.0 0.5 1.0 1.5 2.0 2.5 3.0-1.0

-0.8

-0.6

-0.4

-0.2

0.0

��

�/!0 = 0.1

�/!0 = 0.2

�/!0 = 0.5�/!0 = 1

�/!0 = 0.01

受迫振动：http://v.youku.com/v_show/id_XMzc5MTUzODQ0.html

x+ 2�x+ !

20x =

f0

m

cos(!t)

二阶非齐次线性微分方程：

Prepared by Jiang Xiao

x(t) = A cos(!t� �)

A =f0/mp

(!20 � !2)2 + 4�2!2

� = arctan2�!

!20 � !2

v(t) = x(t) = �!A(!) sin(!t� �)

= B(!) cos(!t� � + ⇡/2)

受迫振动系统的能量分析

P2 = f2v = ��v2 = ��!2A2 sin2(!t� �)

阻尼力的功率：f2 = ��v

P = fv = �!Af0 cos(!t) sin(!t� �)

= �1

2

!Af0 [sin(2!t� �) + sin �]

外力的功率： f = f0 cos(!t)

) P2 =1

T

Z T

0P2(t)dt = �1

2�!2A2

) P =1

T

Z T

0P (t)dt = �1

2!Af0 sin �

=1

2�!2A2

Prepared by Jiang Xiao

受迫振动系统的能量分析

P2 = f2v = ��v2 = ��!2A2 sin2(!t� �)

阻尼力的功率：f2 = ��v

P = fv = �!Af0 cos(!t) sin(!t� �)

= �1

2

!Af0 [sin(2!t� �) + sin �]

外力的功率： f = f0 cos(!t)

) P2 =1

T

Z T

0P2(t)dt = �1

2�!2A2

) P =1

T

Z T

0P (t)dt = �1

2!Af0 sin �

=1

2�!2A2

20 40 60 80

-2

-1

1

2

Prepared by Jiang Xiao

共振系统的品质因子 - Q值

P2 = f2v = ��v2 = ��!2A2 sin2(!t� �)

阻尼力的功率：f2 = ��v

P = fv = �!Af0 cos(!t) sin(!t� �)

= �1

2

!Af0 [sin(2!t� �) + sin �]

外力的功率： f = f0 cos(!t)

) P2 =1

T

Z T

0P2(t)dt = �1

2�!2A2

) P =1

T

Z T

0P (t)dt = �1

2!Af0 sin �

=1

2�!2A2

Q = 2⇡�E0

�Ed

系统的储能

一个周期的能量耗散

品质因子： = 2⇡12kA

2

⇡�!A2=

k

�!=

m!20

�!' !0

2�

Prepared by Jiang Xiao

自激振动

Tacoma大桥：http://v.youku.com/v_show/id_XMTI0NDA1NTMy.html

弹性力：

阻尼力：

mx = f1 + f2 + f(t)

f1 = �kx

f2 = ��v = ��x

f(t) = f0 cos(!t)驱动力：

周期性驱动力

f(t) = f0恒定驱动力：

相位同步：http://v.youku.com/v_show/id_XMzc5NjMyNTUy.html

Prepared by Jiang Xiao

非线性方程

ml¨✓ +mg sin ✓ = 0 ) ¨✓ +g

lsin ✓ ⇡ ¨✓ +

g

l✓ = 0

) ✓(t) = ✓0 cos

✓rg

lt+ �0

◆

dt =

sl

2g

d✓pcos ✓ � cos ✓0

) T = 4

sl

2g

Z ✓0

0

d✓pcos ✓ � cos ✓0

˙✓¨✓ = �g

lsin ✓ ˙✓ )

Z t

0

˙✓¨✓dt = �g

l

Z t

0sin ✓ ˙✓dt

)Z t

0

˙✓d✓ = �g

l

Z t

0sin ✓ d ✓

)Z t

0

˙✓d ˙✓ = �g

l

Z t

0sin ✓ d ✓

) ˙✓2(t)� ˙✓2(0) = ˙✓2(t) =2g

l(cos ✓ � cos ✓0)

Prepared by Jiang Xiao

混沌Prepared by Jiang Xiao

非线性方程和混沌

mx+ k2x+ k1x3= k3 cos t非线性运动方程：

阻尼 非线性弹簧 周期性驱动力

混沌的摆：http://v.youku.com/v_show/id_XMzgxNzQ5NTQw.html

x+ 2�x+ !

20x =

f0

m

cos(!t)

线性运动方程：

阻尼 线性弹簧 周期性驱动力

Prepared by Jiang Xiao

一维同频振动的合成

x1(t) = A1 cos(!t+ �10)

x2(t) = A2 cos(!t+ �20)

实轴

虚轴

�10

A1�20

A2

A

A2= A2

1 +A22 + 2A1A2 cos(�20 � �10)

x(t) = x1(t) + x2(t) = A cos(!t+ �0)

tan�0 =

A1 sin�10 +A2 sin�20

A1 cos�10 +A2 cos�20

10 20 30 40

-1.5

-1.0

-0.5

0.5

1.0

1.5A

max

= A1

+A2

Amin

= |A1

�A2

|

Prepared by Jiang Xiao

一维差频振动的合成 - 拍

x1(t) = A1 cos(!1t+ �10) and x2(t) = A2 cos(!2t+ �20), x(t) = x1(t) + x2(t)

A1 = A2 and �10 = �20

A1 6= A2 and �10 = �20

傅立叶变换 !1 !2

A1 A2

!1 !2

A1

A2

傅立叶变换

Prepared by Jiang Xiao

二维同频振动的合成

x

y

x(t)

y(t)

x(t) = A1 cos(!t+ �1)

y(t) = A2 cos(!t+ �2)

� = �2 � �1

Prepared by Jiang Xiao

二维非同频振动的合成 - 李萨如图形

x

y

x(t)

y(t)

� = �2 � �1

x(t) = A1 cos(!1t+ �1)

y(t) = A2 cos(!2t+ �2)

Prepared by Jiang Xiao

二维非同频振动的合成 - 李萨如图形

x

y

x(t)

y(t)

� = �2 � �1

x(t) = A1 cos(!1t+ �1)

y(t) = A2 cos(!2t+ �2)

!2

!1=

p2

Prepared by Jiang Xiao

三维非同频振动的合成 - 李萨如结

x

y

x(t)

y(t)

x(t) = A1 cos(nx

t+ �

x

)

y(t) = A2 cos(ny

t+ �

y

)

z(t) = A2 cos(nz

t+ �

z

)

z

Prepared by Jiang Xiao

第八章作业

1. 计算有阻尼单摆摆动多少次之后其振幅变为初始振幅的1/e，给出这个次数N与单摆Q值的关系；

2. 根据上面的计算粗略测量单摆Q值：自制一单摆，给出自制单摆的主要参数：如摆长、摆球种类和质量（如没有尺子或秤可估算其长度和质量），可使用钥匙或类似物体做单摆；

习题：8.14

5月2日周三习题课上交

Prepared by Jiang Xiao