Chapter 12 Oscillations
Chapter 12 Oscillations
1,Concepts of Simple Harmonic Motion
2,Expression Methods of SHM
3,Energy in SHM
4,Pendulums
5,Superposition of Oscillations
Chapter 12 Oscillations
Key words:
Oscillation or vibration 振动
Periodic 周期性的
Equilibrium position 平衡位置
Simple harmonic motion (SHM) 简谐振动
Amplitude 振幅
Physical Pendulum 复摆、物理摆
Phase 相位、位相、周相
Angular frequency 角频率
Linear simple harmonic oscillator 线性谐振子
Forced oscillations 受迫振动
Resonance 共振
Chapter 12 Oscillations
Oscillations,motions that repeat themselves,
§ 12-1 Oscillations (振动 ) P297
广义振动：一个物理量在某一定值附近往复变化，
该物理量的运动形式称振动。
振动有各种不同的形式机械振动,位移 x随时间 t的往复变化弹簧振子
X
Chapter 12 Oscillations
电磁振动,Electrical oscillation：
电场、磁场等电磁量随 t的往复变化
+ +
- -
振荡电路
Chapter 12 Oscillations
共振
Resonance
(简谐振动 ） Simple
Harmonic Motion
振动受迫振动
Forced
oscillations
自由振动阻尼自由振动
Damped
无阻尼自由振动无阻尼自由谐振动振动的形式,
Chapter 12 Oscillations
Any Vibrating System for which the net restoring force ( 回复力 )is directly proportional to the negative of the
displacement is said to exhibit simple harmonic motion.
The displacement varies sinusoidally with time,This type of
motion is referred to as Simple Harmonic Motion
(SHM),p299
§ 12-2 Simple Harmonic Motion (SHM) (简谐振动） P299-304
物体运动时，如果离开平衡位置的位移（或角位移）按余弦函数（或正弦函数）的规律随时间变化，
这种运动叫 简谐运动 。
tAx c o s
x是描述位置的物理量,如 y,z 或? 等,
Chapter 12 Oscillations
x
to 2T T 23T 2T
合振动分振动 1 分振动 2
The simple harmonic
motion is the most simple
and basic vibration,Every
complicated vibration can
be considered as composing
of several SHM.
简谐运动 复杂振动合成分解物理上：简谐运动是 最简单、最基本的振动。 一般运动是多个简谐运动的合成。
Chapter 12 Oscillations
kl0
x
m
oA? A
00 Fx
x
x
F? m
o
简谐运动可以用一个 弹簧振子 (spring)
弹簧振子,一个轻质弹簧的一端固定，另一端固结一个可以自由运动的物体，就构成一个 弹簧振子。
Chapter 12 Oscillations
Object,block-spring system;
Origin,equilibrium position
Restoring force,F
1 Motion Function of SHM (P299-300):
一个作简谐运动的质点所受的沿位移方向的合外力与它对于平衡位置的位移成正比而反向，
这样的力称为 回复力 。
Chapter 12 Oscillations
makxF
m
k?2?
let
xa 2
x
x
F? m
o
From Hooke’s Law and N-II law,
0
d
d 2
2
2
x
t
x?
Equation of motion for the
simple harmonic oscillator.
简谐运动的动力学方程
Mathematically it is called a differential equation,
Chapter 12 Oscillations
The solution of this derivation equation is:
—— Oscillation Equation
Feature of Oscillatory Motion:
Dynamics,xaorkxf 2
Kinematics,Motion function is sine or cosine form.
积分常数，根据初始条件确定
)c o s( tAx
Chapter 12 Oscillations
Problem11 on page 317
Chapter 12 Oscillations
Problem17 on page 318
Chapter 12 Oscillations
2 Basic Quantities of SHM,(P301)
Where xm (A),?,
and? are
constants,They
are basic
characteristic
quantities of a
SHM (描述 简谐振动的特征量 ),
Chapter 12 Oscillations
i,Amplitude (振幅 ) xm or A (A > 0),p298
(1) xm is called the amplitude（ 振幅 ） of the motion,it
is a positive constant because the amplitude is the
magnitude of the maximum displacement of the particle
in either direction.
表示质点可能离开原点的最大位移 。
由初始条件决定,表征了系统的能量 。 (A恒?0)
m a xxA?
tx? 图
A
A?
x
T
2
T
t
o
Chapter 12 Oscillations
ii,Period(周期 )T,Frequency(频率 )f and
Angular Frequency (角频率 或圆频率 )?,
周期、频率
π2?T周期：
)c os ( tAx
)co s (
])(co s [
TtA
TtA
tx? 图
A
A?
x
T
2
T
t
o
振动往复一次所需时间 。 The time required for one
complete cycle,Cycle refers to the complete to-and-fro
motion from some initial point back to that same point.
Chapter 12 Oscillations
2
1
Tf
f 2?
频率单位时间内振动的次数。单位,Hz
Where frequency (f ),or number of oscillations
that are completed each second,Its SI unit is
the hertz (Hz)
1 hertz = 1 Hz = 1 oscillation per second = 1 s-1
Is called angular frequency（角频率） of the
motion.
角频率
2?秒内的振 动次数 。 （ 单位,1/S或 rad/S)
Chapter 12 Oscillations
m
k
k
mT?
22
m
k
Tf?2
11
T and f is also called intrinsic period or
frequency (固有周期和固有频率,) ensure by
system property.
They don’t depend on the amplitude.
,,T
都表示简谐运动的周期性,反映振动的快慢 。
Chapter 12 Oscillations
iii,Phase or Phase Angle & Initial ~(相位和初相 p301)
For a certain oscillatory object,A and? are
fixed,At any time,its motion state ( x & v) will
be determined by (?t +?),So (?t +?) is called
phase.
t)c os ( tAx在 中,称为振动的相位。
1）,存在一一对应的关系 ;即其决定质点在时刻的 t的 位置。xt
π2~02） 相位在 内变化，质点 无相同 的运动状态；
) (π2 nn相差 为整数 质点运动状态 全同,（ 周期性）
Chapter 12 Oscillations
(?o is the phase at t = 0 —– initial phase)
3）初 相位 描述质点 初始 时刻的运动状态,)0(?t?
π]20[π]π[( 取 或 )
Chapter 12 Oscillations
*iv,Phase difference (相位差 )
相位差：表示两个相位之差。用来比较简谐调运动的步调。
Phase difference of two SHM with same frequency:
对于两个 同 频率 的简谐运动，相位差表示它们间步调 上的 差异,（解决振动合成问题）
)()( 12 tt 12
)c o s ( 111 tAx
)c o s ( 222 tAx
At any time,their phase difference:
Chapter 12 Oscillations
同相和反相两振动步调相同，称 同相 。
当=?(2k+1)?,( k= 0,1,2,… )，
两振动步调相反,称 反相 。 opposite (out of) phase
x2
T
x
o
A1
-A1
A2
- A2
x1
t
反相
x
o
A1
-A1
A2
- A2
x1x
2
T
t
同相
(a) 两同相振动的振动曲线 (b) 两反相振动的振动曲线
= 2k?,(k=0,?1,?2,?3…),two oscillators
have same phase (in phase);
Chapter 12 Oscillations
,)co s ()()( txdtddt tdxtv m
)2c o s (s i n txtxtv mm
The velocity of oscillating particle is also SHM and varies
between the limits ± vm=±? xm,v(t) is ahead of x(t) for?/2.
3 The Velocity of SHM (p302)
简谐运动的速度函数式
)s i n ( tAv
Chapter 12 Oscillations
4 The Acceleration of SHM,(p302)
dt
tdvta?)(.c o s2 txta
m
The acceleration of the oscillating particle is also
SHM varies between the limits ± am = ±?2 xm.
简谐运动的加速度函数式
)c o s (
)c o s (
2
2
2
2
tA
tA
dt
xd
a
Chapter 12 Oscillations
tx? 图
t?v 图
ta? 图
T
A
A?
2?A
2?A?
x
v
a
t
t
t
A
A?
o
o
o T
T
)c os ( tAx
)
2
πc o s ( tA
)πc o s (2 tAa
振动曲线
Chapter 12 Oscillations
x,?,a
o T t
x
2A
> 0 < 0 < 0 > 0a < 0
< 0 > 0 > 0
减速 加速 减速 加速
AA
-A-?A
-?2A
a
简谐振动的位移,速度和加速度曲线
Chapter 12 Oscillations
讨论
)c os ( tAx
)πc o s (2 tAa
比较：
xtA
dt
xda 22
2
2
)c o s (
简谐运动的加速度和位移成正比而反向。
Chapter 12 Oscillations
2
2
02
0?
v xA
0
0t an
x?
v
常数 和 的确定A?
000 vv xxt
初始条件
co s0 Ax?
s i n0 Av
对于给定的振动系统，周期由系统本身性质决定，
振幅和初相由初始条件决定,
)s i n ( tAv
)c os ( tAx
Example 12-5 on
page 304
Chapter 12 Oscillations
Example 12-3 and 12-4 on page 302 and 303
Chapter 12 Oscillations
5 Expression Methods of SHM
from x=Acos(? t+? )
Given expression? A,T,?
Given A,T, expression
a,Analytical Method:
b,Curve Method:
tx? 图
A
A?
x
T
2
T
t
o
Given curve? A,T,?
Given A,T, curve
Chapter 12 Oscillations
co s0
)co s (
A
tAx
2
π
0s i n0Av?
2
π 0s i n 取
0,0,0 vxt
已知 求 。
x
v?
o
)
2
π c o s ( tAx?
A
A?
x
T
2
T
t
o
Chapter 12 Oscillations
12-4,SHM Related to Uniform Circular Motion (旋转矢量法 ),(P306)
t
用匀速圆周运动表示简谐运动的 位置变化 。
x
o
A?
规定
AA
x
质点在 x轴上的投影式 )c os ( tAx
t
设质点的径矢经过与
x轴夹角为 的位置
t开始计时，则在时刻 t此径矢与 x轴的夹角为设一质点沿圆心在 O点而半径 A的圆周作匀速运动，
其角速度为 。
Chapter 12 Oscillations
)c o s( tAx
旋转矢量 的端点在轴上的投影点的运动为简谐运动,
x
A?
Chapter 12 Oscillations
其与简谐运动的定义公式相同 。
所以,做匀速圆周运动的质点在某一直径上 （ x
轴 ） 的投影的运动就是简谐运动 。
Simple harmonic motion is the projection of
uniform circular motion onto a straight line.
圆周运动的角速度就等于振动的角频率,圆周的半径就等于振动的振幅,初始时刻作圆周运动的质点的径矢与 x轴的夹角就是振动的初位相 。
Chapter 12 Oscillations
x
A
t+?
Relation between the two ways of expressions:
t
x
o
Curve method;
Uniform Circular
Motion (Rotating
Vector) method.
Chapter 12 Oscillations
（旋转矢量旋转一周所需的时间）
π2?T
用旋转矢量图画简谐运动的 图tx?
Chapter 12 Oscillations
用匀速圆周运动表示简谐运动的 速度变化 。
A?mv
做匀速圆周运动的质点的速率是：
在时刻 t,它在 x轴上的投影是
)
2
π
co s (
)s i n (
tA
tmvv
这就是简谐运动的速度公式。
2
πt
mv?
v?
x
y
0
At
)c os ( tAx
Chapter 12 Oscillations
2
n?Aa?
在时刻 t,它在 x轴上的投影是
)c os (
)c os (
2
tA
tnaa
这就是简谐运动的加速度公式。
*用匀速圆周运动表示简谐运动的 加速度变化 。
x
y
0
A?
t
)c os ( tAx
na
a?
Chapter 12 Oscillations
A body (m=10g) is in SHM along x axis,A=20cm,
T=4s,When t = 0,xo= -10cm,and v < 0 (moves
along negative x direction),Find,(1) x(t=1s)=?; (2)
At what time will the body pass the position of
x=10cm first time (何时物体第一次通过 ） (3) How long
does the body need to pass x=10cm second time(再经多少时间物体第二次运动到 x=10cm处 ）?
24
22
T
Chapter 12 Oscillations
mx 173.0)322c o s (2.0
or as the Fig.,mx 1 7 3.0c o s2.0
(2)
sttt o 2
2
(1) When t=1s,
st
3
4
2
3
2
(3)
2A?
0
X
O
6
32
2A
mtx )322c o s (2.0
As the Fig,
3
2
0
Solution:
Chapter 12 Oscillations
物理模型与数学模型比较
A
简谐振动 旋转矢量
t+?
T
振幅初相相位圆频率谐振动周期半径初始角坐标角坐标角速度圆周运动周期
Chapter 12 Oscillations
1 Potential Energy
)(c o s
2
1
2
1)( 222 φ tkxkxtU
m?
12-3 Energy in Simple Harmonic Motion (P304-305)
)s i n (
)co s (
tA
tAx
v
kxF
)(c o s
2
1 22 tkA
Chapter 12 Oscillations
)(s i n
2
1
)(s i n
2
1
2
1
22
2222
tkA
tAmmK v
2 Kinetic Energy:
mk /2
Chapter 12 Oscillations
2
2
1 kAUKE
弹簧振子的总的机械能
3 Total mechanical energy
作 简谐 运动的系统 机械能守恒,线性回复力是 保守力。
弹簧振子的总能量不随时间改变，即作 简谐 运动的系统 机械能守恒。
Total energy of oscillatory system is a constant.
Total mechanical energy of a
simple harmonic oscillator is
proportional to the square of the
amplitude.
Chapter 12 Oscillations
2A?
振幅的动力学意义,振幅不仅给出了简谐运动的运动范围，而且还反映了 振动系统总能量的大小,或者说反映了振动的 强度 。
讨论
2
2
1 kAUKE
Chapter 12 Oscillations
简 谐 运 动 能 量 图
tx?
t?v
2
2
1 kAE?
0
tAx?co s?
tA s i nv
v,x
to
T
4
T
2
T
4
3T
能量
o T t
tkAE?22p c o s21?
tAmE 222k s i n21?
Ek 最大时,Ep最小,Ek,Ep交替变化。
Chapter 12 Oscillations
Graph of potential
Chapter 12 Oscillations
例 质量为 的物体，以振幅作简谐运动，其最大加速度为,求,kg10.0
m100.1 2
2sm0.4
（ 1） 振动的周期；
（ 2） 通过平衡位置的动能；
（ 3） 总能量；
（ 4） 物体在何处其动能和势能相等？
解 （ 1）
2
m a x?Aa?
A
a m a x 1s20
s314.0π2T
Chapter 12 Oscillations
（ 2） J100.2 3
222
m a xm a x,k 2
1
2
1 AmmE v
（ 3）
m a x,kEE?
J100.2 3
（ 4）
pk EE?
时，
J100.1 3pE
由
222
p 2
1
2
1 xmkxE
2
p2 2
m
E
x?
24 m105.0
cm7 0 7.0x
Chapter 12 Oscillations
Example 12-6; p305
Example 12-7 P306
Chapter 12 Oscillations
小结：简谐运动的描述和特征
xa 24） 加速度与位移成正比而方向相反
xt x 22
2
d
d2） 简谐运动的动力学描述
)s i n ( tAv
)c o s( tAx3） 简谐运动的运动学描述
mk弹簧振子
kxF1） 物体受线性回复力作用 平衡位置 0?x
Chapter 12 Oscillations
§ 12-5 The Simple Pendulum单摆 (P307-308)
l
m
o
A
TF
P?
转动正向The cord doesn’t stretch
and that its mass can be
ignored.下端系一可看作质点的重物 (a small object called
pendulum bob)就构成一个单摆 。
mgmgF t s i n
s i n,5?时
The restoring force is the
component of the weight,
mg,tangent to the arc.
Chapter 12 Oscillations
l
m
o
A
TF
P?
转动正向摆球的切向加速度为：
2
2
dt
dla?
mg
t
ml2
2
d
d
由牛顿第二定律
0
d
d
2
2
l
g
t
02
2
x
m
k
t
x
d
d
具有 的形式
Chapter 12 Oscillations
在角位移很小的情况下，单摆的振动是简谐运动。
)c o s (m t
l
g?2?令
glT π2?
l
g
Period does not depend on the mass
of the pendulum bob
Chapter 12 Oscillations
*§ 12-5 The Physical Pendulum (复摆 ):(P308-309)
The torque of physical pendulum to axis O,
s i nm g h
s in m g h
m g h
IT?
22
I
m g h
t
2
2
d
d
0dd 22
2
xt x?
I?
2
Based on rotational law,
,-‖ indicates that the acts to reduce
When is small,?
Chapter 12 Oscillations
Solution:
Two springs of k1 and k2 oscillates on a
frictionless ramp,(a) Proof its motion is SHM;
(b) Find its frequency of oscillation.
2211s i n xkxkmg
)'(s i n 111 xxkmgF )'(s i n 222 xxkmg
When the block at ― O‖,
When the block at ― P‖,
k2
x
o
x k1
P
2211 '' xkxkF (2)
21 '' xxx (1)
xkk kkF )(
21
21
—— It is a SHMFrom (1) & (2),
(a)
mkk
kkf
)(2
1
2 21
21
(b)
U=0
`
Chapter 12 Oscillations
*12-7 Damped (阻尼 ) Harmonic Motion(P310-312)
When the motion of an oscillator is reduced by
an external force,the oscillator and its motion
are said to be damped.
bvF d
,-‖,opposes the
motion.d
F?
The component of damping force 阻尼力
along x is,dF
b,damping constant (阻尼因子 )
Chapter 12 Oscillations
)'c o s (2/ texx mbtm
mabkx v
0
d
d
d
d
2
2
kx
t
xb
t
xm
Chapter 12 Oscillations
2
2
4
'
m
b
m
k
2
2
42
1
2
'
m
b
m
kf
The frequency is lower,and the period longer than
for undamped SHM.Three common cases of
heavily damped system are shown in Fig.
Chapter 12 Oscillations
Curve A represents
underdamped situation.
Curve B represents
critical damping
Curve C represents
overdamped
o
t
x
三种阻尼的比较
220b）过阻尼
220a）欠阻尼
220c）临界阻尼
a
b
c
Chapter 12 Oscillations
*12-8 Forced Oscillations; Resonance共振 p313-315
tFF?c o s0e x t?
tFkxtxb
t
xm?c o s
d
d
d
d
02
2
)c o s(0 tAx d
)c o s ()c o s ( 002/0 tAteAx dmtb
Forced (driven) force 驱动力,
The equation of motion with damping is
The first term approaches zero in time,we need to
be concerned only with
2/1222222
0
0
]/)[( mbm
FA
dd
)/(
t a n
22
01
mbd
d
Chapter 12 Oscillations
The amplitude can become large when the driving
frequency?d is near natural frequency,as
long as the damping is not too large,0d
It is approximately the
condition at which the
displacement amplitude of
oscillations is greatest.
0d
(resonant)
2/1222222
0
0
]/)[( mbm
FA
dd
Chapter 12 Oscillations
共振演示实验
2
3 6
1
4
5
22
0r 2
共振频率
22
0
r
2
fA
共振振幅共振现象在实际中的应用乐器、收音机 ……
单摆 1作垂直于纸面的简谐运动时，单摆 5将作相同周期的简谐运动，
其它单摆基本不动,
Chapter 12 Oscillations
共振现象的危害
1940 年 7月 1日美国 Tocama 悬索桥因共振而坍塌
Chapter 12 Oscillations
补充,Superposition of Oscillations
Superposition of Two SHM in Same Direction With
same frequencies (同方向、同频率简谐运动的合成 ):
x1=A1cos(? t+? 1)
x2=A2cos(? t+? 2)
The resultant oscillation is also SHM,and its frequency
is also?.
x = x1+ x2 x =A cos(? t+?)That is,
)co s (2 12212221 AAAAA
2211
2211
c o sc o s
s ins intg
AA
AA
where
1?2
1x?
2x? x?
x
Chapter 12 Oscillations
The resultant oscillation is related
to the phase difference?2-?1,
),2,1,0(,212 kk If
Two special cases:
A=A1+A2
maximum oscillatory amplitude
(两分振动同相,相互加强 ).
1?2
1x?
2x? x?
x
x
to 2T T 23T 2T
合成振动
)co s (2 12212221 AAAAA
212221 2 AAAA
)co s (2 12212221 AAAAA
Chapter 12 Oscillations
If,..)2,1,0()12(
12 kk
A=|A1-A2|
—–minimum oscillation
amplitude (两分振动反相,相互减弱 )
In general,)( 2121 AAAAA
t
o 2T T 23T 2T
x
2x
1x
合成振动
If A1=A2,then A=0 !
The above results shows us that phase difference
of two SHM plays an important role.
Chapter 12 Oscillations
Summary for Chapter Twelve
See P315 to P316
Chapter 12 Oscillations
Homework:
振动基本量和方程,page 317,4,7; Page
318:20
能量,Page 319,26,29,31
Chapter 12 Oscillations
1,Concepts of Simple Harmonic Motion
2,Expression Methods of SHM
3,Energy in SHM
4,Pendulums
5,Superposition of Oscillations
Chapter 12 Oscillations
Key words:
Oscillation or vibration 振动
Periodic 周期性的
Equilibrium position 平衡位置
Simple harmonic motion (SHM) 简谐振动
Amplitude 振幅
Physical Pendulum 复摆、物理摆
Phase 相位、位相、周相
Angular frequency 角频率
Linear simple harmonic oscillator 线性谐振子
Forced oscillations 受迫振动
Resonance 共振
Chapter 12 Oscillations
Oscillations,motions that repeat themselves,
§ 12-1 Oscillations (振动 ) P297
广义振动：一个物理量在某一定值附近往复变化，
该物理量的运动形式称振动。
振动有各种不同的形式机械振动,位移 x随时间 t的往复变化弹簧振子
X
Chapter 12 Oscillations
电磁振动,Electrical oscillation：
电场、磁场等电磁量随 t的往复变化
+ +
- -
振荡电路
Chapter 12 Oscillations
共振
Resonance
(简谐振动 ） Simple
Harmonic Motion
振动受迫振动
Forced
oscillations
自由振动阻尼自由振动
Damped
无阻尼自由振动无阻尼自由谐振动振动的形式,
Chapter 12 Oscillations
Any Vibrating System for which the net restoring force ( 回复力 )is directly proportional to the negative of the
displacement is said to exhibit simple harmonic motion.
The displacement varies sinusoidally with time,This type of
motion is referred to as Simple Harmonic Motion
(SHM),p299
§ 12-2 Simple Harmonic Motion (SHM) (简谐振动） P299-304
物体运动时，如果离开平衡位置的位移（或角位移）按余弦函数（或正弦函数）的规律随时间变化，
这种运动叫 简谐运动 。
tAx c o s
x是描述位置的物理量,如 y,z 或? 等,
Chapter 12 Oscillations
x
to 2T T 23T 2T
合振动分振动 1 分振动 2
The simple harmonic
motion is the most simple
and basic vibration,Every
complicated vibration can
be considered as composing
of several SHM.
简谐运动 复杂振动合成分解物理上：简谐运动是 最简单、最基本的振动。 一般运动是多个简谐运动的合成。
Chapter 12 Oscillations
kl0
x
m
oA? A
00 Fx
x
x
F? m
o
简谐运动可以用一个 弹簧振子 (spring)
弹簧振子,一个轻质弹簧的一端固定，另一端固结一个可以自由运动的物体，就构成一个 弹簧振子。
Chapter 12 Oscillations
Object,block-spring system;
Origin,equilibrium position
Restoring force,F
1 Motion Function of SHM (P299-300):
一个作简谐运动的质点所受的沿位移方向的合外力与它对于平衡位置的位移成正比而反向，
这样的力称为 回复力 。
Chapter 12 Oscillations
makxF
m
k?2?
let
xa 2
x
x
F? m
o
From Hooke’s Law and N-II law,
0
d
d 2
2
2
x
t
x?
Equation of motion for the
simple harmonic oscillator.
简谐运动的动力学方程
Mathematically it is called a differential equation,
Chapter 12 Oscillations
The solution of this derivation equation is:
—— Oscillation Equation
Feature of Oscillatory Motion:
Dynamics,xaorkxf 2
Kinematics,Motion function is sine or cosine form.
积分常数，根据初始条件确定
)c o s( tAx
Chapter 12 Oscillations
Problem11 on page 317
Chapter 12 Oscillations
Problem17 on page 318
Chapter 12 Oscillations
2 Basic Quantities of SHM,(P301)
Where xm (A),?,
and? are
constants,They
are basic
characteristic
quantities of a
SHM (描述 简谐振动的特征量 ),
Chapter 12 Oscillations
i,Amplitude (振幅 ) xm or A (A > 0),p298
(1) xm is called the amplitude（ 振幅 ） of the motion,it
is a positive constant because the amplitude is the
magnitude of the maximum displacement of the particle
in either direction.
表示质点可能离开原点的最大位移 。
由初始条件决定,表征了系统的能量 。 (A恒?0)
m a xxA?
tx? 图
A
A?
x
T
2
T
t
o
Chapter 12 Oscillations
ii,Period(周期 )T,Frequency(频率 )f and
Angular Frequency (角频率 或圆频率 )?,
周期、频率
π2?T周期：
)c os ( tAx
)co s (
])(co s [
TtA
TtA
tx? 图
A
A?
x
T
2
T
t
o
振动往复一次所需时间 。 The time required for one
complete cycle,Cycle refers to the complete to-and-fro
motion from some initial point back to that same point.
Chapter 12 Oscillations
2
1
Tf
f 2?
频率单位时间内振动的次数。单位,Hz
Where frequency (f ),or number of oscillations
that are completed each second,Its SI unit is
the hertz (Hz)
1 hertz = 1 Hz = 1 oscillation per second = 1 s-1
Is called angular frequency（角频率） of the
motion.
角频率
2?秒内的振 动次数 。 （ 单位,1/S或 rad/S)
Chapter 12 Oscillations
m
k
k
mT?
22
m
k
Tf?2
11
T and f is also called intrinsic period or
frequency (固有周期和固有频率,) ensure by
system property.
They don’t depend on the amplitude.
,,T
都表示简谐运动的周期性,反映振动的快慢 。
Chapter 12 Oscillations
iii,Phase or Phase Angle & Initial ~(相位和初相 p301)
For a certain oscillatory object,A and? are
fixed,At any time,its motion state ( x & v) will
be determined by (?t +?),So (?t +?) is called
phase.
t)c os ( tAx在 中,称为振动的相位。
1）,存在一一对应的关系 ;即其决定质点在时刻的 t的 位置。xt
π2~02） 相位在 内变化，质点 无相同 的运动状态；
) (π2 nn相差 为整数 质点运动状态 全同,（ 周期性）
Chapter 12 Oscillations
(?o is the phase at t = 0 —– initial phase)
3）初 相位 描述质点 初始 时刻的运动状态,)0(?t?
π]20[π]π[( 取 或 )
Chapter 12 Oscillations
*iv,Phase difference (相位差 )
相位差：表示两个相位之差。用来比较简谐调运动的步调。
Phase difference of two SHM with same frequency:
对于两个 同 频率 的简谐运动，相位差表示它们间步调 上的 差异,（解决振动合成问题）
)()( 12 tt 12
)c o s ( 111 tAx
)c o s ( 222 tAx
At any time,their phase difference:
Chapter 12 Oscillations
同相和反相两振动步调相同，称 同相 。
当=?(2k+1)?,( k= 0,1,2,… )，
两振动步调相反,称 反相 。 opposite (out of) phase
x2
T
x
o
A1
-A1
A2
- A2
x1
t
反相
x
o
A1
-A1
A2
- A2
x1x
2
T
t
同相
(a) 两同相振动的振动曲线 (b) 两反相振动的振动曲线
= 2k?,(k=0,?1,?2,?3…),two oscillators
have same phase (in phase);
Chapter 12 Oscillations
,)co s ()()( txdtddt tdxtv m
)2c o s (s i n txtxtv mm
The velocity of oscillating particle is also SHM and varies
between the limits ± vm=±? xm,v(t) is ahead of x(t) for?/2.
3 The Velocity of SHM (p302)
简谐运动的速度函数式
)s i n ( tAv
Chapter 12 Oscillations
4 The Acceleration of SHM,(p302)
dt
tdvta?)(.c o s2 txta
m
The acceleration of the oscillating particle is also
SHM varies between the limits ± am = ±?2 xm.
简谐运动的加速度函数式
)c o s (
)c o s (
2
2
2
2
tA
tA
dt
xd
a
Chapter 12 Oscillations
tx? 图
t?v 图
ta? 图
T
A
A?
2?A
2?A?
x
v
a
t
t
t
A
A?
o
o
o T
T
)c os ( tAx
)
2
πc o s ( tA
)πc o s (2 tAa
振动曲线
Chapter 12 Oscillations
x,?,a
o T t
x
2A
> 0 < 0 < 0 > 0a < 0
< 0 > 0 > 0
减速 加速 减速 加速
AA
-A-?A
-?2A
a
简谐振动的位移,速度和加速度曲线
Chapter 12 Oscillations
讨论
)c os ( tAx
)πc o s (2 tAa
比较：
xtA
dt
xda 22
2
2
)c o s (
简谐运动的加速度和位移成正比而反向。
Chapter 12 Oscillations
2
2
02
0?
v xA
0
0t an
x?
v
常数 和 的确定A?
000 vv xxt
初始条件
co s0 Ax?
s i n0 Av
对于给定的振动系统，周期由系统本身性质决定，
振幅和初相由初始条件决定,
)s i n ( tAv
)c os ( tAx
Example 12-5 on
page 304
Chapter 12 Oscillations
Example 12-3 and 12-4 on page 302 and 303
Chapter 12 Oscillations
5 Expression Methods of SHM
from x=Acos(? t+? )
Given expression? A,T,?
Given A,T, expression
a,Analytical Method:
b,Curve Method:
tx? 图
A
A?
x
T
2
T
t
o
Given curve? A,T,?
Given A,T, curve
Chapter 12 Oscillations
co s0
)co s (
A
tAx
2
π
0s i n0Av?
2
π 0s i n 取
0,0,0 vxt
已知 求 。
x
v?
o
)
2
π c o s ( tAx?
A
A?
x
T
2
T
t
o
Chapter 12 Oscillations
12-4,SHM Related to Uniform Circular Motion (旋转矢量法 ),(P306)
t
用匀速圆周运动表示简谐运动的 位置变化 。
x
o
A?
规定
AA
x
质点在 x轴上的投影式 )c os ( tAx
t
设质点的径矢经过与
x轴夹角为 的位置
t开始计时，则在时刻 t此径矢与 x轴的夹角为设一质点沿圆心在 O点而半径 A的圆周作匀速运动，
其角速度为 。
Chapter 12 Oscillations
)c o s( tAx
旋转矢量 的端点在轴上的投影点的运动为简谐运动,
x
A?
Chapter 12 Oscillations
其与简谐运动的定义公式相同 。
所以,做匀速圆周运动的质点在某一直径上 （ x
轴 ） 的投影的运动就是简谐运动 。
Simple harmonic motion is the projection of
uniform circular motion onto a straight line.
圆周运动的角速度就等于振动的角频率,圆周的半径就等于振动的振幅,初始时刻作圆周运动的质点的径矢与 x轴的夹角就是振动的初位相 。
Chapter 12 Oscillations
x
A
t+?
Relation between the two ways of expressions:
t
x
o
Curve method;
Uniform Circular
Motion (Rotating
Vector) method.
Chapter 12 Oscillations
（旋转矢量旋转一周所需的时间）
π2?T
用旋转矢量图画简谐运动的 图tx?
Chapter 12 Oscillations
用匀速圆周运动表示简谐运动的 速度变化 。
A?mv
做匀速圆周运动的质点的速率是：
在时刻 t,它在 x轴上的投影是
)
2
π
co s (
)s i n (
tA
tmvv
这就是简谐运动的速度公式。
2
πt
mv?
v?
x
y
0
At
)c os ( tAx
Chapter 12 Oscillations
2
n?Aa?
在时刻 t,它在 x轴上的投影是
)c os (
)c os (
2
tA
tnaa
这就是简谐运动的加速度公式。
*用匀速圆周运动表示简谐运动的 加速度变化 。
x
y
0
A?
t
)c os ( tAx
na
a?
Chapter 12 Oscillations
A body (m=10g) is in SHM along x axis,A=20cm,
T=4s,When t = 0,xo= -10cm,and v < 0 (moves
along negative x direction),Find,(1) x(t=1s)=?; (2)
At what time will the body pass the position of
x=10cm first time (何时物体第一次通过 ） (3) How long
does the body need to pass x=10cm second time(再经多少时间物体第二次运动到 x=10cm处 ）?
24
22
T
Chapter 12 Oscillations
mx 173.0)322c o s (2.0
or as the Fig.,mx 1 7 3.0c o s2.0
(2)
sttt o 2
2
(1) When t=1s,
st
3
4
2
3
2
(3)
2A?
0
X
O
6
32
2A
mtx )322c o s (2.0
As the Fig,
3
2
0
Solution:
Chapter 12 Oscillations
物理模型与数学模型比较
A
简谐振动 旋转矢量
t+?
T
振幅初相相位圆频率谐振动周期半径初始角坐标角坐标角速度圆周运动周期
Chapter 12 Oscillations
1 Potential Energy
)(c o s
2
1
2
1)( 222 φ tkxkxtU
m?
12-3 Energy in Simple Harmonic Motion (P304-305)
)s i n (
)co s (
tA
tAx
v
kxF
)(c o s
2
1 22 tkA
Chapter 12 Oscillations
)(s i n
2
1
)(s i n
2
1
2
1
22
2222
tkA
tAmmK v
2 Kinetic Energy:
mk /2
Chapter 12 Oscillations
2
2
1 kAUKE
弹簧振子的总的机械能
3 Total mechanical energy
作 简谐 运动的系统 机械能守恒,线性回复力是 保守力。
弹簧振子的总能量不随时间改变，即作 简谐 运动的系统 机械能守恒。
Total energy of oscillatory system is a constant.
Total mechanical energy of a
simple harmonic oscillator is
proportional to the square of the
amplitude.
Chapter 12 Oscillations
2A?
振幅的动力学意义,振幅不仅给出了简谐运动的运动范围，而且还反映了 振动系统总能量的大小,或者说反映了振动的 强度 。
讨论
2
2
1 kAUKE
Chapter 12 Oscillations
简 谐 运 动 能 量 图
tx?
t?v
2
2
1 kAE?
0
tAx?co s?
tA s i nv
v,x
to
T
4
T
2
T
4
3T
能量
o T t
tkAE?22p c o s21?
tAmE 222k s i n21?
Ek 最大时,Ep最小,Ek,Ep交替变化。
Chapter 12 Oscillations
Graph of potential
Chapter 12 Oscillations
例 质量为 的物体，以振幅作简谐运动，其最大加速度为,求,kg10.0
m100.1 2
2sm0.4
（ 1） 振动的周期；
（ 2） 通过平衡位置的动能；
（ 3） 总能量；
（ 4） 物体在何处其动能和势能相等？
解 （ 1）
2
m a x?Aa?
A
a m a x 1s20
s314.0π2T
Chapter 12 Oscillations
（ 2） J100.2 3
222
m a xm a x,k 2
1
2
1 AmmE v
（ 3）
m a x,kEE?
J100.2 3
（ 4）
pk EE?
时，
J100.1 3pE
由
222
p 2
1
2
1 xmkxE
2
p2 2
m
E
x?
24 m105.0
cm7 0 7.0x
Chapter 12 Oscillations
Example 12-6; p305
Example 12-7 P306
Chapter 12 Oscillations
小结：简谐运动的描述和特征
xa 24） 加速度与位移成正比而方向相反
xt x 22
2
d
d2） 简谐运动的动力学描述
)s i n ( tAv
)c o s( tAx3） 简谐运动的运动学描述
mk弹簧振子
kxF1） 物体受线性回复力作用 平衡位置 0?x
Chapter 12 Oscillations
§ 12-5 The Simple Pendulum单摆 (P307-308)
l
m
o
A
TF
P?
转动正向The cord doesn’t stretch
and that its mass can be
ignored.下端系一可看作质点的重物 (a small object called
pendulum bob)就构成一个单摆 。
mgmgF t s i n
s i n,5?时
The restoring force is the
component of the weight,
mg,tangent to the arc.
Chapter 12 Oscillations
l
m
o
A
TF
P?
转动正向摆球的切向加速度为：
2
2
dt
dla?
mg
t
ml2
2
d
d
由牛顿第二定律
0
d
d
2
2
l
g
t
02
2
x
m
k
t
x
d
d
具有 的形式
Chapter 12 Oscillations
在角位移很小的情况下，单摆的振动是简谐运动。
)c o s (m t
l
g?2?令
glT π2?
l
g
Period does not depend on the mass
of the pendulum bob
Chapter 12 Oscillations
*§ 12-5 The Physical Pendulum (复摆 ):(P308-309)
The torque of physical pendulum to axis O,
s i nm g h
s in m g h
m g h
IT?
22
I
m g h
t
2
2
d
d
0dd 22
2
xt x?
I?
2
Based on rotational law,
,-‖ indicates that the acts to reduce
When is small,?
Chapter 12 Oscillations
Solution:
Two springs of k1 and k2 oscillates on a
frictionless ramp,(a) Proof its motion is SHM;
(b) Find its frequency of oscillation.
2211s i n xkxkmg
)'(s i n 111 xxkmgF )'(s i n 222 xxkmg
When the block at ― O‖,
When the block at ― P‖,
k2
x
o
x k1
P
2211 '' xkxkF (2)
21 '' xxx (1)
xkk kkF )(
21
21
—— It is a SHMFrom (1) & (2),
(a)
mkk
kkf
)(2
1
2 21
21
(b)
U=0
`
Chapter 12 Oscillations
*12-7 Damped (阻尼 ) Harmonic Motion(P310-312)
When the motion of an oscillator is reduced by
an external force,the oscillator and its motion
are said to be damped.
bvF d
,-‖,opposes the
motion.d
F?
The component of damping force 阻尼力
along x is,dF
b,damping constant (阻尼因子 )
Chapter 12 Oscillations
)'c o s (2/ texx mbtm
mabkx v
0
d
d
d
d
2
2
kx
t
xb
t
xm
Chapter 12 Oscillations
2
2
4
'
m
b
m
k
2
2
42
1
2
'
m
b
m
kf
The frequency is lower,and the period longer than
for undamped SHM.Three common cases of
heavily damped system are shown in Fig.
Chapter 12 Oscillations
Curve A represents
underdamped situation.
Curve B represents
critical damping
Curve C represents
overdamped
o
t
x
三种阻尼的比较
220b）过阻尼
220a）欠阻尼
220c）临界阻尼
a
b
c
Chapter 12 Oscillations
*12-8 Forced Oscillations; Resonance共振 p313-315
tFF?c o s0e x t?
tFkxtxb
t
xm?c o s
d
d
d
d
02
2
)c o s(0 tAx d
)c o s ()c o s ( 002/0 tAteAx dmtb
Forced (driven) force 驱动力,
The equation of motion with damping is
The first term approaches zero in time,we need to
be concerned only with
2/1222222
0
0
]/)[( mbm
FA
dd
)/(
t a n
22
01
mbd
d
Chapter 12 Oscillations
The amplitude can become large when the driving
frequency?d is near natural frequency,as
long as the damping is not too large,0d
It is approximately the
condition at which the
displacement amplitude of
oscillations is greatest.
0d
(resonant)
2/1222222
0
0
]/)[( mbm
FA
dd
Chapter 12 Oscillations
共振演示实验
2
3 6
1
4
5
22
0r 2
共振频率
22
0
r
2
fA
共振振幅共振现象在实际中的应用乐器、收音机 ……
单摆 1作垂直于纸面的简谐运动时，单摆 5将作相同周期的简谐运动，
其它单摆基本不动,
Chapter 12 Oscillations
共振现象的危害
1940 年 7月 1日美国 Tocama 悬索桥因共振而坍塌
Chapter 12 Oscillations
补充,Superposition of Oscillations
Superposition of Two SHM in Same Direction With
same frequencies (同方向、同频率简谐运动的合成 ):
x1=A1cos(? t+? 1)
x2=A2cos(? t+? 2)
The resultant oscillation is also SHM,and its frequency
is also?.
x = x1+ x2 x =A cos(? t+?)That is,
)co s (2 12212221 AAAAA
2211
2211
c o sc o s
s ins intg
AA
AA
where
1?2
1x?
2x? x?
x
Chapter 12 Oscillations
The resultant oscillation is related
to the phase difference?2-?1,
),2,1,0(,212 kk If
Two special cases:
A=A1+A2
maximum oscillatory amplitude
(两分振动同相,相互加强 ).
1?2
1x?
2x? x?
x
x
to 2T T 23T 2T
合成振动
)co s (2 12212221 AAAAA
212221 2 AAAA
)co s (2 12212221 AAAAA
Chapter 12 Oscillations
If,..)2,1,0()12(
12 kk
A=|A1-A2|
—–minimum oscillation
amplitude (两分振动反相,相互减弱 )
In general,)( 2121 AAAAA
t
o 2T T 23T 2T
x
2x
1x
合成振动
If A1=A2,then A=0 !
The above results shows us that phase difference
of two SHM plays an important role.
Chapter 12 Oscillations
Summary for Chapter Twelve
See P315 to P316
Chapter 12 Oscillations
Homework:
振动基本量和方程,page 317,4,7; Page
318:20
能量,Page 319,26,29,31
