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Short Version
: 13. Oscillatory Motion 短版: 13.
振盪性運動
(振動)
Disturbing a system from equilibrium results in
oscillatory motion.在平衡狀態中的系统受到干擾後就會進行振盪運動。
Absent friction, oscillation continues forever.
若無摩擦,會一直振盪下去。
穩定平衡
振盪
13.1.
Describing Oscillatory Motion 描述振盪性運動
Characteristics of oscillatory motion
振盪性運動的特徵:- Amplitude A = max displacement from equilibrium.
- Period T = time for the motion to repeat itself.
- Frequency f = # of oscillations per unit time.
[ f ] = hertz (Hz) = 1 cycle / s
赫茲 (赫) 周 / 秒
same period T
同樣的週期 T
same amplitude A
同樣的振幅 A
A, T, f do not specify an
oscillation completely.不能完全指定一個振盪。
13.2. Simple
Harmonic Motion 簡單和諧運動
Simple Harmonic Motion (SHM) :
簡單和諧運動 (簡諧運動) :
Ansatz:
擬設:
angular frequency
角頻
2nd order diff. eq
2 integration
const.
二次微分方程 2個積分常數
A, B determined by initial
conditions由初始條件確定
( t ) 2
x 2A
Amplitude
& Phase 振幅和相位
C = amplitude 振幅
= phase 相位
Note: is independent of amplitude only for SHM.
注:簡諧運動的 與振幅無關
Curve moves to the right for < 0.
< 0 時,曲線往右移。
Velocity &
Acceleration in SHM 簡諧運動的速度和加速度
|x| = max at v = 0 |v| = max at a = 0
位移
速度
加速度
Application: Swaying skyscraper 搖擺的摩天樓
Tuned mass damper 調諧質塊阻尼器 : f damper = f building ,
damper building = .
Taipei 101 TMD : 台北 101 的調諧質塊阻尼器:
41 steel plates, 41 塊鋼板,
660 tonne, d = 550 cm, 660 公噸, d = 550 cm ,
87th-92nd floor. 在 87-92 層樓間
Also used in: 也用於:
- Tall smokestacks 高的煙囪
- Airport control towers. 機場控制塔
- Power-plant cooling towers. 發電廠冷卻塔
- Bridges. 橋樑
- Ski lifts. 滑雪上山吊椅
Example
13.2. Tuned Mass
Damper 調諧質塊阻尼器
The tuned mass damper in NY’s Citicorp Tower consists
of a 373-Mg (vs 101’s 3500 Mg) concrete block that completes one cycle of
oscillation in 6.80 s.紐約花旗銀行大樓的調諧質塊阻尼器是一塊 373-Mg (101 的是 3500 Mg) 的水泥塊,它振盪一週要 6.80 s 。
The oscillation amplitude in a high wind is 110 cm. 在一次強風中它的振幅是110 cm。
Determine the spring constant & the maximum speed & acceleration of the block.
找出彈簧系數,與水泥塊的最高速率和加速度。
The Vertical
Mass-Spring System 垂直式質塊彈簧系统
Spring stretched by x1 when
loaded.負載後彈簧伸長 x1。
mass m oscillates about the new equil. pos.
質塊在新平衡點處振盪
with freq
其頻率為
The Torsional
Oscillator 扭力振盪器
- = torsional constant
Used in timepieces
用於鐘錶
The
Pendulum 單擺
Small angles oscillation:
小角度振盪:
Simple pendulum (point mass
m):
單擺 (質點 m) :
支點
Conceptual
Example 13.1.
Nonlinear Pendulum 非線性鐘擺
A pendulum becomes nonlinear if its
amplitude becomes too large. 鐘擺在振幅太大時會變成非線性。- As the amplitude increases, how will its period changes? 振幅增加時,它的週期有何變化?
- If you start the pendulum by striking it when it’s hanging vertically, 如果鐘擺的起動是當它在正下方時打它,
(a) sin increases slower than sin 增加比 慢
smaller 較小
longer period 週期較長
- If it’s hit hard enough,
打得夠重,運動會變成轉動。
The Physical
Pendulum 物理擺 (複擺)
Physical Pendulum
= any object that’s free to
swing物理擺 = 任何能自由擺動的物體
Small angular displacement SHM
小角度位移 簡諧運動
支點
重心
13.4. Circular
& Harmonic Motion 圓周與諧和運動
Circular motion 圓周運動 : 2 SHO with same A & but = 90
兩互相垂直的簡諧振盪器, A & 相同但 = 90
x = R
x = R
x = 0
GOT IT
懂嗎? 13.3.
The figure shows paths traced out by two pendulums
swinging with different frequencies in the x- & y- directions.圖示兩個 x- 和 y- 頻率不相等的單擺在擺動時劃出的軌跡。
What are the ratios x : y ? x : y 的比值是甚麽?
1 : 2
3: 2
13.5. Energy
in Simple Harmonic Motion 簡諧運動的能量
SHM:
簡諧運動
= constant 能量 時間 位置
平衡點
Potential
Energy Curves & SHM 位能曲線和簡諧運動
Linear force:
線性力:
- parabolic potential energy:
- 拋物線位能
Taylor expansion near local minimum
:
在局部低點附近的泰勒展式:
- Small disturbances near equilibrium points SHM
- 平衡點附近的小干擾 簡諧運動
“最恰”拋物線
位移
位能
13.6. Damped
Harmonic Motion 阻尼諧動
Damping (frictional) force:
阻(摩擦)力:
Damped mass-spring:
阻尼質塊彈簧:
Ansatz 擬設 :
where
sinusoidal oscillation
正弦振盪
Amplitude exponential decay
振幅指數式遞減
Real part 實數部份 :
- At t = 2m / b, amplitude drops to 1/e of max value.
- t = 2m / b 時,振幅掉到最大值的 1/e。
(a) For
- is real, motion is oscillatory ( underdamped )
- 是實數,運動為振盪式(欠阻尼)
(b) For
- is imaginary, motion is exponential ( overdamped )
- 是虛數,運動呈指數式衰減(過阻尼)
(c) For
- = 0, motion is exponential ( critically damped )
- = 0,運動呈指數式衰減(臨介阻尼)
13.7. Driven
Oscillations & Resonance 受驅振盪和共振
External force Driven oscillator外力 受驅振盪器
Let
d = driving frequency
驅動頻率
Prob 75:
習題
= natural frequency 自然頻率
Resonance:
共振:
( long time )( 長期 )
驅動頻率
振幅
Buildings, bridges, etc have natural freq.
建物,橋樑,等都有自然頻率。
If Earth quake, wind, etc sets up resonance, disasters result.
如果地震,風,等形成共振,結果就是災難。
Resonance in microscopic system 微系统的共振 :
- electrons in magnetron microwave oven
- Tokamak (toroidal magnetic field) fusion
- CO2 vibration: resonance at IR freq Green house effect
- Nuclear magnetic resonance (NMR) NMI for medical use.
Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation.
塔科馬橋的倒塌源於自我激發,可以范德蒲方程描述。
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