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Short Version : 13. Oscillatory Motion 短版 : 13. 振 盪性運動 ( 振 動 ). Wilberforce Pendulum. 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 。. 穩定平衡. 振盪. Absent friction, oscillation continues forever . 若無摩擦,會一直振盪下去。. Oscillation.
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Short Version : 13. Oscillatory Motion短版: 13. 振盪性運動 (振動) Wilberforce Pendulum
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。 穩定平衡 振盪 Absent friction, oscillation continues forever. 若無摩擦,會一直振盪下去。 Oscillation
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. 振幅A = 離平衡點的最大位移。 • PeriodT = time for the motion to repeat itself. 週期T = 運動重複自已所需時間。 • Frequencyf = # of oscillations per unit time. 頻率f = 每單位時間的振盪次數 same period T 同樣的週期 T same amplitude A 同樣的振幅 A [ f ] = hertz (Hz) = 1 cycle / s 赫茲 (赫) 周 / 秒 A, T, f do not specify an oscillation completely. 不能完全指定一個振盪。 Oscillation
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) : 2nd order diff. eq 2 integration const. 二次微分方程 2個積分常數 Ansatz: 擬設: angular frequency 角頻
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。 Oscillation
Velocity & Acceleration in SHM簡諧運動的速度和加速度 位移 |x| = max at v = 0 速度 |v| = max at a = 0 加速度
Application: Swaying skyscraper 搖擺的摩天樓 Tuned mass damper 調諧質塊阻尼器 : Damper highly damped , 阻尼器的阻尼值大 Overall oscillation overdamped. 整體擺動為過阻尼型 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 Movie Tuned Mass Damper 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 扭力常數 Used in timepieces 用於鐘錶
The Pendulum 單擺 支點 Small angles oscillation: 小角度振盪: Simple pendulum (point mass m): 單擺 (質點 m) :
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, • will it undergo oscillatory motion no matter how hard it’s hit? 是否不管打多重它都會在擺動? • If it’s hit hard enough, • motion becomes rotational. • 打得夠重,運動會變成轉動。 (a) sin increases slower than sin增加比 慢 smaller 較小 longer period 週期較長
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 支點 Small angular displacement SHM 小角度位移 簡諧運動 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 Lissajous Curves
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 Lissajous Curves
13.5. Energy in Simple Harmonic Motion簡諧運動的能量 能量 時間 位置 SHM: 簡諧運動 = constant 平衡點 Energy in SHM
Potential Energy Curves & SHM 位能曲線和簡諧運動 “最恰”拋物線 Linear force: 線性力: • parabolic potential energy: • 拋物線位能 位能 位移 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
13.6. Damped Harmonic Motion 阻尼諧動 sinusoidal oscillation 正弦振盪 Damping (frictional) force: 阻(摩擦)力: Damped mass-spring: 阻尼質塊彈簧: Amplitude exponential decay 振幅指數式遞減 Ansatz 擬設 : Real part 實數部份 : where
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。 • is real, motion is oscillatory ( underdamped ) • 是實數,運動為振盪式(欠阻尼) (a) For (c) For • is imaginary, motion is exponential ( overdamped ) • 是虛數,運動呈指數式衰減(過阻尼) (b) For • = 0, motion is exponential ( critically damped ) • = 0,運動呈指數式衰減(臨介阻尼) Damped & Driven Harmonic Motion
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 ( long time ) ( 長期 ) Prob 75: 習題 振幅 = natural frequency 自然頻率 Damped & Driven Harmonic Motion Resonance: 共振: 驅動頻率
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。 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。 Tacoma Bridge Resonance in microscopic system 微系统的共振 : • electrons in magnetron microwave oven 磁控管內電子 微波爐 • Tokamak (toroidal magnetic field) fusion 托卡馬克 ( 環形磁場 ) 核融 • CO2 vibration: resonance at IR freq Green house effect 二氧化碳振盪:共振於紅外線頻率 温室効應 • Nuclear magnetic resonance (NMR) NMI for medical use. 核子磁性共振 (核磁共振) 醫療用核磁造像