實驗 9 ：簡諧運動

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PhysicsNTHU MFTai- 戴明鳳

實驗 9 ：簡諧運動

Lab. 9 : Simple Harmonic Motion (SHM)

實驗目的：研究滑車在空氣軌上摩擦力很小的情況下

，因彈簧的恢復力而做的簡諧運動



測量彈簧的靜態彈性係數

k

_s 和動態彈性係數

k

_d



SHM 之週期 T 與運動物體質量 m 的關係



SHM 之週期與彈性係數 k 的關係

Object: Observe the simple harmonic motion of the object

applied by the restoring force of a spring and measure



Both static and dynamic spring constants of the spring, k

_s

& k

_d



Relation of the SHM period T and mass of the motional object



Relation of the SHM period T and the spring constant k

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HyperPhysics

Most fundamental concepts are subtracted from the web site:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

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何謂簡諧運動？



物理上有許多運動情形，如單擺、圓周運動等規律性的運動，可歸類為「簡諧運動」。

圖片來源： http://content.edu.tw/vocation/mechanical/tp_st/top2/ch10/htm

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Simple harmonic motion - SHM



Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring f orce given by Hooke's Law.



The motion is sinusoidal in time and demonstrates a single resonant frequency .

Undamped

spring-mass SHM Damped

spring-mass SHM

Countsey: http://en.wikipedia.org/wiki/Simple_harmonic_motion

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Hook’s Law & Harmonic Oscillating System

An undamped

spring-mass system unde

rgoes simple harmonic m

otion.

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Periodic Motion & Simple

Harmonic Motion (SHM)

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原理

如彈簧伸展量 (x) 不大 , 則彈簧遵守虎克定律 (Hook’s law)

恢復力： F_r

= -kx = -kxx (k: 彈性係數 (spring constant), x  x/

x)

恢復位能： U = kx²

/2

設彈簧的質量可忽略： m

_s

~ 0 (m

_s

<< 滑車質量 m)

滑車的運動方程式為二階微分方程式：

解微分方程式 :

x(t) = Asin[(k/m)

^1/2

t +  ] = Asin[  t +  ] A ： SHM 振幅 (amplitude)

 ：角頻率 (angular frequency)   2f = (k/m)

^1/2

(unit: rad/s) T = 1/f ：週期 (period)

 ：相位 (phase)

Note: 若彈簧質量不可忽略 , m

_s

 0   = [k/(m + m

_s

/3)]

^1/2

k(m

_s

) m t kx

m x  



2 2

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Various Harmonic Oscillating System

An undamped spring-mass system under

goes simple harmonic motion.

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Various SHM Systems



Two SHM Examples

 SHM of Simple pendulum



Circular SHM



Coupled SHM



Damped SHM



Driven SHM

Web site: Acoustics and Vibration Animations

http://www.kettering.edu/~drussell/Demos.html

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Mass-Spring Systems without Damping

Forced Harmonic Oscillator

Mass-Spring Systems

with Damping

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Coupled Oscillators

-Daniel A. Russell, Kettering University



Two mass-spring oscillators are coupled together b y a stretchy cord.

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Mode Shapes for a Hanging Chain

Mode 1

_{Mode 2} _{Mode 3}

Take about 30 paper clips, and connect them end-to-end in a long chain. Hold one end of the chain in your fingers and let the other end dangle. Gently swing (or twirl) the chain and you should find that the chain will "lock" in on a very specific mode shape which occurs at a particular natural (resonance)

frequency.

The shapes of vibration which the chain will "lock" onto are defined by Bessel Functions [More mathematical details to follow soon]

The figures below show the first three mode shapes for a hanging chain.

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實驗步驟

觀察質量為 m 的滑車受彈簧的彈性恢復力作用

，在無摩擦力的空氣軌上做簡諧運動的情形。

1. 先測量彈簧的彈性係數 k

(a) 靜態彈性係數 (static spring constant) k

_s

測量 :

彈簧加砝碼 (m

₁

) 垂直懸掛 , 平衡時 , 伸長值 y

₁

總力 F = F

₁

+ F

_r

= m

₁

g - ky

₁ = 0 k_s = m₁

g/y

₁ ( 測量質量及平衡位移 )

(b) 動態 (dynamic) 彈性係數 k

_d

測量 :

彈簧加砝碼 (m

₁

) 垂直懸掛 ,

伸長 y

₂

作簡諧振盪 ( 振幅 A = y

₂

- y

₁

) 週期 T = 2(m

₁

/k)  k

_d

= 4

²

m

₁

/T

²

 測量質量及週期

 m

_s

修正 ?

2A

不要過份伸張彈簧，以避免彈簧造成彈性疲乏。

y₁

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k

1

k

2

m x

滑車和彈簧的振盪振幅不能太大

2. 耦合振盪 (coupled oscillation) [ 保護儀器 ]

滑車 (m) 左右各繫一根彈簧 (k

₁

, m

_s1

), (k

₂

, m

_s2

) 耦合振盪二彈簧恢復力永遠與位移方向相反 , 為負值 ( 一壓縮 , 另

一伸長 )

md

²

x/dt

²

= - k

₁

x - k

₂

x = -(k

₁

+ k

₂

)x

耦合彈性係數 : k = k

₁

+ k

₂

耦合彈簧位能 : U = (k

₁

+ k

₂

)x

²

/2 (a)改變滑車質量 ( 加砝碼 ) m



求

T vs m 之變化 (b) 換彈簧 / 改變彈性係數 k:



求

T vs k 之變化 (c) 改變振幅 A:



求

T vs A 之變化

(d)

求速度

v(t) vs x(t) 之變化 ( 假設無摩擦不會生熱 )

E = mv

²

/2 + kx

²

實驗 9 ：簡諧運動