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 ks 和動態彈性係數kd
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, ks & kd
Relation of the SHM period T and mass of the motional object
PhysicsNTHU
Relation of the SHM period T and the spring constant
k
MFTai-戴明鳳
HyperPhysics
Most fundamental concepts are subtracted from the web site:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
PhysicsNTHU
MFTai-戴明鳳
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
PhysicsNTHU
MFTai-戴明鳳
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 force given by Hooke's Law.
 The motion is sinusoidal in time and demonstrates a
single resonant frequency.
Undamped
spring-mass SHM
Damped
PhysicsNTHU
spring-mass SHM
MFTai-戴明鳳
Countsey: http://en.wikipedia.org/wiki/Simple_harmonic_motion
Hook’s Law & Harmonic Oscillating System
An undamped springmass system undergoes
simple harmonic motion.
PhysicsNTHU
MFTai-戴明鳳
Periodic Motion & Simple
Harmonic Motion (SHM)
PhysicsNTHU
MFTai-戴明鳳
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：Fr = -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： ms ~ 0 (ms << 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:
2x
m 2  kx
x(t) = Asin[(k/m)1/2t + ] = Asin[t + ]
t
A：SHM振幅(amplitude)
：角頻率(angular frequency)
m
1/2
k(ms)
  2f = (k/m) (unit: rad/s)
T = 1/f ：週期(period)
：相位(phase)
Note:若彈簧質量不可忽略, ms  0
PhysicsNTHU
1/2
  = [k/(m + ms/3)]
MFTai-戴明鳳
Various Harmonic Oscillating System
An undamped spring-mass system
undergoes simple harmonic motion.
PhysicsNTHU
MFTai-戴明鳳
Various SHM Systems

Two SHM Examples

SHM of Simple pendulum

Circular SHM

Coupled SHM

Damped SHM

Driven SHM
Web site: Acoustics and Vibration Animations
PhysicsNTHU
http://www.kettering.edu/~drussell/Demos.html
MFTai-戴明鳳
Mass-Spring Systems
without Damping
Mass-Spring Systems
with Damping
Forced Harmonic
Oscillator
PhysicsNTHU
MFTai-戴明鳳
Coupled Oscillators
-Daniel A. Russell, Kettering University

Two mass-spring oscillators are coupled together
by a stretchy cord.
PhysicsNTHU
MFTai-戴明鳳
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]
PhysicsNTHU
The figures below show the first three mode shapes for a hanging MFTai-戴明鳳
chain.
實驗步驟
不要過份伸張彈簧，以避免彈簧造成彈性疲乏。
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) ks 測量:
彈簧加砝碼(m1)垂直懸掛,平衡時, 伸長值 y1
總力 F = F1 + Fr = m1g - ky1 = 0
ks = m1g/y1 (測量質量及平衡位移)
(b) 動態(dynamic)彈性係數kd 測量:
彈簧加砝碼(m1)垂直懸掛,
伸長y2作簡諧振盪(振幅 A = y2 - y1)
週期 T = 2(m1/k)  kd = 42m1/T2
 測量質量及週期
 ms 修正?
y1
2A
PhysicsNTHU
MFTai-戴明鳳
2. 耦合振盪(coupled oscillation) [保護儀器]
滑車(m)左右各繫一根彈簧(k1, ms1), (k2, ms2)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)
md2x/dt2 = - k1x - k2x = -(k1 + k2)x
耦合彈性係數: k = k1 + k2
耦合彈簧位能: U = (k1 + k2)x2/2
(a) 改變滑車質量(加砝碼) m
x
m k
k
1
2
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
滑車和彈簧的振
(d) 求速度v(t) vs x(t)之變化(假設無摩擦不會生熱)盪振幅不能太大
E=
mv2/2
+
kx2/2
= constant
PhysicsNTHU
MFTai-戴明鳳