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1、PhysicsNTHUMFTai-戴明鳳实验9简谐运动Lab9SimpleHarmonicMotionSHM Still waters run deep.流静水深流静水深,人静心深人静心深 Where there is life,there is hope。有生命必有希望。有生命必有希望PhysicsNTHUMFTai-戴明鳳HyperPhysicsMost fundamental concepts are subtracted from the web site:http:/hyperphysics.phy-astr.gsu.edu/hbase/hframe.htmlPhysicsNTHUM
2、FTai-戴明鳳何謂簡諧運動？何謂簡諧運動？l物理上有許多運動情形，如單擺、圓周運動等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。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/htmPhysicsNTHUMFTai-戴明鳳Simple harmonic motion-SHMlSimple harmonic motion is typified by the motion of a mass on a spring when it is subject to
3、the linear elastic restoring force given by Hookes Law.lThe motion is sinusoidal in time and demonstrates a single resonant frequency.Undamped spring-mass SHMDamped spring-mass SHMCountsey:http:/en.wikipedia.org/wiki/Simple_harmonic_motionPhysicsNTHUMFTai-戴明鳳Hooks Law&Harmonic Oscillating SystemAn u
4、ndamped spring-mass system undergoes simple harmonic motion.PhysicsNTHUMFTai-戴明鳳Periodic Motion&Simple Harmonic Motion(SHM)PhysicsNTHUMFTai-戴明鳳原原 理理如彈簧伸展量如彈簧伸展量(x)不大不大,則彈簧遵守虎克定律則彈簧遵守虎克定律(Hooks law)恢復力：恢復力：Fr=-kx=-kxx (k:彈性係數彈性係數(spring constant),x x/x)恢復位能：恢復位能：U=kx2/2 設彈簧的質量可忽略：設彈簧的質量可忽略：ms 0 (ms 滑
5、車質量滑車質量m)滑車的運動方程式為二階微分方程式：滑車的運動方程式為二階微分方程式：解微分方程式解微分方程式:x(t)=Asin(k/m)1/2t+=Asin t +A：SHM振幅振幅(amplitude)：角頻率角頻率(angular frequency)2 f=(k/m)1/2 (unit:rad/s)T=1/f：週期：週期(period)：相位相位(phase)Note:若彈簧質量不可忽略若彈簧質量不可忽略,ms 0 =k/(m+ms/3)1/2mk(ms)PhysicsNTHUMFTai-戴明鳳Various Harmonic Oscillating SystemAn undampe
6、d spring-mass system undergoes simple harmonic motion.PhysicsNTHUMFTai-戴明鳳Various SHM SystemslTwo SHM ExampleslSHM of Simple pendulum lCircular SHM lCoupled SHM lDamped SHMlDriven SHM Web site:Acoustics and Vibration Animationshttp:/www.kettering.edu/drussell/Demos.htmlPhysicsNTHUMFTai-戴明鳳Mass-Sprin
7、g Systems without Damping Forced Harmonic Oscillator Mass-Spring Systems with Damping PhysicsNTHUMFTai-戴明鳳Coupled Oscillators-Daniel A.Russell,Kettering University lTwo mass-spring oscillators are coupled together by a stretchy cord.PhysicsNTHUMFTai-戴明鳳Mode Shapes for a Hanging Chain Mode 1Mode 2Mod
8、e 3Take 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
9、)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.PhysicsNTHUMFTai-戴明鳳實驗步驟實驗步驟觀察質量為觀察質量為m的滑車受彈簧的彈性恢復力作用，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。在無
10、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先測量彈簧的彈性係數 k(a)靜態彈性係數靜態彈性係數(static spring constant)ks 測量測量:彈簧加砝碼彈簧加砝碼(m1)垂直懸掛垂直懸掛,平衡時平衡時,伸長值伸長值 y1 總力總力 F=F1+Fr=m1g-ky1=0 ks=m1g/y1 (測量質量及平衡位移測量質量及平衡位移)(b)動態動態(dynamic)彈性係數彈性係數kd 測量測量:彈簧加砝碼彈簧加砝碼(m1)垂直懸掛垂直懸掛,伸長伸長y2作簡諧振盪作簡諧振盪(振幅振幅 A=y2-y1)週期週期 T=2(m1/k)kd=4 2m1/T2 測量質量及週期測
11、量質量及週期 ms 修正修正?2A不要過份伸張彈簧，以避免彈簧造成彈性疲乏。不要過份伸張彈簧，以避免彈簧造成彈性疲乏。y1PhysicsNTHUMFTai-戴明鳳k1k2mx滑車和彈簧的振滑車和彈簧的振盪振幅不能太大盪振幅不能太大2.耦合振盪耦合振盪(coupled oscillation)保護儀器保護儀器 滑車滑車(m)左右各繫一根彈簧左右各繫一根彈簧(k1,ms1),(k2,ms2)耦合振盪耦合振盪 二彈簧恢復力永遠與位移方向相反二彈簧恢復力永遠與位移方向相反,為負值為負值(一壓縮一壓縮,另一伸長另一伸長)md2x/dt2=-k1x-k2x=-(k1+k2)x 耦合彈性係數耦合彈性係數:k=k1+k2 耦合彈簧位能耦合彈簧位能:U=(k1+k2)x2/2 (a)改變滑車質量改變滑車質量(加砝碼加砝碼)m(b)求求 T vs m 之變化之變化(b)換彈簧換彈簧/改變彈性係數改變彈性係數 k:求求 T vs k 之變化之變化(c)改變振幅改變振幅 A:求求T vs A 之變化之變化(d)求速度求速度v(t)vs x(t)之變化之變化(假設無摩擦不會生熱假設無摩擦不會生熱)E=mv2/2+kx2/2=constant