Problem 13 Shrieking rod A metal rod is

Problem # 13 • Shrieking rod • A metal rod is held between two fingers and hit. Investigate how the sound produced depends on the position of holding and hitting the rod? 2020/11/27 Reporter: 知 物 達 理 1

Overview • Introduction – Observation – Problem Analysis • Experiment – Experimental Setup – Experiment • Results and Discussion • Conclusions & Summary • References 2020/11/27 Reporter: 知 物 達 理 2

Introduction • Observation • Two obvious modes of vibration: 1. Transverse motion (Lower frequency) video 2020/11/27 2. longitudinal motion (higher frequency) video Reporter: 知 物 達 理 3

Introduction • Problem Analysis 1. Transverse motion compresses air near rod. -> Amplitude and frequency of sound produced thus depends on elastic modulus of rod. • Statics • Dynamics 2020/11/27 Reporter: 知 物 達 理 4

Introduction • • Quantitative Analysis Statics Rod is composed of many parallel fibers Fibers above neutral surface are stretched and fibers below it are compressed. 2020/11/27 Reporter: 知 物 達 理 5

• Definition of stress and strain: • A filament under the neutral surface by distance z, with a cross section stretched a length: • Tensile force of particular filament: is • Torque of particular filament: 2020/11/27 Reporter: 知 物 達 理 6

• The bending moment, which is the amount of force it takes to bend the whole segment by an angle of is: • Also, 2020/11/27 Reporter: 知 物 達 理 7

Introduction • For a circular cross section, • Let: • ( depends only on shape of cross section because is a geometrical quality) Then M can be written as: 2020/11/27 Reporter: 知 物 達 理 8

Introduction • Dynamics • If we consider a shearing force in a state of equilibrium : 2020/11/27 Reporter: 知 物 達 理 on the end of 9

Introduction • The differential equation of the motion of the bar: • General solution: • Predicted frequency: 2020/11/27 Reporter: 知 物 達 理 10

Introduction • For a stainless steel rod (S 347) with a length 1 m and diameter 0. 6 cm, and plugging in actual data, the first harmonic predicted is: • second harmonic predicted: • Third harmonic predicted: 2020/11/27 Reporter: 知 物 達 理 11

Introduction • For an bronze rod of the same specifications, the predicted first harmonic is: • second harmonic predicted: • Third harmonic predicted: 2020/11/27 Reporter: 知 物 達 理 12

Introduction 2. Longitudinal waves: • Frequency depends on • length of rod allowed to vibrate. • The speed of sound in the longitudinal direction. • Unfixed ends: antinodes of the standing wave • Length of the rod is a multiple of a half of the wavelength: . 2020/11/27 Reporter: 知 物 達 理 13

Introduction • Velocity of longitudinal wave , • The shorter the vibrating length, the higher the frequency produced. 2020/11/27 Reporter: 知 物 達 理 14

Experiment Setup • Rods • Mallet • Oscilloscope and computer 2020/11/27 Reporter: 知 物 達 理 15

Experiment • Position of holding rod: • Parameters: • half length • Rod material: • quarter length – Stainless steel • sixth length – Bronze • Method of hitting rod: • Rod diameter: • Hit across rod – 50 cm, 6 mm stainless steel rod • Hit along rod – 50 cm , 12 mm stainless steel rod • Rod length: – 40 cm stainless steel, bronze rods – 50 cm stainless steel , bronze rods – 90 cm stainless steel , bronze rods 2020/11/27 Reporter: 知 物 達 理 16

Introduction 2020/11/27 Reporter: 知 物 達 理 17

Experiment Waveform data of 6 mm bronze rod held at half point 2020/11/27 Reporter: 知 物 達 理 18

Experiment Fourier transform for waveform data, with main peak at 17. 0, representing 3400 Hz 2020/11/27 Reporter: 知 物 達 理 19

Hit along rod: longitudinal waves 50 cm stainless steel rods 110000 9000 8000 7000 6000 5000 4000 3000 2000 10000 9000 bronze theoretical 6 mm bronze Frequency(Hz) 50 cm bronze rod 8000 7000 stainless theoretical 6000 6 mm stainless 5000 12 mm stainless 4000 3000 2000 1 2 holding position 3 1/2 holding position Frequency(Hz) 90 cm stainless steel and bronze rod 110000 9000 8000 7000 6000 5000 4000 3000 2000 1/2 2020/11/27 1/2 holding position • Theory fits well with data in low frequency range • The microphone and stainless theoretical 6 mm stainless oscilloscope cannot pick up bronze theoretical signals over 10 k. Hz, and may 6 mm bronze not be precise in the high range Reporter: 知 物 達 frequency 理 20

Hit across rod: Transverse waves Longitudinal waves in 50 cm rods 700 600 Frequency(Hz) 500 400 6 mm stainless steel 300 12 mm stainless steel 6 mm bronze 200 100 0 1/2 holding position Rod holding position Stainless Steel, d=6 mm 1/2 1/4 1/6 285. 7 Hz 102. 9 Hz 303. 6 Hz Stainless Steel, d=12 mm 600. 6 Hz 219. 8 Hz 232. 9 Hz 501 cm bronze rod: Bronze, d=6 mm 197. 2 Hz 72. 15 Hz 2020/11/27 First harmonic: 7. 9 Hz (too low) Second harmonic: 71. 5 Reporter: 知 物 達 Third harmonic: 198. 6 理 21

• Holding certain points on the rod enforces certain harmonics: Holding half point enforces 1 st harmonic, however, the frequency is too low to hear Holding sixth point can Holding half point can also enforce 3 rd harmonic 700 600 Frequency(Hz) 500 400 3 rd 2 nd 200 Holding quarter point can enforce 2 nd harmonic 2020/11/27 3 rd 6 mm stainless steel 12 mm stainless steel 6 mm bronze 100 0 Reporter: 知 物 達 理 1/2 holding position 22

Longitudinal waves in 50 cm rods 700 3 rd 600 Frequency(Hz) 500 6 mm stainless steel 400 theoretical 6 mm stainless steel 2 nd 12 mm stainless steel 300 theoretical 12 mm stainless steel 6 mm bronze 200 theoretical 6 mm bronze 100 0 1 2 holding position Stainless Steel, d=6 mm theoretical Stainless Steel, d=12 mm theoretical Bronze, d=6 mm theoretical 2020/11/27 3 285. 7 Hz 298. 3 Hz 600. 6 Hz 596. 5 Hz 197. 2 Hz 198. 6 Hz Reporter: 知 物 達 理 102. 9 Hz 107. 4 Hz 219. 8 Hz 214. 8 Hz 72. 15 Hz 71. 5 Hz 303. 6 Hz 298. 3 Hz 232. 9 Hz 596. 5 Hz 202 Hz 198. 6 Hz 23

Results and discussion • 12 mm rod is too thick to completely control with one hand-> more complications • Other data points fall within good range with theory • The theory proves an accurate method to estimate the frequency a thin shrieking rod produces 2020/11/27 Reporter: 知 物 達 理 24

Conclusions & Summary • For longitudinal waves, the frequencies each rod produces are multiples of the first harmonic. The theory for longitudinal waves is in accordance with experimental data. • For transverse waves, our theory considering the bending moment of the rod accurately predicts the harmonics of thin rods in a wide range, including rods of different material, length and holding position. 2020/11/27 Reporter: 知 物 達 理 25