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系統識別號 U0026-2508201117044700
論文名稱(中文) 以共振超音波頻譜同時量測多個材料彈性及消散模數
論文名稱(英文) Simultaneous determination of multiple elastic moduli and loss tangents of materials with resonant ultrasound spectroscopy
校院名稱 成功大學
系所名稱(中) 土木工程學系碩博士班
系所名稱(英) Department of Civil Engineering
學年度 99
學期 2
出版年 100
研究生(中文) 胡志豪
研究生(英文) Chih-Hao Hu
學號 n66981397
學位類別 碩士
語文別 英文
論文頁數 76頁
口試委員 指導教授-王雲哲
口試委員-黃忠信
口試委員-鍾興陽
口試委員-林育芸
口試委員-楊士賢
中文關鍵字 共振超音波頻譜 
英文關鍵字 Resonant ultrasound spectroscopy 
學科別分類
中文摘要 共振超音波頻譜可以在不破壞試體的情況下量測出固體材料的彈性模數,受測的
固體材料藉由兩個超音波壓電傳導器輕壓夾住試體的角點,一端輸入頻率振幅刺激,另一端則接收頻率。藉由試體引起的微小振動轉為振動頻率,量測出訊號再利用電腦快速的從第一共振頻率去推算剪力模數,以及量測共振曲線的寬度,以計算材料的正切消散係數。此外,透過個人電腦Fortran程式運算,可以將受測試體激發的各個共振頻率值計算出材料所有彈性模數,並且與先前利用電腦及第一共振頻率所快速推斷的剪力模數結果比較,兩者間是完美一致的。此分析試驗方法的最大優點在於試驗所需耗費的時間短,操作方式簡單且不會破壞試體可重複試驗,而所求出的材料彈性係數與阻尼係數在金屬材料相當精準,這些均有別於傳統的力學試驗分析方法,利用這些特性可以在品管控制上省下不少成本與時間。本論文中的實驗裡使用多種材料去證實超音波共振頻譜的準確性,以銅試體為例,利用第一共振頻率求得的剪力模數為49.35GPa,再利用LANL RPRcode Ver. 6.0 求得的C44= 48.703 GPa,兩者間的誤差為1.31%。壓克力利用第一共振頻率求得的剪力模數為2.398 GPa,而利用LANL RPRcode Ver. 6.0求得的C44= 3.817 GPa。此外,亦探討試體與超音波壓電傳導器之間擺放的角度,及試體角點的選取不同是否對最後所求取的材料彈性模數有所影響,結果相差不大。
英文摘要 Resonant ultrasound spectroscopy (RUS) has been developed for measuring the elastic constants of solid materials under non-destructive situation. The RUS techniques require slightly corner contact force to mount specimens between two transducers. One transducer inputs an amplitude and frequency excitation to the sample, and another behaves as a receiver. The corner contact provides elastically weak coupling to the transducers, and hence minimal perturbation
to the viberation, minimal shift in resonant frequency and minimum parasitic damping. Using the computer to calculate the shear modulus from the first resonant peak, and measurement the width of the resonant peak to calculate the damping of the sample. Moreover, according to the fortran software, we can take at least 10 resonant peaks to calculate the material elastic moduli.
To compare with the previous results by the fortran performed, basically, they are almost perfect consistency. The analysis experimental method has several advantages: 1. During the experiment, it does not cost much time. 2. The operation of this experiment is very simple to do. 3. The experiment does not cause damage to the specimens. 4. The results from RUS in metal materials has much precision accuracy, all of these advantages are quite different from
the traditional mechanical testing methods. Using these advantages can save much time and money to do the quality control. In this thesis, I use several materials including aluminum, copper, magnet and PMMA to proof the accuracy of the RUS method. Taking copper for example, choosing the first resonant peak to calculate the shear modulus. The result of G is about 49.35 GPa. By the LANL RPRcode Ver. 6.0, Figure 4.9(b) shows that C44= 48.703 GPa, the
error of shear modulus is about 1.31% in copper specimen. Taking PMMA for example, using the first resonant peak to calculate the shear modulus. The result of G is around 2.398 GPa and using the LANL RPRcode Ver. 6.0, Table 4.2 shows that C44= 3.817 GPa. Furthermore, I experiment the same sample but changing the angle of the transducer and specimen, even use the different corner to do the experiment again. The results are not different much.
論文目次 CHINESE ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Goals and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 RUS theoretical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Free vibration of a sphere . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Free vibration of a cylinder . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Resonant frequencies of a cube . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The LANL RPR fortran code . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Lorentzian Curve Fit for Damping calculation . . . . . . . . . . . . . . . . . . 14
3 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 The RUS apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Piezoelectric shear transducer . . . . . . . . . . . . . . . . . . . . . . 15
3.1.2 Homemade shear transducer holder . . . . . . . . . . . . . . . . . . . 15
3.1.3 Lock-In Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.4 National Instrument:PXI(PCI eXtensions for Instrumentation) . . . . . 16
3.1.5 Function generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.6 Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.7 Lock-In amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Data acquisition techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 NI system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Agilent system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Testing samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 Standard RUS results (Aluminum) . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.1 Changing the sample placing angle . . . . . . . . . . . . . . . . . . . 32
4.1.2 Changing the sample placing corner . . . . . . . . . . . . . . . . . . . 35
4.2 Determination of elastic moduli from one scan (Copper) . . . . . . . . . . . . 37
4.3 Determination of damping from multiple peaks (Magnet) . . . . . . . . . . . . 44
4.4 Determination of damping from multiple peaks (PMMA) . . . . . . . . . . . . 52
5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
APPENDICES
Appendix A: Fortran Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
參考文獻 [1] Hiram Brown. Aluminum and its application. Pitman Publishing, New York, 157 (1948).
[2] Julie Capodagli, Roderic Lakes. Isothermal viscoelastic properties of PMMA and LDPE over 11 decades of frequency and time: a test of time-temperature superposition. Published 13 June 2008.
[3] H. H. Demarest. Cube-resonance method to determine the elastic constants of solids. J.Acoust. Soc. Am. 49, 3, 2, 768-775, March (1971).
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[5] J. H. Kinney, J. R. Gladden, G. W. Marshall, S. J. Marshall, J. H. So and J. D. Maynard. Resonant ultrasound spectroscopy measurements of the elastic constants of human dentin. Journal of Biomechanics 37, 4, 437-441, April (2004).
[6] LANL RUS code webpage.
http://www.magnet.fsu.edu/inhouseresearch/rus/index.html [19 June 2011].
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[8] H. Lamb, On the vibration of an elastic sphere, Proc. London Math Soc. 13, 189-212(1882)
[9] T. Lee, R.S. Lakes, A. Lai. Resonant ultrasound spectroscopy for measurement of mechanical damping: Comparison with broadband viscoelastic spectroscopy. Review of Scientific Instruments 71, 7, 2855-2861 (2000).
[10] A.E.H. Love. A Treatise on the Mathematical Theory of Elasticity. Dover Publications (1944).
[11] H.J McSkimin. Ultrasonic Measurement Techniques Applicable to Small Solid Specimens. J. Acoust. Soc. Am. 22, 413-418 (1950).
[12] H.J McSkimin. Pulse Superposition Method for Measuring UltrasonicWave Velocities in Solids. J. Acoust. Soc. Am. 33, 12-16 (1961).
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Review of Scientific Instruments 76, 12, 121301-1-121301-7 (2005).
[14] A.Migliori and J.L.Sarrao. Resonant Ultrasound Spectroscopy. Wiley,New York 1997.
[15] R.D. Mindlin. Simple modes of vibration of crystals. J. Appl. Phys. 27, 1462-1466 (1956).
[16] I. Ohno. Free vibration of a rectangular parallelepiped crystal and its application to determination
of elastic constants of orthorhombic crystals. J. Phys. Earth 24, 355 (1976).
[17] W. H. Press. B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1986.
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analytical refinement. Japanese Journal of Applied Physics 1, 46, 12, 7898-7903, Dec 6 (2007).
[19] J.L. Sarrao, S.R. Chen, W.M. Visscher, Ming Lei, U.F. Kocks, and A. Migliori. Determination of the crystallographic orientation of a single crystal using resonant ultrasound spectroscopy. Review of Scientific Instruments 65, 6, 2139-2140, June (1994).
[20] A. Sommerfeld, Mechanics of deformable bodies, Academic Press, 1950.
[21] W.M. Visscher, A. Migilori, T.M. Bell, R.A. Reinert, “On the normal modes of free vibration of inhomogeneous and anisotropic elastic objects,” J. Acoust. Soc. Am. 90, 2154-2162 (1991)
[22] Y.C.Wang, R.S. Lakes, Resonant ultrasound spectroscopy in shear mode, Rev. Sci. Insts., 74, 1371-1373 (2003)
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