1. Eigenvalues and eigenvectors of the transfer matrix
Nicolae Cretu- Physics Dept, Transilvania University, Brasov, Romania
Ioan- Mihail Pop- Physics Dept., Transilvania University, Brasov, Romania
Ioan-Calin Rosca -Department of the Strength of Materials and Vibrations, Transilvania University, Brasov, Romania
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ICU, Gdansk, September 5-9, Poland

2. Transilvania University Brasov
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3. Transfer matrix approaches
-widely used in computer simulation of wave propagation in finite or infinite media, especially multilayered media-the whole transfer matrix of the multilayered material is obtained as the product of all the layer’s matrices, each layer being considered as homogeneous.
-to measure the acoustical properties of a certain material-if the transfer matrix of a material is known, most of the acoustical properties of a material can be obtained
-to design acoustical structures, very suitable for optimisation algorithms
-to determine the reflection and transmission coefficients, transmission loss
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ICU, Gdansk, September 5-9, Poland

4. Transfer matrix approaches
Transfer matrix depends on the elastic media by the boundary conditions-different kinds of materials have different forms of transfer matrices
Transfer matrix depends on the type of the wave propagating in the sample
Boundary conditions:
For fluid media-continuity of the sound pressure and of the normal acoustic particle velocity
For solid elastic media- the sharp interface
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ICU, Gdansk, September 5-9, Poland

5. Acoustic tube technique
The transmission loss coefficient is defined as the ratio between the amplitude |A| of the incident plane wave and the amplitude |C| of the transmitted plane wave, in case of anechoic termination (i.e. D=0).
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ICU, Gdansk, September 5-9, Poland

6. Two load method for the standing wave tube
Because of the difficulty to obtain a perfectly anechoic termination, two different tube terminations represented by indices a and b must be measured, to obtain four linear equations, that can be used to solve for the four unknown matrix elements in equation
Note that if the numerical value of the difference in the denominator becomes much smaller than the absolute values of the subtracted numbers, then the solution becomes unstable.
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ICU, Gdansk, September 5-9, Poland

7. Intrinsic transfer matrix method
We propose a method based on the intrinsic transfer matrix of a mechanical system which is significant in the resonance context
The method is applied for solid elastic materials and is based on the properties of the eigenvalues and eigenvectors of the intrinsic transfer matrix
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ICU, Gdansk, September 5-9, Poland

8. Starting point-a simple homogeneous rod
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9. Starting point-a simple homogeneous rod
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10. ICU, Gdansk, September 5-9, Poland
Two homogeneous rods
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11. ICU, Gdansk, September 5-9, Poland
A binary system
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12. Condition for real eigenvalues in case of a binary system
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13. The function of the polynomial equation
Binary system
brass and steel,
with dimensions ,
the same cross-section,
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14. Binary system-identical materials
brass-brass,
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15. ICU, Gdansk, September 5-9, Poland
Ternary systems
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16. ICU, Gdansk, September 5-9, Poland
Real eigenvalues
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17. The behavior of the polynomial equation for ternary system
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18. Intrinsic matrix properties
Intrinsic transfer matrix is a complex matrix
The eigenmodes of the system correspond to real eigenvalues of the intrinsic transfer matrix (hermitian matrix)
The frequency of the eigenmodes can be obtained by Fourier analysis of the standing wave signal in the system.
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19. Practical implications
Numerical estimation of the phase velocity in a solid elastic sample
-binary system
-ternary system
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20. Numerical estimation in case of a binary system
Experiment-The eigenmodes of a binary brass-aluminum system, obtained by FFT
Numerical estimation with brass as gauge material
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ICU, Gdansk, September 5-9, Poland

21. Numerical estimation in case of a ternary system
Fourier spectrum for a ternary brass-alumina zirconia ceramic-brass system.
Numerical estimation with brass as gauge material
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ICU, Gdansk, September 5-9, Poland

22. ICU, Gdansk, September 5-9, Poland
Errors
Numerical estimation of the longitudinal wave velocity in a nickel rod using three different experimental setups
The estimated phase velocity in textolite, by using three experimental setups
Error=2%
Error=6.7%
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23. ICU, Gdansk, September 5-9, Poland
Conclusions
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24. ICU, Gdansk, September 5-9, Poland
References
[1]. A. H. Nayfeh, “The general problem of elastic wave propagation in multilayered anisotropic media”, J. Acoust. Soc. Am, 89, ,(1991)
[2]. N. Cretu, G. Nita, “Pulse propagation in finite elastic inhomogeneous media”, Computational Materials Science, 31, , (2004)
[3]. N. Cretu, M. Pop, “Acoustic behavior design with simulated annealing”, Computational Materials Science, 44, , (2009)
[4]. D. N. Johnston, D. K. Longmore, J. E. Drew, “A technique for the measurement of the transfer matrix characteristics of two port hydraulic components”, Fluid Power Sys. And Tech., 1, 25-33, (1994)
[5]. J.S. Bolton, R. J. Yun, J. Pope, D. Apfel, “Development of a new sound transmission test for automotive sealant materials”, SAE Trans., J. Pass. Cars, 106, , (1997)
[6]J. S. Bolton, O. Olivieri, “Measurement of Normal Incidence Transmission Loss and Other Acoustical Properties of Materials Placed in a Standing Wave Tube”, Bruel&Kjaer Technical Review, No1, 1-44, (2007)
[7]A. Anselm, “ Introduction to semiconductor theory: Atomic vibrations in complex three dimensional lattices” MIR Publishers Moscow, (1981)
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25. ICU, Gdansk, September 5-9, Poland
Thank you!
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ICU, Gdansk, September 5-9, Poland