Eigenvalues and eigenvectors of the transfer matrix

1. Eigenvalues and eigenvectors of the transfer matrix Nicolae Cretu-Physics Dept, Transilvania University, Brasov, Romania E-Mail: cretu.c@unitbv.ro Ioan- Mihail Pop-Physics Dept., Transilvania University, Brasov, Romania E-Mail: mihailp@unitbv.ro Ioan-Calin Rosca -Department of the Strength of Materials and Vibrations, Transilvania University, Brasov, Romania E-Mail:icrosca@unitbv.ro ICU, Gdansk, September 5-9, Poland

2. Transilvania University Brasov ICU, Gdansk, September 5-9, Poland

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 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 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). 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. 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 ICU, Gdansk, September 5-9, Poland

8. Starting point-a simple homogeneous rod ICU, Gdansk, September 5-9, Poland

9. Starting point-a simple homogeneous rod ICU, Gdansk, September 5-9, Poland

10. Two homogeneous rods ICU, Gdansk, September 5-9, Poland

11. A binary system ICU, Gdansk, September 5-9, Poland

12. Condition for real eigenvalues in case of a binary system ICU, Gdansk, September 5-9, Poland

13. The function of the polynomial equation • Binary system brass and steel, with dimensions , the same cross-section, ICU, Gdansk, September 5-9, Poland

14. Binary system-identical materials • brass-brass, ICU, Gdansk, September 5-9, Poland

15. Ternary systems ICU, Gdansk, September 5-9, Poland

16. Real eigenvalues ICU, Gdansk, September 5-9, Poland

17. The behavior of the polynomial equation for ternary system ICU, Gdansk, September 5-9, Poland

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. ICU, Gdansk, September 5-9, Poland

19. Practical implications • Numerical estimation of the phase velocity in a solid elastic sample -binary system -ternary system ICU, Gdansk, September 5-9, Poland

20. Experiment-The eigenmodes of a binary brass-aluminum system, obtained by FFT Numerical estimation with brass as gauge material Numerical estimation in case of a binary system ICU, Gdansk, September 5-9, Poland

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

22. The estimated phase velocity in textolite, by using three experimental setups Numerical estimation of the longitudinal wave velocity in a nickel rod using three different experimental setups Errors Error=2% Error=6.7% ICU, Gdansk, September 5-9, Poland

23. Conclusions ICU, Gdansk, September 5-9, Poland

24. [1]. A. H. Nayfeh, “The general problem of elastic wave propagation in multilayered anisotropic media”, J. Acoust. Soc. Am, 89, 1521-1531,(1991) [2]. N. Cretu, G. Nita, “Pulse propagation in finite elastic inhomogeneous media”, Computational Materials Science, 31, 329-336, (2004) [3]. N. Cretu, M. Pop, “Acoustic behavior design with simulated annealing”, Computational Materials Science, 44, 1312-1318, (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, 2651-2658, (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) References ICU, Gdansk, September 5-9, Poland

25. Thank you! ICU, Gdansk, September 5-9, Poland