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 4/20/2019 ICU, Gdansk, September 5-9, Poland

Transilvania University Brasov 4/20/2019 ICU, Gdansk, September 5-9, Poland

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 4/20/2019 ICU, Gdansk, September 5-9, Poland

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 4/20/2019 ICU, Gdansk, September 5-9, Poland

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). 4/20/2019 ICU, Gdansk, September 5-9, Poland

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. 4/20/2019 ICU, Gdansk, September 5-9, Poland

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 4/20/2019 ICU, Gdansk, September 5-9, Poland

Starting point-a simple homogeneous rod 4/20/2019 ICU, Gdansk, September 5-9, Poland

Starting point-a simple homogeneous rod 4/20/2019 ICU, Gdansk, September 5-9, Poland

ICU, Gdansk, September 5-9, Poland Two homogeneous rods 4/20/2019 ICU, Gdansk, September 5-9, Poland

ICU, Gdansk, September 5-9, Poland A binary system 4/20/2019 ICU, Gdansk, September 5-9, Poland

Condition for real eigenvalues in case of a binary system 4/20/2019 ICU, Gdansk, September 5-9, Poland

The function of the polynomial equation Binary system brass and steel, with dimensions , the same cross-section, 4/20/2019 ICU, Gdansk, September 5-9, Poland

Binary system-identical materials brass-brass, 4/20/2019 ICU, Gdansk, September 5-9, Poland

ICU, Gdansk, September 5-9, Poland Ternary systems 4/20/2019 ICU, Gdansk, September 5-9, Poland

ICU, Gdansk, September 5-9, Poland Real eigenvalues 4/20/2019 ICU, Gdansk, September 5-9, Poland

The behavior of the polynomial equation for ternary system 4/20/2019 ICU, Gdansk, September 5-9, Poland

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. 4/20/2019 ICU, Gdansk, September 5-9, Poland

Practical implications Numerical estimation of the phase velocity in a solid elastic sample -binary system -ternary system 4/20/2019 ICU, Gdansk, September 5-9, Poland

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 4/20/2019 ICU, Gdansk, September 5-9, Poland

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 4/20/2019 ICU, Gdansk, September 5-9, Poland

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% 4/20/2019 ICU, Gdansk, September 5-9, Poland

ICU, Gdansk, September 5-9, Poland Conclusions 4/20/2019 ICU, Gdansk, September 5-9, Poland

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, 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) 4/20/2019 ICU, Gdansk, September 5-9, Poland

ICU, Gdansk, September 5-9, Poland Thank you! 4/20/2019 ICU, Gdansk, September 5-9, Poland