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Finite element method for 1D axial periodic structure.pptx
- 1. By-Subhajit Paul FACULTY OFENGINEERING AND TECHNOLOGY M.E. IN CIVIL ENGINEERING ROLL NO: 002210402004 SESSION-2022-24
- 2. Periodic Structure • Aperiodic structure can be defined as a structure or material which is formed by repetition of an identical representative element named a unit cell. The repetition can be in the form of material properties, boundary conditions, geometry, etc. • In periodic structures, the stop bands are generated as a consequence of destructive interference between the incident wave and the reflecting wave due to the change in geometry or material properties • The wavenumber remains constant in the frequency values corresponding to the stop bands which indicates zero group speed. Thus, waves of a certain frequency range cannot propagate through the structure, and undesired vibration can be avoided by proper design • The periodic analysis is independent of the domain and number of unit cells. Fig. Schematic of the periodic unit cell (not to scale) of the 1-D periodic bar made of two different material
- 3. Method of Analysis •The transfer matrix method (TMM) is very popular for analyzing periodic structures • In SEM, the dynamic stiffness matrix is formed, and the dynamic response of the structure is expressed using spectral representations • Compared to conventional finite element, SEM reduces the computational cost substantially • Also there are many other methods available in the literature to calculate the wave propagation characteristics, and researchers have explored them considering different kinds of periodic structures.
- 4. Wave finite element Bloch’stheorem : • Bloch’s theorem was developed to study the electron behaviour in crystalline solids. Later, it was adapted to study the elastic wave propagation in periodic structures • In this method, instead of analyzing the total structure, a small subset of the structure which is one repeating unit cell is analyzed. Thus, the number of degrees of freedom (DOF) reduces dramatically. Fig. Periodic structure and a single cell
- 5. Fig. Schematic ofa periodic unit cell Periodic Cell Representation
- 6. Table : Dimensionsof the single cell of the periodic bar Fig. Schematic of the periodic unit cell (not to scale) of the 1-D periodic bar Periodic Cell Representation
- 7. Type I CellParametric Results Fig. Spectrum relation of the 1-D bar Fig. Frequency response function of the 1-D bar
- 8. Type I CellParametric Results Fig. Propagation and attenuation constant of the 1-D periodic bar Fig. Attenuation constant and FRF of the 1-D bar
- 9. Effect of materialuncertainty on stop bands Fig. Spectrum relation and FRF of 1-D bar
- 10. REFERENCES • Mead, D.J.:Wave propagation and natural modes in periodic systems: I. Mono-coupled systems. J. Sound Vib. 40(1), 1–18 (1975) • Brillouin, L.: Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices. Courier Corporation, North Chelmsford (2003) • Lord Rayleigh, R.S.: On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure. Lond. Edinb. Dublin Philos. Mag. J. Sci. 24(147), 145–159 (1887) • Kittel, C., McEuen, P.: Introduction to Solid State Physics, vol. 8. Wiley, New York (1996) • Heckl, M.A.: Investigations on the vibrations of grillages and other simple beam structures. J. Acoust. Soc. Am. 36(7), 1335–1343 (1964) • Hodges, C.: Confifinement of vibration by structural irregularity. J. Sound Vib. 82(3), 411–424 (1982) • Mead, D.J.: Wave propagation and natural modes in periodic systems: II. Multi-coupled systems, with and without damping. J. Sound Vib. 40(1), 19–39 (1975) • Mead, D.: Wave propagation in continuous periodic structures: research contributions from Southampton, 1964–1995. J. Sound Vib. 190(3), 495–524 (1996) • Lin, Y.K., McDaniel, T.: Dynamics of beam-type periodic structures. J. Eng. Ind. 91(4), 1133–1141 (1969)
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