Symmetry Exploitation in the Natural Vibrations of Rod Systems

Authors: Pavlov А.М., Temnov A.N. Published: 03.08.2017
Published in issue: #4(115)/2017  

DOI: 10.18698/0236-3941-2017-4-28-41

Category: Mechanics | Chapter: Theoretical Mechanics  
Keywords: rod system, symmetry group, irreducible representation, spectral problem, eigenfunction classification, projection operator

The purpose of this work was to study spectral and Cauchy problem for the mechanical system consisting of three rods, two of them being identical and connected with the third one by linear elastic elements. We stated the corresponding spectral problem and studied its spectrum. Findings of the research show that eigenfunctions of the considered spectral problem are classified according to the irreducible representations of the finite group of transformations despite the fact that the initial equations system admits continuous (Lie) transformation groups. We considered the weak solution of Cauchy problem and revealed its simplification in case of special "symmetrical" form of initial conditions and right-hand side of the corresponding operator equation system.


[1] Miller W. Symmetry groups and their applications. New York, Academic Press, 1972. 436 p.

[2] Zlokovic G. Group theory and G-vector spaces in the structural analysis. Chichester, UK, Ellis Horwood, 1989. 283 p.

[3] Zingoni A. Group-theoretic insights on the vibration of symmetric structures in engineering. Phil. Trans. R. Soc. A, 2014, no. 372. DOI: 10.1098/rsta.2012.0037 Available at: http://rsta.royalsocietypublishing.org/content/372/2008/20120037

[4] Healey T., Treacy J. Exact block diagonalization of large eigenvalue problems for structures with symmetry. Int. J. Numer. Meth. Eng., 1991, vol. 31, no. 2, pp. 265-285.

[5] Herzberg G. Infrared and Raman spectra of polyatomic molecules. New York, Van Nostrand, 1945. 632 p.

[6] Kaveh A., Nikbakht M. Decomposition of symmetric mass-spring vibrating systems using groups, graphs and linear algebra. Numer. Meth. Biomed. Eng., 2007, vol. 23, no. 7, pp. 639-664.

[7] Kaveh A., Nikbakht M. Improved group-theoretical method for eigenvalue problems of special symmetric structures, using graph theory. Adv. Eng. Softw., 2010, vol. 41, no. 1, pp. 22-31.

[8] Mohan S., Pratap R. A Group theoretic approach to the linear free vibration analysis of shells with dihedral symmetry. J. Sound Vib., 2002, vol. 252, no. 2, pp. 317-341.

[9] Mohan S., Pratap R. A natural classification of vibration modes of polygonal ducts based on group theoretic analysis. J. Sound Vib., 2004, vol. 269, no. 3-5, pp. 745-764.

[10] Zingoni A., Pavlovic M., Lloyd-Smith D., Zlokovic G. Group-theory considerations of finite-difference plate eigenvalue problems. In: Developments in computational engineering mechanics. 1993. Pp. 243-256.

[11] Bunker P., Jensen P. Molecular symmetry and spectroscopy. Ottawa, NRC research press, 1998. 747 p.

[12] Zingoni A. Group-theoretic exploitations of symmetry in computational solid and structural mechanics. Numer. Meth. Eng., 2009, no. 79, pp. 253-289.

[13] Olver P. Applications of Lie groups to differential equations. New York, Springer-Verlag, 1986. 497 p.

[14] Barut A., Raczka R. Theory of group representations and applications. Singapore, World Scientific, 1986. 717 p.

[15] Asherova R.M., Smirnov Y.F. Projection operators for classical groups. UMN, 1969, vol. 24, no. 3(147), pp. 227-228 (in Russ.).

[16] Ovsiannikov L.V., Ames W. Group analysis of differential equations. New York, Academic Press, 1982. 416 p.

[17] Pavlov A.M., Temnov A.N. System of rods longitudinal vibrations. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2014, no. 6, pp. 53-66 (in Russ.).

[18] Wigner E. Group theory and its application to the quantum mechanics of atomic spectra. New York, Academic Press, 1959. 372 p.

[19] Hamermesh M. Group theory and its application to physical problems. New York, Dover, 1989. 509 p.

[20] Mikhlin S.G. Linear equations of mathematical physics. New York, Holt, Rinehart and Winston, 1967. 318 p.

[21] Zimin V.N., Krylov A.V., Meshkovskii V.E., Sdobnikov A.N., Fayzullin F.R., Churilin S.A. Features of the calculation deployment large transformable structures of different configurations. Nauka i obrazovanie: nauchnoe izdanie MGTU im. N.E. Baumana [Science and Education: Scientific Publication of BMSTU], 2014, no. 10, pp. 179-191 (in Russ.). DOI: 10.7463/1014.0728802 Available at: http://technomag.edu.ru/jour/article/view/702