Abstract
A chain of N rigid bodies with elastic connections transmitting a torque interaction is considered. Similar individual inertial elements of the chain have one degree of freedom. An exact analytical solution of the problem of oscillation eigenfrequencies and eigenmodes is constructed for an arbitrary value of N. The properties of frequencies and modes of such a chain are compared with those of a canonical Newton chain. It is established that the correlation between eigenfrequencies and the properties of the corresponding eigenmodes fundamentally differs from that for a canonical Newton chain. Lower eigenfrequencies of a chain of rigid bodies with inertialess torque connections correspond to modes with a larger number of nodes, while higher eigenfrequencies correspond to the smoother modes. The nontypical correlation between the oscillation eigenfrequencies and eigenmodes discovered based on the exact analytical solution to the problem of free oscillations of a chain of N rotating rigid bodies contradicts the ideas underlying theoretical studies in the field of solid state physics devoted to simulation of mechanical and thermal dynamic processes in crystalline lattices.
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References
L. I. Mandel’shtam, Lectures on Vibration Theory (Nauka, Moscow, 1972) [in Russian].
L. Brillouin and M. Parodi, Wave Propagation in Periodic Structures (Dover, New York, 1953; Inostrannaya Literatura, Moscow, 1959).
M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954; Inostrannaya Literatura, Moscow, 1972).
A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation, Suppl. 3: Solid State Physics (Academic, New York, 1963; Nauka, Moscow, 1965).
V. L. Biderman, Applied Theory of Mechanical Oscillations (Vysshaya Shkola, Moscow, 1972) [in Russian].
S. Timoshenko, Vibration Problems in Engineering (Fizmatgiz, Moscow, 1959; Van Nostrand, Toronto, 1955).
F. Bleich and E. Mellan, Die Gewöhnlichen und Partiellen Differenzengleichungen der Baustatik (Springer-Verlag, Wien, 1927; Nauchno-Tekh. Izd. Ukr., Kharkov, 1936).
A. M. Kosevich, The Crystal Lattice: Phonons, Solitions, Dislocations, Superlattices (Nauka, Moscow, 1988; Wiley, Weinheim, 2005).
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Original Russian Text © D.A. Indeitsev, A.D. Sergeev, 2017, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2017, Vol. 62, No. 2, pp. 106–115.
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Indeitsev, D.A., Sergeev, A.D. Correlation between the properties of eigenfrequencies and eigenmodes in a chain of rigid bodies with torque connections. Vestnik St.Petersb. Univ.Math. 50, 166–172 (2017). https://doi.org/10.3103/S1063454117020066
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DOI: https://doi.org/10.3103/S1063454117020066