ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES |
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Authors: | IV ANDRIANOVVV DANISHEVS'KYY |
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Institution: | a Prydniprovska State Academy of Civil Engineering and Architecture, 24-a Chernyshevskogo St. Dnipropetrovsk, 49600, Ukrainef1igor_andrianov@hotmail.comf1b Institute of General Mechanics, RWTH Aachen, Templergraben 64, Aachen, D-52062, Germanyf2vdanish@ukr.netf2 |
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Abstract: | An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived. |
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