Oscillator with a Sum of Noninteger-Order Nonlinearities
Free and self-excited vibrations of conservative oscillators with polynomial nonlinearity are considered. Mathematical model of the system is a second-order differential equation with a nonlinearity of polynomial type, whose terms are of integer and/or noninteger order. For the case when only one no...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/649050 |
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| Summary: | Free and self-excited vibrations of conservative oscillators
with polynomial nonlinearity are considered. Mathematical model of the
system is a second-order differential equation with a nonlinearity of polynomial type, whose terms are of integer and/or noninteger order. For the
case when only one nonlinear term exists, the exact analytical solution of
the differential equation is determined as a cosine-Ateb function. Based on this solution, the asymptotic averaging procedure for solving the perturbed strong non-linear differential equation is developed. The method does not require the existence of the small parameter in the system. Special attention is given to the case when the dominant term is a linear one and to the case when it is of any non-linear order. Exact solutions of
the averaged differential equations of motion are obtained. The obtained results
are compared with “exact” numerical solutions and previously obtained
analytical approximate ones. Advantages and disadvantages of the suggested
procedure are discussed. |
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| ISSN: | 1110-757X 1687-0042 |