| V.2 No 2 | 35 |
| Investigation of elastic constraint non-linearity | |
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| In order to find the solutions of (20) for the third harmonic, taking into consideration the non-linearity of this system of equations, it is sufficient to present a more general form of solution as a linear superposition of three ancillary solutions for the subsystems of equations; whilst retaining one of the equivalent forces Qi3 in each of them. With it the general solution takes the form | |
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(30) |
In its turn, we can determine the solutions for each subsystem for the investigated elastic line on the basis of results presented in [14]. With it, for the first subsystem containing the force Qi3 we yield |
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(31) |
| For the second subsystem containing the force Q23 the solution has some other form, because in this case the force affects the interior element of the elastic line. Should the considered line have more than three elements, according to [14] the solution would be the system for the right and left parts of the elastic line correspondingly. But since our model has only one interior element, the solution simplifies; it can be written as | |
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(32) |
| Finally, for the third subsystem the solution has the following form: | |
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(33) |
The obtained components (31), (32) and (33) of the general
solution (30) show that the amplitude-frequency characteristic of vibration process for
the third harmonic does not correspond to that of the first harmonic, the same different
are the values of resonance frequencies. This is conditioned not only by the difference of
regularities. After all, the multiplier sin 6 3 determining the location of resonances has the
same form as in (15) for the first
harmonic, but for the third harmonic the parameter  3 determining 3 = arcsin 3
is equal to |
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(34) |
This is three times greater than the related parameter (16) of the first harmonic. Consequently,
the boundary frequency for the third harmonic is one-third of  01 and
determined by the expression |
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(35) |
