Engineering Transactions,

**8**, 2, pp. 213-229, 1960### Parametryczny Rezonans Kombinacyjny w Układach Nieliniowych

The problem of combination resonance (of the second kind) is considered for a model constituting a homogeneous beam with constant double-tee cross-section whose vibration is excited parametrically by a concentrated force in its middle point and in the plane of maximum rigidity, normally to the beam axis. The geometrical non-linearity is accounted for, by assuming that the ends do not approach each other. The axial reaction force is assumed to be due to both the transverse displacement of the points of the axis and the rotation of the cross-sections. For the shaking force it is assumed that (1) it does not change direction, (2) it rotates together with the cross-section, the relative position being not changed.

It is found that the parametric resonance appears in the neighbourhood of the shaking frequency equal to the sum of free frequency in the case where the force preserves its direction during the process of vibration. In the case of rotating load parametric resonance is possible in the neighbourhood of the angular shaking frequency equal to the difference of the angular frequencies of free vibration. From the considerations it follows also that with the same value of P. the width of the resonance region, in the case of the «rotating» force is less than the width of the resonance region for a load preserving its direction during the vibration process. In the case of «rotating» load amplitude increases with decreasing frequency.

It is found that the parametric resonance appears in the neighbourhood of the shaking frequency equal to the sum of free frequency in the case where the force preserves its direction during the process of vibration. In the case of rotating load parametric resonance is possible in the neighbourhood of the angular shaking frequency equal to the difference of the angular frequencies of free vibration. From the considerations it follows also that with the same value of P. the width of the resonance region, in the case of the «rotating» force is less than the width of the resonance region for a load preserving its direction during the vibration process. In the case of «rotating» load amplitude increases with decreasing frequency.

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Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

#### References

[in Russian]

[in Russian]

E. METILER, Allgemeine Theorie der Stabilität erawungener Schwingungen elastischer Körper, Ing. Archiv, Vol. 17, 1949.

E. METTLER, Nichtlineare Schwingungen und kinetische Stabilität bei Saiten und Stäben, Ing. Archiv, Vol. 23, 1955.