Keywords: non-linear vibration, spherical pendulum, auto-parametric systems, quasi-periodic processes, dynamic stability, asymptotic methods.

Auto-parametric systems as mathematical models of strongly non-linear structures are very frequently employed in all engineering branches. Their attractiveness follows from a possibility to characterize the sub-critical as well as post-critical behaviour of particular systems using one universal model. It refers to semi-trivial solution when meaningful solution of the adequate differential system consists of non-trivial part and trivial part. Passing through the bifurcation point non-linear interacting terms take effect. The solution becomes non-trivial in all components as a rule representing the auto-parametric resonance or post-critical state [1].

A double degree of freedom (DDOF) spherical pendulum as an auto-parametric system can be used to demonstrate an effect usually called quasi-periodic response. Sweeping the excitation frequency throughout the auto-parametric resonance interval, various types of quasi-periodic response can be encountered [2,3]. They represent in a horizontal plane some spiral curve, which repeats periodically including higher (thousands) or lower (tens or zero) numbers of cycles.

The response trajectory can be approximated by an ellipse moving around its center with a time dependent angular velocity and changing size of its main axes in "slow" time. All three parameters are functions of a slow time. The angular movement of the ellipse can be very complex having a periodic or orbital character with a separatrix curve between them. The overall period of this quasi-periodic process, as well as the main axes ratio, can be very variable when increasing the excitation frequency from lower to upper limit of the resonance interval. In the upper part of this frequency interval the quasi-periodic response has a toroidal character changing smoothly into a stable limit cycle. It disappears suddenly passing the upper limit of the resonance interval where the semi-trivial regime is regained.

An analytical-numerical approach of these effects is developed using the original non-linear system. This makes it possible to investigate internal processes ruling in auto-parametric systems. A relevant differential system in "slow time" provides periodic, orbital and a few singular solutions separating basic response types. Identification and quantification of the effects discussed have been done using mostly the harmonic balance method in several versions. The general results have been verified using the energy balance approach in a form of an analytical assessment. The direct simulation procedure has been employed as a checking tool. A comparison of the results obtained using above approaches revealed a good agreement. Some hints for engineering applications together with several open problems are outlined in the paper.

1

A. Tondl, T. Ruijgrok, F. Verhulst, R. Nabergoj, "Autoparametric Resonance in Mechanical Systems", Cambridge University Press, Cambridge, 2000.

2

J. Náprstek, C. Fischer, "Auto-parametric semi-trivial and post-critical response of a spherical pendulum damper", Computers and Structures, 87, 19-20, 1204-1215, 2009. doi:10.1016/j.compstruc.2008.11.015

3

W.K. Lee, C.S. Hsu, "A global analysis of an harmonically excited spring-pendulum system with internal resonance", Journal of Sound and Vibration, 171(3), 335-359, 1994. doi:10.1006/jsvi.1994.1125

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