Energy conservative finite element semi-discretization for vibro-impacts of plates on rigid obstacles

Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of vibro-impact of plates between rigid obstacles with non-penetration Signorini’s conditions. To this aim, the dynamical Kirchhoff–Love plate model is considered and an extension to plates of the singular dynamic method, introduced by Renard and previously adapted to beams by Pozzolini and Salaün, is described. A particular emphasis is given in the use of an adapted Newmark scheme in which intervene a discrete restitution coefficient. Finally, various numerical results are presented and energy conservation capabilities of several numerical schemes are investigated and discussed

Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles

Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini’s conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated

Forced System with Vibro-impact Energy Sink: Chaotic Strongly Modulated Responses

AbstractThe paper treats forced response of primary linear oscillator with vibro-impact energy sink. This system exhibits some features of dynamics, which resemble forced systems with other types of nonlinear energy sinks, such as steady-state and strongly modulated responses. However, the differences are crucial: in the system with vibro-impact sink the strongly modulated response consists of randomly distributed periods of resonant and non-resonant motion. This salient feature allows us to identify this type of dynamic behavior as chaotic strongly modulated response (CSMR). It is demonstrated, that the CSMR exists due to special structure of a slow invariant manifold (SIM), which is derived with the help of a multiple-scale analysis of the system. In the considered system, this manifold has only one stable and one unstable branch. This feature defines new class of universality for the nonlinear energy sinks. In the system with the vibro-impact sink, such responses are observed even for very low level of the external forcing. This feature makes such system viable for possible energy harvesting applications

Development of analytical-numerical methods for dynamic analysis of geared transmission systems

The main objective of the present research activity is the study of geared transmission system dynamics, which is basically represented by a system of nonlinear differential equations. First of all, the different approaches to study the nonlinear dynamics of gears are qualitatively presented. Afterwards, the realization of a lumped parameter model is discussed by analyzing two different modeling strategies linked to two different numerical resolution techniques. The first modeling strategy is based on time integration techniques and enhances the employment of a commercial software to speed-up the modeling set-up phase. The proposed method rely on a block diagram technique and it is developed in Simcenter AMESim, a commercial software widely used in industries. By starting from the single gear pair model, detailed guidelines are given to construct any type of ordinary transmission layout by connecting some pre-programmed devices between them. In order to demonstrate the reliability of the approach, an experimental validation on industrial use case is proposed with excellent outcomes. The second modeling strategy rely on a frequency domain solution technique able to capture unstable solution branches in multi-valued frequency response regions. In particular, it proposes the Asymptotic Numerical Method combined to the Harmonic Balance Method as a valuable approach to solve the nonlinear dynamics of gear pairs. Thanks to a quadratic recast of the equation of motion, the Taylor and Fourier series can be computed in a very efficient way and each step produces a continuous representation of the solution branch making the continuation very robust. Effectiveness and reliability of the method are proved by comparing the numerical outcomes with that obtained from the Runge-Kutta time integration scheme. As a result, this technique provides for excellent computational performance despite additional time is needed for the quadratic recast of the equations system. Once a detailed analysis on the modeling strategy has been conducted, rattle noise and whine noise occurrence are investigated. Regarding the rattle noise, the research activity has conducted to the introduction of a new analytical parameter as a novelty to the current state of the art. A rattle index formulation is retrieved by starting from the classical 6-DOFs equation system defining the nonlinear dynamics of a gear pair. The proposed formulation may be applied to single or multiple branch geartrain, both in idle or loaded condions. The reliability of the analytical formulation is proved by numerical experiments which demonstrate the capability of the proposed index to instantaneously describe the vibro-impacts events related to any gear pair of the driveline. In addition its magnitude may be a measure of the tooth impact severity and it is shown to be a proper indicator of the potential presence of mutual interactions between different gear pairs pertaining to the same driveline. Finally, the investigation of whine noise occurrence addresses to an analytical formulation able to forecast the main overall direction and magnitude of bearing reaction forces on idler gear. By starting from the definition of meshing forces by means of Fourier series development, idler gear bearing forces are obtained under the hypothesis of quasi-static motion. This procedure demonstrates that the alternating component of bearing forces on idler gear describes an elliptical trajectory as the prime mover rotates over a pitch angle. The formulation directly links the bearing forces elliptical trajectory with the gear spatial position, the meshing phase and the amplitude of meshing forces. By properly setting the over-mentioned parameters one may be able to control the magnitude and direction of the overall idler bearing reaction forces. Numerical experiments were performed and the obtained results confirm the author intuitionL’obiettivo principale della presente attività di ricerca riguarda lo studio della dinamica non lineare degli ingranaggi che, di fatto, è rappresentata da un sistema di equazioni differenziali. Prima di tutto, viene presentata un ‘analisi qualitativa finalizzata a valutare i diversi approcci per studiare tale fenomeno. Successivamente, viene descritto lo sviluppo di un modello a parametri concentrati analizzando due diverse strategie di modellazione basate su metodi di risoluzione numerica diversi. Il primo approccio propone l’utilizzo di un software commerciale per velocizzare la fase di set-up del modello ed è basato su tecniche di integrazione temporale. Questa strategia di modellazione è sviluppata in Simcenter AMESim, un software commerciale distribuito da Siemens. Partendo dal modello di una singola coppia di ruote, viene dettagliata una procedura per costruire qualsiasi tipo di treno di ingranaggi grazie alla tecnica dei diagrammi a blocchi. Per dimostrare l’efficacia di tale tecnica, il metodo viene applicato ad un caso industriale ottenendo un’ottima correlazione numerico-sperimentale. Il secondo approccio si basa su tecniche di risoluzione numerica nel dominio della frequenza in grado di calcolare i rami instabili della risposta dinamica. Il metodo propone la combinazione del “Asymptotic Numerical Method” con il “Harmonic Balance Method” utilizzando una formulazione quadratica del sistema di equazioni differenziali. Grazie a tale formulazione, sia la serie di Taylor che quella di Fourier possono essere sviluppate in una maniera molto efficiente rendendo la continuazione della soluzione periodica molto robusta. L’affidabilità di questa tecnica è stata dimostrata confrontando i risultati con quelli ottenuti dal metodo di Runge-Kutta, basato sull’integrazione temporale. In più, tale tecnica garantisce performance computazionali eccellenti, anche se la riformulazione quadratica del sistema iniziale non è sempre facile da ottenere. Una volta analizzate le strategie di modellazione e le tecniche numeriche risolutive, lo studio si concentra su i fenomeni di rattle e whine noise. Riguardo il rattle noise, l’attività di ricerca ha portato all’introduzione di un nuovo parametro analitico come novità rispetto allo stato dell’arte. Partendo dal sistema di equazioni che governa il moto di una coppia di ruote dentate, è stato definito un indice analitico denominato “rattle index”. Tale indice può essere applicato a qualsiasi tipo di treno di ingranaggi, a uno o più rami, sia in condizioni di trasmissione di potenza che in folle. La sua affidabilità è supportata da simulazioni numeriche che dimostrano la capacità del “rattle index” di descrivere istantaneamente la perdita di contatto tra qualsiasi coppia di ruote di una trasmissione. Infine, la sua ampiezza è un indice della severità degli urti e permette di identificare l’esistenza di mutue interazioni tra le ruote della driveline. Infine, lo studio del whine noise ha portato ad una formulazione analitica capace di prevedere la direzione e l’ampiezza delle forze sui cuscinetti delle ruote oziose. Tale formulazione viene ottenuta partendo dalla definizione delle forze di ingranamento tramite lo sviluppo in serie di Fourier e ricavando le forze sui cuscinetti sotto l’ipotesi di moto quasi-statico. Questa procedura dimostra che le componenti alterne delle forze sui cuscinetti seguono una traiettoria ellittica quando il movente ruota di un passo angolare. La formulazione mette in relazione la traiettoria delle forze sui cuscinetti con la posizione delle ruote nel piano, la fase di ingranamento e l’ampiezza delle forze di ingranamento. Agendo sui parametri descritti, è possibile pilotare e controllare la direzione delle forze sui cuscinetti delle ruote oziose. La trattazione analitica è supportata da simulazioni numeriche con un ottimo riscontro

Analysis and control of the dynamic response of a higher order drifting oscillator

This work was supported by the EPSRC grant EP/P023983/1 Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.3994266.Peer reviewedPublisher PD