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Dinamica non-lineare e chaos nei rotismi planetari
No full textI riduttori a planetario sono stati largamente impiegati nelle trasmissioni meccaniche con grandi vantaggi, quali l'elevata coppia trasmissibile, il peso ridotto, l'alta affidabilità ed efficienza. Esempi di applicazioni dei rotismi planetari sono presenti nelle trasmissioni automobilistiche, nelle macchine agricole, negli elicotteri e nei motori di impiego aeronautico.
L'analisi dinamica dei riduttori planetari è essenziale per ridurre le problematiche di rumore e vibrazioni. Le forze dinamiche all'interfaccia tra solare-pianeta e corona-pianeta sono la principale fonte di tali problemi a causa della forte connessione tra rumore ed errore di trasmissione dinamico.
Le vibrazioni nei planetari inducono carichi dinamici, provocando la riduzione della vita del meccanismo e producendo rumore. La principale sorgente di vibrazione è l'eccitazione parametrica dovuta alla rigidezza d'ingranamento tempo-variante nelle coppie solare-pianeta e corona-pianeta, poiché il numero di coppie di denti in presa varia durante il funzionamento.
Questa tesi presenta un modello in grado di simulare il comportamento dinamico di un riduttore planetario a singolo stadio in presenza di gioco. La formulazione è generale, così da essere applicabile ad un numero qualsiasi di pianeti. Il modello include la rigidezza tempo-variante e il gioco per tutti gli accoppiamenti presenti; tiene conto altresì della deformabilità dei cuscinetti di supporto.
Nel presente modello, ogni elemento - cioè solare, pianeti, porta-satelliti e corona - possiede tre gradi di libertà: due traslazionali e uno rotazionale, per un totale di diciotto gradi di libertà per il caso di riferimento considerato, che presenta tre satelliti. Le equazioni del moto linearizzate sono state risolte per calcolare le frequenze proprie del sistema - a tal fine si è tenuto conto dei valori medi delle rigidezze di ingranamento coinvolte - e le equazioni non-lineari del moto sono state integrate numericamente per studiare l'effetto della rigidezza tempo-variante e del gioco. Il modello proposto ha poi permesso di descrivere il comportamento non lineare di un riduttore planetario.
È stata studiatala sensibilità delle frequenze proprie ai parametri del sistema: in particolare alla variazione delle rigidezze dei supporti nel caso di pianeti equi spaziati e disposti simmetricamente.
Il distacco dei denti in presenza di vibrazioni elevate introduce una non-linearità nel sistema esaminato. Il presente lavoro descrive la complessa dinamica di un planetario in tali condizioni facendo uso di un modello a parametri concentrati. Tale modello bidimensionale rappresenta le ruote come inerzie concentrate, gli ingranamenti per mezzo di molle non lineari che tengono conto della perdita di contatto tra i denti e della variazione periodica della rigidezza d'ingranamento dovuta al cambiamento del numero di coppie di denti in presa e i supporti come molle lineari.
Il presente lavoro esamina la complessa dinamica non lineare dei planetari facendo uso di metodi numerici all'interno del range di frequenze operative. Si è effettuata un'analisi di biforcazione per studiare la presenza di fenomeni di raddoppio di periodo e di chaos.
La dinamica dei planetari mostra diversi comportamenti non lineari: salti di soluzione, moti caotici e raddoppi di periodo si verificano quando la frequenza di ingranamento o le sue armoniche superiori sono prossime ad una frequenza propria del sistema. In condizioni di risonanza, la vibrazione risultante provoca il distacco dei denti in presa, determinando l'instaurarsi di effetti non lineari, quali il salto di soluzione e le risonanze sub-armoniche.Planetary gear system, have been widely used in power transmission systems owing to their advantages such as high torque to weight ratios, large speed reductions in compact volumes, availability of multiple speed reduction ratios, high reliability and high efficiency. Examples of application of planetary gears are automotive transmissions, tractors, helicopters, and aircraft engines.
The dynamic analysis of planetary gears is essential for reducing noise and vibration problems. The dynamic forces at the sun-planet and ring-planet meshes are the main sources of such problems due to the strong interaction between noise and dynamic transmission error.
Planetary gear vibrations produce dynamic loads, cause reduction in the structural life time and generate noise. The main source of vibration is the parametric excitation due to the periodically time-varying mesh stiffness of each sun-planet and ring-planet gear; because the number of gear tooth pairs in contact changes during gear rotation.
This thesis presents a full model to simulate the dynamic behavior of a single-stage planetary gear system with backlash. The generic nature of the formulation allows the analysis of a planetary gear set with any number of planets. This dynamic model considers time varying mesh stiffness and backlash, for all sun-planets and ring-planets meshes and bearing compliance.
In the present model each element (the sun gear, planets, carrier and the ring gear) has three degrees of freedom; two transverse and one rotational, thus giving a total of eighteen degrees of freedom for the case study system with three planets. The linear averaged equations of motion are solved to obtain the natural frequencies and the nonlinear equations of motion are solved numerically to study the effect of time-varying stiffness and backlash. The proposed model is employed to describe the nonlinear behavior of a planetary gear system.
The natural frequency sensitivity to system parameters is investigated for planetary gears. Parameters under consideration include support stiffnesses for symmetric and equally spaced system.
Tooth separations at large vibrations introduce nonlinearity in geared systems. The present work examines the complex, nonlinear dynamic behavior of spur planetary gears using a lumped-parameter model. The two-dimensional (2D) lumped-parameter model represents the gears as lumped inertias, the gear meshes as nonlinear springs with tooth contact loss and periodically varying stiffness due to changing tooth contact conditions, and the supports as linear springs.
This work examines the complex nonlinear dynamics of planetary gears by numerical method over the meaningful mesh frequency ranges. Bifurcation analysis have been studied to investigate the chaotic and period doubling phenomena.
The dynamics of planetary gears show different nonlinear phenomena. Nonlinear jumps, chaotic motions, and period-doubling bifurcations occur when the mesh frequency or any of its higher harmonics are near a natural frequency of the system. At resonance, the resulting vibration causes tooth separation leading to nonlinear effects such as jump phenomena and sub-harmonic resonance
Complex dynamics of planetary gear systems
No full textA full 2D dynamic model of a single-stage planetary gear system with backlash and time varying stiffness is considered here. This dynamic model is time variant and non-smooth, due to the presence of time varying meshing stiffness and backlash for all sun- planets and ring-planets meshes; moreover, bearing compliance is accounted for.
The linear averaged equations of motion are solved to obtain the natural frequencies; conversely, the fully nonlinear equations of motion are analyzed numerically to study the effect of time varying stiffness and backlash and to point out the nonlinear dynamics of the system. The complex dynamics is deeply investigated over the meaningful mesh frequency ranges. The dynamic scenario is obtained by means of bifurcation analysis and completed with time spectral and topological properties of the response
Performances of Nonlinear Vibration Absorbers for Beams subjected to Moving Loads
No full textThe goal of this work is a general assessment regarding the performances of linear and nonlinear dynamic vibration absorbers (DVAs) applied to the specific problem of moving loads or vehicles. The problem consists of a simply supported linear Euler-Bernoulli beam excited with a moving load/vehicle; a DVA is connected to the beam in order to reduce the vibrations. The moving vehicle is modeled by a single degree of freedom mass spring system. The partial differential equations governing the beam dynamics is reduced to a set of ordinary differential equations by means of the Bubnov-Galerkin method. A parametric analysis is carried out to find the optimal parameters of the DVA that minimize the maximum vibration amplitude of the beam. For the case of a moving vehicle, the energy absorbed by the DVA is evaluated. Comparisons among the performances of different types of linear and DVAs are carried out. The goal is to clarify if the use of nonlinearities in the DVAs can effectively improve their performances. The study shows that the most effective type of DVA for the test cases considered is the piecewise linear elastic restoring force
Symmetry breaking and chaos-induced imbalance in planetary gears
Get PDFThe goal of the present paper was a com- plete analysis of the dynamic scenario of planetary gears. A lumped mass two-dimensional model is adopted; the model takes into account: time-varying stiffness; nonsmooth nonlinearity due to the backlash, i.e., teeth contact loosing; and bearing compliance. The time-varying meshing stiffness is evaluated by means of a nonlinear finite element model, which allows an accurate evaluation of global and local teeth defor- mation. The dynamic model is validated by compar- isons with the most authoritative literature: linear nat- ural frequencies and nonlinear response. The dynamic scenario is analyzed over a reasonable engineering range in terms of rotation speed and torque. The clas- sical amplitude–frequency diagrams are accompanied by bifurcation diagrams, and for specific regimes, the spectral and topological properties of the response are discussed. Periodic, quasiperiodic and chaotic regimes are found; nonsmooth bifurcations lead period one to period two trajectories. It is found that the bearing com- pliance can influence the natural frequencies combina- tion magnifying the modal interactions due to internal resonances and greatly enlarging the chaotic regions. It is evidenced that the chaotic response indices a sym- metry breaking in the dynamical systems. The physical consequence is that the planetary gearbox under inves- tigation, which is perfectly balanced for each position, can suffer of a big dynamic imbalance when chaotic regimes take place; such imbalance gives rise to alter- nate and unexpected high-level stresses on bearings
Experimental identification of FGM shell properties
Get PDFFunctionally gradient materials (FGMs) have attracted a growing interest as advanced
structural materials because of their heat-resistance properties. In this paper, an experimental
study on the vibration of cylindrical shells made of a functionally gradient material
(FGM) composed of Polyethylene terephthalate (PET) is presented: to obtain functional gradient
proprieties the PET shell had been exposed at a thermal temperature gradient in the range
of its glass transition temperature of 79°C. The setting up of the experiment is explained and
deeply described along with the thermal characterisation of the specimen. The linear and the
nonlinear dynamic behaviour have been investigated. The shell behaviour is also investigated
by means of a finite element model, in order to enhance the comprehension of experimental results
Dynamic imbalance of high-speed planetary gears
No full textA non-linear 2D lumped mass model of a single-stage spur planetary gear system with time-varying mesh stiffness, bearing compliance and non-smooth non-linearity due to backlash is taken into account. The time-varying meshing stiffness is evaluated by means of a non-linear finite element model, through an accurate evaluation of global and local tooth deformation. The non-linear dynamic behaviour of the system is analysed over a reasonable range of rotation speed and torque. The possibility of occurrences of different dynamic phenomena and instability of the system with respect to the bearing compliance and operating parameters are also evaluated. The possibility of dynamic imbalance of equally-spaced planetary gears in the presence of chaotic regimes is discussed. Such imbalance may lead to unexpected high-level stresses on bearings and gears.The effect of tooth profile modification at the sun-planet and ring-planet meshes on the vibration behaviour of the planetary gear system is also investigated in this paper. In order to avoid modification on the ring gear, both tip and root reliefs are considered for sun and planet gears
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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