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This thesis investigates the characteristics of static torsional mesh stiffness, load sharing ratio, and transmission errors of gears in mesh with and without a localised tooth crack.Gearing is perhaps one of the most critical components in power transmission systems. The transmission error of gears in mesh is considered to be one of the main causes of gear noise and vibration. Numerous papers have been published on gear transmission error measurement and many investigations have been devoted to gear vibration analysis. There still, however, remains to be developed a general non-linear Finite Element Model capable of predicting the effect of variations of gear torsional mesh stiffness, transmission error, transmitted load and load sharing ratio. The primary purpose of this study was to develop such a model and to study the behaviour of the static torsional mesh stiffness, load sharing ratio, and transmission error over one completed cycle of the tooth mesh.The research outlined in this thesis considers the variations of the whole gear body stiffness arising from the gear body rotation due to tooth bending deflection, shearing displacement, and contact deformation. Many different positions within the meshing cycle were investigated and then compared with the results of a gear mesh having a single cracked tooth.In order to handle contact problems with the finite element method, the stiffness relationship between the two contact areas must be established. Existing Finite Element codes rely on the use of the variational approach to formulate contact problems. This can be achieved by insertion of a contact element placed in between the two contacting areas where contact occurs. For modelling of gear teeth in mesh, the penalty parameter of the contact element is user-defined and it varies through the cyclic mesh. A simple strategy of how to overcome these difficulties is ++ / also presented. Most of the previously published finite element analysis with gears has involved only partial teeth models.In an investigation of gear transmission errors using contact elements, the whole body of the gears in mesh must be modelled, because the penalty parameter of the contact elements must account for the flexibility of the entire body of the gear not just the local stiffness at the contact point.

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Gear transmission systems are considered one of the critical aspects of vibration analysis, and it contains various potential faults such as misalignment, cracks, and noise. Therefore, it requires vibration monitoring to ensure the system is operating properly. Case mounted accelerometers are frequently used to monitor frequencies in a system. However, it is not a simple task to identify and interpret the acceleration data since there are many gear mesh frequencies present. One of the approaches utilized by researchers to perform gear diagnostic is Finite Element Modeling. This study focuses on stiffness cycle and meshing stiffness of non-linear quasi-static finite element modeling. The comparisons of meshing stiffness will concentrate on the type of elements, the integration methods, the meshing quality, plane stress and plane strain analysis, sensitivity of model tolerance, and crack modeling. The results show that the FEA approach is extremely sensitive to tolerance, mesh density, and element choice. Also, the results indicate that a complete sensitivity and convergence studies should be carried out for a satisfactory stiffness match.

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This work investigates steady deformations, measured spectra of quasi-static ring deformations, natural frequencies, vibration modes, parametric instabilities, and nonlinear dynamics of high-speed planetary gears with an elastically deformable ring gear and equally-spaced planets. An analytical dynamic model is developed with rigid sun, carrier, and planets coupled to an elastic continuum ring. Coriolis and centripetal acceleration effects resulting from carrier and ring gear rotation are included. Steady deformations and measured spectra of the ring deflections are examined with a quasi-static model reduced from the dynamic one. The steady deformations calculated from the analytical model agree well with those from a finite element/contact mechanics (FE/CM) model. The spectra of ring deflections measured by sensors fixed to the rotating ring, space-fixed ground, and the rotating carrier are much different. Planet mesh phasing significantly affects the measured spectra. Simple rules are derived to explain the spectra for all three sensor locations for in-phase and out-of-phase systems. A floating central member eliminates spectral content near certain mesh frequency harmonics for out-of-phase systems. Natural frequencies and vibration modes are calculated from the analytical dynamic model, and they compare well with those from a FE/CM model. Planetary gears have structured modal properties due to cyclic symmetry, but these modal properties are different for spinning systems with gyroscopic effects and stationary systems without gyroscopic effects. Vibration modes for stationary systems are real-valued standing wave modes, while those for spinning systems are complex-valued traveling wave modes. Stationary planetary gears have exactly four types of modes: rotational, translational, planet, and purely ring modes. Each type has distinctive modal properties. Planet modes may not exist or have one or more subtypes depending on the number of planets. Rotational, translational, and planet modes persist with gyroscopic effects included, but purely ring modes evolve into rotational or one subtype of planet modes. Translational and certain subtypes of planet modes are degenerate with multiplicity two for stationary systems. These modes split into two different subtypes of translational or planet modes when gyroscopic effects are included. Parametric instabilities of planetary gears are examined with the analytical dynamic model subject to time-varying mesh stiffness excitations. With the method of multiple scales, closed-form expressions for the instability boundaries are derived and verified with numerical results from Floquet theory. An instability suppression rule is identified with the modal structure of spinning planetary gears with gyroscopic effects. Each mode is associated with a phase index such that the gear mesh deflections between different planets have unique phase relations. The suppression rule depends on only the modal phase index and planet mesh phasing parameters (gear tooth numbers and the number of planets). Numerical integration of the analytical model with time-varying mesh stiffnesses and tooth separation nonlinearity gives dynamic responses, and they compare well with those from a FE/CM model. Closed-form solutions for primary, subharmonic, superharmonic, and second harmonic resonances are derived with a perturbation analysis. These analytical results agree well with the results from numerical integration. The analytical solutions show suppression of certain resonances as a result of planet mesh phasing. The tooth separation conditions are analytically determined. The influence of the gyroscopic effects on dynamic response is examined numerically and analytically. / Doctor of Philosophy / Planetary gears in aerospace applications have thin ring gears for reducing weight. These lightweight ring gears deform elastically when transmitting power. At high speed, Coriolis and centripetal accelerations of planetary gears become significant. This work develops an analytical planetary gear model that takes account of an elastically deformable ring gear and speed-dependent gyroscopic (i.e., Coriolis) and centripetal effects. Steady deformations, measured spectra of quasi-static ring deformations, natural frequencies, vibration modes, parametric instabilities, and dynamic responses of planetary gears with equally-spaced planets are investigated with the analytical model. Steady deformations refer to quasi-static deflections that result from applied torques and centripetal acceleration effects. These steady deformations vary because of periodically changing mesh interactions. Such variation leads to cyclic stress that reduces system fatigue lives. This work evaluates planetary gear steady deformations with the analytical model and studies the effects of system parameters on the steady deformations. Ring deflections measured by sensors fixed to the rotating ring gear (e.g., a strain gauge), space-fixed ground (e.g., a displacement probe), and the rotating carrier have much different spectra. The planet mesh phasing, which is determined by gear tooth numbers and the number of planets, significantly influences these spectra. Simple rules are derived that govern the occurrence of spectral content in all the three measurements. Understanding these spectra is of practical significance to planetary gear engineers and researchers. Planetary gears have highly structured modal properties due to cyclic symmetry. Vibration modes are classified into rotational, translational, and planet modes in terms of the motion of central members (sun and carrier). The central members have only rotation for a rotational mode, only translation for a translational mode, and no motion for a planet mode. Translational modes have two subtypes, rotational modes have only one subtype, and planet modes may not exist or have one or more subtypes depending on the number of planets. For each subtype of modes, all planets have the same motion with a unique phase relation between different planets and the elastic ring gear has unique deformations. Understanding this modal structure is important for modal testing and resonant mode identification in dynamic responses. Sun-planet and ring-planet mesh interactions change periodically with mesh frequency. These mesh interactions are modeled as time-varying stiffnesses that parametrically excite the planetary gear system. Parametric instabilities, in general, occur when the mesh frequency or one of its harmonics is near twice a natural frequency or combinations of two natural frequencies. Closed-form expressions for parametric instability boundaries that bound the instability region are determined from the analytical model. Certain parametric instabilities are suppressed as a result of planet mesh phasing. Near resonances, vibration can become large enough that meshing teeth lose contact. The analytical model is extended to include the tooth separation nonlinearity. Closed-form approximations for dynamic responses near resonances are determined from the analytical model, and these analytical results compare well with those from numerical simulations of the analytical model. Tooth separation conditions are analytically determined. The influences of planet mesh phasing and Coriolis acceleration on dynamic responses near resonances are investigated numerically and analytically.

This work investigates the dynamics and vibration in gear systems, including spur and helical gear pairs, idler gear trains, and planetary gears. The spur gear pairs are analyzed using a finite element/contact mechanics (FE/CM) model. A hybrid analytical-computational (HAC) model is proposed for nonlinear gear dynamics. The HAC predictions are compared with FE/CM results and available experimental data for validation. Chapter 2 investigates the static and dynamic tooth root strains in spur gear pairs using a finite element/contact mechanics approach. Extensive comparisons with experiments, including those from the literature and new ones, confirm that the finite element/contact mechanics formulation accurately predicts the tooth root strains. The model is then used to investigate the features of the tooth root strain curves as the gears rotate kinematically and the tooth contact conditions change. Tooth profile modifications are shown to strongly affect the shape of the strain curve. The effects of strain gage location on the shape of the static strain curves are investigated. At non-resonant speeds the dynamic tooth root strain curves have similar shapes as the static strain curves. At resonant speeds, however, the dynamic tooth root strain curves are drastically different because large amplitude vibration causes tooth contact loss. There are three types of contact loss nonlinearities: incomplete tooth contact, total contact loss, and tooth skipping, and each of these has a unique strain curve. Results show that different operating speeds with the same dynamic transmission error can have much different dynamic tooth strain. Chapters 3, 4, and 5 develops a hybrid-analytical-computational (HAC) method for nonlinear dynamic response in gear systems. Chapter 3 describes the basic assumptions and procedures of the method, and implemented the method on two-dimensional vibrations in spur gear pairs. Chapters 4 and 5 extends the method to two-dimensional multi-mesh systems and three-dimensional single-mesh systems. Chapter 3 develops a hybrid analytical-computational (HAC) model for nonlinear dynamic response in spur gear pairs. The HAC model is based on an underlying finite element code. The gear translational and rotational vibrations are calculated analytically using a lumped parameter model, while the crucial dynamic mesh force is calculated using a force-deflection function that is generated from a series of static finite element analyses before the dynamic calculations. Incomplete tooth contact and partial contact loss are captured by the static finite element analyses, and included in the force-deflection function. Elastic deformations of the gear teeth, including the tooth root strains and contact stresses, are calculated. Extensive comparisons with finite element calculations and available experiments validate the HAC model in predicting the dynamic response of spur gear pairs, including near resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC model is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy. Chapter 4 investigates the in-plane motions in multi-mesh systems, including the idler chain systems and planetary gear systems, using the HAC method that introduced in Chap. 3. The details of how to implement the HAC method into those systems are explained. The force-deflection function for each mesh is generated individually from a series of static finite element analyses before the dynamic calculations. These functions are used to calculated the dynamic mesh force in the analytical dynamic analyses. The good agreement between the FE/CM and HAC results for both the idler chain and planetary gear systems confirms the capability of the HAC model in predicting the in-plane dynamic response for multi-mesh systems. Conventional softening type contact loss nonlinearities are accurately predicted by HAC method for these multi-mesh systems. Chapter 5 investigates the three-dimensional nonlinear dynamic response in helical gear pairs. The gear translational and rotational vibrations in the three-dimensional space are calculated using an analytical model, while the force due to contact is calculated using the force-deflection. The force-deflection is generated individually from a series of static finite element analyses before the dynamic calculations. The effect of twist angle on the gear tooth contact condition and dynamic response are included. The elastic deformations of the gear teeth along the face-width direction are calculated, and validated by comparing with the FE/CM results. / Ph. D.