**8. Conclusion**

102 Mechanical Engineering

In the case of wide-faced gears, gear body deflections (especially those of the pinion) cannot be neglected and the torsion/bending distortions must be modelled since they can strongly affect the contact conditions between the teeth. For solid gears, one of the simplest approaches consists in modelling gear bodies by two node shaft finite elements in bending, torsion and traction as described in Ajmi and Velex (2005) which are connected to the same mesh interface model as that described in section 3 and Fig. 6. Assuming that any transverse section of the pinion or gear body originally plane remains plane after deformation (a fundamental hypothesis in Strength of Materials), gear bodies can then be sliced into elemental discs and infinitesimal gear elements using the same principles as those presented in section 2. The degrees of freedom of every infinitesimal gear element are expressed by using the shape functions of the two-node, six DOFs per node shaft element. By so doing, all the auxiliary DOFs attributed at every infinitesimal pinion and gear are condensed in terms of the degrees

The approach in 6.1 is valid for solid gears but is irrelevant for deformable structures such as thin-rimmed gears in aeronautical applications for example where the displacement field cannot be approximated by simple polynomial functions as is the case for shafts. Most of the attempts rely on the Finite Element Method applied to 2D cases (Parker et al., 2000), (Kahraman et al., 2003) but actual 3D dynamic calculations are still challenging and do not lend themselves to extensive parameter analyses often required at the design stage. An

of freedom of the shaft nodes leading to a (global) gear element with 24 DOFs.

Fig. 18. Example of hybrid model used in gear dynamics (Bettaieb et al., 2007).

**7. Towards continuous models** 

**7.1 Pinion, gear distortions** 

**7.2 Thin-rimmed applications** 

A systematic formulation has been presented which leads to the definition of gear elements with all 6 rigid-body degrees-of-freedom and time-varying, possibly non-linear, mesh stiffness functions. Based on some simplifications, a number of original analytical results have been derived which illustrate the basic phenomena encountered in gear dynamics. Such results provide approximate quantitative information on tooth critical frequencies and mesh excitations held to be useful at the design stage.

Gear vibration analysis may be said to have started in the late 50's and covers a broad range of research topics and applications which cannot all be dealt with in this chapter: multimesh gears, power losses and friction, bearing-shaft-gear interactions, etc. to name but a few. Gearing forms part of traditional mechanics and one obvious drawback of this long standing presence is a definite sense of déjà vu and the consequent temptation to construe that, from a research perspective, gear behaviour is perfectly understood and no longer worthy of study (Velex & Singh, 2010). At the same time, there is general agreement that although gears have been around for centuries, they will undoubtedly survive long into the 21st century in all kinds of machinery and vehicles.

Looking into the future of gear dynamics, the characterisation of damping in geared sets is a priority since this controls the dynamic load and stress amplitudes to a considerable extent. Interestingly, the urgent need for a better understanding and modelling of damping in gears was the final conclusion of the classic paper by Gregory et al. (1963-64). Almost half a century later, new findings in this area are very limited with the exception of the results of Li & Kahraman (2011) and this point certainly remains topical. A plethora of dynamic models can be found in the literature often relying on widely different hypotheses. In contrast, experimental results are rather sparse and there is certainly an urgent need for validated models beyond the classic results of Munro (1962), Gregory et al. (1963), Kubo (1978), Küçükay (1984 &87), Choy et al. (1989), Cai & Hayashi (1994), Kahraman & Blankenship (1997), Baud & Velex (2002), Kubur et al. (2004), etc. especially for complex multi-mesh systems. Finally, the study of gear dynamics and noise requires multi-scale, multi-disciplinary approaches embracing non-linear vibrations, tribology, fluid dynamics etc. The implications of this are clear; far greater flexibility will be needed, thus breaking down the traditional boundaries separating mechanical engineering, the science of materials and chemistry.
