Kobayashi [9] also co-authored a later paper, Shimizu et al. [31], investigating the dynamic behaviour of the metal pushing V-belt. The theory models the variator mechanism as a whole using Finite Element Analysis (FEA). The aim of the research was to study the forces acting on the belt components and quantify the stresses occurring within the components. The downside of this complex modelling was CPU run times of approximately 100 hours per operating condition. The simulation work showed a reasonably good fit to experimental data for the band tensions. The authors then added to the simulation the effects of belt misalignment as described by Robertson & Tawi [32] and Hendriks [2]. The authors used this to investigate the effects of the misalignment on the band tension distribution and the belt torque capacity. The results of the simulation indicated that with increased misalignment the maximum tension in the bands would increase. The simulation also indicated that the active arc, in which segment compression force changes, on the primary pulley increases as the
belt misalignment is increased. This indicates that increased belt misalignment effectively reduces the torque capacity of the belt drive.
A number of the papers discussed previously have shown that the addition of a simple pulley bending model can improve the fit of a model to the experimental data. A number of authors have investigated this aspect of the metal V-belt CVT, using pulley deflection methods to model the transmission and understand its function. One such paper is the analysis provided by Sorge [33], who investigates the effect of pulley bending on V-belt mechanics. The author applies the virtual displacement approach to give approximations to belt trajectories, tension distributions, axial thrust and slip. Unlike rubber belts the elastic deformation of metal belts is small, and generally may be neglected in comparison to the flexural deformation of the pulley, especially for large wrap radii. The paper analyses the bending of the pulley halves, as annular plates fixed at the inner edge, and assumes the belt to have infinite rigidity in both the longitudinal and transverse directions. The author concludes that the influence of pulley deflection is important when the belt trajectory is close to the outer radius. The influence decreases near the middle radius and becomes negligible near the inner clamped radius. Therefore at large radii the assumption of high belt stiffness is apt, but at smaller radii a mixed model should be adopted accounting for the deformation of both components. Although the analysis is an interesting approach to the problem from a new angle, the results and significance of the findings appear to be lost, with insignificant variables, such as sliding direction vs. pulley angle, being the validated output of the simulation.
A similar investigation of pulley deformation was performed by Sattler [34], analysing the mechanics of the belt drive considering both longitudinal and transverse belt stiffness, belt misalignment and pulley deformation. The author however opted to use an FEA approach to the modelling of the pulley deformations. The pulley is considered to deform in two ways, purely axial deformation and a skew deformation. The belt radius within the pulley is then determined by the addition of a local transverse compression of the chain and the shape of the pulley groove. Power losses in the wrap angle were predicted although, validation to experimental data was difficult as no internal losses in the belt were modelled. The sliding motion on the pulley is predicted to cause a 2% power loss at rated torque using ideal clamping forces, but this can double with some of the clamping pressures actually implemented. Under optimal force conditions the pulleys of the variator should exceed 97% efficiency, although additional losses exist associated with events within the belt itself.
Gerbert [35 & 36] also investigated pulley deformations. The author identifies three types of pulley flexibility, namely local deflections, plate deflections and skew pulleys. Local deflections are defined as local elastic deflections of the pulley surface and belt, while plate deflections are those associated with the properties of the pulleys, which produce global deflections not associated with localised forces. Finally the skew pulley effect is due to tolerances in the manufacture of the two pulley halves allowing them to move relative to each other. The author indicates the difficulty in modelling the final phenomena, as it is possible for these tolerances to change throughout the life of the transmission. Thus a pulley pair originally assembled such that only small skewness levels existed could end up displaying high levels of skewness later in its life.
On a similar subject, Robertson & Tawi [32] produced a model to describe the misalignment of the belt and pulleys over the ratio range of the transmission. They investigated the effects of the ratio at which zero misalignment occurs and how this ratio can be optimised to reduce maximum misalignment over the transmission ratio range. Hendriks et al. [2] developed a simple approximation to this misalignment problem and the complexity of 3D modelling used by the authors of this paper seems excessive in an effort to achieve very similar results. It is possible that these misalignment effects could influence the efficiency of the transmission. However implementing a model of the misalignment would be difficult due to the difficulty of measuring the position of zero misalignment, and effects are likely to be minimal compared to those of pulley deflection and skewness.