Tension variations in pliable material in production machinery

Many production machines are used in which flexible material is processed. In these machines, cyclic tension variations can be induced in the material by interaction with the machine components. At resonance, these variations can be large and may result in breakage or distortion of the material. It is important to determine these resonant frequencies, and their dependence on the machine parameters, to help avoid these problems at the machine design stage. A mathematical model of such machines has been developed. Materials with linear elastic and linear visco-elastic tensile properties have been considered. The effects of friction between material and guide rollers have been included. A numerical method has been used to solve the resulting differential equations. This method gives accurate results even when adhesion between the material and some machine components is lost. By assuming the tension perturbations proportionately small, an approximate linear model was derived. It was found to give resonant frequencies close to those predicted by the non-linear model, but gave less accurate estimates of the amplitudes at resonance. For the linear model orthogonality relations for the eigenvectors were derived. Using these relations expressions were obtained which are first order approximations to the amplitudes of oscillation at resonance. The amplitudes are related to the angles of wrap round the rollers and, in the visco-elastic case, to the loss factor of the material. In addition, the expressions determine how the amplitudes are related to the position of the corresponding span in the system. Expressions were also found using perturbation theory. They give the local variation of the resonant frequencies with respect to the span lengths and moments of inertia of components of the system. These relations are used to carry out an analysis of certain types of system. Experimental work has been carried out, using a closed loop of material, to test the predictions of the mathematical model. A variety of materials was used with several different systems of rollers. In general, the results show that the resonant frequencies detected corresponded closely with those predicted, but the amplitudes tended to be over-estimated. Finally a programme of work was carried out on a production machine, and the results compared with model predictions.