Vibrations of the toothed belt of a linear drive can affect the dynamic positioning accuracy and smoothness which is important for practical applications. An analytical, one-dimensional model of a belt section is derived, taking into account the preload and flexural rigidity, as well as the non-linear cable effect and the acceleration forces due to the belt mass. In addition to the case of constant belt length, the temporal change of the length as a result of the axial motion is an essential part of the modeling. The solution of the equation of motion is limited to the dominant transverse deflection of the belt. For the longitudinal extension, the assumption of a quasi-static solution is taken. A FEM program is implemented to calculate the solution for given operation conditions. The evaluations are made for selected motions according to a path planning with trapezoidal velocity profiles. A modification of individual parameters, based on a practical parameter set, enables the analysis of the respective effects.

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The dynamic positioning accuracy and the smoothness are key aspects of linear belt drives where vibrations have a negative impact. A FE-Model, based on a one-dimensional analytical model of the belt, is derived to study the respective effects for different parameters. The model considers the dominant transverse deflection and a quasi-static assumption for the longitudinal deformation. The possibility to consider various time-varying belt lengths, the belt mass, stiffness and tension force is an essential part of the modeling. The non-linear cable effect is demonstrated by comparison with the linearized solution.Vibrations for the cases of constant and variable belt length are investigated and compared to common in literature used approximation formulas for the resonance frequencies of such systems. The influence on positioning deviations and the use of vibration reduction techniques considering the different belt sections of the linear drive can be derived from the computed results and are topics for future research.

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