Dynamic Modeling Approach for a Continuous Moving Belt

Belt drives are used in numerous applications, such as automotive engines, household applications or industrial engines, to transmit power between different machine elements. Because of their simple installation and low maintenance together with their ability to absorb shocks, they are frequently used instead of chain or geared transmission systems. However, they can exhibit complex dynamic behaviors, such as the transverse vibrations of the belt spans, sliding of the belt over the pulley, etc. All this phenomena impact the belt life and also the acoustic comfort. It is therefore of interest to predict the dynamic response of such systems using numerical models. Examples of related simulation models and references to this class of problems can be found in [1] and [2]. In this work a mathematical model for the non-synchronous belt drive is presented, which is able to describe the entire belt (free spans as well as wrapped arc sections), and which makes the need of dissections unnecessary. In order to simulate the dynamic behavior of the belt, the longitudinal movement and the transversal vibrations are modeled. Furthermore, the belt-pulley interaction is incorporated and modeled by two functions, which define the belt-pulley repulsion force, modeled by elastic contact conditions, and the transferred torque, modeled by a slip approach. Model The movement of the belt drive can be separated into longitudinal motion and lateral vibration, where the lateral displacement describes the belt configuration as offset from a reference state. This reference configuration agrees to the unloaded ideal geometric (static) shape of the drive. Hence, the reference configuration of the belt can be described by ideal straight segments and arcs wrapped around the pulleys, and which as a whole describe a closed curve in two dimensions:   2 0 0 0 0 0 0 : 0, with (0) ( ) and 1 x L x x L x    (1) The parameter 0 L is the length of the belt in the reference configuration. The position of a moving belt is distorted with respect to its reference state and can be described by the lateral displacement ( , ) w w s t  (2) which is a function of arc length (of the reference state) s and time t . The curve of the belt at time t is given as: 0 0 ( , ) ( ) ( , ) ( ) x s t x s w s t Rx s    (3) where R denotes the 90° counter clockwise rotation matrix: 0 1 : 1 0 R         The lateral displacement of the belt w is represented as a linear combination of N time invariant basis functions i w : 0 ( , ) ( ) ( ) ( ) ( ) T i i i N w s t q t w s q t w s      (4) The local longitudinal dynamics of the belt is neglected, therefore only one global uniform longitudinal displacement long q and global transport velocity v of the belt is taken into consideration.