Stochastic analysis of the critical velocity of an axially moving cracked elastic plate

Abstract

In this study, a probabilistic analysis of the critical velocity for an axially moving cracked elastic and isotropic plate is presented. Axially moving materials are commonly used in modelling of manufacturing processes, like paper making and plastic forming. In such systems, the most serious threats to runnability are instability and material fracture, and finding the critical value of velocity is essential for efficiency. In this paper, a formula for the critical velocity is derived under constraints for the probabilities of instability and fracture. The significance of randomness in different model parameters is investigated for parameter ranges typical of paper material and paper machines. The results suggest that the most significant factors are variation in the crack length and tension magnitude.

Reliability analysis of processes with moving cracked material

“…The optimal average tension was sought for the maximum crack length by considering a productivity function which takes into account both instability and fracture. Moreover, an attempt to take the stochasticity of systems with moving material into account was made in the study by Tirronen et al [22] in which the safe transition of elastic and isotropic material with initial cracks was analyzed by modelling the problem parameters as random variables. In [22], critical regimes for the tension and velocity of the material were sought by considering the probabilities of fracture and instability.…”

“…Although tension in a printing press is known to change in time due to draw variations [24] and tension fluctuations have been suggested to cause web breaks [23], the tension in the system was regarded as constant in [25,21]. In [22], the tension was assumed to be constant while a crack travels through an open draw although the constant value was assumed to include uncertainty. In [2], only deterministic variations of tension were considered although the draw variations contain white noise in addition to specific high/low frequency components [24].…”

On reliability of systems with moving material subjected to fracture and instability

“…In studies by Tirronen et al [10,11] and Banichuk et al [12], an axially moving elastic and isotropic cracked plate was studied in the case in which parameter values include uncertainty.…”

“…However, in [10,12,11] longevity of the system was not considered, although it is of interest from the practical point of view. Temporal variations were not addressed in [10,12,11] although, e.g., in a printing press tension is known to vary with respect to time [13].…”

“…The present paper extends [10][11][12] by modeling randomly varying problem parameters as stochastic processes, which provide natural models for temporal stochastic variations of the system and enable examination of its longevity. In this paper, the tension variations are described by a stationary Ornstein-Uhlenbeck process.…”