Analytical prediction for multiple roll systems with negative gaps

Multiple roll coaters with deformable rolls are widely spread for the coating of thin liquid films with a wet film thickness of only a few microns and below on continuous substrates. The presented analytical solution for multiple roll systems offers for the first time the possibility to predict the performance of multiple roll coaters with a nip feed system and negative gaps. The solution is based on a previously presented solution of the authors for roll pairs and extended to multiple roll systems. The results are validated with experimental data from literature and the main influencing factors of multiple roll systems are investigated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of roll grinding system dynamics with rotor equations and speed control

A nonlinear dynamical model for paper machine roll grinding process is investigated through a group of delay differential equations with one constant time-delay. In this model, the time-delay effect is originated from shape error traces on the surface of the roll. The contact interaction of the roll and the grindstone is based on the wear theory, and the lateral deformations of the roll as a simply supported continuous beam element inside a rotational coordinate frame and the rotational rigid body vibration system are considered. The PD-controllers of the roll and the grindstone drives are also included. The numerical simulations for time-history responses provide a view of the stability of this grinding process for the design, analysis and verification of industrial roll grinding measurements in future.     

Modelling and experimental validation of the deflection of a leveller for hot heavy plates

In order to successfully automate levelling processes, in particular for heavy plates, the deflection of the leveller has to be compensated based on a deflection model. In this work, a detailed mathematical deflection model of a hot leveller with bending mechanism and its experimental validation are presented. The roll intermesh profiles are calculated based on the deflection of the work rolls that are elastically supported by support rolls, frames, posts and adjustment screws. The deflection model is suited to compensate the effect of deflection on the roll intermesh and the plate flatness as well as to assess the loads of critical parts, for example the support rolls. A new experimental design to measure the deflection of a leveller is presented and successfully applied for model validation. The work roll deflection is measured directly by means of displacement sensors that are inserted in cut-outs of test plates. These test plates are modelled as linear elastic stripes. For normal load levels, the relative accuracy (repeatability) of the roll intermesh prediction of the model is better than 0.08 mm.     

Convection patterns near a lateral boundary

In Rayleigh–Bénard convection between rigid conductingplanes, a roll pattern parallel to a lateral boundary is unstableto cross-rolls in the vicinity of the boundary. Normal modesof the instability are determined and a stable finite amplitudestructure which consists of a combination of rolls paralleland perpendicular to the boundary is identified in the weaklynonlinear regime. The theory incorporates the presence of asmall boundary imperfection and assumes that the roll patternin the bulk of the fluid is parallel to the lateral boundary.The extent of the perpendicular cross-rolls is determined asa function of and a nonlinear interaction parameter µwhich depends on the Prandtl number of the fluid. In the standardRayleigh–Bénard system, where µ > 1, rollsare the preferred weakly nonlinear equilibrium state in thebulk of the fluid. However, solutions are also examined herefor µ < 1, where the roll pattern is replaced by apattern of square cells as time evolves; such patterns are relevantin a variety of related physical systems. For µ > 1,solutions are also found for the case where the roll patternin the bulk of the fluid is perpendicular to the lateral boundary.     

Prediction of Extreme Ship Motions in Irregular Waves

The occurrence of indirectly excited large amplitude roll motions of ships is investigated under the influence of a time-varying righting lever curve in waves. In order to account for the irregular character of ocean waves appropriately, the ship motions are described within the framework of nonlinear dynamics and stochastic process theory. The analysis of roll motions in regular waves provides a qualitative understanding of the occurrence of unpredictable large amplitude roll motions as observed in ocean waves. However, quantitative information on the rolling behavior in irregular waves may be retrieved only from a probabilistic analysis. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Localized Roll-Solutions in the Ekman-Couette-System

In the Ekman-Couette-System, where the usual Couette-System is additionally rotated about its normal axis, localized single roll solutions have been known for some time. By varying different system parameters, a new localized solution emerging by saddle-node bifurcations from these known solutions has now been found. This new solution is of the same localized nature like the old solution but with an additional roll. Further bifurcations lead then to an increasing number of rolls, still localized. This behavior is kind of analogous to the so called ‘homoclinic snaking’ which has recently been investigated in conjunction with the Swift-Hohenberg equation and binary fluid convection. It might link the unstable localized single roll solutions with stable multi roll solutions or even with stable periodic roll solutions, which has to be shown yet. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Almost Sure Stability Analysis of Parametric Roll in Random Seas Based on Top Lyapunov Exponent

Stability analysis of the upright position of a ship in random head or following seas is presented. Such seas lead to parametric excitation of roll motion due to periodic variations of the righting lever. The development of simple criteria for the occurrence of parametric induced roll motion in random seas is of major interest for improvement of the international code on intact stability provided by the International Maritime Organization. The stability analysis in random seas is based on the calculation of the top Lyapunov exponent using the fact, that a negative top Lyapunov exponent yields no roll motion. With this findings, roll motion can be excluded for specific sea states. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Roll Wave Structure in Long Tubes with Compliant Walls

We consider a flow of a fluid in a long vertical tube with elastic walls and show that, for certain parameters of the flow, small perturbations of the flow at the inlet section of the tube give rise to roll waves. Depending on the properties of the closing relation, either regular or anomalous roll waves are formed. In the latter case, a roll wave is characterized by two strong discontinuities that connect regions of continuous flow. We present the results of numerical simulations of the development of a pulsatile flow mode for convex and nonconvex closing relations that demonstrate the formation of regular and anomalous roll waves. We also construct a two-parameter class of exact periodic solutions and obtain existence diagrams for roll waves.     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

Numerical Computation of Parametric Induced Roll Motions in Random Seas

For a vessel in open seas, the sudden appearance of roll motions due to waves from the front or rear leads to dangerous situations up to capsizing. The equations of motion used to analyze the roll motion include the righting lever curve. This curve is set up by means of hydrostatic calculations and approximated by polynomials for further analysis. The irregular waves are modeled in terms of a continuous-time ARMA process. The resulting model of stochastic differential equations is investigated numerically by Local Statistical Linearization. The necessary stochastic moments and their derivatives are computed using Itô's differential rule and Gaussian closure. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical model for coupled roll and yaw motions of a floating body in regular waves under resonant and non-resonant conditions

The prediction of resonance is very important with respect to the vessels stability in the early stages of design. In this paper, an efficient modeling approach is presented to determine coupled roll and yaw motions of a symmetric and slender floating body when the influences of small amplitude regular waves are dominant. The angular motions described in time domain by considering all internal and external forces are transformed into frequency domain to obtain motion characteristics. We adopt a semi-analytical treatment to obtain roll and yaw motions and derive system instability due to roll resonance. To compute hydrodynamic forces, we employ strip theory method where frequency dependent sectional added-mass, damping and restoring coefficients are derived from the Frank’s close-fit curve. Numerical experiments carried out for a vessel of mass 19,190 ton under the action of wave of frequencies 0.56 and 0.76 rad/s with zero and non-zero initial conditions are reported and the effect of various parameters on system stability is investigated. Model results indicate that damping factor (ς) plays a pivotal role when wave encountering frequency (ω) and undamped natural frequency (β) are nearly equal. The essence of this study lies in the efficient modeling technique to evaluate damping factor and critical encountering frequency regime for a given ship particulars when experimentally derived resonance zone is absent.     

Linear Stability of Viscous Roll Waves

The purpose of this paper is to study the linear stability of “viscous” roll waves. These are periodic continuous traveling waves solutions of viscous perturbations of inhomogeneous hyperbolic systems. We first study the scalar case for the Burgers equation and for an inhomogeneous hyperbolic equation. Then we analyze the stability of roll waves, solutions of the shallow water equations with a real viscosity. In both cases, we first analyze the Evans function and compute an asymptotic expansion in the low frequency regime. Under a strong spectral stability condition, we prove the linear stability of viscous roll waves, solutions of the Saint Venant equations, with pointwise estimates on the Green functions.     

Analytical model for deformable roll coating with nip feed

The presented work deals with an analytical solution for a roll coater with deformable rolls, as commenly used in industry. The focus is on the calculation of the nip feed system in a forward coating mode. The calculation is done with thin film theory and includes Hook's law for elasticity for modelling the elasticity behaviour of the deformable roll. The new model has been validated by experimental data from literature. The agreement is sufficiently good. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled mathematical model for the rollgap and interstand regions in a hot strip mill

In this paper we consider the process of hot-rolling steel strip.The thickness is reduced by passing the strip through a sequenceof rollers with reductions varying from 40% at the first rollpair to about 10% at the last pair. Between the roll pairs,the strip deflection curve is controlled by a looper roll. Amathematical model of the coupled roll gap and interstand regionis required to assist in the initial setup so as to spread theloads and torques evenly through the roll stands. The modelis of potential value in monitoring the transient behaviourof the rolling process. A review of the various models for eachregion and a coupled complete rolling model is described.     

The structure of roll waves in two-layer flows

The structure of non-linear waves in a two-layer flow of an incompressible fluid in extended channels is investigated. Periodic discontinuous solutions, describing roll waves of finite amplitude, are constructred for the equations of two-layer shallow water. “Anomalous” waves of limited amplitude are found which correspond to the transition from stratified to slug flow conditions.     

A multi-objective programming approach to 1.5-dimensional assortment problem

In this paper we study a 1.5-dimensional cutting stock and assortment problem which includes determination of the number of different widths of roll stocks to be maintained as inventory and determination of how these roll stocks should be cut by choosing the optimal cutting pattern combinations. We propose a new multi-objective mixed integer linear programming (MILP) model in the form of simultaneously minimization two contradicting objectives related to the trim loss cost and the combined inventory cost in order to fulfill a given set of cutting orders. An equivalent nonlinear version and a particular case related to the situation when a producer is interested in choosing only a few number of types among all possible roll sizes, have also been considered. A new method called the conic scalarization is proposed for scalarizing non-convex multi-objective problems and several experimental tests are reported in order to demonstrate the validity of the developed modeling and solving approaches.     

The Compaction of Granular Materials by Rolling II. The Effect of Material Compressibility

In an earlier paper a theory for the compaction of an incompressiblegranular material into a cohesive strip by rolling was developed.The present paper deals with the extension of this theory tothe case of a compressible material. The investigation is basedon the model of the plane-strain deformation of an isotropiccontinuum satisfying the Coulomb-Mohr flow criterion in whichthe material density is related to the pressure by a power law.General equations for the determination of the roll pressureare derived and are used to compute the roll force and torquefactors for a particular physical process. The results of thesecomputations are compared with values obtained experimentally.     

Three-dimensional electronic display

Three-dimensional pictures are of potential importance to many users of graphical displays. One of the most versatile 3D-effects is achieved by giving the viewer access to a rotating device by means of which he can roll the figure arbitrarily in space. This paper discusses both software and hardware realizations of three-dimensional rotating.     

Switching Stability of Automotive Roll Dynamics Subject to Interval Uncertainty

In this paper we apply some recent results on the stability of switched systems with interval uncertainty for analyzing the roll stability of automotive vehicles subject to bounded parametric uncertainties and switching. The application is motivated by the fact that the roll dynamics of an automotive vehicle is affected in a significant and a nonlinear fashion due to possible sudden switches in the vehicle's center of gravity (CG) height, as well as the uncertainty in the suspension parameters. Utilizing a recent stability result, it is possible to model such parametric variations as bounded interval uncertainties for this problem, and obtain easily verifiable conditions for analyzing the stability of the resulting switched/uncertain system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

新型纺纱胶辊的优化设计和现场生产管理

本文介绍应用实验设计法对纺纱工艺中主要牵伸元件胶辊配方进行抗干扰最优化设计及胶辊生产现场管理