Analysis of the lamination stack influence on the stiffness of stator active component

For electric motors light weight construction is becoming increasingly important. Therefore the mechanical behavior of the lamination stack is necessary to be known for the simulation. The stacking of a lot of sheets has a relevant influence onto the behavior of the whole motor, because of the significant contact characteristics between the sheets. Quasi-static tests are performed to identify this. In these tests an axial load, representing the packaging process, is applied onto the stacked sheets and thereafter a cyclic load is superposed. With these quasi-static tests elastic and plastic effects and a hysteretic behavior are detected. For modeling this stiffness behavior nonlinear springs and frictional elements, containing the Coulomb's law, are assembled. The nonlinearities and the hysteresis are dependent on the sheet's roughness. Moreover the behavior of the stack is influenced by the coreplate varnish viscoelasticity. With modeling these effects, the measured values can be well simulated. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastic-plastic halfspace simulation

Due to the roughness of technical surfaces only the surface peaks are in contact for moderate contact pressures. Thus, the real contact area is smaller than the apparent contact area. Contact forces can only occur in the real contact area. Consequently it is necessary to determine the deformation of surface asperities in order to analyse the tribological properties of surfaces. The real contact area is usually small in initial contact. This leads to large contact pressures which in turn lead to the plastic deformation of surface roughness peaks. Therefore an elastic-plastic model is necessary. The halfspace model seems to be beneficial because there is only a system of equations on a surface mesh to be solved and not on a volume mesh like in the Finite-Element-Method. This leads to a much smaller system of equations which should allow reasonable calculation times even for large contact surfaces. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material modelling of a sheet-layered lamination stack by homogenization

In electrical machines, sheet-layered lamination stacks play an important role for the mechanical behavior of the system. Especially the interlayer between individual sheets and their interaction have a severe influence on the structure. In the context of performance and computational effort, it is desirable to avoid a full FE simulation of a lamination stack with every single sheet. Therefore, homogenization techniques are presented to identify a transversely isotropic surrogate material model, while Zero-Thickness elements are utilized during this process to cover the interplay of single sheets. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Two Dimensional Model for the Dynamical Contact of Elastic Rough Surfaces

To model the contact behavior including dynamical effects, a two dimensional mechanical model of elastic rough contact is developed. This model can simulate the contact behaviour between two rough surfaces depending on normal pressure, sliding speed and roughness profiles. The contact between two rough surfaces is reduced to a rough rigid and a rough elastic layer. The elastic layer is modeled by point masses connected by spring-damper elements. The total system is described by coupled ODEs. The number of ODEs and thus the degree of freedom of the model depends on the varying contact conditions. The contact conditions are monitored during the simulation and the simulation interrupts, in case the contact conditions change. The equations of motion are then adapted with respect to the contact constraints. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolutional contact with Coulomb's friction on a periodic microstructure

We consider the elasticity problem in a heterogeneous domain with an ε-periodic micro-structure, ε ≪ 1, including a multiple micro-contact in a simply connected matrix domain with inclusions completely surrounded by cracks, which do not connect the boundary, or a textile-like material. The contact is described by the Signorini and Coulomb-friction contact conditions. In the case of the Coulomb friction, the dissipative functional is state dependent, like in [2]. A time discretization scheme from [2] reduces the contact problem to the Tresca one (with prescribed frictional traction or state independent dissipation) on each time-increment. We further look for the spatial homogenization. The limiting energy and the dissipation term in the stability condition were obtained for the contact with Tresca's friction law in [4] for closed cracks and can be extended to textile-like materials. Using these results and the concept of energetic solutions for evolutional quasi-variational problems from [2], for a uniform time-step partition, the existence can be proved for the solution of the continuous problem and a subsequence of incremental solutions weakly converging to the continuous one uniformly in time. Furthermore, the irreversible frictional displacements at micro-level lead to a kind of an evolutional plastic behavior of the homogenized medium. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stacks in a two-level store

The total number of stack pushing operations that occur for each stack size in the set of all stack histories when compiling or evaluating an expression of size n is calculated. The expected fraction of operations that use the slow store when implementing a stack in a two-level store is then obtained. Two different paging strategies are then examined, and the fraction of stack pushing operations that are paging operations is obtained for both cases. The results are an application of the techniques of P. Flajolet, J. Françon, and J. Vuillemin for analyzing data structures that are subjected to sequences of operations.     

Frictional contact of elastomer materials on rough rigid surfaces

The frictional behavior of elastomer materials is still unexplored, but very important for many industrial applications. Special attention is turned to rubber friction on rough road tracks. Due to the non‐rigid material characteristics of the rubber it is not sufficient to use a constant friction coefficient like Coulombs law. The frictional qualities depend on many different influences like sliding velocity, applied normal stress, surface roughness, material properties and the temperature in the contact zone. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model of a layered plate considering nonideal contact between layers

We propose a mathematical model for doing calculations for layered plates, allowing for both rigid and sliding contact in the presence of frictional forces between the sliding layers. The model takes into account the distribution of tangential and normal displacements across the thickness of the sliding layered stack, and also the distribution of transverse normal stresses. The strain tensor is obtained using the Cauchy relations; the stress tensor is obtained based on Hooke's law. Tne Lagrange variational principle allows us to obtain the resolvent system of differential equations and the corresponding boundary conditions. The spatial model for deformation of a layered plate has a number of special features compared with familiar models. The system of differential equations has operators no higher than second order. It is described relative to displacements on the faces of the stack. This is convenient in solving problems involving sliding of layers with and without friction.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 671–676, September–October 1995.     

Non-linear response of an electrode–electrolyte interface impedance with the frequency

In this paper we analyze the frequency response of a distributed parameters non-linear circuit. This circuit is a fractal model of an electrode–electrolyte interface. The results show that the series equivalent resistance R_eq decreases as the number of fractal levels rises. Its frequency behavior is similar to the results described by other authors that used a constant phase angle element. At low frequency range, the series equivalent reactance X_eq decreases with frequency and R_eq shows a flat response in this range. The model also explains how interface geometry modifies the values of some electrochemical parameters. As roughness increases the Argand diagram shows smaller semicircles. The model behavior is like a single dispersion system with its central frequency increasing with the electrode roughness.     

Investigations of the influence of geometrical irregularities and roughness on the normal contact stiffness of joints

Conducting experiments is necessary to determine the normal contact stiffness of rough surfaces. In compression tests the interface of specimen pairs is loaded in normal direction and the deflection is measured by extensometers with strain gages. The results can be used for example to describe the contact behaviour of joints in finite element analyses. It is shown that geometrical irregularities which are superposed by the surface roughness have a large influence on the elastic deformation of a joint and on the determination of the contact stiffness. They can be characterized by a skewed and a waved form of the surface with wavelengths which are larger than the surface roughness. In the experimental setup several different surfaces are compressed. In addition, the influence of the size of the contact area on the deformation of a joint is analysed. The results of the normal contact tests are compared with numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The bending of a circular membrane on a linearly deformed foundation

The axisymmetric problem of the bending of a circular transversely-loaded membrane (i.e., a thin plate having no flexural stiffness), which lies without friction on a linearly deformed foundation, where there is contact over the whole area of the membrane, is considered. The problem is reduced to the combined investigation of a differential equation for the bending of the membrane and an integral equation of the first kind with an irregular kernel in the unknown contact pressure. The method of special orthonormalized polynomials and the regular asymptotic “large λ” method are used to solve the problem.     

An External Meniscus on a Thin Fiber Whose Profile Has Separate Rectification Points

We clarify the limits of applicability of the previously developed asymptotic method for defining the configuration of an external meniscus of a liquid on a thin fiber. As was found out earlier, if the fiber material is fully wet, while the fiber profile has rectilinear parts, then the mentioned method gives unsatisfactory results when being applied near the fiber, because it leads to the infinite growth of the contact line. We prove that in the case when a smooth convex fiber profile contains only separate rectification points, the asymptotic approach predicts the meniscus shape, as a whole, satisfactory. However, it inadequately describes the behavior of the contact line near rectification points of the profile, namely, it gives a line with breaks instead of an expected smooth one.     

Solvability of a dynamic contact problem between a piezoelectric body and a conductive foundation

We consider a mathematical model which describes the dynamic process of contact between a piezoelectric body and an electrically conductive foundation. We model the material’s behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the normal compliance condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove a continuous dependence result. Then, we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the contact by using a penalized approach and a version of Newton’s method. We implement this scheme in a numerical code and, in order to verify its accuracy, we present numerical simulations in the study of two-dimensional test problems. These simulations provide a numerical validation of our continuous dependence result and illustrate the effects of the conductivity of the foundation, as well.     

Some Convergence Results for a Class of Nonlinear Phase-Field Evolution Equations

Two heat diffusion problems in the framework of the parabolic phase-field model are presented. The first problem is related to a single isotropic fluid and the other describes the heat transmission between two different substances in contact. Some known existence and uniqueness results are briefly recalled. Then, an asymptotic analysis of both situations is carried out as the kinetic equation collapses to a temperature-phase relation of Stefan type, in the first case in the whole material, and in the second in only one of the substances. In both cases, a convergence result for the solutions is proved. The second problem shows some more mathematical difficulties that are due to the presence of nontrivial terms on the common boundary. In order to control the latter, some tools are used from the Γ-convergence theory for convex functionals.     

Efficient Calculation of Tread Block Vibrations

A model is presented that allows to calculate the dynamics of tyre tread blocks. The numerical efficiency is achieved by the combination of static and important modal ansatz functions, also known as Hurty/Craig/Bampton transformation. Nonlinear single point contacts provide the contact forces and consider the surface roughness. The non-smooth friction characteristic is approximated an arctan function and a parameter study of the slope parameter is conducted. Typical stickslip limit cycles of the tread block are shown. These phenomena can have a part in the tyre/road noise. Experimental results qualitatively agree with the modelled tread block vibrations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model of a spatially inhomogeneous one-dimensional active medium

We investigate the dynamics of one-dimensional discrete models of a one-component active medium analytically. The models represent spatially inhomogeneous diffusively concatenated systems of one-dimensional piecewise-continuous maps. The discontinuities (the defects) are interpreted as the differences in the parameters of the maps constituting the model. Two classes of defects are considered: spatially periodic defects and localized defects. The area of regular dynamics in the space of the parameters is estimated analytically. For the model with a periodic inhomogeneity, an exact analytic partition into domains with regular and with chaotic types of behavior is found. Numerical results are obtained for the model with a single defect. The possibility of the occurrence of each behavior type for the system as a whole is investigated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 506–519, September, 2000.     

Plastic deformation of rough surfaces

Friction forces are only transferred within the the real area of contact A_real, which is usually smaller than the apparent area of contact A_o. The maximum friction stress τ_fric is therefore determined by the shear limit τ_max in the area of real contact and the fraction of the real area of contact (τ_fric = τ_max (A_real/A_o)). For rough surfaces the size of A_real is governed o by the plastic deformation of the surface roughness. We present a fully elasto-plastic halfspace contact formulation based on the work of Jacq et al. [1]. Linear elastic-plastic material behavior is modeled based on v. Mises plasticity with isotropic hardening. The algorithm gives the residual stress as well as the full plastic deformation field due to a frictionless normal contact. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiscale simulation for description of rubber friction

The interaction between tire and road generates the transferable forces, which are necessary for driving dynamics and safety. These forces are based on friction between rubber material and pavement surface and depend on the roughness of the pavement, the slip velocity, the contact pressure and the temperature. Based on the finite element method, the friction coefficient is calculated by numerical simulation. The roughness of the pavement surface is described by the height difference correlation function (HDCF), which allows partitioning into different length scales. This multiscale approach is suitable to understand and to evaluate friction phenomena. These phenomena are hysteresis friction based on dissipation inside the rubber material and adhesion friction, which describes the direct bonding between two materials. Given, that the material parameters of rubber highly depend on temperature and the frictional dissipation leads to a warming of the rubber, the provision for these effects is necessary for a realistic desciption of friction. The method allows an understanding of friction phenomena on the micro-scale like the real contact area or the microscopic contact pressure. Also, the temperature distribution inside the tire cross-section can be illustrated. The resulting coefficient of friction is validated by experimental data based on linear friction tests and compared to analytical solutions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)