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This paper provides a thorough investigation of a heat conduction problem that pertains to tolerance modelling in layered materials made up of multiple components. These media are functionally graded materials and thus have varying properties that affect their effectiveness. The proposed equations explain the conduction of heat in layered composites. The formulation involves partial differential equations, which utilise smooth and slowly varying functions. Notably, an extension of the unified tolerance modelling procedure is presented generalising existing models for two-component step-wise functionally graded materials (FGMs). This extension allows for the analysis of specific issues related to heat conduction in multi-component stratified composites with a transversal gradation of effective properties. This is the most important novelty achievement of the present paper because it will contribute to advancing knowledge and allows researchers, engineers, and practitioners to use the method in a broader context, addressing a more extensive set of real-world situations not limited to the number of component materials. [ABSTRACT FROM AUTHOR]

Dynamic problems of elastic non-periodically laminated solids are considered in this paper. It is assumed that these laminates have a functionally graded structure on the macrolevel along the x1-axis and non-periodic structure on the microlevel. However, along the other two directions, i.e., x2 and x3, their properties are constant. The effects of the size of a microstructure (the microstructure effect) on the behaviour of the composites can play a significant role. This effect can be described using the tolerance modelling method. This method allows us to derive model equations with slowly varying coefficients. Some of these terms can depend on the size of the microstructure. These governing equations of the tolerance model make it possible to determine formulas describing not only fundamental lower-order vibrations related to the macrostructure of these composite solids, but also higher-order vibrations related to the microstructure. Here, the application of the tolerance modelling procedure is shown to lead to equations of the tolerance model that can be used for non-periodically laminated solids. Then, these model equations are mainly used to analyse a simple example of vibrations for functionally graded composites with non-periodically laminated microstructure (FGL). Similar problems were investigated in the framework of the homogenised (macrostructural) model (Jędrysiak et al. 2006); the resulting equations neglect the microstructure effect. [ABSTRACT FROM AUTHOR]

Some considerations of slender elastic nonperiodic beams are shown in this paper. These beams have a functionally graded structure on the macro-level along the x-axis, and a nonperiodic structure on the micro-level. The effect of the size of the microstructure on the behavior of the beams can play a crucial role. This effect can be taken into account by applying the tolerance modelling method. This method leads to model equations with slowly varying coefficients, some of which depend on the microstructure size. In the framework of this model, formulas of higher order vibration frequencies related to the microstructure can be determined, not only for the fundamental lower-order vibration frequencies. Here, the application of the tolerance modelling method was mainly shown to derive the model equations of the so-called general (extended) tolerance model and standard tolerance model, describing dynamics and stability for axially functionally graded beams with the microstructure. A simple example of free vibrations of such a beam was presented as an application of these models. The formulas of the frequencies were determined using the Ritz method. [ABSTRACT FROM AUTHOR]

Sandwich structures are commonly used in many branches of modern engineering, such as aerospace or naval constructions. In this work, a vibration analysis of such structures is performed with the use of an anlytical model based on a zig-zag hypothesis. Due to the assumed periodic microstructure, which may occure in any layer of the structure, the initial governing equations describing its dynamic behaviour may contain periodic, non-continuous coefficients. The main aim of the presented paper is to show an analytical solution to the issue of the vibration analysis of the mentioned structures. With the use of the tolerance averaging technique, the initial governing equations are transformed to the form with constant coefficients, which is convenient to solve using well-known mathematical methods. The derived model is a versatile solution for any type of periodically inhomogeneous sandwich plate, including sandwich plates with a honeycomb core. Eventually, in the calculation example, the application of the derived averaged model in the analysis of vibrations of such structures is presented and discussed. The convergence of results of the tolerance model and FEM analysis proves the correctness and superiority of the proposed solution. [ABSTRACT FROM AUTHOR]

Thin elastic periodic plates are considered in this paper. Since the plates have a microstructure, the effect of its size on behaviour of the plates can play a crucial role. To take into account this effect, the tolerance modelling method is applied. This method allows us to obtain model equations with constant coefficients, which involve terms dependent of the microstructure size. Using the model equations, not only can formulas of fundamental lower-order vibration frequencies be obtained, but also formulas of higher-order vibration frequencies related to the microstructure. In this paper, the effect of the material periodic microstructure on free vibration frequencies for various boundary conditions of the plates was analysed. To obtain proper formulas of frequencies, the Ritz method is applied. Moreover, some results are compared to the results calculated using the FEM. [ABSTRACT FROM AUTHOR]

Copyright of Acta Scientiarum Polonorum: Architectura is the property of Wydawnictwo SGGW and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract Considered are free and forced transverse vibrations of slender periodic beams of finite length. It is assumed that the vibration amplitude is of the order of cross-section dimensions, still much smaller than the beam length. An averaged non-asymptotic model is proposed as a tool in analysis. The description is based on the tolerance approach to averaging of differential operators, using the concept of weakly slowly-varying function. The resulting differential equations with constant coefficients involve the effect of periodicity cell length. The model is verified by comparison of linear frequencies and mode shapes with Finite Element Method results, and then applied in analysis of free and forced vibrations of beam with variable cross-section. The method employed in obtaining the solution involves Galerkin orthogonalization and Runge–Kutta (RKF45) method. The results of nonlinear vibrations analysis are presented by backbone and amplitude-frequency response curves, time series, Poincare sections and bifurcation diagrams. [ABSTRACT FROM AUTHOR]

The present study aims to analyse a two-dimensional problem of displacements in theory of thermal stresses for multicomponent, multi-layered periodic composites. The model equations are obtained within the framework of the tolerance modelling procedure. These equations allow to determine the distribution of displacements caused by the temperature field in the theory of thermal stresses. The paper presents an example of a solution of a boundary value problem. [ABSTRACT FROM AUTHOR]

The well-known parabolic Heat Transfer Equation is a simplest recognized description of phenomena related to the heat conductivity in solids with microstructure. However, it is a tool difficult to use due to the discontinuity of coefficients appearing here. The purpose of the paper is to reformulate this equation to the form that allows to represent solutions in the form of Fourier’s expansions. This equivalent re-formulation has the form of infinite number of equations with Fourier coefficients in expansion of the temperature field as the basic unknowns. The first term in Fourier representation, being an average temperature field, should satisfy the well-known parabolic heat conduction equation with Fourier coefficients as fields controlling average temperature behavior. The proposed description takes into account changes of the composite periodicity accompanying changes in the variable perpendicular to the surfaces separating components, concerning FGM-type materials and can be treated as the asymptotic version of Heat Transfer Equation obtained as a result of a certain limit passage where the cell size remains unchanged. [ABSTRACT FROM AUTHOR]

In this paper the problem of linear thermoelasticity in a laminate with functional gradation of properties is considered. In micro level this laminate is made of two different materials, microlaminas, distributed non-periodically but also not randomly along one of directions, what in macro level results in aforementioned functionally gradation of laminate properties. In order to describe behavior of such structure, equations of two models are here presented – the tolerance and the tolerance-asymptotic model. Both are obtained by the tolerance averaging technique. The basic aim of this work is to analyse the influence of some terms from these averaged equations on the distribution and the values of the displacements and the temperature functions. To solve the equations of two proposed models the finite difference method is used. [ABSTRACT FROM AUTHOR]

Boundary effect behavior understood as near-boundary suppression of boundary fluctuation loads is described in various ways depending on the mathematical representation of solutions and the type of the center. In the case of periodic composites, the homogenization method is decisive here. In the framework of the Tolerance Averaging Approach, developed by prof. Cz. Woźniak leading to an approximate model of phenomena related to periodic composites this effect is described by a homogeneous part of differential equation for fluctuation amplitudes and usually this approximate description of the boundary effect behavior is restricted to a single fluctuation. In this paper, contrary to the previous elaborations, the boundary effect is developed in the variant of the tolerance thermal conductivity model in which the temperature field is represented by the Fourier expansions composed by an average temperature with infinite number of Fourier terms imposed on the average temperature as tolerance fluctuation suppressed in the framework of the boundary effect. [ABSTRACT FROM AUTHOR]

The object of analysis is a heat conduction problem within the framework of tolerance modelling in laminated media with a functional gradation of effective properties. In contrast to the known asymptotic models (based on the homogenization technique), the characteristic feature of the tolerance model equations is that they make it possible to analyze the effect of the microstructure size on the overall behavior of the laminate. The proposed model equations describe heat conduction in laminates by mean of partial differential equations with smooth and slowly varying functions. This paper describes a certain extension of the unified tolerance modelling procedure, which makes it possible to analyze specific problems of heat conduction in laminates with a transversal gradation of effective properties. [ABSTRACT FROM AUTHOR]

Non-nominal visualisation is used to evaluate how geometrical variation influences the appearance of split-lines between components on a car body. Depending on the simulation method, visualised variation can be represented in an unrealistic manner, which can affect the perception of the result. This paper presents a comparative study in which eye-tracking equipment has been used in order to explore whether the interpretation of variation differs between rigid- and non-rigid-based variation simulations during the evaluation of visualised models. Subjects from the automotive industry were asked to evaluate two virtual models where the same amount of variation was represented by rigid and non-rigid models. Eye tracking was used to record how the subjects performed the assessment. The result shows that there is a significant difference in how simulation results are interpreted using rigid versus non-rigid simulation. [ABSTRACT FROM PUBLISHER]

Thin periodic plates with large (i.e. of the order of plate thickness) deflections are considered. In this note the tolerance and the asymptotic models of these plates are presented. As an example of applications, these models are used to analyse a bending of periodic plates under various loadings. [ABSTRACT FROM AUTHOR]

Copyright of Archives of Civil & Mechanical Engineering (Oficyna Wydawnicza Politechniki Wroclawskiej) is the property of Oficyna Wydawnicza Politechniki Wroclawskiej and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Archives of Civil & Mechanical Engineering (Oficyna Wydawnicza Politechniki Wroclawskiej) is the property of Oficyna Wydawnicza Politechniki Wroclawskiej and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

An optimal approach for process tolerance balancing is presented. The new approach is based on process capabilities and is to be used in the stage of process planning. A nonlinear programming model is used to simultaneously optimise process tolerances of required operations. In the optimisation model, the objective function is to minimise the total manufacturing cost with different weighting factors. Using the estimated standard deviations of the dimensions and the manufacturing cost-tolerance functions, the constraint equations for the process tolerance chains and the manufacturing capability indices are established, together with a model for the economical tolerance bounds of machine tools. A practical example was used to verify the usefulness of the proposed approach. The results of the comparative study show that the proposed approach is more advantageous in relaxing tolerance requirements and in reducing scrap rates generally, compared with the existing methods. [ABSTRACT FROM AUTHOR]

This paper deals with the linear natural vibrations analysis of beams where the geometric and material properties vary periodically along the beam axis. In contrast with homogeneous prismatic beams, the frequency spectra of such beams are irregular as there exist enlarged intervals between some adjacent frequencies. Presented here are two averaged models of beams based on the tolerance modelling approach. The assumptions of classical Euler–Bernoulli and Timoshenko–Ehrenfest beam theories are adopted as the foundations. The resulting mathematical models are systems of differential equations with constant, weight-averaged coefficients. This makes it possible to apply any classical method of solution suitable for homogeneous beams, such as Galerkin orthogonalization. Here, emphasis is placed on the comparison of natural frequencies neighbouring the frequency band-gaps that are obtained from these two theories. Two basic cases of material and geometric property distribution in a periodicity cell are studied, and the natural frequencies and mode shapes are obtained for a simply supported beam. The results are supported by a comparison with the finite element method and partially exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, the problem of the stability of functionally graded thin plates with a microstructure is presented. To analyse this problem and take into consideration the effect of microstructure, tolerance modelling is used. The tolerance averaging technique allows us to replace the equation with non-continuous, tolerance-periodic, highly oscillating coefficients of the system of differential equations with slowly-varying coefficients, which describes also the effect of the microstructure. As an example, the buckling of a microstructured functionally graded plate band on a foundation is investigated. To obtain results, the tolerance model and the asymptotic model combined together with the Ritz method are used. It is shown that the tolerance model allows us to take into account the effect of microstructure on critical forces. [ABSTRACT FROM AUTHOR]

The mathematical modelling of certain problems of vibrations and stability for periodic slender visco-elastic beams is presented in this note. To consider these problems and take into account the effect of the microstructure, the tolerance modelling approach is proposed. Using this technique, the equation with non-continuous, periodic, highly oscillating coefficients is replaced by a system of differential equations with constant coefficients. Moreover, these governing equations describe the effect of the microstructure on the overall behavior of the beams under consideration. The tolerance modelling can lead to equations of two different tolerance models—the standard and the general, under weakened assumptions. This averaging tolerance method was assessed by comparison with the asymptotic homogenization, the governing equations of which omit this effect. My considerations were limited to proposing and presenting only mathematical models describing investigated beams. In a simple analytical example, the application of the presented average models is shown. [ABSTRACT FROM AUTHOR]