Important indications for builders of newly manufactured Machines using fem

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PIOTR SMOLIēSKI DAMIAN RYDZ BOGDAN ĩÓŁTOWSKI

University of Technology and Life Sciences Faculty of Mechanical Engineering

IMPORTANT INDICATIONS FOR BUILDERS OF NEWLY MANUFACTURED MACHINES USING FEM

Summary

This publication contains useful information and tips from the perspective of a newly manufactured machine designer. There is a brief description of the Finite Elements Method and its exemplary use of practical engineering applications. Then, a typical pattern of conduct in FEM calculations was presented. In summary, signifi-cant shortcomings, advantages and possible problems and limitations of that method were marked.

Keywords: FEM, Finite Element Method, diagnostics 1. Description of the method

This chapter will be about theoretical issues concerning the substance of the method and its possible use.

1.1. What is FEM?

Finite element method (FEM) is now a widely used tool for engineering calculations. A finite element method is constantly developing, alongside computer technology. This is an advanced mathematical and physical computation method based on the split of the so-called discretization, called mesh, mostly of area or space, with finite elements of the physical body averaging and performing the actual calculations only for the nodes of this division. A determined value is approximated on the base of the values in the closest nodes [1].

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Piotr SmoliĔski, Damian Rydz, Bogdan ĩółtowski

Important indications for builders of newly manufactured machines using FEM 137

1.2. Possibilities of using the method

The method is used for examining design strength; it simulates deformation, stresses, dis-placements, heat transfer and fluid flow in the mechanics of computer engineering (CAE). Also dynamics, kinematics and statics of machines, as well as electrostatic, magnetic and electromag-netic interactions are examined.

FEM calculations can be carried out in two-dimensional area (2D), where the discretization is mostly within the division of the area into triangles. This solution allows to calculate the values appearing within a given system. However, some limitations are associated with it due to the nature of the problem to be solved (e.g. the direction of the flow only penetrates modelled surface, etc.)

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Fig. 1. Types of finite elements

Due to the progress of computer technology in recent years, most simulation packages are equipped with the possibility of solving problems in three dimensional space (3D). Discretization usually involves a split of tetrahedrons. Such modelling is deprived of fundamental limitations of 2D technology, but it is much more demanding in terms of memory and computing power [2]. 2. Finite element method – the use of engineering

In this chapter, issues relevant to the builders of newly manufactured machines are discussed. Examples of algorithms using FEM and advantages and disadvantages of that method are given, too. Then, possibilities of using FEM are discussed on the basis of the ANSYS software package and its individual programs.

2.1. An example diagram for designing and using FEM

A solution to the problem of using finite element method in the classical model can be divided into the following stages:

1. Analyzed area is divided mentally into a finite number of geometrically simple elements, the so called finite elements.

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2. It is assumed that these are connected with one another in a finite number of points on the circuits. Most often they are the corner points. They are called nodes. The sought values of physical quantities are a basic layout of unknowns.

3. Certain functions clearly defining the distribution of the analyzed physical quantity inside the finite elements are taken, in dependence on the values of these variables in the nodes. These functions are called nodal functions or the functions of the shape.

4. Differential equations that describe the phenomena under investigation become, by applying Weighting functions, equations of the finite element method. They are algebraic equations. 5. The assembling of the system of equations facing the values of the coefficients of the unknowns and the corresponding values of the right sides is carried out, i.e. it is calculated on the basis of equations of the finite element method. If a task to be solved is not stationary, initial conditions in the calculation of the right-hand sides are also used. The number of equations in the system is equal to the number of nodes multiplied by the number of degrees of freedom of nodes, i.e. the number of unknowns occurring in a single node.

6. Boundary conditions are amended to this set of equations. An introduction to these conditions takes place by making appropriate modifications of the matrix of coefficients of equations and a right side vector.

7. The equations are solved by obtaining desired physical quantities in the nodes.

8. Depending on the type of the problem being tackled or needs, an additional volume is calcu-lated.

9. If a task is non-stationary, then the operations described in paragraphs 5, 6, 7 and 8 are repeated until the end of the calculation condition is completed. It may be i.e. the physical size of the specified value in any of the nodes, the time course of the events or some other parameters [1].

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Fig. 2. A block diagram of a procedure using FEM

Modern CAE engineering applications in which FEM is used consist of three mutually co-operating modules, which are:

a) post processor (to import or prepare geometry, the selection of the types of finite elements, discretization (by area) of the continuum (surface), and the application of boundary conditions) is a unit used to support the presentation and the interpretation of results.

b) solver (the module designed to build and solve equations on the basis of which the re-quested data values of physical quantities are received)

From a practical point of view, before the discretization of a CAD model, it needs appropriate simplification, in which the items not relevant to the analyzed phenomena such rays, phase, holes, tilt, etc should be removed.

Geometry of the analyzed systems can differ significantly. They may be one-dimensional ob-jects (beams), two-dimensional (thin discs, membranes) and three-dimensional (solids). Therefore, when preparing an FEM analysis, many types of finite elements are available, and the criteria for their allocation are:

– The number of dimensions that can describe an item, – Geometric shapes,

– The type and degree of the polynomial assumed shape function of a finite element, – The number of nodes in an element,

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Fig. 3. The procedure of preparing CAD geometry for FEM calculations a) to build a precise CAD model, b) to simplify a CAD geometry model,

c) discretization of a simplified model.

During the discretization of a model it may be useful to condense the mesh of elements, par-ticularly in the areas loaded by boundary conditions. It should, however, be remembered that the so-called "endless grid compacting", i.e. bringing to generate very small finite elements in the data areas could even imply the distortion of wanted unknowns. It should be mentioned that the divi-sion of the geometric continuum into finite elements can be carried out in a manual or semi-automatic way (i.e. Automesh).

Fig. 4. Boundary conditions of the upper cylinder sleeve damper geometry

While preparing a FEM calculation model, it is important to determine the type and number of degrees of freedom (DOF) in the nodes. Such features as: movement, pressure, temperature, magnetic potential and electrical voltage may belong to DOF. A model of a cylinder, gear shock with a prescribed boundary condition liner is shown in Fig. 4:

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– anchoring (to cylindrical front sides of the socket in which the pins mounting chassis to chassis hull recess aircraft are fixed),

– extortion (the burden of the shock absorber piston rod to the cylinder bumper when landing with a speeding vertical drop).

Solving a task by the solver is done in most studies in an “invisible” way to the user. When analyzing the results using the postprocessor, there are many opportunities to present the results of research. In summary, it is important to notice that the application of Finite Element Method in computer-aided engineering analysis enables quick and relatively accurate achievement of results, which, in an analytical way, would be extremely difficult or impossible to obtain.

Using FEM to verify the correctness of the operation of a device enables gradual or exact op-timization of selected features from early stages of product development. Therefore, a possibility of a radical reduction in the duration of starting the production of a new product or modifying of the product that is already in production is obtained.

It should be noticed that the results of FEA describe the behaviour of the system in an ap-proximate manner; they are always burdened with some error, which, in the case of proper conduct of CAE analysis, can be considered negligible [3].

2.2. Advantages and disadvantages and possible problems and limitations of the method The main advantage of FEM is obtaining the results for complex shapes, for which the im-plementation of analytical calculations is not possible. This means that the problem can be simu-lated in computer memory without the necessity to build a prototype, which meaningfully simpli-fies the process of design.

The division of the area into smaller parts usually gives more accurate results of calculations, but it causes an increased demand for computing power of the computer.

Additionally, you should reckon with overlapping errors resulting from the calculation of the value of many approximations of the processed values. If the area is made up of hundreds of thousands of elements in the calculation, the property must be properly modified in successive iterations, so that the final solution is correct.

Therefore, in exceptional circumstances, incremental errors in the calculations cannot be over-looked. In order to minimize these errors between different versions of the same problem (e.g. changes in material parameters with the same dimensions) the same discretization of the problem will be applied, so that the rounding errors are identical, and possible differences indeed result from the changes in material properties.

FEM simulations cannot be carried out in real time because very complex systems as a solution to the problem can be lengthy (depending on their complexity and the complexity of computing power, it can last from several seconds to several days or even longer). In addition, the values calculated by FEM may be flawed, the value of which depends on the assumptions made when formulating the problem to be solved, and available data and accuracy of the material. Therefore, if possible, verify the data calculated with data measured on the actual device or sys-tem.

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Major advantages of FEM as compared to other methods:

1. Material properties of elements do not have to be the same. This gives a possibility to use FEM multiphase materials, as well as materials whose properties are a function of temperature.

2. The centre with a complicated shape can be approximated with high accuracy by using curved elements.

3. Dimensions of elements may be different in volume. This makes it possible to enlarge or reduce dimensions of elements in certain zones of that volume.

4. With MES, you can include non-linear boundary conditions [4].

Examples of software for modelling and calculating with the use of FEM:

ABAQUS – a package used for the analysis of nonlinear circuits using finite element method for complex engineering studies. It is used in the problems of mechanics of solids and liquids and to assess the strength of machine elements and structures, taking into account load, temperature, switching points, potential collisions and other environment conditions. ABAQUS is used suc-cessfully during seismic and geotechnical studies in acoustics, automotive, etc.

ANSYS is the world's leading FEA package to allow complex calculations to simulate almost every field of science and industry. Easy to use program and comfortable graphical user interface enables even an inexperienced user to do the first analysis after a short introduction. It allows optimal matching of required options to suit your needs. The application of finite element method (FEM) has already turned cost-effective after a short time, with its benefits far outweighing the costs incurred. It is possible to make an optimal design in many respects (i.e. minimum weight, efficiency, etc.); the number of expensive prototypes dramatically decreased, and the time of introducing the product on the market was significantly reduced.

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Bibliography

1. http://icis.pcz.pl/~roman/mat_dyd/prz_rown/mac_rzadkie/4_1.html.

2. http://www.naukowy.pl/encyklopedia/Metoda_element%C3%B3w_sko%C5%84czonych. 3. BudzyĔski A. T.: Krótki wstĊp do zastosowania Metody Elementów Sk oĔczonych (MES)

do numerycznych obliczeĔ inĪynierskich, http://www.knse.pl/publikacje/65.pdf.

4. Skibicki D.: Techniki wirtualne w projektowaniu i konstruowaniu modelowych konstrukcji maszyn z zastosowaniem oprogramowania CAD. Prace POIG, UTP, WIM, 2009.

5. ĩółtowski B.: Machine diagnostics basics. [in Polish] Wyd. ATR, Bydgoszcz, 1996. 6. ĩółtowski B.: Multidimensional monitoring of the track-vehicle interface of a railway

system. Besanson, 2007.

7. Żółtowski B, Castañeda L: Monitoreo Multidimensional de la Interfase Vía-Vehículo de un Sistema Ferroviario Congreso Internacional de Mantenimiento – ACIEM – Marzo 2007, Bogotá, Colombia.

FEM W KONSTRUOWANIU NOWOCZESNYCH MASZYN Streszczenie

W tej pracy przedstawiono uĪyteczne informacje o sposobach konstruowania nowoczesnych maszyn. Zwrócono uwagĊ na przydatnoĞü metody elementów skoĔ-czonych w praktyce inĪynierskiej. Zaprezentowano podstawowe sposoby wykorzy-stywania FEM w praktyce. W podsumowaniu zwrócono uwagĊ na sposoby, moĪliwo-Ğci i rozwój metod komercyjnych.

Słowa kluczowe: FEM, metoda elementów skoĔczonych, diagnostyka, efektywnoĞü

*This paper is a part of WND-POIG.01.03.01-00-212/09 project. Piotr SmoliĔski

Damian Rydz Bogdan ĩółtowski

University of Technology and Life Sciences Faculty of Mechanical Engineering