fem modelling of a bellows and a bellows- based micromanipulator

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FEM modelling of a bellows and a bellows-based micromanipulatorI. IntroductionCompared to the real manipulator, the model is simplified to check only the tripod-likebellows structure – not the piezoactuators – where one end of the bellows are fixed. One ofthe first simulation was performed in ADAMS 1 by Mikael Lind [Lind98], using beams toassume the bellows. It shows that their bending should not cause any insuperable problems incontrolling the movement of the manipulator. But the created models can’t be used as exactkinematic model, because the used beam statement doesn’t take into account that diffractionline of the beam is actually curved, and it's not known exactly how good the approximation ofthe bellows as elastic beams is. In practice, the non-linear pressure differences inside thebellows will cause some variations during the motion.That’s the reason why a more accurate model using finite elements method should be usedto simulate the bellows behaviour under distinct situations, and under different pressures. Theparameters that will arise from that simulation should be used in the kinematic model of themicromanipulator, to perform an accurate open-loop control.1.2 Finite Element Method1.2.1 FEMThe finite element method (FEM) was developed more by engineers using physical insightthan by mathematicians using abstract methods. Its first application was in stress analysis, butwas since applied to other problems, as temperature flux, electronics and fluidics. In our case,it will be used structural analysis.A naive way to describe the FE method is that it involves cutting a structure into severalelements – kinds of little pieces of the whole structure – describing the behaviour of eachelement in a simple way, and then, reconnecting elements together by their corners, the socalled“nodes”, as if they were pins or drops of glue that hold elements together. This processresults in a set of simultaneous algebraic equations. In stress analysis, these equations areequilibrium equations of the nodes. It means that at each node, the addition of all forcesapplied on it should be equal to zero. This may generate hundreds or thousands of suchequations, which means that computer implementation is compulsory.In all applications, the analyst seeks to calculate a field quantity. In stress analysis, it is thedisplacement field or the stress field; in thermal analysis it is the temperature field or the heatflux – which is also a field. In fluidics, it is the stream function or the velocity potentialfunction, and so on. Usually, the interest lies in peak values of either the field quantity or itsgradients. The FEM is a way to get a numerical solution to a specific problem. A FE analysisdoes not produce a formula as a solution, nor does it solve general problems. Also, thesolution is only an approximation since the way to get it is numerical.A more involved description of the FEM considers it as piecewise polynomial interpolation.That is, over an element, a field quantity such as displacement is interpolated from values ofthe field quantity at nodes. By connecting elements together, the field quantity becomesinterpolated over the entire structure in piecewise fashion, by as many polynomial expressionsas there are elements.The best values of the field quantity at nodes are those that minimise some function such asthe total energy contained in the stress. The minimisation process generates a set ofsimultaneous algebraic equations for values of the field quantity at the nodes.1 ADAMS is a world-wide used simulation and modelling software, especially for mechanical systems.3
I. Introduction FEM modelling of a bellows and a bellows-based micromanipulatorMatrix symbolism for this set of equations isK ⋅ D = R(1.1)where K is the matrix of known constants, D is a vector of unknowns – values of the fieldquantity at each node – and R the vector of known constants. In stress analysis, K is known asthe stiffness matrix.The power of the FE method is its flexibility. The analysed structure may have arbitraryshape, arbitrary supports and arbitrary loads. Such compliance does not exist in analyticalmethods.1.2.2 PRE- AND POST- PROCESSINGThe theory of FEM includes matrix manipulations, numerical integration, equation solvingand other procedures, all this carried out automatically by commercial software, such asANSYS ® . The user may see only allusions of these procedures as the software processes data.But he will mostly deal with preprocessing – which means describing loads, supports andboundary conditions, material properties, generating FE mesh – and postprocessing – thatconsists of sorting output, listing and plotting results. In a large software package, the analysisportion is accompanied by the preprocessor and the postprocessor portions of the software.There also exist stand-alone pre- and postprocessors that can communicate with other knownprograms.The most relevant questions when beginning with a new FE problem are : what kinds ofelement should be used, where should the meshing be fine and where can it be coarse, can themodel be simplified, how much physical details must be represented, and how accurate willthe answer be in function of these choices. To answer these preliminary questions, it is notnecessarily needed to understand the mathematics of FE. However, it is very important tounderstand the problem, and to solve it mentally first, so that an imprecise idea of theexpected result can be available. And also, it is significant to understand how elements behavein order to choose suitable kinds, sizes and shapes of them. So it is conceivable to guardagainst misinterpretations and unrealistically high expectations.1.2.3 ELEMENTS AND NODESFinite elements correspond to fragments of structure. Nodes appear on element boundariesand serve as connectors that fasten elements together. In the next figure, twelve elements froma very simple structure are showed. They can be triangular or quadrilateral. Except thoseFATypical elementTypical nodeFigure 3.A coarse-mesh of a simple 2D bracket model. Nodes areindicated by dots and boundaries between elements by lines.A force F is applied on node A, and a none-displacementboundary condition is set to the nodes B, C and D at thebottom of the structure.BCDlocated on the border of the structure, all nodes – indicated by dots – act as connectorsbetween two or more elements. All of them that share the same node have the samedisplacement components at that node. Lines on this figure indicate boundaries betweenelements. Thus we see elements with corner nodes only and other with side nodes as well.Such a combination of element types is not common but serves the present example. In real4
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fem modelling of a bellows and a bellows- based micromanipulator

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