fem modelling of a bellows and a bellows- based micromanipulator

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