fem modelling of a bellows and a bellows- based micromanipulator

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FEM modelling of a bellows and a bellows-based micromanipulatorII. Bellows' dimensionsThe first conclusion is that, corresponding to figures, following balance exits:A B= + t + r2+ m + r1+ t(2.1)2 2Actually, because of the measurement imprecisions, the expected result is not observed:3.175 ≠ 3.1173. This means an error of 1,85%. But it is to be noted that the thickness of thewall is not guaranteed by the manufacturer, it is only an average! It signifies that the samplecan have a different thickness. The cutting-manipulations may also have been the origin ofsome deformations.However, the simulations showed that the result is not substantially affected by those radii.One pressure was applied inside the bellows, with different radii, and had a look on the y-displacement. The results didn’t change more than about 0,2% when changing the radius r 1and r 2 by around 20%.It can be deduced that the assumption that r 1 = r 2 holds. For both of them, the followingvalue is used:D − 2tr1= r2= r = = 0,1778 [mm] (2.3)4Identical conclusion for m implies the subsequent equation to be used:A BA Bm = − − t − r2 − r1− t = − − 2t− 2r= 0,8382 [mm] (2.4)2 22 2The result differs only by 4% of the one measured.11
III. ModellingFEM modelling of a bellows and a bellows-based micromanipulator3. MODELLING3.1 Procedure3.1.1 WAY TO ANALYSE ONE BELLOWSThe first step of this analysis is to check if the results as given by the manufacturer areobtained by simulation. It will be later on to confirm that the radial displacement of thebellows’ wall is insignificant in comparison with the axial one when applying differentpressures into the bellows. After that, the spring ratio obtained by simulation is comparedwith that given by the designer. This phase will certify that the model behaves in a correctway.The second phase will be the most important one. It is desired to obtain the bendingbehaviour of the bellows, when applying different forces on top at different pressures. Twodifferent graphs are determined. One is the function of bending displacement versus appliedforce, under one constant pressure. The other is a check how the bellows' stiffness increaseswhen applying various internal pressures, with a constant bending-force.The third phase is the axial displacement of the bellows when applying various pressures. Inthis way, the exact relationship between pressure applied by the piezoactuator and axialmovementof the bellows is obtained.To perform these simulations, a simplified solid model is used. Indeed, the whole bellowsneeds not to be used, since it can be recovered using smaller parts placed in series or inparallel, depending on the analysis. The subsystem will consist of a single ring – also calledconvolution – of the bellows, as showed in figure 8. The boundary condition on its bottom hasYwholebellows single ring Figure 8The whole bellows can be modelled bysimulating the behaviour of one single ring, andextend the results to the bellows afterwards.to be set, so that it can’t move in y-direction. The upper surface can move without anyconstraint. Then, the spring-like behaviour of the total bellows can be calculated from that ofthe single-ring.In order to spare CPU-time, the ring behaviour is also modelled in only two dimensions,each time while axisymmetric loads – such as the pressure – and boundary conditions areapplied. It can effectively run up to hundred times faster than the three dimensional problem.But to check if the 2-D model is accurate enough, it is first compared with the 3D one, andonly afterwards the real simulations are run.3.1.2 CHOOSING RIGHT 2-D ELEMENTSA solid of revolution is generated by revolving a plane figure about an axis in the plane.Loads and supports may or may not have axial symmetry. Initially, the case where geometry,elastic properties, loads and supports are all axisymmetric will be considered. Consequently,nothing varies with the circumferential coordinate θ, material points displace only radiallyand axially, and shear stresses τ rθ and τ rθ are both zero. Thus the analysis problem ismathematically two-dimensional. Axisymmetric finite elements are often pictured as planetriangles or quadrilaterals. These plane shapes are actually referred as cross sections ofannular elements, and what appears to be nodal points are actually circles (figure 9).12
Page 1: Helsinki University of Technology C
Page 4: PrefaceThe author wishes to thank t
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fem modelling of a bellows and a bellows- based micromanipulator

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