fem modelling of a bellows and a bellows- based micromanipulator

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FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingThe three pictures display the displacement in y-direction. Indeed, what is wanted at the endis an axial displacement, when applying a pressure with the piezoactuator. The bands ofdifferent grey represent the range of y-displacement. It is not very useful, because in this case,only the peak value is interesting. Anyway, it proves that the models behave axisymmetrically.It can also be noted that the bands correspond in each model.The three peak values – showed by DMX – are very close:• PLANE83 : 2.85·10 -6 [mm]• SHELL43 : 2.86·10 -6 [mm]• SOLID45 : 2.82·10 -6 [mm]The maximum mismatch is 1,4%, which could certainly be reduced by using more elements– and hence use more CPU-time – in the three different meshes. But it is absolutely useless,since it was observed in the previous chapter that values of the different geometricalparameters between two bellows can deviate up to 50% from the nominal value.Consequently it can be concluded that the models are all equivalent, and therefore that thetwo-dimensional model can be exploited, which consumes the least computer resources.Of course, in each new problem, in which different types of loads and boundary conditionsare applied, the first simulation result will be checked with at least one of the two 3D-models,to be on the safe side and validate it. But anyway, the opportunity to go ahead with the 2Dmodelwill spare quite a lot of time and computer-resources, and can allow more simulationsto be executed, and more legitimate results will be obtained.19
III. ModellingFEM modelling of a bellows and a bellows-based micromanipulator3.2 Spring Ratio3.2.1 GETTING THE SPRING-RATENow that it is known that the ring-models are consistent, one has to check if they correspondwith the real behaviour of the real bellows. To that effect, the spring ratio of one single ring iscomputed, and compared to that given by the manufacturer.Servometer gives the following formula derived from known spring equations and combinedwith empirical data:⋅ E ⋅( A + B)3( A − B − t)⋅34.3 ⋅tk == 5.9 [lb./in] ≈ 1'033 [N/m] (3.4)NThis value is to be considered very carefully, since it is provided by Servometer with a largetolerance of ±30%. Considering that the thickness t is very important in the formula – raisedto the third power two times – it can easily be understood why the results can vary so much.As mentioned in Section 3.1.1, the bellows is composed of N individual rinks placed in series.These rings are springs with a spring ratio of k i . It is known from the theory, that the springrateof one spring composed by two of them is1 1 1= +(3.5)k k k12The rings are all similar. Hence, the spring-rate k of the whole bellows iskik = (3.6)NSo, for the simulation, the expected spring-rate k i of one single ring is equal to N · k, which isaround 24'800 [N/m], since N = 24. The well-known spring formula F = k · x permits to obtainthat spring-rate by applying a force F – e.g. 10 [mN] – on the ring, and measure thedisplacement 10 x.The succeeding boundary conditions are to be applied to the single ring, to warranty that itbehaves exactly as the bellows. On the bottom line – or surface if it is a three-dimensionalmodel – the nodes are allowed to move only in the radial-direction. In the axis-direction theycan’t, because the following ring of the stack would thwart that movement. On top of it, theaction of the previous results in a simple pressure on the wall, as shown in figure 19.There are two ways to apply uniformly a force on mathematical models. The first one is toy10 [mN]xsolid plateprevious ringmodellised single-ringFigure 19Schema showing the boundary conditions that have to beapplied to the single-ring to simulate the correct behaviourof the total bellows. The solid plate is not real; it is just arepresentation that shows that the load must be adjusteduniformly on the bellows’ top face.following ring10 It is important to position the location of the maximum displacement, since every node of the mathematicalmodel will move, in a range spreading from zero to x, the correct value.20
Page 1: Helsinki University of Technology C
Page 4: PrefaceThe author wishes to thank t
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