fem modelling of a bellows and a bellows- based micromanipulator

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FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingQ parameter [mN mm 2 ]82762.1082762.0582762.0082761.9582761.9082761.85Bending parameter Q for the whole bellowsConstant pressure p = 20 [mN/mm 2 ]82761.8010 15 20 25 30 35 40 45 50Force [mN]The Q value doesn't vary more than 0,002% in the full range of bending force. It means thatthe bellows' inclination depends in a very linear way on the applied force. The mean value ofthose results – which is 82,77·10 3 [mN·mm 2 ] – differs less than 0,6% of the value determinedearlier.3.4 Pressure vs axial displacementThe previous parameters acquired were so called passive parameters, since they don't reallyact on the bellows, but are just some kind of response when applying external loads – asbending force, or as a mass. They are necessary to settle the correct behaviour of the bellowsacting together, and to build the whole micromanipulator beam-based model.From the viewpoint of the command, there are also other important elements that have to bedetermined. Since the micromanipulator is only controlled by the voltage applied on the threepiezoelectric actuators, it is important to get now the different parameters entering at stakethrough the whole motion-process of one bellows:y cControly c ë VvoltagePiezosV ë ppressureBellowsp ë yyFigure 22Motion-process of themicromanipulator: y c is the desiredposition of the bellows' top, and yits reached position.In this project, the point of interest is of course only the bellows. Therefore, the lastparameter to be found now is the conversion constant Z between the pressure applied by thepiezoactuator and the axial motion of the bellows.Hence, the next simulation will be the same as the one used for the comparison between thethree different models (figure 18). For a more accurate answer, the 3D model with SOLID45-elements is used. But this time, a time-varying internal pressure is applied into the ring, sothat it can also be checked if Z is really constant over the entire pressure range. It can also beseen if the radial displacement is truly negligible, as the manufacturer suggests.Inasmuch as only a lone ring simulation was considered, care has to be taken about whichnode should be observed during the measurements. The upper-part of the ring can movefreely, and that's actually not exactly the case of each ring constituting the bellows. It meansthat when applying an internal pressure, the top surface can have very little twist, as shown infigure 23.31
III. ModellingABFEM modelling of a bellows and a bellows-based micromanipulatorFigure 23yxThe twist effect of the free top surface that can be observedeven in the 3D solid-model, when applying an interiorpressure p. The y-displacement of node A is thereforelarger that that of node B. The real undergoes a totaldisplacement equal to the mean of both of them. The effectis of course intensely scaled for illustration.This effect can even be observed in the 3D solid-model used here. The consequence is thatthe node labelled A and located in the internal perimeter of the top surface undergoes a largerdisplacement as that one positioned on the outer perimeter. Of course the difference is verytiny, but the mean value of both of them is computed in order to plot the result.One could ask if there is any way to avoid this torque effect. Since in ANSYS ® , any degreeof freedom for any node in the mathematical model can be fixed, no rotation to all topsurface's nodes could effectively be assigned. But this would probably induce more errorsthan it would elude. Considering that all the previous theory is based on a ring with free topsurface, introducing these new constraints would affect it, whereas the so called free-end ofthe beam wouldn't be anymore unrestricted.The extension of simulations' results to the entire bellows is very easy, since one ring's y-displacement can just be multiplied by N (18) . The function of y-displacement versus pressureis displayed in next figure:y -displacement [mm]1.41.21.00.80.60.40.20.0y -displacement of the whole bellows0 25 50 75 100 125 150 175 200Interior pressure [mN/mm 2 ]As it can be observed, the relationship between the pressure and the displacement is verylinear. It might be very good in control! Let's now compute the behaviour of the Z parameterin function of pressure:6.7E-03Z -parameter of the bellows6.6E-03Z [mm 3 /mN]6.5E-036.4E-036.3E-036.2E-036.1E-0325 50 75 100 125 150 175 200Interior pressure [mN/mm 2 ]18 N is the number of rings constituting the bellows, and equals 24.32
Page 1: Helsinki University of Technology C
Page 4: PrefaceThe author wishes to thank t
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