fem modelling of a bellows and a bellows- based micromanipulator

FEM modelling of a bellows and a bellows-based micromanipulatorIII. Modelling3.3.1 CONSTANT BENDING FORCEFor the same reasons as mentioned in the spring-rate simulation, a force cannot be applied atonce on the bellows, because it could initiate errors in the starting K matrix. Hence, the forcehas to be applied gradually, in several steps in the beginning, and only after that, the interiorvarying pressure load can be administered:Time-varying applied loadsLoads [%]1009080706050403020100ForcePressure1 2 3 4 5 6 7 8StepsSo, at step 3, the initial conditions and the correct parameters are reached, which canafterwards be used with beams-formulas. In the following steps, Q may increase because ofthe increasing stiffness, so that the obtained δx displacement should decrease. To get theexpected curve, care must be taken to interpret the simulation's result .Effectively, the δx a displacement, induced by the augmentation of the bellows' stiffness,decreases in a very small magnitude. But the axial growth δy of the bellows generated by theinternal pressure will be much more significant. Because of the angle α on top of the bellows,δx b displacement is also induced, in an increasing way, as shown by figure 21.Bellows' growthBending at p i ≠ 0Initial bending at p i = 0hδx 2δyαδx 1Bellows' growthFigure 21The effective decreaseof δx 1 has to becalculated, taking intoaccount δx 2 , that isinduced by δy, whenapplying a constantforce F, and a varyinginterior pressure p.Bending at p i ≠ 0Initial bending at p i = 0The software will not give us directly δy, but only the h displacement at the very top of thebellows, and the rotation α of the element located at that end-point. Hence, δx b can becalculated:δx b = h·tan(α) (3.9)Since the value of the initial bending δx at step 3 is known, the correct δx a decrease of thebellows' bending can easily be deduced, when adding an interior pressure p i ≠ 0.27