fem modelling of a bellows and a bellows- based micromanipulator

More documents

Recommendations

Info

FEM modelling of a bellows and a bellows-based micromanipulatorIII. Modellingstrain concentration at those nodes. Even substantial strain-gaps would appear between somenodes, so that the result would then be less accurate, or maybe wrong. Hence, to avoid theseproblems and to make the aspect ratio of the elements close to 1, meshing has to be very fine.As shown in next figure, the number of elements will become enormous.(a)(b)Figure 13Mesh (a) is too coarse and couldinduce errors. So the mathematicalmodel should be more accurate,corresponding to the mesh (b).3.1.4 CHOOSING RIGHT SHELL 3D-ELEMENTBecause of the thin walls, this bellows should properly be classed as a shell of revolution forstress and displacement analyse purpose. The geometry of a shell is defined by its thicknessand its midsurface, which is a curved surface in space. Load is carried by a combination ofmembrane action and bending action. A thin shell can be very strong if membrane actiondominates, in the same way that a wire can carry great load in tension but only small load inbending. A shell of a given shape can carry a variety of distributed loadings by membraneaction alone.However, no shell is completely free of bending stresses. They appear at or near point loads,line loads, reinforcements, junctures, changes of curvature and supports. In short, anyconcentration of load or geometric discontinuity can be expected to produce bending stresses,often much larger than membrane stresses, but usually quite localised in a “boundary layer”near the load or discontinuity.Flexure stress and bending moment in a shell are related in the same way as for a plate:t / 2Mx= ∫σ x⋅ z ⋅ dz ,y= ∫−t/ 2t / 2M σy⋅ z ⋅ dz , Mxy= ∫τ xy⋅ z ⋅ dz(3.3)−t/ 2where t is the given thickness of the shell element.A quadrilateral shell element can be produced by combining quadrilateral plane and plateelement. A four-node “flat” quadrilateral is in general a warped element because its nodes arenot all coplanar. A modest amount of warping can seriously degrade the performance of anelement. Commercial software may allow only a very small amount of deformations.Curved elements based on shell theory avoid some shortcomings of flat elements, butintroduce other difficulties. More data are needed to describe the geometry of a curvedelement. Formulation is complicated, as it invokes a shell theory – of which there are many.Membrane and bending actions are coupled within the element, so it is harder to avoidmembrane locking, that is, harder to avoid great overstiffness in bending because details ofthe element formulation cause membrane strains to appear in association with bending action,and membrane stiffness is far greater than bending stiffness if the shell is thin.t / 2−t/ 2±I®L−¬°JXK¯ZYFigure 14Explanation picture of the SHELL43 element. It ismade of four nodes, I, J, K and L, and six differentfaces. Each of these surfaces can admit loads. At eachnode a different thickness t can be defined. Thiselement has plasticity, creep, stress stiffening, largedeflection and strain capabilities.15
III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatorIn respect to that theory, ANSYS ® SHELL43 plastic large strain shell-element is used (seefigure 14). This element is well suited to model linear, warped, moderately-thick shellstructures, as the bellows is. The element has six degrees of freedom at each node:translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes.The deformation shapes are linear in both in-plane directions. For the out-of-plane motion, ituses a mixed interpolation of tensorial components. The element 8 has plasticity, creep, stressstiffening, large deflection, and large strain capabilities.3.1.5 COMPARING ELEMENTS' RESULTSIn order to check out if the three different mathematical models are accurate enough, theywill be subjected to the same loads and boundary conditions, as a first simulation. If theresults are the same, it can be deduced that the models are correct, since it is infrequent toperpetrate the same mistakes three times in a different way and obtain identical false answers.But first, these various models have to be meshed. To make sure that the behaviour of thesingle-ring will be perfectly axisymmetric, a map-meshing has to be operate. This implies thatthe most regular possible mesh is desired, and degenerated elements are avoided.Degenerated elements are elements whose characteristic face shape is quadrilateral, but ismodelled with at least one triangular face, as for example the prism option seen in figure 11.These elements are often used for modelling transition regions between fine and coarsemeshes, for modelling irregular and warped surfaces, etc. Degenerated elements formed fromquadrilateral and brick elements without midside nodes are much less accurate than thoseformed from elements with midside nodes and should not be used in high stress gradientregions.After a few trials, the meshes shown in following figures have been obtained:yxFigure 15Two-dimensional axisymmetricmathematical model of the singlering.The reduced number (270) ofelements – and therefore alsoreduced number (2'529) of nodes– allows a very low consumptionof CPU-time. The axis ofsymmetry is y.Figure 16Three-dimensionalmodel with SHELL43element. The numberof elements reachesexactly 12'000, andthis model has only12'124 nodes.8 For more details, see ANSYS ® Element Manual, section 4.43 and section 14.43 of the ANSYS ® TheoryReference Manual.16
Page 1: Helsinki University of Technology C
Page 4: PrefaceThe author wishes to thank t
Page 8 and 9: FEM modelling of a bellows and a be

fem modelling of a bellows and a bellows- based micromanipulator

Create successful ePaper yourself

Delete template?

Save as template?