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 3D model is axisymmetric, but all the loads to be applied are not. It means the 2D finiteelements have to handle axisymmetric boundary conditions, but the bending behaviour of thebellows having also to be simulated, the element should as well manage non-axisymmetricloads.θθFigure 9An eight-node axisymmetric element, thatappears usually only as a two-dimensionalelement in representations. What appears to benodal points are actually nodal circles.Except for having to account for circumferential strain, axisymmetric elements are verysimilar to plane elements. A special class of ANSYS ® axisymmetric elements – calledharmonic elements – allows a non-axisymmetric load. For these elements, the load is definedas a series of harmonic functions – Fourier series. For example, a load F is given by:F(θ) = A 0 + A 1·cos(θ) + B 1·sin(θ) + A 2·cos(2θ) +B 2·sin(2θ) + … (3.1)Each term of the above series must be defined as a separate load step. A term is defined bythe load coefficient, A x or B x , the number of harmonic waves x and the symmetry condition,cos(xθ) or sin(xθ). θ is the circumferential coordinate implied in the model. The loadcoefficient is determined from the standard boundary condition input. The input value forforce should be a number equal to the peak value per length-unit times the circumference. Thedeflections and stresses are output at the peak value of the sinusoidal function. For our model,PLANE83 is used (figure 10).LPY (axial)IX (radial)OMKL,O,KNPNJ I MTriangular optionJFigure 10Representation of the twodimensionalelement PLANE83, thathas 8 nodes, one at each corner, I, J,K and L, and four sidewise, M, N, Oand P. This element supportsaxisymmetric geometry and nonaxisymmetricloads.This element has three degrees of freedom per node: translation in the nodal x, y and zdirections. These directions correspond respectively to the radial, axial and tangentialdirections. It can tolerate irregular shapes without loss of accuracy. It has 8 nodes, one at eachcorner and one in the middle of each side 5 . This element of course also accepts axisymmetricloads, and will be used for that purpose. But for non-axisymmetric loads, the results given bythis element have to be checked with another 3D element.3.1.3 CHOOSING RIGHT SOLID 3D-ELEMENTThe term “3D solid” is used to mean a three-dimensional solid that is unrestricted withrespect to shape, loading, material properties and boundary conditions. A consequence of thisgenerality is that all six possible stresses – three normal and three shear – must be taken intoaccount. Also the displacement field involves all three possible components, x, y and z.Typical finite elements for 3D solids are tetrahedra and hexahedra, with three translationaldegrees of freedom (DOF) per node.5 For more details about this element, see ANSYS ® Element Manual, chapter 4, section 4.83.13
III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatorThe SOLID45 3D structural element will therefore be used. It is used for three-dimensionalmodelling of solid structures. It is defined by eight nodes, each having three DOF; translationsin the x, y and z directions. The element has plasticity, creep, swelling, stress stiffening, largedeflection and large strain capabilities 6 .XZMIYLONJNKMINJPrism optionN,OK,LFigure 11Representation of the three-dimensionalelement SOLID45, that has 8 nodes, oneat each corner, I, J, K, L, M, N, O and P.This element supports also prism option.Problems of beam bending, plane stress, plates, and so on, can all be regarded as specialcases of a 3D solid. Why then not simplify FE analysis by using 3D elements to modeleverything? In fact, it would not be a simplification. 3D models are the hardest to prepare, themost tedious to check for errors and the most demanding for computer resources. Also, some3D elements would become quite elongated in our problem.Effectively, because of the very thin shape of the bellows, the 3D solid model would bequite difficult to mesh. Accordingly to the FE theory, the aspect ratios 7 of the 3D elementshould not be to much different from 1. To explain the problem, an example with a quadrilateral2D-element with only four nodes is discussed. The solid hexahedron element is a directextension of the quadric planar element.(a)A (b) DFigure 12(a) Correctθ 1θ 1 θ 2 θ 2deformation mode ofa rectangular block .M 1bM 1 M 2 bM 2(b) Deformationmode of the bilinearaquadrilateralB aCelement.BlockQuardic elementThe principal defect of these elements is their overstiffness in bending, which can beillustrated by comparison of the bending moments in figures 12-a and 12-b. Let therectangular block and the element have the same dimensions, elastic modulus E, and Poissonratio ν. Then apply whatever bending moments M 1 and M 2 are necessary to make verticalsides of the block and the element include the same angle, θ 1 = θ 2 . Moment M 1 is the correctvalue. It can be demonstrated that M 2 isM21 ⎡ 1= ⋅ ⎢ +1+ν ⎢⎣1−ν12⎛ a ⎞⋅⎜⎟⎝ b ⎠2⎤⎥ ⋅ M⎥⎦1(3.2)where a and b are dimensions of element and block. If aspect ratio a/b increases without limit,so does M 2 , which means that the quadrilateral element becomes infinitely stiff in bending.This phenomenon is called locking. In practice, elements of large aspect ratio should beavoided. If it is the case, nodes A and B will be so distant from C and D, that there will be6 For more details about this element, see ANSYS ® Element Manual, section 4.45 and section 14.45 of theANSYS ® Theory Reference Manual.7 The aspect ratios are measured between the height, the length and the width.14
Page 1: Helsinki University of Technology C
Page 4: PrefaceThe author wishes to thank t
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