fem modelling of a bellows and a bellows- based micromanipulator

Helsinki University of Technology Control Engineering LaboratoryEspoo 1999 Report 113FEM MODELLING OF A BELLOWS AND A BELLOWS-BASED MICROMANIPULATORSébastien RobyrTEKNILLINEN KORKEAKOULUTEKNISKA HÖGSKOLANHELSINKI UNIVERSITY OF TECHNOLOGYTECHNISCHE UNIVERSITÄT HELSINKIUNIVERSITE DE TECHNOLOGIE D´HELSINKI

Distribution:Helsinki University of TechnologyControl Engineering LaboratoryP.O. Box 5400FIN-02015 HUT, FinlandTel. +358-9-451 5201Fax. +358-9-451 5208E-mail: control.engineering@hut.fiISBN 951-22-4471-3ISSN 0356-0872Picaset OyHelsinki 1999

PrefaceThe author wishes to thank the Helsinki University of Technology, and more precisely theControl Engineering Laboratory to have received him to perform his Master Thesis. Thewhole laboratory lead by Prof. Heikki Koivo was very kind, and never balked at helping. Aspecial thanks for having been able to participate in ANSYS ® Introductory Course, providedby Anker-Zemer Engineering. I like to thank also the Tampere University of Technology andparticularly Pasi Kallio and Quan Zhou for their participation. The Electrical Workshop atTUT was also helpful for the measurements, and have also to be thanked.Finally, I would like to thank the Centre for Scientific Computing for their providedservices and CPU-time, and especially want to tell my acknowledgement to Reijo Lingren,for his precious advises and his fruitful collaboration in ANSYS ® comprehension. He helpedme to resolve various difficult problems by using FEM, and his assistance was very profitablefor this project.ESPOO, February 1999Sébastien ROBYR

FEM modelling of a bellows and a bellows-based micromanipulatorI. SummarySummary1 INTRODUCTION.............................................................................................................. 11.1 BELLOWS BASED MICROMANIPULATOR – OBJECTIVE OF THIS PROJECT ......................... 11.1.1 Background ............................................................................................................ 11.1.2 The Bellows............................................................................................................ 21.1.3 Aim of this project.................................................................................................. 21.2 FINITE ELEMENT METHOD ............................................................................................ 31.2.1 FEM........................................................................................................................ 31.2.2 Pre- and post- processing ....................................................................................... 41.2.3 Elements and nodes................................................................................................ 41.2.4 Modelling ............................................................................................................... 51.2.5 Discretization and approximation .......................................................................... 51.3 ANSYS ® ....................................................................................................................... 61.3.1 What ANSYS ® is ................................................................................................... 62 BELLOWS' DIMENSIONS AND PROPERTIES ............................................................ 72.1 PRELIMINARIES ............................................................................................................. 72.2 GEOMETRICAL DATA..................................................................................................... 72.2.1 Servometer ® data-sheets......................................................................................... 72.2.2 Measurements......................................................................................................... 82.3 SOLID MODEL................................................................................................................ 92.3.1 Consistent units ...................................................................................................... 92.3.2 Units conversion..................................................................................................... 93 MODELLING .................................................................................................................. 123.1 PROCEDURE ................................................................................................................ 123.1.1 Way to analyse one bellows ................................................................................. 123.1.2 Choosing right 2-D elements................................................................................ 123.1.3 Choosing right solid 3D-element ......................................................................... 133.1.4 Choosing right shell 3D-element.......................................................................... 153.1.5 Comparing elements' results ................................................................................ 163.2 SPRING RATIO............................................................................................................. 203.2.1 Getting the spring-rate.......................................................................................... 203.2.2 Constancy of the spring ratio ............................................................................... 223.3 BENDING..................................................................................................................... 253.3.1 Constant bending force......................................................................................... 273.3.2 Constant pressure ................................................................................................. 303.4 PRESSURE VS AXIAL DISPLACEMENT ........................................................................... 313.5 TOLERANCES............................................................................................................... 333.5.1 Spring-rate tolerances........................................................................................... 343.5.2 Bending parameter's tolerances............................................................................ 353.5.3 Internal pressure vs displacement ........................................................................ 363.5.4 Consequences....................................................................................................... 364 OUTLOOK....................................................................................................................... 384.1 EXPANSION OF THE BELLOWS' MODEL TO THE ENTIRE MICROMANIPULATOR............... 384.1.1 Preliminaries......................................................................................................... 384.1.2 Choosing right 3d-beam element ......................................................................... 38I

FEM modelling of a bellows and a bellows-based micromanipulatorI. Introduction1. INTRODUCTION1.1 Bellows based micromanipulator – objective of this project1.1.1 BACKGROUNDTampere University of Technology and Helsinki University of Technology have developedin co-operation a 3 degrees of freedom (DOF) piezohydraulic micromanipulator. Theactuation system consists of a piezoelectric actuator, a small tank, a connecting pipe and abellows. The piezoelectric actuator is placed in the tank which is connected to the bellowsPiezoelectricactuatorBellowsFluidFigure 1.Overview of the actuation system. It consists of a piezoelectric actuator, asmall tank, a connecting pipe and a bellows. The piezoelectric actuator isplaced in the tank which is connected to the bellows using the pipe.Tankusing the pipe, as shown in figure 1. When a voltage is applied to the piezoelectric actuator, itdeforms. While the actuator buckles, oil flows from the tank to the bellows which elongates,and vice versa: when the actuator gets straightened, oil flows from the bellows to the tank.Since the effective area of the bellows is smaller than that of the actuator, the displacement ismagnified. The bellows is pre-elongated when hydraulic fluid is filled in. This facilitates notonly elongation, but also shortening of the bellows by deforming the piezoelectric actuatoreither upwards or downwards. The results of displacement experiments have shown that themovement range of the actuator is about ± 250 [µm] – maximum displacement 500 [µm].41Figure 2.Overview of the actuation system. The figure shows one ofthe bellows (1), one of the piezoelectric actuators (2), theplatform (3) and the end effector (4).32The major purpose in the structural design was to provide an actuation system whose shapefacilitates the construction of a compact Stewart platform type of structure, as showed infigure 2. Moreover, the inside volume of the tank was made as small as possible in order to1

I. Introduction FEM modelling of a bellows and a bellows-based micromanipulatorlimit the amount of liquid and thus, to decrease problems arising from temperature changes.The bending of the bellows facilitates a joint-free structure. Using the bending character of thebellows instead of ball joints simplifies the structure and the manufacturing process. Tests andmeasurements have proven that it is possible to control a bellows-based parallel manipulator.By changing the lengths of the bellows the orientation of the mobile platform, and thus theposition of the end-effector can be controlled.The objective of this thesis was to develop a computer-controlled micromanipulator thatfacilitates automatic and semi-automatic operations. Its main application are currently inbiotechnology where needs for automating operations have emerged. For example in celltoxicology, hundreds of cells are typically injected and therefore automatic micromanipulatorwould speed up the process remarkably. Since the size of biological objects vary frommicrometers to hundreds of micrometers, the micromanipulator should be capable of bothsubmicrometer resolution and work space of several hundreds of micrometers.1.1.2 THE BELLOWSThe bellows is a spring type of passive component, i.e. force required to deform the bellowsis directly proportional to the displacement. The bellows is manufactured using electrodeposition,where spring-quality nickel – nickel plus cobalt at 99,85% and maximum 0,04%of sulphur – is first deposited onto an aluminium mandrel. In the end of the process, themandrel is dissolved out.Electro-deposited metal bellows have the following advantages. Because the bellows wallcan be thinner than other types – up to 8 [µm] – they are extremely sensitive, which makesthem excellent for very accurate instrument applications requiring a high degree of sensitivity.They can provide large deflections with only very tiny forces. They are up to 25 times assensitive as hydroformed bellows in the same size range. Servometer ® manufactures bellowswhich can be fully compressed by a force as small as 4 grams.Their stroke can be 60% of their extended length. They can be designed for infinite lifeexpectancy and have a normal minimum life of 100’000 cycles. They are seamless and nonporous.So, they are fully impervious and can be filled with oil without losses. Therefore, theycan be used in micromanipulations of cells without any contamination of the environment.That’s why they were chosen for this application.1.1.3 AIM OF THIS PROJECTThe micromanipulator can be controlled using either open or closed-loop control. In teleoperation,open-loop control is typically sufficient, since the operator closes the loop. Inapplications that are not critical to imperfections of the manipulator, open-loop control cantherefore be used. However, an increasing number of micromanipulation applications,especially those that involve automatic operations, require high absolute accuracy, high speedand elimination of drift.Actually, to satisfy these requirements, closed-loop control must be applied. Thedisplacement of the actuators are measured using strain gages, to compensate non-linearity.The position of the end-effector is controlled at a second level using Hall sensors. Of course,this hardware implementation increases remarkably the costs. Thus, it should be moreproficient to enhance the precision of the closed-loop control. But the bellows have alsobending characters when non-axial forces are applied to them. This might be a problem incontrolling the manipulator’s movement. Therefore an accurate model that shows the bendingand behaviour of the used bellows was needed.2

FEM modelling of a bellows and a bellows-based micromanipulatorI. IntroductionCompared to the real manipulator, the model is simplified to check only the tripod-likebellows structure – not the piezoactuators – where one end of the bellows are fixed. One ofthe first simulation was performed in ADAMS 1 by Mikael Lind [Lind98], using beams toassume the bellows. It shows that their bending should not cause any insuperable problems incontrolling the movement of the manipulator. But the created models can’t be used as exactkinematic model, because the used beam statement doesn’t take into account that diffractionline of the beam is actually curved, and it's not known exactly how good the approximation ofthe bellows as elastic beams is. In practice, the non-linear pressure differences inside thebellows will cause some variations during the motion.That’s the reason why a more accurate model using finite elements method should be usedto simulate the bellows behaviour under distinct situations, and under different pressures. Theparameters that will arise from that simulation should be used in the kinematic model of themicromanipulator, to perform an accurate open-loop control.1.2 Finite Element Method1.2.1 FEMThe finite element method (FEM) was developed more by engineers using physical insightthan by mathematicians using abstract methods. Its first application was in stress analysis, butwas since applied to other problems, as temperature flux, electronics and fluidics. In our case,it will be used structural analysis.A naive way to describe the FE method is that it involves cutting a structure into severalelements – kinds of little pieces of the whole structure – describing the behaviour of eachelement in a simple way, and then, reconnecting elements together by their corners, the socalled“nodes”, as if they were pins or drops of glue that hold elements together. This processresults in a set of simultaneous algebraic equations. In stress analysis, these equations areequilibrium equations of the nodes. It means that at each node, the addition of all forcesapplied on it should be equal to zero. This may generate hundreds or thousands of suchequations, which means that computer implementation is compulsory.In all applications, the analyst seeks to calculate a field quantity. In stress analysis, it is thedisplacement field or the stress field; in thermal analysis it is the temperature field or the heatflux – which is also a field. In fluidics, it is the stream function or the velocity potentialfunction, and so on. Usually, the interest lies in peak values of either the field quantity or itsgradients. The FEM is a way to get a numerical solution to a specific problem. A FE analysisdoes not produce a formula as a solution, nor does it solve general problems. Also, thesolution is only an approximation since the way to get it is numerical.A more involved description of the FEM considers it as piecewise polynomial interpolation.That is, over an element, a field quantity such as displacement is interpolated from values ofthe field quantity at nodes. By connecting elements together, the field quantity becomesinterpolated over the entire structure in piecewise fashion, by as many polynomial expressionsas there are elements.The best values of the field quantity at nodes are those that minimise some function such asthe total energy contained in the stress. The minimisation process generates a set ofsimultaneous algebraic equations for values of the field quantity at the nodes.1 ADAMS is a world-wide used simulation and modelling software, especially for mechanical systems.3

I. Introduction FEM modelling of a bellows and a bellows-based micromanipulatorMatrix symbolism for this set of equations isK ⋅ D = R(1.1)where K is the matrix of known constants, D is a vector of unknowns – values of the fieldquantity at each node – and R the vector of known constants. In stress analysis, K is known asthe stiffness matrix.The power of the FE method is its flexibility. The analysed structure may have arbitraryshape, arbitrary supports and arbitrary loads. Such compliance does not exist in analyticalmethods.1.2.2 PRE- AND POST- PROCESSINGThe theory of FEM includes matrix manipulations, numerical integration, equation solvingand other procedures, all this carried out automatically by commercial software, such asANSYS ® . The user may see only allusions of these procedures as the software processes data.But he will mostly deal with preprocessing – which means describing loads, supports andboundary conditions, material properties, generating FE mesh – and postprocessing – thatconsists of sorting output, listing and plotting results. In a large software package, the analysisportion is accompanied by the preprocessor and the postprocessor portions of the software.There also exist stand-alone pre- and postprocessors that can communicate with other knownprograms.The most relevant questions when beginning with a new FE problem are : what kinds ofelement should be used, where should the meshing be fine and where can it be coarse, can themodel be simplified, how much physical details must be represented, and how accurate willthe answer be in function of these choices. To answer these preliminary questions, it is notnecessarily needed to understand the mathematics of FE. However, it is very important tounderstand the problem, and to solve it mentally first, so that an imprecise idea of theexpected result can be available. And also, it is significant to understand how elements behavein order to choose suitable kinds, sizes and shapes of them. So it is conceivable to guardagainst misinterpretations and unrealistically high expectations.1.2.3 ELEMENTS AND NODESFinite elements correspond to fragments of structure. Nodes appear on element boundariesand serve as connectors that fasten elements together. In the next figure, twelve elements froma very simple structure are showed. They can be triangular or quadrilateral. Except thoseFATypical elementTypical nodeFigure 3.A coarse-mesh of a simple 2D bracket model. Nodes areindicated by dots and boundaries between elements by lines.A force F is applied on node A, and a none-displacementboundary condition is set to the nodes B, C and D at thebottom of the structure.BCDlocated on the border of the structure, all nodes – indicated by dots – act as connectorsbetween two or more elements. All of them that share the same node have the samedisplacement components at that node. Lines on this figure indicate boundaries betweenelements. Thus we see elements with corner nodes only and other with side nodes as well.Such a combination of element types is not common but serves the present example. In real4

FEM modelling of a bellows and a bellows-based micromanipulatorI. Introductionmodels the meshing would also be finer than this, because such an assemblage would be weakand unrepresentative.In figure 4, the difference between a mathematical model (4b) composed by those elementsand nodes, and the solid model (4a), that has the exact geometry can be seen. Nearly all finiteelement analysis models today are built using a solid model. This CAD-type mathematicalrepresentation of the structure defines the geometry to be filled with nodes and elements, andcan also be used to facilitate applying loads or other analysis data. However, the solid modeldoes not participate in the finite element solution! All analysis information must ultimatelybe transferred to the FE model for solution. The process of creating a finite element modelfrom a solid model is called meshing.(a)meshing(b)Figure 4.Figure (b) shows the mathematical modelof the solid model (a). It is important tonote that they are not equal.1.2.4 MODELLINGModelling is the simulation of a physical structure or physical process by means of asubstitute analytical or numerical construct. It is not simply preparing a mesh of nodes andelements. Modelling requires that the physical action of the problem be understood wellenough to choose suitable kinds of elements, and enough of them, to represent the physicalaction adequately.Badly shaped elements and elements too large to represent important variations of the fieldquantity should be avoided. At the other extreme, we should avoid the waste of analysis timeand computer resources associated with over-refinement, that is, using many more elementsthan needed to adequately represent the field and its gradients. Later, when the computer hasdone the calculations, the results must be checked to see if they are reasonable. Checking isvery important because it is very easy to make mistakes in describing the problem to thesoftware.The FE method calculates nodal displacements, then – in present ANSYS ® software too –uses the displacement information of the field to calculate strains and finally stresses. Ifdisplacements are incorrect, stresses will of course also be erroneous. Accordingly, we shouldexamine the computed displaced shape first. If it is substantially different from what isexpected, an error in the model is suspected. The software will permit the display of thedisplaced shape superposed on the original shape, with displacement scaled up if necessary,so that they are easily visible. Additionally, the displacement shape can be animated, so thatthe model appears to be vibrating slowly between its deformed and unreformed positions.Thus, it can easily be seen if it coincides with the real motion or not.As for stresses, the software will plot them as bands of different colours. ANSYS ® allowsusers to select an option that calls for averaging of stresses. This means that stresses fromindividual elements are averaged at nodes before plotting, so that stress contours have nodiscontinuities between elements. This is poor practice, because it removes information usefulto analyse. A stress plot that displays significant interelement discontinuities warns that afiner mesh is needed.1.2.5 DISCRETIZATION AND APPROXIMATIONWhatever the analysis method, the actual physical problem is not analysed, but themathematical model of it. Thus, we introduce modelling errors. For example, it is regularly5

I. Introduction FEM modelling of a bellows and a bellows-based micromanipulatornecessary to make some simplifications. A problem cannot be modelled taking into accountevery kind of geometrical incertitude – the geometrical model always differs from the truth,since the real model has constantly imperfections, as small as they can be – or all true materialproperties. The model is invariably only an approximation of the real problem. When meshingthe solid model, a discretization is introduced, as small as the elements can be.Finally, it must be admitted that software almost certainly contains errors. Commercialsoftware packages, like ANSYS ® , are large, versatile and under continual revision. It ispractically impossible to get everything right. Many errors either make a software featureinoperable or cause the program to crash, but some can lead to erroneous results. It istempting to blame all strange results on the software, but it is far more often the case that wehave blundered in modelling or in describing the model to the software. Strange results areobtained so often, that it is vital that, once more, the analyst be able to recognise that resultsare strange.1.3 ANSYS ®1.3.1 WHAT ANSYS ® ISThe finite element analysis (FEA) program that will be used in this project is ANSYS ® . Thissoftware is well-known and used by a lot of design-companies around the world. Ansys Incheadquarters is based in Canonsburg, south of Pittsburgh. It has 250 employees, of whichsixteen are developers. Over seventy-five offices are based world-wide.The father of ANSYS ® is Dr. John Swanson, very eminent in the FEA-field. He wrote thefirst code-lines of that program. The first release came out in the early seventies, giving proofthat in more than twenty-five years of being used by several companies, this software shouldbe quite reliable.The current release of ANSYS ® is 5.5 and came out in the beginning of November, butrelease 5.4 is used in this study, because of the minor differences in importing solid modelfiles from other CAD-packages. ANSYS ® capabilities are: structural, thermal, electromagneticand fluid analysis. It can also make coupled-field analysis. All these differentabilities are contained in different products. ANSYS ® Faculty Research is going to be used,which contains the same package as ANSYS ® Multiphysics, aside from some little restrictionson the element amount. But it will be largely enough for the simulations.ANSYS ® offers also a very large panel of about 150 different element types. It providesstructural beam elements, shell elements, pipe elements, for plastic or elastic materialproperties, linear or non-linear behaviour, and all that either for two or three dimensionalanalysis. It has also a comprehensive graphical user interface (GUI) that gives usersinteractive access to program functions, commands, documentation and reference material.An intuitive menu system helps users to navigate through the program. Users can input datausing the GUI, or by producing an input file that the software can thereafter read, so that hecan re-run his simulation very easily by changing just some parameters, without having torestart from the beginning.6

FEM modelling of a bellows and a bellows-based micromanipulatorII. Bellows' dimensions2. BELLOWS' DIMENSIONS AND PROPERTIES2.1 PreliminariesThe bellows are manufactured by SERVOMETER ® Electroforms. They guarantee thefollowing tolerances accordingly to their data sheets:• Inner diameter B : ±0,13 [mm]• Wall thickness t is only the average metal thickness of the entire bellows.• Outer diameter A varies with wall thickness and size of the bellows.• Length of convolutions length D : ±0,25 [mm]• Spring rate k : ±30%The standard metal is nickel. Normally, the bellows have a 0.0025 [mm] lamination ofcopper between equal thickness' of nickel. Servometer ® bellows have about the samechemical analysis as commercial “A” nickel with the exclusion of the copper lamina.The composition of the metal is:• Ni + Co 99.80%• Sulphur < 0.05%• Interstitially deposited impurities 0.05% (Oxygen and carbon)The bellows normally have a bright corrosion resistant surface and are leak tested 100% ona helium mass spectrometer leak detector. No bellows is shipped if its leak rate exceeds1·10 -9 [cc/sec]. This rate amounts to one cubic centimetre of helium in 32 years.The following properties are given for this nickel :• Young’s modulus23'350'000 [psi]• Yield strength125'000 [psi]• Tensile strength110'000 [psi]• Hardness270 [Vickers]2.2 Geometrical data2.2.1 SERVOMETER ® DATA-SHEETSBecause the details of the way to manufacture these bellows are maintained secret, the onlygeometrical parameters available are the following:• Outer diameter A 0.250 [in.]• Wall thickness0.0015 [in.]• Inner diameter B 0.150 [in.]• Spring ratio k 5.9 [lb./in.]• Convolution length l o 0.740 [in.]• Number of convolution 24• Effective area S 0.0292 [in. 2 ]• Maximum pressure for ½ stroke 290 [psi]Of course, this information does not give all the basics, to create the entire solid model. Theradius of the convolution curves is missing, for example, as well as the length between thetwo curves. Servometer ® didn’t want to give any further informations. Hence this had to bemeasured in this project.7

II. Bellow's' dimensionFEM modelling of a bellows and a bellows-based micromanipulator2.2.2 MEASUREMENTSThe parameters to be determined are (figure 7) : the inner and outer radii, respectively r 1 andr 2 , of the convolutions, and the straight length m between the inner and outer curves. Theycan be deducted from the given parameters, but only if symmetricity assumption is made, i.e.r 1 = r 2 . These measurements have to be made to be on the safe side.The first step was to cut carefully the bellows axially into two equal parts, in a way asprecise as possible. If the FE analysis is made with inexact geometrical basis, the results willbe erroneous, and it is useless to proceed.bellowsPower drillwooden maintaining-toolCircular cuttingplatezxFigure 5Overview of the cutting system.The power-drill is fixed andguarantees a non-displacementin Z-direction. The X- and Y-direction movements areperformed by moving thewooden piece, which lays onthe table.Consequently, a small oblong hole was built in a wooden cube, so that the bellows has justenough place to be put in, without moving it too much. A thin circular cutting-tool was fixedon a power drill, and adjusted the elevation so that it will cut the bellows just in its axis, asshowed in figure 5.To study at the imprecision ratio, two bellows are cut in order to compare if similardimensions are obtained, and so that an average value for each needed measurement can beselected. is more accurate in this manner, because there are four different parts to measure.Figure 6Photo taken under a microscope,showing a part of onehalf-bellows. The ruler on theleft side is in millimetres. Itmeans that the picture heightrepresents about 0.9 [mm].The dimensions of the bellows are so tiny, that some photographs had to be taken under amicroscope. A ruler is set on the bottom of the bellows, so that it can be compared it with the8

FEM modelling of a bellows and a bellows-based micromanipulatorII. Bellows' dimensionspixels of the pictures. Hence, a factor converting dimensions measured by numbering pixelsin millimetres is determined. One example is exhibited in figure 6. The conversion factorfrom pixels into millimetres is 49 – but not in this figure, that has been re-sized – and thepicture is orthogonal, i.e. the same factor suits for both x- and y-directions.2.3 Solid model2.3.1 CONSISTENT UNITSANSYS ® is an unitless program. This means that it does not take into account any units forthe parameters it calculates with. It is up to the operator to be consistent with the units heutilises. Therefore, one basis has to be chosen, and all units have to be adapted to it. TheI.U.S. 2 involves meters for length, kilograms for weight, seconds for time, amperes forcurrent, Kelvins for temperature, moles for substance quantity and candelas for luminousintensity. With those seven units, it is possible to build all other, accordingly to theirdefinitions.2.3.2 UNITS CONVERSIONBecause of the bellows’ dimensions, millimetres are used for the geometrical description.Consequently, the unit basis will be the following (only these are needed):Length = [mm] Mass = [kg] Time = [s]Let’s now deduce from the I.U.S., the units of different parameters to be applied in thissimulation, such as velocity, acceleration, force, pressure, and density. The followingcalculations are shown, to make the reader attentive of the unit-problem:• 1 [m] = 1 [m] · [m/m] = 1 [m] · (10 3 [mm/m]) = 10 3 [m·mm/m] = 10 3 [mm] (3)• 1 [m/s] = 1 [m/s] · (10 3 [mm/m]) = 10 3 [mm/s]• 1 [m/s 2 ] = 1 [m/s 2 ] · (10 3 [mm/m]) = 10 3 [mm/s 2 ]• 1 [N] = 1 [m·kg/s 2 ] = 1 [m·kg/s 2 ] · (10 3 [mm/m]) = 10 3 [mm·kg/s 2 ] = 10 3 [mN]• 1 [N/m 2 ] = 1 [N/m 2 ] · (10 -3 [m/mm]) 2 = 10 -6 [N·m 2 /m 2·mm 2 ] = 10 -3 [mN/mm 2 ]• 1 [kg/m 3 ] = 1 [kg/m 3 ] · (10 -3 [m/mm]) 3 = 10 -9 [kg·m 3 /m 3·mm 3 ] = 10 -9 [kg/mm 3 ]The following conversion table can be given:{ [m] ; [kg] ; [s] } { [mm] ; [kg] ; [s] }Length 1 [m] = 10 3 [mm]Velocity 1 [m/s] = 10 3 [mm/s]Acceleration 1 [m/s 2 ] = 10 3 [mm/s 2 ]Force 1 [N] = 10 3 [mN]Pressure 1 [Pa] = [N/m 2 ] = 10 -3 [kPa] = 10 -3 [mN/mm 2 ]Density 1 [kg/m 3 ] = 10 -9 [kg/mm 3 ]2 International Units System3 This result is evident, but shows the correct method which should be applied for all other units.9

II. Bellow's' dimensionFEM modelling of a bellows and a bellows-based micromanipulatorIt is very important to consider these units definitions when settling values in ANSYS ® .Otherwise, it can induce misinterpretation of the results given by the software – because whenshowing results, the postprocessor doesn’t furnish any unit – or inject wrong data to its solver.Now that the units are known as being consistent, the results of the measurements 4performed on the pictures taken under the microscope, as well as the data furnished by themanufacturer, can be resumed in the subsequent table, accordingly to figure 7.For the inner and outer radii r 1 and r 2 , 6 measures were took each time, and the mean ofthem was put as value in that table. For the straight part m, it was needed to take themeasurements between one reference point on two different pictures, since that dimension isto big to fit entirely under one microscope photo. The tolerances are the minimum andmaximum values measured.PartValue in US systemValue in I.U.S.Outer diameter A 0.250 - [in] 6.3500 - [mm]Inner diameter B 0.150 ± 0.005 [in] 3.8100 ± 0.13 [mm]Mating part C 0.246 - [in] 6.2484 - [mm]Convolution pitch D 0.032 ± 0.010 [in] 0.8128 ± 0.254 [mm]Total convolution l 0 0.740 - [in] 18.7960 - [mm]Thickness of wall t 0.0015 - [in] 0.0381 - [mm]Inner radius r 1 75.375 ± 3.25 [pixels] 0.151 ± 0.007 [mm]Outer radius r 2 90.000 ± 7.00 [pixels] 0.1800 ± 0.014 [mm]Straight part m 402.625 ± 6.00 [pixels] 0.8053 ± 0.012 [mm]Number of pitches N 24 - - 24 - -Max. compression stroke 0.15 - [in] 3.7846 - [mm]Max. c. stroke per conv. 0.0067 - [in] 0.1702 - [mm]Effective area S 0.03 - [in 2 ] 18.7096 - [mm 2 ]Spring ratio k 5.9 ± 1.77 [lb./in] 1033 ± 310 [N/m]Spring ratio k per conv. 141 ± 42.3 [lb./in] 24'693 ± 7407 [N/m]Density of Ni/Co 0.03197 - [lb./in 3 ] 8.85 10 -7 - [kg/mm 3 ]Young's modulus E 23.35 10 6 - [psi] 162 10 6 - [mN/mm 2 ]Yield strength 110 10 3 - [psi] 760 10 3 - [mN/mm 2 ]Tensile strength 125 10 3 - [psi] 862 10 3 - [mN/mm 2 ]Hardness 270 - [Vickers] 270 - [Vickers]DBAl 0LDr 1r 2tB/2Figure 7Drawings that provideparameter location, andfurthermore, parametersdefinitions.mCA/24 The measured values are shown in the table with darker background colour.10

FEM modelling of a bellows and a bellows-based micromanipulatorII. Bellows' dimensionsThe first conclusion is that, corresponding to figures, following balance exits:A B= + t + r2+ m + r1+ t(2.1)2 2Actually, because of the measurement imprecisions, the expected result is not observed:3.175 ≠ 3.1173. This means an error of 1,85%. But it is to be noted that the thickness of thewall is not guaranteed by the manufacturer, it is only an average! It signifies that the samplecan have a different thickness. The cutting-manipulations may also have been the origin ofsome deformations.However, the simulations showed that the result is not substantially affected by those radii.One pressure was applied inside the bellows, with different radii, and had a look on the y-displacement. The results didn’t change more than about 0,2% when changing the radius r 1and r 2 by around 20%.It can be deduced that the assumption that r 1 = r 2 holds. For both of them, the followingvalue is used:D − 2tr1= r2= r = = 0,1778 [mm] (2.3)4Identical conclusion for m implies the subsequent equation to be used:A BA Bm = − − t − r2 − r1− t = − − 2t− 2r= 0,8382 [mm] (2.4)2 22 2The result differs only by 4% of the one measured.11

III. ModellingFEM modelling of a bellows and a bellows-based micromanipulator3. MODELLING3.1 Procedure3.1.1 WAY TO ANALYSE ONE BELLOWSThe first step of this analysis is to check if the results as given by the manufacturer areobtained by simulation. It will be later on to confirm that the radial displacement of thebellows’ wall is insignificant in comparison with the axial one when applying differentpressures into the bellows. After that, the spring ratio obtained by simulation is comparedwith that given by the designer. This phase will certify that the model behaves in a correctway.The second phase will be the most important one. It is desired to obtain the bendingbehaviour of the bellows, when applying different forces on top at different pressures. Twodifferent graphs are determined. One is the function of bending displacement versus appliedforce, under one constant pressure. The other is a check how the bellows' stiffness increaseswhen applying various internal pressures, with a constant bending-force.The third phase is the axial displacement of the bellows when applying various pressures. Inthis way, the exact relationship between pressure applied by the piezoactuator and axialmovementof the bellows is obtained.To perform these simulations, a simplified solid model is used. Indeed, the whole bellowsneeds not to be used, since it can be recovered using smaller parts placed in series or inparallel, depending on the analysis. The subsystem will consist of a single ring – also calledconvolution – of the bellows, as showed in figure 8. The boundary condition on its bottom hasYwholebellows single ring Figure 8The whole bellows can be modelled bysimulating the behaviour of one single ring, andextend the results to the bellows afterwards.to be set, so that it can’t move in y-direction. The upper surface can move without anyconstraint. Then, the spring-like behaviour of the total bellows can be calculated from that ofthe single-ring.In order to spare CPU-time, the ring behaviour is also modelled in only two dimensions,each time while axisymmetric loads – such as the pressure – and boundary conditions areapplied. It can effectively run up to hundred times faster than the three dimensional problem.But to check if the 2-D model is accurate enough, it is first compared with the 3D one, andonly afterwards the real simulations are run.3.1.2 CHOOSING RIGHT 2-D ELEMENTSA solid of revolution is generated by revolving a plane figure about an axis in the plane.Loads and supports may or may not have axial symmetry. Initially, the case where geometry,elastic properties, loads and supports are all axisymmetric will be considered. Consequently,nothing varies with the circumferential coordinate θ, material points displace only radiallyand axially, and shear stresses τ rθ and τ rθ are both zero. Thus the analysis problem ismathematically two-dimensional. Axisymmetric finite elements are often pictured as planetriangles or quadrilaterals. These plane shapes are actually referred as cross sections ofannular elements, and what appears to be nodal points are actually circles (figure 9).12

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

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

FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingFigure 17Three-dimensionalmodel withSOLID45 element.The number ofelements in thatcase reachesexactly 12'960,and that of thenodes 26'400.When solving a problem, the CPU-time consumption depends naturally on the number ofequations the computer has to solve. The number of loads and boundary conditions don’t alterthe number of equations. At each node, there are as many equations as degrees of freedom,and they just aggregate the force equilibrium at that node. So each force applied on the solidmodel will just add a term to these equations, but that doesn’t augment significantly thecomputing duration. The number of equations is therefore directly 9 proportional to that one ofthe nodes constituting the mathematical model [Cook89].Accordingly to the three different meshes obtained with those solid models, it is evident thatone should work as much as possible with the two-dimensional one. Effectively, the amountof nodes is multiplied by the number of degrees of freedom per node in each case, thefollowing results are obtained:• 2'529 [nodes] · 3 [DOF] ≈ 7'600 equations for PLANE83-model• 12'124 [nodes] · 6 [DOF] ≈ 72'750 equations for SHELL43-model• 26'400 [nodes] · 3 [DOF] ≈ 79'200 equations for SOLID45-modelThe total expected difference in the CPU-time consumption between PLANE83 andSOLID45 models is about ten times. However approximately all needed loads and boundaryconditions can be applied in the two-dimensional problem. Consequently this point must bevalidated by applying the same conditions to each of the three mathematical models andcheck if the same results are obtained.To test every model correctly, including the one with shell elements, an homogenouspressure to the inner surface equal to 0,01 [mN/mm 2 ] is applied, with simple boundaryconditions to bottom surface: nodes can not move in y-direction, as shown by the next figure.yxFigure 18Boundary conditions and pressure applied to all threemodels, to check if the results are corresponding.9 Actually, that is not exact. Each boundary condition also adds some equations to the mathematical model, butin comparison with the amount of elements – and therefore of nodes – this small number of added equationsis negligible.17

III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatorThe following results have been obtained :18

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

FEM modelling of a bellows and a bellows-based micromanipulatorIII. Modellingcreate a rigid region by generating constraint equations to related nodes in that region. Thesegenerated constraint equations are based on small deflection theory. Six equations aregenerated for each pair of constrained nodes in three-dimensional space, and three of them intwo-dimensional space. This part can then be assimilated as having an infinite Young’smodulus. So apply the centralised force F equal to 10 [mN] on the master node of that region,and that force will be propagated on all other slave nodes.The second method is to calculate the area S of the top surface of the ring, and to apply apressure p equivalent to the force F on that surface:Fp = (3.4)SIn the studied case, the surface on which the pressure is applied is a hollow circle. The innerdiameter equals B and the outer diameter (B+2t). So the surface is:2( B + 2t)− B ) ⋅ = 0. 4611 2S = π [mm 2 ] (3.5)4and the pressure p is about 21.71 [mN/mm 2 ].The simulation was computed with three different linear models: the first one in twodimensions, with the rigid-region method; the second also in two dimensions, but with thepressure applied to the top face of the ring; the third is in three dimensions, with solidelements, by applying also a pressure on the upper surface. Once more, consistent values witheach simulation are obtained:2D rigid-region21

III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatorIn the display of the 2D-models, the displacement scale has been augmented, so that one canfeel if the shape behaves as expected, but this appearance does not match with thecalculations. The true value is then only showed by DMX. In the three dimensional model, thedisplay has the correct appearance.It can now be checked which spring-rate has been obtained:• 2D-model, with pressure applied : 3,27·10 -4 [mm] ⇒ 30'211 [N/m] for one ring, and1'276 [N/m] for the whole bellows.• 2D-model, with force applied : 3,34·10 -4 [mm] ⇒ 29'940 [N/m] for one ring, and1'247 [N/m] for the whole bellows.• 3D-model, with pressure applied : 3,28·10 -4 [mm] ⇒ 30'672 [N/m] for one ring, and1'278 [N/m] for the whole bellows.The differences between the three models are less than 2,3%, which is quite agreeable. Theexpected value of k was 1033 [N/m]. It means that the mismatch between the theoretical valueand the mean of the three computed (30'274 [N/m]) is about 18%. As has been seen in theprevious pages, the analytical value is given only with ±30% tolerance. Hence the errorobtained by the simulation stays in that tolerance. Since it is known that the formula given byServometer is not only theoretical, but was obtained by experiments 11 , and that the thicknessof the bellows can vary quite a lot between two different bellows of the same type, a moreaccurate result is useless.Accordingly to foregoing assertions, the first simulation can be considered to validate thethree mathematical models. The shell-elements model was not used to calculate this result,because it was probably not very useful in that configuration. But inasmuch as the mismatchof previous test was only 1,4%, it can be assumed that this proof also validates the SHELL43-model.3.2.2 CONSTANCY OF THE SPRING RATIONow, it should be checked if this spring ratio is really constant as expected. To do that, avery useful method provided by ANSYS ® is used: Multiple Load Steps by array parametermethod. It is a powerful and elegant method that requires knowledge of array parameters anddo-loops, which is part of APDL 12 . The method involves building tables of load versus time.In our case, this table is very simple:Time [s] Force [mN]0.0 0.0100.0 100.0The load history has now been defined, so this force can be applied by using a loop 13 thatwill automatically complete the table at each time-step, so that the load will increase linearlyfrom zero [mN] at starting time to 100 [mN] at ending time. With this method, the timeincrement can be very easily changed, to get more or fewer steps.Time-increment of 10 [s] can be selected, so that eleven steps are obtained. These will beenough to check the linearity of the spring-rate of our model. Once more, two models are usedto perform this simulation: one 2D and one 3D.11 The constant 4,3 in formula (3.4) was obtained experimentally, explained Servometer.12 ANSYS ® Parametric Design Language13 See annexe, section 6.1.3, Applying time-varying load for the spring k.22

FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingWith the 2D-model, the force F is replaced by an equivalent pressure p, acting on the topsurface of the bellows. To be sure that the simulation can take into account any non behaviourof the bellows, a non-liner analysis has to be carried on. The next table shows the obtainedresults, the displacement being in absolute value 14 .k [mN/mm]StepForce F [mN]Displacement dy1 ring [µm] Bellows[µm]1 0 0 0 -2 10 0,327 7,838 1'2763 20 0,654 15,685 1'2754 30 0,981 23,541 1'2745 40 1,309 31,405 1'2746 50 1,637 39,277 1'2737 60 1,965 47,159 1'2728 70 2,294 55,048 1'2729 80 2,623 62,946 1'27110 90 2,952 70,853 1'27111 100 3,282 78,769 1'270127712761275127412731272127112701269Spring-rate k in fonction of the displacement-80 -70 -60 -50 -40 -30 -20 -10Displacement dy [µm]The mobile platform of the micromanipulator weighs about four grams, which equalsroughly 40 [mN], and the purpose of the manipulator indicates that it should not handle withmore than few grams. Hence the applied force in this simulation should embody the full rangeof all loads. It can be noted that the spring-rate k varies linearly, and less than 0,5%.With the 3D-model, a pressure equivalent to a force varying between -100 [mN] and+100 [mN] is applied. So it can be seen that the spring-rate varies when the bellows iscompressed or extended. A third model has been created with a solid region on top of thering, and the same varying force is applied on its middle node. But between these two 3Dmodels,the results varied quite a lot, as well as in the previous simulation, when linearanalysis was applied.The model using solid region can be ignore in that precise case, since ANSYS ® warns, thatconstraint equations added by these solid region may not suit to models undergoing largek14 A compressing force results in a negative displacement, since the same coordinate system as shown in figure18, at page 17 is used.23

III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatordeflection in non-linear analysis. The exact “value” of such a large deflection is not welldefined. In the studied case, the maximum displacement of nodes in the models is about80 [µm], and the height of the bellows is 18,8 [mm]. It seams that a deflection of 0,4% cannotbe considered as large. But anyway, only the result of the 3D-model without solid region iskept.The non-linear analysis permits the software to re-calculate the K matrix 15 of knownmaterial constants at each step, since the shape of the mathematical model may changebetween two altered steps because of the applied loads. To guarantee the correct K matrix, theload of 100 [mN] should not be reached at once, but in several steps. That’s why thefollowing varying force F is applied on the bellows:Force [mN]100806040200-20-40-60-80-100It means that the grey part of the graph doesn’t interest us. And if the results showed in nexttable are investigated, it can be seen that this precaution is worthwhile, since the k value atstep 2 is not included 16 between the steps 7 and 8, as it should be. That shows how importantthe gradually increasing load during the first three steps is.StepForce F [mN]Time-varying extending and compressing force1 2 3 4 5 6 7 8 9 10 11 12 13 14StepsDisplacement dx1 ring [µm] Bellows[µm]1 0 0 0 -2 33 -1,088 -26,106 1'2643 67 -2,179 -52,308 1'2624 100 -3,275 -78,604 1'2725 80 -2,617 -62,815 1'2746 60 -1,961 -47,060 1'2757 40 -1,306 -31,339 1'2768 20 -0,652 -15,653 1'2789 0 0 0 -10 -20 0,651 15,618 1'28111 -40 1,300 31,203 1'28212 -60 1,948 46,753 1'28313 -80 2,595 62,270 1'28514 -100 3,246 77,914 1'282k15 See equation (1.1)16 Effectively,1'264∉[1'276;1'278]24

FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingNot taking into account the first three steps, less than 1% difference for k over the full loadrange, accordingly with the 2D-model is obtained:Finally, the value of the computed spring-rate k can now be determined by calculating theaverage between all the values obtained with the 2D and 3D non linear analysis’:k = 1'277 [N/m] (3.6)For all succeeding analysis, this calculated value, which differs about 20% of that valuegiven by the manufacturer will not be used. This result was only a validation of the models,since it stays in the tolerances given by Servometer. From this point on, the spring-ratio kgiven by the data-sheets will be used.3.3 BendingOne of the most important parameter for the control of the micromanipulator is the bendingcomportment of the bellows’. For this, one bellows is to be identified with a beam. Therefore,δxαFk [mN/mm]1288128612841282128012781276127412721270=Spring-rate k in function of the displacement-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80Displacement dy [µm]δxFαFigure 20Bent beam, which behaviour equals that of one bellows underconstant inside pressure. The applied force F induces adisplacement δx of the free-end of it, that can be easily calculatedby beam formulas. The angle α at the same extremity can also becalculate.for one bellows under constant inside pressure and one load at the free-extremity (figure 20),the displacement δx of the free end should be given by the following formula:3F ⋅lδ x = (3.6)3 ⋅ E ⋅ Iwhere F is the applied force, l the length of the bellows, E its Young’s modulus and I theinertial momentum of its surface. These two last parameters, E and I, can be chosen as25

III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatordesired, but they just need to give the same constant result when multiplied. It means that anew parameter Q = E·I can be chosen, and put it into the beam formula.Equation (3.6) thus becomes:3QF = ⋅δxl 3(3.7)which has exactly the same kind as the spring formula F = k·δx. The physical length l of thebellows is known, so applying some load on it and simulating the situation can determine thelone unknown parameter Q.The angle α at the end of the bellows can also easily be calculated by the beam formula:22F ⋅lltan( α ) = − = −F⋅(3.8)2⋅E ⋅ I 2QThis furnishes a good confirmation, to verify if formula (3.8) gives the same angle α as thatone ensued by the simulation, taking into account the computed parameter Q. But accordingto Calladine, the internal pressure should have some effect [Call83].When a curved tube is subjected to pure bending, its originally circular cross-section isdistorted into a slightly elliptic configuration, without an appreciable change of perimeter.Therefore there will in general be a small change in the enclosed cross-sectional area – andhence of the inertial momentum I – of the tube, since of all figures having a given perimeter,the circle encloses the maximum area. It follows that if the tube which is to be bent sustainsan interior pressure, the overall flexure stiffness of it must be expected to be somewhat greaterthan in the absence of pressure.It is not known how exactly the curve of the function bending vs pressure looks like, buttwo certain facts are known. First, without any internal pressure, one definite bending isobtained for each applied force. Secondly, the bending must decrease for a constant force,when the pressure increases. According to Calladine, it is clear that this stiffening effect isdirectly proportional to the pressure. But in this case, the pressure changes are so tiny, that itseffect should not be so significant. So the curve that will result from the simulations shouldlook like that :δx 0Bending [mm]Full-range working region0Pressure [mN/mm 2 ]For a constant pressure, the bending should be directly proportional to the applied force, asshown by equation (3.7). The force to be applied to get a constant displacement by differentpressures should also be directly proportional to that pressure.In order to find out this bending parameter Q, time-varying loads are applied on the solidmodels. First, a constant bending force is used, with a pressure varying from zero to themaximum that is applied in the bellows. Then, exactly the opposite is done: a constantpressure is applied with a time-varying force. In this way, the expected curves can beconfirmed.∞26

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

III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatorThe model is only that of one single ring. The simulations' results have to be extended to theentire bellows to get the total bending constant Q. For that, one lone ring is approximated witha very short beam of length l 1 , and the following reflection according to equation (3.7) isobtained to get the bending δx of the bellows:F 3δ x1= ⋅l1(3.10)3QF 3 F3 F 3 33δ x = ⋅l0 = ⋅( 24⋅l1) = ⋅l1⋅24= δx1⋅24(3.11)3Q3Q3QThe constant bending force F equals 50 [mN]. The internal pressure varies from 0 up to100 [mN/mm 2 ]. The following graph shows how the total bending of the entire bellowsincreases, when the δy growth-effect is no taken into account.δx -displacement [µm]1.351.341.341.341.341.341.33UX-displacement directly given by the softwareConstant bending Force F = 50 [mN]0 20 40 60 80 100Interior pressure [mN/mm 2 ]This graph shows the true displacement of the bellows, which can be observed in the reality.But if one wouldn't know that the calculation has to take care of the axial bellows' growth, hecould be quite surprised that the bending increases when applying an internal pressure p. Itwould mean that it would be possible to bend the bellows without applying any bending force,which is of course absolutely impossible.The next plot takes into account the bellows' growth and shows that the bending δxeffectively decreases when the internal pressure increases. This means that the stiffnessincreases as expected by Calladine.δx -displacement [µm]1.3361.3341.3321.3301.3281.3261.324Corrected δ x -displacement, taking into account δ y -growthConstant bending Force F = 50 [mN]0 20 40 60 80 100Interior pressure [mN/mm 2 ]28

FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingThe bending δx is very small, only a few micrometers, which is not very significant. So thesimulation has been repeated several times more, with various bending forces, and always thesame upshot is observed.The bending parameter Q is determined next. The bending value δx at step 3 – without anyinternal pressure – is taken into account, because it varies less than 0,72% on the wholepressure range. Equation (3.7) implies:3F ⋅lQ =3⋅δx( 18.8)50⋅=3⋅1.3353= 83.27·10 3 [mN·mm 2 ] (3.12)Next diagram shows the bending-angle α at the top of the entire bellows, for the same50 [mN] bending force. The angle doesn't need any correction, because there is no differencedue to the growth of the bellows.z -rotation [°]5.525.505.485.465.445.425.405.385.36Angle α at the top of the entire bellowsConstant bending Force F = 50 [mN]0 20 40 60 80 100Pressure [mN/mm 2 ]To check the correctness of the found Q parameter, let's see if equation (3.8) gives the samebending angle as the simulation.( 18.8)22⎛ F ⋅l⎞ ⎛ 50⋅⎞α = arctan⎜−⎟ = arctan⎜⎟−= 6.06º (3.13)3⎝ 2⋅Q⎠ ⎝ 2⋅83.27⋅10⎠This calculated value is 9% off to the value given by the simulation – which was 5.5º. Thisvery small mismatch corroborates that the beam-model can be used to simulate the completemicromanipulator later. It also confirms that previously suggested way to calculate thebending δx of the entire bellows is correct.Let's calculate now the same Q parameter and α angle, with those results directly generatedby ANSYS ® for one lone ring, without any data manipulation:3F ⋅lQ3⋅δxand−13( 8.128⋅10) =50⋅=3⋅9.6554⋅10=−592.689·10 3 [mN·mm 2 ] (3.14)−12( 8.128⋅10) ⎞⎟ =2⎛ F ⋅l⎞ ⎛⎜ 50 ⋅α = arctan⎜−⎟ = arctan −0.00102º (3.15)⎝ ⋅ ⎠⎜32 Q⎟⎝2 ⋅92.689⋅10⎠29

III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatorThis time, the mismatch with simulated value is 6% only 17 . Since this comes from nonmanipulateddata, it can be deduced that the few simplifications used previously to extend thisresult to the whole bellows do not introduce much error.3.3.2 CONSTANT PRESSURENow, it is the interior pressure that will keep the same value, and the bending-force that isgoing to increase linearly. The same assumptions as in the previous analysis about the way toextend the result of one lone ring simulation to the whole bellows are made. The pressurewon't change during all the steps. The calculation is thus a little bit easier, since care has notto be taken about the bellows' growth δy – because it won't grow at all.For the same reasons as mentioned previously, these pattern-like loads during the multi-stepsimulation have to be applied:10080Time-varying applied loadsLoads [%]6040200ForcePressure1 2 3 4 5 6 7 8StepsThe constant pressure p will be set as 20 [mN/mm 2 ], and the maximum value of the force Fwill be 50 [mN]. As expected, the next plot shows that the displacement is effectively directlyproportional to the applied bending-load:1.75δx bending at the toppoint of the whole bellows1.50x -displacement [ m]1.251.000.750.500.250.00Constant pressure p = 20 [mN/mm 2 ]0 10 20 30 40 50Force [mN]Let's once more check the parameter bending Q obtained from formula (3.7). It belongs tothe constant of proportionality of the previous graph, and should be quite equal to that onefound in previous constant bending-force simulation.17 The value issued by the simulation is 0.0096º.30

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

FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingThe pressure range is much larger than the pressure that is used in the micromanipulator.Effectively, the measurements 19 showed that the typical maximum displacement of onebellows was around 150 [µm], which corresponds on our graph to an applied pressure ofroughly 20 [mN/mm 2 ]. In that case, Z does not vary more than approximately 1,20%. Hence,once more, a constant value for Z can be assumed by computing an average.Z = 6,6·10 -3 [mm 3 /mN] (3.16)After this, it is time to take a look on the radial displacement occurring when applying aforce inside the bellows. The consequence is effectively expected to be minor, since theeffective surface on which the pressure applies in radial direction is very small comparing onthat on which the pressure is applied in the axial direction, as illustrated by figure 24. Tomeasure the radial displacement, the node having the largest motion is needs to be checked.Actually, that node E is showed on figure 24 and corresponds to the external shape of thewhole bellows. There is no need to extend the simulation result of one ring's to the entirebellows, since it is exactly the same.Figure 24Illustration of the amount ofpressure effectively acting inradial direction, incomparison with thatbehaving in axial direction.Next table shows the comparison between axial and radial displacement:EPressure[mN/mm 2 ]y-displacement[mm]Radial motion[µm]Ratio25 0,202584 0,1667918 82350 0,406616 0,3297720 81175 0,611811 0,4889844 799100 0,817910 0,6444888 788125 1,024680 0,7963584 777150 1,231900 0,9446724 767175 1,439400 1,0895208 757200 1,647000 1,2309972 747The mean value of axial versus radial displacement is about 783. It means that the radialdisplacement is almost three order of size smaller than the axial one, and can therefore freelybe neglected in further simulations.3.5 TolerancesIt is known from the manufacturer's data sheets that the bellows' proportions have sometolerances, as for instance the inner diameter B, which can for example vary up to ±3,5%.These dimension mismatches will certainly influence some parameters, as the spring-rate,19 Made in Tampere University of Technology33

III. ModellingFEM modelling of a bellows and a bellows-based micromanipulatorwhich is given by Servometer with ±30% accuracy. That's why it is important to find out howmuch previous simulation results will be affected by them.Unfortunately the exact tolerance of all geometrical data of the bellows are not known. Ithas also been previously noticed that not all of them affect the behaviour significantly. Analteration of 20% of the radii r 1 and r 2 , for example, instigates an axial displacementmodification of only 0.2%. That's why a choice was made, and only the most relevantparameters were selected, namely the wall thickness t, the convolution pitch D and the innerdiameter B.Next tolerances have been fixed for the three parameters:• D: ±70% D = 0.783 ±0.254 [mm]• B: ±3% B = 3.810 ±0.13 [mm]• T: ±10% t = 38.100 ±3.8 [µm]The manufacturer doesn't give any tolerance for t. But since it is a very sensitive parameter,it was decided anyway to include this parameter in the following tables. A large tolerance of±10% was fixed for it. In each of the next sections, the simulation was performed six times,changing at each stage only one of the three parameters to its upper or lower tolerance value.3.5.1 SPRING-RATE TOLERANCESA force of 10 [mN] on the top of the bellows was applied, with the same conditions as in theforegoing spring-ratio simulation 20 .t [µm] B [mm] D [mm] k [mN/mm]38,1 3,81 0,7832 1259 (21)34,3 3,81 0,7832 961 (-23%)41,9 3,81 0,7832 1599 (+27%)38,1 3,68 0,7832 1101 (-13%)38,1 3,94 0,7832 1465 (+16%)38,1 3,81 0,5588 24'505 (-19%)38,1 3,81 1,0668 40'493 (+34%)The tolerance of the convolution pitch D cannot be extended to the entire bellows.Effectively, D is a mean value of the constituting rings of the bellows. It is impossible that allpitches have, for instance, the upper tolerance, because it would signify that the resultantbellows' length would be 25.6 [mm]! This is unrealistic, and would of course never passServometer's quality control! It wouldn't be therefore consistent to take the entire bellows'spring-rate with extreme convolutions pitches' value. That's why in the table, the spring ratiois computed and compared to only one ring's value 22 .One can notice through these results that the spring ratio is affected by all of the parameters.A 3%-variation of the inner diameter B changes the spring rate up to 16% of its nominalvalue. But the thickness t doesn't change so much as the simulated spring-rate, as expectedreferring to formula (3.4) given by Servometer. Effectively, when t changes up to 10%, thespring rate is only affected by 27%.20 See section 3.2.21 This first table row refers to the result found with nominal values.22 The spring-rate k of one ring is 30'211 [mN/mm].34

FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingIf the percentage error is calculate with the formula, following outcomes are obtained:4,3⋅E=3⋅( A + B) ⋅( 0,0381)( A − B − t)⋅ N3⋅( A + B) ⋅( 0,0343)( A − B − t)⋅ N3⋅( A + B) ⋅( 0,0419)( A − B − t)⋅ Nk38,134,3⋅E=k34,334,3⋅E=k41,93= 1033 [mN/mm] (3.17a)= 751 [mN/mm] -27% (3.17b)= 1383 [mN/mm] +34% (3.17c)That means that the formula gives also a larger difference when the wall's thickness t isincreased than when it's decreased. This agreeing fact validates once more our simulations.Since these calculation give approximately a tolerance of ±30% for k, it is probable that the10%-tolerance chosen for the wall's thickness is not that wrong.3.5.2 BENDING PARAMETER'S TOLERANCESA bending force of 100 [mN] and an interior pressure of 10 [mN/mm 2 ] are applied to thebellows. For that purpose, the solid 3D-model was used. Here also, the changes induced bythe convolution pitch's tolerance on a lone ring only have to be looked at, for the same reasonsmentioned sooner. The conclusions based on simulations of Q are resumed in next table:t [µm] B [mm] D [mm] Q [mN·mm 2 ]38,1 3,81 0,7832 82'81734,3 3,81 0,7832 63'374 (-24%)41,9 3,81 0,7832 105'432 (+27%)38,1 3,68 0,7832 70'566 (-15%)38,1 3,94 0,7832 97'692 (+18%)38,1 3,81 0,5588 49'943 (-40%)38,1 3,81 1,0668 143'823 (+74%)As expected, the length of a beam – which corresponds to the convolution pitch – is veryrelevant for the bending. As short the beam is, as ample is its stiffness – that can be assumedto be proportional to Q. But since this is an average of those constituting the entire bellows, Dshouldn't be so significant in bellows' mismatches.Not taking into account D, it can be concluded that the bending stiffness of the bellowswon't vary so much between various different specimen. The maximum mismatch shouldnever reach ±27%, since quite extreme tolerances were considered for t 23 . This result isabsolutely very interesting for the future open-loop control.Let's check with beam formula (3.6) which difference can this tolerance of Q induce on thebending δx of the bellows when applying a nominal bending force F = 100 [mN] and nointerior pressure 24 p:( 18.8)3100⋅δx 83k== 2.67 [µm] (3.18a)3⋅82'81723 The metal deposition on a mandrel, for instance, can nowadays be quite good controlled, and ±10% oftolerance is too large for an automated manufacture!24 It would not be very pertinent to take care of p, since it was previously noticed that it doesn't influencesignificantly the bellows' stiffness.35

III. Modelling( 18.8)3FEM modelling of a bellows and a bellows-based micromanipulator100⋅δx 60k== 3.47 [µm] +38% (3.18b)3⋅60'2023100⋅( 18.8)δx 105k== 2.17 [µm] -21% (3.18c)3⋅105'432These outcomes are a little bit worse. For an open-loop control, it would mean that betweentwo different micromanipulators, the behaviour could be quite different. And maybe, it shouldbe necessary to make one short calibration for each of them before operation.3.5.3 INTERNAL PRESSURE VS DISPLACEMENTThe same tolerances experiments have been made for the pressure vs y-displacement Zparameter. But this time, the 2D-model was used, which needs less CPU-time. Naturally, thisfirst result was checked with the 3D solid model, and inasmuch as it was agreeing, the 2Dmodelonly was freely continued to be used.Next table shows the influence of the parameters on the axial displacement, and on the Zparameter:t [µm] B [mm] D [mm] Z [mm 3 /mN] δy [µm]38,1 3,81 0,7832 6,668·10 -3 33334,3 3,81 0,7832 8,709·10 -3 435 (+31%)41,9 3,81 0,7832 5,221·10 -3 261 (-22%)38,1 3,68 0,7832 7,836·10 -3 392 (+18%)38,1 3,94 0,7832 5,556·10 -3 278 (-17%)38,1 3,81 0,5588 0,324·10 -3 16 (+17%)38,1 3,81 1,0668 0,214·10 -3 11 (-23%)The last two rows' results are once more calculated for only one lone ring, and compared tothose of one ring with nominal geometry. Here also, the maximum mismatch is due to thewall thickness' tolerances. But the biggest effect is still for B: one difference of only 3% of theinner diameter induces up to 18% more growth of the bellows!3.5.4 CONSEQUENCESThe spring-rate was from the beginning known as being given with only ±30% accuracy.But the other parameters, which are also important for the behaviour of the micromanipulator,have surprisingly large tolerances too. This can affect the open-loop control with a large lossof precision. All these geometrical parameters being independent, one bellows may have veryunfavourable stiffness, if all appear to have the value acting in that way, for instance.Bellows 3End-effector, k 3y c,3y = f(y 1 , y 2 , y 3 )y 1Z2 , Q 2 , k 2y c,2Bellows 1 Z 1 , Q 1 , k 1y c,1y c = f c (y c,1 , y c,2 , y c,3 )y 2Bellows 2Figure 25Diagram of the open-loop controlsystem. The desired position y c ofthe end-effector depends mostlyon the three bellows, and canlargely differ from its realposition y, depending on bellows'mismatches.Controller36

FEM modelling of a bellows and a bellows-based micromanipulatorIII. ModellingAccordingly to notation of figure 25, the controller may calculate the individual y c,i positionof each constituting bellows, throughout the inverse kinematic model (IKM), to attain adesired position y of the micromanipulator's end-effector. This IKM will be constructedespecially with data obtained from the simulations performed during this project, butassuming a nominal 25 bellows.However, mismatches between different bellows cannot be taken into account in the IKM.As calculated, one bellows' bending δx and axial growth δy can consequently vary up to 30%,depending on its effective geometry. This would of course affect the reached position y i of thevarious bellows, and surely, the resulting y of the end-effector.Consequently, the bellows' should be selected in a more accurate way, in order to avoidlarge mismatches. Servometer proposes for instance bellows with smaller tolerances. Anotherpossibility could be to enable a calibration in the control system.Figure 26Photography of a prototype of themicromanipulator built inUniversity of Technology ofTampere.25 I.e. a bellows which geometrical parameters have their nominal value.37

IV. OutlookFEM modelling of a bellows and a bellows-based micromanipulator4. OUTLOOK4.1 Expansion of the bellows' model to the entire micromanipulator4.1.1 PRELIMINARIESThe main point of this project was modelling of one bellows, in order to be able to buildafterwards the model of the whole micromanipulator. Therefore, the time allowed for thischapter was very short. The aim of it was only to build the beam-based model, but all thesimulations that have to be performed in order to build the kinematic model are to becontinued in an upcoming project.Consequently, the following paragraphs are only a starting point and a short overview of theprospective work, and have no pretension to cover all.4.1.2 CHOOSING RIGHT 3D-BEAM ELEMENTThe very next step is now the expansion of the bellows' model, in order to build a model ofthe entire micromanipulator. ANSYS ® proposes various beam-like elements and pipeelement.The first idea would be to choose a pipe element, since the bellows behaves mostlyas a pipe with very thin walls and small axial stiffness filled with water. But the completecomportment of those elements is quite complicated, and all the needed parameters are notavailable. The simulations have been oriented in a way to replace the bellows in future modelby a beam, and our artificial bending parameter Q for instance can only be used with beams 26 .BEAM44-element is proposed to be used for this model. It is a tapered uniaxial elementwith tension, compression, torsion and bending capabilities. The element has six degrees offreedom at each node: translations in the nodal x, y and z directions and rotations about thoseaxes. Stress stiffening and large deflection 27 capabilities are also included (figure 27).zziyiIyxixjyjJzjFigure 27Representation of the three-dimensional elementBEAM44, that has two principal nodes, I and J.This element is tapered, on the contrary of otherusual beam-elements, which are not axisymmetric.xThis element requires additional parameters, such as the momentum of inertia I and beam'sthickness t x and t y in y and x-direction. Since the Q parameter is known, this permits to choosethe inertial momentum I and Young's modulus E as desired. It is proposed to keep the originalmaterial's Young's modulus, and to compute I with formula Q = E·I.QI = = 5,11·10 -7 [mm 4 ] (4.1)EAs the corresponding beam's cross-section is not known, the bellows' wall thickness t is firsttried as input for the beam's widths t x and t y .26 ANSYS ® 's pipe are beam-like elements with modified parameter, to take into account the internal cavity.But our parameters are made for non-hole beams.27 What allows non-linear analysis.38

FEM modelling of a bellows and a bellows-based micromanipulatorIV. OutlookIn order to examine if the selected element, and its inputs are valid, it is first wanted to builda simple beam-based model of only one bellows. Comparative load is applied, to get resultsthat are known from previous simulations. Hence, all degrees of freedom of one beam's leftborder are fixed, and a force F = 50 [mN] is applied to its free edge, as shown below:Bending at the free-edge of the beam in this model equals 1.337 [µm], which is quite closeto the result obtained with the antecedent ring-based model, 1.335 [µm]. But this does notprove that the foregoing results were exact 'till now, since the parameter Q was calculated sothat it will be valid for the beams. It can only be inferred that, according to previoussimulations, the correct beam parameters have been found for the beam-based model of theentire micromanipulator.4.2 Beam-based model of the entire micromanipulator4.2.1 MODELThe micromanipulator model can now be constructed. It might be quite simple, only threebeams, and a platform binding them together are needed. But one has to be very careful in theway the geometry is erected.Considering that software can only compute numbers with limited decimals, rounding offerrors can generate misbehaving of the model. Therefore, it is better to put parameters in thelisting, and let the program make the calculations. For example in the case studied, thegeometry is "120º axi-symmetrical". So, instead of calculating the location of keypoints "byhand", it is preferable to copy one bellows three times, to guarantee a perfectly"axisymmetric" model.In a previous study [Lind98] the whole micromanipulator just collapsed when a load wasapplied on it, because the model was not perfectly symmetric. It took a long time to discoverthat this strange bearing came from numerical errors occurring because of such round offerror.39

IV. OutlookFEM modelling of a bellows and a bellows-based micromanipulatorThe purpose of the platform is only to fasten the three bellows together. It can be consideredas having infinite inflexibility. In that case, it's useless to build a solid structure for thesupport. One easy way suggested to model it is to create a rigid region 28 with a middle masselement29 , which can support further loads.Figure 28 shows an isometric view of the model created, with three beams and one rigidregion. The three top lines represent the rigid region that binds the three beams – in bluecolour – together. On the beam's base, the boundary conditions are shown with the blue andorange arrows. It means that those nodes can neither move in any direction, and nor rotatearound any of their axes.Figure 28Isometric view of the beam-based mathematical model ofthe micromanipulator. Pink coloured is the so called rigidregionbinding the three beams – in blue – together. Themodel is perfectly symmetric, to avoid any wrongbehaviour.This very simple model can now serve as a tool to simulate any of the behaviours of themicromanipulator. As previously explained, in this project, all of these simulations are notcarried out. Nevertheless, two examples may show the correct comportment of this model.The first one will be horizontal force F z applied on the middle node of the platform (figure29). Since its purpose is only to show the general carriage of the model, no numerical valuesare shown in this first example.Figure 29General comportment of the beam-based model. Themovement of middle point A is scaled fordemonstrative purpose. A further study, it should benecessary to compare such result with measurements.28 See page 21 for its definition.29 A mass-element is constituted by only one node, and simulates a virtual mass-point.40

FEM modelling of a bellows and a bellows-based micromanipulatorIV. OutlookThe next example proves that the model is consistent with its expected behaviour. A verticalload F = 100 [mN] – equivalent to a platform mass of 10 grams, for instance – is applied andlook at the platform y-displacement.The movement is magnified, so that its correct appearance can be inspected. But the result isin fact very small, since it is about 1.05·10 -5 [µm]. There are no measurement at this time tosupport the correctness of that assumption, but the micromanipulator should effectively bevery inflexible in the y-direction, since the bellows are filled with incompressible oil.It can also be noted that the micromanipulator behaves in a very symmetric way. The beampart with the same colour on the plot sustain the same removal. So as can be seen, eachbellows seems to undergo an identical compression.4.3 PerspectivesThis chapter showed one way to use results of chapter 3. The construction of our model wasexplained step by step. At each level, it was checked if the intermediate results were suitable.Hence, the model should be able to furnish all necessary data to complete the kinematicmodel of the micromanipulator.But there are many ways to construct a beam-based model. One could for instance stackseveral beam-elements for each of the bellows, as suggested by ANSYS ® ' VerificationManual. The solid region may be a very simple way to model it, but the platform can as well41

IV. OutlookFEM modelling of a bellows and a bellows-based micromanipulatorbe modelled by many different element types. However, its stiffness should in any case bemuch bigger than that of the beams.42

FEM modelling of a bellows and a bellows-based micromanipulatorV. Completion5. COMPLETION5.1 Project conclusions5.1.1 RELIABILITYIn this work, the most observed principle has been mistake forethought. At each phase, stepafter step, any result was carefully checked before proceeding through the next phase. Indeed,a non-negligible part of the allowed task duration was devoted to understand the FE methods,in order to select the most fitting elements for each case. Different models were experimentedfor a same problem, so that the results given by different methods compare could becompared.After having chosen the elements, they were checked by setting up three models. Thosevery unlike models were built with completely distinct elements. One was constructed in twodimensioned space with plane element accepting axial symmetry. Another was made of 3Dsolid elements, and the third composed by shell elements. The theory on which these elementsare based differ radically. Hence, if all three results for one corresponding concur, theoutcome is very likely reliable.If two different models are based on too similar elements, the way to construct them may beso identical that the same error can be committed in both cases, and lead to the identical butfalse answer. On the contrary, if the models are completely unlike, the way to establish themmay also be utterly in contrast, and ensure that the same mistake cannot occur in both models.This is actually exactly our case. The way to construct a 2D-model absolutely differs from a3D one. Moreover, solid elements fill a volume-based model, but a shell-based model can befabricated with areas exclusively. These changed building methods ensure that it is nearlyimpossible that one identical mistake appears in two models, and make an accordant resultbeing even more reliable.It is finally to note, that there have been no previous studies of this sort of microcomponent,which made the simulations and modelling much harder.5.1.2 BELLOWS' PERFORMANCESThis project showed that the bellows has very linear characteristics. Its bending varies in aperfectly proportional fashion to the applied load. Moreover, some parameters are quasiconstantover the full work range of the micromanipulator. The bellows' stiffness varies nomore than 0,002% for bending forces varying from 10 [mN] to 50 [mN].The spring ratio varies a few percentages when applying varying compressing or elongatingloads. But even then, it varies in a very linear way, and can easily be taken into account forthe open-loop control, if necessary.The axial expansion of the bellows when applying interior pressure also varies less than0,3% over the full pressure range, and in an entirely linear way.Finally, it can be concluded that the high linearity of the bellows' motion constitute anexcellent connection between the piezoactuators and the end-effector, and that it can improvea new type of accurate join-free parallel structure, where the joints are integrated into theactuation links.Furthermore, once the geometrical parameters exactly checked for each bellows through anelementary calibration method, and the kinematic precisely adapted, the open-loop control of43

V. Completion FEM modelling of a bellows and a bellows-based micromanipulatorthe micromanipulator may be very accurate, since the bellows' will act in a perfectly linearway, and without any hysteresis.5.1.3 PROSPECTSThis project developed of course a considerable understanding of bellows' behaviour. Themagnitude of its movements when undergoing various loads can at this time be easilyevaluated. It is known now which parameters are more relevant for one property, and whichare not for another coveted characteristic. It is consequently easier to judge which applicationcan support which type of bellows.These simulations and conclusions will permit to create a model of the actuating level of thewhole micromanipulator, and understand better its behaviour, as well as its limits. This futuremodel will then be seriously useful to empower the kinematic model of the mechanism, andsupply the numerical data needed in the open-loop control of the telemanipulation, what wasactually the primary aim of this project.This enhanced tool can even be used for design purposes. With the beam-like model takinginto account the real performances of the bellows', one can smoothly weave a newconfiguration of bellows based micromanipulator, and have an immediate notion of itscomportment, without having first to manufacture a prototype. The optimal arrangement ofbellows to achieve a compact and reliable micromanipulator, with more degrees of freedomcan be chosen by designer.One idea at the Tampere University of Technology was for instance to conceive amicromanipulator with six bellows arranged in a Stewart platform shape, to fulfil a join-freeactuator with six degrees of freedom. This can now very simply be examined with thissimulation tool.44

FEM modelling of a bellows and a bellows-based micromanipulatorV. Completion5.2 References[Call88] Calladine C. R., Theory of shell structures, Cambridge University Press,Cambridge, First paperback edition, 1988, pp 456-461.[Tim56] Timoshenko Stephen, Strength of Materials, D. Van Nostrand Company Canada,Toronto, Third Edition, March 1956, pp 117-133, 145-153.[Har63] Harvey John F., Pressure Vessel Design: Nuclear and Chemical Applications,, D.Van Nostrand Company New Jersey, New-York, 1963, pp 28-41, 123-126.[Cook89] Cook Robert D., Malkus David S. and Plesha Michael E., Concepts andapplications of Finite Element Analysis, John Wiley & Sons, New York, ThirdEdition, 1989, page 574.[Cook95] Cook Robert D., Finite Element Modelling for Stress Analysis, John Wiley &Sons, New York, 1995.[Kall96] Kallio P., Lind M., Kojola H., Zhou Q., Koivo H., An Actuation System forParallel Link Micromanipulators, Proceedings of the Intelligent Robots andSystems, IROS'96 Japan, Osaka, November 1996, pp 856-862.[Kall97] Kallio P., Lind M., Zhou Q., Koivo H., A Parallel Piezohydraulic Micromanipulator– Mechanics Aspects, CCAM, Germany, Illmenau, 1997, pp 195-200.[Kall98] Kallio P., Lind M., Zhou Q., Koivo H., A 3 DOF Piezohydraulic ParallelMicromanipulator, International Conference on Robotics and Automation,Leuven, Belgium, May 1998.[Lind98] Lind Mikael, Modelling a bellows with a beam-like component in ADAMS,Eurosim'98 Simulation Congress, Proceedings Volume 3, pp 513-519.[Cran57] Crandall S. H. and Dahl N. C., The influence of pressure on the bending of curvedtubes, Proceedings of the 9 th International Congress of Applied Mechanics,University of Brussels, Vol. 6, 1957, pp 101-111.[Lau62] A. Laupa, N. A. Weil, Analysis of U-shaped expansion joints, Journal of AppliedMechanics, March 1962, pp 573-574.45

VI. AnnexFEM modelling of a bellows and a bellows-based micromanipulator6. ANNEX6.1 Principal Listings6.1.1 GEOMETRYa) PLANE83 2D-model!***************** PLANE83 Model **********************!* *!***************** December 1998 **********************FINISH! Exits the previous processor/CLEAR! Clears the database/PREP7! Enters the PREP7 pre-processorET,1,PLANE83! Defines a new element type, PLANE83KEYOPT,1,3,1! Sets different flags for axisymmetric behaviour!***** Material properties *****MP,ex,1,161e+6 ! Young's Modulus E = 161·10 6 [mN/mm 2 ]MP,nuxy,1,0.3 ! Poisson's Ratio σ = 0,3!****** Scalar parameters ******A=6.35! Outer diameter AB=3.81! Inner diameter BD=0.7832! Convolution pitch Dl0=18.7960 ! Bellows' length L 0t=0.0381! Wall's thickness tr1=(D-2*t)/4 ! Inner convolution's radius r 1r2=r1 ! Outer convolution's radius r 2m=(A/2)-r2-r1-2*t-(B/2)! Straight part mpi=acos(-1)! Calculation of π!********** Geometry ***********K, ,,,, ! Rotation's axis keypoint 1K, ,,1,, ! Rotation's axis keypoint 2K, ,B/2+r2+t+m,t+r1,, ! Centre of top circlesK, ,B/2+r2+t,t+2*r1+t+r2,, ! Centre of middle circlesK, ,B/2+r2+t,-r2,, ! Centre of bottom circlescircle,3,r1,,,360,! Draws top inner circlecircle,3,r1+t,,,360,! Draws top outer circlecircle,4,r2,,,360,! Draws middle inner circlecircle,4,r2+t,,,360,! Draws middle outer circlecircle,5,r2,,,360,! Draws bottom inner circlecircle,5,r2+t,,,360,! Draws top outer circleLDELE,9,10,1 , ,1! Deleting of unused arcsLDELE,12,16,4 , ,1 ! " " " "LDELE,13,14,1 , ,1 ! " " " "LDELE,2,3,1 , ,1 ! " " " "LDELE,6,7,1 , ,1 ! " " " "LDELE,19,20,1 , ,1 ! " " " "LDELE,23,24,1 , ,1 ! " " " "LDELE,17,21,4 , ,1 ! " " " "KDELE,14,15,1! Deleting of unneeded keypointsKDELE,18,19,1 ! " " " "KDELE,3,5,1 ! " " " "KDELE,8,12,4 ! " " " "KDELE,25,26,1 ! " " " "KDELE,22,29,7 ! " " " "LSTR,16,20! Building straight walls of the bellowsLSTR,11,17 ! " " " " " "LSTR,21,7 ! " " " " " "LSTR,9,27 ! " " " " " "LSTR,23,13 ! " " " " " "LSTR,28,24 ! " " " " " "LSTR,17,21 ! " " " " " "LSTR,27,23 ! " " " " " "LSTR,7,11 ! " " " " " "LSTR,6,10 ! " " " " " "LSTR,9,13 ! " " " " " "LSEL,ALL! Selecting all lines46

FEM modelling of a bellows and a bellows-based micromanipulatorVI. AnnexLSEL,S,,,12,14,1! Selects various linesLSEL,A,,,16,17,1 ! " " "LSEL,A,,,2,10,8 ! " " "LESIZE,ALL,,,4! Specifies 4 divisions for those linesLSEL,S,,,3,6,3! Selects various linesLSEL,A,,,7,9,2 ! " " "LESIZE,ALL,,,50! Specifies 50 divisions for those linesLSEL,ALL! Selects all linesLESIZE,ALL,,,20! Specifies 20 divisions for all other linesLSEL,S,,,11! Selects various linesLSEL,A,,,2 ! " " "LSEL,A,,,15 ! " " "LSEL,A,,,12 ! " " "AL,ALL! Creates an area with those lines (to improve the meshing)LSEL,S,,,3! Selects various linesLSEL,A,,,12 ! " " "LSEL,A,,,6 ! " " "LSEL,A,,,14 ! " " "AL,ALL! Creates an area with those lines (to improve the meshing)LSEL,S,,,1! Selects various linesLSEL,A,,,14 ! " " "LSEL,A,,,5 ! " " "LSEL,A,,,16 ! " " "AL,ALL! Creates an area with those lines (to improve the meshing)LSEL,S,,,4! Selects various linesLSEL,A,,,16 ! " " "LSEL,A,,,8 ! " " "LSEL,A,,,17 ! " " "AL,ALL! Creates an area with those lines (to improve the meshing)LSEL,S,,,7! Selects various linesLSEL,A,,,9 ! " " "LSEL,A,,,17 ! " " "LSEL,A,,,13 ! " " "AL,ALL! Creates an area with those lines (to improve the meshing)LSEL,S,,,13! Selects various linesLSEL,A,,,22 ! " " "LSEL,A,,,10 ! " " "LSEL,A,,,18 ! " " "AL,ALL! Creates an area with those lines (to improve the meshing)LSEL,ALL! Selects all linesASEL,ALL! Selects all areasAGLUE,ALL! Connecting those areas togetherMSHAPE,0,2D! Specifies the element shape to be usedMSHKEY,1! Specifies that map-meshing should be used to mesh the modelAMESH,ALL! Meshing the model!! ************************************************! * Creating Rigid region *! ************************************************!n,3000,0,t+2*r1+t+r2,0! Creates node 3'000 for the rigid region's master nodeET,2,beam3! Defines element type Nº2 as BEAM3R,2,.0001,.00001, ! Sets the material properties for REAL Nº2TYPE,2 ! Sets the element types attribute pointer to 2REAL,2 ! Sets the element real constant attribute pointer to 2E,3000,2 ! Defines mass-element by node 3'000NSLA,S,,,2! Selects slave nodeNSEL,A,,,3000! Selects also node 3'000 (master)ALLSEL,ALL! Reselect allSAVE,meshed2d,db,! Saving the database to 'meshed2d.db'FINISH! Exits the pre-processor47

VI. AnnexFEM modelling of a bellows and a bellows-based micromanipulatorb) 3D solid45 model, with optional rigid regionFINISH! Exits previous processor/CLEAR! Erases the database/PREP7! Enters pre-processorET,1,SOLID45! Defines element type no.1 to SOLID45MP,ex,1,161E+6! Sets Young's Modulus to 1'160 PaMP,nuxy,1,0.3! Sets Poisson’s ratioA=6.35! Outer diameter AB=3.81! Inner diameter BD=0.7832! Convolution pitch Dl0=18.7960 ! Total convolution's length L 0t=0.0381! Wall thickness tr1=(D-2*t)/4 ! Calculated inner radius r 1r2=r1 ! Calculated outer radius r 2m=(A/2)-r2-r1-2*t-(B/2)! Straight part mpi=acos(-1)! Definition of πK, ,,,, ! Rotation's axis keypoint 1K, ,,1,, ! Rotation's axis keypoint 2K, ,B/2+r2+t+m,t+r1,, ! Centre of top circlesK, ,B/2+r2+t,t+2*r1+t+r2,, ! Centre of middle circlesK, ,B/2+r2+t,-r2,, ! Centre of bottom circlescircle,3,r1,,,360,! Draws top inner circlecircle,3,r1+t,,,360,! Draws top outer circlecircle,4,r2,,,360,! Draws middle inner circlecircle,4,r2+t,,,360,! Draws middle outer circlecircle,5,r2,,,360,! Draws bottom inner circlecircle,5,r2+t,,,360,! Draws top outer circleLDELE,9,10,1 , ,1! Deletes unneeded circle partsLDELE,12,16,4 , ,1 ! " " "LDELE,13,14,1 , ,1 ! " " "LDELE,2,3,1 , ,1 ! " " "LDELE,6,7,1 , ,1 ! " " "LDELE,19,20,1 , ,1 ! " " "LDELE,23,24,1 , ,1 ! " " "LDELE,17,21,4 , ,1 ! " " "KDELE,14,15,1! Deletes unneeded keypointsKDELE,18,19,1 ! " " "KDELE,3,5,1 ! " " "KDELE,8,12,4 ! " " "KDELE,25,26,1 ! " " "KDELE,22,29,7 ! " " "LSTR,16,20! Builds straight wallsLSTR,11,17 ! " " "LSTR,21,7 ! " " "LSTR,9,27 ! " " "LSTR,23,13 ! " " "LSTR,28,24 ! " " "AL,ALL! Generates an areas by previously defined linesLSTR,17,21! Builds straight wallsLSTR,23,27 ! " " "LSTR,7,11 ! " " "LSTR,9,13 ! " " "LSTR,6,10 ! " " "LSEL,S,,,12,14,1 ! Selects lines 12-14LSEL,A,,,16,17,1 ! Also selects lines 16 and 17ASBL,1,ALL,,DELETE,KEEP! Subtracts selected lines from area! so that the meshing will be! more efficientLSEL,ALL! Selects all linesLSEL,S,,,11,18,7 ! Selects new set of line 11 and 18LESIZE,ALL,,,6! Specifies 6 division for those linesLSEL,ALL! Selects all linesLSEL,S,,,1,4,3 ! Selects new set of line 1 and 4LESIZE,ALL,,,6! Specifies 6 divisions for those lines also! (kept, so that it can be changed)LSEL,ALL! Selects all lines againLESIZE,ALL,0.06! Specifies the line-divisions' length! for all other linesASEL,ALL! Selects all areas48

FEM modelling of a bellows and a bellows-based micromanipulatorVI. Annexlplovrotat,all,,,,,,1,2,360,8LSEL,ALLLESIZE,ALL,,,30! Plots lines (to control the shape)! Creates the 3D-model by rotating the areas! Selects all lines! Specifies 30 divisions! for all new created lines!! ************************************************! * Meshing *! ************************************************!MSHAPE,0,3D! Specifies the element shape to be used! (allows mapped meshes)MSHKEY,1! Specifies mapped meshingVMESH,ALL! Generates nodes and volumes elements within volumesASEL,ALL! Selects all areasASEL,S,LOC,Y,t+2*r1+t+r2! Selects areas located on the very topCM,bordsup,AREA! Creates a area-component called 'bordsup'asel,all! Selects all areasasel,s,,,14,164,25! Selects various areas belongingasel,a,,,188! to the extern surface of the bellowsasel,a,,,18,168,25 ! "asel,a,,,191 ! "asel,a,,,9,159,25 ! "asel,a,,,20,170,25 ! "asel,a,,,24,174,25 ! "asel,a,,,28,178,25 ! "asel,a,,,184 ! "asel,a,,,192,198,3 ! "CM,EXTERN,AREA! Creates a area-component called 'extern'asel,all! Selects all areasasel,s,,,12,162,25! Selects various areas belongingasel,a,,,17,167,25! to the intern surface of the bellowsasel,a,,,32,150,25 ! "asel,a,,,22,182,25 ! "asel,a,,,26,186,25 ! "asel,a,,,30,190,25 ! "asel,a,,,1 ! "asel,a,,,157 ! "asel,a,,,194,200,3 ! "asel,a,,,182,190,4 ! "CM,INTERN,AREA! Creates a area-component called 'intern'ASEL,ALL! Selects all areasASEL,S,LOC,Y,-r2! Selects areas located on the very bottomCM,bordinf,AREA! Creates a area-component called 'bordinf'!! ************************************************! * Creating Rigid region *! ************************************************!N,30000,0,t+2*r1+t+r2! Creates node 30'000 for the rigid regionET,2,MASS21! Defines element type no.2 to MASS21KEYOPT,2,3! Sets element to make the element be only a massR,2,0.000001! Sets the mass of that element to quasi zeroTYPE,2 ! Sets the element types attribute pointer to 2REAL,2 ! Sets the element real constant attribute pointer to 2E,30000,2 ! Defines mass-element by node 30'000CMSEL,S,intern,AREA! Selects 'intern' areasNSLA,S,1! Selects corresponding nodesNSEL,R,LOC,Y,t+2*r1+t+r2! Reselects only those at the top of the bellows (slave)NSEL,A,,,30000! Selects also node 30'000 (master)CERIG,30000,ALL! Defines rigid region by those selected nodesSAVE,meshed3d,db,! Saves database to 'meshed3d.db'49

VI. AnnexFEM modelling of a bellows and a bellows-based micromanipulatorc) 3D shell43 modelFINISH! Exits previous processor/CLEAR! Erases the database/PREP7! Enters pre-processorET,1,SHELL43! Defines element type no.1 to SHELL43MP,ex,1,161E+6! Sets Young's Modulus to 1'160 PaMP,nuxy,1,0.3! Sets Poisson’s ratioA=6.35! Outer diameter AB=3.81! Inner diameter BD=0.7832! Convolution pitch Dl0=18.7960 ! Total convolution's length L 0t=0.0381! Wall thickness tr1=(D-2*t)/4 ! Calculated inner radius r 1r2=r1 ! Calculated outer radius r 2m=(A/2)-r2-r1-2*t-(B/2)! Straight part mpi=acos(-1)! Definition of πR,1,t,t,t,t,0,! Setting shell elements nodes' thickness to tK, ,,,, ! Rotation's axis keypoint 1K, ,,1,, ! Rotation's axis keypoint 2K, ,B/2+r2+t+m,r1,, ! Centre of top circleK, ,B/2+r2+t,2*r1+r2,, ! Centre of middle circleK, ,B/2+r2+t,-r2,, ! Centre of bottom circlecircle,3,r1,,,360,! Draws top circlecircle,4,r2,,,360,! Draws middle circlecircle,5,r2,,,360,! Draws bottom circleLDELE,5,8,3 , ,1! Deletes unneeded linesLDELE,9,12,3 , ,1 ! " " "LDELE,2,3,1 , ,1 ! " " "LDELE,6,11,5,,1 ! " " "KDELE,3,5,1,,1! Deletes unneeded keypointsKDELE,1,2,1,,1 ! " " "LSTR,7,13! Builds straight wallsLSTR,9,15! Builds straight wallsLSEL,S,,,6,7,1! Selects some linesLSEL,A,,,10,11,1 ! " " "LSEL,A,,,1,4,3 ! " " "LESIZE,ALL,,,5! Specifies 5 division for those linesLSEL,S,,,2,3,1! Selects some linesLESIZE,ALL,,,15! Specifies 15 division for this set of linesAROTATE,ALL,,,,,,1,2,360,8! Creates the 3D model by rotating areasLESIZE,ALL,,,30! Specifies 30 division for each new created lineMSHAPE,0,2D! Specifies the element shape to be usedMSHKEY,1! Specifies that map meshing should be uses for meshingAMESH,ALL! Generates nodes and shell elementsENORM, 68! Specifies the elements' orientation to be! the same as element 68, because during the AROTATE,! some elements are created with different orientation,! and this can be wrong when applying pressure to all! elements together!FINISH! Exits the pre-processorSAVE,shell43,db,! Saves the database to 'shell43.db'50

FEM modelling of a bellows and a bellows-based micromanipulatorVI. Annex6.1.2 PRESSURE APPLIED INSIDE THE BELLOWSa) 2D-modelFINISH! Exits previous processor/CLE! Erases current databaseRESUME,meshed2d,db! Resumes database from 'meshed2d.db'pressure = 0.01 ! Sets the variable pressure to 0,01 [mN/mm 2 ]/TITLE, 2D-Bellows, under 0,01 [mN/mm^2] pressure ! Sets plot's title/SOLU! Enters SOLU processorKSEL,S,,,24,28,4! Selects some keypointsD,ALL, UY,,,,, ROTX,ROTY,ROTZ! Fixes some degrees of freedom for bottom nodes! The result is that these nodes can still move radiallyLSEL,S,,,1,4,3! Selects some nodesLSEL,A,,,6,7,1 ! " " "LSEL,A,,,15,22,7 ! " " "CM,intern,LINE! Creates component called 'intern', composed by lines,! and corresponding to the internal wall of the bellowsSFL,ALL,PRES,pressure, ,! Applies the pressure on those selected linesALLSEL,ALL! Selects allSOLVE! Runs the simulation/POST1! Enters the post-processor POST1/DSCALE,1,0! Imposes no scaling for the layoutPLNSOL,U,Y,2! Displays results by continuous lineFINISH! Exits postprocessingb) 3D Solid-modelFINISH! Exits previous processor/CLE! Erases the actual databaseRESUME,meshed3d,db,,! Resumes from databasePRESSURE=0.01 ! Sets pressure to 0,01 [N/mm 2 ]/PREP7! Enters pre-processorNSEL, S, LOC, Y, -r2! Selects bottom-nodesCLOCAL,11,1,0,0,0,0,-90! Creates a new coordinate-systemCSYS,11! New coordinate-system activeNROTAT,ALL! Rotates bottom-nodes' c.-s.D,ALL,UY! No movement allowed in angular directionD,ALL,UZ! No movement allowed in axial directionALLSEL,ALL! Selects all nodesCSYS,0! Cartesian coordinate system activated/SOLU! Enters Solution processorCMSEL,S,INTERN! Selecting intern surface of the bellows (areas only)NSLA,S,1! Selecting corresponding nodesSF,ALL,PRES,PRESSURE! Applies the internal pressureALLSEL,ALL! Selecting allSOLVE! Initiates solution calculationsFINISH! Exits postprocessingc) 3D Shelled-modelFINISH! Exits previous processor/CLE! Erases the actual databaseRESUME,shell43,db,,! Resumes from database/SOLU! Enters Solution processorPRESSURE=0,01 ! Sets pressure to 0,01 [N/mm 2 ]SFE, ALL, 2, PRES,,PRESSURE! Applies the internal pressure on face Nº2 of shell elementsNSEL, S, LOC, Y, -r2! Selects bottom nodesCLOCAL,11,1,0,0,0,0,-90! Defines new coordinate systemCSYS,11! New coordinate-system activeNROTAT,ALL! Rotates bottom-nodes' c.-s.D,ALL,UY! No movement allowed in angular directionD,ALL,UZ! No movement allowed in axial directionALLSEL,ALL! Selects all nodesCSYS,0! Cartesian coordinate system activatedSOLVE! Initiates solution calculationsFINISH! Exits postprocessing51

VI. AnnexFEM modelling of a bellows and a bellows-based micromanipulator6.1.3 APPLYING TIME-VARYING LOAD FOR THE SPRING-RATE Ka) 2D model with corresponding pressureFINISH! Exits previous processor/CLE! Clears current databaseRESUME,meshed2d,db! Resumes database 'meshed2d.db'/TITLE, Bellows under time-varying force ! Title for the layoutF=100 ! Maximum force F = 100 [mN]Tend=130! Time at the end of the simulationIncr=13! Number of time incrementsT1=Tend/Incr*3! Time at the second simulation stepS=(-(B/2)**2+(B/2+t)**2)*pi! Computation of the effective surface! on which the force F acts/PREP7! Enters the pre-processorCSYS,0! Cartesian coordinate system activatedNSEL, S, LOC, Y, -r2! Selects bottom nodesD,ALL, UY,,,,, ROTX,ROTY,ROTZ! No movement allowed in axial direction, and no rotationALLSEL,ALL! Selects all nodes/SOLU! Enters Solution processorNLGEOM,ON! Enables non-linear simulation*DIM,FORCE,TABLE,3,1! Defines array parameter FORCEFORCE(1,1)=0,F,-F ! Force values in column 1FORCE(1,0)=0,T1,Tend ! Corresponding time value in column 0FORCE(0,1)=1! Zeroth rowTM_START=1E-6 ! Starting time (must be > 0)TM_END=Tend! Ending time of the transientTM_INCR=Tend/Incr! Time increment*DO,TM,TM_START,TM_END,TM_INCR! Do for TM, from TM_START to TM_END in TM_INCR stepsTIME,TM! Time valueF=FORCE(TM)! Force value at time TMPressure=F/S! Computes the pressure induced by FNSEL, S, LOC, Y,t+2*r1+t+r2! Selects top nodesSF,ALL,PRES,Pressure! Applies the pressure to these nodesALLSEL,ALL! Selects allSOLVE! Initiates the simulation*ENDDO! End of the DO-loopSAVE,loads,db! Saves the database to 'loads.db'b) 2D model with solid regionFINISH! Exits previous processor/CLE! Clears current databaseRESUME,meshed2d,db! Resumes database 'meshed2d.db'/TITLE, Bellows under time-varying force ! Title for the layoutF=100 ! Maximum force F = 100 [mN]Tend=130! Time at the end of the simulationIncr=13! Number of time incrementsT1=Tend/Incr*3! Time at the second simulation stepS=(-(B/2)**2+(B/2+t)**2)*pi! Computation of the effective surface! on which the force F acts/PREP7! Enters the pre-processorCSYS,0! Cartesian coordinate system activatedNSEL, S, LOC, Y, -r2! Selects bottom nodesD,ALL, UY,,,,, ROTX,ROTY,ROTZ! No movement allowed in axial direction, and no rotationALLSEL,ALL! Selects all nodes/SOLU! Enters Solution processorNLGEOM,ON! Enables non-linear simulation*DIM,FORCE,TABLE,3,1! Defines array parameter FORCEFORCE(1,1)=0,F,-F ! Force values in column 1FORCE(1,0)=0,T1,Tend ! Corresponding time value in column 0FORCE(0,1)=1! Zeroth rowTM_START=1E-6 ! Starting time (must be > 0)TM_END=Tend! Ending time of the transientTM_INCR=Tend/Incr! Time increment*DO,TM,TM_START,TM_END,TM_INCR! Do for TM, from TM_START to TM_END in TM_INCR stepsTIME,TM! Time value52

FEM modelling of a bellows and a bellows-based micromanipulatorVI. AnnexSF,3000,FORCE,FORCE(TM)ALLSEL,ALLSOLVE*ENDDOSAVE,loads,db! Applies the force to the middle node of the solid region! Selects all! Initiates the simulation! End of the DO-loop! Saves the database to 'loads.db'c) 3D solid modelFINISH! Exits previous processor/CLE! Clears current databaseRESUME,meshed3d,db! Resumes database/TITLE, Bellows under time-varying force (spring-rate check) ! Title for the layoutF=100 ! Maximum force F = 100 [mN]Tend=130! Time at the end of the simulationIncr=13! Number of time incrementsT1=Tend/Incr*3! Time at the second simulation stepS=(-(B/2)**2+(B/2+t)**2)*pi! Computation of the effective surface! on which the force F acts/PREP7! Enters the pre-processorCSYS,0! Cartesian coordinate system activatedNSEL, S, LOC, Y, -r2! Selects bottom nodesCLOCAL,11,1,0,0,0,0,-90! Defines new coordinate systemCSYS,11! New coordinate-system activeNROTAT,ALL! Rotates bottom-nodes' c.-s.D,ALL,UY! No movement allowed in angular directionD,ALL,UZ! No movement allowed in axial directionALLSEL,ALL! Selects all nodesCSYS,0! Cartesian coordinate system activated/SOLU! Enters Solution processorNLGEOM,ON! Enables non-linear simulation*DIM,FORCE,TABLE,3,1! Defines array parameter FORCEFORCE(1,1)=0,F,-F ! Force values in column 1FORCE(1,0)=0,T1,Tend ! Corresponding time value in column 0FORCE(0,1)=1! Zeroth rowTM_START=1E-6 ! Starting time (must be > 0)TM_END=Tend! Ending time of the transientTM_INCR=Tend/Incr! Time increment*DO,TM,TM_START,TM_END,TM_INCR! Do for TM, from TM_START to TM_END in TM_INCR stepsTIME,TM! Time valueF=FORCE(TM)! Force value at time TMPressure=F/S! Computes the pressure induced by FCMSEL,S,bordsup! Selects the upper part areas of the ringNSLA,S,1! Selects the corresponding nodesSF,ALL,PRES,Pressure! Applies the pressure to these nodesALLSEL,ALL! Selects allSOLVE! Initiates the simulation*ENDDO! End of the DO-loopSAVE,loads,db! Saves the database to 'loads.db'53

VI. AnnexFEM modelling of a bellows and a bellows-based micromanipulator6.1.4 MICROMANIPULATOR MODELFINISH! Exits previous processor/CLE! Clears current database/PREP7! Enters the PREP7 pre-processorNLGEOM,ON! Large deflection onET,1,beam44! Defines a new elementR,1,18.8,5.11216E-4,5.11216E-4,.0381,.0381 ! Beam-element material propertiesMP,EX,1,161E6! Sets Young's Modulus to 1'160 [Pa]MP,DENS,1,8.85E-7 ! Sets the material density to 8,85·10 -7 [kg/mm 3 ]MP,NUXY,1,.3 ! Sets the Poisson's ratio to 0,3pi=acos(-1)! Defines the value of πR=9 ! Radius of the base "circle"L=18.796! Length of the bellowsplatform=10! Weight of the platform (if needed)alpha=8! Angle between one bellows' axis and the verticalalpharad=alpha*pi/180! Same angle in radianH=COS(alpharad)*L! Height of the actuation system (only the bellows part)TYPE,1 ! Sets the element types pointer to 1REAL,1 ! Sets the material real constant pointer to 1WPOF,-R! Offsets the working plane of the basis' radiusWPRO,-alpha! Rotates the workplane of -alpha degreesCSYS,4! Sets the active coordinate system to the workplaneN,1,! Creates the basis node of the first bellowsN, ,,L, ! Creates the free-end node of the first bellowsNSEL,s,,,1,2,1! Selects both last created nodesNROTAT,ALL! Rotates the nodal coordinate system to the active systemE,1,2! Defines a beam element between for the first bellowsCSYS,0! Sets the active coordinate system to the Cartesian oneWPRO,alpha! Re-rotates the working plane of alpha degreesWPOF,R! Re-offsets the working plane to its previous originWPSTYLE,,,,,,,,0! Controls the display and style of the working planeWPRO,,,120! Rotates the working plane of one 120º for the second bellowsWPOF,-R! Offsets the working plane of the basis' radiusWPRO,-alpha! Rotates the workplane of -alpha degreesCSYS,4! Sets the active coordinate system to the workplaneN,,! Creates the basis node of the second bellowsN, ,,L, ! Creates the free-end node of the second bellowsNSEL,S,,,3,4,1! Selects both last created nodesNROTAT,ALL! Rotates the nodal coordinate system to the active systemE,3,4! Defines a beam element between for the first bellowsWPRO,alpha! Re-rotates the working plane of alpha degreesWPOF,R! Re-offsets the working plane to its previous originWPRO,,,120! Rotates the working plane of one 120º for the second bellowsWPOF,-R! Offsets the working plane of the basis' radiusWPRO,-alpha! Rotates the workplane of -alpha degreesN,,! Creates the basis node of the second bellowsN, ,,L, ! Creates the free-end node of the second bellowsNSEL,S,,,5,6,1! Selects both last created nodesNROTAT,ALL! Rotates the nodal coordinate system to the active systemE,5,6! Defines a beam element between for the first bellowsWPRO,alpha! Re-rotates the working plane of alpha degreesWPOF,R! Re-offsets the working plane to its previous originWPRO,,,120! Rotates the working plane of one 120º for the second bellowsCSYS,0! Sets the active coordinate system to the Cartesian oneN,,,H! Creates the platform's middle nodeET,2,mass21! Defines element type nº 2 to be MASS21R,2,platform! Sets the mass of that element type to that of the platformTYPE,2 ! Sets the element types attribute pointer to 2REAL,2 ! Sets the element real constant attribute pointer to 2E,7,2 ! Defines mass element by node nº7NSEL,S,LOC,Y,H! Selects all free-end nodes of the bellows (slaves)NSEL,A,,,7! Selects also the middle node of the platform (master)CERIG,7,ALL,UXYZ! Defines a rigid region by those selected nodesWPSTYLE,,,,,,,,0! Controls the display and style of the working planeNSEL,S,LOC,Y! Selects all basis nodes of the bellows'D,ALL,ALL! Fixes all those nodes' degrees of freedomNSEL,ALL! Selects all nodesSAVE,beams,db! Saves current database as 'beams.db'54

FEM modelling of a bellows and a bellows-based micromanipulatorVI. Annex6.1.5 BENDING LOAD ON THE MICROMANIPULATORFINISH! Exits previous processor/CLE! Clears current databaseRESUME,beams,db! Resumes database 'beams.db'EPLO! Plots elements of that model/PREP7! Enters pre-processor PREP7F,7,FY,-1000! Specifies applied force on platform's middle nodeALLSEL,ALL! Selects everything/SOLU! Enters processor SOLUSOLVE! Solves current problemFINISH! Exits solving processor/POST1! Enters the post-processor/DSCALE,1,0! Defines scaling of the next plot to automatic/PSF,DEFA, ,1! Specifies which surface loads should be displayed/PBF,DEFA, ,1! Shows body force loads as contour on next display/PSYMB,CS,0! Shows various symbols on display/PSYMB,NDIR,0 ! " " " " "/PSYMB,ESYS,0 ! " " " " "/PSYMB,LDIR,0 ! " " " " "/PSYMB,ECON,0 ! " " " " "/PSYMB,DOT,1 ! " " " " "/PSYMB,PCONV, ! " " " " "/PSYMB,LAYR,0 ! " " " " "/PBC,ALL,,1! Shows various boundary conditions and values on display/PBC,NFOR,,0 ! " " " " " " " "/PBC,NMOM,,0 ! " " " " " " " "/PBC,RFOR,,0 ! " " " " " " " "/PBC,RMOM,,0 ! " " " " " " " "/PBC,PATH,,0 ! " " " " " " " "/REP! Re-plots current displayPLNSOL,U,Y,2,1! Displays result as continuous contour for the y-displacement55

HELSINKI UNIVERSITY OF TECHNOLOGY CONTROL ENGINEERING LABORATORYReport 100Report 101Report 102Report 103Report 104Report 105Report 106Report 107Report 108Report 109Report 110Report 111Report 112Report 113Hrúz, B.,Control of Discrete Event Dynamic Systems in the Context of Flexible Manufacturing Systems and PetriNets. June 1995.Blomberg, H., Ylinen R.,Descriptor Representation of Linear Constant Systems - Impulse Modes and Critical InterconnectionConstraints. October 1995.Lin, Ju,Design of Active Dampers for Mechanical Vibrations. April 1996.Hyötyniemi, H.,Supplement to Self-Organizing Artificial Neural Networks in Dynamic Systems Modeling and Control.December 1996.Jäntti, R.,Automaatiojärjestelmän dynamiikan vaikutus PI-säätimen viritykseen. January 1997.Herto, M-L, Sipari, P. (eds.),Triennial Activity Report 1994-1996Hyötyniemi, H., Koivo H. (eds.),Multimedia Applications in Industrial Automation. June 1997.Pulkkinen, J.,Capacitance and Diameter Control in Telephone Cable Insulation Process. October 1997.Ojala, P.,Design and Control of the Piezo Actuated Micro Manipulator. December 1997.Niemi, A. J., Berndtson, J., Karine, S.,Automatic Control of Paper Machine by Dry Line Measurement. December 1997.Ylén, J-P, Nissinen, A. S.,Sumean logiikan sovelluksen kehistysprosessi ja sen soveltaminen fuzzyTECH-ohjelmiston evaluointiin.December 1997.Hyötyniemi, H.,Mental Imagery: Unified Framework for Associative Representations. August 1998.Hyötyniemi, H., Koivo, H. (eds.),Multivariate Statistical Methods in Systems Engineering. December 1998.Robyr, S.,FEM Modelling of a Bellows and a Bellows-Based Micromanipulator. February 1999.ISBN 951-22-4471-3ISSN 0356-0872