NONLINEAR SIMULATION OF A LEVER MECHANISM

Transcription

1 NONLINEAR SIMULATION OF A LEVER MECHANISM Sergio E. Adeff The MacNeal-Schwendler Corporation 2975 Red Hill Ave. Costa Mesa, California sergio.adeff@macsch.com Abstract Many device designs include shaped lever mechanisms that must function reliably through a variety of load conditions. A lever shape is often optimized to carry out their function with optimal use of the material and processes involved in their fabrication. Levers are present in all types of devices, commonly in computer hard drives, car engines, airplane control mechanisms, medical instruments, et cetera. The design discussed in this paper is a hypothetical yet realistic one. The purpose of this work is to serve as a demonstration and training device for Engineers interested in learning how to create an adequate model and carry out complex calculations involving large displacements and contact between rigid surfaces and a three-dimensional representation of the deformable bodies of two levers and other parts of a mechanism. The model will be available to any interested party in the form of an MSC/PATRAN database (1) with the MSC/ADVANCED FEA preference [2] in the World Wide Web. The paper focuses on one aspect of the more general analysis required for a complete design, namely the calculation of stress and strain distributions, particularly in areas of contact and impact, nonelastic deformations, velocities, and driving and reaction forces under transient conditions. Problems of convergence often faced when doing a nonlinear analysis of this type are also discussed. The design better understood by looking into figures in the paper- includes four distinctive pieces, designated as hammer, trigger, anvil, and band. Both the hammer and trigger are levers pivoting about fulcrum pins extruded from the anvil. Likewise the hammer and trigger have each an extruded pin to which the band is tightly attached pulling the two pieces together. The disposition is such that when the trigger is actuated, the hyperelastic band first accumulates strain energy and the hammer remains in place, and then the band suddenly transfers the stored strain energy to the hammer, which is accelerated and then impacts a protrusion of the anvil. The hammer carries a seal, which impacts a stamp lying on the anvil. The stamp undergoes plastic deformation as a result of the impact. 1

2 Introduction The modest engine presented in this paper (its general dimensions measured in inches are 4 by 8 by 2) presents several challenges on account of the various nonlinearities involved. There are very large displacements, both translational and rotational large accelerations both due to driving boundary conditions and due to impacts. There is accumulation of strain energy taking advantage of (rubber) material hyperelastic nonlinear properties. There is a sudden release of this accumulated energy (converted into momentum) on account of the structure going through a state of impossible static equilibrium. The phase of energy accumulation involves the development of large strains and the release produce large strain rates. A driving displacement is enforced and reaction forces are computed producing the necessary work that converts into strain energy. The amount of work needed is one of the original unknowns of the system resolved by the analysis. There are contact conditions between the different parts of the mechanism both with large sliding and low impact and with small sliding and high impact. (With the slave region purposely undergoing plastic deformation). There are seven widely separated general areas of contact with contact areas involved in large relative rotations of about 60 degrees, all pieces are threedimensional and with fairly complicated shapes. It was desired to use 8-node hexahedral finite elements only, and appropriately meshing these pieces was an additional challenge. The need of a sufficient number of elements to make the computations possible from the point of view of numerical stability and deformation and stress distribution representational quality had to be carefully balanced with the potentially awesome amount of computational processing time and data storage. Despite every effort, the amount of computing time used for the analysis discussed in this paper is significantly large. Although not intended as a regular study exercise, the model is also proposed as a self-paced, hard-worked, advanced instructional tool for engineers wanting an in-depth, realistic example. As such, a delicate balance between simplicity and complexity had to be struck. The example had to have relevance across the various industries typically using MSC/PATRAN and MSC/ADVANCED FEA. Finally the model is likewise intended as a demonstration tool, so that engineers making could, with animations and detailed discussions, illustrate the impressive modeling and analysis capabilities of the software employed. This paper exhibits this model for the first time in the 1998 MSC Americas Users Conference. Currently, the model and this paper are available on the web pages of the MacNeal-Schwendler Corporation with its illustrations in full color. (Please the author to comment on your experience with the model, as this would encourage future similar developments). 2

3 Problem definition The model is made of four parts shown assembled in Figure 1. The only part fixed to the ground is the anvil, shown in Figure 2. The anvil is made mostly of an aluminum alloy but is has a protuberance designated as seal in Figure 1, which includes a layer of a magnesium alloy capable of sustaining sizable plastic stresses. The seal is the part of the anvil acting as such because it is struck by the stamp attached to the hammer shown in Figure 3. There is a trigger, displayed in Figure 4, and a band, shown in Figure 5. Refer to these figures for the description that follows. The thickness of anvil, hammer, tape, pins, and trigger is equal to 1 /8 inch. The distance between centers of the hammer action pin and hammer fulcrum hole in Figure 3, as well as that between the centers of the stop pin and hammer pin in Figure 2 is equal to 2 inches. The width of the tape is _ inch. The outer diameter of all pins is _ inch. The seal has dimensions of 2.58 by 1.16 by _ measured in inches. The stamp has similar measures than the seal but its free surface has the MSC characters represented in bass-relief. (by way of depressing selected nodes a distance of 1 /32 inch normal to the originally flat surface). The seal is made of a flat surface and should deform plastically under the impact of the stamp as intended by the design. Taking advantage of the somewhat humanlike shape of the anvil, notice that it has three cylindrical pins. One of these pins is on the anvil face and acts as a fulcrum for the (lever) hammer; another is in its risen right hand and easily fits inside the curved guide in the hammer and acts as a stop when the hammer pivots about its fulcrum. This limits the total rotation of the hammer to 60 degrees. The third anvil pin is on its lowered left hand and acts as a fulcrum for the (lever) trigger. Both hammer and trigger have a similar single pin, which together are used to stretch out the (hyperelastic) band. In the original resting position the band is positioned so inclined that its pulling keeps the hammer up against the lower surface of the stop. When the trigger revolves clockwise in Figure 1- about its fulcrum, the band is extended and likewise rotated clockwise until it goes under the hammer fulcrum, so that eventually pulls the hammer down, which will thus be sent counterclockwise against the seal on the hammer. The hammer has in fact a stamp that will impact the seal at great speed while concentrating much of the whole impulse on account of its small size. All parts are designed directly in MSC/PATRAN as geometric surfaces from which the three-dimensional hex elements are (swept) extruded (The surfaces are first paved with quad elements). All boundaries of the original surfaces are made of either straight, circular or cubic spline segments, thus achieving geometric smoothness and manufacturability. The contact definition on and about each pin is done with slide lines and ISL elements. These have some characteristics that make them very suitable for this problem. They allow for large sliding displacements of the (slave) ISL elements along the (master) slide lines which form rings around the pins, thus 3

4 guiding the pivoting motion about the virtual fulcrum at the axis of the pins. Each pin is allotted four parallel rings of slide lines, so that three-dimensional effects are fully accounted. The pins do wobble and pitch as a result of impacts and deformation of the various parts, and also undergo deformations out of their original circular cross-sections. In fact, the slide lines do accompany these deformations dynamically and nonlinearly. Thus, the fulcrum moves as well and if perceived not as an ideal fulcrum but rather as an axis about which the threedimensional lever revolves, then the fulcrum wobbles and pitches as well. MSC/PATRAN with the MSC/ADVANCED FEA module creates these slide line/isl arrangements by means of special one-dimensional elements with contact properties stitched to the three-dimensional elements. Instead, the impact between the hammer stamp and the anvil seal is resolved with a contact pair which defines the contact directly between the faces of the three-dimensional elements. As a practical matter, MSC/PATRAN with the MSC/ADVANCED FEA module creates this type of contact definition as a special type of boundary condition that makes no apparent use of special elements. This is most appropriate for the stamp-seal contact because the nodes on the (slave) seal will move on different directions when impacted by the (master) stamp on account of the seal hex elements undergoing plastic strains. The model is made of three different materials. Isotropic elasticity and plasticity as well as hyperelasticity were defined with the help of MSC/PATRAN capabilities to enter material stress-strain data as field functions. Note that MSC/ADVANCED FEA uses plasticity constitutive laws defined in terms of true stress and strains instead of engineering or nominal values. The hyperelastic law for the rubber band is the same used by Oden (3) in terms of the Biderman strain energy function U = C10 ( I1 3 ) + C01 ( I2 3 ) + C20 ( I1 3 ) 2 + C30 ( I1 3 ) 3 where I1 and I2 are the first and second deviatoric strain invariants. The values reported by Oden are: C10 = psi, C01 = 1.42 psi, C20 = psi, and C30 = psi. With these coefficients, Biderman s function closely matches the experimental data by Treolar (4), which includes measurements, made on uniaxial, biaxial, and planar tension laboratory tests. The anvil (but the upper half of the seal), the hammer (including the stamp), and the trigger are made of the aluminum alloy The seal therefore has a backing of aluminum taking half of its thickness while the upper half which free surface is struck by the stamp- is made of the magnesium alloy ZE4. The actuator is represented by four nodes, each the center to the respective circle of nodes on the actuator hole Figure 4. Four (one for each actuator node) rigid pinned MPC (multiple-point constrain) are created for the 4

5 MSC/ADVANCED FEA model taking advantage of the MPC creator in MSC/PATRAN. Each MPC defines a pinned rigid link between the actuator nodes and each of the nodes on the circle around it. The actuator is moved a distance of 2 inches down the vertical or Y direction in 1 second and this is the only external action on the model. (The out-of-plane or Z displacement of the actuator is forced to be nil and the horizontal or X direction displacement of the actuator is let free). Three small node sets on the base of the anvil are fixed and this is the only restriction on displacements enforced on the entire model other than the displacement of the actuator. Aside from the contact definition discussed above, no restrictions are imposed on the displacement of hammer and hyperelastic band. This is not expected to slide out in the out-of-plane direction Z as a result of the forces into play. The small distance of 1 /200 inch initially separates the contact surfaces. The typical size of an element in the area of contact is 1 /20 inch. The model has 43,069 hex8 elements and 64,149 nodes. Analysis A nonlinear transient analysis is performed to produce a realistic simulation of the final impact of the stamp-hammer on the seal-anvil with the subsequent plastic deformation of the seal into the desired MSC shape. This paper includes results for the very beginning of the analysis when there is an initial impact due to initial separation between the contact surfaces. Figure 6 exhibit Mises stresses after seconds, shortly after initial impact. The largest stresses clearly happen in the walls of the trigger fulcrum hole. Stresses are also relatively sizable on the walls of the actuator hole. The first are due to the impact between the trigger and anvil s trigger pin. The second are accountable to the use of MPCs to simulate the actuator. Although not apparent in Figure 6, contact between the trigger and the rubber band has been made at the time discussed. Since the band is highly deformable this second impact develops much lesser stresses and is therefore hidden by the scale of the fringe plot. By changing the scale of the stresses to the lower range in the previous plot, Figure 7 displays the Mises distribution on the trigger action pin that results from impact with the rubber band. In fact, the rubber band at this time of seconds is already strained up to the point of producing a similar though lesser impact with the hammer action pin. The hammer is therefore set in movement and it has impacted both the anvil hammer s pin and the anvil s stop pin. This is evident in the following Figures 8 and 9. Figure 8 shows the velocity magnitude, which already exhibits a nonlinear distribution on all parts affected. The maximum velocity at seconds is 3.75 in/sec. Clearly the right end of the rubber band is moving more rapidly than the left end and therefore the band is being elongated, thus accumulating strain energy. Eventually much of the strain energy will be 5

6 converted into kinetic energy in the hammer and stamp, and when this impacts the seal much of this kinetic energy will in turn produce a plastic deformation of the seal. Figure 9 displays the contact pressure computed on the slave surfaces located on the anvil pins. Note that all these surfaces have developed contact pressures at this time of seconds. In fact, as intended, the stop pin is preventing the hammer from moving up on account of the rubber band pulling the hammer pin up, explaining the small velocities exhibited by the hammer in Figure 8. As explained, there is no perfect fit between pin and holes, so the hammer impacts the anvil s hammer pin, as shown in Figure 9, and develops some small speed. When the trigger would move down sufficiently, the rubber band will pull the hammer down sending the hammer and its stamp in a collision course with the seal. Figures 10 and 11 are fringe plots on deformed shapes of the right end of the rubber band immediately after impact and shortly after impact respectively. The times are 0.01seconds (Figure 10) and seconds (Figure 11). The undeformed shape is shown in wireframe form as well. Displacements are exaggerated by a factor of 6 to make more evident the difference between the plots. The velocities immediately after impact are bigger than (shortly) later. This is because the impact produces an acceleration that is subsequently countered by the development of hyperelastic strains. Notice that the difference of velocities between the lower and upper arms of the rubber band is reduced from 1 in/sec to 0.5 in/sec from one time to the next. This difference would later vanish and even reverse. The velocity of the lower arm is greater than the one of the upper arm because the trigger pin impacts only the former at first. The rubber band develops strains and as a consequence the right end of the band pulls down the upper band. While the distance between the arms has widened on impact in Figure 10, this distance is reduced some in Figure 11. It would next be reduced to the diameter of the trigger pin when the upper arm of the band impacts the trigger pin. Because of these impacts, the MSC/ADVANCED FEA automatic timeincrementation algorithm will fail to produce a convergent solution for hard contact definitions. In order to achieve convergence, the contact definition was chosen to be of the softened type. Using a clearance at zero pressure of 0.01 inch and a pressure at zero clearance of 1 lb, and selecting a halftol value of 1,000, the automatic algorithm reduced the time increment to values typically between and seconds shortly after impact. At worst the time increment was reduced to seconds. This allows the analysis to progress steadily without lack of convergence and without need of recurring to additional alterations such as the use of incompatible mode elements as needed in other situations see Adeff (5). The time increments grow as the analysis progresses once the initial impact has been lessened by the development of a stable contact interaction and by the straining of the rubber band, allowing a more rapid progress of the calculations. Additional results of the analysis will be reported in a subsequent paper. 6

7 Discussion The simulation discussed in this paper represents a combination of challenges rarely found in a single problem, not because they do not happen in real-life engineering problems, but precisely because such a situation has always be found difficult to tackle. The typical engineering breaking down of a complex problem is indeed still highly advisable; but on occasion, an analyst needs to put together the pieces of the ensemble and let the complexity play its role. This is true for nonlinear problems such as the one discussed in this paper. The actual pieces are made of not-so-perfectly-matching pieces which are sent striking each other by an element the rubber band- exhibiting a definitely nonlinear behavior. The contact happens not on an idealized location (the fulcrum of each lever) but on rapidly changing areas over wobbling cylindrical surfaces that introduce quickly changing three-dimensional momentum impulses to the various components. The impact between the acting head (the imprinting stamp attached to the hammer) and the passive seal (attached to the anvil piece) does not really occur as a head-on collision but happens truly at an angle and while the surfaces are pitching, wobbling and rolling. The velocity and acceleration of encounter is also not perfectly known and the plastic deformation under impact and sudden deacceleration both at a not well known rate- must be accounted for; all these effects are simulated in this work. Other than making a large number of costly laboratory tests, no simplistic analysis will render a reliable design. Of course some physical test will still be needed given the inaccuracies needed to make the analysis; such physical tests may however be minimized in quantity, time, material consumption, and cost by making some analyses of the type presented in this paper. Only recently such an analysis can be undertaken, and the question whether or not was even possible to realize it did not have a clear answer before the work done to produce this paper. This is particularly true when making use of an implicit solver capable of accurately reproducing the nonlinear mechanical behavior of materials involved making use of hex elements as MSC/ADVANCED FEA does. An implicit solver is not well suited for impact-related problems, because these involve high frequencies of oscillation associated with local deformations- which are better dealt with by an explicit solver such as MSC/DYTRAN (6). However, in a case like the present one, the impact only dominates a small area around the surface of contact and therefore low frequencies still dominate the behavior. In such case the implicit solver would still behave better even when material representation would not be at issue which in this case it does. The matter remains to achieve results quickly enough without an unacceptable degradation of quality. The quick improvement in hardware capabilities should make us very optimistic though. 7

8 Conclusions This paper demonstrates the possibility of doing analyses of complex mechanical works exhibiting the simultaneous presence of various nonlinear behaviors, including hyperelasticity, plasticity under impact, three-dimensional and multiple contact surfaces, all combined with imperfect fitting and passing through instability conditions. Although such analysis still require sizable computer resources, it already is feasible with an advanced workstation in a reasonable time, provided the analyst would take care of producing an adequate mesh and representation of materials, loads, boundary conditions, and definition of contact conditions. Likewise a careful tailoring of leading to convergence techniques must still be considered, all of which requires experience from the analyst. The demonstrated capability of the MSC software to resolve such complex problem is of obvious benefit to MSC software users, as it should remove doubts about the possibility of effecting such complicate calculations. The use of the software in uncountable designs should produce sizable savings in the process of developing designs into manufacturable products. Acknowledgements I warmly extend my thanks to the MSC customers calling for support to the MSC/PATRAN Hotline ( ). By exchanging insights and involved discussions with them, I have found the motivation and I have honed the skills needed to produce this paper. References (1) MSC/PATRAN User s Manual, Version 8.0, The MacNeal-Schwendler Corporation, Los Angeles, CA, (2) MSC/ADVANCED FEA Module User s Guide, Version 8.0, The MacNeal- Schwendler Corporation, Los Angeles, CA, (3) Oden, J.T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York, (4) Treolar, L.R.G., Stress-Strain Data for Vulcanized Rubber Under Various Types of Deformation, Trans. Faraday Soc., pp , (5) Adeff, S.E., Nonlinear Analysis of a Pin Insertion, 1998 ABAQUS Users Conference, Newport, RI, 1998 (6) MSC/DYTRAN User s Manual, Version 3, The MacNeal-Schwendler Corporation, Los Angeles, CA,

9 Y X Z Figure 1: View of entire mechanism, made of anvil (with seal attached), hammer (with stamp attached), trigger, and rubber band. 9

10 Stop pin Hammer pin Anvil seal ANVIL Trigger pin Figure 2: View of Anvil showing the finite element mesh, seal and pins. Stamp Guide hole HAMMER H. Action pin H. Fulcrum hole Figure 3: View of the hammer FEM mesh with guide, pin, and stamp. 10

11 T. Action pin TRIGGER Actuator hole T. Fulcrum hole Figure 4: View of Trigger with action pin, and fulcrum and actuator holes. BAND T. Action pin H. Action pin Figure 5: View of Band FEM mesh set about the hammer and trigger pins. 11

12 Mises [psi] seconds Figure 6: Mises stresses distribution shortly after initial impact. Mises [psi] seconds Figure 7: Range changed to exhibit impact effects on cylindrical pin. 12

13 Velocity [in./sec.] seconds Figure 8: Velocity magnitudes shortly after initial impact. Pressure [psi] Lower Stop (Anvil-to- Hammer s guide) Anvil-to- Hammer s fulcrum Anvil-to-Trigger s fulcrum seconds Figure 9: Contact pressure on (anvil) slave surfaces after initial impact. 13

14 Vy Velocity [in./sec.] Vy= in/sec seconds front view displacements * 6 undeformed mesh Vy= in/sec Figure 10: Vertical velocity component immediately after initial contact. Vy Velocity [in./sec.] undeformed mesh Vy= in/sec seconds front view displacements * 6 Vy= in/sec Figure 11: Vertical velocity component shortly after initial contact. 14