Similar Pulley Wheel Description J.E. Akin, Rice University

Transcription

1 Similar Pulley Wheel Description J.E. Akin, Rice University The SolidWorks simulation tutorial on the analysis of an assembly suggested noting another type of boundary condition that is not illustrated in any other tutorial cyclic symmetry. Consider the construction of just the pulley wheel in that tutorial. Its construction involved the use of a circular pattern. When that construction feature is useful in building a major portion of a part then the analyst should be alert to see if a cyclic symmetry condition can be used in the simulation(s) of the part. A cyclic symmetry requires that all aspects of the simulation repeat themselves n times around the circumference. In other words each aspect repeats every 360 n degrees. The aspects include the materials properties, the shape, the essential boundary conditions, and the applied loads, as shown in Figure 1. Furthermore, the cyclic cutting surfaces can be curved as illustrated for the turbine impeller shown in Figure 2. The computer storage requirement is reduced by a factor of n, and the run time is generally reduced by a factor on n 3. Figure 1 Cyclic symmetry cut surfaces (AB and ab) of a part with two materials Figure 2 Common cyclic symmetry parts have curved cutting surfaces If the pulley had been shrunk fit to the center shaft and loaded in pure torsion, then it would be a candidate for a one-sixth cyclic symmetry simulation. Assume that is the case, and return to the construction details rolled back to before the circular pattern execution sequence shown in Figure 3. If the center shaft opening transfers a pure torque and the boundary condition region repeats every 60 then the simulation can utilize the very efficient cyclic symmetry condition, which requires the sparse solver. Figure 4 shows that the retained pie slice can be taken beginning at any reference angle, but usually the best choice makes it easier to visualize the full part, as illustrated in Figure 5. Page 1 of 9

2 Figure 3 The pulley has cyclic symmetry of its geometry and materials Figure 4 Extruded cut at an arbitrary cyclic section of the pulley wheel Figure 5 A clearer cut through the rib mid-planes Page 2 of 9

3 Once the geometric segment is available the simulation can begin. First, the axis of the cyclic symmetry must be identified along with the two surfaces on which the cyclic constraint equations are to be applied. That provides the information to the mesh generator to assure that both selected surface have the same identical surface mesh. In a thermal study the unknown temperature at each node on the first surface is required (constrained) to be the same as the unknown temperature at each corresponding node on the second face. In a solid stress analysis the unknown displacement components, relative to the first surface, are constrained to be equal to the (rotated) displacement components, relative to the second surface. Figure 6 shows the selection of the two cyclic symmetry surfaces for the pulley wheel segment, and the region(s) of imposed displacements (anywhere in the slice). Figure 6 Define the cyclic planes and then impose the known displacements (right) and loads (bottom) Page 3 of 9

4 At this point the stress simulation proceeds as usual, except that it is more likely that individual components of the stress tensor may be needed relative to a cylindrical coordinate system (as illustrated later). Create the mesh and run the analysis. The displacements should be checked first to verify that the deformed shape is the same on both cyclic surfaces. That is illustrated in Figure 7. Figure 7 Verifying cyclic symmetry of the boundary displacements By default, most finite element systems, including SolidWorks, displays the von Mises effective stress (the measure of the distortional energy) material failure criterion since it is valid for most ductile materials with equal yield stresses in tension and compression. However, it is not valid for this application because this, and other, cast iron parts have a compressive yield stress that is three or four times larger than the tensile yield stress. Thus, the von Mises stress is show for reference only in Figure 8 (for future comparison to the Burzynski values). Here, the material failure is more likely to be predicted by the maximum shear stress (Intensity?=? tensile yield), or the maximum tensile stress (positive part of the First Principal Stress (P1?=? tensile yield), or the maximum compressive stress (negative part of the Third Principal Stress (P3?=? compressive yield), or a material failure theory like the Burzynski criterion (σ B ) which includes the ratio of the yield stresses ratio. Figure 8 Von Mises values, for reference only Page 4 of 9

5 The Intensity, shown in Figure 9 with a zoomed detail, shows irregular contour lines which are indicative of a poor mesh (large element sizes) in the most important region. The tensile stress (P1 > 0) values in Figure 10 occur in about the same region. In cyclic symmetry parts often the radial or circumferential stress components are easier to understand, so the components of the stress tensor will also be checked in a centered cylindrical coordinate system which is established in Figure 11 by selecting the center axis of the original full part. The part s radial stress values are given in Figure 12 while the directions of the surface circumferential stresses are depicted in Figure 13. All of the above stress studies indicate that a refined stress should be constructed before adding a Burzynski failure criterion calculation. The Apply Mesh Control feature was used to create the finer mesh shown in Figure 14. That mesh gave more accurate stress values, of each type, and smoother contour lines in the regions of peak values. The improved Intensity stress contours are shown in a zoomed view in Figure 15. Figure 9 The intensity plot reveals an inadequate mesh in the region of the maximum values Figure 10 Tensile stress values Page 5 of 9

6 Figure 11 Selecting cylindrical coordinates for the radial (SX) and circumferential (SY) stress recovery Figure 12 Radial stress values (SX in cylindrical coordinates) Page 6 of 9

7 Figure 13 Circumferential stress (SY in cyl. coord.) directions and relative magnitudes Figure 14 Refined cyclic symmetry mesh Page 7 of 9

8 Figure 15 Smoother Intensity stress peak values with the refined mesh This cyclic symmetry model required 1 6 as much storage and would run about 216 times faster than using a full pulley wheel model. Those values can be improved by another factor of two by recognizing that there is another plane of symmetry at the thickness in the axial direction. That regional cut and resulting final cyclic symmetric part is given in Figure 16. Page 8 of 9

9 Figure 16 Enhancing the study with a final mid-plane of symmetry Page 9 of 9