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1 1. Summary Finite element modeling has been used to determine deflections and stress levels within the SRC planar undulator. Of principal concern is the shift in the magnetic centerline and the rotation of the girders about the longitudinal axis under the magnetic load. These uncorrected movements affect all aspects of the device s performance including the photon energy, phase error, integrals and multipoles. Magnetic loads used in the model were calculated using Radia [1]. Forces arising from the action of the springs, which are used to maintain a preload on the magnet girders, were included in the simulation. The compliance of the bearings and lead-screws were also included in the model. Uncorrected deflections of up to 11.7 µm and rotations of up to 8.21 µrad are predicted when operating over gaps ranging from 20 to 35 mm. Displacement feedback eliminates any changes in the nominal gap but cannot correct for deformation of the girders, which can be as large as 3.14 microns after shimming. In addition, the centerline can shift by 3.57 µm over the operating range. The effects of centerline shift and girder rotation will be evaluated using SRW at a later date but the magnitude of these deformations is not sufficient to prevent the undulator from meeting its specifications. 2. Model description The finite element model was initially generated in ANSYS Workbench directly from the solid model within Inventor. However, Workbench does not have sufficient flexibility to allow the user to accurately simulate bearing stiffness. Much of the predicted deformation results from movement within the bearings and deformation of the lead-screw. A simplified solid model was therefore built within native ANSYS. Most holes, fillets, radii and other small features were removed, as these have no discernable effect on the structure s behavior. As a result the model size and complexity are significantly reduced. The model was generated using quadratic elements throughout, with the vast majority of elements parallelepipeds. Some tetrahedral elements were required in hard-to-mesh regions but the higher order shape functions used with quadratic elements ensured that numerical stiffening did not degrade the solution accuracy. Each node had three degrees-of-freedom. Figure 1: Bearing stiffness of Rexroth roller rail systems. 3

2 A right-hand Cartesian coordinate system has been used in the model has its x-direction aligned with the electron path through the ID and z vertical. Symmetry about the y-z-plane has been used to reduce the model size. Bearings were simulated using 1D springs that were directly generated (i.e., not associated with any solid body). These springs connect two nodes and are affected only by their displacements along a single global axis. Each spring s stiffness is defined explicitly by a constant and has no relation to the initial distance between its nodes. Thus two nodes, connected by a spring defined to act in the global x-direction, transmit a load proportional to the displacements in that direction only. They can move by any amount in the y- and z-directions without transmitting any load. In the case of the linear bearings attaching the girder to the undulator frame eight springs are needed to correctly represent each of the bearing blocks - two at each corner of the block acting in the global x- and y-directions. Note that there is no z-axis constraint acting through these bearings. Stiffness of the Rexroth roller bearings used in the undulator is shown in Figure 1. The initial slope of the stiffness curve has been used to establish the stiffness of the springs used in the model. Since compression and extension in the normal direction differ the lower value has been used for stiffness in the y-direction. 1D springs have also been used to represent the ball-nuts. In this case a single spring with its stiffness defined in the z-direction has been used to attach each girder to the lead-screw. Spar elements have been used for the lead-screw, with their cross-sectional area equal to that of the lead-screw at the root of the thread. Two spars have been used the first connecting the undulator frame to a point midway between the operating limits of the upper girder and the second connecting this latter point to a point midway between the operating limits of the lower girder. Spars and springs have also been combined to represent the undulator s counterforce springs. Spar elements can have an initial strain applied that can be used to apply a preload to the springs. Each spar has been given a length of 100 mm so that an initial strain of produces a deflection of 2.5 mm at its end. By assigning an arbitrary, large cross-sectional area to each spar its stiffness becomes much greater than that of the attached spring and a spring extension of 2.5 mm is produced. Using springs and spars in a solid model produces local deformations and stresses that are unreasonably large, however, the effects at points far from these points can be ignored (St. Venant s Principal). There is no observable effect from these point loads in the model at the magnet mounting surfaces. Components have been defined to simplify the task of modifying the model for various gaps. The volumes representing lower and upper girders have been assigned the names lowergdr and uppergdr. The corner nodes on the magnet mounting surfaces have been assigned to component corners. The model may be modified to simulate a particular gap as follows: 1. With a viewing angle set by /view,1,1,0,0 move the volumes associated with one of the girders using the vgen command or menu selections. All nodes and elements associated with the selected volumes will be moved. Note that the 4

3 2. To move girders select side view (/view,1,1,0,0) then pick modeling/movemodify/volumes. Select box in pop-up menu and pick volumes in either upper or lower girder. Enter displacement in meters. Or, select component uppergdr or lowergdr and then use VGEN command with IMOVE set to 1. Don t neglect 93.9 mm stand-off. Local CS 11 is at center of gap By using springs and spars as described above the process of simulating various gaps is simplified. By using these springs and spars it becomes very easy to modify the model to represent various gaps. Simulating guide bearing stiffness constrained bearing blocks to motion in one direction moved girders to correct spacing attached 2, 1D springs (X and Y) to each corner of the four bearing blocks (total of 32) spring stiffnesses 2.92x10 8 N/m normal (k y ) and 4.38x10 8 N/m lateral (k x ) girders and bearing blocks set up as two components (volumes) components moved using VGEN et2 combin14, z-axis longitudinal option et3 combin14, y-axis longitudinal option et4 combin14, x-axis longitudinal option real 1 stiffness of Belleville springs, z-axis, use with ET2 (K=79.5x10 3 N/m) real 2 stiffness of lead-screw bearings, z-axis, use with ET2 real 3 stiffness of bearing blocks, y-axis, use with ET3 real 4 stiffness of bearing blocks, x-axis, use with ET4 Simulating counterforce stiffness 2 elements used to represent each spring spring with stiffness kz=1.59x10 5 (ET2, real 1) constant length link8 with kz = 1x10 9 N/m (E=220x10 9, L = 0.1m, a = 4.55x10-4 m 2 use ISTRN to provide initial strain link connected to all nodes around spring seat in frame using CE spring length changes as girders move but K remains constant (real) Simulating ball-nuts Ball-nut stiffness is 4.25x10 6 lb f /in = 7.44x10 8 N/m Extra space between girders allows for magnet height (magnet = 0.030, magnet holder = , block holder = 0.042) x 2 Select component corners then use the prdisp (undocumented command) to list displacements at corners of both girders.. 5

4 Forces within SRC planar Force (N) Magnetic attraction Upper array Lower array Spring force Gap (mm) Figure 2: Forces acting within the SRC planar undulator (gravity, pre-load springs and magnetic attraction). Real # Value Type # Associated object 1 K=7.94x (spring) Belleville springs 2 K=7.44x (spring) Leadscrew ball-nut 3 K=2.92x (spring) Bearing block, y-stiffness 4 K=2.18x (spring) Bearing block, x-stiffness 5 A=4.55x10-4 Provides preload to counterforce springs (spar) initial strain 6 A=1 (spar) Leadscrew Lead-screw modeled in two segments, not changed between runs. Use gap of 28mm to define position of nodes. Node on springs between these nodes and nodes on the volumes representing girders automatically moved by vgen command. surface. Pressures are scaled for the element area and are treated as nodal forces within Workbench. The loads carried by the lead-screws were treated as pressures applied to the bearing seats. The reaction force of the pre-load springs were dealt with similarly. The net force applied by the three pressures was zero. Gravitational loading was also included although any sag that it produces is unvarying and is eliminated during shimming. The loading over gaps from 20 to 80 mm is shown in Figure 2. 6

5 Material properties used in the analyses are given in Table 1. All materials are linear elastic and have no temperature dependence. In order to include the bearing stiffness in the simulation it was necessary to transfer the model from workbench to the full ANSYS program. Only nodes, elements, constraints and nodal forces can be transferred between the two programs. Nodal forces transferred from Workbench can be scaled within ANSYS using the FSCALE command so deflections at various gaps can be simulated if necessary. RexRoth Star size 35 roller rails are to be used in this undulator to constrain the motion of the girders to the vertical members of the strongback. Stiffness curves for these bearings are given in the manufacturer's web-site and are reproduced in figure 1 above. For modeling purposes a linear stiffness is used, calculated from the initial slope of these data. This is a conservative estimate, especially for tensile loads (those that tend to open a gap). Stiffness values estimated from the data were 1.3e9 N/m (normal, compressive) 1.1e9 N/m (normal, tensile) and 8.50e8 N/m (lateral). Each bearing block is treated as a anisotropic material with a modulus calculated to give the appropriate stiffness. Assuming a contact area of.0042 m 2 for each each bearing block and a thickness of XXX m, moduli of YYY and ZZZ were used. Once again, this is conservative as the lower tensile stiffness is used rather than the higher, compressive stiffness. Although the pressures applied to the magnet girders, lead-screw bearing seats and spring seats produced a net load of zero (with gravitational loads) it was necessary to add a spring with very small stiffness to prevent rigid-body motion of the girders in the vertical, or unrestrained, direction. With this addition the model was complete and ready to run. A pre-conditioned Jacobigradient solver was used to produce a solution. 3. Results Table 1: Materials properties used in FEA. All properties are evaluated at 28 C. 304L SS Al-6061 Mild steel Modulus of Elasticity (GPa) Poisson's Ratio Mass Density (kg/m 3 ) 8,000 2,710 7,850 CTE (ppm/ C) Thermal Conductivity (W/m C) Specific Heat (J/kg C) Tensile Yield Strength (MPa) Tensile Ultimate Strength (MPa)

6 The loads present in the undulator produce three distinct types of deformation that can affect brilliance and phase error: bending of the frame moves the magnet girders vertically relative to one another; curvature in the upright frame members introduces a rotation to the magnet girders; and bending of the magnet girders themselves produces local changes in the gap. The first of these is the bending of the frame, which causes the magnet girders to move in the vertical (z-axis) direction. This motion is detected by the linear encoders and can be corrected by the control network, however, a shift in magnetic centerline also occurs. An arbitrary limit of 10 µm deflection is used as a design criterion for the strongback. Figure 3 shows the vertical (zaxis) deflection of the undulator at Figure 3: Deflections... minimum gap. Deflections are greatest at the cantilevered supports for the lead-screws. At the bottom this deflection is about two microns and results solely from local deformation. At the top the deflection is much larger microns due not only to local deformation but also bending of the frame. This bending can be seen clearly in Figure 4. Deflections of the magnet girders at minimum gap can be effectively eliminated by shimming the magnets, even if these deflections are on the order of 100 µm. Of more concern is the reduction of this deflection as the gap is increased to the maximum. While phase errors are generally reduced as the gap is increased a large change in deformation can result in an increase in phase error at some intermediate gap. Limiting the change of deflection to less than 5 µm has been effective in meeting phase, and brilliance, specifications. Figure 4: Deflections uz 8

7 Deformation in three regions of the undulator affect its performance. Of more concern is the bending of the strongback at the girder linear bearing rails. This bending produces a rotation of the girders that can have an effect on the photon beam. Deflections of the undulator strongback and magnet girder are shown in figures 3 and 4. More about this effect after I have modeled it! Figure 5: Deflection of magnet girders uz, from front and end. due to unequal periods, due to variations in pole strength or direction and pole spacing, [1] Blomqvist, Ingvar, Report on SRC undulator, private report, 5/5/

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