By Arsalan Jamialahmadi Girder Kinematics Modeling By Arsalan Jamialahmadi

Aim of the Study To provide a model to study: Static deformation of the Micro-Control girder for the Main Beam of the CLIC two-beam prototype module. Maximum possible displacement of the beam axis on the maximum master movement(s). The parametric actuation of the conceptual design.

Figure 1 – Master-Slave movement Modelling Maximum vertical and lateral static deformation of 10 μm Maximum girder weight of 240 kg Maximum girder length is almost 2 m Maximum sustainable dead weight of 400 kg/m Maximum cross section of 320 mm × 150 mm Maximum master actuation of ±0.3 mm Maximum slave travel of ±3 mm Micro-Control Technical Requirements: Figure 1 – Master-Slave movement

Figure 2 – Two-Girder system Modelling Girders and V-supports are integrated parts which are glued to each other and to the cradles. Cradles and actuators have multiple parts glued to each other. Actuators, flexural joints and supports Dummy load as accelerating structure Z-direction movement at the end cradles suppressed Roller compensated by frictionless contact Figure 2 – Two-Girder system

Yield’s Strength (MPa) Modelling Table 1 – Material Properties Material/Component Young’s Module (GPa) Density (kg/m3) Poisson Ratio Yield’s Strength (MPa) SiC 250 3215 0.16 3440 Structural Steel 200 7850 0.3 Dummy Acc. Stru. (Cu properties) 100 39706 0.34 69

Figure 3 – Actuator modelling Cylindrical joint for actuator Supporting the structure Flexural joints bear stress Frictionless contact simulates rotation Figure 4 – Compensation of rotation by frictionless contact Figure 3 – Actuator modelling

Modelling Table 2 – Performed studies Analysis Type Assemblies Purpose Static deflection – no actuation 1-Girder To control the static deflection for comparison with the real model 2-Girder 3-Girder – maximum actuation 1-Girder with spring To extract the extreme cases of deflection 2-Girder with spring 3-Girder with spring Modal Analysis 2-Girder fixed To find the resonance frequencies Parametric Study To give a tool for alignment Table 3 – Number of Elements for different configurations System 1-Girder 2-Girder 3-Girder Number of Elements 36350 74446 98888 Note: Girder with spring points out the girder system in which spring serves as the master-slave movement provider for actuators.

Figure 5 – Static deflection with no actuation Results Static deflection – no actuation Table 4 – Static deflection results System Maximum Stress (MPa) Maximum Deflection (μm) 1-Girder 37.4 27.38 2-Girder 68.6 30.6 3-Girder 32.4 Note: The load/actuator and the Z-direction movement suppression are the contributors to the increase of deflection and stress. The values of deflection are lower compared to the values given by Micro-Control without pre-stress. Figure 5 – Static deflection with no actuation

Figure 6 – Displacement b1p-c1n Results Static deflection – maximum actuation Applied abbreviations: a,b,c Actuator position on cradle 1,2,3 Cradle number p,n Positive or negative F,R Front and rear Figure 6 – Displacement b1p-c1n

Results Static deflection – maximum actuation Table 5 – Deflection values for One-Girder system with spring a1 b1 c1 a2 b2 c2 f1x f1y f1z r1x r1y r1z teta-x(Rad) teta-y(Rad) c1n -9.7091 9.20725 -300 -0.26954 2.35335 -20.156 -323.15 -9.2261 -4.2624 -16.4815 -7.2197 -3.7117 0.0063 0.0824 c1p 4.633 -6.348 300 2.59505 0.22516 21.3565 314.29 -9.8721 -3.8705 16.098 -6.8258 -3.32065 -0.0029 -0.0743 a1n -300.06 -17.3605 -147.19 -15.5442 -2.1398 -111.695 384.88- -170.92 39.201 -117.785 -13.691 39.7665 0.0185 0.1299 a1p 299.93 20.6405 145.445 18.1155 4.77395 114.5 377.97 154.55 -48.002 119.166 -0.014 -47.4585 -0.0153 -0.1229 b1n -21.179 152.5 -2.3065 -15.1778 106.395 385.1 -172.9 40.004 110.3115 -13.527 40.565 -0.0449 -0.0539 b1p 15.7695 -156.53 4.55075 17.588 -104.5 -395.86 152.1 -47.616 -109.872 -0.362 -47.0815 0.0479 0.0628 a1n-c1n -300.065 -299.995 -16.0752 -1.5493 -120.995 -544.44 -168.75 38.564 -124.718 -13.7035 39.1305 0.0207 0.1679 a1n-c1p -13.9893 -3.86843 -84.4615 82.062 -177.27 41.064 -97.497 -13.6505 41.625 0.0121 0.0167 a1p-c1n 16.569 6.497 87.3845 -87.153 160.87 -49.864 98.9535 -0.075 -49.328 -0.0090 -0.0101 a1p-c1p 18.652 4.17575 123.92 539.35 152.35 -47.355 126.1635 0.008 -46.812 -0.0175 -0.1614 b1n-c1n -32.8685 -4.63109 -14.1996 78.842 -86.66 -178.85 41.438 89.776 -14.0995 42.013 -0.0397 0.0637 b1n-c1p -17.3672 -1.54855 -15.4971 115.38 538.88 -170.96 39.537 116.994 -13.339 40.098 -0.0466 -0.0921 b1p-c1n 12.06232 299.935 3.81255 17.898 -113.24 -545.44 150.22 -47.16 -116.3585 -0.536 -46.622 0.0496 0.0999 b1p-c1p 27.5622 6.8976 16.603 -76.6945 80.1 158.1 -49.068 -89.1575 0.195 -48.5275 0.0426 -0.0558 a1n-b1n 7.162 -17.8015 -17.4295 -5.3666 6.399 -315.17 78.072 -6.7429 -19.653 78.651 -0.0264 0.0679 a1n-b1p -321.3 -13.0356 15.0024 -231.095 -824.91 -9.0255 -4.3466 -242.44 -6.7325 -3.7981 0.0688 0.1984 a1p-b1n 320.73 15.5177 -12.6092 235.1 822.04 -9.107 -3.7446 245.07 -6.202 -3.19255 -0.0662 -0.1922 a1p-b1p -7.7522 20.3565 19.8955 9.4197 -9.3043 297.06 -86.474 9.44605 5.61285 -85.9415 0.0291 -0.0614 a1n-b1n-c1n -18.9545 -16.692 -21.101 -307.29 -315.16 77.962 -17.5005 -19.8635 78.5455 -0.0234 0.1445 a1n-b1n-c1p -16.702 -18.133 9.6356 305.46 -315.18 78.178 3.5112 -19.4495 78.7505 -0.0293 -0.0059 a1n-b1p-c1n -12.9557 14.9514 -230.005 -803.16 -9.0264 -4.339 -241.695 -6.718 -3.79025 0.0686 0.1932 a1n-b1p-c1p -10.7021 13.51045 -199.255 -190.42 -9.0517 -4.1236 -220.72 -6.303 -3.5767 0.0626 0.0443 a1p-b1n-c1n 299.925 13.1884 -11.1177 203.325 188.12 -9.0808 -3.9691 223.355 -6.6375 -3.40495 -0.0601 -0.0380 a1p-b1n-c1p 15.43985 -12.5593 234.065 800.86 -9.1061 -3.752 244.37 -6.2165 -3.19985 -0.0660 -0.1872 a1p-b1p-c1n 19.2595 20.597 -5.5497 -307.76 297.07 -86.579 -0.7844 5.41225 -86.054 0.0320 0.0122 a1p-b1p-c1p 21.512 19.1555 25.194 304.99 297.05 -86.363 20.189 5.8263 -85.832 0.0260 -0.1382 Note: Displacements are in micrometer

Figure 7 – Two-Girder system maximum actuation a2p-b2p-c2p Results Static deflection – maximum actuation Table 5 gives the following information: Slave movement of actuators with respect to maximum actuation of the master movement(s). Beam axis movement with respect to maximum actuation of the master movement(s). Angle of rotation of beam axis with respect to its initial position. Figure 7 – Two-Girder system maximum actuation a2p-b2p-c2p

Results Modal Analysis Table 6 – Resonance frequencies for Two-Girder system Mode Number Frequency (Hz) with spring with fixed actuators 1 58.2 45.5 2 60.8 47.7 3 69.5 55.4 4 92.5 60.3 5 99.0 103.0 Note: The resonance values of the system with spring might be used for comparison only For this system, the first resonance frequency estimate from Micro-Control analysis is 49.8 Hz.

Results Modal Analysis Frequency 45.5 Hz, Max 1.83 mm In-phase bending b) Frequency 47.7 Hz, Max 1.85 mm Anti-phase bending c) Frequency 55.4 Hz, Max 1.55 mm First girder shear d)Frequency 60.4 Hz, Max 1.58 mm Second girder shear Figure 8 – First 4 resonance frequencies and mode shapes of the Two-Girder system with fixed actuators

Results Parametric Study Overview: Number of input variables: 3 Number of output variables: 9 (1-Girder) or 18 (2-Girder) The range for input variables are ±0.3 mm. The 3 input variables are the two vertical and one horizontal actuator movements of one cradle. Output variables give the changes in x, y and z coordinates of the beam axis ends for each girder. Results (outputs) are shown as variation diagrams of two input variables while the third input variable remains constant.

Results Parametric Study Two vertical actuators of the first cradle are moving while the horizontal actuator is set to be fixed at zero. By having the same amount of actuation for the vertical actuators, front point of the beam axis will not have any displacement component in x-direction. Figure 9 – Parametric study of One-Girder system. F1x is a function of three variables a1,b1 and c1. Here c1=0

Results Parametric Study The second vertical (b1) and the horizontal actuator (c1) from the first cradle are moving while the first vertical actuator (a1) is set to be fixed at 0.156 mm displacement. By having the b1 constant, rear point of the beam axis will not have any displacement component in y-direction. Figure 10 – Parametric study of One-Girder system. Ry1 is a function of three variables a1, b1 and c1. Here a1=0.156

Conclusions Static deformation values are relative values as the pre-stress option was not possible with ANSYS. If pre-stress is considered, then only static deflection values are to be changed. Worst case deflection is not passing the ±3 mm limits. Lowest resonance frequency is 45.5 Hz. Parametric study is a suitable tool to locate the beam axis

Further Work The mechanism of master-slave movement needs to be studied more thoroughly. The snake system kinematics is governing. Modal analysis can be done again with accelerating structure for comparison purpose. The more the number of girders, the more precise results but heavier model at the same time! Number of input variables of the parametric study can be increased to consider 6 actuator movements for alignment study. A thorough report of this work will be written for through description of the results.

Thank You!