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Girder Kinematics Modeling

By Arsalan Jamialahmadi

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Aim of the StudyTo 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.

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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

Modelling

Figure 1 – Master-Slave movement

Micro-Control Technical Requirements:

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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

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Modelling

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 0.3 250

Dummy Acc. Stru. (Cu properties) 100 39706 0.34 69

Table 1 – Material Properties

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Modelling Cylindrical joint for actuator Supporting the structure Flexural joints bear stress Frictionless contact simulates

rotation

Figure 3 – Actuator modelling

Figure 4 – Compensation of rotation by frictionless contact

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ModellingAnalysis Type Assemblies Purpose

Static deflection – no actuation

1-Girder To control the static deflection for comparison with the real

model2-Girder3-Girder

Static deflection – maximum actuation

1-Girder with springTo extract the extreme cases of

deflection 2-Girder with spring3-Girder with spring

Modal Analysis2-Girder fixed To find the resonance

frequencies2-Girder with spring

Parametric Study1-Girder with spring

To give a tool for alignment2-Girder with spring

Table 2 – Performed studies

System 1-Girder 2-Girder 3-GirderNumber of Elements 36350 74446 98888

Table 3 – Number of Elements for different configurations

Note: Girder with spring points out the girder system in which spring serves as the master-slave movement provider for actuators.

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Results

SystemMaximu

m Stress (MPa)

Maximum Deflection

(μm)

1-Girder 37.4 27.38

2-Girder 68.6 30.6

3-Girder 68.6 32.4

Table 4 – Static deflection results

Figure 5 – Static deflection with no actuation

Static deflection – no actuation

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.

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ResultsStatic deflection – maximum actuation

Applied abbreviations:

a,b,c Actuator position on cradle1,2,3 Cradle number p,n Positive or negativeF,R Front and rear

Figure 6 – Displacement b1p-c1n

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ResultsStatic deflection – maximum actuation

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 -300.06 152.5 -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 299.93 -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 -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 -300.065 -300.065 300 -13.9893 -3.86843 -84.4615 82.062 -177.27 41.064 -97.497 -13.6505 41.625 0.0121 0.0167

a1p-c1n 299.93 299.93 -300 16.569 6.497 87.3845 -87.153 160.87 -49.864 98.9535 -0.075 -49.328 -0.0090 -0.0101

a1p-c1p 299.93 299.93 300 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 -300.065 -300 -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 -300.06 300 -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 -300 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 299.935 300 6.8976 16.603 -76.6945 80.1 158.1 -49.068 -89.1575 0.195 -48.5275 0.0426 -0.0558

a1n-b1n -300.065 -300.065 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 -300.065 299.93 -321.3 -13.0356 15.0024 -231.095 -824.91 -9.0255 -4.3466 -242.44 -6.7325 -3.7981 0.0688 0.1984a1p-b1n 299.935 -300.065 320.73 15.5177 -12.6092 235.1 822.04 -9.107 -3.7446 245.07 -6.202 -3.19255 -0.0662 -0.1922

a1p-b1p 299.935 299.935 -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 -300.065 -300.065 -300 -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 -300.065 -300.065 300 -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 -300.065 299.935 -300 -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 -300.065 299.935 300 -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 -300.06 -300 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 299.93 -300.06 300 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 299.935 299.935 -300 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 299.935 299.935 300 21.512 19.1555 25.194 304.99 297.05 -86.363 20.189 5.8263 -85.832 0.0260 -0.1382

Table 5 – Deflection values for One-Girder system with spring

Note: Displacements are in micrometer

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ResultsStatic deflection – maximum actuation

Table 5 gives the following information:

1. Slave movement of actuators with respect to maximum actuation of the master movement(s).

2. Beam axis movement with respect to maximum actuation of the master movement(s).

3. Angle of rotation of beam axis with respect to its initial position.

Figure 7 – Two-Girder system maximum actuation a2p-b2p-c2p

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ResultsModal Analysis

Mode NumberFrequency (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

Table 6 – Resonance frequencies for Two-Girder system

Note: The resonance values of the system with spring might be used for comparison onlyFor this system, the first resonance frequency estimate from Micro-Control analysis is 49.8 Hz.

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ResultsModal Analysis

a) Frequency 45.5 Hz, Max 1.83 mmIn-phase bending

b) Frequency 47.7 Hz, Max 1.85 mmAnti-phase bending

c) Frequency 55.4 Hz, Max 1.55 mmFirst girder shear

d)Frequency 60.4 Hz, Max 1.58 mmSecond girder shear

Figure 8 – First 4 resonance frequencies and mode shapes of the Two-Girder system with fixed actuators

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ResultsParametric 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.

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ResultsParametric Study

Figure 9 – Parametric study of One-Girder system. F1x is a function of three variables a1,b1 and c1. Here c1=0

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.

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ResultsParametric Study

Figure 10 – Parametric study of One-Girder system. Ry1 is a function of three variables a1, b1 and c1. Here

a1=0.156

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.

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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

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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.

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Thank You!