1. By Arsalan Jamialahmadi
Girder Kinematics Modeling
By Arsalan Jamialahmadi

2. 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.

3. 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

4. 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

5. 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

6. 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

7. 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.

8. 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

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

10. 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
-300
0.0063
0.0824
c1p
4.633
-6.348
300
314.29
16.098
a1n
39.201
0.0185
0.1299
a1p
299.93
114.5
377.97
154.55
-0.014
b1n
152.5
385.1
-172.9
40.004
40.565
b1p
17.588
-104.5
152.1
-0.362
0.0479
0.0628
a1n-c1n
38.564
0.0207
0.1679
a1n-c1p
82.062
41.064
41.625
0.0121
0.0167
a1p-c1n
16.569
6.497
160.87
-0.075
a1p-c1p
18.652
123.92
539.35
152.35
0.008
b1n-c1n
78.842
-86.66
41.438
89.776
42.013
0.0637
b1n-c1p
115.38
538.88
39.537
40.098
b1p-c1n
17.898
150.22
-47.16
-0.536
0.0496
0.0999
b1p-c1p
6.8976
16.603
80.1
158.1
0.195
0.0426
a1n-b1n
7.162
6.399
78.072
78.651
0.0679
a1n-b1p
-321.3
0.0688
0.1984
a1p-b1n
320.73
235.1
822.04
-9.107
245.07
-6.202
a1p-b1p
9.4197
297.06
0.0291
a1n-b1n-c1n
77.962
0.1445
a1n-b1n-c1p
9.6356
305.46
78.178
3.5112
a1n-b1p-c1n
-4.339
-6.718
0.0686
0.1932
a1n-b1p-c1p
-6.303
0.0626
0.0443
a1p-b1n-c1n
188.12
a1p-b1n-c1p
800.86
-3.752
244.37
a1p-b1p-c1n
20.597
297.07
0.0320
0.0122
a1p-b1p-c1p
21.512
25.194
304.99
297.05
20.189
5.8263
0.0260
Note: Displacements are in micrometer

11. 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

12. 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.

13. 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

14. 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.

15. 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

16. 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 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

17. 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

18. 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.

19. Thank You!