Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT
Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT
of freedom in the model. The stiffness and mass matrices for one beam element are: ⎡ Ke = ⎢ ⎣ EA L 0 0 − EA 0 12EI L L 0 0 3 6EI L2 0 − 12EI L3 6EI L2 0 6EI L2 4EI L 0 − 6EI L2 − 2EI L EA L 0 0 − 0 EA L 0 0 12EI L3 − 6EI L2 0 12EI L3 − 6EI L2 0 6EI L2 2EI L 0 − 6EI L2 ⎤ ⎥ ⎦ 4EI L Me = ρAL ⎡ 175 ⎢ 0 ⎢ 0 ⎢ 420 ⎢ ⎣ 0 156 22L 0 22L 4L 35 0 0 54 0 −13L 2 0 −13L −3L2 35 0 0 54 0 13L 175 0 0 156 0 −22L 0 13L3L20 −22L 4L2 ⎤ ⎥ ⎦ (2.2) (2.3) where E and ρ are the Young’s Modulus and material density, A and I are the cross- sectional area and inertia of the beam, and L is the element length. The mass matrix is of the same form as that used by the finite element program NASTRAN. The element matrices are assembled into global mass and stiffness matrices, K and M. The optical elements are simply lumped masses without inertial or optical properties. They are rigidly fixed to the beam elements and are incorporated by adding the appropriate mass to the x and y degrees of freedom of the corresponding node in the global mass matrix. The mass breakdown of the model is given in Table 2.1. Table 2.1: Mass breakdown of SCI development model. Component Mass [kg] Truss 791.7 Combiner 200 Collector (−x) 200 Collector (+x) 200 Total 1391.7 42
The equations of motion of the undamped system are written as: M ¨ ˆx + K ˆx = BˆxwF (2.4) where Bˆxw is a mapping matrix between the disturbance forces, F and the physical degrees of freedom, ˆx. The disturbances enter at the combiner node, where the spacecraft bus and reaction wheels are located and include force and torque in all three directions, Fx, Fy, Tz. Therefore, Bˆxw is a sparse matrix with fifteen rows and three columns. The axial force, Fx, andthetorque,Tz, disturbances are modeled as unit-intensity white noise, while the y-force, Fy, is white noise with an intensity of N 2 0.01 . The y-force intensity is a fraction of the torque intensity to ensure that the Hz system response from the two disturbances are of similar scale. Equation 2.4 is transformed to modal coordinates through the eigenvalue problem: � −Ω 2 M + K � Φ = 0 (2.5) where Ω is a diagonal matrix of natural frequencies and Φ contains the associated mode shapes. The natural frequencies and mode shape descriptions for select modes are listed in Table 2.2. The system dynamics are written in a modal state-space Table 2.2: Natural frequencies and mode shapes of nominal SCI model, unconstrained and with model of Attitude Control System (ACS) included. unconstrained with ACS Model Description Mode # Freq (Hz) Mode # Freq (Hz) Rigid X-translation 1 0 N/A N/A Rigid Y-translation 2 0 N/A N/A Rigid Z-rotation 3 0 N/A N/A ACS mode, θz rotation N/A N/A 1 0.082 1st bending mode, symmetric 4 0.197 2 0.197 2nd bending mode, asymmetric 5 0.708 3 0.813 3rd bending mode, symmetric 6 1.294 4 1.294 4th bending mode, asymmetric 7 2.848 5 3.030 1st axial mode, asymmetric 12 47.81 10 47.81 2nd axial mode, symmetric 13 83.83 21 83.83 43
- Page 1: Dynamic Tailoring and Tuning for Sp
- Page 4 and 5: Acknowledgments This work was suppo
- Page 6 and 7: 3.2 RPT Formulation . . . . . . . .
- Page 9 and 10: List of Figures 1-1 Timeline of Ori
- Page 11 and 12: List of Tables 1.1 Effect of simula
- Page 13 and 14: Nomenclature Abbreviations ACS atti
- Page 15: dk optimization search direction f
- Page 18 and 19: 1.1 Space-Based Interferometry NASA
- Page 20 and 21: unfettered by the Earth’s atmosph
- Page 22 and 23: the SCI, both the size and flexibil
- Page 24 and 25: maybethatitbecomescertain that the
- Page 26 and 27: Table 1.1: Effect of simulation res
- Page 28 and 29: Table 1.2: Effect of simulation res
- Page 30 and 31: has been found that structural desi
- Page 32 and 33: precision telescope structure for m
- Page 34 and 35: attractive, and more conservative a
- Page 36 and 37: to solve the performance tailoring
- Page 39 and 40: Chapter 2 Performance Tailoring A c
- Page 41: ometer (SCI). In the following sect
- Page 45 and 46: The frequency response functions fr
- Page 47 and 48: the output covariance matrix, Σz,
- Page 49 and 50: where the subscript indicates the i
- Page 51 and 52: 2.3.3 Design Variables The choice o
- Page 53 and 54: and then, by inspection, the inerti
- Page 55 and 56: algorithms begin at an initial gues
- Page 57 and 58: at least locally optimal, and the s
- Page 59 and 60: initial design variable state, x =
- Page 61 and 62: and the RMS OPD is computed using E
- Page 63 and 64: # Designs 25 20 15 10 5 Accepted, b
- Page 65 and 66: does not provide information on why
- Page 67 and 68: energy is distributed almost evenly
- Page 69 and 70: also symmetric as seen in the figur
- Page 71 and 72: Chapter 3 Robust Performance Tailor
- Page 73 and 74: through careful and experienced mod
- Page 75 and 76: described above. However, one can r
- Page 77 and 78: ic, σz(�x, �p), that is depend
- Page 79 and 80: Magnitude, OPD/F x [µm/N] Magnitud
- Page 81 and 82: % Energy 100 90 80 70 60 50 40 30 2
- Page 83 and 84: metric to the cost function. Note,
- Page 85 and 86: tion: ∂hi (z,�x, �pi) ∂�x
- Page 87 and 88: values are chosen from their statis
- Page 89 and 90: Table 3.3: Algorithm performance: a
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of freedom in the model. The stiffness and mass matrices for one beam element are:<br />
⎡<br />
Ke =<br />
⎢<br />
⎣<br />
EA<br />
L 0 0 − EA<br />
0 12EI<br />
L<br />
L 0 0<br />
3<br />
6EI<br />
L2 0 − 12EI<br />
L3 6EI<br />
L2 0 6EI<br />
L2 4EI<br />
L 0 − 6EI<br />
L2 −<br />
2EI<br />
L<br />
EA<br />
L<br />
0<br />
0<br />
−<br />
0 EA<br />
L 0 0<br />
12EI<br />
L3 − 6EI<br />
L2 0 12EI<br />
L3 − 6EI<br />
L2 0<br />
6EI<br />
L2 2EI<br />
L 0 − 6EI<br />
L2 ⎤<br />
⎥<br />
⎦<br />
4EI<br />
L<br />
Me = ρAL<br />
⎡<br />
175<br />
⎢ 0<br />
⎢ 0<br />
⎢<br />
420 ⎢<br />
⎣<br />
0<br />
156<br />
22L<br />
0<br />
22L<br />
4L<br />
35<br />
0<br />
0<br />
54<br />
0<br />
−13L<br />
2 0 −13L −3L2 35<br />
0<br />
0<br />
54<br />
0<br />
13L<br />
175<br />
0<br />
0<br />
156<br />
0<br />
−22L<br />
0 13L3L20 −22L 4L2 ⎤<br />
⎥<br />
⎦<br />
(2.2)<br />
(2.3)<br />
where E and ρ are the Young’s Modulus and material density, A and I are the cross-<br />
sectional area and inertia of the beam, and L is the element length. The mass matrix<br />
is of the same form as that used by the finite element program NASTRAN. The<br />
element matrices are assembled into global mass and stiffness matrices, K and M.<br />
The optical elements are simply lumped masses <strong>with</strong>out inertial or optical properties.<br />
They are rigidly fixed to the beam elements and are incorporated by adding the<br />
appropriate mass to the x and y degrees of freedom of the corresponding node in the<br />
global mass matrix. The mass breakdown of the model is given in Table 2.1.<br />
Table 2.1: Mass breakdown of SCI development model.<br />
Component Mass [kg]<br />
Truss 791.7<br />
Combiner 200<br />
Collector (−x) 200<br />
Collector (+x) 200<br />
Total 1391.7<br />
42
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