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Douglas R. Neill 8 / 19 / 05

M1. M2. M6. M3. M5. M4. Determination of the Stroke and Stiffness Requirements for the SOAR Primary Mirror Active Tangent Links. Douglas R. Neill 8 / 19 / 05. G. Telescope Front View. PM Tangent Link Arrangement. Summary.

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Douglas R. Neill 8 / 19 / 05

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  1. M1 M2 M6 M3 M5 M4 Determination of the Stroke and Stiffness Requirements for the SOAR Primary Mirror Active Tangent Links Douglas R. Neill 8 / 19 / 05 G Telescope Front View PM Tangent Link Arrangement

  2. Summary • The present stiffness of 9.2e5 Lb/in could be reduced to 4.0e5 Lb/in with minimal effects on the dynamic characteristics of the SOAR primary mirror. (The results for this stiffness are highlighted in green, in the results section) • If the stiffness was reduced below 2.0 e5 Lb/in, the natural frequencies of the primary mirror would decrease rapidly which is undesirable. (The results for this stiffness are highlighted in red, in the results section) • If the stiffness is decreased to the lower value of 2.0 e5 Lb/in, the required stroke to counteract the entire load is still minimal. • Stroke Requirement for K = ~2.0e5 lb / in : D = 0.026 inches.

  3. Introduction • The effective stiffness of every individual link system is a combination of the tangent link stiffness and structural stiffness (the stiffness of the mirror, etc is included in the structural stiffness). These stiffness are added in series, consequently, every link system stiffness must be lower than the lowest series component. • Although all six links are nominally identical, there are two distinct system stiffness. • The load bearing link systems M2, M3, M5 and M6 attach to the piers of the mirror cell assembly near the elevation bearing assemblies. Although all 6 links are nominally identical, the structure near the elevation bearing assemblies is stiffer, consequently, the system stiffness at these locations is stiffer. • The positioning links, M1 and M4, attach to piers midway between the elevation bearing assemblies. The structure is less stiff at these location, consequently, the system stiffness must be less at these locations. • This study only uses classical closed form methods and measured stiffness. • Parallel methods, based on FEA were also used in a separate study.

  4. List of Symbols • D1 = The required stroke (actuator induced displacement) of the positioning links, M1 and M4. • D2 = The required stroke of the load bearing links, M2, M3, M5 and M6. • Fx = Vibration frequency of the primary mirror in the X direction. • Fy = Vibration frequency of the primary mirror in the Y direction. • Kx = The total combined system level stiffness, of all six tangent links systems in the X direction. • Ky = The total combined system level stiffness, of all six tangent links systems in the y direction. • K1 = The nominal system level stiffness of the positioning links, M1 and M4. • K2 = The nominal system level stiffness of the load carrying links, M2, M3, M5 and M6. • KS1 = The nominal stiffness of the structure (and mirror) at the M1 and M4 locations. • KS2 = The nominal stiffness of the structure (and mirror) at the M2, M3, M5 and M6 locations. • L1 = The load requirement for the positioning links, M1 and M4. • L2 = The load requirement for the load carrying links, M2, M3, M5 and M6 • M = The mass of the primary mirror. • X = The direction parallel to the elevation axis. • Y = The direction perpendicular to both the elevation axis and optical axis.

  5. Procedure • The link system level stiffness were measured on the telescope by Mike Warner. Stiffness: Link1 = 4.03e6 Kg/m Link2 = 6.04e6 Kg/m Link3 = 6.29e6 Kg/m Link4 = 4.06e6 Kg/m Link5 = 5.57e6 Kg/m Link6 = 6.52e6 Kg/m

  6. Procedure • Nominal values for the system level stiffness were determined by averaging for the two locations. • K1 for the locations of M1 and M4 • K2 for the locations of M2, M3, M5 and M6 • The system level stiffness is a series combination of the links and structure stiffness. • K1 = 1 / { 1/KL + 1/KS1} • K2 = 1 / { 1/KL + 1/KS2} • Since the links stiffness (KL = 9.2e5 lb/in) and the nominal system stiffness (K1 and K2) are known, the equations must be rearranged to determine the structural stiffness. • KS1 = 1 / { 1/K1 - 1/KL} • KS2 = 1 / { 1/K2 - 1/KL}

  7. Procedure • Once the structural stiffness (KS1 and KS2) have been determined through the tangent links stiffness and the measured system stiffness, new values of system stiffness can be determined by reusing the previous equations, but with a variable tangent link stiffness. • K1 = 1 / { 1/KL + 1/KS1} As a function of KL • K2 = 1 / { 1/KL + 1/KS2} As a function of KL • Load variation requirements (770 Kg) are provided in the initial specs for the tangent link actuators. Load Variations per link in Normal Operation

  8. Procedure • Total load requirements (1720 Kg), which include the weight of the mirror, are also provided in the initial specs for the tangent link actuators. • Although the specs only require that the strokes (D1 and D2) correct the variable load, it would be preferable that the load bearing actuators can correct the total load. As the tangent links become less stiff, the gravity induced decenter will increase proportionally. Consequently, it would be better if the tangent links can recenter the primary mirror. Contributions to Total Link Force (per link)

  9. Procedure • Stiffness, by definition, is force over displacement. • KS1 = L1 / D1 • KS2 = L2 / D2 • Since the system stiffness is known as a function of tangent link stiffness, the load requirement can be used to determine the stroke requirement as a function of system stiffness and, therefore, tangent link stiffness. • D1 = L1 / KS1 • D2 = L2 / KS2 • For the load carrying links, M2, M3, M5 and M6, the stroke (D2) was determined for correcting both the load variation, and the total load.

  10. Results • Values in pounds/in and inches.

  11. Results • Values in Kg/M and mm

  12. Conclusion • The stroke requirements determined in this investigation are the minimum values required to provide the necessary force correction. • There are reasons for using actuators with longer strokes than the minimum values determined in this report. • Excess stroke allows the automated alignment of the primary mirror. • There is some uncertainty in all the analysis and measured values. There are reasons for using actuators with longer strokes than the minimum values determined in this report.

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