Future extremely large telescopes require both improving imaging resolution and enhancing field of view (FoV), that calls for complex multiple-mirror layouts and advanced adaptive optic (AO) systems. The 39m European Extremely Large Telescope (E-ELT) proposes a novel five-mirror configuration design where the Deformable Mirror (DM) M4 and the Tip-Tilt Mirror (TTM) M5 are embedded within the main structure (Figure 1) for compensating high temporal frequency disturbances induced by atmospheric turbulence and wind buffeting. This built-in AO system exhibits more flexibility, simplifies the instrument complexity and can be operated with other post-focal AO components. Other examples of built-in AO components can be found in deformable secondary mirror of VLT UT4, adaptive secondary mirror module in the Large Binocular Telescope (LBT) and the adaptive secondary mirrors of the Giant Magellan Telescope (GMT).
Despite the known benefits of embedded AO systems, the interaction between the AO control loop and the main structures should be a special concern, since the active components with large inertia (DM M4 and TTM M5) are attached mechanically to the telescope structure (unlike traditional lightweight AO mirrors in post focal platforms), and the resonance frequencies of the telescope structure are significantly lower than that of smaller telescopes; the resonance frequencies scale according to f1 ~ 1/D (Preumont et al., 2009). When compensating the external disturbances, the reactions of the control efforts excite the main structure and may introduce parasitic vibration sources which contaminate the wavefront. Figure 1 (right) gives a schematic view of the tip/tilt control loop and how the reaction torque is applied onto the main structure supporting M1. Notice the mechanical amplification by Iω2.
Figure 1: Left: Five-mirror configuration of the E-ELT telescope.
Right: Interaction of the tilt compensation control system with the telescope structure.
Another distinctive feature of E-ELT as compared to other smaller telescopes is that it will be a wind-limited telescope instead of a seeing-limited telescope. This is because the wind excitation of the telescope structure produces a tip-tilt mode that is more than one order of magnitude larger than the atmospheric tilt. Figure 2 shows both the atmospheric and the wind tilt Power Spectral Density (PSD). The corresponding root mean square (RMS) in arcsec on sky are 1.003 arcsec for the wind and 0.0947 arcsec for the atmospheric turbulence.
Figure 2: PSD of the wind shaking and the atmospheric tilt.
This study discusses the field stabilization control of the M5 unit under wind disturbances. The first part contains an investigation on the spectral content of the atmospheric tilt and the response of large structures to wind loads, which are disturbances observed on the wavefront sensor; the uncertainty on the high frequency decay rate is of particular interest. The second part considers a model of the full telescope; it is used to justify a decoupled design of the M5 control system, and to analyze the response of the primary mirror M1 to the disturbance generated by the control torques of the M5 unit. The strong dependency on the high frequency decay rate of the wind tilt response is pointed out.
It is not clear how the wind disturbance PSD of Figure 2 was obtained; the high frequency decay rate is of particular importance because of the mechanical amplification of the reaction forces driving the high frequency components of the angular correction. This justifies a quick survey of the literature on wind buffeting. The most popular PSD models describing the turbulent wind velocity at a point are the von Karman spectrum, originally developed for airplanes fatigue analysis, and the Davenport spectrum, based on meteorological data. The von Karman spectrum assumes
where γ = urms/Um is the turbulence intensity, ratio between the RMS turbulent velocity urms and the mean velocity Um, and Lu is the turbulence eddy scale (average correlation length). The spatial coherence plays a crucial role in the wind response of very large structures, because the forces acting on the various parts of the structure are gradually uncorrelated as the distance between these parts grows; there seems to be a significant uncertainty on the asymptotic decay rate of the wind disturbance, between f−2.2 and f−13/3 . Its impact on the disturbance torque applied to the primary structure will be investigated below.
In order to estimate the impact of the reaction torque of the TTM (M5) on the telescope structure, a finite element (FE) model (see Figure 3) has been built based on the information available in ESO Official Document; although not accurate, this model may be regarded as sufficiently representative of a 40 m telescope such as the E-ELT for the purpose of this discussion. The primary mirror consists of 714 independent segment units represented by a node array related to the upper surface of the truss system by springs; every node holds a lumped mass of 300 kg and has a single d.o.f. normal to the truss. The spring stiffness is such that the piston mode of the segments is 60 Hz. No tilt of the segments is allowed; they are considered as fully co-phased, in such a way that the surface of M1 is the best fit of the node displacements. The secondary mirror M2 is modeled as a lumped mass (12000 kg) supported by the spider structure with only three degrees of freedom (piston, tip, tilt). The relay mirror M3, the deformable mirror M4, and the TTM M5 are modeled similarly with lumped masses of 12000 kg, 2000 kg, and 500 kg, respectively; they are mechanically connected with the tower structure, which is modeled as a truss. The altitude structure weighs 1500000 kg and the center of gravity. height is 21 m. In the FE model, the truss system is modeled with beam elements with a tube cross section; the Nasmyth platform is modeled using Mindlin shell elements, and non-structural mass is added to obtain 95000 kg on each side. The rotor drive is locked in the zenithal position.
Figure 3: Structural model of a 40m-class ELT telescope.
The material is steel, and the modal damping is assumed to be 0.01 on all modes. The FE model has been developed in SAMCEF. A Craig-Bampton reduction has been performed to produce a reduced model of 1623 d.o.f. (including the 714 vertical displacements of the M1 segments). Figure 4 shows the first few global modes, the results seem to be in line with those available in ESO Official Document.
Figure 4: Typical global modes, deformed sub-structure is highlighted in red. Mode 1: Cross elevation mode (f1 = 3.04 Hz), Mode 2: Locked rotor mode (f2 = 3.17 Hz),Mode 3: Spider rotation mode (f3 = 3.4 Hz), Mode 4: Support frame extension (f4 = 4 Hz), Mode 5 and 6: Support frame distortion modes (f5 = 4.36 Hz, f6 = 4.92 Hz), Mode 7: Vertical pumping mode (f7 = 5.42 Hz), Mode 8: Support frame shear mode (f8 = 5.75 Hz).
The control torques Ms necessary to move M5 act as a disturbance to the telescope, and the spectral content of the disturbing torque depends strongly on the control bandwidth and on the high frequency attenuation rate a of the wind-shaking disturbance. The dynamic response of the primary mirror M1 may be estimated by a least-squares fit of the 714 nodes representing the segments; the wavefront error is twice the mirror surface figure RMS error. The results are summarized in Table 1.
Table 1: Residual tilt error σe (arcsec) and M1 RMS wavefront error σw (nm) and central frequency fw (Hz), for various high frequency decay rate a of the wind tilt excitation (normalized to 1 arcsec) and various control bandwidth fc.
The foregoing analysis shows that if the target of a residual tilt error of 0.07 arcsec is to be met and the decay rate of the wind tilt disturbance is relatively low (a > −3), the wavefront aberration induced by the field stabilization control loop may become significant, with frequency components in the range 30 - 100Hz which may not be easy to eliminate by the Adaptive Optics control loop.
Relevent Papers
K. Wang, A. Preumont, Field stabilization control of the European Extremely Large Telescope under wind disturbance. Applied optics, 58(4), 1174-1184, 2019.
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