Experimental Modal Analysis on the supporting structure of
the secondary mirror of the Large Binocular Telescope

Luciano Miglietta

Osservatorio Astrofisico di Arcetri

L.go Enrico Fermi, 5 - 50125 Florence (Italy)

migliett@arcetri.astro.it

Large Binocular Telescope Project

Technical Memo OAA - 95 - 01 July 14, 1995

ABSTRACT
1. INTRODUCTION
2. PREPARATION OF THE TEST SETUP
3. FREQUENCY RESPONSE FUNCTION (FRF) TESTS SETUP
4. FRF COLLECTION
5. MODAL PARAMETERS EXTRACTION
6. CONCLUSIONS
7. ACKNOWLEDGMENTS
8. REFERENCES

List of tables and figures

ABSTRACT

The final mechanical structure of the Large Binocular Telescope (LBT) has an innovative design in order to achieve the strict scientific and technical requirements. Some of the most critical subassemblies of the telescope are the supporting structures of the secondary and tertiary optics. In detail the supports of the latter mirrors have been drawn as swinging arms mounting the container of the optics at the end of it. The rotating structure of the M2-f15 swinging arm, as that of the entire telescope, has been optimized by extensive finite element analyses to achieve the better compromise between a high modal performances and the minimum optical obstruction and emissivity. In order to validate the finite element model (FEM) results in the presence of real mechanical joints the LBT technical office decided to manufacture a full scale prototype of this swinging arm for testing by an experimental modal analysis . This experimental modal analysis was carried out on a simplified geometry on which we performed a new finite element analysis (FEA) optimization especially for the modal response. In this paper we show the experimental methods and the measurement equipment used in the tests and a brief comparison between different techniques are discussed. The spider in its test configuration has been prepared by Osservatorio Astrofisico di Arcetri (Florence) and the tests have been performed at Dipartimento di Meccanica e Tecnologie Industriali (DMTI - University of Florence) from June to November 1994.

1. INTRODUCTION

The goal of this job is to identify the dynamic response of the structure of the spiders and the positioning linkage with an accuracy sufficient for the LBT project requirements. The main mechanical specification for the spider is devoted to guarantee the stability of the supported optical elements within optical tolerances in the frequency domain between 0.01 and 10 Hz where the wind loads can be the cause of vibration phenomena. Displacements at lower frequency, due to the gravity and to thermal deformations of the spider and of the entire structure of the telescope, can be sensed and corrected by the "active optics" alignment system. The adopted specification is that the lowest resonant frequency should be above 25 Hz for all the spiders, meaning, for instance, that the displacement due to the deflection moving from zenith to horizon should be less than 0.36 mm. Referring to the wind load it is equivalent to a deflection less than 3.6 microns for a mirror of 0.5 ton mass and a diameter of 1.128 m under a wind pressure change of 50 Pa, even if this kind of pressure change should be avoided by windshields under most observing conditions.

The tool used to obtain the dynamic response of the structure has been experimental modal analysis (EMA). As usual for this kind of analysis, the job has been logically subdivided into consecutive separate phases:

2. PREPARATION OF THE TEST SETUP

 In order to reduce its weight the tested structure has been simplified removing most of the horizontal trusses on the lower and upper planes of the frontal section. In Fig. 1a and in Fig. 1b the differences are shown. The structure on which all the dynamic tests have been performed is shown in Fig. 1b. In Fig. 1c the final proposed structure layout is shown: some pretensioned vaned have been introduced in the structure in order to have high local modal frequency with low optical transverse section and low mass ( see ref. [4] ).

The telescope spider, equipped with the motorization, arrived dismantled on May 1994 and has been assembled in an open area behind the DMTI (see pictures) . To reproduce the constraints to which the spider will be subject, it has been fixed to a heavy steel welded structure by means of some spherical joints. The structure obtained has been pulled up by using hydraulic jacks and then supported with very soft air springs that almost reproduce free support boundary conditions. Once balanced with an additional mass (sand sacks) on the part opposite the spider, the structure exhibited all rigid natural oscillations at frequencies around 1 Hz, well below the first elastic natural frequency, which has been found in the following analyses to be around 25 Hz.

3. FREQUENCY RESPONSE FUNCTION (FRF) TESTS SETUP

Two different test setups, with different excitations, have been prepared to verify the existence of sufficient signal (observability problem) to perform the following analyses:

In both cases the accelerations on the structure have been measured by using a set of three mutually orthogonal monoaxial accelerometers mounted onto a small steel cube. A small magnet, with a mass negligible with respect to the structure's one, has been screwed on the cube, thereby allowing a fast and reliable mounting even on painted and not always very regular surfaces. Flat metal adaptors have been glued on to the curved surfaces of the structure, always taking extreme care of the alignment to the chosen reference coordinate system. Since the accelerometers are preamplified with built-in electronics (PCB 303A02 and 353B16, with about 10 mV/g sensitivity), their signals have been fed directly to the instrument (front-end) that is in fact capable to provide the proper power supply.

All signals have been generated (in the case of the shaker) and measured by a 4-channel spectrum analyzer Hewlett & Packard HP 35650 driven by the modal analysis software Leuven Measurement Systems LMS release 3.2 running on an HP 715/50 workstation. When the hammer excitation has been used, acceleration analog signals have been exponentially windowed to reduce leakage effects. The exponential decay has been maintained fixed in all such tests, allowing the direct comparison of the results without any further correction. In the case of shaker excitation no windows have been used since all signals vanish in the acquisition time.

Signals have been digitized and treated with the main acquisition parameters listed hereinafter:

test "hammer" test "shaker" test "stiff"
force type impact burst random impact
force window force uniform force
window parameter 5 % length 100 % length 5 % length
response window exponential uniform exponential
window length 0.749 s-1 exp. decay 100 % length 0.749 s -1 exp. decay

The following time and frequency functions have been observed and computed to guarantee a sufficient quality of the measurements:

Strictly speaking, only FRFs are needed to perform modal parameter estimation and should be stored. Nevertheless, coherence functions are good indicators of the quality of the measurements and have been stored too.

4. FRF COLLECTION

To obtain a sufficient accuracy and readability of modal shapes, the structure has been discretized in 45 measurement points, taking care particularly of the points before and after the spherical joints (Fig.2) .

In the excitation point only the response along the input axis has been measured, while in the remaining 44 points all the three orthogonal accelerations have been measured, giving a total of 133 FRFs and 133 coherences. No attempts have been made to place the accelerometer triad parallel to global reference system, since the rotation through the Euler's angles is much easier via software. In Tab. A1.1 the coordinates and Euler's angles for each point are listed.

Anticipating somewhat the conclusions more extensively illustrated in the next paragraph, the first two analyses made respectively with the hammer and the shaker excitation highlighted a poor (not satisfactory) dynamic behavior of the structure, hence suggesting some structural modifications. These have been simulated via a FEM analysis and then effectively made by welding stiffening bars (trusses) in several points of the spider. A third measurement set has been made to observe the effect of such modifications, but in this case a reduced set of measurement points - located only on the spider - proved to be sufficient to obtain desired information.

Summarizing, the following measurements' sets have been made (in parentheses the name of the test section in the LMS software):

  1. hammer excitation of the original structure - 133 responses (44 xyz + point FRF) (hammer)
  2. shaker excitation of the original structure - 133 responses (44 xyz + point FRF) (shaker)
  3. hammer excitation of the modified structure - 57 responses (19 xyz FRF) (stiff)
The more interesting parameter to be determined is dynamic flexibility, expressed as the ratio of the displacement at a response point divided by the force at an input point at any frequency. Since measurements are made with accelerometers, FRFs have been integrated twice to give the correct x/F ratios. Prior to processing, these FRFs have been stored in three new tests in the LMS project and called respectively hammer2int', shaker2int' and stiff2int' where the suffix 2int' clearly reveals the operation performed. Obviously coherences have not been integrated at all, and they have been simply copied to the new tests.

FRF measurements have been qualitatively and quantitatively compared prior to extract modal parameters. In particular point FRFs show a very high dynamic range. Measurements made with hammer excitation proved to be better than corresponding made with shaker excitation (Figs. 3 and 4), probably since the energy given to the structure was higher in the first case. Even if useful dynamic range was lower, measurements made with shaker excitation have sufficiently high coherence in correspondence of FRF's peaks, thus allowing a good parameter extraction, as verified later. Obviously point FRFs are different, since the excitation point is not the same for hammer and shaker. Another comparison has been made for the same response d.o.f. (namely 2: +Z) where peak levels are comparable. The level of the highest peak is around 1 2 m/N.

An interesting comparison between several FRFs made with hammer excitation is shown in Fig.5 . Here are compared three FRFs at the spider's tip (1:+X), in the middle of an upper arm (6:+Z) and close to a spherical joint (5:+X). The response direction in the same for the three measurements, since d.o.f.s' are expressed in the relative coordinate system, obviously rotated through Euler's angles. The amplitude of the FRF close to the joint is from 1 to 2 order of magnitudes lower then the others, and so we can conclude that this point almost does not move. The coherence for this FRF is clearly not satisfactory, certainly due to an extremely low signal/noise ratio.

A comparison of the imaginary parts of the FRFs are related (hence the one responsible of the modal displacement for light damped structures) for two similar points (2: +Z and 6: +Z, on the two upper arms) with both kind of the excitations. Comparing the FRFs for the same excitation is possible to predict the modal shape at that frequency, while comparing the FRFs for different excitations show the very good agreement of peak levels for the two excitations, even if tests have been made in different times: the structure proved to be time invariant and the experimental procedures proved to be consistent.

Points on the ballast tank that simulates the telescope frame almost show the same level of the spider's joints. Comparisons of FRFs made on the upper arm of the spider (6: +Z), close to a joint but on movement linkage (15: +X) and on the telescope frame (138:+X), showing the last two to be 10 to 100 times lower than the first one.

As previously mentioned, dynamic behavior of the spider were considered not satisfactory, and then structural modifications were planned. After bracing with steel trusses, FRFs measurements were made again and compared to the original ones. Figs. 6 and 7 clearly show that the bracing greatly enhanced the dynamic stiffness of the structure, leaving almost unchanged the global mass. The first peak moved from about 25 Hz up to about 37 Hz, and what's more the first important peak moved at about 47 Hz. Clearly the peak amplitude remained almost the same, since it's governed only by damping, that was not modified at all. This behavior has been considered excellent, since real wind excitation will never have frequency content up to these frequencies.

5. MODAL PARAMETERS EXTRACTION

Modal analysis is the process of obtaining a dynamic model of a structure in terms of a reduced number of information, usually called modal parameters. These consist of natural frequencies, modal shapes and damping coefficients, and are obtained from the FRF measurements fitting' them with a certain number of different algorithms. Since in this work the main goal was to detect with a good accuracy just the first structural modes, no attempt has been made to obtain a full description of the structures in modal terms (for example rigid body modes have been neglected).

Extraction of frequencies and damping ratios has been made by using Least Squares Complex Exponential (LSCE) method that works in the time domain anti-transforming FRFs. Modal shapes (often called residues') are obtained with the Least Squares Frequency Domain (LSFD) method. No details are given here on these algorithms; nonetheless we emphasize that all the results have been obtained with the highest possible accuracy, for example by reducing the bandwidth of FRF considered or by increasing the number of samples in the covariance matrix building process. A great effort has been made to eliminate the so-called computational modes', namely those that are not related to the physics of the system but are just numerical unstable solutions. To do this, visual-numerical tools have been carefully and extensively used, for example the Error Chart and the Stabilization Diagram.

Modal frequencies for the three test setups are shown in Tab. A1.2, while some modal shapes are shown in Figs. 8 and 9. Frequencies are reported here in a more compact form:

hammer shaker stiff
mode # freq
[Hz]		 damp
[%]		 mode # freq
[Hz]		 damp
[%]		 mode # freq
[Hz]		 damp
[%]
01 24.65 0.51 01 24.88 0.66 01 37.85 1.69
02 30.93 1.89 02 31.43 0.72 02 43.82 1.57
03 35.59 0.55 03 36.18 0.78 03 46.73 1.65
04 35.89 0.47 04 39.87 0.24 04 49.45 0.84
05 39.51 0.34 05 42.80 1.25 05 59.83 0.41
06 40.90 0.80 06 53.19 0.84 06 62.70 0.34
07 42.43 0.69 07 60.58 0.60 07 64.71 0.32
08 51.25 0.80 08 65.20 0.99
09 52.97 0.69 09 67.46 0.59
10 60.39 0.34 10 69.08 0.97
11 63.29 1.08
12 64.58 0.26
13 65.66 0.33
14 67.01 0.35
15 68.76 0.43

The analysis clearly evidences that the first mode is a global one in which the whole spider elastically bends, even if the optical axis (represented by the spider's tip) rotates and does not move significantly. The first frequency at which the optical axis significantly deflects is 35.89 Hz. Shaker excitation analysis confirmed this behavior, and results are almost identical.

Despite of this, bracing was planned and made. Extraction of modal shapes showed that the first frequencies are now relative to local modes of the rear spider cross and of the positioning linkage. The first frequency at which the spider's tip rotate around a horizontal axis is now about 50 Hz. Arms bending appears only at 58.9 Hz, proving the effectiveness of bracing solution.

6. CONCLUSIONS

Results obtained with experimental modal analysis proved to be qualitatively and quantitatively excellent. The observability of the structure is good, the information content of excitation and response signal is abundantly sufficient, the reliability of modal parameter extraction methods is accepted, the structure's constraints responds to good modal analysis practice requirements, the structure proved to be time invariant and linear.

Spherical joints used in the linkage and that support the spider behave as rigid connections both on the original and on the modified structure, resulting in equal displacement before and after them in the whole range of interest. The dynamic response demanded in the technical specification have been satisfied with lower resonant frequency higher than 25 Hz required.

Obviously the high quality of the results opens a whole new class of design aspects, typically those that concern with the Finite Element model that could be enhanced just based on these experimental data.

7. ACKNOWLEDGMENTS

This Experimental Modal Analysis was performed on the structure designed by ADS Italia s.r.l. where all the drawings were prepared by the experience of Mr. W. Gallieni. The entire strutture was manufactored in the workshop of Bergomi s.p.a. -Vobarno - ( Brescia - Italy) where the preliminary tests were performed. The modal analysis measurements were carried out under the supervision of Dr. Andrea Bracciali and Prof. Paolo Rissone at the DMTI of the University of Florence. The author wishes to thanks all the people mentioned above and everyone who took part in the discussions of the results during and after the tests.

8. REFERENCES

  1. P.Salinari, C.Del Vecchio, J.M.Hill - "Spiders and Swing Arms: a Summary" - LBT PROJECT Technical Memo OAA-93-02
  2. J.M.Hill - "Telescope Specifications for the Large Binocular Telescope" - Draft of May 12 , 1995
  3. DMTI - University of Florence - Modal Analysis of Large Binocular Telescope Spider - Firenze, December 20th, 1994.
  4. ADS ITALIA srl - Technical Specification of the secondary and tertiary spiders with rotation mechanisms -
    (b401a005 ) - June 30th, 1995

TABA1.1 TABA1.2 Figure 1a Figure 1b Figure 1c Figure 2
Figure3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8a
Figure 8b Figure 9a Figure 9b Picture 1 Picture 2 Picture 3