December 1994
Revised March 1995
1. What is the Question?
2. Summary of the Approach
3. What is the Model?
4. Which Type of Analysis?
5. What is the Forcing Function?
6. Results
7. Conclusions
1. What is the Question?
The main question which we set out to answer was: Does the pier of the
Large Binocular Telescope (LBT) need buttresses or not? But here we
only try to give a part of the answer -- as the exploration turned
into a more general exploration of the dynamical properties of a
telescope mounted on a pier which is not perfectly rigid. In fact the
deflection of the pier due to the wind blowing directly on the
telescope and that due to the wind blowing on the enclosure and
affecting the pier by coupling through the ground has been already
estimated in earlier studies. At the wind speed specified for pointing
and tracking (24 km/hour) the wind low frequency components, say <1
Hz, do not pose requirements on the pier that cannot be fulfilled
by a well designed cylindrical pier. But the wind, especially after
interaction with a complex building, can originate small perturbations
at relatively high frequency (a few, maybe several Hz). We therefore
like to have a control system with a sufficiently wide control band to
compensate this perturbation. The bandwidth of the control system is
affected by the open--loop dynamic response of the whole system,
including telescope and pier. This is why we started this exercise of
computing the open--loop response of the entire system,
telescope + pier.
2. Summary of the Approach
This is an attempt to reorganize in a more logical pattern the evolution of our
ideas, of course without going through all the mistakes and contradictions
that we experienced. Some of the facts became more clear to us only recently.
Warren Davison from UA, Roberto Pozzi and Louis Genovese from ADS,
Gianfranco Rigato and Giorgio Conte from EIE and Ciro Del Vecchio from OA
used in the recent months various models of the pier and telescope to
understand the influence of the pier on the telescope. In these the pier was
represented often by a few beam elements (but in some cases the pier was
fully modeled) and the telescope was represented by various combinations of
beams, lumped masses and moments of inertia, depending on the author and
on the aspects of interest. The interest was in most cases oriented to
estimate the change of the first relevant resonant frequency of the elevation
structure, identified normally with the ``C mode'', assuming that the control
system bandwidth scales directly with this frequency. We all had several
debates on the validity of this assumption. In particular, we
gradually convinced ourselves that this type of simulations risked to be of
little meaning because of the excessive simplification of the telescope.
The objection could be summarized like this: when one excites the telescope
the resulting response depends on the actual distribution of mass and
stiffness, not only on the single parameter ``resonant frequency of the C
mode''. There are many modes, and the frequency of each one depends in a
different way on the rigidity of the connection to the ``ground''. Moreover, the
importance of each mode depends on how each mode is coupled to the
excitation, therefore there is no way of evaluating what happens when we
add the pier to the telescope using only the resonant frequencies. One must
use an appropriate excitation.
It became gradually clearer that two different cases could be considered: one
with El and Az platforms free of rotating about their axes and another one
with the El platform rigidly connected to the Az, and this rigidly connected to
the pier. The two conditions are were very different for two reasons.
200 T, while no special account was taken
of the mass of the Az track.
The pier is restrained at all its bottom nodes, therefore the ground rigidity is
assumed to be infinite.
The pier alone has a first resonant frequency of 14 Hz and a total mass of
2054 T. This makes it very similar to the pier models done by ADS and EIE
with the same geometry.
The whole pier was used as a ``superelement'' in the complete model, because
the wave front limit of 1500 that can be used would be exceeded otherwise.
4. Which Type of Analysis?
A harmonic response analysis was done with the
Ansys® code. More precisely, the
method used is the one called ``Mode Superposition'', that uses the
eigenvectors previously computed by a modal run to evaluate the system
response in terms of amplitude and phase of the XYZ displacements and
rotations at a number of nodes. The Mode Superposition method was
selected because was by far the fastest one once the modal analysis results
were available. In all the analyses a damping factor of 0.02 was used for all
the models. The first twelve resonant modes were used in all cases.
The monitored nodes were five for each side of the telescope, one at the
position of M2 F/15, and four around the central hole of M1. Of the four nodes
of M1 we computed average displacements and rotations. In the last runs we
also monitored four additional nodes on the Az platform and four located at
the top of the pier to disentangle the motion of the pier from the motion of the
mirrors. To reduce the wave front, the pier and azimuth elements have no
rotation degree--of--freedom ( dof), therefore for azimuth and pier
nodes we only report translation dof. A very brief description of
the ``not--rigid--body--motion'' modes at frequencies lower than 10 Hz
is reported in Table 1.
5. What is the Forcing Function?
To evaluate the accuracy of the harmonic analysis we did the static analysis in two ways, one direct and one very indirect, i.e., by taking the zero frequency case of the harmonic analysis. We can then compare the static case with the f=0 one, and this with the one at f=1 Hz, where there should be, and there is, little variation due to dynamics (see Table 2, ``static wind'', and Table 4, ``0 Hz wind'', ``1 Hz wind'', and ``4 Hz wind'', for the points representing the mirrors and Table 3 for the two set of points that reports UX, UY, and UZ at the top of the pier and on the Az platform).
The indirect static analysis is close to the direct one for the largest deformations and rotations (UY and RX in this load case) although differences of several per cent are present, while it is unreliable at least for those cases where there are significant asymmetries for values of deformation/rotation that should be physically the same (e.g., RZ of M1 in this load case).
Appendix 1.
The ratios between quantities computed with
and without pier can be significantly different from 1 even at 0
and 1 Hz, and therefore the effect of the dynamics at 4 Hz must be evaluated
for each significant quantity by using the ratio between the amplitude
ratios at 4 Hz and the the amplitude ratios at 0 Hz, when available, or
at 1 Hz. For
example the largest deformation -- the displacement UY of M2 -- has the same ratio
1.18 at 0 and 1 Hz, while it changes only by 10% at 4 Hz (see
Table 4).
Pier and azimuth platform contribute only 10% of the M2 UY between 0
and 4 Hz. M1 has UY in this load case much less than M2 ( 1/5 with pier,
1/10 without) and a ratio that only changes by 30% between 0 and 4 Hz.
Table 3 and Table 4 show that about
half of the displacement UY of M1 is due to azimuth platform and pier deformation.
The rotation RX of M1 at 4 Hz (with pier) is a factor 1.5 more than at 1 or 0 Hz. If one considers amplitude ratios as a function of frequency, the discrepancy between ``low'' and ``high'' frequency becomes a factor of two, because the amplitude of M1 rotation (without pier) decreases at 4 Hz. This makes this last evaluation of the change, in part, ``virtual''.
Of the four basic numbers that we considered above, more than else to illustrate the way we are looking at these data, three are right and one is marginal or out of range, but probably affected by a larger error. Although it is early to draw firm conclusions based on the agreed rule of ``no more than a factor 1.5 degradation'', the ``simple'' pier seems to affect in a tolerable way the dynamic response to this type of excitation, that is dominated by deformations of the top of the telescope.
200 nm. Such a value has to be compared with
500 nm of M2 and 200 nm
of M1 (radius is similar for the three
sets of points). We have to do something to the telescope rather than to the
pier. In particular we are dealing with a telescope resonant mode that
was never subject to optimization, as we always optimized the locked axes
model.Also in this case a significant loss of symmetry can be noticed. Even if the explanation given in Sec.6.3 can be applied also to this case, a further complication comes from the fact that the first two modes of vibration -- the rigid--body motions -- are not computed as a free rotation around the elevation axis and a free rotation around the azimuth axis, but both of them are a linear superposition of these two modes, so that they are both neither symmetric nor antisymmetric.
Also in this case the preliminary conclusion is therefore that the pier modeled in the present exercise is adequate.
7. Conclusions
We have discussed the frequency response of the system
telescope + pier in two different conditions. In the first
case, we have
applied an ``external'' excitation, simulating the wind, to a
``locked'' telescope -- the elevation and the azimuth axes
are assumed to be kept in their positions by very rigid
drives. In the second case,
we studied the excitation of the drives themselves by
leaving one of the two axes (and also both of them together)
free to rotate.
The first condition represents a situation in which the
telescope is tracking open--loop and a wind affects the
tracking. The second represents a situation in which the
tracking loop is closed on a guiding star and the drives provide
a variable torque accelerating the axes.
The above excitation cases were applied to a telescope
mounted on a perfectly rigid pier and on a pier of the
expected geometry and stiffness. We adopted the ratio
between the ratios of the amplitudes of the two responses
with and without pier to evaluate the effect of the pier. We
assumed that a degradation of performances by a factor of
1.5 with an excitation at 4 Hz was acceptable.
When an external excitation is applied, the response is
dominated by deformations of the top the telescope, and the
pier model considered is entirely adequate. The drive
excitation is more delicate, because the numerical accuracy
is barely sufficient to evaluate the internal deformations
of the telescope, which are much smaller than the rigid--body
rotation of the excited axis. Moreover, the telescope
model that was used had never been optimized with free axes
and consequently the telescope performances were likely to
be not well representative of the final telescope. In spite
of this problems, we can state that the effect of the pier
on the dynamic response is still acceptable.
The worst case is when both axes are free and the excitation
is applied to the azimuth axis. Here, the ratio of the
deformations exceeds our specification of 1.5 by nearly a
factor of two, but we attribute this poor performance more
to a non optimum design of the azimuth platform and
to numerical inaccuracies rather than to the effect of the
torsional rigidity of the pier which is extremely high.
In summary, the pier we modeled,
a cylindrical concrete structure 20 m high, with
stepped wall thickness (five thicknesses, 1 m at
the bottom, 0.6 m at the top), seems to be
adequate to preserve the dynamic response of the telescope,
for both internal and external excitations.
As a byproduct of the pier analysis, we realized that we
have to perform a further optimization of the telescope azimuth
platform.
i = [ uxi/uxi';uyi/uyi';uzi/uzi';rxi/rxi';ryi/ryi';rzii/rzi' ], i = 1, 2, ......... , N/2,
where [uxi ; uyi ; uzi ; rxi ; ryi ; rzi]
are the displacements and
rotations of the node belonging to the i--th pair at X > 0 and
[ uxi' ; uyi' ; uzi' ; rxi' ; ryi' ; rzi' ]
are the displacements and rotations of the node symmetric
to the previous.
The symmetric modes should be the ones having i = [ -1;+1;+1;+1;-1;-1]
while the antisymmetric modes are the ones
whose i is opposite to the previous, i.e.
equal to [ +1; -1; -1; -1; +1; +1].
We define them as S and A, respectively.
The modes defined as S* and A* are the ``quasi--symmetric'
and ``quasi--antisymmetric'' modes, respectively. They are
relative to the two rigid--body motions originated by
leaving free (to rotate) the elevation axis and the azimuth
axis, respectively. In these two cases, the ``theoretical'' i
are [ 0/0;+1;+1;+1;0/0;0/0 ] for S* and
[ +1;-1;0/0;0/0;0/0;+1] for A*.
Modes defined as U are all the others (such modes occur
only when both axes are free, because the
first two modes, both at 0 Hz, are computed as a linear
superposition of the S* mode and of the A* mode).
If we take the average of i for all the N/2 pairs
for all the models, we obtain values that do not match well any
of the above defined theoretical values A, S, A*, and S*. Only
if we heavily filter the computation of the mean value
(i.e., computing the mean value only of those nodes whose
difference with the average is lower than one
half of the standard deviation), or if we take the median
instead of the mean, we obtain values reasonably
close to either 1 or -1. The mean values of the ratios
i are summarized in Table 13,
for each case and for each mode.
Tables 14, 15, 16, 17, 18, 19, 20, 21, 22, and 23 contain the modal contributions
for each excitation, for each type of restraint system (both axes
locked, one axis locked, and both axes locked, respectively) at
1 Hz and 4 Hz. All tables are limited to the first 4
contributions out of 12 for readability.
After a long analysis of the results and many attempts to define a subset of ``relevant nodes'', we have identified the reason of the asymmetry of the results of the harmonic analysis. The key fact is that the autovectors of the dynamic analysis (modes), that are used for the harmonic analysis, are not quite symmetric (S) or antisymmetric (A). They are the sum of a truly S or A autovector and of a ``non random numerical noise'' vector (NN). Such a vector has no well defined symmetry, therefore it is in turn a linear combination of a ``symmetric noise'' (SN) and an ``antisymmetric noise'' (AN).
No problem if the NN vector has components that are much
smaller than those of the autovectors, but in our case
there is a group of components, in particular those
relative to the rotations of the nodes of the primary
mirrors, for which the NN vector components are often
(depending on the particular mode) only one or two orders
of magnitude smaller than the corresponding components of
the ``clean'' modes. This was the origin of the difficulty
in assigning a precise symmetry to the modes.
A good way to visualize the problem is that
of plotting on a semilog scale for each mode the absolute
value of the sum (or the difference) of the displacements
of corresponding nodes versus the node pair number.
Take for example a mode that is supposed to be symmetric.
The sums uxi+uxi', ryi+ryi', rzi+rzi' should be zero, while
uyi+uyi', uzi+uzi', rxi+rxi' should be, in general, different
from zero. If one plots, for instance, the absolute values of the
components of the vector
=
[ uxi+uxi';uyi+uyi';uzi+uzi';rxi+rxi';ryi+ryi';rzi+rzi' ], i = 1, 2, ........., N/2,
one would expect to find one set of horizontal noisy--looking lines
(uyii+uyi', uzi+uzi', and rxi+rxi')
at the top of the plot, while a set of
noisy lines should be found at the bottom of the plot (the
remaining three components). Where one sees big ``jumps'' on
any of the bottom lines, there is a violation of the mode symmetry.
These symmetry violations are present at
various level of significance in essentially all modes, are
always well visible for all the nodes belonging to the
primary mirrors (but not only those), in particular for the
node rotations and are large enough to justify our
results.
The 8th mode of vibration of the telescope (without pier) free
to rotate around the Az axis was selected as a typical
example to illustrate this situation.
Fig.1 shows the absolute values of either the
sums of the displacements (top) and the sum of the rotations (bottom)
versus the pair number (totally, N/2). The jump is clearly
distinguishable. Fig. 2 is
``zoom'' of the previous one in the range of the jump, between
50 and 300 -- the pairs representing the
nodes on the primary mirror. It shows only the sums of
displacements and rotations of
the four monitored nodes on the primary mirrors and on the
secondary mirror, according to the definitions given in Sec 4.
In particular, the bottom plot of Fig. 2 shows that
the three curves, instead of being separated by several
orders of magnitude, are confined in a narrow range, because
they are of the same order of magnitude.
Where and why are these symmetry violation important? After all they have at most a few percent of the amplitude of the ``real'' mode. In fact, in the case where both axes are locked (``wind excitation'') the NN components affect the results by very little, and the expected symmetry is obtained with a sufficient approximation. When one of the axes is free to rotate, and the excitation is directly coupled with this free rotation, typically 99% of the resulting response is in the free rotation. For this reason, we want to detect the response of other modes that are excited at much lower level. In this case, the violation of the mode symmetry can easily affect the quantities we are interested in. A mode orthogonal to the excitation appears to be excited at the ``very significant'' (in this case) level of 1% of the total displacement because of its spurious NN component with, say, 1% amplitude that is not orthogonal to the excitation.
Although all modes, with free Az, free El and with locked axes, are affected by similar NN components that have particularly ``large'' values for the rotational degrees of freedom of the primary mirrors, the effects are different for the different excitations. The locked axis case has been already discussed; why the free Az and the El cases differ so much from each other? Looking at modal excitation table for El one can see that, except for one line, in all cases the main contributions to the amplitude of the dynamic response for all degrees of freedom come from modes of the appropriate symmetry (S modes for S excitation). This is not at all true for the free Az case. Our way of answering the above question is the following. The Az and El cases differ because the coupling between excitations and NN components is significantly different in the two cases. The primary mirrors are very close to the El rotation axis and very far from the Az rotation axis. For a similar angular acceleration (the moments of inertia of the telescope about the X and Z axes are similar, the exciting moments are equal), the moments and the forces acting on the mirrors are very different. In the free El case with RX excitation the NN modes of the primary are coupled with a coefficient that depends on the M1 moment of inertia about its diameter. In the Az case with RZ excitation the coupling depends fundamentally on the moment of inertia of the primary mirror about the azimuth axis, that is much larger than the moment of inertia about its diameter. Therefore, we must expect that the response, with similar NN components, is larger for the Az excitation.
