ik / kmax) * ( fk* / f k) 
] (i=1,.......... ,n k=1, ........,m) [1]
C. Del Vecchio
Arcetri Astrophysical Observatory
Florence, Italy
Proceedings of the ESO conference on Progress in Telescope and Instrumentation Technologies, held in Garching, Germany, ed. M-H. Ulrich, p. 83 (1992.)
Abstract
1. Introduction
2. Description of the Method
3. An Application to a Large Telescope
References
Abstract
A simple iterative procedure for improving static and
dynamic performances of complex mechanical structures is
described. This method uses the results of the stress
analysis of a frame built with only tension and compression
elements to modify the cross sections of
each element in order to better achieve the
complex goal. An example relative to the Columbus
Telescope is presented.
1. Introduction
More and more strict requests are coming from the astronomers regarding
structural stiffness of astronomical instrumentation. In particular, the
dynamic specifications for a telescope are severe, in order to
guarantee that the telescope is stiff enough for the astronomical
use (see [2]). The Finite Element Analysis (FEA) -- the
most versatile analysis technique in engineering -- allows us to
achieve very accurate results when evaluating structural stresses and/or
the dynamic behavior of a structure.
We present here a method called dynamic optimization, that can be used to adjust the characteristics of the finite elements of a model in order to increase the eigenfrequency values, while, if possible, preventing the increase of the structural mass. The main restriction of the method concerns the element type we can work with, because only tension and compression elements, such as trusses and/or membranes, are suitable for this procedure.
The method is presented after a very brief discussion of its basic assumptions. The example reported in the last section demonstrates an application of this algorithm.
of the mass
m equal to 
k/m, where k
is the elastic constant of the spring.
In the FEA of a multidegree system, both springs kj and masses
mj of the j-th element are assumed to be proportional
to the element cross section. The whole idea is therefore
that of finding a criterion selecting, in a given mode, elements
acting mainly as a spring and elements acting mainly as a dead
mass. The criterion we adopted is that of the element
modal strain, i.e. the strain of the element when the
structure is deformed according to the considered modal
shape.In general, FEA codes do not produce strains as output. Nevertheless, they always compute the element stresses. Within the linear theory -- the one usually adopted in the dynamic analysis -- the strain is proportional to the stress. When the elements are simple, e.g. in truss and in membrane cases, the strain is inversely proportional to the cross section, so that the information about the stress/strain can be directly related to the cross area for trusses and the thickness for membranes.
ik
are the modal stress of the i-th element
at the k-th mode of frequency fk.
If f k* is the desired value for the frequency of
the k-th mode, we define a matrix S' as
ik / kmax) * ( fk* / f k) ] (i=1,.......... ,n k=1, ........,m) [1]
where kmax is the maximum value of the k-th
column of S. The matrix S' defined by Eq.[1] identifies the
normalized stresses
| _ | ||
| at frequency f k*. If *imax is the maximum value of each row of S',
and | * | is the mean value of
the n stresses *kmax, |
| _ | ||
| M=[ *imax / |
*] | (i=1, ........ ,n). [2] |
The weights (f k* / f k)
are somewhat arbitrary, and the optimization has a higher
probability of producing improved frequencies for the lowest
frequency modes if one allows a reduction of the frequency for
the highest considered modes.
Because the iterative process is not necessarily convergent, and needs to be continuously monitored by the FE analyst, it terminates as soon as satisfactory results are achieved. The procedure can be summarized as follows:
3. An Application to a Large Telescope
When applying the above procedure to a telescope FE model,
some further complications may occur. As a telescope
rotates, it must be analyzed at several angular
positions -- the restraint locations vary with respect to
the structure. Furthermore, the dynamic response may be
heavily affected by moving loads -- a modern telescope
is a very versatile instrument, accommodating a number of
focal stations, with some components changing their positions on
the structure.
For these reasons, the astronomical application of the method requires that the matrix S in Eq.[1] contains all the geometrical configurations and all the load cases, and that the small variations deriving from Eq.[2] are used as input of a further modal analysis in order to iterate the process.
What follows is an example using only the core algorithm in a single iteration, and takes into account the first three modes of vibration at four zenith angles of an alt-az instrument -- the Columbus Telescope. The FE model is a simplified version of the one described in [1], and is shown in Fig. [1].
Figure 1:The FE model to be optimized.
Similarly, also the frequencies f k of Eq.[1] are defined by the mode number and the zenith angle.
Eqs [1] and [2]
were obtained by means of a Matlab [4]
script file in two different ways, in order to show the
arbitrary choice of the weights defined in Sec. 2.2.
In the first run, = 2 was used in Eq. [1],
in the second one, = 1.
In both cases, the optimization was performed with a goal
of 12 Hz for all modes and positions of the telescope, i.e.
it was imposed f k*=12 for each k in Eq.[1].
Fig. 2 shows the results of the
optimization program using = 1 in the top half
plot and = 2 in the bottom half plot.
The original first three frequencies
(before the optimization) are plotted as dotted lines,
while the optimized ones are plotted as solid lines.
Figure 2: Results of the Optimization.
