ER-85 Wind Force Spectrum for EOST Telescope

Bruce Truax (brucet@pcnet.com)
Frank Scinicariello (franks@eost.com)

Introduction

In order to set the requirements for tower mass and stiffness and servo bandwidth it is necessary to model the telescope pointing errors caused by various disturbances. The most significant disturbance is the force of the wind. The wind force has two components; a steady state force generated by the pressure of the DC component of the wind on the telescope, and a time varying component. The force generated by the steady state component of the wind is given by the equation:

Kolmogorov Model

The time varying portion of the force has been modeled in two ways. For frequencies above approximately 0.10 Hz the Kolmogorov model can be used. This model states that the force generated by the wind on an object is proportional to the frequency to the -7/6 power. Studies on the MMT have verified this model. For a 20 mph wind, Ulich and Woolf found that the torque on the elevation axis obeyed the equation :

The MMT study also took data at three different wind speeds and showed that the torque function was proportional to the square of the mean wind velocity. In addition the paper lists the MMT exposed surface area at 16 square meters and the radius of oscillation of this are as 2.5 meters. Using these pieces of information we can derive an equation for the frequency dependent force per unit area as a function of velocity.

(Sorry about the mixing of units but telescope guys seem to like forces in metric units but wind speeds in miles per hour.)

The wind applies two types of driving forces on the EOST telescope. First, it pushes the telescope and tends to bend the pier. The force spectrum for this force can be found simply by multiplying equation 3 by the exposed area of the telescope which is 5 square meters. The resulting force function is given by equation 4.

The second type of force generated by the wind is a torque on both the elevation and azimuth axes. When the telescope is pointing at the zenith, the sum of the exposed area of the elevation axis times the radius of oscillation is 2.917 . When the telescope is pointing a the horizon a similar number is computed for the azimuth axis. Therefore the worst case torque on each axis created by the force of the wind can be given by Equation 5.

Modified Davenport Model

The problem with the Kolmogorov spectrum is that it goes to infinity at 0 Hz. This can cause problems in some models where the wind spectrum is used. An alternative model is the davenport spectrum. It has the functional form given by Equation 6.

This equation has a breakpoint at V/Ka. Below this frequency the wind power begins to head back down. One fact which is immediately obvious with this equation is that above the frequency break point the force is proportional to . Since the MMT data demonstrated that in this range the wind force is proportional to we have to modify equation 6 to fit the data. We propose the following form for the modified Davenport spectra.

This equation provides the proper scaling of force with velocity while maintaining the proper form of the force spectrum as a function of f. We can now compute C and derive the wind force and torque equations for the EOST telescope. C is found by setting the Kolmogorov and Davenport spectra equal at a particular wind velocity and frequency (above the breakpoint). The resulting equations are:

The graph below shows F(f,V) and FD(f,V) for both 20 and 35 mph.