Jorge Esguerra - 20 January, 2010
DXM2 NCR Feedback
Summary:
The unscheduled increase of the mass of the adaptive secondary mirror will likely result in a decrease in
frequency of the bending and twisting modes of the M2 structure, as well as significant decrease in
damping.
With regards to natural frequencies: Assuming the current first bending mode is between 11-Hz and 15Hz, the expected first bending mode of the M2 structure with the heavier DM will be between 9.8-Hz
and 13.4-Hz. This is not a large change. However, decreasing frequency of the structural modes (by
increasing mass) requires larger forces to add damping or to counter-act disturbances. For example:
Most commercial reaction-mass-based active damping devices do not operate below 10-Hz because of
mass and displacement requirements to generate sufficient forces. Similarly, a passive tuned-mass
damper would require a significant mass itself in order to have a matching low natural frequency (10%,
or 80-kg in this case).
Decreasing the M2 swing-arm first-mode is also problematic in that it is now closer to the predicted (but
never measured) modes of the C-ring extension structure, meaning there are opportunities for greater
amplification of disturbances in either direction.
On the other hand, there is a chance that the observed oscillations in the M2 swing-arm are not the
bending modes of the swing-arm; they could be reaction to what are referred to in the commissioning
documents as “local modes.” (Even so-called “local modes” can have global effects.) For example: The
11-Hz and 13-Hz modes could be the bending modes of the deployment arms, which support the M2
swing-arm and mirror. If this is the case, additional mass of the M2 will not be as detrimental to
vibration performance. But this scenario cannot be confirmed until our measurement hardware arrives.
With regards to damping: The current damping ratio of the M2 structure was calculated from the
LCROSS mission coverage. Damping is estimated to be about 1%, which is extremely low. With additional
mass, the damping will likely drop to lower levels (<1%), and the transient response will last about 26%
longer.
First mode
Rigid M2
  0.008
  0.012
  0.015

5 (<1%)
1.75-s
8.73-s
1.23-s
6.14-s
0.98-s
4.92-s
Deformable M2
  0.0074
  0.0105
  0.0131

2.21-s
11.03-s
(+26.2%)
1.55-s
7.76-s
(+26.2%)
1.24-s
6.21-s
(+26.2%)
5 (<1%)
Increase in
response length
K
3.04e6-N/m
5.66e6-N/m
Mass
804-kg
804-kg
Predicted new first
mode
9.8-Hz
13.4-Hz
Motion Measured on Current SX Secondary Structure in the OPD direction (LBT global Z)
Measurements from the OVMS system indicate that M2 swing arm first mode may already be as low as
10.6-Hz. Vibration measurements by LBTI and LBT indicate that there is a cluster of resonant frequencies
between 10-Hz and 20-Hz. (This accounts for about 90% of measured displacement of M2 in the OPD
direction, as measured by accelerometer A, on the 5-Hz to 100-Hz band. Accelerometer “A” is the M2
OVMS accelerometer located furthest from the C-ring extension measuring along the M2 optical axis.)
Figure 1 Orientation of OVMS accelerometers, from a point in front of LBT, starting through the SX secondary mirror towards
the SX primary mirror.
Band
0-100-Hz
1-100-Hz
5-100-Hz
5-30-Hz
10-20-Hz
10-30-Hz
RMS over band
(from PSD)
81.2-um
57.3-um
48.3-um
46.8-um
43.5-um
45.6-um
Contribution over
1-100-Hz band
100%
84.3%
81.6%
76.0%
79.6%
Table 1 Distribution of motion measured on SX M2 by band.
Contribution over
5-30-Hz band
100%
93.0%
99.5%
Figure 2 Power spectral density plot derived from OVMS SX M2 measurement, and associated forward sum (integral) of PSD.
Figure 3 10000-point FFT (no window) of transient response(s).
Mass Estimates
Mass of the current rigid M2 is estimated at 512-kg, +/- 5%. However this was not measured, as it is not
a facility instrument.
Mass of ARGOS surrogate mass: 125-kg (Matt Rademacher)
Lumped-mass of rigid M2 structure is 512-kg + 125-kg = 637-kg.
Mass of deformable M2: 679-kg, as opposed to 550-kg design mass.
Lumped-mass of deformable M2 structure is 679-kg + 125-kg = 804-kg.
Mass difference, not including swing-arm structure, is 167-kg, or a 26.2% increase over the existing M2
mass.
Assumptions for Napkin-Level Model
1)
2)
3)
4)
5)
The first bending mode is between 11-Hz and 15-Hz and is represented in the data.
Bending modes impacted more severely than twisting modes by additional mass.
First twisting mode is higher frequency than bending modes.
Assumes measured motion is due to swing-arm deflection, and not C-ring extension.
Peaks identified in PSD and FFT plots are assumed to be structural resonances, and not response
to periodic excitation.
6) Lumped mass model; does not take into account mass of swing-arm, so total mass is likely
underestimated--(thus stiffness is underestimated) .
7) Various forces on system lumped to single representative force on mass.
Analysis
1) Use OVMS measurements to estimate first bending mode(s)
a. Currently 11-Hz – 15-Hz based on various measurements by LBTI and by LBT. (Figure 2)
2) Use mass estimate of current rigid M2 and ARGOS surrogate to estimate swing-arm stiffness
a. Structure is modeled as a simple single degree-of-freedom system, with a single pair of
complex Eigenvalues.
b. Representative differential equation (for one pair of Eigenvalues): x 
c
M
x  Mk x  f t 
, where M is total mass (kg), c is viscous damping coefficient (N/m/s), k is stiffness
(N/m), and f(t) is an unknown but representative force disturbance.
c. Define standard damping and natural frequency (rad/s) such that: 2 n 
n 
k
M
c
M
and
d. Using total mass estimates for current M2 (637-kg), and observed first mode (11-15-hz),
estimate stiffness (range) of M2 structure:
First mode estimate
Mass
k
11-Hz
637-kg
3.04e6-N/m
15-Hz
637-kg
5.66e6-N/m
3) Use swing-arm stiffness to estimate static sag for rigid and deformable M2:
K
Mass
Sag, assuming
11-Hz first mode
3.04e6-N/m
637-kg
2.1-mm
5.66e6-N/m
804-kg
2.6-mm
Sag, assuming
15-Hz first mode
1.1-mm
1.4-mm
4) Use swing-arm stiffness to estimate bending modes with deformable M2:
K
Mass
Predicted new first
mode
3.04e6-N/m
804-kg
9.8-Hz
5.66e6-N/m
804-kg
13.4-Hz
5) Estimate damping ratio from OVMS LCROSS measurements using log-decrement method.

x(t ) 
 , where T is the period of one cycle of observed response.
x
(
t

T
)


a. Define   ln 
Figure 4 Transient response of SX M2 as measured by OVMS during LCROSS mission coverage (Oct 2009)
b. From LCROSS coverage, δ is estimated to be between 0.3 and 0.5. In this case, because
of beating in the response, T was assumed to be the period of the beat.
Figure 5 Transient response to mount pointing adjustments during LCROSS mission coverage. Beat frequency is about 2-Hz.
c. Sanity check of damping ratio calculates indicates that damping between 6.5% and 8% is
probably more accurate. (See Figure 5.) However this value is for the combined beat
response only. The individual modes have their own damping.
d. It is reasonable to assume that the transient beating response and the individual
transient modal responses decay at the same rate. Thus damping of the individual
observed modes may be estimated using:  nn   beatbeat
e. Damping ratios of 11-Hz and 13-Hz are very small—about 1%. However as can be seen
in Figure 6 and Figure 7, simulating a simple 4-pole model with 11-Hz and 13-Hz natural
frequencies and 1% damping yields very good comparison with the measured LCROSS
coverage data.
δ
0.289
0.411
0.514
Beat response
damping ratio (ζbeat)
0.046
0.065
0.082
 1 @11-Hz
0.008
0.012
0.015
Table 2 Estimated damping ratio of M2 structure based on step response of LBT.
 2 @13-Hz
0.007
0.010
0.012
Figure 6 Check of estimated damping ratio for beat-response. This is measured data.
Figure 7 Simple 4-ploe model (11-Hz and 13-Hz) with 1% damping exhibits behavior consistent with observed transient
response from LCROSS coverage (Figure 5). The beat decay rate is consistent between the model and the measured data,
which means that the very small damping ratios associated with the Eigenvalues are reasonable.
6) Estimate new damping coefficient using mass estimate of rigid M2 and ARGOS surrogate
a. Using the relationship between the viscous damping coefficient and the damping ratio,
2 n 
c
M
, the viscous damping coefficient c can be estimated. The damping
coefficient is assumed to be a property of the swing-arm structure and materials.
Viscous damping
coefficient, c
 beat  0.046
 beat  0.065
 beat  0.082
729.1-N/(m/s)
1036.7-N/(m/s)
1294.6-N/(m/s)
Table 3 Note that c can also be calculated using the ~1% damping values, along with their respective natural frequencies. The
results will be the same.
7) Estimate damping for deformable M2 using deformable M2 mass, predicted new first modes,
and viscous damping coefficient. (Use expression in 6.a)
C=729.1-N/(m/s)
C=1036.7-N/(m/s)
C=1294.6-N/(m/s)
Predicted first mode
(low, high estimates)
9.8-Hz
  0.0074 , (-11%)
  0.0105 , (-11%)
  0.0131 , (-11%)
13.4-Hz
  0.0054 , (-23%)
  0.0077 , (-23%)
  0.0096 , (-23%)
8) Calculate time constants to estimate decay times. Define  n 
seconds, a signal will have mostly decayed away (<1%).
9)
First mode
Rigid M2

  0.008
  0.012
  0.015
5 (<1%)
1.75-s
8.73-s
1.23-s
6.14-s
0.98-s
4.92-s
Deformable M2
  0.0074
  0.0105
  0.0131

2.21-s
11.03-s
(+26.2%)
1.55-s
7.76-s
(+26.2%)
1.24-s
6.21-s
(+26.2%)
5 (<1%)
Increase in
response length
1
 nn
and note that after 5 n