An approach for augmenting vibration damping characteristics in space structures with large panels is presented. It is based on generation of bending moments rather than forces. The moments are generated using bimetallic strips, suitably mounted at selected stations on both sides of the large panels, under the influence of differential solar heating, giving rise to thermal gradients and stresses. The collocated angular velocity sensors are utilized in conjunction with mini-servos to regulate the control moments by flipping the bimetallic strips. A simple computation of the rate of dissipation of vibrational energy is undertaken to assess the effectiveness of the proposed approach

Cochran, J. E., Jr.

Kumar, K.

English

NASA Technical Reports Server

N87-22743
A NEW APPROACH FOR VIBRATION CONTROL
IN LARGE SPACE STRUCTURES
K. Kumar* and J. E. Cochran, Jr.¢
Auburn University, Alabama
Abstract
A new approach for augmenting vibration damping characteristics in
space structures with large panels is presented. It is based on
generation of bending moments rather than forces. The moments are
generated using bimetallic strips, suitably mounted at selected stations
on both sides of the large panels, under the influence of differential
solar heating, giving rise to thermal gradients and stresses. The
collocated angular velocity sensors are utilized in conjunction with
minl-servos to regulate the control moments by flipping the bimetallic
strips. A simple computation of the rate of dissipation of vibrational
energy is undertaken to assess the effectiveness of the proposed
approach.
*Visiting Professor, Department of Aerospace Engineering, on leave from
I.I.T., Kanpur, India.
CProfessor, Department of Aerospace Engineering.
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https://ntrs.nasa.gov/search.jsp?R=19870013310 2020-03-20T10:32:45+00:00Z
Introduction
Recent trends in space technology have been toward significantly
larger and lighter weight structures. I-3 The successful experimental
mission of Communications Technology Satellite (Hermes) which employed a
pair of solar panels measuring ii x 7.3 m each is already behind us.
One of the Large Space Structures (LSS) currently under development is
the wrap-rlb mesh deployable antenna _,5 with a diameter of I00 m and
surface thickness of 0.0254 mm. Another example is the spinning sall 6
which consists of a spacecraft propelled by photon light pressure
impinging upon 12 blade-like surfaces, each measuring 8 m x 4000 m x
0.00254 mm and made of extra-light material. With Space Transportation
System already in operation, several other space missions demanding
varying sizes of LSS are under consideration for technological
applications such as power generation and transmission, earth-resources
observations and large scale communications related to mass education
and electronic mail systems. 7
Space structures with large dimensions and minimal weight charac-
terized by distributed mass and elasticity properties are extremely
flexible and, thus, are highly susceptible to on-board disturbances
causing surface deformations and vibrations. Since, achieving adequate
performance of LSS generally relies on precise pointing and/or precise
structural configuration, it is necessary to minimize their structural
response to disturbances through judicious design and by incorporating
suitable control devices.
The dynamic analysis, design and control of an LSS poses a
formidable technological challenge, calling for a major all-out effort.
1080
The structural flexibility may interact with attitude control systems of
spacecraft in a variety of ways. A NASA special report 8 has presented
an excellent review of the earlier efforts in thls area. It is
interesting to note that what began as isolated efforts to understand
the anomalous behavior of some specific orbiting spacecrafts by
accounting for flexibility has emerged into a new field of
astroelastlclty. 9 In the early era, there had been considerable
emphasis on developing adequate models 10 of the LSS which could account
for the flexibility of the appendages. Even concise formulations of the
spacecraft structures treating them as "hybrid" assembly of rigid and
flaxibla components seem to represent a significant achlevement.lt, t2
Simulations of the dynamics of spinning and non-spinning systems with
_lexlDie arrays were attempted in several specific cases. The results
of some of these analyses have also been verified by ground testing.
During the last ten years, the emphasis seems to have shifted to
investigations related to the problems involved in controlling the LSS,
the stability of controls and the interaction of dynamics and control
functions.
The methods most often proposed for controls are active ones
although some passive approaches not requiring direct use of energy have
also been suggested. For example, stiffening of an LSS can be utilized
to improve its dynamic characteristics in a number of situations but
only at the expense of severe weight penlaties imposed. Besides,
additional stiffness does not automatically add to structural damping.
A feasible damping technique considered in the past is to incorporate
mechanical dampers such as linear viscous dashpots into the structure.
1081
Even this approach appears to be inadequate and has not gained much
acceptance with spacecraft designers. This is perhaps due to only
marginal improvements in damping characteristics associated with the
rather large weight penalties involved.
Suggested here is a new approach for achieving positive damping
action based on the use of moments rather than forces. In order to
provide for the development of the controlling bending moments, it is
proposed to have a number of moment generating units suitably spread on
both sides of the large spacecraft appendages (Fig. I). A suitable
combination of strips of two different metals forms these units with
constraints at the ends to obstruct any possible relative elongation or
contraction of the strips under differential solar heating. This in
turn generates the tensile and compressive stresses in the two members
giving rise to a bending moment My at the pin through which the whole
unit is mounted on the panel.
For control implementation, the use of an "on-off" controller is
proposed here. The sensors, essentially collocated with the bimetallic
strips are required to measure the local angular velocities of the
appendages. The mini-servo units are utilized to rotate/flip the strips
in "on-off" positions depending upon the direction of the local angular
velocity as indicated by the sensor. While in "on" position, the strips
generate the bending moment through their differential thermal heating
under the influence of the solar radiations. When the positive Z face
of the strips is exposed to the sun and the local angular velocity of
the panel about Y-axis is positive, the corresponding moment about this
axis remains negative causing dissipation of vibrational energy. On the
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other hand, during the vibration phase of negative local angular
velocity, the strips are turned into "off" position by flipping them
through 90° about x-axis when the controlling momentvirtually
disappears. In this sun-orientation, the strips on the other side of
the panel remaining unexposedto the sun do not generate any momentand
may be kept in "off" position. However, when the sun moves to the other
side of the panel, heating the negative Z face of the strips, the moment
generated about the Y-axis becomespositive in the "on" position, thus
requiring the control law to be reversed. Now, the strips are kept in
the "on" position, generating positive momentwhile the local angular
velocity of the panels remains negative. At other times, the strips are
flipped into the "off" position.
Rates of Energy Dissipation
The success of the proposed control mechanism depends upon the
level of rates of vibrational energy dissipation that can be achieved.
The computation of these is the objective of the analysis undertaken
here. It may be emphasized that since the objective is to establish the
feasibility of the proposed concept, only an approximate analysis is
attempted.
The feasibility study is based on a simple cantilever beam model of
an LSS. Several moment unlts/packets are assumed distributed along the
length of the beam. Both the metallic pieces in the packets are assumed
to have rather small cross-sectional dimensions as compared to the
length. Therefore, l-d analysis permitting temperature variation only
along the thickness would be sufficient. In steady thermodynamic state,
the differential equation governing the temperature distribution across
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the thickness of any of the two pieces can be written as (Fig. 2)
d2T
- 0 (i)
d_ 2
where T is the temperature at position _. In order to maximize the
temperature differences, an insulating material is introduced into the
gap between the two pieces. Then, the boundary conditions for the "hot"
piece can be stated as follows:
_ _ _T _ _aS - _eO T_ = 0 at _ : - a t (2)
where
K1 = Thermal conductivity of material for the "hot" piece.
ea,ee = Absorptivity and emislvlty of the metallic
surface, respectlvely.
S = Rate of solar energy received per unit area of the
exposed surface; is same as the solar constant at normal
incidence.
= Stefan-Boltzmann constant.
From Eq. (2), the temperature of the "hot" piece is given by
S I/_
T = ( a
-_'T') (31
c
Since the other piece virtually does not receive any heat
radiations, it can be assumed to be at 0° absolute. Under the
temperature differences thus set up and neglecting the effect of
insulating material on the thermal stresses, the resulting tensile and
compressive forces in the two members can be obtained using geometric
compatibility condition
el AT
F =
(AIEI)-I + (A2E2)-I
(4)
1084
where
AT = (_aS/_eo) I/4
_i = coefficient of thermal expansion; i = 1,2
A i = area of cross section; i ffi1,2
E 1 = modulus of rigidity; i = 1,2
The subscripts I and 2 refer to the "hot" and "cold" members of the
assembly. At 0 ° absolute, the two members in the assembly are assumed
to have been kept in "free-force" condition.
The resulting moment in the steady state is given by
al_(_aS/_eo)I/4
My = (AIEI)_ I + (A2E2)_ l
where 6 = mean distance between the two pieces.
O
Yhe local angular rate of ___,,=_=-_I.... panel to_.,v,_....._o expr_=qed
as follows:
0
8 = _2W
8XSt
(5)
where W ffideflection along Z-axis at the station X.
0
The instantaneous rate of energy dissipation (-E) is then given by
0
(-_.) = - My 8 (6)
The standard modal analysis of the appendage vibrations leads to the
O
following expression for 8.
1085
where
_j = c I (slnh u - sin u) + c2 (cosh u - cos u)
u = (i_/L)X
L = length of the appendage
_j = _jmax sin (_jt+B)
_jmax = amplitude of the jth mode of panel vibrations
_j = (12_2/L 2) a
a --¢El-i7_
i = 0.597 for j=l; 1.49 for j=2; (n-I/2) for j=3,4,...
(El),m = section modulus and mass/length of the appendage,
respectlvely.
On isolating the rate of energy dissipation corresponding to the
jth mode and integrating it, one finds that the average loss of kinetic
energy for this mode per vibration cycle is given by
AE. = 2_16(aaS/ae°)114 (d_j_l _jmax _j--_ (7)
3 (AIEI)_ I + (A2E2)_ i du " X=Li
where Li specifies the position of the ith station where the moment-unlt
is located. This now must be compared to the maximum kinetic energy of
vibrations corresponding to the jth mode (Ej). It is easy to show that
X=L
Ej = (I/2) m_ma x _2j X=0f _ du 4a/mj (8)
On dividing (7) by (8), the loss factor nj is obtained as
4a 1 6(aaS/aeo)i/4 i i (d_j/du) ]X=L i
nj =
(AIEI)-I + (A2E2)-I _jmax _.4EI m X=L
3 X_=O,_ du
108.6
In case the moment generating units distributed along the length of the
appendage are at distances LI, L2,... from the clamped end of the
appendage, the above results modify to:
nj = noj fj (9)
where
noj -- pm/[Ooj _jmax _/Elm]
4al6(aaS/aco)I/4
Pm = (AIEI)-I + (A2E2)-l '
the moment parameter
X--L
fj = [(dSj/du)x=L! + (dSj/du)x=L2 + ---]/X]O¢___=_ du
(to)
(11)
(12)
Results and Discussion
The approximate analytical results developed here are of
considerable significance. Mere observation suggests the effects of
various system and design parameters on the loss factor, denoting the
fraction of kinetic energy dissipated per cycle in various vibrational
modes. The loss coefficient rapidly increases with increase in length
of the appendage and hence decreasing frequency. In fact, this factor
increases in proportion to L2. It also suggests that the damping
effectiveness progressively declines for higher modes of vibrations, as
is usually the case with other damping techniques as well.
It is evident that the vibrational energy dissipation can be
maximized by maximizing the design parameters Pm and fj. The maximiza-
tion of Pm demands the use of materials which provide the highest values
of thermal coefficients, al, and modulil of rigidity, E 1 and E2.
1087
Further increase in Pmcan be achieved by moving the two pieces apart to
maximize the momentarm and of course by taking the higher areas of
cross-section.
The parameters fj merit special attention. By changing the
longitudinal distribution of the moment-strips, it is possible to
augment the damping effects in certain chosen/critical modes. However,
it will result in compromising with dissipation rates associated with
other modes. Rather simple algebraic calculations can be performed to
determine the "best" locations. It is interesting to note that this
approach enables attaining fairly significant values of loss factor even
for higher modes. Since the control effectiveness increases with
decreasing frequency, it appears particularly attractive for LSS.
Conceptually, the approach suggested here is a simple one.
However, it involves several challenging problems at the design stage.
The problems of mounting of the moment-strip units at various stations
on a flexible llght-weight structure, providing for the measuring
sensors and the mini-servos which can control the orientation of the
moment-strlp unit according to the control policy proposed have to be
tackled. The loss factors undergo periodic variation becoming zero
during the periods of grazing solar incidence on the strips. Thus, in
this phase, the dampingmechanismappears to be ineffective. However,
this problem can be overcometo a large extent simply by flipping the
moment-strips through small angles (~30-45°) so as to ensure a
significant differential solar heating even in this situation.
No doubt, the proposed control mechanisminvolves significant
weight penalty associated with the use of the moment-strips and other
1088
accessories; however, it is hoped that a judicious design can keep it to
within acceptable limits. Besides, the addition of this mass may also
serve the secondary purpose of stiffening the system. Furthermore, the
power required to drive the servos is expected to remain at low level.
In a nutshell, this investigation brings out several interesting and
useful features of the proposed scheme to augment the system damping.
1089
l.
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i0.
ll.
12.
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1091
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0
0
C
C
.,H
E
0
<
,-4
1092
I
| |
!
T+J
M
A 1 E 1
A 2 E2
METAL STRIP 1
METAL STRIP 2
Fig. 2. (a) Coordinate System for Thermal Analysis
(b) Geometry of forces and moments in the moment-packet
1093


A new approach for vibration control in large space structures

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