N.C. Formica/ L. Balis Crema,* C. Galeazzi,* F. Morganti*
"Department of Aerospace Engineering, University
"La Sapienza", Via Eudossiana 16,1-00184 Roma, Italy
*>Alenia Spazio, Via Saccomuro 24,1-00131 Roma, Italy
Abstract
In the present paper a numerical study is conducted on a sandwich panel of atypical communication satellite (ARTEMIS), aiming to investigate thefeasibility of vibration reduction by means of piezo elements. A process ofoptimization of the placement of piezoelectric actuators and sensors has beenconducted on the satellite's sandwich panel. Several analysis cases have beenconsidered in the study, by varying the number of sensors/actuators (4 or 6) andof the modes under control (3, 6 or 8). In order to verify the quality of theoptimization process several numerical simulations have been performed byvarying the sensors/actuators location and feedback gain.
1 Introduction
The need to reduce the weight for the space systems leads, in general, to veryflexible structures. The design solutions and the materials used to minimize theweight usually have the drawback of significant amplification factors, in thedynamic response, due to the low value of the structural damping. Typicalexample are the sandwich panels, which are used both for the primarystructures (which are in charge to carry the loads in the launch phase) and thesecondary structures (which are in charge to accommodate the payload). Dueto the fact that for this kind of structures the structural damping is usually verylow, the necessity to preserve the equipment from excessive environmental(sine/random) levels during the launch phase or to reduce the transmission ofmicrovibrations during the on-orbit phase becomes a driver factor in the systemdesign. In order to reduce the g-levels and hence the mass of the supportingstructures, the structural damping shall be increased; systems based on the use
of piezoelectric sensors and piezoceramic actuators are perfectly suited for thispurpose.
In the first part of the paper, a static FEM analysis is performed on thespacecraft central body, so as to derive the stiffness at the interface (I/F) withthe panel; once extracted these boundary conditions, the modal analysis for thestand-alone panel is performed, so as to identify its dynamic behaviour.
The second part is devoted to the optimization of the placement ofpiezoelectric actuators and sensors on the panel; two cases are studied, for fourand six active elements respectively. The developed methodology, based on thecomputation of the system controllability gramian, is independent from theselected control law. A (Guyan) reduction of the mathematical model wasrequired to reduce the computational effort. The optimal positions have beendetermined in the cases of control of three, six and eight bending modes.
In the third part, a numerical verification of the optimization process isperformed by using a Direct Output Feedback Control (DOFC) law. A set ofnumerical simulations has been performed, by varying either the sensor/actuatorlocation or the feedback gain; the achieved damping is then computed toevaluate the effectiveness of the proposed solution. A new model reduction hasbeen needed for this purpose.
Every reduction process has been verified by comparison between the fulland reduced model modal characteristics. The numerical results of the study arepresented and discussed; they show that the proposed system can besuccessfully used to reduce the vibration environment on lightweight spacestructures.
2 FEM models and analysis of the ARTEMIS satellite
2.1 The global analysis
The application selected for the present study is ARTEMIS (Advanced Relayand TEchnology MISsion), a geostationary communication satellite that willprovide earth mobile and inter-satellite laser communication. The presence ofan optical payload imposes extremely stringent requirements in terms on-boardmicro-dynamic environment. The spacecraft is composed by a main body (M/B)which is in charge to sustain the quasi-static and dynamic loads during thelaunch phase, and by a secondary structure which houses the payloads and theservice subsystems. A view of ARTEMIS is shown in figure 1.
The objective of the study was to increase the structural damping in thoseareas where the high environmental g-levels during the launch could damagesome critical equipment; nevertheless, the proposed strategy is general, so thatit could also be used to absorb the microvibrations generated during the on-orbit operative life. To this end, the South panel (which houses most of theequipment) was selected.In order to reduce the computational efforts, a local analysis of the stand-alonepanel was performed. The first step was therefore to extract from the study ofthe complete M/B the mechanical boundary conditions for the panel. A large
Finite Element Model (FEM) was built using the MSC/NASTRAN 67.5structural analysis program and the PDA/PATRAN 2.5 as pre- and post-processor, running on a mainframe VAX 7000.
Figure 1: The ARTEMIS satellite.
The model is representative of the actual M/B design, and is assembledfrom the sub-models provided from several subcontractors. The main elementswhich can be identified are:• the primary structure, constituted by an Al-C sandwich central cylinder
connected to three horizontal platforms and four vertical shear webs (Al-Al sandwich);
• the secondary structure, constituted by four A1-A1 sandwich verticalpanels;
• two solar array wings of four Al-C sandwich panel each;• two large antenna reflectors in Al-C sandwich;• the Silex optical payload.The total number of degrees of freedom is about 17.500. The South panel isconnected to the other satellite panels and platforms by means of 36 cleats,which ensure high linear and rotational stiffness. In order to evaluate thestiffness in all these locations, a static analysis of the M/B (after removal ofSouth panel) was performed; the M/B has been clamped at the launcher I/F, inorder to simulate the boundary conditions during launch. The equation systemto be solved is:
[K]{u}={P} (1)where{P} is the vector of the applied external nodal forces[K] is the stiffness matrix{u} is the vector of the induced displacementsIn figure 2 the numerical model used for this purpose is shown.
Figure 2: The ARTEMIS satellite Finite Element Model.
2.2 The FEM local analysis
The South panel is a large (1.86m x 2.455m) A1-A1 sandwich plate devoted tosupporting most of the payload and platform equipment, such as one of thebatteries and the thermal control units. It is connected by means of cleats to theEast and West panels along its length, to the Earth Facing panel on its top, andto the South shear web and Main platform in its internal; finally, an horizontalstrut connects its lower edge to the central cylinder.
A Finite Element Model (FEM) was built using the MSC/NASTRAN 67.5code, representative of the actual S/C design, including the mass distribution.The properties and the materials used for the core and the skins are reported inthe table 1 below. The total number of degrees of freedom is 1550; a view ofthe model is reported in figure 3 (the shading correspond to regions with equalmass density).
Table 1 : The ARTEMIS South panel properties and materials.
CoreEi [Pa]100
E,[Pa]100
Vp0.3
Gn[Pa]100
Giz[Pa]138E+6
G,z[Pa]138E+6
FacesE[Pa]71E+9
V0.33
Once extracted the boundary conditions, a modal analysis was performed on theSouth panel in the range [0-200] Hz. The very low damping (£=0.5%) allowsto run a normal mode analysis, producing real eigenvalues and eigenvectors.Thirty-three bending modes were found, with a total effective mass (in direction
Figure 3: The ARTEMIS South panel Finite Element Model.
perpendicular at the plane of the South panel) 89% of the rigid one; anyway,the first four modes dominate the dynamic behaviour of the system; also, theeffective mass (out of the plane) associated to the modes above 100 Hz neverreaches the 1.3% of the rigid one. Table 2 summarizes the results of theanalysis, while in figures 4 to 5 we report two examples of the main modalshapes.
Figure 4: South panel 1st mode. Figure 5: South panel 2nd mode.
3 Optimal placement for the actuators
Following this preliminary characterization phase, the problem of optimalplacement of piezo actuators on the bare structure was addressed; the systemdegree of controllability was chosen as the driving parameter for the optimality;e.g. Hac & Liu* We will assume that the controllability is independent from theselected control law; the system is known to be fully controllable when thecontrollability gramian is non-singular.In the following, only actuators will be mentioned; the sensors are assumed tobe pairwise collocated, so as to avoid spillover and instability problems.
The aim is to determine the optimal positions for the actuators in presenceof a stationary disturbance, e.g. Hac & Liu* . The chosen approach is based onthe evaluation of the amount of energy which is either dissipated or transferredfrom the actuator into the structure.From the operative point of view, the number of modes on which we intend toact needs to be defined as a first step; then, we can define the objective functionto be maximized as the statistical expected value of the total energy (potential +kinetic) to be managed by the actuators:
9=1 (4)In the above equations, the symbols denote respectively:n is the number of modes to be controlledCi is the structural damping (£,= 0.5%)
is the natural frequency of the i-th modeis the total number of actuators to be usedis the position of the q-th actuator (the allotted locations are thoseof the FEM mesh)is the value attained by the Oj eigenmode in the location Pq
The operators sum (I) and product (IT) in eqn (2) ensure the maximization ofthe global energy and the controllability of the single modes, respectively. Ofcourse, the objective function is to be considered effective only for the nselected modes.In the study we have considered several analysis cases, by varying the numberof actuators (4 or 6 for each of the sandwich skins) and of the modes undercontrol (3, 6 or 8).In order to reduce the computational efforts associated with the solutionprocedure, the mathematical model was reduced by means of a standard Guyanstatic condensation, in which only the out-of-plane displacements were retained.In order to verify this reduction process, the modal characteristics associated tothe reduced model were compared to the complete ones.The possibility to have more than one pair of actuators in a node of the meshhas been excluded; the optimal positions achieved in the different cases arereported in the figures 6 to 11.
In order to verify the effectiveness of the actuator placement for the purpose ofthe vibration structural transmission reduction, a Low Authority Control (LAC)has been used; in particular, Direct Output Feedback Control (DOFC) ispreferred, due to its effectiveness and robustness (damping increase andspillover minimization).
The simulations have been performed under the hypothesis of pairwisecollocated sensors and actuators on both faces (internal and external) of thepanel. For what concerns their location, two configurations are investigated,with four and six pairs respectively (see figures 12 and 13).
The FEM mesh has been modified in order to match the active elements withthe plate elements. A Guyan reduction was necessary, retaining 5 dofs (threetranslations, two bending rotations) for all the corner nodes of the activeelements. This new reduction process has been properly checked, too.The numerical simulations have been performed on a PC486, making use of theMATLAB commercial software.The material selection for the sensors and actuators was driven by theirsensitivity, operating frequency range, strain capabilities and Young modulus;piezoelectric (PVDF) and piezoceramic (PZT) are chosen, respectively. Themain characteristics are reported in table 3. The bonding between the plies(sensor-actuator and actuator-skin) has been considered ideal.
Table 3: Sensors (PVDF) and Actuators (PZT) characteristics.
Young modulus En=E]2Shear modulus Gi2
Poisson coefficient 12Density pMax electric field E axStrain piezo constant d^\=d^Stress piezo constant gi3=g23Electric permittivity TJ 11=1122 33
A thickness of 0.25mm has been assumed for both the sensors and theactuators; the dimension of the "active" plate elements is lOxlOcm.
4.1 The open-loop system
AgarwaP) as:We can write the plant and the sensor equations (e.g. Chandrashekhara &
(5)
Q" = {d)le]k[H]{q} ^
and
[H] = L f[H][4f]dAe-lA. (7)
in which:• [M] is the system mass matrix• [C] is the system damping matrix• [K] is the system stiffness matrix• {q} contains all the N dofs• {F} is the consistent external force vector• {Fp} is the consistent piezoelectric force vector• Ng is the number of mesh (plate) elements equipped with sensors/actuators• {d} is the piezoelectric constant vector• Qk is the electric charge generated by the sensor in the laminate k-th ply• [c]k is the stress-strain tensor for the sensor in the laminate k-th ply• [H] is the partial derivative matrix• [0c] is the shape functions matrixWe must now find the expression for the damping and shape function matricesand for the piezoelectric forces and electric charge vectors.
The damping matrix
If we indicate with [T] and with [Q] the eigenvector and eigenvalues matricesfor the undamped system, and we assume a uniform, low damping (£=0.5%) forall the modes, we obtain:
The actuator piezoelectric force vector
The global piezoelectric force vector is given by, e.g. Chandrashekhara &AgarwaP :
A (9)
where the sum is extended to the total number of the model elements. [B]grepresents the matrix relating the nodal vector displacements to the generalized
strain vector; (Np}g represents the vector of the mechanical and piezoelectriccharacteristics of the active elements. Denoting with
: £y Yyz Yxz Yxy %x Zy Zxy]
the generalized strain vector, the following relation holds:
For an N-ply laminate, if we indicate with {V}g and [Z] the vector of thegenerated electric charges and the matrix of the piezo-mechanic characteristicsof the plies, we can write:
N
JF U I J[B]J[Z],{V},dA = [
in which [Y] is the matrix which distributes the piezoelectric forces over thenodes. If the FEM number of dofs is n, the order of the [Y] matrix will be nx
20N&, while the [Fp] and {V} vectors are 20N x2N and 20N&, respectively.
The sensor electric charge vector
With reference to the eqns (6) and (7), we have now to find the relation[H][O ]between the nodal displacements to the strains. If we denote with z° thedistance from the sandwich neutral plane, with {e°} the neutral plane strain andwith (x) the curvature vector, we can draw the expression [H][O] by:
{G)e={G"}e+z\{%},=[H][(De]{%} (13)
By derivation of the charge generated at the sensor electrodes, we obtain thecurrent:
(14)This expression gives, for the two sensors:
i'.i-ft}-41L NJ I Jg (15)
where [L] is 2x20. The global vector can therefore be written as:
(16)
The assembly matrix [E] is 2N x20N [Y], which extracts from the global
vector (q j those relevant to the sensors, is 20N%xn.
Collocated sensors and actuators have been used; the control law is a simplerate feedback, so as to increase the damping of the system. The input controlvariable for the actuators is the tension V(t), which is linearly related to thesensors output current:
(17)
in which [G] is the (diagonal) feedback gain matrix. From this, together withthe other equations:
[M] qL[C] q +[K]{qHF}+[Y][Fp]{V}L > ^ > (18)
(19)the closed-loop system equation can be written as:
[M] q(20)
or, denoting with [CJ the equivalent damping matrix generated by thefeedback:
(21)
It is useful to remind here that the [M] and [K] matrices are computed takinginto account the presence of the sensors and actuators.
5 Numerical results
5.1 State space equations
The study of the system described by the eqn (21) is easily performed by usingthe state space notation for the equations:
x =[A]{x}
in which:
• {x}={q q }T is the state vector
• {y}={ V} is the output vector• {u}={F} is the input vectorIn order to verify the capability of the proposed sensor/actuator disposition tomodify the structural behaviour (mainly, by increasing its damping), thehomogeneous system:
has been studied.Once extracted the system eigenvalues:
(23)
(24)the damped frequencies and the damping factors can be easily computed as:
f co
-a
(25)The main objective of the numerical analyses was to verify the effectiveness ofthe above described optimization study for the sensors and actuators location.
5.2 System equipped with four sensors/actuators pairs
The dynamic parameters achieved by the system equipped with four pairs ofsensors/actuators are shown in table 4 and in figure 14, where frequencies anddamping factors are presented as a function of the feedback gain.The damping coefficient increases with the feedback gain, as expected. In orderto verify that the selected locations are "optimal" for what concerns the increaseof damping, the analysis was repeated for three other dispositions of the activeelements, as shown in figure 15. For simplicity, the feedback gain was fixed to1000 V/A.
Table 4: Dynamic parameters for the system with optimally placedsensors/actuators.
No contr. sfrt D Contr. sist (Gt-SOO V/A) I No contr. *kt DContr.sist{Gi-1000 V/A)
| |j
O 1'J,#ldl^l5i; lifllir r r 4* s* r r r
Mod*• No contr. sist D Contr.ttrMGN2500 V/A)|
i$
411-52 1,211 0.5T-1 O.V-, O.S1,>W 1 ]»
r r 3' 4* s* r r @Mode
No contr. sist O Contr.*kt(OMOOO V/A)|
Figure 14: Dynamic parameters for the system with optimally placedsensors/actuators.
The comparison between the "optimal" and the other configuration is given intable 5, where we report the percentage differences of the damping factor.These results confirm that the optimization process is effective, apart from the2nd and 4th modes for the 2nd and 3rd configurations, respectively (in grey inthe table). However, the global results (in particular focusing on the first 4modes) show that the optimal configuration is really superior to the alternativeones.
Configuration 1 Configuration 2 Configuration 3
Figure 15: Alternative sensor/actuator dispositions.
Table 5: Damping for optimal and alternative configurations: percentagedifferences.
Configuration 1
;A(%)
0.95-23
0.86-17.3
0.78-22.
0.55-23.6
0.96-57.1
0.79-22.5
0.51-1.9
0.57
-20.8
Configuration 2
;A(%)
0.84-31.7
.1.76+69,
0.78
-22.
0.56
-22.2
1.00
-55.3
0.89-12.7
0.53
+ 1.9
0.58
-19.
Configuration 3
;A(%)
0.90-26.8
0.70-32.7
0.85
-15.0.92+27.7
1.21-46.
1.54
-50.10.56+7.7
0.63-12.5
5.3 System equipped with six sensors/actuators pairs
The table 6 shows the results achieved applying six pairs of sensors/actuatorswhich are disposed according to the optimal and alternative (figure 16)configuration; in the table are riported the percentage differences between theoptimal and alternative configuration, too. In the numerical simulation we haveutilized a feedback gain of 1000 V/A. With special care for the mode withgreater associate modal mass, we can observe that the results show again thesuperiority of the optimal configuration.
Figure 16: Alternative disposition with six sensors/actuators pairs.
Table 6: Damping for optimal and alternative configuration: percentagedifferences (6 pairs).
Optimal configurationFreq.
C
32.41.21
33.31.53
38.81.75
47.40.79
57.52.2
61.21.07
66.80.52
88.10.83
Alternative configuration
;A(%)
0.83
-31.4
1.23-19.6
0.86
-50.8
0.83
+5.0
0.95
-56.8
1.88
+75.7
1.17
+125
1.12
+34.9
6 Conclusions
In the paper a strategy to address the topics related to the vibrations of 2-D,low-damped structures is presented. The study case is a typical sandwich panelof a geostationary satellite (ARTEMIS), for which the management of bothhigh- and low-level vibrations is a key issue.
Starting from standard FE analyses (static and dynamic), the optimallocations for active damping elements is determined, based on the systemintrinsic dynamic behaviour; piezoelectric sensors and actuators are chosen forthis scope, connected in a simple and robust control loop which allows toevaluate the effectiveness of the selected locations. Several analyses areperformed, by varying the number of actuator pairs (4 or 6), the number ofmodes under control (3, 6 or 8) and the feedback gain; the results confirm thatthe structural damping can be significantly increased (up to 4 times), withminimum impacts in terms of added mass.
The main limitations of the proposed study are those relevant to thedimension of the analysis case: by increasing the number of modes to becontrolled or of actuators to be placed, the computational effort associated withthe processes of optimal locations identification and with solution of the closed-loop system equations becomes unpractical. The study considers a direct outputfeedback law; if more complex control laws are to be used, it can be foreseenthat strong restrictions could apply to the size of the problems to be solved.
References
1. A. Hac, L. Liu. Sensor and Actuator Location in Motion Control of FlexibleStructures, Journal of Sound and Vibration (1993) 167(2), 239-261.
2. K. Chandrashekhara, A. N. Agarwal. Active Vibration Control of LaminatedComposite Plates Using Piezoelectric Device: A Finite Element Approach,Journal of Intelligent Material System and Structures Vol.4-October 1993.
3. C. Galeazzi, F. Morganti, C. Arduini and P. Gaudenzi. Analysis and Controlof Microvibrations on ARTEMIS Satellite, VI International Conference onAdaptive Structures ICAS '95 - Key West - Florida, November 1995.
4. N. C. Formica, L. Balis Crema, C. Galeazzi, F. Morganti. StruttureIntelligent!: Applicazione sul satellite ARTEMIS, Thesis in AeronauticsEngineering, July 1995.
