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Parameter Estimation for Distributed Parameter Models of Complex, Flexible Structures
Copyright © IFAC Identification and System Parameter Estimation, Budapest, Hungary 1991
PARAMETER ESTIMATION FOR DISTRIBUTED PARAMETER MODELS OF COMPLEX, FLEXIBLE STRUCTURES L. W. Taylor, Jr. NASA Lang/ey Research Center. Hampton. VA 23665. USA
Abstract. Distributed parameter modeling of structural dynamics has been limited to simple spacecraft configurations because of the difficulty of handling several distributed parameter systems linked at their boundaries. Because of this limitation the computer software PDEMOD is being developed for the purposes of modeling. control system analysis. parameter estimation and control/structure optimization, PDEMOD is capable of modeling complex. flexible spacecraft which consist of a 3-dimensional network of flexible beams and rigid bodies. Each beam element has bending in two planes. torsion and axial deformation. PDEMOD was used for parameter estimation for the Mini-MAST truss based on matching experimental modal frequencies and static deflection test data. thereby reducing significantly the instrumentation requirements for on-orbit tests. Keywords. Parameter estimation: distributed parameter systems: modeling. Sup erscripts
NOMENCLATURE
T -1
Symbols A* A. B. C. D B* c C* EA E1x. Ely
stability matrix coeffiCients of functions control matrix model parameter vector observation matrix longitudinal stiffness bending stiffness GA lateral shear G1'I' torsional stiffness F - force distribution function F0 - axial. steady force I - inertia matrix L - length of beam m - mass per unit length PF - force coefficient matrix PM - moment coefficient matrix Qu - deflection coeffiCient matrix Qs - angular deflection matrix r - distance vector t - time T - direction cosine ux. Uy. Uz - beam deflection components U'l' - beam deflection in torsion W - error covariance matrix x. y. z - coordinates in x. y. and z axes ~ - eigenvalue divided by length /) - difference vector between measured and model values cr - real part of the roots e - state vector. coeffiCients of sinusoidal and hyperbolic shape basis functions (() - modal frequency . imaginary part of the roots . angular velOCity vector
1155
- transpose - inverse - differentiation with respect to t differentiation with respect to z
Subscripts c.g.
n x.y. z 'l'
-
mode index center-of-gravity general index x. y. z axis torsional axis. z
INTRODUCTION
Accurate models of the structural dynamics of large . flexible spacecraft are crucial to ensure the stability and performance of flight control systems (Likins. 1971). The common practice is to construct complex finite element models of the structure based on material properties . Enhancements of the models are needed as additional information is obtained from ground and on-orbit tests. Because of the complexity of finite element models of structural dynamics. however. it is of interest to capitalize on the advantages of distributed parameter models by using continuum elements to characterize the structure. One of the reasons for using continuum elements is that distributed parameter models are more suitable for parameter estimation because of the greatly reduced number of model parameters. The theoretical basis on which this homogenization process is based is due in large measure to the work of Noor (1978), Williams (1978). Blankenship (1988), and Sun (1986). Another advantage of distributed parameter models is the elimination of the need to reduce the order of the system for closed-loop control system
generalized beam equations a nd the wave equations for torsion and elongation. The model state variables will be the coefficients of the trigonometric and hyperbolic functions which make up the solution of the system of partial differential equations.
stability analysis. The advantages of distributed parameter models of spacecraft structures have long been recognized but were not practical because of the difficulties in constructing models of complex configurations. Recently. modeling software has been written to ease the task of constructing distributed parameter models of even complex configurations. Poelaert (1983) h a s written the software DISTEL and has used it to model t.he Spacecraft Control Laboratory Experiment (SCOLE) and several satellites. Anderson and Wil!iams (1987) have writted the software BUNVIS-RG and have used it to model threedimensional structures wit.h repetitive geometry. What these software packages lack. however. is the ability to perform parameter estimation from experimental data. The techniques that have been used to date for analyzing on-orbit data have been based on lumped parameter models (Taylor and Williams. 1988). Shen (1990) gives an explicit formulation for a distributed parameter estimator for a cantilevered beam exa mple. Balakrishnan formulates a distributed parameter model for the Spacecraft Control Laboratory Experiment (SCOLE) in reference 11 . A point is soon reached. however. where spacecraft configurations impose a model complexity that is beyond closed form solutions. It is the necd for distributed parameter estimation for complex configurations that PDEMOD was developed.
The beam equations represent (1) Euler bending Stiffness. (2) Timoshenko shear. and (3) axial force stiffness for lateral deflections in the x -z and y-z planes. For bending in the x-z plane: m'll X + Elxux.zzzz + GAux.zzll +Foux.zz = Fx(z.t)
(1)
The equation for bending in the y-z plane is similar. Wave equations are used to represent torsion and elongation.
(2) (3)
The solutions of these partial differential equations for zero damping produce the sinuSOidal and hyperbolic spatial equations which comprise the mode shape functions. For the case that F 0 = O. the bending mode shape in the x-z plane is:
This papcr describes the formulation and the computer program which can model complex three-dimensional flexible structurcs and discusses how distributed parametcrs are estimated based on experimental d a ta for the NASA Mini-MAST truss . A priori information and experimental. frequency response and static deflection data are combined into a single likelihood function using Bayes Rule. The process produces model parameter values which are the most consistent possible given both types of experimental data. An upgrading process of this type which uses modal frequencies instead of time histories has the advantage of not requiring extensive and costly instrumentation.
+ DxcoshJ32xz
(4 )
The mod e shape for bending in the y-z plane has the same form. The mode shape functions for torsion and elongation about the z axiS are: (5) (6)
These undamped mode shapes are expected to be good approximations to the exact solutions for low level of damping. The mode shape of the entire configuration consists of these functions. repeated for each beam element. Because bending in two directions. torsion and elongation are conSidered. a total of 12 coefficients are needed. The vector of coeffiCients is the state vector of the structural dynamics. A vector of the coefficients of these sinusoidal and hyperbolic functions will serve as the state vector.
BASIC FORMULATION
PDEMOD is intended for modeling complex. flexible spacecraft which consist of a threedimensional network of flexible beams and rigid bodies. Each beam has bending in two directions. torSion. and elongation degrees of freedom. The rigid bodies can attach to the beam ends at any angle or body location. The full six degrees-of-freedom are allowed at either end of the beam. There are. of course. unlimited degrees of freedom internal to the boundaries. Any point or angle of attachment is allowed. Although it would be possible. for example. to make a distributed parameter model of the elements in a truss. it is often better to approximate such a truss with a single element by taking advantage of the homogenization process of Noor (1978) and Blankenship (1988).
Under conditions of applied forces it is necessary to include rigid body modes. Their coefficients will expand the state vector accordingly. All deflections. forces. moments. and accelerations will be expressed in terms of such state vectors . The motion of each rigid body is put in terms of the deflection at the point of attachment of a particular reference beam element. The linear and angular deflection vectors can be expressed as:
u· = Qs(z)6
Partial Differential Equations The force and moment vectors for both ends of a single-beam element can be described in terms of the spatial derivatives of the solutions of the 1156
Next. it is necessary to express the forces and moments at either end of the beam elements. The force and moment vectors are:
F attach
Equations of motion are written for each rigid body and the forces and moments imparted by the beams are taken into account. In each case it is necessary to account for the different frames of reference and pOints of attachment. Equations of motion for the linear and angular degrees of freedom for all of the bodies are assembled into a single matrix. In the frequency domain the linear and angular equations of motion are the basis for each block of elements:
Maltach It is also necessary to account for changes in axes from each beam to the body to which it is attached. and for pOints of attachment at some distance away from the center of gravity. The force and moment that a beam-i applies to a body-j are:
= Tbody-j TTbeam-i P F.i(Z)8
Fbody-j Mbody-j
= Tbody-/(Tbeam-iPM.i(Z)
+
Rbeam-i(Z)Tbeam-iPF(Z)J8 The partial differential equations provide the relationships between the modal frequency and the eigenvalues for the mode shape equations. The Euler and wave equations can be solved for the zero damping cases to produce the following relationships between the modal frequency and the wave numbers in the mode shape functions .
For each case in which a rigid body has more than one beam element attached. a constraint equation is added to the system of equations. To account for the continuity in deflection. the constraint equation must be satisfied:
For bending in the x-z plane: The constraint equation for ensuring continuity in the angular deflection is:
where b
=
mco2 /GA
Assembly of the equations of motion and the constraint equations yields the system matrix from which we get the characteristic equation:
+ Fo/EIx.
The case for bending in the y-z plane is similar. For torsion and elongation:
I A(a+jco) I
=D
For cases in which both damping is zero and control is not present. the modal frequencies will be those values of co for which the determinant of the entire system matrix equals zero. COi
Equations of Motion
For the general case which involves damping and/ or control . the modal characteristics are given by the complex roots. s:
A Newtonian or inertial frame of reference is used for the motion of all beam elements and rigid bodies. For example. the point of attachment in the Newtonian axis of a reference. undeflected beam is: Raltach.O
= Rc.g .. o + Tbeam r PARAMETER ESTIMATION FOR THE MINI-MAST TRUSS
For the deflected beam: Rattach.t = Rallach.O + Tbeam u
The Mini-MAST truss is deployable. 66.24 feet long. cantilevered at one end. and has a tip mass at the other end. Figures 5a and 5b show pictures of the truss before and after complete deployment. There are 162 major structural elements not counting joints and corner bodies. Finite element models of the truss involve thousands of elements. The distributed parameter model of the Mini-MAST truss is depicted in Fig. 6. Only two beams and three bodies are needed. The reduced number of unknown parameters give distributed parameter models a distinct advantage for parameter estimation using experimental data. The cost function will. in general. have the form
= Rc.g .. o + Tbeam r + Tbeam u The position of the body center-of-gravity due to beam deflection is:
Rc.g .. t = Rc.g.,o + Tbeam r + Tbeam u- Tbody r For small angular deflections Tbody
= Roots( I AGm) I = DJ
= Tbeam
+ Tilda(Tbeam u')
Substituting and differentiating. we get for the acceleration of the body center-of-gravity:
J
= I((Ymeas.n-Ymodel.n)'IW- 1 (Ymeas,nYmodel.n))
1157
at its base. (2) beam # 1 which runs from the bas. to bay 10. (3) body #2 and bay 10, (4) beam #2 which runs from bay 10 to bay 18. and (5) body #3 at the tip, bay 18. The input file for PDEMO[ requires information concerning the topology of the configuration. The first column of matrix CONFIG identifies the beam. The second columJ identifies the inboard body, and the third columJ identifies the outboard body. A negative sign is used to deSignate the beam which a body uses as its refe rence axis. Le.:
and can be interpreted either as a least-squares or likelihood function . The modified NewtonRaphson technique can be used to determine, iteratively, estimates of the model parameters, Le.: cn+ 1 = Cn+[L((iaYmodell ac)'IW- l (aYmodell ac))J-l *[L((aYmodel/ dC )'IW- l (8 nlll The sensitivity functions were determined by using numerical differences and formulations.
Beam ~
CONFIG =
The model response can often be expressed in terms of an infinite series, partial fraction expansion, Le.: Ymodel(t)
1 2
Inboard Body
Outboar. Body
-1
-2 -3
2
Body 1 uses the inboard end (z=O) of beam 1 as a reference axis . Bodies 2 and 3 use the outboard ends of beams 1 and 2, respectively.
L(CneAn(t-t)x(O) + n=l
In addition. the phySical characteristics of each body and each beam element is required. English units are used. For the first and second beams (#1 and #2, respectively) are:
which is the solution of Xn = An"xn + Bn"Fn
Ll = 66.24"10/18 EIxl = 27565000. Ely! = 27565000 . EAzl = 10530000. GI'l'1 = 2163465 . miLl = .1075 I'l'/LI = .27
The sensitivity function, ay IOc, can be generated by solving similar equations, Le .:
L2 = 66.24"8/18 Elx2 = 27565000 . EIY2 = 27565000. EAz2 = 10530000 . Gi'l'2 = 2163465 m/L2 = .1075 I'l'/LI = .27
The mass characteristics of the first body represent the cantilevered condition at the base of the first beam.
It is necessary. of course . to link the distributed parameters to An and Bn.
mI Ixxl lyy I la I
Because of the high cost. instrumentation may be quite limited for on-orbit tests. Modal frequencies can be determined by only a minimum of instrumentation. For example. it is useful to merge results from static deflection tests and frequency response tests . The cost function can be expressed as :
=
999999999 . 1000000000000000 . = 1000000000000000. = 1000000000000000.
The mass characteristics of the second body located at bay 10 and the third body at bay 18. the tip, are :
J = 8'IW- l 8
m3 = 10.36 Ixx3 = m3 * 1.5 2 Iyy3 = m3*1 .5 2 i zz.3 = ixx3 + Iyy3
m2 = 3.391 Ix x2 = m2* .5 2 lyY2 = m2 * .5 2 Izz2 = ixx2 + Iyy2
am -2(m lO.meas.-m lO.model)2 +
Modal Frequencies of the A Priori Model
am- 2 (ml8.meas.-m l8.model)2 +
PDEMOD was first used to generate the modal frequencies for the a priori model of the MiniMAST truss .
am/L-2 (m/Lmeas.- m / L.model)2
Mode I, 2 3 4, 5 6 7,8 9 10, II 12 13 14, 15
In this way. information is used from static deflection tests and frequency response tests to determine the distributed parameters. El. m/L. mlO. and ml8. The software PDEMOD will be used to estimate these parameters and several other parameters for the Mini-MAST Truss experiment. Input for the A Priori Mini-MAST Model The model of Mini-MAST will consist of (1) body # 1 to represent the cantilevered condition 1158
Jype Bending Torsion Bending Torsion Bending Torsion Bending Torsion Torsion Bending
#1 #1 #2 #2 #3 #3 #4 #4 #5 #5
Frequency-Hz .7577 3 .941 6.506 20.41 28 .05 42.61 46 .27 60 .30 82.80 85 .65
TABLE 1.
Parameter Estimation Values of El were derived from static deflection tests and were treated as measured parameters. The modal frequencies derived from frequency response tests were also treated as measured values. The cost function consisted of a weighted sum of differences squared, i.e.:
I, 2
3 4,5 6 7,8 9 10, 11 12 13 14, 15
where:
I)
Wla priori
W12meas El meas EAmeas GI'I'meas m/Lroeas m18meas mlOmeas I'I'/Lroeas r18meas QOmeas
W12a prtorl Ela priori EAa priori GI'I'a priori m/La priori ml8a priori m lOa priori I'I'/La priori rl8a priori QOa priori
W2a priori W3a priori W4a priori
The inverse error covariance, W- l , is a diagonal matrix with elements of the following form : Wii- l = l/(.Ol·~i,a priori)2
In other words, a standard deviation of 1 percent was used for the modal frequencies and mass characteristics, 2 percent for El and GI'I', and 4 percent for fI8 and rlO .
TABLE 2.
cbest
Modal Frequencies of the Estimated Model The modal frequencies of the distributed parameter model using the estimated parameters are listed below:
Freq. Res .86 3 .9 6.2 22.7 30.9. 43 .5 40. 67.2 67. 69.5
Model .79 3 .86 6 .64 20 .7 28.6 43.7 47.2 61.5 84.8 87.6
Comparison of Model and Measured Modal Frequencies.
Mode I, 2 3 4,5 7 8,9 10 11, 12 13 15 16, 17
Cbest = Ca priori +[dl)/dCTw-ldl)/dC]-ldl)/dCTw-ll) 28620000. 13120000. 1988000 . .1077 9.915 3.397 .235 1.512 .4916
Bending #1 Torsion #1 Bending #2 Torsion #2 Bending #3 Torsion #3 Bending #4 Torsion #4 Torsion #5 Bending #5
Although there is fair agreement in the modal frequencies, the RM.S. error of 14.2% indicates a need for improvement. The errors of the 1st, 4th. and 5th bending modes indicate the need of adding the shear term of the Timoshenko beam equation . The inclusion of shear causes frequencies of the higher modes to be reduced compared to those for the Euler beam. It also became apparent that a condition, perhaps peculiar to the Mini-MAST, was at work. The diagonal struts have hinges which make them oscillate at about 15 Hz. Because the mass of the hinges plus the diagonals represent about 37.5 percent of the total weight of the truss structure. this effect cannot be ignored. Secondary mass equations were coupled to the bending, axial. and torsion equations . The effect of these changes was that a Significant improvement in the agreement with experimental measurements of modal frequencies. The experimental values were also improved through the application of the Eigensystem Realization Algorithm (Pappa, 1990) . Table 2. compares the modal frequencies.
Next, PDEMOD was used to generate the sensitivity matrix, dl)/ik, by calculating the changes in the modal frequencies for incremental changes in the model parameter. Then the "best" model parameter values were obtained from :
El EA GI'I' m/L ml8 mlO I'I'/L rl8 rlO
We
Mode
J = 1)1M- l l)
Wlmeas W2meas W3meas W4meas
Comparison of Model and Measured Modal Frequencies.
we Bending #1 Torsion #1 Bending #2 Torsion #2 Bending #3 Torsion #3 Bending #4 Torsion #4 Torsion #5 Bending #5
Freq. Res .864 4.19 6.15 22.89 31.77 38.06 41.80 51.55 67.27 67.09
Model .8 1 4.20 6 .85 21.57 27 . 18 36.68 43.7 53.39 70.32 73 .5
ThiS experience makes it clear that unless the distributed parameter model has the appropriate structure, no amount of sophistication in parameter estimation can produce an accurate model. Unless the distributed parameter model has little model error, the contention that more accurate models result, may not be the case. The revised equations resulted in improved bending modal frequencies, and the RM .S. error was reduced to 7 ,5 %. The error for the first 5 torsion modes was reduced from 8.5 % to 4.0 %. By comparison the corresponding RM.S . error of a finite element model is 6.4 %. It is clear that the distributed parameter model is somewhat more accurate than the finite element model.
1159
Likins, Peter W.; and Bouvier, H. Karl (1971) . Attitude Control of Nonrigid Spacecraft. Astronautics and AeronautiCS.
CONCLUDING REMARKS
Distributed parameter models of structural dynamics offer considerable advantage over finite element models for parameter estimation because of the smaller number of model parameters. Until recently the work needed to generate distributed parameter models of complex configurations which were also flexible, was prohibitive .
Noor, Ahmed K.; Anderson, M. S.; and Greene, W. H. (1978). Continuum Models for Beamand Plate-Like Lattice Structures. ATAA Journal, 16, pp. 1219-1228. Pappa, Richard; Miserentino, Bob; Bailey, Jim; Elliott, Ken; Perez, Cooper, Paul; and Williams , Boyd (1990). Mini-MAST Control Structure Interaction Testbed: A User's Guide . NASA TM - 102630, March.
Recently, a computer program, PDEMOD, has been developed to aid in generating distributed parameter models of flexible spacecraft. Any configuration which can be modeled by a network of flexible beam elements and rigid bodies can be modeled using PDEMOD. The same software was used for parameter estimation for the NASA Mini-MAST truss using experimental data .
Pappa, Richard S., A1ex Schenk, and Christopher Noli (1990) . ERA Modal Identification Experiences with Mini-MAST. 2nd and Health Monitoring of Precision Space Structures. Poelaert, D. (1983). DISTEL, A Distributed Element Program for Dynamic Modelling and Response Analysis of Flexible Spacecraft. Dynamics and Control of Large Structures. Fourth VPI&SU /AlAA Symposium. B1acksburg, Virginia.
Maximum likelihood estimates of the distributed parameter model were determined using PDEMOD and were based on (1) a priori design information, (2) static deflection test results, and (3) frequency response measurement of the modal frequencies. The resulting distributed parameter model of the Mini-MAST truss approximated both the static deflection data and measured modal frequencies. The accuracy of the model was improved significantly, however, with the use of a Timoshenko beam equation and the inclusion of secondary equation to represent the effects of the diagonal elements and their hinges. The resulting improvement emphasizes the importance of complete model equations.
Shen, Ji Yao; Taylor, Lawrence W., Jr. ;and Huang, Jen-Kuang (1990). An Algorithm for Maximum Likelihood Estimation for Distributed Parameter Models of Large Beam-Like Structures. 2nd USAF/NASA Workshop on Monitoring of Precision Space Structures. Pasadena, California. Sun, C. T.; and Juang, J. N. (1986). Modeling Global Structural Damping in Trusses Using Simple Continuum Models. AIAA Journal. VoJ. 24, No. I , January, Page 144.
The accuracy of the distributed parameter model of the Mini-MAST truss is competitive with a finite element model. Distributed parameter models support optimal parameter estimation because of their greatly reduced number of model parameters.
Taylor, Lawrence W., Jr. (1985). On-Orbit Systems Identification of Flexible Spacecraft. 7th TFAC Symposium on Identification and System Parameter Estimation. York, England.
REFERENCES
Adams, Louis R. (1987). Design, Development and Fabrication of a Deployable/Retractable Truss Beam Model for Large Space Structures Application. NASA Contractor Report, 1 78287. June.
Taylor, Lawrence W., Jr. (1988) . Maximum Likelihood Estimation for Distributed Parameter Models of Flexible Spacecraft. TFAC/IFORS Symposium on Identification and Parameter Estimation. Beijing, China.
Anderson, Melvin S.; and Williams, F . W. (1987). BUNVIS - RG: Exact Frame Buckling and Vibration Program, with Repetitive Geometry and Sub structuring. Journal of Spacecraft and Rockets , Vo!. 24, No . 4, July-August, page 353 .
Taylor, Lawrence W., Jr.; and Tlif£. Kenneth W. (1972). Systems Identification Using a Modified Newton -Raphson Method - A FORTRAN Program. NASA TN D-6734 . WilIiams, F. W.; and Howson, W. P. (1978). Concise Buckling, Vibration and Static AnalySiS of Structures Which Include Stayed Columns. International Journal of Mechanical Sciences, Vo!. 20, No. 8, pp . 513-520.
Balakrishnan, A. V. (1985). A Mathematical Formulation of the SCOLE Problem, Part l. NASA Contractor Report 172581, May. Blankenship, Gilmer L (1988). Applications of Homogenization Theory to the Control of Flexible Structures. IMA Volume 10, Stochastic Differential Systems. Stochastic Control Theory and Application, SpringerVerJag, New York.
1160
PARAMETER ESTIMATION FOR DISTRIBUTED PARAMETER MODELS OF COMPLEX, FLEXIBLE STRUCTURES L. W. Taylor, Jr. NASA Lang/ey Research Center. Hampton. VA 23665. USA
Abstract. Distributed parameter modeling of structural dynamics has been limited to simple spacecraft configurations because of the difficulty of handling several distributed parameter systems linked at their boundaries. Because of this limitation the computer software PDEMOD is being developed for the purposes of modeling. control system analysis. parameter estimation and control/structure optimization, PDEMOD is capable of modeling complex. flexible spacecraft which consist of a 3-dimensional network of flexible beams and rigid bodies. Each beam element has bending in two planes. torsion and axial deformation. PDEMOD was used for parameter estimation for the Mini-MAST truss based on matching experimental modal frequencies and static deflection test data. thereby reducing significantly the instrumentation requirements for on-orbit tests. Keywords. Parameter estimation: distributed parameter systems: modeling. Sup erscripts
NOMENCLATURE
T -1
Symbols A* A. B. C. D B* c C* EA E1x. Ely
stability matrix coeffiCients of functions control matrix model parameter vector observation matrix longitudinal stiffness bending stiffness GA lateral shear G1'I' torsional stiffness F - force distribution function F0 - axial. steady force I - inertia matrix L - length of beam m - mass per unit length PF - force coefficient matrix PM - moment coefficient matrix Qu - deflection coeffiCient matrix Qs - angular deflection matrix r - distance vector t - time T - direction cosine ux. Uy. Uz - beam deflection components U'l' - beam deflection in torsion W - error covariance matrix x. y. z - coordinates in x. y. and z axes ~ - eigenvalue divided by length /) - difference vector between measured and model values cr - real part of the roots e - state vector. coeffiCients of sinusoidal and hyperbolic shape basis functions (() - modal frequency . imaginary part of the roots . angular velOCity vector
1155
- transpose - inverse - differentiation with respect to t differentiation with respect to z
Subscripts c.g.
n x.y. z 'l'
-
mode index center-of-gravity general index x. y. z axis torsional axis. z
INTRODUCTION
Accurate models of the structural dynamics of large . flexible spacecraft are crucial to ensure the stability and performance of flight control systems (Likins. 1971). The common practice is to construct complex finite element models of the structure based on material properties . Enhancements of the models are needed as additional information is obtained from ground and on-orbit tests. Because of the complexity of finite element models of structural dynamics. however. it is of interest to capitalize on the advantages of distributed parameter models by using continuum elements to characterize the structure. One of the reasons for using continuum elements is that distributed parameter models are more suitable for parameter estimation because of the greatly reduced number of model parameters. The theoretical basis on which this homogenization process is based is due in large measure to the work of Noor (1978), Williams (1978). Blankenship (1988), and Sun (1986). Another advantage of distributed parameter models is the elimination of the need to reduce the order of the system for closed-loop control system
generalized beam equations a nd the wave equations for torsion and elongation. The model state variables will be the coefficients of the trigonometric and hyperbolic functions which make up the solution of the system of partial differential equations.
stability analysis. The advantages of distributed parameter models of spacecraft structures have long been recognized but were not practical because of the difficulties in constructing models of complex configurations. Recently. modeling software has been written to ease the task of constructing distributed parameter models of even complex configurations. Poelaert (1983) h a s written the software DISTEL and has used it to model t.he Spacecraft Control Laboratory Experiment (SCOLE) and several satellites. Anderson and Wil!iams (1987) have writted the software BUNVIS-RG and have used it to model threedimensional structures wit.h repetitive geometry. What these software packages lack. however. is the ability to perform parameter estimation from experimental data. The techniques that have been used to date for analyzing on-orbit data have been based on lumped parameter models (Taylor and Williams. 1988). Shen (1990) gives an explicit formulation for a distributed parameter estimator for a cantilevered beam exa mple. Balakrishnan formulates a distributed parameter model for the Spacecraft Control Laboratory Experiment (SCOLE) in reference 11 . A point is soon reached. however. where spacecraft configurations impose a model complexity that is beyond closed form solutions. It is the necd for distributed parameter estimation for complex configurations that PDEMOD was developed.
The beam equations represent (1) Euler bending Stiffness. (2) Timoshenko shear. and (3) axial force stiffness for lateral deflections in the x -z and y-z planes. For bending in the x-z plane: m'll X + Elxux.zzzz + GAux.zzll +Foux.zz = Fx(z.t)
(1)
The equation for bending in the y-z plane is similar. Wave equations are used to represent torsion and elongation.
(2) (3)
The solutions of these partial differential equations for zero damping produce the sinuSOidal and hyperbolic spatial equations which comprise the mode shape functions. For the case that F 0 = O. the bending mode shape in the x-z plane is:
This papcr describes the formulation and the computer program which can model complex three-dimensional flexible structurcs and discusses how distributed parametcrs are estimated based on experimental d a ta for the NASA Mini-MAST truss . A priori information and experimental. frequency response and static deflection data are combined into a single likelihood function using Bayes Rule. The process produces model parameter values which are the most consistent possible given both types of experimental data. An upgrading process of this type which uses modal frequencies instead of time histories has the advantage of not requiring extensive and costly instrumentation.
+ DxcoshJ32xz
(4 )
The mod e shape for bending in the y-z plane has the same form. The mode shape functions for torsion and elongation about the z axiS are: (5) (6)
These undamped mode shapes are expected to be good approximations to the exact solutions for low level of damping. The mode shape of the entire configuration consists of these functions. repeated for each beam element. Because bending in two directions. torsion and elongation are conSidered. a total of 12 coefficients are needed. The vector of coeffiCients is the state vector of the structural dynamics. A vector of the coefficients of these sinusoidal and hyperbolic functions will serve as the state vector.
BASIC FORMULATION
PDEMOD is intended for modeling complex. flexible spacecraft which consist of a threedimensional network of flexible beams and rigid bodies. Each beam has bending in two directions. torSion. and elongation degrees of freedom. The rigid bodies can attach to the beam ends at any angle or body location. The full six degrees-of-freedom are allowed at either end of the beam. There are. of course. unlimited degrees of freedom internal to the boundaries. Any point or angle of attachment is allowed. Although it would be possible. for example. to make a distributed parameter model of the elements in a truss. it is often better to approximate such a truss with a single element by taking advantage of the homogenization process of Noor (1978) and Blankenship (1988).
Under conditions of applied forces it is necessary to include rigid body modes. Their coefficients will expand the state vector accordingly. All deflections. forces. moments. and accelerations will be expressed in terms of such state vectors . The motion of each rigid body is put in terms of the deflection at the point of attachment of a particular reference beam element. The linear and angular deflection vectors can be expressed as:
u· = Qs(z)6
Partial Differential Equations The force and moment vectors for both ends of a single-beam element can be described in terms of the spatial derivatives of the solutions of the 1156
Next. it is necessary to express the forces and moments at either end of the beam elements. The force and moment vectors are:
F attach
Equations of motion are written for each rigid body and the forces and moments imparted by the beams are taken into account. In each case it is necessary to account for the different frames of reference and pOints of attachment. Equations of motion for the linear and angular degrees of freedom for all of the bodies are assembled into a single matrix. In the frequency domain the linear and angular equations of motion are the basis for each block of elements:
Maltach It is also necessary to account for changes in axes from each beam to the body to which it is attached. and for pOints of attachment at some distance away from the center of gravity. The force and moment that a beam-i applies to a body-j are:
= Tbody-j TTbeam-i P F.i(Z)8
Fbody-j Mbody-j
= Tbody-/(Tbeam-iPM.i(Z)
+
Rbeam-i(Z)Tbeam-iPF(Z)J8 The partial differential equations provide the relationships between the modal frequency and the eigenvalues for the mode shape equations. The Euler and wave equations can be solved for the zero damping cases to produce the following relationships between the modal frequency and the wave numbers in the mode shape functions .
For each case in which a rigid body has more than one beam element attached. a constraint equation is added to the system of equations. To account for the continuity in deflection. the constraint equation must be satisfied:
For bending in the x-z plane: The constraint equation for ensuring continuity in the angular deflection is:
where b
=
mco2 /GA
Assembly of the equations of motion and the constraint equations yields the system matrix from which we get the characteristic equation:
+ Fo/EIx.
The case for bending in the y-z plane is similar. For torsion and elongation:
I A(a+jco) I
=D
For cases in which both damping is zero and control is not present. the modal frequencies will be those values of co for which the determinant of the entire system matrix equals zero. COi
Equations of Motion
For the general case which involves damping and/ or control . the modal characteristics are given by the complex roots. s:
A Newtonian or inertial frame of reference is used for the motion of all beam elements and rigid bodies. For example. the point of attachment in the Newtonian axis of a reference. undeflected beam is: Raltach.O
= Rc.g .. o + Tbeam r PARAMETER ESTIMATION FOR THE MINI-MAST TRUSS
For the deflected beam: Rattach.t = Rallach.O + Tbeam u
The Mini-MAST truss is deployable. 66.24 feet long. cantilevered at one end. and has a tip mass at the other end. Figures 5a and 5b show pictures of the truss before and after complete deployment. There are 162 major structural elements not counting joints and corner bodies. Finite element models of the truss involve thousands of elements. The distributed parameter model of the Mini-MAST truss is depicted in Fig. 6. Only two beams and three bodies are needed. The reduced number of unknown parameters give distributed parameter models a distinct advantage for parameter estimation using experimental data. The cost function will. in general. have the form
= Rc.g .. o + Tbeam r + Tbeam u The position of the body center-of-gravity due to beam deflection is:
Rc.g .. t = Rc.g.,o + Tbeam r + Tbeam u- Tbody r For small angular deflections Tbody
= Roots( I AGm) I = DJ
= Tbeam
+ Tilda(Tbeam u')
Substituting and differentiating. we get for the acceleration of the body center-of-gravity:
J
= I((Ymeas.n-Ymodel.n)'IW- 1 (Ymeas,nYmodel.n))
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at its base. (2) beam # 1 which runs from the bas. to bay 10. (3) body #2 and bay 10, (4) beam #2 which runs from bay 10 to bay 18. and (5) body #3 at the tip, bay 18. The input file for PDEMO[ requires information concerning the topology of the configuration. The first column of matrix CONFIG identifies the beam. The second columJ identifies the inboard body, and the third columJ identifies the outboard body. A negative sign is used to deSignate the beam which a body uses as its refe rence axis. Le.:
and can be interpreted either as a least-squares or likelihood function . The modified NewtonRaphson technique can be used to determine, iteratively, estimates of the model parameters, Le.: cn+ 1 = Cn+[L((iaYmodell ac)'IW- l (aYmodell ac))J-l *[L((aYmodel/ dC )'IW- l (8 nlll The sensitivity functions were determined by using numerical differences and formulations.
Beam ~
CONFIG =
The model response can often be expressed in terms of an infinite series, partial fraction expansion, Le.: Ymodel(t)
1 2
Inboard Body
Outboar. Body
-1
-2 -3
2
Body 1 uses the inboard end (z=O) of beam 1 as a reference axis . Bodies 2 and 3 use the outboard ends of beams 1 and 2, respectively.
L(CneAn(t-t)x(O) + n=l
In addition. the phySical characteristics of each body and each beam element is required. English units are used. For the first and second beams (#1 and #2, respectively) are:
which is the solution of Xn = An"xn + Bn"Fn
Ll = 66.24"10/18 EIxl = 27565000. Ely! = 27565000 . EAzl = 10530000. GI'l'1 = 2163465 . miLl = .1075 I'l'/LI = .27
The sensitivity function, ay IOc, can be generated by solving similar equations, Le .:
L2 = 66.24"8/18 Elx2 = 27565000 . EIY2 = 27565000. EAz2 = 10530000 . Gi'l'2 = 2163465 m/L2 = .1075 I'l'/LI = .27
The mass characteristics of the first body represent the cantilevered condition at the base of the first beam.
It is necessary. of course . to link the distributed parameters to An and Bn.
mI Ixxl lyy I la I
Because of the high cost. instrumentation may be quite limited for on-orbit tests. Modal frequencies can be determined by only a minimum of instrumentation. For example. it is useful to merge results from static deflection tests and frequency response tests . The cost function can be expressed as :
=
999999999 . 1000000000000000 . = 1000000000000000. = 1000000000000000.
The mass characteristics of the second body located at bay 10 and the third body at bay 18. the tip, are :
J = 8'IW- l 8
m3 = 10.36 Ixx3 = m3 * 1.5 2 Iyy3 = m3*1 .5 2 i zz.3 = ixx3 + Iyy3
m2 = 3.391 Ix x2 = m2* .5 2 lyY2 = m2 * .5 2 Izz2 = ixx2 + Iyy2
am -2(m lO.meas.-m lO.model)2 +
Modal Frequencies of the A Priori Model
am- 2 (ml8.meas.-m l8.model)2 +
PDEMOD was first used to generate the modal frequencies for the a priori model of the MiniMAST truss .
am/L-2 (m/Lmeas.- m / L.model)2
Mode I, 2 3 4, 5 6 7,8 9 10, II 12 13 14, 15
In this way. information is used from static deflection tests and frequency response tests to determine the distributed parameters. El. m/L. mlO. and ml8. The software PDEMOD will be used to estimate these parameters and several other parameters for the Mini-MAST Truss experiment. Input for the A Priori Mini-MAST Model The model of Mini-MAST will consist of (1) body # 1 to represent the cantilevered condition 1158
Jype Bending Torsion Bending Torsion Bending Torsion Bending Torsion Torsion Bending
#1 #1 #2 #2 #3 #3 #4 #4 #5 #5
Frequency-Hz .7577 3 .941 6.506 20.41 28 .05 42.61 46 .27 60 .30 82.80 85 .65
TABLE 1.
Parameter Estimation Values of El were derived from static deflection tests and were treated as measured parameters. The modal frequencies derived from frequency response tests were also treated as measured values. The cost function consisted of a weighted sum of differences squared, i.e.:
I, 2
3 4,5 6 7,8 9 10, 11 12 13 14, 15
where:
I)
Wla priori
W12meas El meas EAmeas GI'I'meas m/Lroeas m18meas mlOmeas I'I'/Lroeas r18meas QOmeas
W12a prtorl Ela priori EAa priori GI'I'a priori m/La priori ml8a priori m lOa priori I'I'/La priori rl8a priori QOa priori
W2a priori W3a priori W4a priori
The inverse error covariance, W- l , is a diagonal matrix with elements of the following form : Wii- l = l/(.Ol·~i,a priori)2
In other words, a standard deviation of 1 percent was used for the modal frequencies and mass characteristics, 2 percent for El and GI'I', and 4 percent for fI8 and rlO .
TABLE 2.
cbest
Modal Frequencies of the Estimated Model The modal frequencies of the distributed parameter model using the estimated parameters are listed below:
Freq. Res .86 3 .9 6.2 22.7 30.9. 43 .5 40. 67.2 67. 69.5
Model .79 3 .86 6 .64 20 .7 28.6 43.7 47.2 61.5 84.8 87.6
Comparison of Model and Measured Modal Frequencies.
Mode I, 2 3 4,5 7 8,9 10 11, 12 13 15 16, 17
Cbest = Ca priori +[dl)/dCTw-ldl)/dC]-ldl)/dCTw-ll) 28620000. 13120000. 1988000 . .1077 9.915 3.397 .235 1.512 .4916
Bending #1 Torsion #1 Bending #2 Torsion #2 Bending #3 Torsion #3 Bending #4 Torsion #4 Torsion #5 Bending #5
Although there is fair agreement in the modal frequencies, the RM.S. error of 14.2% indicates a need for improvement. The errors of the 1st, 4th. and 5th bending modes indicate the need of adding the shear term of the Timoshenko beam equation . The inclusion of shear causes frequencies of the higher modes to be reduced compared to those for the Euler beam. It also became apparent that a condition, perhaps peculiar to the Mini-MAST, was at work. The diagonal struts have hinges which make them oscillate at about 15 Hz. Because the mass of the hinges plus the diagonals represent about 37.5 percent of the total weight of the truss structure. this effect cannot be ignored. Secondary mass equations were coupled to the bending, axial. and torsion equations . The effect of these changes was that a Significant improvement in the agreement with experimental measurements of modal frequencies. The experimental values were also improved through the application of the Eigensystem Realization Algorithm (Pappa, 1990) . Table 2. compares the modal frequencies.
Next, PDEMOD was used to generate the sensitivity matrix, dl)/ik, by calculating the changes in the modal frequencies for incremental changes in the model parameter. Then the "best" model parameter values were obtained from :
El EA GI'I' m/L ml8 mlO I'I'/L rl8 rlO
We
Mode
J = 1)1M- l l)
Wlmeas W2meas W3meas W4meas
Comparison of Model and Measured Modal Frequencies.
we Bending #1 Torsion #1 Bending #2 Torsion #2 Bending #3 Torsion #3 Bending #4 Torsion #4 Torsion #5 Bending #5
Freq. Res .864 4.19 6.15 22.89 31.77 38.06 41.80 51.55 67.27 67.09
Model .8 1 4.20 6 .85 21.57 27 . 18 36.68 43.7 53.39 70.32 73 .5
ThiS experience makes it clear that unless the distributed parameter model has the appropriate structure, no amount of sophistication in parameter estimation can produce an accurate model. Unless the distributed parameter model has little model error, the contention that more accurate models result, may not be the case. The revised equations resulted in improved bending modal frequencies, and the RM .S. error was reduced to 7 ,5 %. The error for the first 5 torsion modes was reduced from 8.5 % to 4.0 %. By comparison the corresponding RM.S . error of a finite element model is 6.4 %. It is clear that the distributed parameter model is somewhat more accurate than the finite element model.
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Likins, Peter W.; and Bouvier, H. Karl (1971) . Attitude Control of Nonrigid Spacecraft. Astronautics and AeronautiCS.
CONCLUDING REMARKS
Distributed parameter models of structural dynamics offer considerable advantage over finite element models for parameter estimation because of the smaller number of model parameters. Until recently the work needed to generate distributed parameter models of complex configurations which were also flexible, was prohibitive .
Noor, Ahmed K.; Anderson, M. S.; and Greene, W. H. (1978). Continuum Models for Beamand Plate-Like Lattice Structures. ATAA Journal, 16, pp. 1219-1228. Pappa, Richard; Miserentino, Bob; Bailey, Jim; Elliott, Ken; Perez, Cooper, Paul; and Williams , Boyd (1990). Mini-MAST Control Structure Interaction Testbed: A User's Guide . NASA TM - 102630, March.
Recently, a computer program, PDEMOD, has been developed to aid in generating distributed parameter models of flexible spacecraft. Any configuration which can be modeled by a network of flexible beam elements and rigid bodies can be modeled using PDEMOD. The same software was used for parameter estimation for the NASA Mini-MAST truss using experimental data .
Pappa, Richard S., A1ex Schenk, and Christopher Noli (1990) . ERA Modal Identification Experiences with Mini-MAST. 2nd and Health Monitoring of Precision Space Structures. Poelaert, D. (1983). DISTEL, A Distributed Element Program for Dynamic Modelling and Response Analysis of Flexible Spacecraft. Dynamics and Control of Large Structures. Fourth VPI&SU /AlAA Symposium. B1acksburg, Virginia.
Maximum likelihood estimates of the distributed parameter model were determined using PDEMOD and were based on (1) a priori design information, (2) static deflection test results, and (3) frequency response measurement of the modal frequencies. The resulting distributed parameter model of the Mini-MAST truss approximated both the static deflection data and measured modal frequencies. The accuracy of the model was improved significantly, however, with the use of a Timoshenko beam equation and the inclusion of secondary equation to represent the effects of the diagonal elements and their hinges. The resulting improvement emphasizes the importance of complete model equations.
Shen, Ji Yao; Taylor, Lawrence W., Jr. ;and Huang, Jen-Kuang (1990). An Algorithm for Maximum Likelihood Estimation for Distributed Parameter Models of Large Beam-Like Structures. 2nd USAF/NASA Workshop on Monitoring of Precision Space Structures. Pasadena, California. Sun, C. T.; and Juang, J. N. (1986). Modeling Global Structural Damping in Trusses Using Simple Continuum Models. AIAA Journal. VoJ. 24, No. I , January, Page 144.
The accuracy of the distributed parameter model of the Mini-MAST truss is competitive with a finite element model. Distributed parameter models support optimal parameter estimation because of their greatly reduced number of model parameters.
Taylor, Lawrence W., Jr. (1985). On-Orbit Systems Identification of Flexible Spacecraft. 7th TFAC Symposium on Identification and System Parameter Estimation. York, England.
REFERENCES
Adams, Louis R. (1987). Design, Development and Fabrication of a Deployable/Retractable Truss Beam Model for Large Space Structures Application. NASA Contractor Report, 1 78287. June.
Taylor, Lawrence W., Jr. (1988) . Maximum Likelihood Estimation for Distributed Parameter Models of Flexible Spacecraft. TFAC/IFORS Symposium on Identification and Parameter Estimation. Beijing, China.
Anderson, Melvin S.; and Williams, F . W. (1987). BUNVIS - RG: Exact Frame Buckling and Vibration Program, with Repetitive Geometry and Sub structuring. Journal of Spacecraft and Rockets , Vo!. 24, No . 4, July-August, page 353 .
Taylor, Lawrence W., Jr.; and Tlif£. Kenneth W. (1972). Systems Identification Using a Modified Newton -Raphson Method - A FORTRAN Program. NASA TN D-6734 . WilIiams, F. W.; and Howson, W. P. (1978). Concise Buckling, Vibration and Static AnalySiS of Structures Which Include Stayed Columns. International Journal of Mechanical Sciences, Vo!. 20, No. 8, pp . 513-520.
Balakrishnan, A. V. (1985). A Mathematical Formulation of the SCOLE Problem, Part l. NASA Contractor Report 172581, May. Blankenship, Gilmer L (1988). Applications of Homogenization Theory to the Control of Flexible Structures. IMA Volume 10, Stochastic Differential Systems. Stochastic Control Theory and Application, SpringerVerJag, New York.
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