1

Design of Structures for Proportional DampingApproximation Using the Energy Gain

Cornel Sultan1

Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

The problem of designing structures such that accurate proportional damping approximationmodels are obtained is addressed. For this purpose an error system is defined using the differencebetween the generalized coordinates of the non-proportionally damped system and its proportionally

damped approximation and the structure is designed such that the energy (or 2L ) gain of the error

system is minimized. An algorithm which combines linear matrix inequalities techniques and stochastic

approximation is proposed to solve the 2L gain minimization problem and an example of its

application to a tensegrity structure is presented.

NomenclatureM = mass matrixC = damping matrixK = stiffness matrixq = generalized coordinates vectorf = external actions/signal vectorU = modal matrixqm = modal coordinates vectorqp = proportionally damped system coordinates vectorc1 = tendon damping coefficientmi = inertial parameterski = elastic parameters

I. IntroductionNLIKE inertial and stiffness characteristics, which can be easily measured in static conditions, damping, adynamic characteristic, is more difficult to quantify. Hence, in many cases the artificial Rayleigh damping

model, which assumes that the damping matrix is a linear combination of the mass and stiffness matrices, is used.Rayleigh damping (or a generalization of it1) is preferred because it leads to the ideal situation of a proportionallydamped linear model of the structure’s dynamics, but it is neither a physics based nor a data based model.

When the source of damping can be identified and accurately modeled using physics principles the Rayleighdamping assumption (or any artificial damping model for that matter) is not recommended. One example is that oftensegrity structures: the major damping sources can be easily identified, the joints and the tendons2-4, and for theseelements reliable physics based damping models can be built. However, in most cases the resulting linearizeddynamics models are not proportionally damped3-4. In general, the likelihood of obtaining non-proportionallydamped models will increase due to our enhanced ability to accurately model damping using physics principles.

Even if a system is not proportionally damped, one would still like to be able to approximate it with aproportionally damped system. For structures, usually described using models with many degrees of freedom, suchmodels are very advantageous because they allow the replacement of non-proportionally damped models withdecoupled models, which can be easily used for control design, fast computations, etc.5

This paper, which is strongly related to Ref. 6, pursues the idea of designing the structure such that it yields alinearized dynamics model that is “close” to a proportionally damped one. In Ref. 6 the design of structures forproportional damping approximation was investigated by exploiting only one, indirect factor that influences theaccuracy of the approximation, namely the separation between natural frequencies. It was ascertained that separation

1 Assistant Prof., Aerospace and Ocean Engineering, 215 Randolph Hall, AIAA Senior Member.

U

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th4 - 7 May 2009, Palm Springs, California

AIAA 2009-2427

Copyright © 2009 by Cornel Sultan. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2

between natural frequencies might be a misleading criterion, because increased separation does not necessarily leadto more accurate approximation. In this paper a global, direct measure of the accuracy of the approximation is used.For this purpose, a proportionally damped linear system is built by removing the off-diagonal terms from the non-proportionally damped modal damping matrix and an error system is defined, which uses the difference between thegeneralized coordinates of the two systems. The “distance” between the two systems is characterized using

the 2L gain (also called the energy gain), which is directly related to the approximation error. The 2L gain has a

major advantage in that it can be computed using Linear Matrix Inequalities (LMI), for which efficient numericalsolutions exist. An algorithm that uses LMI and Simultaneous Perturbation Stochastic Approximation (SPSA) is

proposed for the minimization of the 2L gain of the error system and applied to a tensegrity structure.

II. Proportionally Damped SystemsThe linearized dynamics of many structures is described by

0,0,0, >≥>=++ KCMfKqqCqM &&& (1)

where M, C, K are the mass, damping, stiffness matrices, respectively, q is the n-dimensional vector of generalizedcoordinates, and f is a vector due to the external actions (e.g. generalized forces). A transformation from the physical(q) to the modal (qm) coordinates is performed using the modal matrix, U, as follows:

0.,, >=Λ=Λ= diagIUUUUM MTMM

TMMM

0,,211>=Ω=Ω=ΛΛ

−−diagIUUUUKUU T

KKTKKMM

TMM

.,1

mKMM UqqUUU =Λ=−

(2)

Using (2), in modal coordinates (1) becomes

fUqqCq Tmmmm =Ω++ 2&&& (3)

where CUUC Tm = , )(diag 22

lω=Ω , and lω are the structure’s natural frequencies obtained by solving

0)det( 2 =− MK lω . (4)

The system (1) is proportionally damped if mC is diagonal. There are tremendous benefits if mC is diagonal.

For example the equations of motion (3) decouple and they can be easily solved independently. Proportionaldamping models also lend themselves easily to computationally efficient identification, model order reduction andcontrol design tools5. Previous research1 indicated that if the damping matrix is

∑−

=

−=1

0

1 )(n

i

ii KMMaC (5)

where ia are real numbers, the modal damping matrix is diagonal; (5) is a generalization of the Rayleigh damping.

If mC is not diagonal it is desired to obtain a sufficiently accurate proportionally damped approximation of the

initial system. The simplest approach5 is to neglect the off diagonal terms in mC , thus defining

).( mp CDiagC = (6)

The proportional and non-proportional damping models are then

3

02 =Ω++ pppp qqCq &&& (7)

0)( 2 =Ω+++ mmnpm qqCCq &&& (8)

where 0)(, =−= npmn CDiagCCC .

Previous research focused considerably on identifying the individual factors related to the modal system (8) thatenable accurate proportional damping approximation, like the natural frequencies and the modal damping matrix.Over the years sufficient separation between natural frequencies has been identified as a crucial factor for suchapproximation7-9. However, several publications6,10-14 indicate that the natural frequencies separation might be amisleading criterion, especially if quantification of the quality of the approximation is desired: in some situations, ifthe separation increases the approximation error also increases6. This is due to the approximation error’s highlynonlinear dependency on the natural frequencies separation. Other factors (e.g. the modal damping matrix) also playa role in the approximation error. Hence designing structures for accurate proportional damping approximation usingthe natural frequencies separation as the only design criterion is not recommended6.

III. System GainsIn this paper a different approach is investigated. The idea is to design the structure such that a quantity related

directly to the approximation error is optimized rather than an indirect measure like the separation between naturalfrequencies. For this purpose the error system is built using (7) and (8):

[ ]TTp

Tp

Tm

Tm qqxExyBfAxx &&& εε==+= where,, , ),()()( tqtqt pmm −=ε

[ ]000,0

0

0

,

00

000

0

000

2

2

IE

U

B

C

I

CC

I

A

Tp

nm =

=

−Ω−

−−Ω−= . (9)

System (9) defines a linear operator between the external actions, f, and the approximation error, my ε= . For

accurate proportional damping approximations it is desired that the external actions are “diminished” by this linearoperator. In order to quantify this property, the system gains, defined in the following, can be used.

In system theory there are two common ways to measure the size of a signal, f(t). In one framework, signal size

is measured by its peak norm (or ∞L norm), defined by

( ) 2/1

0)()(sup tftff T

t≥∞

= , (10)

and in the other framework signal size is measured by its energy, also called the 2L norm, and defined as2/1

02

)()(

= ∫

∞

dttftff T . (11)

Corresponding to these signal measures the ∞L and 2L gains of the linear system (9) are defined as

,supgain,supgain2

2

)(2

)( f

yL

f

yL

tftf

==∞

∞∞ (12)

4

where f(t) are all nonzero signals of finite ∞L and 2L norms, respectively. In this paper it is assumed that the initial

conditions are zero, but definitions (12) are also valid for nonzero initial conditions. Note that the 2L gain of a

linear time invariant (LTI) system is actually the ∞H norm of its transfer matrix, defined as

[ ])(max ωσω

jGG =∞

(13)

where )]([ ωσ jG is the maximum singular value of the transfer matrix )( ωjG . It can be easily checked using

the Laplace transform that the transfer matrix of (9) for zero initial conditions is

( ) ( ) Tmnp UsCIsCsCIssG 22122)( Ω++Ω++= −

. (14)

Here ωjs = and the error in frequency domain can be expressed as )()()( sfsGsm =ε .

The major advantage of the 2L (or energy) gain over the ∞L gain is that for an LTI system it can be computed

using Linear Matrix Inequalities (LMI). Specifically, by solving the following eigenvalue problem (EVP) for β

0,0

0

:subject tomin

>>

≤

−++

ββ

β

P

IPB

PBEEAPPAT

TT

(15)

the 2L gain of (9) is obtained as βγ = . LMI solving techniques have become so reliable during the 1990s that

casting a problem as a LMI is considered an almost closed form solution15. On the other hand the ∞L gain is

difficult to compute. An upper bound on the ∞L gain, the *-norm, can be computed using LMI but it is not tight15.

The minimization of the energy gain will be pursued next for accurate proportional damping approximation design.

IV. Energy Gain Minimization for Proportional Damping Approximation DesignIn many cases the mass, damping, and stiffness matrices are linear combinations of inertial, damping, and elastic

characteristics of the structure and can be written in terms of free scalar design parameters iii kcm ,, as

∑∑∑===

+=+=+=E

iii

D

iii

I

iii KkKKCcCCMmMM

10

10

10 ,, . (16)

Matrices M0, C0, K0 are constant. Let the vector of design parameters be [ ]TEDI kkcmmx ......c... 111= . These

parameters can be used to minimize the energy gain of the error system (9). Specifically the problem of interest is:

.0,0,0

0)(

)()()(

:subject tomin

2

>>>

≤

−++

i

T

TT

x

xP

IPxB

xPBEEPxAxPA

γγ

γ

(17)

5

Solving (17) for a fixed value of x is easy using LMI tools (“mincx” in Matlab), but the objective function’sderivatives with respect to x, γ(x), cannot be computed analytically. Moreover, γ(x) might not be differentiableeverywhere because bifurcated solutions of the EVP (15) might occur as x varies. However, the objective is ingeneral almost everywhere differentiable, which makes application of certain stochastic approximation techniquespossible. Specifically, a simultaneous stochastic perturbation approximation16 (SPSA), which has proven effective inthe optimization of other non-differentiable functions17, has been selected to solve (17). The advantage of SPSA isthat it is robust, and under certain conditions16,17, convergences in probability to a global minimum. SPSA emergedas a strong competitor to genetic algorithms or simulated annealing for global optimization16 because it is muchmore economical in terms of the number of objective function evaluations.

V. A Stochastic Optimization AlgorithmThe algorithm used in this paper to solve (17) combines LMI and SPSA and it is described next.

Step 1: Initialization: Set the initial values of the design parameters, 0xx = , and the iteration index k=1. Compute

the A, B matrices using (9) and (16).

Step 2: Nominal 2L gain evaluation: Solve the EVP problem (15) and compute the nominal 2L gain, βγ = .

Step 3: Calculation of the direction of movement, g: Use a typical SPSA method16, in which g is given by

np

np

xxg

−

−=

γγ, (18)

where the division is performed component-wise. Here pγ and nγ are the 2L gains of the perturbed systems

obtained for “simultaneous perturbations”, ∆−=∆+= dxxdxx np , . These gains are obtained by solving (15)

with matrices A, B updated for np xx , using (9) and (16). Each element of ∆ is extracted from the Bernoulli

distribution, (+/-1), while d is computed as )min(2

1,min 1 x

k

dd

r= to avoid negative elements in nx or px .

Step 4: Prediction of x: Predict hgxx −=+ , where

( )

+= )min(

2

1,min

1

1 wkA

hh α (19)

and the elements of w are computed asi

i

g

xfor each nonzero gi.

Step 5: Return/Stop: If xhg δ<|||| , where xδ is the minimum allowed variation of x, or the number of iterations

is greater than the maximum allowed, exit, else set += xx , k=k+1 and return to Step 2.

Scalars h1, d1, A1, r, α have been selected according to specific rules16,17: h1=100, d1=10, A1=20, r=0.101, α=0.602.An important remark is that if the nominal gain thus obtained, γf, is smaller than 1, then for any external action

f(t) of finite energy the error between the non-proportionally damped system and its proportionally damped

approximation obtained for the final value of x is attenuated by at most γf (i.e. fm

fγ

ε≤

2

2 ). Thus, the structure is

designed for conditions which it may never experience since the entire space of vector-functions of finite 2L norm is

considered. However, if there is no information on the external actions, considering the entire set of finite energysignals is a good strategy.

Another remark is that this algorithm is significantly slower than the one presented in Ref. 6, which investigatedproportional damping approximation design by guaranteeing certain separation between the natural frequencies.This is so because the LMI solvers are slow for large scale problems and because, unlike in Ref. 6, the gradients arenumerically (and not analytically) computed.

6

VI. Example: A Tensegrity Structure’s Design

A. Tensegrity Structure DescriptionTensegrity structures are special prestressed assemblies of “soft” elements that can carry only tensile forces (e.g.

elastic cables) and “hard” elements (e.g. rigid bodies). These structures are capable of yielding stiff equilibriumconfigurations under no external forces and torques and with all soft members in tension called “prestressableconfigurations”2. In the following the previous algorithm will be applied to such a structure.

Consider a tensegrity structure composed of 6 bars, AijBij, a top (B12B22B32), a base (A11A21A31), and 18 tendons(Fig. 1). The tendons are massless, viscoelastic Voigt elements. The base is fixed, the top and bars are rigid, and thebars are attached to the top and base via frictionless spherical joints. The rotational degree of freedom around thelongitudinal axis of each bar is ignored and the system has 18 generalized coordinates (see Ref. 2 for more details).

Fig. 1: Tensegrity Structure.

Linearized dynamics models around certain equilibria called “symmetrical prestressable configurations”2, havebeen derived. If the bars are identical and the tendons have the same damping coefficients, matrices M, C, K are

0,0,0,,, 1

4

111

5

1

>>>=== ∑∑==

iii

iii

ii kcmKkKCcCMmM (20)

where m1 is the mass of the top, m2-4 are its principal moments of inertia, m5,6 the mass and longitudinal moment ofinertia of a bar, k1-3 the stiffness of three classes of tendons called “S” (Ai2Bj1), “V” (Ai1Bj1 and Bi2Aj2), “D” (Ai1Aj2

and Bi2Bj1), k4 the pretension coefficient (see Ref. 2), and c1 is the damping coefficient for all tendons. These will bethe design parameters. The symmetrical prestressable configuration analyzed here is characterized by

deg,60,75.0,67.0,1 ===== δαHbl (21)

where l is the length of a bar, b is the length of the side of the base and top equilateral triangles, H is the height ofthe structure (all in meters),δ is the angle made by each bar with the vertical symmetry axis (OOt) and α is the

angle made by the projection of A11B11 on the horizontal plane (A11A21A31) with 1b . Matrices Mi, C1, Ki, which

depend only on l, b,α andδ , have been computed using general formulas presented in Ref. 18.

B. Application of the AlgorithmConsider the following values for the design parameters (this design is called the “arbitrary” design):

7

.10,10,10,50,40,30,10,10 165432141 ========− cmmmmmmk (22)

The 2L (energy) gain of the corresponding error system (9) computed by solving (15) is γ = 5.82. This is a very

poor design for accurate proportional damping approximation, if finite energy inputs are considered. To improve thisdesign the previous algorithm has been applied for various initializations. One design, which yielded γf = 0.81 hasbeen selected for future analysis. This design corresponds to the following values of the design parameters:

.94.46,75.0,30.9,85.57,54.10,71.47

,59.2,90.45,97.17,44.0,80.38

165432

14321

===========

cmmmmm

mkkkk(23)

For evaluation of the design from the point of view of the accuracy of the approximation, consider that theexternal signal is oscillatory exponentially decaying,

( )teftf t ωσ cos)( 0= (24)

where σ <0. Consider also that this signal is of unit L2 norm, which leads to the following:

)2(

)(422

222

0 ωσωσσ

++−=f . (25)

In addition, for simplicity, in the following numerical experiments it is assumed that all elements of f0 are equal. Fig.2 shows the time histories of the Euclidean norm of the error for the arbitrary design and for the optimized design,(23), for σ = -0.8 and ω = 1.13 rad/s. Clearly the performance of the optimized design is superior for this signal.

Fig. 2: Approximation Error Time History for a Particular Exponentially Decaying Signal.

Consider now the variation of the L2 norm of the approximation error, mε , with ω. Note that this norm can be

easily computed using a Lyapunov equation by building an “extended” system as follows. Firstly, f(t) given by (24)can be written as the output of a linear system with zero initial conditions,

8

ffffff xCftBxAx =+= ),(δ& , (26)

−

=σωωσ

fA ,

=

0

1fB , [ ]00fC f = (27)

where δ(t) is the Dirac impulse. An extended system is created using (9) and (26) as

eeeeee xCytBxAx =+= ),(δ& , (28)

[ ]0,0

,0

ECB

BA

BCAA e

fe

f

f

e =

=

= , (29)

where [ ]TTf

Te xxx = . In the above 0 represents matrices of appropriate dimensions. The L2 norm of y is

)(trace)()(

2/1

02

Teee

T CXCdttytyy =

= ∫

∞

(30)

where Xe is given by the following Lyapunov equation:

0=++ Tee

Teeee BBAXXA . (31)

Fig. 3 shows the variation of the L2 norm of the approximation error with ω for σ = -0.8 for the arbitrary andoptimized designs, indicating the superiority of the optimized design across the entire frequency spectrum. Note thatthe forcing signal, (24), has a L2 norm equal to one and all elements of f0 are equal.

Fig. 3: Variation of the L2 Norm of the Error with ω for σ = -0.8.

9

However, if the damping of the external excitation decreases, it may so happen that the arbitrary design yieldsbetter results than the optimized one over a certain range of frequencies. Fig. 4 shows such a situation for σ = -0.2.The explanation is found in the natural frequencies distribution for the two systems, shown in Fig. 5. The arbitrarydesign’s natural frequencies are restricted to the [0, 2.3] rad/s region and this leads to poor approximate dampingapproximation in that range for ω, especially where the natural frequencies are clustered. The optimized design’snatural frequencies are spread over a larger range. On one hand this has a beneficial influence for low ω’s becausethe natural frequencies are not so densely clustered like the ones of the arbitrary design. One the other hand, becausesome of the natural frequencies of the optimized design have larger values than the maximum natural frequency ofthe arbitrary design, this leads to relatively poorer performance of the optimized design for higher ω’s.

A last remark before concluding this paper is that the previous considerations are based on a particular form ofthe external signals, given by (24). This form is only a subset (yet very representative) of the finite energy signalsthat where implicitly considered in the design procedure presented herein.

Fig. 4: Variation of the L2 Norm of the Error with ω for σ = -0.2.

10

Fig. 5: Natural Frequencies Distribution.

VII. ConclusionsDesign of structures for accurate proportional damping approximation is pursued using minimization of the L2

gain of the error system. The 2L gain is selected because it has physical significance, as the energy gain of the error

system, and because it can be easily computed using LMI. An algorithm which combines LMI and SPSA isproposed to solve the resulting optimization problem. One issue with this algorithm is its slowness due to the size ofthe LMI and to the numerically approximated gradients. However, its application to the design of a tensegrity

structure indicates its effectiveness: the 2L gain of an initial, arbitrary design, is dramatically reduced. Furthermore,

numerical investigation of the resulting optimized design shows that the error between the non-proportionallydamped system and its proportionally damped approximation decreases dramatically as compared to the initial,arbitrary design, especially if the system is subjected to highly damped, oscillatory exponentially decayingexcitations. On the other hand, if the external excitations are lightly damped, the optimized design yields slightlypoorer performance than the arbitrary design in certain frequency ranges related to the natural frequencies of theoptimized design.

References1Caughey, T.K., “Classical Normal Modes in Damped Linear Dynamic Systems”, Journal of Applied Mechanics Vol. 27,

1960, pp. 269-271.2Sultan, C., Corless, M., Skelton, R.E., “The Prestressability Problem of Tensegrity Structures. Some Analytical Solutions”,

International Journal of Solids and Structures Vol. 38-39, 2001, pp. 5223-5252.3Sultan, C., Skelton, R.E., “Linearized Dynamics of Tensegrity Structures”, Engineering Structures Vol. 26, No. 6, 2002, pp.

671-685.4 Sultan, C., Ingber, D.E., Stamenovic, D., “A Computational Tensegrity Model Explains Dynamic Rheological Behaviors of

Living Cells”, Annals of Biomedical Engineering, Vol. 32, No. 4, 2004, pp. 520-530.5Gawronski, W. K., Advanced Structural Dynamics and Active Control of Structures, Springer-Verlag, New York, 2004.6Sultan, C, “Designing Structures for Dynamic Properties via Natural Frequencies Separation; Application to Tensegrity

Structures Design”, Mechanical Systems and Signal Processing Vol. 23, No. 4, pp. 1112-1122.7Shahruz, S.M., “Comments on An Index of Damping Non-proportionality for Discrete Vibrating Systems”, Journal of

Sound and Vibration, Vol. 186, No. 3, 1995, pp. 535-542.8Gawronski, W.K., Sawicki, J.T., “Response Errors of Non-proportionally Lightly Damped Structures”, Journal of Sound

and Vibration, Vol. 200, No. 4, 1997, pp. 543-550.

11

9Adhikari, S., “Optimal Complex Modes and an Index of Damping Non-proportionality”, Mechanical Systems and SignalProcessing, Vol. 18, 2004, pp. 1-24.

10Park, S., Park, I., Ma, F., “Decoupling Approximation of Non-classically Damped Structures”, AIAA Journal, Vol. 30, No.9, 1992, pp. 2348-2351.

11Shahruz, S.M. and Packard, A.K. “Approximate Decoupling of Weakly Damped Linear Second-order Systems UnderHarmonic Excitations”, Proceedings of the 31st IEEE Conference on Decision and Control, Tucson, AZ, 1992.

12Shahruz, S. M. and Langari, G., “Closeness of the Solutions of Approximately Decoupled Damped Linear Systems to TheirExact Solutions”, Transactions of ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 114, 1992, pp. 369-374.

13Park, S., Kim, I. and Ma, F., “Characteristics of Modal Decoupling in Non-classically Damped Systems Under HarmonicExcitation”, Journal of Applied Mechanics, Vol. 61, 1994, pp. 77-83.

14Shahruz, S. M. and Packard, A. K., “Approximate Decoupling of Weakly Non-classically Damped Linear Second-orderSystems Under Harmonic Excitations”, Transactions of ASME Journal of Dynamic Systems, Measurement, and Control, Vol.115, 1993, pp. 214-218.

15Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, SIAM,Philadelphia, 1994.

16Maryak, J.L., Chin, D.C., “Global Random Optimization by Simultaneous Perturbation Stochastic Approximation”,Proceedings of the American Control Conference, Arlington, VA, 2001.

17He, Y., Fu, M.C., Marcus, S.I., “Convergence of Simultaneous Perturbation Stochastic Approximation forNondifferentiable Optimization”, IEEE Transactions on Automatic Control, Vol. 48, No. 8, 2003, pp. 1459-1463.

18Sultan, C., Modeling, Design, and Control of Tensegrity Structures with Applications, Ph.D. Thesis, Purdue University,West Lafayette, IN, 1999.