Sultan, Cornel - Designing Structures for Dynamical Properties

7/31/2019 Sultan, Cornel - Designing Structures for Dynamical Properties

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Published inMechanical Systems and Signal Processing 23 (2009) 1112-1122

Designing structures for dynamical properties via natural frequencies separation.

Application to tensegrity structures design

Cornel SultanAerospace and Ocean Engineering, Randolph Hall 215,

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

[email protected], Phone: 1-540-231-0047, Fax: 1-540-231-9632

Abstract

The design of structures for dynamic properties is addressed by placing conditions on the separation between natural

frequencies. Additional constraints, like lower and upper bounds on the natural frequencies, are also included. A fast

numerical algorithm that exploits the mathematical structure of the resulting problem is developed. Examples of the

algorithms application to tensegrity structures design are presented and the connection between natural frequencies

separation and proportional damping approximation is analyzed.

Keywords: dynamic design; natural frequencies separation; proportional damping approximation; tensegrity

structures

1. Introduction

Structures design using numerical methods has been initiated by Dorn [1] (see [2,3] for

detailed reviews) and most of the research has been focused on static requirements satisfaction(e.g. constraints on displacement, stress, strain in equilibrium conditions). A limited number of

articles deal with dynamic requirements (see [4-7] and the references therein). There are two

major reasons for the limited interest in designing structures for dynamic properties. Firstly,structures were traditionally designed for static operating conditions (arches, bridges, domes).

Secondly, because many classical structures are heavily damped, the dynamic transitory regime

decays rapidly and it is not considered important. However, with the advent of new technologiesin the area ofcontrollable structures like morphing structures [8], adaptive buildings [9], flexible

manipulators [10], these facts will no longer hold true. Firstly, controllable structures will

operate in conditions for which the dynamic regime will play an important role in their design.

Secondly, for these applications, lightly dampedstructures will be preferred in order to dissipateless energy and make their shape control cost effective and efficient.

This article approaches the design of structures from the dynamic perspective. The main

design requirement is represented by the natural frequencies location. Placing restrictions on the

natural frequencies is justified because of their importance in structures dynamics. Firstly, thesefrequencies are crucial for dynamic response characteristics like the rise time, peak time, and

settling time [11]. Secondly, in many cases, separation between natural frequencies is crucial in

enabling accurate proportional damping approximation [12]. Thirdly, as remarked in [4], theamount of degeneration in structures is reduced if constraints on the natural frequencies are

imposed.

A fast algorithm is presented which guarantees prescribed separation between natural

frequencies. The algorithm also considers constraints on the minimum and maximum naturalfrequencies values. The algorithm is fast because it exploits the structure of the mass and

stiffness matrices, which are linear in the design parameters, employs active set methods in order


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to reduce the number of computations, and the gradients used in the solution process areanalytically computed. Examples of the algorithms application to the dynamic design of

tensegrity structures are given. Analysis of the correlation between natural frequencies and

proportional damping approximation confirms the assertion that separation between naturalfrequencies mightbe a misleading criterion for accurate approximation.

2. Dynamics

2.1. Physical and modal systems

The linearized dynamics of many structural systems is described by

0,0,0, KCMfKqqCqM (1)

where M, C, Kare the mass, damping, stiffness matrices, respectively, q is the n-dimensionalvector of generalized coordinates, and f is the vector of external loads, respectively. A

transformation from the physical (q) to the modal (qm) coordinates is performed using the

modal matrix, U, which is constructed as follows:

0.,, diagIUUUUM MT

MM

T

MMM

0.,,2

11

diagIUUUUKUUT

KK

T

KKMM

T

MM

KMM UUU1

. (2)

Using mUqq in modal coordinates (1) becomes

fUqqCqT

mmmm 2

(3)

where CUUC Tm , )(diag22

l , and l are the natural frequencies obtained by solving

0)det( 2 MK l . (4)

Because natural frequencies are crucial in the dynamic response, dynamic design problemsusually include constraints on their values [4]. The first condition imposed here is that the natural

frequencies are separated. There are several reasons for which this is considered an importantrequirement as discussed next.

2.2. Natural frequencies separation and proportional damping approximation

System (1) is proportionally damped if mC is diagonal. There are big benefits if mC is

diagonal. For example, the equations of motion (3) decouple and they can be easily solved. Thisis very important, because the number of equations used in structures dynamics, n, is usually

large. Proportional damping models are also desired because they lend themselves easily to

computationally efficient identification, model order reduction, and control design tools [12].

A general form of the damping matrix, which results in a diagonal modal damping matrix, is


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1

0

1)(

n

i

i

i KMMaC (5)

where ia are real numbers [13]. Expression (5) is a generalization of the Rayleigh damping

model, in which the damping matrix is a linear combination of the mass and stiffness matrices.

One can easily verify that ifUsimultaneously diagonalizesMand Kit also diagonalizes Cof (5).

If mC is not diagonal it is desired to determine under what conditions it can be

approximated by a diagonal matrix, i.e. when proportional damping approximation is possible.The most popular approach to obtain such an approximation is to reduce the modal damping

matrix to a diagonal one by neglecting its off diagonal terms (see [12]), thus writing

).( mp CDiagC (6)

The proportional and non-proportional damping models are then

fUqqCqT

pppp 2

and (7)

0)(,where,)( 2 npmnT

mmnpm CDiagCCCfUqqCCq , (8)

respectively.

In order to determine when accurate proportional damping approximation is possible,

various non-proportionality indices involving the modal damping matrix, natural frequencies,and external excitations have been proposed. For example Tong et al. [14] proposed a measure of

damping non-proportionality based only on the damping matrix. Shahruz [15] showed through

an example that this measure is insufficient and indicated that the distribution of the systemsnatural frequencies must be taken into account. Gawronski [12] and Gawronski and Sawicki [16]

showed that neglecting the off diagonal terms in the modal damping matrix yields good results in

terms of the relative error, which is negligible if the natural frequencies are sufficiently separated

and the damping is small. Adhikari [17] introduced a non-proportionality index based on the

normal modes of the system and not on the system matrices. He shows that for viscously damped

systems with small damping this index is inversely proportional to the separation betweennatural frequencies, thus clustered natural frequencies should be avoided because they lead to

large values of the index. In [18] a measure of non-proportional damping which only depends on

the matrix of complex eigenvectors, and it is independent of the natural frequencies, has been

proposed. Over the years it has become generally accepted that sufficient separation of thenatural frequencies is crucial for accurate proportional damping approximation. However, when

the structure is subjected to harmonic excitation, this condition might not be sufficient, as

discussed next.

Park et al. [19] showed that if the input, f, is harmonic, the error between the response ofthe system and the response of its proportional damping approximation depends strongly on the

excitation frequency and may be significant for excitation frequencies which are close to the

natural ones. Shahruz and Packard [20] showed that if the system is lightly damped and theexcitation frequency is close to some of the lightly damped natural frequencies, the error might

be big even if the off-diagonal terms of the modal damping matrix, mC , are small. However, if

the corresponding proportionally damped system is reasonably damped and the off-diagonal

terms of mC are small, then the error is small [21]. Park et al. [22] gave several examples which


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show that, in the case of external harmonic input, neither the diagonal dominance of the modaldamping matrix nor the separation between natural frequencies is sufficient for accurate

proportional damping approximation; the location of the excitation frequency with respect to the

natural frequencies is an important factor in the error. Moreover, the approximation errorincreases substantially when there are natural frequencies which are clustered and the excitation

frequency is close to these clustered natural frequencies (see also [23]).It is remarkable that so far the discussion focused on accurate proportional damping

approximation in modal coordinates. Even more remarkable is the fact that accurate

approximation in the modal space is not sufficient for accurate approximation in the physical

space (see [24]). The following lemma connects the errors in modal and physical coordinates.

Lemma: The error in the physical space is smaller than the error in the modal space if

and only ifthe minimum eigenvalue of the mass matrix, )(M , is greater than one.

Proof: Let )(tm and )(t denote the error in the modal and physical space, respectively:

)()()()()()()()( tUttUttqtqt mmpmm (9)

where . is the Euclidean norm and )(U the maximum singular value of U. Since the upper

bound in Eq. (9) is tight, )()( tt m is equivalent to 1)( U . From Eq. (2)

KMM UUU1

and, since MU and KU are unitary,2/1)()( MU . Thus 1)( U is

equivalent to 1)( M .

2.3. Natural frequencies separation and computational efficiency and accuracy

From the computational point of view, it is also advantageous to have separated natural

frequencies. Firstly, the sensitivities of repeated natural frequencies and of the associatedeigenvectors with respect to various parameters are difficult to compute, both analytically andnumerically [25]. Secondly, in the case of repeated natural frequencies numerical computations

might lead to unacceptable accumulation and propagation of numerical errors.

On the other hand, if the natural frequencies are adequately separated, the likelihood of

having repeated eigenvalues in the corresponding first order linear modal system,

,,,0

,0

,2

fuq

qx

UB

C

IABuAxx

m

m

T

m

(10)

is reduced. This is important because, ifA has repeated eigenvalues it may be defective, hence

not diagonalizable, and the response of Eq. (10) may include secular terms. This is definitely notdesirable. Note that in the proportional damping case, A is diagonalizable and secular terms do

not appear, even if there are repeated natural frequencies (see Appendix A). In general, defective

systems represent exceptions and have been rarely reported in practical structures.


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3. The natural frequencies allocation problem

3.1. Problem formulationAs indicated before, a crucial dynamic design requirement for structures should be

sufficient separation between the natural frequencies. This requirement can be enforced in two

ways: through equality constraints, when all natural frequencies are prescribed fixed values to

achieve adequate separation, or through inequality constraints, when only lower bounds on theseparations are enforced via inequalities. These two options are discussed next.

The equality constraints category can be embedded in the larger class of ideal

dynamic designs, when the natural frequencies and the corresponding eigenvectors are

specified. Research conducted in this area indicated two major deficiencies. Firstly, solutions tothis problem, usually called the inverse spectral problem, require considerable freedom in the

structure ofM, C, K, and, secondly, the solutions are restrictive with respect to the specificationof the modal data [5-7]. For example in [26], the inverse spectral problem was solved only for

lumped conservative systems (i.e. C=0) modeled using tri-diagonal matrices. Likewise, in [5] theinverse spectral problem was solved when C and K are singular but the solution restricts the

eigenvalues to having complex values and does not even preserve the eigenvectors. The

interested reader may consult [7] for a review of results in the area.

A less constrained approach is to require that only the natural frequencies are allocated todesired locations. In practice even this approach is not usually possible, because the design space

is limited [4]. In many cases the mass, damping, and stiffness matrices are linear combinations of

inertial, damping, and elastic characteristics of the individual elements (e.g. bars, dampers,

springs), and can be written in terms offree scalar design parameters 0,0,0 iii kcm as

.,,1

0

1

0

1

0

E

i

ii

D

i

ii

I

i

ii KkKKCcCCMmMM (11)

Here I, D, and Eare the numbers of free design parameters and, in general, are small, because

many characteristics are fixed by other considerations (e.g. specifications on the materials). InEq. (11), matricesM0, C0, K0 account for the fixed parameters.

In this article, the exact placement requirement is relaxed and prescribed separations, lj ,

between natural frequencies are enforced through inequality constraints:

nljnlljjl ,...,1,1,...,1, . (12)

Additional constraints on the natural frequencies are included:

nll ,...,1,maxmin (13)

where min and max are such that jl

lj

,

minmax . These constraints can be easily justified

for controllable structures as follows. The structure should be designed to an upper bound on the

maximum natural frequency, because it is desired that the high frequency modes are measurable


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and the structural sensing devices sampling rate, which must be at least twice the structuresmaximum natural frequency, is limited (see [27]). The minimum natural frequency is lower

bounded in order to avoid slow modes. Low values of natural frequencies also correspond to a

soft (not sufficiently stiff) structure, which is not desirable.

To complete the problem formulation, limits on the design parameters are included:

EikkkIimmm iiiiii ,...,1,,,...,1, maxminmaxmin . (14)

Thus, the problem of interest is to find ii km , subject to Eqs. (12) - (14).

The major advantage this inequality constraints approach has is that it guarantees

increased flexibility in the design process. The exact placement problem might not have any

solution, whereas the inequality constraints problem might have many solutions. This factfacilitates the incorporation of the inequality constraints approach in more complex design

problems which include static constraints on stress, strain, displacement and optimization

requirements like the minimization of the structures mass.

The main disadvantage of this approach is that it leads to a nonlinear problem withinequality constraints for which closed form solutions are not possible. However, an efficient

algorithm has been designed which exploits the mathematical structure of the problem. This

algorithm is described next.

3.2. An iterative solution algorithm

The algorithm proposed herein to solve Eqs. (12-14) is inspired by active set methods.

The idea underlying these methods is to partition inequality constraints into two groups: thosethat are to be treated as active and those that are to be treated as inactive. The constraints treated

as inactive are ignored, decreasing the number of computations. This is especially useful for

problems with a large number of constraints, like structural design ones (see [28]).

The algorithm proceeds as follows: at the current iteration step, for a known value of thevector of design parameters, called x, the natural frequencies are computed by solving Eq. (4)

and ordered: n ...21 . Satisfaction of constraints (12-14) is evaluated. The constraints

which are violated are chosen as the active ones and a penalty function, P(x), is built:

),()()()( xPxPxPxP bbxs (15)

where

.)(2

1)(

2

1)(

,)(2

1

)(2

1

)(,)(2

1

)(

2

max

2

min

2

max

2

min

2

11

l

ll

l

llbx

ll

llb

llllls

xxxxxP

xPxP (16)

Here Ps(x) is associated with the separation constraints (12), Pb(x) with the constraints on the

boundaries of, Eq. (13), and Pbx(x) with the constraints on the boundaries ofx, Eq. (14). In all

these sums, only the violating pairs appear. For example, in Ps(x), only the indices l for


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which llll 11 are considered. This particular choice of a quadratic penalty function is

advantageous because it leads to convexification of the problem, thus facilitating the use of

gradient or Newton based iterative procedures for fast convergence (see [28] for more details).

Next, the penalty function is driven to zero using a gradient method. The advantage ofusing gradients is that, for this problem, they can be easily computed. Indeed:

)()()()( xPxPxPxP bbxs (17)

where

l

lll

l

lllb xP )()()( maxmin (18)

l

lllllls xP ))(()( 111 (19)

.)()()( maxmin l

lll

l

lllbx exxexxxP (20)

Here el is a vector with the l component equal to 1 and all the other components equal to 0.

The gradient of l is

T

w

lll

lxxx

...

21

(21)

where w=I+E is the dimension ofx. If l is single,j

l

x

can be easily computed using

ljl

j

l

lj

lj

T

lj

l

xxx

M

x

K

x

2/

2

2

2

(22)

where l is the corresponding mass-normalized eigenvector: IM lT

l . Using Eq. (11) the

following formulas are obtained:

.,

,

2

2

jjlj

T

ll

jjlj

T

l

j

l

mxifM

kxifK

x

(23)

Note that zero natural frequencies cannot appear because of the condition that K> 0 (see Eq.(1)). In the case of repeated natural frequencies, complex formulas have been derived for the

computation ofj

l

x

(see [25]), but complicated formulas are not necessary here because the

solution will be such that there are no repeated natural frequencies. Thus, if l is not single, the

corresponding value ofx will be randomly perturbed by a small amount.

Next, a line search method is used, in which a change inx is made along 0 Pg ,

,gxx (24)


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can also be computed analytically. The penalty function just has to be modified accordingly.

Some of the examples shown next will include this constraint.

4. Examples: tensegrity structures design

4.1. Tensegrity structures description

Tensegrity structures are assemblies of soft elements which can carry only tensile

forces (e.g. elastic cables), and hard (e.g. rigid bodies), and which are capable of yieldingequilibrium configurations under no external forces and torques and with all soft members in

tension. These configurations are called prestressable configurations [29]. Tensegrity

structures feasibility in controllable structures applications has been demonstrated [30-33]. In thefollowing, the previous algorithm will be applied to a tensegrity structure.

Consider a tensegrity structure composed of six bars, labeled AijBij, a top (B12B22B32), a

base (A11A21A31), and 18 tendons (Fig. 1). For mathematical modeling the tendons are

considered massless, viscoelastic Voigt elements, which consist of a linear elastic spring in

parallel with a linearviscousdamper. The base is fixed and the top and the bars are rigid. Thebars are axially symmetric. For each bar the rotational degree of freedom around the longitudinal

axis of symmetry is ignored. No external forces act on the structure (see [29] for details).

Fig. 1. Tensegrity structure.

Linearized dynamics models around certain equilibria called symmetrical prestressable

configurations (see [29] for details on these configurations), have been derived. If the bars are

identical and the tendons have the same damping coefficients matricesM, C, Kare given by

6 4

1 1 0

1 1

, ,i i i ii i

M m M C c C K K k K

(25)

where m1 represents the mass of the top, m2-4 its principal moments of inertia, m5,6 the mass and

longitudinal moment of inertia 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 [29]


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for details), and c1 the damping coefficient for all tendons. These will be the design parameters.

The symmetrical prestressable configuration analyzed here is characterized by

1, 0.67, 0.75, 60l b H (26)

where l is the length of a bar, b the length of the side of the base and top equilateral triangles,H

the height of the structure (all in meters), the angle made by each bar with the verticalsymmetry axis (OOt) and the angle made by the projection of A11B11 on the horizontal plane

(A11A21A31) with the fixed direction 1b . The corresponding matrices Mi, C1, Ki, which depend

only on l, b, , and , have been computed using the general formulas presented in [34].

4.2. Natural frequencies separation and proportional damping approximation

Consider the following ad-hoc values for the design parameters (the arbitrary design):

.1,1,1,5,4,3,1,1 165432141 cmmmmmmk (27)

All quantities are given in SI units. The natural frequencies distribution, shown in Fig. 2,indicates regions in which these frequencies are clustered: for 8 pairs of neighboring natural

frequencies the separation is less than 0.02. This is not a good dynamic design. For example ifthe responses to initial conditions of the proportionally and non-proportionally damped models -

(7) and (8) withf=0 - are computed, the approximation error is unacceptably large. Fig. 2 shows

the Euclidean norms of the modal, )(tm , and physical, )(t , error, for unity initial conditions:

10000 ppmm qqqq . Hence proportional damping approximation cannot be used. Redesign

of the structure to achieve separation of the natural frequencies must be pursued.

The algorithm presented before has been applied for various prescribed separations, lj ,

to solve (12)-(14). In some cases an upper bound constraint on the singular value norm of the

modal matrix, max)( U , has also been considered, as indicated next. In all cases only lowerbounds on the design parameters were enforced: lxl ,0min . Convergence of the algorithm,

implemented in Matlab, was very fast (miliseconds to seconds on a standard desktop computer).

Fig. 3 shows the results obtained for 2.0lj , 10,8.0 maxmin (rad/s), and a hard

constraint on the maximum singular value of the modal matrix: 15.0)( U . Analysis of

responses to initial conditions confirms that the proportional damping approximation can be

applied. Fig. 3 gives the errors for 10000 ppmm qqqq (similar patterns were observed for

other initial conditions) and shows that transformation in the physical space dramatically reduces

the error due to the hard constraint on the modal matrix norm, whose final value is 12.0)( U .


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Fig. 2. Natural frequencies distribution and initial conditions response errors

for the arbitrary design


for the hard modal constraint design ( 12.0)( U , 0.2lj )


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for the soft modal constraint design ( 8.0)( U , 0.2lj )

Fig. 4 corresponds to a design obtained for 2.0lj , 18,8.0 maxmin and a soft

constraint on the modal matrix norm: 8.0)( U . The error norms, shown in Fig.4 for

10000 ppmm qqqq , increase and the modal and physical errors are noticeable closer

because of the soft constraint. Similar results were obtained for other initial conditions. The

design is still good and proportional damping approximation can be used.

The next set of results reveals very interesting features. Firstly, even if the prescribedseparation is increased it may so happen that the results are worse than the ones obtained for a

smaller separation. Fig. 5 corresponds to such a design, in which 4.0lj , 18,2 maxmin

and the constraint on )(U was eliminated. It can be easily ascertained that the natural

frequencies range is similar to the one in Fig. 4, but the error in modal coordinates is much

larger, even though the minimum separation between natural frequencies doubled (the initial

conditions considered in Fig. 5 are the same as before). Secondly, because the constraint on

)(U has been removed, the error in the physical space is hugely amplified (the maximumsingular value of the modal matrix is 55.1)( U ). This is not a good design if proportional

damping approximation is thought after. However it is a good dynamic design for other purposes

because, for example, sufficient separation between the natural frequencies is achieved.


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for the no modal constraint design ( 55.1)( U , 0.4lj )


for the no modal constraint design with small damping (c1=0.1)


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These results indicate that separation between natural frequencies may not be sufficient

for accurate proportional damping approximation. They complement similar results obtained

when harmonic excitations, rather than nonzero initial conditions, were considered (see [19-24]).

Another interesting result is obtained if the damping is substantially reduced. Fig. 6corresponds to the same design as before (i.e. same design parameters) but when c1=0.1 (a very

lightly damped structure). It is remarkable that the errors are substantially smaller. This is inagreement with Gawronskis [12] observation, that for lightly damped structures separation ofnatural frequencies is sufficientfor accurate proportional damping approximation in modal space.

Nevertheless, the constraint on the modal matrix should be introduced in the design for accurate

approximation in the physical space.

The design parameters associated with Figs. 2-6 are given in Table 1. All the resultspresented in this article correspond to situations in which the algorithm converged to solutions at

which some of the separation constraints (12) are tight hence the minimum prescribed separation

is achieved. Except for the solution corresponding to Fig. 4, when the modal matrix norm

constraint was also tight ( 8.0)( U ), none of the other constraints were tight.

k1 k2 k3 k4 m1 m2 m3 m4 m5 m6 c1 lj

Fig.

1 1 1 1 1 3 4 5 1 1 1 0.0 2

640.40 747.40 2055.5 1974.4 706.5 63 320.3 258.3 7.2 104.6 1 0.2 3

104.41 88.86 0.23 127.88 48.04 1.61 23.45 22.08 0.49 1.77 1 0.2 4

13.14 31.45 51.85 42.77 1.43 0.36 1.11 1 0.65 0.49 1 0.4 5

13.14 31.45 51.85 42.77 1.43 0.36 1.11 1 0.65 0.49 0.1 0.4 6

Table 1. Design parameters values for Figs. 2-6.

These results indicate that if proportional damping approximation is the main goal of the

design, one should not count on obtaining increasingly accurate approximations just byincreasing the separation between natural frequencies. This is due to the fact that the approximationerrors dependence on the natural frequencies separation is nonlinear. Also other factors, except for the

natural frequencies, like the modal damping matrix, play a role in the approximation error.

A better approach to the design of structures for proportional damping approximation isto consider design requirements directly related to the approximation error like properties of the

transfer matrices between the initial conditions or the inputs and the approximation error. These

matrices can be easily obtained using the Laplace transform. For nonzero initial conditions,000 pm qq

and 000 pm qq , and non-zero external input,f(t), Eqs. (7) and (8) yield

2 2

0 0 0( ) ( ( ) ) ( ) ( )T

m m m m m m ms q s sq q C sq s q q s U f s (28)

2 2

0( ) ( ) ( ( ) ) ( ) 0m p m n m m ms s C s s C sq s q s (29)

where js . From Eqs. (28) and (29), it easily follows that


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20 0

1( ) ( ) ( )Tm m ms G s U f s q q

s

(30)

where

ssCIsCsCIssG mnp122122

)(

. (31)

One approach is to design the structure such that the error norm is minimized wheninputs or initial conditions with certain properties are considered. This is a topic of future

research.

5. Conclusions

For future controllable structures dynamic requirements will play an important role,hence constraints on their dynamic characteristics should be considered in the design process.

Specifically, constraints on the natural frequencies should be imposed due to these frequenciesinfluence on the dynamic response. A key dynamic design requirement is that the natural

frequencies are sufficiently separated, which is crucial for simple and exact computations, andaccurate proportional damping approximation. Other requirements, which are especiallyimportant for controllable structures, are that the natural frequencies are lower and upper

bounded. Thus the dynamic design problem formulated in this article includes separation

constraints on the natural frequencies and lower and upper bounds on their values.

A numerical algorithm for the solution of this problem was proposed. The algorithm isvery fast because of two key features: it relies on active set methods and the gradients used in the

iterative solving process are analytically computed. The algorithm can be easily extended to

solve problems which include other constraints whose gradients can be analytically computed; an

upper bound constraint on the maximum singular value of the modal matrix is particularly

important for accurate proportional damping approximation in the physical space. The algorithmwas evaluated on tensegrity structures design and in all cases convergence was obtained very fast

(milliseconds to seconds).

Analysis of the relation between natural frequencies separation and the accuracy ofproportional damping approximation indicated that separation of these frequencies is essential

but it is may be a misleading design criterion. For example the approximation error might

increase when the separation between natural frequencies increases, even when only the responseto nonzero initial conditions is considered. Hence, if the main goal of the design is accurate

proportional damping approximation, a problem which includes constraints directly related to the

approximation error norm should be formulated and solved.

Appendix A: Proportional damping yields non-defective systems.

Consider the case of a diagonal modal damping matrix which can be written as

10),(diag,2 iiT

m ZZCUUC . (32)


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It can be shown using linear algebra that

Z

IA

2

0

2is diagonalizable:

1 2 0.50

, , [ ]0

I IE AE E Z j I Z

(33)

where the inverse ofEis

11

11

1

)()(

)()(IE . (34)

The state transition matrix is

2 2 0.5 2 1 2 0.5 2

2 0.5 2 2 2 0.5 2

[cos( ) [ ] sin( )] [ ] sin( )

[ ] sin( ) [cos( ) [ ] sin( )]

At

Z t Z t

Z t Z t

e

e I Z t Z I Z I Z t e I Z I Z t

e I Z I Z t e I Z t Z I Z I Z t

(35)

and the response of the system does not include secular terms.

References

[1] W. Dorn, R. Gornery, M. Greenberg, Automatic design of optimal structures, Journal de Mecanique 3 (1964) 25-

52.

[2] M.P. Bendsoe, A. Ben-Tal, J. Zowe, Optimization methods for truss geometry and topology design, Structural

Optimization 7 (1994) 141-159.

[3] Special issue on multidisciplinary design optimization, Journal of Aircraft 36 (1) (1999).

[4] N.L. Pedersen, A.K. Nielsen, Optimization of practical trusses with constraints on eigenfrequencies,

displacements, stresses and buckling, Structural and Multidisciplinary Optimization 25 (2003) 436-445.

[5] L. Starek, D. J. Inman, A. Kress, A symmetric inverse vibration problem, Transactions of the ASME Journal ofVibration and Acoustics 114 (1992) 564-568.

[6] L. Starek, D.J. Inman, A symmetric inverse eigenvalue vibration problem with overdamped modes, Journal of

Sound and Vibration 181 (5) (1995) 893-903.

[7] L. Starek, D.J. Inman, Symmetric inverse eigenvalue vibration problem and its application, Mechanical Systems

and Signal Processing 15 (1) (2001) 11-29.

[8] R.W. Wlezien, G.C. Horner, A.R. McGowan, S.L. Padula, M.A. Scott, R.J. Silcox, J.O. Simpson, The Aircraft

Morphing Program, AIAA paper AIAA-1998-1927, in Proceedings of the AIAA/ASME/ASCE/AHS/ASC

Structures, Structural Dynamics, and Materials Conference and Exhibit, Long Beach, CA, 1998.

[9] B.F. Spencer Jr., S. Nagarajaiah, State of the art of structural control, Journal of Structural Engineering 129 (7)

(2003) 845-856.

[10] S.K. Dwivedy, P. Eberhard, Dynamic analysis of flexible manipulators, a literature review, Mechanism and

Machine Theory 41 (7) (2006) 749-777.

[11] G. Franklin, J.D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, 5 th ed., Prentice Hall,

Englewood Cliffs, NJ, 2005.

[12] W.K. Gawronski, Advanced Structural Dynamics and Active Control of Structures, Springer, New York, NY,

USA, 2004.

[13] T.K. Caughey, Classical normal modes in damped linear dynamic systems, Journal of Applied Mechanics 27

(1960) 269-271.

[14] M. Tong, Z. Liang, G.C. Lee, An index of damping non-proportionality for discrete vibrating systems, Journal

of Sound and Vibration 174 (1994) 37-55.

[15] S.M. Shahruz, Comments on an index of damping non-proportionality for discrete vibrating systems, Journal of



17/17

[16] W.K. Gawronski, J.T. Sawicki, Response errors of non-proportionally lightly damped structures, Journal of


[17] S. Adhikari, Optimal complex modes and an index of damping non-proportionality, Mechanical Systems and

Signal Processing 18 (2004) 1-24.

[18] U. Prells, M.I. Friswell, A measure of non-proportional damping, Mechanical Systems and Signal Processing

14 (2) (1999) 125-137.

[19] S. Park, I. Park, F. Ma, Decoupling approximation of non-classically damped structures, AIAA Journal 30 (9)

(1992) 2348-2351.

[20] S.M. Shahruz, A.K. Packard, Approximate decoupling of weakly damped linear second-order systems under

harmonic excitations, in: Proceedings of the IEEE Conference on Decision and Control, Tucson, AZ, USA, 1992.

[21] S. M. Shahruz, G. Langari, Closeness of the solutions of approximately decoupled damped linear systems to

their exact solutions, Transactions of ASME Journal of Dynamic Systems, Measurement, and Control 114 (1992)

369-374.

[22] S. Park, I. Kim, F. Ma, Characteristics of modal decoupling in non-classically damped systems under harmonic

excitation, Journal of Applied Mechanics 61 (1994) 77-83.

[23] S.M. Shahruz, A.K. Packard, Approximate decoupling of weakly non-classically damped linear second-order

systems under harmonic excitations, Journal of Dynamic Systems, Measurement, and Control - Transactions of

the ASME 115 (1993) 214-218.

[24] S.M. Shahruz, P.A. Srymatsia, Approximate solutions of non-classically damped linear systems in normalized

and physical coordinates, Journal of Sound and Vibration 201 (2) (1997) 262-271.

[25] J.S. Jensen, N.L. Pedersen, On maximal eigenfrequency separation in two material structures: the 1D and 2Dscalar cases, Journal of Sound and Vibration 289 (2006) 967-986.

[26] G.M.L. Gladwell, Inverse Problems Vibrations, Martinus Nijhoff, Boston, MA, USA, 1986.

[27] C. Sultan, Tensegrity structures: from avant-garde art to next generation controllable structures, in: Proceedings

of the World Conference on Structural Control and Monitoring, San Diego, CA, USA, 2006.

[28] D.G. Luenberger, Linear and Nonlinear Programming, Addison-Wesley, Reading, MA, USA, 1984.

[29] C. Sultan, M. Corless, R.E. Skelton, The prestressability problem of tensegrity structures. Some analytical

solutions, International Journal of Solids and Structures 38-39 (2001) 5223-5252.

[30] C. Sultan, M. Corless, R.E. Skelton, Peak to peak control of an adaptive tensegrity space telescope, in:

Proceedings of the SPIE International Symposium on Smart Structures and Materials, Newport Beach, CA, USA,

1999.

[31] C. Sultan, M. Corless, R.E. Skelton, Tensegrity flight simulator, Journal of Guidance, Control, and Dynamics

23 (6) (2000) 1055-1064.

[32] C. Sultan, R.E. Skelton, Deployment of tensegrity structures, International Journal of Solids and Structures 40(18) (2003) 4637-4657.

[33] C. Sultan, R.E. Skelton, A force and torque tensegrity sensor, Sensors and Actuators: A - Physical 112/2-3

(2004) 220-231.

[34] C. Sultan, Modeling, design, and control of tensegrity structures with applications, Ph.D. Thesis, Purdue

University, West Lafayette, IN, USA, 1999.

Documents

Sultan, Cornel - Designing Structures for Dynamical Properties