passive control applied to structural optimization and...

Passive control applied to structural optimization and dy-namic analysis of a space truss considering uncertainties

F. A. Pires 1, P. J. P. Gonçalves 1

1 UNESP - Bauru/SP - Department of Mechanical EngineeringAvenida Engenheiro Luiz Edmundo Carrijo Coube, 14-01, Bauru, Brazile-mail: [email protected]

AbstractDue to the need for monitoring Earth’s natural systems, fleets of satellites will be launched in the next fewyears. In order to minimize shuttle launch costs, space structures should have the shape of trusses because ofthe significant weight reduction by being assembled with elements made of light materials (e.g. aluminum).For this reason, a control method must be applied to maintain the requirements of vibration levels in thesetypes of structures. This work applies a passive vibration control to a space lattice structure utilizing theFinite Element Method (FEM) in order to predict and compare the system’s frequency response functions(FRFs). The optimization technique simulated annealing (SA) is also applied to the structure to find thecoordinates of each node of the space truss that minimize the value of norm H2 for each mode of vibration.The Monte Carlo technique is also applied so it is possible to come up with an envelope function to showwhether or not the FRFs are inside it.

1 Introduction

Because the increasing demand for spatial structures, it is of fundamental importance to understand howthis type of system behaves. Besides, with the objective of reducing launch costs, these structures mustbe assembled with elements made of light materials, such as aluminum. However, this weight reductionbrings flexibility to the system as a result of the components containing low structural damping [1]. Con-sequently, the system presents undesireable responses, known as vibrations, owing to disturbances from theenvironments where these structures will be exposed.

In order to maintain the vibration levels in small scales, a control method should be applied. The vibrationscan be controlled utilizing both passive and active control techniques. Different strategies are used to modeland design the control of a flexible structure, as seen in [2] who studies a passive control technique inmechanical systems making use of dynamic vibration absorbers. [3] uses an active control of a tridimensionaltruss utilizing stacked piezoelectric actuators. [4], applies the norm H2 technique to the vibration control ofa truss containing piezoelectric actuators whereas [5] utilizes the norm H∞ technique for the robust controlof a MEMS (micro electro mechanical system) gyroscope.

This work applies the conventional Finite Element Method (FEM) to model a space lattice structure. Suchmethod is one of the most used computational tools to analyse structures under vibrations. [6] utilizes thistechnique for a vibration analysis of variable geometry trusses, and [7] makes use of this modelling methodto do a fatigue and free vibration analysis of a space truss.

During the past few years, the way of how different types of uncertainties are handled with this modellingtechnique has awaken the concern among researchers. These uncertainties might emerge because of theunknown values of physical properties (for example Young’s Modulus, density and geometry). One way

4419

to account for the uncertainties is to use the Monte Carlo simulations in order to treat the system with aprobabilistic modelling since this approach has become practical due to advances in computational technolo-gies. The basic idea behind Monte Carlo simulations is to carry out repeated random simulations to obtainnumerical results which will represent the uncertainties in the model’s response. This technique treats theparameters as random variables, rather than considering them as deterministic quantities. In the determin-istic approach, the values of the parameters are constant so that there is only a single frequency responsefunction (FRF) calculated for the structure. However, according to [8], when the values of the properties areset randomly, the behavior of the system and the FRFs can be predicted using the range of values for thoseparameters.

Admittedly, the use of an optimization tool becomes feasiable to study the behavior of the system’s FRF dueto a disturbunce. The usage of the optimization technique Simulated Annealing (SA) is an alternative to thisanalysis. According to [9], this procedure draws an analogy between the energy of physical systems andthe minimization of the objective function of structural systems. The SA’s main idea is to provide a way toavoid being trapped in local minima and being able to explore globally for better solutions. [10] uses thistechnique to optimize the design of distinct types of lattice structures by varying the number of elements,while [11] utilizes this methodology in order to find an optimum design for trusses depeding on size, formand topology.

This paper focuses on a vibration analysis by applying the optimization tool Simulated Annealing to find thebest nodes coordinates to minimize the parameter norm H2. By the finite element (FE) model, there will bea set of FRFs which were optimized in order to lower the amplitude of each resonance peak and to be ableto compare them to the original FRF. This methodology has shown that the passive control method appliedto the system was effective since the amplitudes of vibration have indeed diminished. The work also puts touse the Monte Carlo simulation. It represents a way to assess whether or not the found FRFs are within therange of frequency responses of the model. The approach demonstrated that the FRFs remained inside theenvelope function generated by the Monte Carlo method.

2 The Structure

This work studies the structure shown in figure 1. This truss contains 93 elements made of aluminum witha diameter of 7 mm, length of 150 mm and 33 nodes. The system will not be clamped which simulates thefree-free boundary condition. This structure has 99 degrees of freedom (dofs) since each node can movearound the 3 directions in the cartesian plane (x-y-z). Consequently, the system owns 99 natural frequenciesand therefore, 99 vibration modes.

y

xz

3 47

1013

1619

2225

2831

25

811

1417

2023

2629

32

1

69

3033

Figure 1: Space truss utilized in the finite element model.

4420 PROCEEDINGS OF ISMA2016 INCLUDING USD2016

3 Finite Element Method Applied to the Structure

Consider the truss illustrated in figure 1. The system’s motion is governed by the following equation ofmotion in a matrix form

M q + K q = F(t) (1)

where K and M are the structure’s stiffness and mass matrices, respectively, of order 99 x 99, q the displace-ment vector and F(t) the external load vector, both of order 99 x 1, obtained by the finite element modelusing truss elements [12].

As mentioned in the previous section, the structure’s boundary condition is free-free and there is no externalload acting on the system at first, because in the section the pursuit is to show the natural frequencies andvibration modes of the structure. Thus, equation 1 is transformed into

M q + K q = 0 (2)

Once the system’s stiffness and mass matrices are computed for the truss, it is possible to analyse the trussthrough the eigenvalue problem. With the goal of computing the structure’s natural frequencies, it is as-sumed a harmonic motion for the displacement in the form q=Qsen(ωt). The equation that results from theeigenvalue problem is given by equation 3.

(K− ω2M) q = 0 (3)

where ω is the natural frequency and q the vector of vibration modes. These parameters are represented bythe eigenvalues and engenvectors, respectively.

The natural frequencies can be acquired by the square root of the diagonal of the engenvalues matrix and eachvibration mode can be computed by the column of the engenvectors matrix referent to its natural frequency.For instance, if the pursuit is to analyse the seventh natural frequency, its vibration mode will be representedby the seventh column of the engenvectors matrix, in which each row denotes the displacement of eachdegree of freedom (dof).

In this simulation, it is set the theoretical values for the material properties so that the aluminum’s Young’sModulus and density are E = 70 GPa and ρ = 2700 kg/m3, respectively.

The present work considers only the frequency range between 100 and 600 Hz, which embraces 6 vibrationmodes of the truss. It is important to point out that the paper will not consider the rigid body modes ofvibration which actually are the first 6 modes of vibration so that the ones in study represent from the 7thto 12th vibration modes. These modes of vibration are the first bending (7th), second bending (8th), firsttorsion (9th), second torsion (10th), third bending (11th) e fourth bending (12th).

Figure 2 shows the vibration modes as well as their respective natural frequencies. It’s worth pointing outthat they could be gotten by the finite element model assuming the principle that there is no external loadacting on the system.

USD - UNCERTAINTY IN THE AEROSPACE SECTOR 4421

First Bending - 231.2 Hz Second Bending - 231.2 Hz First Torsion - 251.5 Hz

Second Torsion - 500.0 Hz Third Bending - 555.2 Hz Fourth Bending - 555.2 Hz

Figure 2: Modes of vibration and natural frequencies of the space lattice structure obtained by the FEM.

4 State-Space Representation

A linear time-invariant (LTI) system of finite dimensions is described by the equations below

x = A x + B u,

y = C x(4)

with state space initial condition x(0) = x0. In the equations above, x is the state vector of dimension N, x0

is the state space initial condition, u is the system’s input vector of dimension s and y the system’s outputvector of dimension r. The matrices A, B and C are real and constant matrices of dimensions N x N ,N x sand r x N , respectively.

Moreover, the state-space representation of a system can be represented alternatively by its transfer function.The transfer function G(s) is defined in equation 5

G(s) = C(sI −A)−1B (5)

4.1 State-Space Representaion of the Truss

With the objective of carrying out structural dynamic simulations and control analysis, it is convenient torepresent the equations of flexible structures in state-space representaion, as seen in equation 4.

By assuming that the system possesses the following equation of motion,

q + M−1D q + M−1K q = M−1B0 u

y = Cq q + Cv q(6)

It is assumed that,

x =

{x1

x2

}=

{qq

}(7)


Thus, the state-space representaion of the system will be,

x1 = x2

x2 = −M−1K x1 −M−1D x2 + M−1B0 u

y = Coq x1 + Cov x2

(8)

where M e K are the mass and stiffness matrices, respectively, represented in equation 1. Matrix D isthe proportional damping matrix, which is proportional to the stiffness matrix. This study assumed thatD = 10−6 K.

It is known that (A,B,C) are the state-space parameters of equation 4 and by combining the above equations,it is possible to obtain state-space equations,

A =

[0 I

−M−1K −M−1D

], B =

[0

M−1B0

], C =

[Coq Cov

], (9)

where matrices A, B and C have dimensions 198 x 198, 198 x 1 and 99 x 198, respectively.

In addition, B is the input matrix and C the output matrix of the system. This analysis assumes that thereis an external load so that the parameter B0 is equivalent to the vector F(t) from equation 1, because in thiscase, the input will be a load. Also, Coq and Cov are the output displacement and output velocity matrices,respectively, and both possess dimension 99 x 99. These matrices have values only in their diagonals, withthese values being either 0 or 1.

5 Norms of a System

The norms of a system are utilized to measure the intensity of a structure’s response to standard excitationssuch as a load, white noise and unit impulse. The use of this tecnique allows the comparison betweendifferent systems. There are three types of system norms: H2, H∞ e Hankel. This work suggests a studythat verifies whether with the minimization of norm H2, it is possible to optimize the structure’s FRF bydecreasing the amplitudes of vibration. The system illustrated in figure 1 is cosidered to be discrete inthe frequency domain. For structures in modal representation, theory encountered in [13], by considering(Ami, Bmi, Cmi) the modal representation in state-space, each mode of vibration is independent so that thenorms of the vibration modes are independent as well.

5.1 The Norm H2

It is defined ∆ωi as being a half-power frequency at the ith resonance, ∆ωi = 2ζiωi, see [14]. Let Gi(ω) =Cmi(jωI − Ami)

−1Bmi, be the transfer function of the ith vibration mode of the structure, where I is theidentity matrix. It is known that the transfer function is in frequency domain. The norm H2 of the ith modeof vibration is given by equation 10

||Gi||2 ∼=||Bmi||2||Cmi||2

2√ζiωi

=||Bmi||2||Cmi||2√

2∆ωi

∼= γ√

2∆ωi (10)

5.2 Norm H2 of the Structure

The norm H2 of a structure is expressed in terms of norms of each vibration modes, as shown previously.By following the theory presented by [13], the norm H2 of the system is approximately the sum rms of the


norms of each mode of vibration, as shown in equation 11

||G||2 ∼=

√√√√ n∑i=1

||Gi||22 (11)

where n is the number of vibration modes.

For the simulation, the first six vibration modes of the truss will not be considered, since these are thestructure’s rigid body modes of vibration. In this section, the system will be considered as a forced system sothat it is possible to obtain the frequency response of the structure. It is assumed that there is an external loadof 1 N acting at node 1 of the truss on the positive direction of y, in other words, B0(2,1) = 1, see equation9. Furthermore, the output matrix C will be set so that the response due to the applied load at node 1 will beverified at the 3 last nodes of the trusss, at nodes 31, 32 and 33, respectively. This work just intends to getthe displacement reponse, so Cov will be a matrix entirely of zeros, 99 x 99. Besides, Coq will be built upso that there will only be values in the diagonals that refer to the degrees of freedom of those 3 nodes whoseresponse aims to be acquired.

Therefore, the norm H2 of the system, considering all of the modes of vibration but the rigid body ones,from 7 to 99, owns the value of Hsystem

2 = 4.8259 10−4.

As mentioned before, this work’s aim is the study of the frequency range between 100 Hz and 600 Hz,which contains the 6 natural frequencies of interest. Hence, the norm H2 related to this frequency range, thatrepresents the vibration modes 7 to 12, possesses the value of H(7−12)

2 = 4.8091 10−4.

It is worth mentioning that the value of the system’s norm H2 and the one related to the 6 vibration modes instudy are close. It shows that the first modes of vibration are the ones that have the greatest modal values ofnorm H2. This characterizes them as the most significant ones related to the system. So, it has been chosento study the frequency range that contains these vibration modes which is from 100 Hz to 600 Hz.

Thus, the analysis will be carried out by studying the modal values of norm H2 for each of these modes.Table 1 illustrates them.

Vibration Mode Norm H2

7 3.3024 10−4

8 2.2875 10−4

9 2.5137 10−4

10 6.3553 10−5

11 2.3521 10−5

12 4.5917 10−5

Table 1: Norm H2 of each vibration mode

6 Simulated Annealing

Simulated Annealing is a method used to solve optimization problems of unconstrained and bound-constrainedsystems. At each iteration of the simulated annealing algorithm, a new point is randomly generated. Thealgorithm accepts all of the new points that lower the objective, but also, with a certain probability, pointsthat raise the objective. By accepting points that raise the objective function, the algorithm avoids beingtrapped in local minima and is able to globally look for better solutions. The procedure programs a way tosystematically decrease the objective function as the code runs. As the objective diminishes, the algorithmreduces the extention of the search in a way to converge to a minimum value.


6.1 Simulated Annealing Applied to the Structure

This method was used in this work so that it could be posible to obtain a minimum value of norm H2 for the6 vibration modes mentioned above by varying the coordinates of the 33 nodes of the truss. It is possibleby assuming an initial value besides an upper and lower bounds for each nodal coordinate. It has beencarried out one simulation for each mode of vibration utilizing diferent values in between the upper andlower bounds. Let X0 be the matrix of the original nodal coordinates of the truss illustrated in figure 1.

In this simulation, it was used a standard deviation (SD) of 2% for the upper and lower bounds, in otherwords, LB = X0 − 0.02 X0 and UB = X0 + 0.02 X0, where 0.02 represents the SD. The values of theminimized normH2 of each vibration mode is represented in table 2. Appendix A illustrates the convergenceof the algorithm for the norms H2 shown in the table below.

Vibration Mode Norm H2

7 4.5720 10−9

8 2.7798 10−9

9 6.8567 10−5

10 3.1917 10−5

11 2.5906 10−10

12 2.5460 10−9

Table 2: Norm H2 minimized for each vibration mode

7 Uncertainty Analysis

When a deterministic model is considered, the parameters of the model are constants. So, if repeated runsare computed with the same inputs the model will return the same outputs. In this case, the uncertainties arenot being taken into account.

In order to treat the uncertainties of a deterministic model properly, the values of the uncertain input pa-rameters have to be considered as multivariate random variables so that the outputs of the model are alsomultivariate random variables. Thus, with the range of solutions obtained, statistics are made and the mainresult of the problem is the distribution of probability of the results. The aim of this work is also to propagatethe uncertainties through the computer model to characterize the distribution.

A solution to this problem is to use the Monte Carlo procedure. In this case, a vast sample is drawn from theinput distribution, running the model at each sampled input configuration. The resut is a sample of outputsfrom each any summary of the uncertainty distribution can be estimated by using the corresponding summarystatistic.

As follows, Monte Carlo simulations are presented to analyse the 6 modes of vibration in consideration.The study aims to verify the uncertainty distribution of the FRF of the structure. For this analysis, the codewas run 10000 times keeping the material properties constant so that the nodal coordinates (inputs) of thetruss were set randomly in accordance with a range by considering a SD of 2% as well and, consequently, adifferent FRF (output) could be computed after each iteration. As a result, it was possible to come up withan envelope function from the 10000 simulations, shown in figure 3.

It is well known that in an actual structure, the position of each node is uncertain and keeping in mind that FEmodel has the premise to pursue as close as possible the real solution, it is important to take into account theuncertainties of the model so that the likelihood of the actual value of the nodal coordinates, and consequentlythe actual FRF, being in the generated range is high. According to [15], the accuracy of this estimate isdetermined by the size of the sample. This explains why it was chosen to run the code a considerable amountof times. There are some techniques to evaluate the error on the Monte Carlo simulation’s estimates [16],however, the work does not aim to perform an error evaluation.


Figure 3: Envelope function of the FRFs.

8 Results and Analysis

This section carries out an analysis of the FRFs by comparing the curve of the original FRF to those foundfor the optimized nodal coordinates when the norm H2 is minimized for each mode of vibration consideringan SD of 2%. These curves of frequency response are obtained from the finite element model. The FRFs aregenerated by utilizing a cost function or objective function. This cost function is based on the square of aquantity such as displacement and velocity. In this case, the objetive function is then determined as the sumof the linear squared displacements at nodes 31, 32 and 33. The values of this cost function are converted todecibels (dB) and then the frequency curves are plotted. The y-axis represents the values of amplitudes in(dB) and the x-axis allocates the values of frequencies in the range 100-600 Hz. In figure 4, the solid linesrepresent the FRFs of the non-modified truss while the dashed lines are the FRFs of the modified truss withnodal coordinates optimized and norm H2 minimized for the 6 vibration modes in consideration.

100 200 300 400 500 600−160

−140

−120

−100

−80

−60

−40

−20

0

20

Frequency [Hz]

dB

Original FRFFRF for the Norm H

2 minimized

a)

100 200 300 400 500 600−160

−140

−120

−100

−80

−60

−40

−20

0

20

Frequency [Hz]

dB


2 minimized

b)

100 200 300 400 500 600−160

−140

−120

−100

−80

−60

−40

−20

0

20

Frequency [Hz]

dB


2 minimized

c)

100 200 300 400 500 600−160

−140

−120

−100

−80

−60

−40

−20

0

20

Frequency [Hz]

dB


2 minimized

d)

100 200 300 400 500 600−160

−140

−120

−100

−80

−60

−40

−20

0

20

Frequency [Hz]

dB


2 minimized

e)

100 200 300 400 500 600−160

−140

−120

−100

−80

−60

−40

−20

0

20

Frequency [Hz]

dB


2 minimized

f)

Figure 4: Comparison between the original FRF and the FRF for the minimized norm H2 of the a) 7thvibration mode. b) 8th vibration mode. c) 9th vibration mode. d) 10th vibration mode. e) 11th vibrationmode. f) 12th vibration mode.

The figures above show that the original FRF’s resonance peaks within the range 100-600 Hz dropped in allof the situations when compared to the FRFs plotted from the minimization of norm H2 of each vibrationmode. This confirms that the passive control method adopted of finding the optimal nodal coordinates that


minimize the parameter norm H2 of each mode of vibration was effective. It is also worth mentioningthat there were dramatic decreases in the amplitudes of the resonance peaks between the range 200-300 Hzwhereas for the range 450-600 Hz the amplitudes decreased slighter. Note also that in figures 4 c) and d)for the range 450-600 Hz, there was a more significant change in the natural frequencies than for the range200-300 Hz. It was chosen to analyse as well how the 3 last modes of vibration behave in the two simulationsillutrated in figures 4 c) and d). These new vibration modes are represented in figures 5 and 6

a) b) c)

Figure 5: New coordinates of each node considering the minimization of norm H2 of the 9th vibration modefor the a) Second Torsion. b) Third Bending. c) Fourth Bending.

a) b) c)

Figure 6: New coordinates of each node considering the minimization of norm H2 of the 10th vibrationmode for the a) Second Torsion. b) Third Bending. c) Fourth Bending.

Appendix B illustrates the FRFs shown in figure 4 and it points out that the FRFs are indeed inside theenvelope function. It indicates that the new coordinates of the nodes as well as the FRFs are convenient.

9 Conclusion

This work suggests the study of a passive control method applied to a space truss. In this scenario, the tra-ditional finite element method has been used. The concept of norm H2 has been introduced. This parameterserves as a measure of intensity of a structure’s response to standard excitations. Together with the FEM, theoptimization technique simulated annealing has been utilized to find the optimal coordinates of each node ofthe truss that minimize the value of norm H2 of 6 modes of vibration within the range 100-600 Hz.

Then, for every new nodal coordinates, a couple of new frequency responses could be obtained. Admittedly,it was shown that the resonance peaks of the original FRF decreased in all of the situations when comparedto the FRFs obtained by the minimization of norm H2 of each vibration mode. It shows that the passivecontrol method used in the study was effective.

Moreover, the paper applies the Monte Carlo simulation in order to take into account the uncertaties relatedto the the position of each node. By these means, it was possible to check if all of the FRFs remained insidethe envelope function of FRFs. The analysis indicated that the new coordinates of the nodes were convenientfor the study since all of the new FRFs stayed inside the envelope function.


Acknowledgements

The authors acknowledge and thank CAPES (Coordenação de aperfeiçoamento de pessoal de nível superior)for the financial support during the research.

References

[1] P. J. P. Gonçalves, M. J. Brennan, S. Elliott, Active Vibration Control of Space Truss Structures: PowerAnalysis and Energy Distribution, Proceedings of ISMA-International Conference on Noise and Vibra-tion, (2006)

[2] V. Steffen Jr, D. A. Rade, D. J. Inman, Using passive techniques for vibration damping in mechanicalsystems, Journal of the Brazilian Society of Mechanical Sciences, Vol. 22, No. 3,(2000), pp. 411-421.

[3] M. Moshrefi-Torbati, A. J. Keane, S. J. Elliott, M. J. Brennan, D. K. Anthony, E. Rogers, Active vibrationcontrol (AVC) of a satellite boom structure using optimally positioned stacked piezoelectric actuators,Journal of Sound and Vibration, Vol. 292, No. 1, (2006), pp. 203-220.

[4] G. L. C. M. De Abreu, V. Lopes Jr, H2 optimal control for smart truss structure,(2010).

[5] Y. Fang, J. Fei, S. Wang, H-Infinity control of mems gyroscope using ts fuzzy model, IFAC-PapersOnLine,Vol. 48, No. 14, (2015), pp. 241-246.

[6] I. O. De Zarate, J. Aguirrebeitia, R. Avilés, I. Fernández, A finite element approach to the inverse dy-namics and vibrations of variable geometry trusses, Finite Elements in Analysis and Design, Vol. 47,No. 3, (2011), pp. 220-228.

[7] K. Koohestani, A. Kaveh, Efficient buckling and free vibration analysis of cyclically repeated space trussstructures, Finite Elements in Analysis and Design, Vol. 46, No. 10, (2010), pp. 943-948.

[8] S. W. Doebling, C. R. Farrar, Estimation of statistical distributions for modal parameters identified fromaveraged frequency response function data, Journal of Vibration and Control, Vol. 7, No. 4, (2011), pp.603-624.

[9] W. A. Bennage, A. K. Dhingra, Single and multiobjective structural optimization in discrete-continuousvariables using simulated annealing, International Journal for Numerical Methods in Engineering, Vol.38, No. 16, (1995), pp. 2753-2773.

[10] L. Lamberti, An efficient simulated annealing algorithm for design optimization of truss structures,Computers & Structures, Vol. 86, No. 19, (2008), pp. 1936-1953.

[11] O. Hasançebi, F. Erbatur, Layout optimisation of trusses using simulated annealing, Advances in Engi-neering Software, Vol. 33, No. 7, (2002), pp. 681-696.

[12] Y. W. Kwon, H. Bang, The Finite Element Method Using MATLAB, CRC press,(2000).

[13] W. Gawronski, Advanced Structural Dynamics and Active Control of Structures, Springer Science &Business Media,(2004).

[14] R. W. Clough, J. Penzien, Dynamics of Structures, Tech. report,(1975).

[15] T. E. Fricker, J. E. Oakley, N. D. Sims, K. Worden, Probabilistic uncertainty analysis of an FRF of astructure using a gaussian process emulator, Mechanical Systems and Signal Processing, Vol. 25, No.8, (2011), pp. 2962-2975.

[16] J. Hammersley, Monte Carlo Methods, Springer Science & Business Media,(2013).


Appendix A: Convergence of the Simulated Annealing

100

101

103

104

105

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Fun

ctio

n va

lue

Best Function Value: 4.572e−009

Iteration

a)

100

101

103

104

105

10−9

10−8

10−7

10−6

10−5

10−4

Best Function Value: 2.77984e−009

Iteration

Fun

ctio

n va

lue

b)

100

101

103

104

105

10−5

10−4

10−3

Iteration

Fun

ctio

n va

lue

Best Function Value: 6.85678e−005c)

100

101

103

104

105

106

10−4.4

10−4.3

Iteration

Fun

ctio

n V

alue

Best Function Value: 3.1917e−005d)

100

101

103

104

105

10−10

10−9

10−8

10−7

10−6

10−5

10−4

Iteration

Fun

ctio

n V

alue

Best Function Value: 2.59061e−010e)

100

101

103

104

105

10−9

10−8

10−7

10−6

10−5

10−4

Iteration

Fun

ctio

n V

alue

Best Function Value: 2.54604e−009f)

Figure 7: Convergence of normH2 using Simulated Annealing for the a) 7th vibration mode. b) 8th vibrationmode. c) 9th vibration mode. d) 10th vibration mode. e) 11th vibration mode. f) 12th vibration mode.


Appendix B: Monte Carlo technique application

Figure 8: Envelope functions with FRFs of the a) 7th vibration mode. b) 8th vibration mode. c) 9th vibrationmode. d) 10th vibration mode. e) 11th vibration mode. f) 12th vibration mode.


passive control applied to structural optimization and...

Documents