**5. Multi-objective optimization simulated annealing – MOSA**

Due to the success obtained by the SA in different science and engineering applications, their extension to the multi-objective context is desirable. In this work, the Multi-objective Optimization Simulated Annealing (MOSA) algorithm is proposed. This approach is based on the classical SA associated with the so-called Fast Non-Dominated Sorting operator and has the following structure:


The steps presented are repeated until a determined stopping criterion is reached. The operators used in the MOSA are described below.

### **5.1. Fast non-dominated sorting**

204 Simulated Annealing – Single and Multiple Objective Problems

local Pareto-optimal sets in multi-objective optimization.

Basically, the main features of these MOEAs are [20,21]:

allow for maximum front expansion.

to produce the offspring.

solutions.

optimal solutions in the case of single-objective optimization, there could be global and

In the multi-objective context, various Multiple-Objective Evolutionary Algorithms (MOEAs) can be found. This group of algorithms conjugates the basic concepts of dominance described above with the general characteristics of evolutionary algorithms.

• **Mechanism of adaptation assignment in terms of dominance** - between a nondominated solution and a dominated one, the algorithm will favour the non-dominated solution. Moreover, when both solutions are equivalent in dominance, the one located in a less crowded area will be favoured. Finally, the extreme points (e.g. the solutions that have the best value in one particular objective) of the non-dominated population are preserved and their adaptation is better than any other non-dominated point, to

• **Incorporation of elitism** - the elitism is commonly implemented using a previously stored secondary population of non-dominated solutions. When performing recombination (selection-crossover-mutation), parents are taken from this file in order

In the literature, various multi-objective algorithms based on SA have been proposed. Basically, the first extensions were proposed by Serafini [24,25] and by Ululgu and Teghem [26], where various ways of defining the probability in the multi-objective framework and how they affect the performance of SA based multi-objective algorithms. Czyzak et al. [27] combined mono-criterion SA and genetic algorithm to provide efficient solutions for multi-criteria shortest path problem. Ulungu et al. [28] designed a MOSA (Multi-objective Optimization Simulated Annealing) algorithm and tested its performance using multi-objective combinatorial optimization problems. Suppapitnarm et al. [29] used the neighbourhood perturbation method to create a new point around an old point using MOSA. In this algorithm, the single objective SA is modified to give a set of non-dominated solutions by using archiving of solutions generated earlier, and using a sorting procedure (based on non-dominance and crowding). Kasat et al. [30] used the concept of jumping genes in natural genetics to modify the binary-coded non-dominated sorting genetic algorithm (NSGA-II) to give NSGA-II-JG. Smith et al. [31] compared the candidate to the current solution according to the cardinalities of their dominant subsets in the file. Marcoulaki and Papazoglou [32] proposed a new multiple objective optimization approach by using a Monte Carlo-based algorithm stemmed from SA. Since the expected result in a multiple objective optimization task is usually a set of Paretooptimal solutions, the optimization problem states assumed here are themselves sets of

**5. Multi-objective optimization simulated annealing – MOSA** 

Due to the success obtained by the SA in different science and engineering applications, their extension to the multi-objective context is desirable. In this work, the Multi-objective The so-called Fast Non-Dominated Sorting operator was proposed by Deb et al. [21] in order to sort a population of size N according to the level of non-domination. Each solution must be compared with every other solution in the population to find if the solution is dominated. This requires O(MN) comparisons for each solution, where M is the number of objective functions. When this process is continued to find the members of the first non-dominated class for all population members, the total complexity is O(MN2). At this point, all individuals in the first non-dominated front are found. In order to obtain the individuals in the next front, the solutions of the first front are temporarily discarded and the above procedure is repeated. In the worst case, the task of obtaining the second front also requires O(MN2) computations. The procedure is repeated so that subsequent fronts are found.

#### **5.2. Crowding distance operator**

This operator describes the density of solutions surrounding a vector. To compute the Crowding Distance for a set of population members the vectors are sorted according to their objective function value for each objective function. To the vectors with the smallest or largest values, an infinite Crowding Distance (or an arbitrarily large number for practical purposes) is assigned. For all other vectors, the Crowding Distance (*distxi*) is calculated according to [20,21]:

$$dist\_{x\_i} = \sum\_{j=1}^{m} \frac{f\_{j,i+1} - f\_{j,i-1}}{\left| f\_{j\max} - f\_{j\min} \right|} \tag{8}$$

where *f*j corresponds to the *j*-th objective function and *m* equals the number of objective functions. This operator is important to avoid many points close together in the Pareto's Front and to promote the diversity in terms of space objectives [21].

#### **5.3. Consideration of constraints**

In this work, the treatment of constraints is made through the Static Penalization Method, proposed by Castro [33]. This approach consists in assigning limit values to each objective function to play the role of penalization parameters. According to Castro [33], it is guaranteed that any non-dominated solution dominates any solution that violates at least one of the constraints. In the same way, any solution that violates only one constraint will dominate any solution that presents two constraint violations, and so on. For a constrained problem the vector containing the objective functions to be accounted for, is given by:

$$f(\mathbf{x}) \equiv f(\mathbf{x}) + r\_p n\_{\rm void} \tag{9}$$

Design and Identification Problems of Rotor Bearing Systems Using the Simulated Annealing Algorithm 207

**External Diameter (mm)** 

min 5 <sup>1</sup> inf 1 *<sup>c</sup> f x ROT a v* = − (10)

min 6 <sup>2</sup> *<sup>c</sup>* 2 sup *f x v a ROT* = − (11)

**Thickness (mm)** 

The material used for the shaft and disks is the steel-1020 (density = 7800 Kg/m3, Elasticity modulus = 2.1E11 N/m2 and Poisson coefficient = 0.3). The shaft geometry is so that the diameter and length are 10 mm and 552 mm, respectively. The geometric characteristics of

**Inertia (Kg m2)** 

1 0.818 0.0008 90 16.0 2 1.600 0.0045 150 11.2 3 0.981 0.0018 120 10.6

( ) ( ) ( ) <sup>2</sup>

() () ( )<sup>2</sup>

where *vc* is the critical speeds vector, *ROT*i are the permissible rotations (*i*=inf or sup), *a*1=1.3

For evaluating the methodology proposed in this work, some practical points regarding the

1. The design variables are the following: radius of bar elements (*xi*), where the design

• Non-dominated Sorting Genetic Algorithm (NSGAII) parameters [20,36]: population size (50), crossover probability (0.8), mutation probability (0.01). For the considered parameters, the number of objective function evaluations is 12550. • Multi-objective Optimization Differential Evolution (MODE) parameters [37]: population size (50), perturbation rate (0.8), crossover probability (0.8), DE/rand/1/bin strategy for the generation of potential candidates, reduction rate (0.9) and number of pseudo-curves (10). For the considered parameters, the

• Multi-objective Optimization Simulated Annealing (MOSA) parameters [14]: population size (50), initial temperature (5.0), cooling rate (0.75), number of temperatures (20), number of times the procedure is repeated before the temperature is reduced (25), and tolerance (10-6). For the considered parameters,

3. To solve the optimization problem the following heuristics are used:

number of objective function evaluations is 15050.

the number of objective function evaluations is 12550.

4. Stopping criterion: maximum number of generations (250).

the disks are presented in Table 2.

**Table 2.** Geometric characteristics of the disks.

and *a*2=1.3.

**Disc Mass (Kg) Moment of** 

Mathematically, the optimization problem can be formulated as:

application of this procedure should be emphasized:

space is given by: 0.4 mm ≤ *xi* ≤ 0.8 mm. 2. *ROT*inf = 1400 Hz and *ROT*inf = 1900 Hz.

where *f*(*x*) it is the vector of objective functions, *rp* it is the vector of penalty parameters that depends on the type of problem considered, and *nviol* is the number of violated constraints.

## **6. Applications**

#### **6.1. Rotor-dynamics design**

Modern design of rotor-bearing systems usually aims at increasing power output and improved overall efficiency. The demanding requirements placed on modern rotating machines, such as turbines, electric motors, electrical generators, compressors, turbo-pumps, have introduced a need for higher speeds and lower vibration levels [34]. This problem can be formulated as a multi-objective problem aiming at minimizing, for instance, the total weight of the shaft, the transmitted forces at the bearings and the positions of the critical speeds [35]. In this context, the present application considers the maximization of the difference between the 6th and 5th critical speeds for the system whose finite element model is composed of rigid disks with seventeen elements, two bearings and two additional masses, as shown in Figure 3.

**Figure 3.** Finite element model of the rotor-bearing system.

The material used for the shaft and disks is the steel-1020 (density = 7800 Kg/m3, Elasticity modulus = 2.1E11 N/m2 and Poisson coefficient = 0.3). The shaft geometry is so that the diameter and length are 10 mm and 552 mm, respectively. The geometric characteristics of the disks are presented in Table 2.


**Table 2.** Geometric characteristics of the disks.

206 Simulated Annealing – Single and Multiple Objective Problems

In this work, the treatment of constraints is made through the Static Penalization Method, proposed by Castro [33]. This approach consists in assigning limit values to each objective function to play the role of penalization parameters. According to Castro [33], it is guaranteed that any non-dominated solution dominates any solution that violates at least one of the constraints. In the same way, any solution that violates only one constraint will dominate any solution that presents two constraint violations, and so on. For a constrained

problem the vector containing the objective functions to be accounted for, is given by:

where *f*(*x*) it is the vector of objective functions, *rp* it is the vector of penalty parameters that depends on the type of problem considered, and *nviol* is the number of violated constraints.

Modern design of rotor-bearing systems usually aims at increasing power output and improved overall efficiency. The demanding requirements placed on modern rotating machines, such as turbines, electric motors, electrical generators, compressors, turbo-pumps, have introduced a need for higher speeds and lower vibration levels [34]. This problem can be formulated as a multi-objective problem aiming at minimizing, for instance, the total weight of the shaft, the transmitted forces at the bearings and the positions of the critical speeds [35]. In this context, the present application considers the maximization of the difference between the 6th and 5th critical speeds for the system whose finite element model is composed of rigid disks with seventeen elements, two bearings and two additional

() () *<sup>p</sup> viol f x f x rn* ≡ + (9)

**5.3. Consideration of constraints** 

**6. Applications** 

**6.1. Rotor-dynamics design** 

masses, as shown in Figure 3.

**Figure 3.** Finite element model of the rotor-bearing system.

Mathematically, the optimization problem can be formulated as:

$$\min \, f\_1(\mathbf{x}) = \left( ROT\_{\text{inf}} - a\_1 v\_c \, \text{(5)} \right)^2 \tag{10}$$

$$\min \, f\_2\left(\mathbf{x}\right) = \left(v\_c\left(\mathbf{6}\right) - a\_2 ROT\_{\text{sup}}\right)^2 \tag{11}$$

where *vc* is the critical speeds vector, *ROT*i are the permissible rotations (*i*=inf or sup), *a*1=1.3 and *a*2=1.3.

For evaluating the methodology proposed in this work, some practical points regarding the application of this procedure should be emphasized:

	- Non-dominated Sorting Genetic Algorithm (NSGAII) parameters [20,36]: population size (50), crossover probability (0.8), mutation probability (0.01). For the considered parameters, the number of objective function evaluations is 12550.
	- Multi-objective Optimization Differential Evolution (MODE) parameters [37]: population size (50), perturbation rate (0.8), crossover probability (0.8), DE/rand/1/bin strategy for the generation of potential candidates, reduction rate (0.9) and number of pseudo-curves (10). For the considered parameters, the number of objective function evaluations is 15050.
	- Multi-objective Optimization Simulated Annealing (MOSA) parameters [14]: population size (50), initial temperature (5.0), cooling rate (0.75), number of temperatures (20), number of times the procedure is repeated before the temperature is reduced (25), and tolerance (10-6). For the considered parameters, the number of objective function evaluations is 12550.

5. Each algorithm was run 20 times by using 20 different seeds for the random generation of the initial population.

Design and Identification Problems of Rotor Bearing Systems Using the Simulated Annealing Algorithm 209

*x*1 (mm) 0.55186 0.40015 0.50692 *x*2 (mm) 0.49994 0.42153 0.52587 *x*3 (mm) 0.53139 0.40623 0.50733 *x*4 (mm) 0.79128 0.40000 0.50851 *x*5 (mm) 0.49810 0.40000 0.50726 *x*6 (mm) 0.55709 0.40000 0.50025 *x*7 (mm) 0.41059 0.40000 0.50811 *x*8 (mm) 0.71018 0.40000 0.50683 *x*9 (mm) 0.57777 0.40151 0.50713 *x*10 (mm) 0.44676 0.41028 0.50192 *x*11 (mm) 0.79883 0.40000 0.49997 *x*12 (mm) 0.67159 0.40000 0.50671 *x*13 (mm) 0.57684 0.40000 0.50592 *x*14 (mm) 0.46109 0.40000 0.51012 *x*15 (mm) 0.52452 0.40000 0.50257 *x*16 (mm) 0.61355 0.40399 0.50747 *x*17 (mm) 0.57261 0.40281 0.50533 *f*1 (Hz2) 0.00364 1110741.5 479693.69 *f*2 (Hz2) 1842938.0 225219.04 602024.24

**Table 3.** Results obtained using MOSA (all the algorithms were executed 20 times so that average

function between experimental and simulated data is minimized.

4

**Figure 5.** Rotor system finite element model.

Furthermore, identification procedures try to establish an unequivocal relation in between the damage and specific mechanical parameters, based on a suitable model and can be used to fault detection and machinery diagnosis as in Seibold and Fritzen [40]. On a simple manner parameter identification of rotor-bearing systems can be performed as follows: *i*) the frequency response function (or unbalance response) is measured for different operation speeds; *ii*) the design variables (unknown parameters) are initialized; *iii*) and an error

0.4 m

12 3 5 6 8 9 10

7

values were calculated).

**OFA OFB OFC**

6. Objective Function (OFA) is the best value of objective function considering the first objective proposed. Objective Function (OFB) is the best value of objective function considering the second objective function. Objective Function (OFC) is calculated using the origin of the coordinated axes as a reference, e.g., the point (0,0) is used to obtain the distance between this point and each one of the solution points along the Pareto's Front. Thus, the smallest distance obtained was defined as the choice criterion.

Figure 4 shows the Pareto's Front obtained by NSGA II, MODE and MOSA algorithms.

**Figure 4.** Pareto's Front.

In this figure it is possible to observe that all evolutionary algorithms are able to obtain, satisfactorily, the Pareto's Front for a similar number of objective function evaluations.

Table 3 present some points of Pareto's Front obtained by the MOSA algorithm by considering the criteria specified earlier.

### **6.2. Identification**

As mentioned earlier, the identification of unknown parameters in rotating machinery is a difficult task and optimization techniques represent an important alternative for this goal [3,38,39]. The machine parameters are needed to perform the dynamic analysis and prediction of rotor-bearing systems: Campbell diagram, stability, critical speeds, excitation responses [15]. Another important aspect is when one desires to tune a finite element model to match experimental data generated by tests of an actual rotor system [10].


**Table 3.** Results obtained using MOSA (all the algorithms were executed 20 times so that average values were calculated).

Furthermore, identification procedures try to establish an unequivocal relation in between the damage and specific mechanical parameters, based on a suitable model and can be used to fault detection and machinery diagnosis as in Seibold and Fritzen [40]. On a simple manner parameter identification of rotor-bearing systems can be performed as follows: *i*) the frequency response function (or unbalance response) is measured for different operation speeds; *ii*) the design variables (unknown parameters) are initialized; *iii*) and an error function between experimental and simulated data is minimized.

**Figure 5.** Rotor system finite element model.

208 Simulated Annealing – Single and Multiple Objective Problems

of the initial population.

**Figure 4.** Pareto's Front.

**6.2. Identification** 

considering the criteria specified earlier.

0

0.5

1

*f*

*2(x)*

1.5

<sup>2</sup> x 10<sup>6</sup>

5. Each algorithm was run 20 times by using 20 different seeds for the random generation

6. Objective Function (OFA) is the best value of objective function considering the first objective proposed. Objective Function (OFB) is the best value of objective function considering the second objective function. Objective Function (OFC) is calculated using the origin of the coordinated axes as a reference, e.g., the point (0,0) is used to obtain the distance between this point and each one of the solution points along the Pareto's Front.

Thus, the smallest distance obtained was defined as the choice criterion.

Figure 4 shows the Pareto's Front obtained by NSGA II, MODE and MOSA algorithms.

In this figure it is possible to observe that all evolutionary algorithms are able to obtain, satisfactorily, the Pareto's Front for a similar number of objective function evaluations.

0 2 4 6 8 10 12

NSGA II MODE MOSA

*f <sup>1</sup>(x)*

x 10<sup>5</sup>

Table 3 present some points of Pareto's Front obtained by the MOSA algorithm by

As mentioned earlier, the identification of unknown parameters in rotating machinery is a difficult task and optimization techniques represent an important alternative for this goal [3,38,39]. The machine parameters are needed to perform the dynamic analysis and prediction of rotor-bearing systems: Campbell diagram, stability, critical speeds, excitation responses [15]. Another important aspect is when one desires to tune a finite element model

to match experimental data generated by tests of an actual rotor system [10].

In this application, a simple flexible rotor containing two disks and two bearings is studied. The Figure 5 shows the finite element model of the system with 10 nodes, 2 disks and two bearings.

The characteristics of bearing and disks are given in Table 4. It can be observed that damping parameters are also taken into account in this application.


**Table 4.** Parameters of bearings and disks.

The goal of this application is to identify the unknown parameters of the rotor-bearing system, e.g., the stiffness and damping parameters. For this purpose the following steps are established:

1. The objective function consists in the determination of the stiffness and damping values through the minimization of the difference between the experimental and calculated values given by the solution of the direct problem. To mimic real experimental data, sets of synthetic experimental data were generated from eq. (12):

$$\boldsymbol{\theta}\_{i}^{\text{exp}} = \boldsymbol{\theta}\_{i}^{\text{cal}} \left( \boldsymbol{Z}\_{\text{exact}} \right) + \mathbf{\varkappa} \boldsymbol{\aleph} \tag{12}$$

Design and Identification Problems of Rotor Bearing Systems Using the Simulated Annealing Algorithm 211

• Genetic Algorithm (GA) parameters [41]: population size (50), type of selection (normal geometric in the range [0 0.08]), type of crossover Arithmetic, 2), type of

• Differential Evolution (DE) parameters [42]: population size (25), perturbation rate

• Particle Swarm Optimization (PS) parameters [43]: population size (25), maximum velocity (100), upper limits (2.0), and a linearly decreasing inertia weight starting at

• Simulated Annealing (SA) parameters [14]: initial design (generated randomly in the design space), initial temperature (5.0), cooling rate (0.75), number of temperatures (20), number of times the procedure is repeated before the

and crossover probability both equal to 0.8 and DE/rand/1/bin strategy.

3. Stopping criterion: maximum number of objective function evaluations equal to 5000. 4. Each algorithm was run 20 times by using 20 different seeds for the random generation

Table 5 presents the results obtained by the algorithms considered (pristine condition and

Considering κ=0 (see Table 5), all the optimization strategies were able to estimate the parameters satisfactorily as shown by the values obtained for the objective function. However, the SA algorithm shows to be very competitive, in averege, with the smallest standard deviation of the objective function. When noise is taken into account (κ=0.002, e.g., error corresponding to 5%), all the algorithms were able to obtain good estimates, as

> *kxz*  (N/m)

0 % 984674.3 1802257.4 24197.8 1157.4 1953.4 168.4 4.38

5 % 962861.5 1817105.5 18378.3 1046.6 2843.4 35.8 7.01

0 % 998204.7 2030908.5 3589.1 1078.8 2068.1 149.7 4.42

5 % 986445.7 2114381.8 8695.9 1010.5 2164.4 25.8 07.11

0 % 1000380.9 2005635.1 17449.8 1102.4 2092.8 183.7 4.35

5 % 970336.4 1964371.6 46943.8 1037.2 2347.5 84.5 6.96

0 % 987254.1 2098581.6 57858.5 1079.1 1425.6 119.8 4.40

5 % 990895.2 2042193.6 15630.8 1118.9 1937.8 189.7 7.09

*cxx*  (Nm/s) *czz* 

(Nm/s) *cxz* (Nm/s) OF

2. To solve the optimization problem the following heuristics are used:

temperature is reduced (25), and tolerance (10-6).

*kzz*  (N/m)

mutation (non-uniform [2 100 3]).

0.7 and ending at 0.4 was used.

of the initial population.

noisy data).

SA

GA

DE

PS

presented in Figure 6.

Error *kxx* 

**Table 5.** Estimation Results.

(N/m)

were *θ* represents the calculated values of the unbalance response by using known values of the physical properties *Zexact* (*kxx*, *kzz*, *kxz*, *cxx*, *czz*, *cxz*). In real applications these values are not available (they can be obtained through the solution of the corresponding inverse problem). κ simulate the standard deviation of the measurement errors, and *λ* is a pseudo-random number from the interval [-1, 1].

$$\boldsymbol{\Theta}\_{i}^{\text{exp}} = \boldsymbol{\Theta}\_{i}^{\text{cal}} \left( \boldsymbol{Z}\_{\text{exact}} \right) + \boldsymbol{\kappa} \boldsymbol{\mathcal{X}} \tag{13}$$

In order to examine the accuracy of the inverse problem approach for the estimation of the physical parameters, the influence of noise (κ =0.02, e.g., corresponding to 5% error) was compared to the case without noise (κ =0).

The design variables considered to generate the synthetic experimental data are presented in Table 5. The following ranges for the design space are considered: 5.0E05 N/m ≤ *kxx* ≤ 1.2E07 N/m, 1.0E06 N/m ≤ *kzz* ≤ 2.4E07 N/m, 5.0E03 N/m ≤ *kxz* ≤ 1.2E05 N/m, 5.0E02 Nm/s ≤ *cxx* ≤ 1.2E04 Nm/s, 1E03 Nm/s ≤ *czz* ≤ 2.4E04 Nm/s, 1E01 Nm/s ≤ *cxz* ≤ 1.2E03 Nm/s.

Design and Identification Problems of Rotor Bearing Systems Using the Simulated Annealing Algorithm 211

2. To solve the optimization problem the following heuristics are used:

210 Simulated Annealing – Single and Multiple Objective Problems

**Table 4.** Parameters of bearings and disks.

number from the interval [-1, 1].

compared to the case without noise (κ =0).

bearings.

established:

In this application, a simple flexible rotor containing two disks and two bearings is studied. The Figure 5 shows the finite element model of the system with 10 nodes, 2 disks and two

The characteristics of bearing and disks are given in Table 4. It can be observed that

**Bearing Stiffness(N/m) Damping (Nm/s)**

A=B 1E06 2E06 1E04 1E03 2E03 1E02

Disk External Diameter (m) Thickness (m) 1 0.5 0.005 2 0.3 0.005

The goal of this application is to identify the unknown parameters of the rotor-bearing system, e.g., the stiffness and damping parameters. For this purpose the following steps are

1. The objective function consists in the determination of the stiffness and damping values through the minimization of the difference between the experimental and calculated values given by the solution of the direct problem. To mimic real experimental data,

( ) exp *cal*

( ) exp *cal*

In order to examine the accuracy of the inverse problem approach for the estimation of the physical parameters, the influence of noise (κ =0.02, e.g., corresponding to 5% error) was

The design variables considered to generate the synthetic experimental data are presented in Table 5. The following ranges for the design space are considered: 5.0E05 N/m ≤ *kxx* ≤ 1.2E07 N/m, 1.0E06 N/m ≤ *kzz* ≤ 2.4E07 N/m, 5.0E03 N/m ≤ *kxz* ≤ 1.2E05 N/m, 5.0E02 Nm/s ≤ *cxx* ≤

 θ

were *θ* represents the calculated values of the unbalance response by using known values of the physical properties *Zexact* (*kxx*, *kzz*, *kxz*, *cxx*, *czz*, *cxz*). In real applications these values are not available (they can be obtained through the solution of the corresponding inverse problem). κ simulate the standard deviation of the measurement errors, and *λ* is a pseudo-random

 κλ

 κλ

*i i exact* = + *Z* (12)

*i i exact* = + *Z* (13)

 θ

sets of synthetic experimental data were generated from eq. (12):

θ

θ

1.2E04 Nm/s, 1E03 Nm/s ≤ *czz* ≤ 2.4E04 Nm/s, 1E01 Nm/s ≤ *cxz* ≤ 1.2E03 Nm/s.

*xx k zz k xz k xx c zz c xz c*

damping parameters are also taken into account in this application.


Table 5 presents the results obtained by the algorithms considered (pristine condition and noisy data).

Considering κ=0 (see Table 5), all the optimization strategies were able to estimate the parameters satisfactorily as shown by the values obtained for the objective function. However, the SA algorithm shows to be very competitive, in averege, with the smallest standard deviation of the objective function. When noise is taken into account (κ=0.002, e.g., error corresponding to 5%), all the algorithms were able to obtain good estimates, as presented in Figure 6.


**Table 5.** Estimation Results.

Design and Identification Problems of Rotor Bearing Systems Using the Simulated Annealing Algorithm 213

The authors acknowledge the financial support provided by FAPEMIG and CNPq (INCT-EIE). The fourth author is grateful to the financial support provided by CNPq and FAPERJ.

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**Author details** 

Antônio José da Silva Neto

**Acknowledgement** 

**8. References** 

1-10.

2005, 1-10.

885-897.

Fran Sérgio Lobato, Elaine Gomes Assis and Valder Steffen Jr

*Universidade Federal de Uberlândia, Brazil* 

*Universidade do Estado do Rio de Janeiro, Brazil* 

**Figure 6.** Boxplots showing the influence of different optimizaion strategies to solve the inverse problem.
