**1. Introduction**

The study of rotating machinery appears in the context of machines and structures due to the significant number of phenomena typical to their operation that impact their dynamic behavior and maintenance. Consequently, rotor bearing systems face numerous problems that affect a wide variety of machines, e.g., compressors, pumps, motors, centrifuge machines, large and small turbines. This type of machine finds various applications in the industry, such as, automotive, aerospace and power generation. In most applications an unpredictable stoppage can lead to considerable financial losses and risks. Therefore, there is an evident need for the complete modelling of rotating systems, including the components of the interface between fixed and moveable parts, such as the hydrodynamic bearings. Bench-scale experimental analyses provide more complete models of the main components of the rotor, with strong emphasis on the modelling of the bearings of rotary machines, since they constitute the rotor-foundation structure connecting elements.

The machinery parameters are needed to study the dynamic behavior of the system, namely the Campbell diagram, stability analysis, critical speeds, excitation responses, control and health monitoring. The determination of unknown parameters in rotating machinery is a difficult task. To overcome this difficulty, the use of optimization techniques to solve the inverse problem represents an important alternative approach.

In the literature, various works have been proposed to determine unknown parameters of dynamic systems. Edwards et al. [1] presented a procedure to determine unbalance and support parameters simultaneously based on the least-squares method. Xu et al. [2] proposed a rotor balancing method by using optimization techniques, which does not need

trial weights. Assis and Steffen [3] developed strategies in order to use optimization techniques for determining the parameters of gyroscopic systems and they commented about the difficulties that arise in using classical optimization algorithms due to their difficulty in avoiding local minima. The properties of the supports located at the ends of the rotor were considered as variables in the optimization procedure. An inverse problem was developed by using a hybrid cascade-type optimization scheme considering a single unbalance distribution. Castro et al. [4] proposed an optimization method based on genetic algorithms to tune displacements of the rotor supported by hydrodynamic bearings. Castro et al. [5] applied a hybrid algorithm based on genetic algorithm and simulated annealing to tune the orbits of the rotary system in the critical region. In this search algorithm, the genetic algorithm is applied in order to make an approximation of the optimal result, while the simulated annealing refines this result. Tiwari and Chakravarthy [6] presented an identification algorithm for simultaneous estimation of the residual unbalances and the bearing dynamic parameters by using the impulse response measurements for multi-degreeof-freedom flexible rotor-bearing systems. Kim et al. [7] presented a bearing parameter identification of rotor–bearing system using clustering-based hybrid evolutionary algorithm. Castro et al. [8] applied multi-objective genetic algorithm to identify unbalance parameters. Nauclér and Söderstöm [9] consider the problem of unbalance estimation of rotating machinery based on the development of a novel method which takes disturbances into account, leading to a nonlinear estimator. More recently, Saldarriaga et al. [10] proposed a methodology for the experimental determination of the unbalance distribution on highly flexible rotating machinery using Genetic Algorithms. Modal analysis techniques were previously performed to obtain an initial guess for the unknown parameters. A pseudo-random optimization-based approach was used first to identify the parameters of the system in such a way that a reliable rotor model was obtained. Satisfactory results encouraged the use of the proposed approach in the industrial context. Sudhakar and Sekhar [11] proposed a method dedicated to fault identification in a rotor bearing system by minimizing the difference between equivalent loads estimated in the system due to the fault and theoretical fault model loads. This method has a limitation since the error found in the identified fault parameters increases when decreasing the number of measured experimental data.

Design and Identification Problems of Rotor Bearing Systems Using the Simulated Annealing Algorithm 199

Metropolis' algorithm can be used to generate sequences of configurations in a combinatorial optimization problem. SA is seen as a sequence of Metropolis algorithms, executed with a decreasing sequence of the control parameter. The temperature (control parameter) is continually reduced after a certain number of neighbourhood searches in the

It is worth mentioning that although the SA is a powerful and important optimization tool, often it is not applied according to strict adherence to sufficiency conditions, permitting the researcher to truly claim that the optimal solution has been (statistically) found. According to Ingber [13], the reason typically given is simply that many variants of this technique are considered to be too consuming of resources to be applied in such strict fashion. There exist faster variants of SA canonical, but these apparently are not as quite easily coded and so they are not widely used. Many modifications of SA are really quenching, and should aptly

In the present contribution, the canonical SA, e.g., based on the algorithm proposed by Kirkpatrick et al. [14] to include a temperature schedule for efficient searching, is used for the design and identification of rotor bearing systems. The goal for the first problem presented is to increase the difference between two critical speeds of a rotor-bearing system that was previously modelled by using the finite element method. In this case, the design variables are the parameters of the rotor-bearing system. To solve this multi-criteria optimization problem a methodology based on a combination of SA, non-dominated sorting strategy and crowding distance operator for guaranteeing convergence and diversity of potential candidates in the population is proposed. The second problem studied is related to the identification of unknown parameters of flexible rotor-bearing systems, modelled mathematically by using the finite element method. The difference between the unbalance experimental responses of the rotor and the simulated unbalance responses (obtained by using the mathematical model) is used to write the objective function to be minimized, so that the damping and stiffness parameters are found. For illustration purposes, the *experimental* (synthetic) data used were generated by using the solution of the direct problem

This chapter is organized as follows. The rotor bearing formulation is revisited in Section 4. In Sections 5 and 6 the main characteristics of the SA and multi-objective optimization are briefly presented, respectively. The Multi-objective Optimization Simulated Annealing (MOSA) proposed in this work is described in Section 7. The results and discussion are presented in Section 8. Finally, the conclusions and suggestions for future work complete the chapter.

The mathematical model used to calculate the unbalance forces, natural frequencies and vibration mode shapes is obtained by using the Finite Element Method. The discrete rotor model is composed of symmetric rigid disc elements, symmetric Timoshenko beam elements, nonsymmetric coupling elements, and nonsymmetric viscous damped bearings,

current state.

be called simulated quenching (SQ).

to which artificial noise was added.

**2. Rotor bearing modelling** 

as presented in Figure 1.

In this context, the present chapter discusses the possibility of using the Simulated Annealing algorithm (SA) for the identification of unknown parameters of a rotor model from the unbalanced response of the system. Basically, the SA algorithm exploits the analogy between the search for a minimum in the optimization problem and the process of gradual cooling of a metal in a crystalline structure of minimal energy. A desirable characteristic of a minimum search method is the ability to avoid the convergence to a local optimal point, e.g., in terms of the physical process of annealing a meta-stable structure is obtained in the end. Thus, the paradigm of SA is to offer means of escaping from local optima through the analysis of the neighbourhood of the current solution, which can assume, within a given probability, worse solutions, but makes the finding of a new path to the global optimum possible. Metropolis et al. [12] presented an algorithm that simulates the evolution of a crystalline structure in the liquid state up to its thermal equilibrium. Metropolis' algorithm can be used to generate sequences of configurations in a combinatorial optimization problem. SA is seen as a sequence of Metropolis algorithms, executed with a decreasing sequence of the control parameter. The temperature (control parameter) is continually reduced after a certain number of neighbourhood searches in the current state.

It is worth mentioning that although the SA is a powerful and important optimization tool, often it is not applied according to strict adherence to sufficiency conditions, permitting the researcher to truly claim that the optimal solution has been (statistically) found. According to Ingber [13], the reason typically given is simply that many variants of this technique are considered to be too consuming of resources to be applied in such strict fashion. There exist faster variants of SA canonical, but these apparently are not as quite easily coded and so they are not widely used. Many modifications of SA are really quenching, and should aptly be called simulated quenching (SQ).

In the present contribution, the canonical SA, e.g., based on the algorithm proposed by Kirkpatrick et al. [14] to include a temperature schedule for efficient searching, is used for the design and identification of rotor bearing systems. The goal for the first problem presented is to increase the difference between two critical speeds of a rotor-bearing system that was previously modelled by using the finite element method. In this case, the design variables are the parameters of the rotor-bearing system. To solve this multi-criteria optimization problem a methodology based on a combination of SA, non-dominated sorting strategy and crowding distance operator for guaranteeing convergence and diversity of potential candidates in the population is proposed. The second problem studied is related to the identification of unknown parameters of flexible rotor-bearing systems, modelled mathematically by using the finite element method. The difference between the unbalance experimental responses of the rotor and the simulated unbalance responses (obtained by using the mathematical model) is used to write the objective function to be minimized, so that the damping and stiffness parameters are found. For illustration purposes, the *experimental* (synthetic) data used were generated by using the solution of the direct problem to which artificial noise was added.

This chapter is organized as follows. The rotor bearing formulation is revisited in Section 4. In Sections 5 and 6 the main characteristics of the SA and multi-objective optimization are briefly presented, respectively. The Multi-objective Optimization Simulated Annealing (MOSA) proposed in this work is described in Section 7. The results and discussion are presented in Section 8. Finally, the conclusions and suggestions for future work complete the chapter.
