**2. Rotor bearing modelling**

198 Simulated Annealing – Single and Multiple Objective Problems

experimental data.

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

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.

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, as presented in Figure 1.

Design and Identification Problems of Rotor Bearing Systems Using the Simulated Annealing Algorithm 201

**Physical system Optimization problem** State Feasible Solution Energy Cost Function Ground state Optimal solution Rapid quenching Local search Careful annealing Simulated annealing

The basic steps of canonical SA are presented in Figure 2 and described in the following

**Table 1.** Analogy between simulated annealing and optimization.

**Figure 2.** Simulated Annealing algorithm flowchart (*NI* is the number of iterations).

In this iterative technique, an initial guess is randomly generated according to the design space. It should be emphasized that other forms of generating the initial population can be

The control of the 'temperature' parameter must be carefully defined since it controls the acceptance rule defined by the Boltzmann distribution. *T* has to be large enough to enable the algorithm to move off a local minimum but small enough not to move off a global minimum. According to Chibante et al. [18], the value of *T* should be defined in an

subsections [18].

**3.1. Initial population** 

**3.2. Initial temperature** 

used to initialize the optimization process.

**Figure 1.** Rotor references frames.

Two reference systems are considered, namely the inertial frame (*X*,*Y*,*Z*) and the frame (*x*,*y*,*z*) that is fixed to the disk [3]. By using the Lagrange's equations in steady-state conditions, the rotor model is represented by the following matrix differential equation [15]:

$$M\ddot{q} + \mathcal{C}\dot{q} + Kq = F\_1 + F\_2 \sin(\Omega t) + F\_3 \cos(\Omega t) + F\_4 \sin(a\Omega t) + F\_5 \cos(a \,\Omega t) \tag{1}$$

where *q* is the *N* order generalized coordinate displacement vector; *K* is the stiffness matrix which takes into account the symmetric matrices of the beam and the nonsymmetric matrices of the bearings; *C* is the matrix containing the antisymmetric matrices due to gyroscopic effects and the nonsymmetric matrices due to bearings viscous damping; *F*1 is the constant body force such as gravity; *F*2 and *F*3 are the forces due to unbalance; *F*4 and *F*<sup>5</sup> are the forces due to the nonsynchronous effect; and *a* is a constant.
