**3.1. Initial population**

200 Simulated Annealing – Single and Multiple Objective Problems

**Figure 1.** Rotor references frames.

**3. Simulated annealing** 

be reduced at a slow rate [16].

in Table 1 [17].

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]:

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>

SA resembles the cooling process of molten metal through annealing (slow cooling process). At high temperature (*T*), the atoms in the molten metal can move freely with respect to each other, but as the temperature is reduced, the movement of the atoms gets restricted. The atoms start to get arranged and finally form crystals having the minimum possible energy which depends on the cooling rate. If the temperature is reduced at a very fast rate, the crystalline state may not be achieved at all and, instead, the system may end up in a polycrystalline state, which may have a higher energy state than the crystalline state. Therefore, in order to achieve the absolute minimum energy state, the temperature should

From the optimization point of view, this physical process is analogous to the determination of near-global or global optimum solutions. The energy of the atoms represents the objective function and the final ground state corresponds to the global minimum of the objective function. The analogy between the physical system and the optimization problem is shown

*q Cq Kq F F t F t F a t F a t* + + = + Ω+ Ω+ Ω+ Ω sin( ) cos( ) sin( ) cos( ) (1)

12 3 4 <sup>5</sup> *M*

are the forces due to the nonsynchronous effect; and *a* is a constant.

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 used to initialize the optimization process.
