The setting of these limits on each dimension of the problem would depend on the analyst's understanding of the problem. The types of constraints are also very much dependent on the problem, for example some particle dimensions might be known to be constrained to be positive values, and in that case the absolute |Mmax| or |Vmax| might be applicable [10,11, 25]. 

A modification to the original PSO algorithm by [24] introduced wk, the inertia weight variable. It controls the influence of the previous velocity on the current velocity value. An adaptive inertia weight is often used to improve the algorithm's search from an initially explorative (global) search to a more local search as this variable decreases. This also has a tendency to improve the algorithm's convergence rate [11, 24]. This variable is specified by the starting weight (wstart), wf is the fraction of iterations over which the inertia weight is decreased and wend the final inertia weight value The initial wk in equation 5 is wstart and it is decrease by w w w *k k dec* = − where 

 start end
$$w_{d e c}={\frac{w_{s t a r t}-w_{e e d}}{N-w_{f}}}$$
=− (7) 
from the first iteration up to iteration N x wf, thereafter wk is wend. The (pik- mik) term in equation 5 measures how far each particle is currently from its personal best position (local) and (pgk- mik) measures how far each particle is from the global (social) best particle in the swarm. This means the middle term in equation 6 tends to control the particle's velocity based on the particles own best position while the last term allows the particle to be influenced by the best performing particle in the swarm. 

In the next section we present the finite element models and propose a particular representation of these in the particle swarm context. 

## 3. Finite Element Models

The finite element models updated in this paper were all developed from the unsymmetrical h-beam structure In [20] and previously used in [19]. The details of the modelled beam are described in the next section. 

## 3.1 Unsymmetrical H-Beam

A simple unsymmetrical h-beam shown in figure 1 is modelled. This unsymmetrical h-beam is suspended on rubber bands (see [19] for more details on the structure and experimental set-up). The measured natural frequencies of interest of this structure occur at; 53.9Hz, 117.3Hz, 208.4Hz, 254Hz and 445Hz which correspond to modes 7, 8, 10, 11 and 13 respectively. The aluminium beam material has a Young's Modulus of 7.2x1010 Pa, the beam length is 600mm with a width of 32.2mm and a section thickness of 9.8 mm. The left edge has a length of 400mm, the right edge length 200mm and a density of 2793 kg/m3. 

Figure 1 shows that the beam is divided into elements numbered from one to twelve. Each cross-sectional area is 9.8mm by 32.2 mm. Each finite element model used standard isotropic material properties and Euler Bernoulli beam elements to approximate the beam sections of the structure. The beam is free to move in all six degrees. 

![0_image_0.png](0_image_0.png)

$\left(7\right)$. 