PSI - Issue 83

Amal Lahrizi et al. / Procedia Structural Integrity 83 (2026) 162–170

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3.2. Natural Frequency Modelling with Cubic Regression Response Surface Methodology (RSM), developed over six decades ago by Box and his colleagues, represents an ingenious combination of mathematical and statistical concepts that enables the creation of highly relevant empirical models. Its fundamental aim is to link natural frequencies to essential parameters such as the depth and location of cracks. This link is achieved through the clever use of numerical analysis combined with a regression equation. In particular, the integration of PSO algorithms into the process provides the ability to minimise the objective function, leading to the determination of optimal crack depths and locations. A crucial step in this methodology involves establishing response surface models for the first three natural frequencies. To accomplish this, normalized natural frequencies are computed for various depths and normalized locations of cracks. The outcomes of these calculations are visualized in a 3D context, illustrating the influence of crack depth and location on frequency variations. This observation implies that a specific frequency can correspond to different crack locations and depths. Figure 3 further illustrates that, for all three vibration modes, frequencies, at a given crack depth, exhibit fluctuations as the crack location shifts from the fixed end to the other fixed end of the beam. In the figures (refer to Figure 3(a), (b), and (c)), the number of oscillations increases with the number of modes since oscillations are linked to the excitation frequency of each mode. To develop response surface models for the first three natural frequencies, a central composite design is employed, complemented by a second-order model. The resulting quadratic regression equation takes the fundamental form:   , i j y f x y  , where ' y ' represents the natural frequencies, while ' i x ' and ' j y ' denote crack depth and location, respectively. This equation essentially characterizes the response surface model and its capability to predict natural frequencies as a function of the relevant parameters. (a)

(b)