PSI - Issue 52

Dong Xiao et al. / Procedia Structural Integrity 52 (2024) 667–678 Dong Xiao et al. / Structural Integrity Procedia 00 (2023) 000–000

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where p ( t ) is the impact loading force as a function of time, p max represent the peak force, f l and f u denote the loading and unloading frequency respectively. The force parameterization method allows for the control of the severity of impacts through the peak force p max , while the loading and unloading speeds are controlled by f l and f u respectively. Figure 2 illustrates the realizations of this force parameterization method using these three parameters. It can be observed that the peak force p max determines the amplitude of the force, while f l and f u govern the loading and unloading speeds. Decreasing f l or f u results in a longer impact contact duration, as the analytical contact duration is given by T = ( f l + f u ) / 2 f l f u . By employing this force parameterization method, the impact force can be e ff ectively characterized with 3 parameters, allowing for more e ffi cient and accurate estimation using model-based methods.

Fig. 2: Impact force parameterization

2.5. Loading and unloading frequency constraints

In the optimisation process, determining the appropriate constraints for the design variables is crucial. In the context of impact identification using model-based methods, the design variables include impact location variables ( x , y ) and impact force parameters ( p max , f l , f u ) as per the force parameterization method. To bound the loading and unloading frequencies ( f l , f u ), CWT is applied to the impact responses of target impact. The dominant frequency f d , which represents the most significant frequency component of the response, is estimated as ˆ f d by CWT. And the dominant frequency is recognized as the harmonic average of loading frequency f l and unloading frequency f u . 2 f d = 1 f l + 1 f u (13) For practical impacts, the loading frequency f l and the unloading frequency f u are usually not significantly di ff erent. Assuming f l / C ≤ f u ≤ Cf l ,where C > 1, and combining it with Equation 13, the bounds for loading frequency f l and unloading frequency f u can be determined. The bounds are given by: C + 1 2 C ˆ f d ≤ f l ≤ C + 1 2 ˆ f d , C + 1 2 C ˆ f d ≤ f u ≤ C + 1 2 ˆ f d (14) Considering the uncertainties in the estimated dominant frequency, the equality constraint in Equation 13 is trans formed into an inequality constraint with respect to the estimated dominant frequency ˆ f d .Assuming f d ∈ [ c l ˆ f d , c u ˆ f d ], where c l ≤ 1 and c u ≥ 1, this inequality constraint for loading and unloading frequencies can be expressed as: 2 c u ˆ f d ≤ 1 f l + 1 f u ≤ 2 c l ˆ f d (15) Equation 14 represents the bound constraint, and Equation 15 represents the nonlinear constraint for loading fre quency f l and unloading frequency f u . To expedite the optimisation process, the nonlinear constraint is transformed into a scaled linear constraint. The scaled linear constraint is deduced as: 2 C l ˆ f d ≤ f l + f u ≤ 2 C 2 2 C − c u ˆ f d (16)