PSI - Issue 81

Andrejs Kovalovs et al. / Procedia Structural Integrity 81 (2026) 388–395

390

      K x M 0  2

det

(1)

where K ( x ) is the stiffness matrix parameterized by the vector of unknown elastic constants x , M is the mass matrix, ω = 2π f is the circular frequency.

Fig. 1. Identification procedure.

The inverse problem seeks to determine the vector x such that the predicted frequencies match the measured ones. Thus, the frequency spectrum { f i } represents a nonlinear mapping F : x → { f i }. Because this mapping is highly nonlinear and partially correlated, a direct inversion is impossible; instead, a least-squares functional is minimized:

  2  x

exp

FEM

  

f

f



m 

    x

i

i

min



 

,

(2)

exp

f

1

i



i

The goal is to find x* minimizing Φ , which yields the set of elastic constants that best reproduces the experimentally observed modal characteristics. 2.1. Design of virtual experiment To assess the performance of the proposed identification methodology, a virtual experiment was carried out. The modal data of a free rectangular plate, computed using known material properties, were considered as the “measured” quantities for the inve rse problem. The experimental design included six identification parameters for each material system and consisted of 101 design points uniformly distributed within the search domain. The upper and lower bounds of each parameter were defined based on literature sources and manufacturer data, typically within ±20% of the nominal values. For certain p arameters, these ranges were subsequently refined to improve the precision of the identification process. A minimal-mean-squared-distance criterion ensured optimal space filling of the design points. For every point, a FE modal analysis produced the first twelve eigenfrequencies. The reference geometry was a rectangular plate with dimensions a = 400 mm and b = 250 mm, consisting of 8 layers. The thickness of one unidirectional CFRP layer is 0.2 mm, giving a total laminate thickness of 1.6 mm, with a material density of 1552 kg/m 3 . For the twill-weave CFRP, the thickness of one layer is 0.28 mm, resulting in a total thickness of 2.24 mm and a density of 1552 kg/m 3 . The FE model was implemented in ANSYS Mechanical APDL using layered SHELL281 elements, which provide full coupling between membrane, bending, and transverse-shear effects and are therefore suitable for thin and moderately thick laminates. A 400×250 element mesh ensured frequency convergence. Completely free boundary conditions were simulated by releasing all translational and rotational degrees of freedom, thereby replicating the experimental configuration of a plate suspended on thin strings. The influence of structural damping on the modal frequencies was neglected.