PSI - Issue 81

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

392

Table 3. Natural frequencies for twill-weave CFRP completely free plate.

No. Mode ( m , n )

Frequency (Hz)

Mode ( m , n )

Frequency (Hz)

Shape

No.

Shape

1.

(1,1)

47.85

7.

(3,1)

308.80

2.

(2,0)

99.25

8.

(2,2)

335.27

3.

(2,1)

138.84

9.

(3,2)

474.23

254.01 10.

(4,0)

535.29

4.

(0,2)

5.

(1,2)

269.89 11.

(4,1)

566.87

(3,0)

274.71 12.

(0,3)

698.68

6.

The numerical results for the nominal (reference) material properties are summarized in Tables 2 and 3 for the unidirectional and twill-weave CFRP laminates, respectively. For the unidirectional completely free plate, the fundamental bending mode (1,1) occurs at approximately 30 Hz, while for the twill-weave laminate, the first mode appears at about 48 Hz, reflecting the higher in plane isotropy of the woven architecture. Subsequent modes up to the twelfth involve combined bending and torsional motions, with frequencies extending to 330 Hz for the unidirectional laminate and 700 Hz for the twill-weave. Comparison of the modal spectra highlights the influence of material anisotropy on the dynamic response of the composite plates. The unidirectional laminate, characterized by a high stiffness ratio ( E 1 ≫ E 2 ), exhibits distinct separation between longitudinal and transverse bending frequencies, as well as stronger directional dependency of torsional modes. In contrast, the twill-weave composite, with E 1 ≈ E 2 and more uniform shear stiffness, demonstrates a nearly symmetric modal spectrum with closely spaced pairs of frequencies. This characteristic trend agrees well with experimental observations reported for balanced woven CFRP laminates (Kovalovs et al., 2025; Li et al., 2018) and validates the FE model. The obtained frequency datasets were subsequently used to construct polynomial response surfaces describing the dependence of each modal frequency on the selected elastic constants. These surfaces served as an analytical response surface for the FE model during optimization, enabling fast and stable identification of material parameters without the need for repeated numerical simulations. 2.4. Approximation, statistical analysis and optimization procedure To efficiently explore the multidimensional space of elastic parameters, a Latin Hypercube Sampling (LHS) design was employed. After defining the boundaries of the elastic characteristics of the material under study, a Latin hypercube-type experimental plan was constructed. This model-independent approach, first introduced in classical design-of-experiments studies (Rikards, 2003; Rikards and Auzins, 2004), ensures statistically uniform coverage of the parameter domain. In contrast to traditional full-factorial designs, which require a large number of simulations, the LHS method distributes sampling points quasi uniformly across the multidimensional space, guaranteeing that each parameter range is evenly explored. This strategy provides a statistically representative coverage of the design domain while minimizing the number of FE analyses required to construct accurate response surface models. The discrete modal data obtained from the virtual experiment (FE simulations) were subsequently approximated using RSM in order to establish continuous functional relationships between the modal frequencies and the elastic constants of the material. This approximation step replaces direct FE computations during the optimization process and substantially reduces the overall computational cost. For each vibration mode, a second-order polynomial model was constructed in the following general form:

2      k k kk k b x

f b b x  

b x x

,

(3)

0

i

km k m

k m 