*5.3. Comparison of Free-Vibration Characteristics*

The first vibration mode and natural frequency of 2D-PWL and 3D-TCP under different boundary conditions in Figure 12 are compared as shown in Table 17. The vibration modes of 3D-TCP are consistent with those of 2D-PWL, while the natural frequency of 3D-TCP is smaller than that of 2D-PWL, which is consistent with the obtained equivalent stiffness.

**Table 17.** Comparison of the first vibration modes and natural frequencies (Hz) predicted by 2D-PWL and 3D-TCP under different boundary conditions.

#### **6. Conclusions**

The VAM-based equivalent model (2D-EPM) of 3D textile composite plate is established for bending, buckling and free-vibration analysis. The following conclusions can be obtained:

(1) The maximum errors of bending displacement and buckling load between 3D-FEM and 2D-EPM are within the range of engineering accuracy, and the displacement distributions along the analysis path predicted by two models have the same trend with small differences. Furthermore, the local stress, strain and displacement distributions within the recovered unit cell are well captured. In addition, the computational efficiency of 2D-EPM is greatly improved by reducing the number of nodes and elements.

(2) The width of binder/warp yarn mainly affect the stiffness components *A*<sup>11</sup> and *D*11. That is, *A*<sup>11</sup> and *D*<sup>11</sup> increase with the increasing width of warp or weft yarn, while decrease with increasing width of binder yarn, which may due to the fact that the increasing width of binder yarn will decrease the yarn content.

(3) Compared with 2D plain-woven laminate with the same thickness and yarn content, the 3D textile composite plate has smaller equivalent bending stiffness and larger torsional stiffness, resulting in large displacement and small natural frequency. This may be because the 3D textile composite plate is only constrained by the binding yarns in the thickness direction, while the warp yarns and weft yarns are intertwined closely in the 2D-PWL.

**Author Contributions:** Individual contributions are as follows: conceptualization, S.X. and Y.Z. Investigation and writing; analysis, Y.Z. and Z.S. Revision, Q.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (Grant Numbers: 51778088, 52073036), and the Natural Science Foundation of Chongqing, China (cstc2021jcyjmsxmX0035).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available on request due to restrictions, e.g., privacy or ethical. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to subsequent analyzes and publications.

**Conflicts of Interest:** The authors declare no conflict of interest.
