Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core
Abstract
:1. Introduction
2. Modelling Foundations
3. Derivation of Initial Governing Equations
4. Basics of Tolerance Averaging Technique
- ,
- ,
- ,
- ,
5. Governing Equations of the Tolerance Model
6. Free Vibration Analysis of Sandwich Plate with Honeycomb Core
- Size I ()—
- Size II ()—
- Size III ()—
- Mode I—
- Mode II—
- Mode III—
- Mode IV—
- Mode V—
- Mode VI—
- The relative errors between the results tend to decrease as the parameter a (representing the thickness of the walls of honeycomb in the periodicity cell) raises. It should be noticed that the initially assumed in-plane displacement field, cf. (1), is dedicated to sandwich structures with cores filling the whole available space between faces. In the case of ’thick’ honeycomb core, characterised by high values of parameter a, this assumption is still applicable, which results in generally satisfactory convergence of results. In the case of ’thin’ honeycomb, this assumption is slowly corrupting. As a consequence, the relative errors between averaged solution and FEM can reach up to 25% (cf. ).
- There are significant differences in the results of the averaged models in Case I and Case II. The only difference between those cases is a set of fluctuation shape functions. It can be noticed that for structures with lower thickness of the core , the assumption of negligibly small fluctuations of vertical displacements (Case II) is applicable. Hence, it produces results, which in general are convergent with the FEM analysis. Meanwhile, for higher values of the thickness , it seems that the fluctuations of vertical displacements are more influential, hence they cannot be neglected (Case I).
- In order to obtain precise results within the averaged model, the considered structure should be made of a sufficiently large quantity of periodicity cells. It can be noticed that for relatively small periodic structures (), the convergence of results in Case II is usually worse than in the case of larger structures (). It can be caused by boundary conditions, which produce disturbances in displacement fields on a considerable span of the plate. A similar remark can be made for Case I, excluding structures with a low thickness of the core , for which such a set of fluctuation shape functions is not applicable.
- Another reason for lower accuracy of results in the case of small periodic structures () can be the issue of slowly varying functions. Based on the definitions, a slowly varying function is a function, which is ’almost’ constant on any periodicity cell, with respect to a certain tolerance parameter . In the case of small periodic structures, this assumption can be satisfied only for higher values of parameter , which results in lower accuracy of the proposed solution.
- It can be noticed that in several cases the discrepancies in the results between the averaged model and FEM analysis tend to raise higher modes of vibrations. The reason for this phenomenon can be also connected with difficulties in satisfying the condition of nearly constant values of slowly varying functions on a basic periodicity cell. In the case of higher modes of vibrations, this condition must yield higher values of tolerance parameter and, consequently, lower accuracy of the obtained results. Nevertheless, with a properly adjusted fluctuation shape functions, the averaged model can be used to estimate several basic free vibration frequencies of any of the analysed sandwich structures.
7. Final Remarks
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Mode | |||||||
---|---|---|---|---|---|---|---|
100 | I | 10.7% | 16.9% | 2.0% | 8.8% | −1.7% | 5.4% |
II | 14.1% | 20.0% | 5.6% | 12.1% | 3.5% | 10.2% | |
III | 16.9% | 22.7% | 9.3% | 15.6% | 8.3% | 14.7% | |
IV | 16.1% | 22.0% | 9.1% | 15.4% | 2.0% | 8.8% | |
V | 16.8% | 22.6% | 10.5% | 16.7% | 5.7% | 12.2% | |
VI | 18.8% | 24.5% | 14.6% | 20.5% | 10.7% | 16.9% | |
50 | I | 4.6% | 11.8% | −3.1% | 4.7% | −5.6% | 2.4% |
II | 8.1% | 15.1% | −0.5% | 7.2% | −1.9% | 5.8% | |
III | 11.7% | 18.4% | 2.9% | 10.3% | 2.2% | 9.6% | |
IV | 11.5% | 18.2% | 3.2% | 10.6% | −3.1% | 4.8% | |
V | 11.9% | 18.6% | 4.4% | 11.7% | −0.4% | 7.3% | |
VI | 14.1% | 20.7% | 9.2% | 16.1% | 4.4% | 11.7% | |
25 | I | −1.5% | 6.6% | −6.8% | 1.7% | −8.4% | 0.2% |
II | 1.6% | 9.4% | −5.4% | 3.0% | −5.6% | 2.8% | |
III | 5.6% | 13.1% | −2.6% | 5.6% | −2.4% | 5.8% | |
IV | 5.4% | 12.9% | −1.9% | 6.2% | −6.7% | 1.7% | |
V | 5.2% | 12.7% | −1.6% | 6.5% | −5.2% | 3.1% | |
VI | 7.3% | 14.7% | 2.6% | 10.4% | −1.8% | 6.3% |
Mode | |||||||
---|---|---|---|---|---|---|---|
100 | I | 4.7% | 10.2% | 0.2% | 6.0% | −1.3% | 4.5% |
II | 7.0% | 12.3% | 1.9% | 7.6% | 0.9% | 6.7% | |
III | 9.0% | 14.3% | 3.9% | 9.4% | 4.4% | 9.9% | |
IV | 9.5% | 14.7% | 4.1% | 9.7% | 0.1% | 5.9% | |
V | 10.0% | 15.3% | 4.8% | 10.3% | 1.8% | 7.5% | |
VI | 12.6% | 17.8% | 8.0% | 13.4% | 4.9% | 10.4% | |
50 | I | −0.3% | 6.0% | −3.7% | 2.8% | −4.7% | 1.8% |
II | 1.7% | 7.8% | −2.6% | 3.8% | −3.1% | 3.3% | |
III | 4.0% | 10.0% | −1.1% | 5.2% | −1.0% | 5.3% | |
IV | 4.9% | 10.9% | −0.6% | 5.7% | −3.7% | 2.7% | |
V | 5.3% | 11.2% | −0.2% | 6.0% | −2.7% | 3.7% | |
VI | 8.2% | 14.0% | 3.0% | 9.1% | −0.2% | 6.1% | |
25 | I | −4.6% | 2.1% | −6.4% | 0.5% | −7.1% | −0.2% |
II | −3.7% | 3.0% | −6.1% | 0.7% | −5.8% | 1.1% | |
III | −1.7% | 4.9% | −5.0% | 1.8% | −4.3% | 2.4% | |
IV | −0.2% | 6.2% | −4.2% | 2.6% | −6.4% | 0.5% | |
V | −0.7% | 5.8% | −4.6% | 2.2% | −6.2% | 0.7% | |
VI | 1.7% | 8.1% | −2.5% | 4.1% | −4.8% | 2.0% |
Mode | |||||||
---|---|---|---|---|---|---|---|
100 | I | 1.4% | 6.7% | −1.4% | 4.0% | −2.1% | 3.4% |
II | 3.2% | 8.3% | −0.4% | 5.0% | −0.9% | 4.5% | |
III | 4.9% | 10.0% | 0.9% | 6.2% | 0.6% | 5.9% | |
IV | 4.6% | 9.7% | 0.7% | 6.0% | −1.4% | 4.0% | |
V | 5.2% | 10.3% | 1.3% | 6.5% | −0.4% | 5.0% | |
VI | 7.1% | 12.1% | 3.5% | 8.6% | 1.5% | 6.7% | |
50 | I | −2.5% | 3.3% | −4.1% | 1.8% | −4.5% | 1.4% |
II | −1.2% | 4.5% | −3.7% | 2.2% | −3.8% | 2.1% | |
III | 0.4% | 6.1% | −2.8% | 3.0% | −2.8% | 3.0% | |
IV | 0.0% | 5.7% | −2.9% | 2.9% | −4.2% | 1.7% | |
V | 0.5% | 6.1% | −2.7% | 3.1% | −3.7% | 2.2% | |
VI | 2.0% | 7.5% | −1.1% | 4.6% | −2.5% | 3.2% | |
25 | I | −5.6% | 0.5% | −6.2% | 0.0% | −6.6% | −0.4% |
II | −5.2% | 0.9% | −6.2% | −0.1% | −5.7% | 0.5% | |
III | −3.9% | 2.1% | −5.5% | 0.6% | −4.9% | 1.2% | |
IV | −4.0% | 2.0% | −5.2% | 0.9% | −6.2% | 0.0% | |
V | −4.4% | 1.7% | −5.7% | 0.5% | −6.3% | −0.1% | |
VI | −4.1% | 1.9% | −5.1% | 1.0% | −5.8% | 0.3% |
Mode | |||||||
---|---|---|---|---|---|---|---|
100 | I | 1.2% | 3.7% | −0.5% | 2.0% | −1.1% | 1.4% |
II | 2.1% | 4.5% | -0.2% | 2.4% | −0.6% | 1.9% | |
III | 3.0% | 5.5% | 0.5% | 3.0% | 0.2% | 2.7% | |
IV | 4.0% | 6.4% | 1.1% | 3.5% | −0.5% | 2.0% | |
V | 4.2% | 6.6% | 1.3% | 3.7% | −-0.2% | 2.3% | |
VI | 6.1% | 8.5% | 2.9% | 5.4% | 1.0% | 3.5% | |
50 | I | −1.6% | 1.3% | −2.4% | 0.5% | −2.7% | 0.3% |
II | −1.1% | 1.8% | −2.3% | 0.7% | −2.3% | 0.7% | |
III | −0.3% | 2.6% | −1.9% | 1.1% | −1.9% | 1.1% | |
IV | 0.3% | 3.1% | −1.5% | 1.4% | −2.4% | 0.5% | |
V | 0.3% | 3.2% | −1.6% | 1.4% | −2.3% | 0.7% | |
VI | 1.7% | 4.5% | −0.6% | 2.3% | −1.7% | 1.2% | |
25 | I | −3.9% | −0.7% | −4.0% | −0.8% | −4.4% | −1.1% |
II | −4.1% | −0.9% | −4.3% | −1.0% | −2.7% | 0.5% | |
III | −3.5% | −0.3% | −3.9% | −0.7% | −3.2% | 0.0% | |
IV | −2.7% | 0.5% | −3.2% | 0.0% | −4.0% | −0.8% | |
V | −3.5% | −0.3% | −3.8% | −0.6% | −4.3% | −1.1% | |
VI | −3.2% | 0.0% | −3.6% | −0.4% | −4.2% | −1.0% |
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Marczak, J. Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core. Materials 2022, 15, 7611. https://doi.org/10.3390/ma15217611
Marczak J. Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core. Materials. 2022; 15(21):7611. https://doi.org/10.3390/ma15217611
Chicago/Turabian StyleMarczak, Jakub. 2022. "Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core" Materials 15, no. 21: 7611. https://doi.org/10.3390/ma15217611
APA StyleMarczak, J. (2022). Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core. Materials, 15(21), 7611. https://doi.org/10.3390/ma15217611