# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 4 Conclusions  
This study achieved the following results.  
1. The dynamic stability of an orthotropic viscoelastic rectangular plate of variable thickness, considering the geometric nonlinearity under the action of periodic loads, was described by a nonlinear system of integrodifferential equations.
2. The application of Bubnov-Galerkin method, based on a polynomial approximation of the deflection and displacements, with the following discretization of spatial variables at each moment of time, reduces the problem of dynamic stability of an orthotropic viscoelastic rectangular plate of variable thickness to solving a nondecaying system of ordinary nonlinear integrodifferential equations with weakly singular kernels with variable coefficients.
3. The proposed numerical method for solving a nondecaying system of nonlinear integrodifferential equations with variable coefficients with weakly singular kernels, using the algorithm for eliminating singularities of integrodifferential equations with a singular kernel of the Koltunov-Rzhanitsyn type, based on the use of quadrature formulas, is effective for solving problems of dynamic stability of orthotropic viscoelastic plates.
4. It was found that an increase in the value of the rheological parameter \(\alpha\) of the Koltunov-Rzhanitsyn core leads to an increase in the vibration amplitude. In this case, \(\alpha\) plays an essential role in comparison with the parameter \(\beta\). Parameter \(\beta\) has no significant effect on the change in the amplitude of the oscillation.
5. It was found that the results of the viscoelastic problem obtained using the exponential relaxation kernel almost coincide with the results of the elastic problem. Using the Koltunov-Rzhanitsyn kernel, the differences between elastic and viscoelastic problems turn out to be very significant and amount to more than 40%.
6. The proposed method can be used for various viscoelastic thin-walled structures such as plates, panels, and shells of variable thickness.
7. The developed technique for studying vibrations of a viscoelastic orthotropic plate of variable thickness is easily extended to other types of laws of thickness variation, in the case of specifying in an analytical form the law of thickness variation in one or two directions of the coordinate axes.  
Conceptualization, R.A. and D.K.; methodology, R.A. and M.A.; software, R.A. and D.K.; validation, R.A., D.K., and M.A.; formal analysis, R.A. and M.A.; investigation, R.A. and M.A.; data curation, R.A. and D.K.; writing--original draft preparation, R.A. and D.K.; writing--review and editing, N.I.V.; visualization, D.K.; supervision, R.A. and M.A.; funding acquisition, N.I.V. All authors have read and agreed to the published version of the manuscript.  
The research was partially funded by the Ministry of Science and Higher Education of the Russian Federation as part of the World-Class Research Center program for Advanced Digital Technologies (contract No. 075-15-2020-934 dated 17 November 2020).  
Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.