... are analytically solved by directly applying the model. Numerical results reveal that the microstructure and flexoelectric effects lead to enhanced extensional and bending stiffnesses of the shell. In addition, the first and second natural frequencies of the shell predicted by the new model are found to be higher...

... and buckling displacement. Solution is provided for the flexural rigidity and the natural frequency. It is shown that, for each level of axial loading, there may exist up to five flexural rigidities satisfying the governing differential equation and boundary conditions. buckling load vibration graded...

... frequencies are systematically investigated. Finally, the frequency differences, due to the different foundation models, and the possible modal interaction of the beam are discussed. beam elastic foundation natural frequency cut-off frequency mode shape nonlinear interaction In structural...

... by the balance of linear momentum. The natural frequencies for the in-plane and out-of-plane vibrations are calculated by both the Galerkin method and differential quadrature method. It can be found that natural frequencies of the in-plane vibrations are much higher than those in the out-of-plane case, which...

... and periodic solutions for nonlinear evolution equations arising in mathematical physics using symbolic computation. By using this method, natural frequency and dynamic responses of the wood-composite beams are obtained. The existing data obtained from our published work, the grindosonic instrument, and also...