Dynamic stability of a size-dependent micro-beam

Dynamic instability characteristics of graphene‐reinforced cellular sandwich plates using a three‐variable isogeometric approach

“…In this study, we adopt Bolotin's technique [47] in order to obtain the solutions of this equation, allowing us to identify the instability zones of the plate structures subjected to time-dependent loading at the mid-plane. Readers are encouraged to refer to [48] for a detailed procedure of this method. Accordingly, the fundamental displacements of sandwich plates can be calculated using the Fourier series corresponding to the period of 2𝑇, where 𝑇 = 2𝜋∕𝜔, as follows…”

“…herein, 𝐚 𝑗 and 𝐛 𝑗 illustrate the two vectors of unknown constants. By considering the first-order approximation (𝑗 = 1), the first dynamic instability areas of cellular three-layer plates are obtained by solving the following determinant [46,48].…”

“…In this study, we adopt Bolotin's technique [47] in order to obtain the solutions of this equation, allowing us to identify the instability zones of the plate structures subjected to time‐dependent loading at the mid‐plane. Readers are encouraged to refer to [48] for a detailed procedure of this method. Accordingly, the fundamental displacements of sandwich plates can be calculated using the Fourier series corresponding to the period of 2 T , where $$T=2\pi /\omega$$, as follows $$$$\begin{equation} {\bf d}=\sum \limits _{j=1,3,5\ldots }^{\infty }{{\left[ {{\bf a}_j}\sin {\left(\frac{j\omega t}{2}\right)}+{{\bf b}_j}\cos {\left(\frac{j\omega t}{2}\right)}\right]}}, \;\;{\ddot{\bf d}} = -{\left(\frac{j\omega }{2} \right)}^2{\bf d}, \end{equation}$$$$herein, $${\bf a}_j$$ and $${\bf b}_j$$ illustrate the two vectors of unknown constants.…”

“…Accordingly, the fundamental displacements of sandwich plates can be calculated using the Fourier series corresponding to the period of 2 T , where $$T=2\pi /\omega$$, as follows $$$$\begin{equation} {\bf d}=\sum \limits _{j=1,3,5\ldots }^{\infty }{{\left[ {{\bf a}_j}\sin {\left(\frac{j\omega t}{2}\right)}+{{\bf b}_j}\cos {\left(\frac{j\omega t}{2}\right)}\right]}}, \;\;{\ddot{\bf d}} = -{\left(\frac{j\omega }{2} \right)}^2{\bf d}, \end{equation}$$$$herein, $${\bf a}_j$$ and $${\bf b}_j$$ illustrate the two vectors of unknown constants. By considering the first‐order approximation ($$j=1$$), the first dynamic instability areas of cellular three‐layer plates are obtained by solving the following determinant [46, 48]. $$$$\begin{equation} \text{det}{\left[ -\frac{\omega ^2}{4}\mathbf {M} - \lambda _{cr}{\left(\alpha \pm 0.5\beta \right)}{\mathbf {K}_g}+\mathbf {K} \right]}=0.$$…”

Dynamic Stability of the Periodic and Aperiodic Structures of the Bernoulli-Euler Beams

“…in the works [16,17], the dynamic stability of a beam built of materials of different stiffness was investigated. The dynamic stability of micro and nanobeams was also investigated [18][19][20]. in [21] the material with axially continuous gradual variation properties was considered.…”