MS02.3: Asymptotic Methods

In this paper, the transverse vibrations of a one-dimensional string attached to a parabolic-shaped obstacle at one end are studied. Gravity and axial vibrations are taken into account. The changing position of the attachment point leads to a moving boundary problem. A boundary fixing transformation is applied to transform the problem to a fixed domain, leading to additional nonlinearities in the problem formulation. Different cases are studied. If the tension in the string and the gravity balance in a certain sense, then the effects of the nonlinearities become apparent, and lead to nonstandard eigenvalue problems. Otherwise, a standard multiple-time scales perturbation approach and a characteristic coordinate transformation can be applied to analyze the problem and its solution.

Dynamical Energy Analysis (DEA) is an operator based ray-tracing approach used to compute the vibro-acoustic response of complex structures to external forcing in the high-frequency limit. So far, DEA has been applied to model uncorrelated point force excitations. In this work, we will show how to adapt DEA to incorporate correlated input force excitations distributed over large regions. An example of such an excitation is the pressure field given by a Turbulent Boundary Layer (TBL) acting, for example, on an airplane wing. The properties of these excitations are captured by a source ray density function, which is calculated from an appropriate correlation function representing the applied pressure field via Wigner transformation. The source density is used for a structure-borne sound computation using standard DEA software. Results are presented for the vibrational energy distribution across a flat plate excited by a fully formed, stationary, TBL under a variety of flow and material properties, and boundary conditions.

Oscillations of the electrically actuated micro and nano beams and plates are considered, described by strongly nonlinear PDEs. One of the essentially nonlinear effects is the pull-in phenomenon, i.e., the transition of the oscillatory regime to the attraction one. We propose a simple and physically justified algorithm to determine the voltage values at which the system collapses. It is based on the detection of voltage leading to the merging of stable (central) and unstable (saddle) equilibrium points. Comparison with the results of calculations based on other methods as well as with experimental results shows sufficient accuracy of the proposed algorithm. The electrically activated oscillations are considered taking into account the geometric nonlinearity within the framework of the Kirchhoff’s and Berger’s models. Neglecting of geometrical nonlinearity leads to large quantitative error. It is shown that van der Waals forces have a greater influence on the pull-in value than Casimir forces. The effect of compression/extension on the pull-in value has been studied. Common errors when taking this factor into account are indicated. Dynamic scenarios for the collapse of the considered systems are studied. Analytical estimation for the initial values which guarantee oscillatory character of motion is obtained.

This presentation considers a one-dimensional problem of a passage of linearly coupled pair of particles over the potential well. In the conservative case, due to Liouville theorem, one can encounter only scattering (transmission or reflection) of the system. The introduction of damping results in energy dissipation, potentially causing the system to become entrapped within the well. The maps of the responses reveal zones of regular and complex (chaotic-like) behavior.