Size-dependent axial dynamics of magnetically-sensitive strain gradient microbars with end attachments

Abstract

The multi-parameter Sturm–Liouville eigenvalue problems, associated with the size-dependent longitudinal vibration of a finite micro-scale bar embedded in orthogonal transverse magnetic fields, are addressed in this study. Derived as a fourth-order partial differential equation, and augmented by enriched higher-order boundary conditions, the mathematical model of the micro-scale bar is predicated on the duo of the strain gradient theory of elasticity and the extended Hamilton׳s principle. The derived model is tackled with the computation scheme of the power series method. A thorough validation of the simplified form of the model is presented with benchmark results published in the literature. Presented along with the validation study is a comprehensive parametric examination of the influence of the aspect ratio, the material length scale, the mass and stiffness ratios of attachments and the magnetic field strength. Results from the analyses affirm that in the case of a lightweight mass attached to the end of the microbar, the axial resonant frequencies approach that of a microbar with a fixed–free boundary condition. However, in the case of an attachment with a heavy mass the fundamental resonant frequency of the microbar tends to zero. A Pareto analysis of the order of influence of the model׳s variables unmasks the ratio of the stiffness of the microbar and an attached elastic spring as having the most significant effect on the fundamental axial natural frequency, in the absence of the magnetic field. In the presence of the magnetic field, however, the effects of the end attachments are totally overshadowed by the influence of the magnetic field strength.