Surface effect on dynamic stability of microcantilevers on an elastic foundation under a subtangential follower force

Abstract

For micro/nano structures, surface elasticity, surface stress and surface mass strongly affect mechanical behaviors of 1D beam-columns. This article studies dynamic stability of microcantilevers on an elastic foundation or embedded in an elastic matrix when subjected to a subtangential follower force, where the surface effects are taken into account. An exact characteristic equation is derived for clamped–free end supports. For differential tangency coefficients, the force–frequency interaction diagram is displayed and the influences of surface elasticity modulus, residual surface tension, surface mass and the elastic foundation are analyzed for conservative and non-conservative compressive forces. When the tangency coefficient vanishes, a cantilever column subjected to a conservative tip force is reduced, and conventional Euler buckling for a compressive axial load is recovered. When the tangency coefficient does not vanish, a generalized Beck’s column with the surface effects is tackled. When the tangency coefficient exceeds certain critical value, flutter instability take places. For a fixed frequency, the critical divergency and flutter loads as a function of the tangency coefficient are given for various surface influences from residual surface tension, surface elasticity, surface mass and the stiffness of the elastic foundation. The boundary map of stability, divergence and flutter domain is shown.