In the present study, a novel geometrically exact nonlinear beam model based on the nonlinear displacement field, centerline-inextensibility condition, Euler-Bernoulli beam hypothesis, and modified couple stress theory is established to investigate the extremely large deformations and stability characteristics of size-dependent microcantilevers subjected to a tip-concentrated non-conservative (follower type) load with arbitrary inclination angle. The newly developed mathematical formulation explains the size effect phenomenon in the microcantilevers with only one material length scale parameter. The nonlinear shooting method for two-point boundary value problems is utilized to assess the deformed configuration of microsystem based on the quasistatic form of nonlinear mathematical model. Afterwards, the linearized dynamical mathematical model around the nonlinear deformed configurations of microsystem is numerically solved using the Galerkin technique to examine the stability of obtained responses. A comprehensive investigation is then conducted to highlight the role of size-dependency in the critical follower force at which the microsystem undergoes a flutter instability, as well as the nonlinear large deformation of microsystem in the pre-flutter region.

在本研究中，基于非线性位移场、中心线不可延展条件、Euler-Bernoulli 梁假设和修正的耦合应力理论，建立了一种新的几何精确非线性梁模型，以研究尺寸相关的超大变形和稳定性特性。微悬臂梁承受具有任意倾角的尖端集中非保守（跟随器类型）载荷。新开发的数学公式仅用一个材料长度尺度参数来解释微悬臂梁中的尺寸效应现象。基于非线性数学模型的准静态形式，利用两点边值问题的非线性射击方法来评估微系统的变形构型。然后，使用伽辽金技术对微系统非线性变形构型周围的线性化动力学数学模型进行数值求解，以检验所得响应的稳定性。然后进行了全面的研究，以强调尺寸依赖性在微系统经历颤振不稳定性的临界跟随力中的作用，以及微系统在预颤振区域的非线性大变形。