We have developed a general method, dubbed the split beam method, to solve Euler–Bernoulli equations for cantilever beams under multiple loading conditions. This kind of problem is, in general, a difficult inhomogeneous eigenvalue problem. The new idea is to split the original beam into two (or more) effective beams, each of which corresponds to one specific load and bears its own Young’s modulus. The mode shape of the original beam can be obtained by linearly superposing those of the effective beams. We apply the split beam method to simulating mechanical responses of an atomic force microscope probe in the “dynamical” operation mode, under which there are a stabilizing force at the positioner and a point-contact force at the tip. Compared with traditional analytical or numerical methods, the split beam method uses only a few number of basis functions from each effective beam, so a very fast convergence rate is observed in solving both the resonance frequencies and the mode shapes at the same time. Moreover, by examining the superposition coefficients, the split beam method provides a physical insight into the relative contribution of an individual load on the beam.

我们开发了一种通用方法，称为分裂梁法，用于求解多种载荷条件下悬臂梁的 Euler-Bernoulli 方程。这类问题一般来说是一个困难的非齐次特征值问题。新的想法是将原始梁拆分为两个（或多个）有效梁，每个有效梁对应一个特定载荷并承担自己的杨氏模量。原始光束的模式形状可以通过线性叠加有效光束的模式形状来获得。我们应用分束法模拟原子力显微镜探针在“动态”操作模式下的机械响应，在这种模式下，定位器有稳定力，尖端有点接触力。与传统的解析或数值方法相比，分裂梁方法仅使用来自每个有效梁的少数几个基函数，因此在同时求解谐振频率和模式形状时观察到非常快的收敛速度。此外，通过检查叠加系数，拆分梁方法提供了对梁上单个负载的相对贡献的物理洞察力。