剪切变形和非局部梁板中尺度不变频率比下的边界条件识别
Boundary-condition identification from a scale-invariant frequency ratio in shear-deformable and nonlocal beams and plates
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中文总结 AI 辅助
本文利用结构前两阶频率比与边界条件相关、与尺寸和材料无关的特性，通过测量频率比位移识别小尺度梁板的边界条件变化，无需已知材料参数，并验证了方法的有效性。
中文摘要 AI 辅助
结构前两阶固有频率之比取决于其边界条件，但不取决于其尺寸、材料密度或绝对刚度，因为这些因素对每个频率的影响是等倍的，在比值中相互抵消。我们利用这一特性，通过两次比值测量来识别小尺度梁或板的边界条件，而无需知道其精确尺寸或材料。本文首先建立了比值 $r=\omega_2/\omega_1$ 以及当边界条件改变时其经历的分数位移 $\Delta r$ 如何响应小尺度下两个重要效应：横向剪切变形（通过 Timoshenko 梁理论和 Mindlin 板理论）以及小尺度弹性（通过 Eringen 的非局部 Timoshenko 模型）。使用一个无锁定的 Rayleigh-Ritz 梁求解器和一个有限元 Mindlin 板求解器（两者均经过基准验证），我们发现了一种不对称性。绝对比值随厚度变化明显，在剪切变形范围内变化 12% 至 14%，随正交各向异性变化 6% 至 9%，但随非局部性变化较弱；相比之下，位移 $\Delta r$ 在整个非局部范围内变化小于 0.2%，在厚度变化下变化几个百分点。由于 $\Delta r$ 在很大程度上独立于这些细节，测量的位移可以识别出有意改变支撑所产生的边界条件转变，其分辨率我们进行了量化，并针对分子动力学数据进行了验证。随后，两个弹簧连续体将方法扩展到估计支撑柔度，并明确说明了尖端支撑情况的非唯一性。该方法不需要小尺度参数或材料常数的值。
英文摘要
The ratio of a structure's first two natural frequencies depends on its boundary conditions but not on its size, material density, or absolute stiffness, which multiply every frequency equally and cancel in the quotient. We use this to identify the boundary conditions of a small-scale beam or plate from two ratio measurements, without knowing its exact dimensions or material. The paper first establishes how the ratio $r=ω_2/ω_1$, and the fractional shift $Δr$ it undergoes when a boundary condition changes, respond to the two effects that matter at small scale: transverse shear deformation, through Timoshenko beam and Mindlin plate theory, and small-scale elasticity, through Eringen's nonlocal Timoshenko model. Using a locking-free Rayleigh-Ritz beam solver and a finite-element Mindlin plate solver, each checked against benchmarks, we find an asymmetry. The absolute ratio moves appreciably with thickness, by 12 to 14 per cent over the shear-deformable range, and with orthotropy, by 6 to 9 per cent, but only weakly with nonlocality; the shift $Δr$, by contrast, varies by less than 0.2 per cent over the full nonlocal range and by a few per cent under thickness change. Because $Δr$ is largely independent of these details, a measured shift identifies which boundary-condition transition a deliberate change of support has produced, with a resolution we quantify and verify against molecular-dynamics data. Two spring continua then extend the method to estimating support compliance, with the non-uniqueness of the tip-support case stated explicitly. The method needs no value of the small-scale parameter or the material constants.
发表机构
- Alanya Alaaddin Keykubat University（阿拉尼亚阿拉丁凯库巴特大学）
机构由 AI 辅助整理，请以论文原文为准。
