Abstract One-dimensional vibrating nanostructures show remarkable performance in detecting small adherent masses added to a referential configuration. The mass sensing principle is based on measuring the resonant frequency shifts caused by the unknown attached masses. In spite of its important application in several fields, few studies have been devoted to this inverse eigenvalue problem. In this paper we have developed a distributed mass reconstruction method for initially uniform nanobeams based on measurements of the first lower resonant frequencies of the free bending vibration. Two main inverse problems are addressed. In the first problem, the mass variation is determined by using the first lower eigenfrequencies of a supported nanobeam, under the a priori assumption that the mass variation has support contained in half of the axis interval. In the second problem, we show that the a priori assumption can be removed, provided that the spectral input data include an additional set of first lower eigenfrequencies belonging to a second spectrum associated to different end conditions. The nanobeam is modelled using the modified strain gradient elasticity accounting for size effects. The reconstruction is based on an iterative procedure which takes advantage of a closed-form solution when the mass change is small, and shows to be convergent under this assumption and for smooth mass variation. The accuracy of the reconstruction deteriorates in presence of discontinuous mass variation. For these cases, a constrained least-squares optimization filtering shows to be very effective to reduce the spurious oscillations around the target coefficient. Numerical simulations show that the identification method performs well even for not necessarily small mass changes and it is stable in presence of errors on the data. An experimental validation of the method has provided encouraging results, despite the fact that only the first four eigenfrequencies under cantilever end conditions were used.

中文翻译：

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纳米束谐振器中的听觉分布质量

摘要 一维振动纳米结构在检测添加到参考配置中的小粘附质量方面表现出卓越的性能。质量传感原理基于测量由未知附着质量引起的共振频移。尽管它在多个领域有重要应用，但很少有研究致力于这个逆特征值问题。在本文中，我们基于对自由弯曲振动的第一较低共振频率的测量，开发了一种用于初始均匀纳米梁的分布式质量重建方法。解决了两个主要的逆问题。在第一个问题中，在质量变化具有包含在轴间隔的一半中的支撑的先验假设下，通过使用支撑的纳米束的第一较低特征频率来确定质量变化。在第二个问题中，我们表明可以去除先验假设，前提是频谱输入数据包括一组附加的第一较低特征频率，这些第一较低的特征频率属于与不同结束条件相关联的第二频谱。使用考虑尺寸效应的修正应变梯度弹性对纳米梁进行建模。重建基于迭代过程，当质量变化很小时，该过程利用封闭形式的解决方案，并在此假设下收敛，质量变化平滑。在存在不连续质量变化的情况下，重建的准确性会下降。对于这些情况，约束最小二乘优化滤波显示对于减少目标系数周围的虚假振荡非常有效。数值模拟表明，该识别方法即使对于不一定很小的质量变化也能很好地执行，并且在数据存在错误的情况下它是稳定的。尽管仅使用了悬臂末端条件下的前四个特征频率，但该方法的实验验证提供了令人鼓舞的结果。