微纳米谐振器在传感领域有着诸多应用，包括用于表面材料探测的原子力显微镜、电路中核心振动元件以及用于粒子质量探测的纳米机械质谱仪。微纳米谐振器用作粒子质量探测的原理是共振频率会由于粒子附着而发生偏移，当粒子质量越大，共振频率偏移也越大。基于此，微纳米谐振器可以实现测量单个原子质量的分辨率，也可以用于探测聚合的蛋白质分子。尽管微纳米谐振器在粒子探测领域拥有巨大的潜力，但用于粒子质量识别面临的挑战在于：工作状态对环境要求很高，微纳尺度下极易受到热、电、光等效应干扰，从而影响测量精确度；灵敏度和分辨率需要进一步提升，同时需要保持系统的稳定性，降低噪声的干扰；需要发展出高精度的粒子质量估计算法，以应对复杂的测量数据。在此背景下，本论文主要围绕两大科学问题展开研究：一是微纳米谐振器的测量机理，二是粒子质量估算的反问题算法。具体而言包括：发展考虑多种因素的微纳米谐振器振动模型，提出多粒子测量策略，并以提升灵敏度为目标函数优化谐振器形状；发展高可靠性的粒子识别反问题求解算法，构造以振动模态为基的变形将反问题线性化，并且利用深度神经网络方法对附着粒子质量进行估计。主要研究工作如下：

(1) 提出了多粒子质量探测模型和反问题求解方法。基于Timoshenko 梁模型，提出了用于解释多个粒子附着引起频率偏移的振动模型和近似解析解。该模型在高阶共振、粘性阻尼下的振动和轴向张力等场景下均表现出更好的准确性。通过迭代算法解决了多粒子识别的反问题，并对该方法在抵抗噪声能力和不同粒子吸附区域方面的表现进行了评估。多个粒子的质量识别比单个粒子场景显示出更好的灵敏度和鲁棒性，对于质量探测是有意义的，为多粒子的质量识别提供了理论优势基础。

(2) 提出了变厚度谐振器优化设计模型。灵敏度和分辨率是质量谐振器的两个关键性能指标。通过设计优化梁谐振器沿轴向的厚度分布，实现了质量谐振器灵敏度的显著提升。给出了非均匀谐振器灵敏度的解析表达式和厚度分布的优化流程，通过形状优化获得了悬臂梁和两端固支梁谐振器的最优形状。通过计算发现最优形状的梁谐振器灵敏度比均匀形状的高出三个数量级。还讨论了轴向载荷对于谐振器厚度优化结果的影响。

(3) 发展了一种适用于粒子和梁谐振器的通用接触理论模型。为了研究梁结构的接触问题，基于经典Hertz 接触理论的位移近似假设，并融合了梁的位移模型，提出了适用于梁结构弹性体之间的通用接触理论模型。本工作中将梁结构的变形纳入Hertz 位移假设中，从而发展了一个适用于具有非均匀梁形状和功能梯度涂层的弹性体的综合接触力学模型。该问题通过基于Gauss-Chebyshev 多项式的Cauchy 型奇异积分方程的数值方法来解决。本研究深入探讨了梁压痕与弹性体相互作用中的接触压力和接触刚度，考虑了简支和悬臂两种边界条件。还研究了结构变形与功能梯度涂层对接触应力分布和接触刚度的影响。

(4) 发展了基于模态偏移的附着多粒子谐振器振动模型及机器学习识别方法。基于大质量粒子附着会导致谐振器的振动模态发生偏移的前提，提出了适用于多个大质量粒子附着谐振器上的振动模型，并利用传递矩阵方法获得频率偏移的结果。研究表明多粒子附着会破坏频率偏移对粒子附着位置的对称性，大大增加反问题求解的难度。通过理论模型的数值演绎结果训练深度神经网络对反问题进行求解，从而获得了单粒子与多粒子附着时的粒子质量附着结果，验证了神经网络方法对于复杂场景反问题求解的适用性。

本文全面研究了微纳米谐振器在粒子质量估计中的动力学机理及反问题求解，为微纳米谐振器的应用提供了理论基础与设计参考。

Micro- and nanoresonators are widely used in the field of sensing, including atomic force microscopy for material detection, vibrating elements at the core of electrical circuits, and nanomechanical mass spectrometrys for particle mass detection. The mechanism of the micro- and nanoresonators for particle mass detection is that the resonance frequency shifts due to particle attachment, and the larger the particle mass, the greater the resonance frequency shift. Based on this, micro- and nanoresonators can achieve the resolution to measure the mass of individual atoms and can also be used to detect aggregated protein molecules. Although micro- and nanoresonators have great potential in the field of particle detection, there are many challenges for particle mass identification: Micro- and nano-scale devices are highly susceptible to environmental factors such as heat, optical, and electrical influences, which affects the accuracy of measurements; the sensitivity and the resolution need to be further improved, and at the same time, it is necessary to maintain the stability of the system and to reduce the influence of the noise; it is necessary to develop a high-precision particle mass estimation algorithms to cope with complex measurement data. \par

This dissertation focuses on two major scientific problems, the measurement mechanism of micro- and nanoresonator and the inverse problem algorithm for mass estimation: Developing a more general vibration model of the micro- and nanoresonator, proposing a measurement strategy of multiple particles and optimizing the shape of the resonator with the objective function of improving the sensitivity; developing a reliable algorithm for solving the inverse problem of particle identification, constructing a deformation based on vibrational modes to linearize the inverse problem, and also utilizing the deep neural network approach to estimate the mass of attached particles. The main research work is as follows:

(1) A multi-particle mass detection model and an inverse problem solving method are proposed. Based on the Timoshenko beam model, the theoretical model and approximate analytical solution for explaining the frequency shift caused by the attachment of multiple particles are proposed. The model shows a good accuracy in scenarios of higher-order resonance, vibration with viscous damping, and axial tension. The inverse problem of multiple particle identification is solved by an iterative algorithm, and is evaluated with the robustness against noise and different particle adsorption regions. The mass identification of multiple particles shows better sensitivity and robustness than the scenario of single particle, which is meaningful for mass detection and provides a advantageous basis for mass identification of multiple particles.

An optimized design model is obtained for variable thickness resonators. Sensitivity and resolution are two key performance indices of mass resonators. By designing and optimizing the thickness distribution of the beam resonator along the axial direction, a significant improvement in the sensitivity of the mass resonator is achieved. The analytical expression for the sensitivity of the nonuniform resonator and the optimization flow of the thickness distribution are given, and the optimal shapes of the cantilever beam and the doubly clamped beam resonators are obtained through shape optimization. It is found that the sensitivity of the optimally shaped beam resonator is three orders of magnitude higher than that of the uniform one. The effect of axial load on the optimization results of resonator is also discussed.

(2) A generalized contact theory model for particle and beam resonators is developed. In order to study the contact problem of beam structures, a generalized contact theory model applicable between beam structures is proposed based on the displacement approximation assumptions of the classical Hertz contact theory and the displacement model of beams. The deformation of the beam structure is incorporated into the Hertz displacement assumptions, leading to the development of a comprehensive contact mechanics model applicable to the elastic bodies with non-uniform beam shapes and functional gradient coatings. The problem is solved by a numerical approach based on the Gauss-Chebyshev polynomials for singular integral equations of the Cauchy type. The contact pressure and contact stiffness in the interaction of beam indentations and elastic bodies are investigated, considering the boundary conditions of the cantilever and simply supported beams. The effects of structural deformation and functional gradient coating on the contact stress distribution and contact stiffness are also investigated.

(3) A vibration model of the resonator with multiple adsorbate is developed based on the shift of mode shape and a machine learning identification method. Based on the mechanism that the attachment of massive particles will cause the vibration modes of the resonator to be shifted, a vibration model applicable to multiple massive particles attached to the resonator is proposed, and the transfer matrix method is utilized to obtain the results of the frequency shift. It is shown that the attachment of multiple particles destroys the symmetry of the frequency shift to the particle attachment position due to the change of mode shapes, which greatly increases the difficulty of solving the inverse problem. The deep neural network is trained to solve the inverse problem through the numerical simulation results of the theoretical model, so as to obtain the results of the particle mass attachment when a single particle is attached to multiple particles, which verifies the applicability of the neural network method for the solution of the inverse problem in complex scenes.

In this dissertation, the dynamics mechanism and inverse problem solving of micro and nano resonators in particle mass estimation are comprehensively investigated, which provides a theoretical basis and design reference for the application of micro and nano resonators.