基于非局部弹性理论的微纳米质量传感器振动特性研究
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摘要
碳纳米管、石墨烯等纳米尺度下的材料和结构具有尺度小，密度小，刚度大，强度高，对周围环境特别敏感等特性，可用于细菌、病毒、气体分子等原子级质量的探测，在微纳米质量传感器方面具有广阔的应用前景。碳纳米管/石墨烯作为原子级质量传感器，主要利用其固有频率的改变量来探测附加质量，因此，研究其振动特性具有重要的理论意义和实际应用价值。本文研究工作的重点是建立适用于弹性模量具有尺度效应的碳纳米管/石墨烯等微纳米质量传感器振动特性分析的非局部弹性理论模型，并采用分布参数传递函数方法（以下简称传递函数方法）或Galerkin方法对其进行求解。主要研究内容如下：
建立了单壁碳纳米管（SWCNT）质量传感器振动特性分析的非局部Euler梁模型。利用携带集中质量的非局部Euler梁模型模拟含附加质量的SWCNT质量传感器，考虑悬臂和两端固支两种支撑方式，得到了含附加质量的SWCNT振动特性分析的传递函数解。进一步，考虑轴向预应力的影响，建立了两端固支SWCNT屈曲分析和振动特性分析的传递函数求解模型。讨论了非局部效应、附加质量、支撑方式以及轴向预应力对SWCNT固有频率和频率漂移的影响。研究表明，碳纳米管质量传感器的灵敏度可以达到10-21g。
建立了双壁碳纳米管（DWCNT）质量传感器振动特性分析的非局部Euler梁模型。利用带集中质量的两根相互嵌套的非局部Euler梁模拟含附加质量的DWCNT质量传感器，内外管之间的范德华力由线弹簧模拟，分别得到了含附加质量的内外管长度不同的悬臂DWCNT、以及两端固支DWCNT振动特性分析的传递函数解。此外，对内外管受不同轴向预应力的两端固支DWCNT进行了稳定性分析，在此基础上，讨论了内外管不同预应力对两端固支DWCNT固有频率和频率漂移的影响。
建立了SWCNT质量传感器振动特性分析的非局部Timoshenko梁模型。考虑横截面剪切变形和转动惯量的影响，通过含集中质量的非局部Timoshenko梁模拟携带附加质量的SWCNT质量传感器，采用传递函数方法得到了悬臂和两端固支SWCNT的固有频率。分析比较了Timoshenko梁模型和Euler梁模型在SWCNT振动特性分析中的差异性。研究表明，Timoshenko梁模型能更精确地模拟SWCNT的振动，对短粗SWCNT或SWCNT高阶模态的影响尤为突出。
建立了多壁碳纳米管（MWCNT）质量传感器振动特性分析的非局部Timoshenko梁模型。基于非局部Timoshenko理论，推导了含附加质量的内外管长度不同悬臂DWCNT以及两端固支DWCNT振动特性分析的传递函数解。考虑端部附加质量转动惯量的影响，利用传递函数方法，得到了悬臂MWCNT传感器的固有频率，结果可应用于生物传感器的病毒辨识。比较分析了单壁和双壁Timoshenko模型，以及Timoshenko梁模型和Euler梁模型在DWCNT振动特性分析中的差异性。研究表明，附加质量转动惯量降低了悬臂MWCNT的基频。
建立了单层石墨烯（SLGS）质量传感器振动特性分析的非局部Kirchhoff板模型。基于非局部Kirchhoff板理论，分别在直角坐标系和柱坐标系下推导了含附加质量的矩形和圆形SLGS质量传感器的控制方程，考虑简支和固支两种支撑方式，利用Galerkin近似方法进行求解。研究了非局部参数、附加质量以及SLGS形状对系统基频和频率漂移的影响。
由于纳米尺度下结构和材料的弹性模量具有尺度效应，经典理论不能诠释这一现象。本文将非局部弹性理论成功用于碳纳米管/石墨烯微纳米质量传感器的振动特性分析，并将传递函数方法和Galerkin近似方法的应用范围拓展到了微纳米力学领域。相关结论可为微纳米质量传感器的设计提供理论依据和参考价值。
Materials and structures at nano scale such as carbon nanotubes and graphenesheets, have small scale, low weight, extremely high rigidity as well as high intensity,and are highly sensitive to their environment change. Therefore, they are very promisingcandidates as atomic-resolution mass sensors that are used to detect the micro-masssuch as bacteria, virus or molecule of some gas. The principle of mass detection usingthe nanosized mass sensor from a vibration analysis is in quantifying the change in itsresonant frequency due to added mass. Therefore, the study of the vibrationcharacteristic of carbon nanotubes/graphenes based mass sensor is significant both intheory and in practical application. The emphasis of this dissertation is to establishtheoretical models suiting for dynamic behaviors of micro/nano-mass sensor. Themodels are based on nonlocal elasticity theory, which captures scale-dependent elasticmodulus of carbon nanotubes/graphenes. The transfer function method(TFM) orGalerkin method is used to solve the governing equations. The main achievements aresummarized as follows:
The nonlocal Euler beam model is established to analyze the vibrationcharacteristics of single-walled carbon nanotube(SWCNT) based mass sensor. ASWCNT mass sensor carrying an attached micro-mass is modeled as a nonlocal Eulerbeam with a concentrated mass. Taking into account the support conditions ofcantilevers and clamped-clamped beams, the governing equations are solved by TFM.Furthermore, the critical buckling stress and natural frequencies of the SWCNT basedsensors under initial axial stress are computed using TFM. The effects of nonlocalparameter, attached mass, support condition and axial stress on the natural frequenciesand frequency shifts are discussed. The obtained results show that the mass sensitivityof the carbon nanotube based mass sensor can reach at least10-21g.
Nonlocal Euler beam model for vibration characteristic analysis of double-walledcarbon nanotube (DWCNT) based mass sensor is established. The DWCNT mass sensorcarrying a nanoparticle is modeled as two nonlocal Euler-Bernoulli beams, and theinteraction between two tubes is governed by van der Waals force. The nonlocal Eulerbeam theory and TFM are used to analyze the vibration characteristic of cantilever andclamped-clamped DWCNT with different inner and outer wall lengths. Moreover, theeffect of different initial axial stress in inner tubes and outer tubes on the naturalfrequencies and frequency shift of clamped DWCNT based sensor is discussed.
A nonlocal Timoshenko beam model is established to analyze the dynamicbehavior of SWCNT based mass sensor. Taking into account the effects of sheardeformation and rotary inertia, the nonlocal Timoshenko beam model with aconcentrated mass is used to model a SWCNT sensor with an attached micro-mass. The natural frequencies are computed by the TFM. The difference of the vibrationcharacteristic of SWCNT according to the theories of Timoshenko beams and of Eulerbeams is analyzed. The results show that Timoshenko beam model is more adequatethan Euler beam model, especially for short beams or higher-order vibration modes.
Nonlocal Timoshenko beam model for vibration characteristic of multiwall-walledcarbon nanotubes(MWCNT) based mass sensor is established. Based on the nonlocalTimoshenko beam model, TFM is used to obtain the natural frequencies of theclamped-free and clamped-clamped DWCNT with different inner and outer wall lengths.The natural frequencies of a MWCNT-based biosensor carrying a spherical nanoscalebio-object at the free end are calculated using the nonlocal Timoshenko beam theoryand TFM. The influence of the rotary inertia of the bio-object itself is considered. Thedifference between single-walled and double-walled Timoshenko model, as well as theTimoshenko and Euler beam theory in vibration of DWCNT based mass sensor isanalyzed. The results show that the rotary inertia decreases the fundamental frequenciesof MWCNT based biosensor.
The nonlocal Kirchhoff plate model is established for vibration characteristic ofsingle-layered graphene sheet (SLGS) based mass sensor. Using the nonlocal Kirchhoffplate theory, the dynamic governing equations of the rectangle and circular SLGS withan attached micro-mass is derived under Cartesian coordinate system and cylindricalcoordinate system, respectively. The Galerkin method is used to obtain the naturalfrequencies. In the analysis, the clamped and simply supported support conditions areboth considered. The effect of nonlocal parameter, attached mass and the shape ofSLGS on the natural frequency are discussed.
The elastic modulus of materials and structures at nano scale exhibits strongscale-dependent property, and this property cannot be interpreted by the classicalcontinuum mechanics approach. In the present dissertation, the nonlocal elasticitytheory is developed to study the vibration of CNT/grapheme based micro/nano-masssensor. Moreover, the classical TFM and Galerkin method are extended tomicro-/nano-mechanics. The obtained results provide a theoretical foundation and arehelpful to the design of CNT/SLGS based resonator as nanomechanical mass sensor.
建立了单壁碳纳米管（SWCNT）质量传感器振动特性分析的非局部Euler梁模型。利用携带集中质量的非局部Euler梁模型模拟含附加质量的SWCNT质量传感器，考虑悬臂和两端固支两种支撑方式，得到了含附加质量的SWCNT振动特性分析的传递函数解。进一步，考虑轴向预应力的影响，建立了两端固支SWCNT屈曲分析和振动特性分析的传递函数求解模型。讨论了非局部效应、附加质量、支撑方式以及轴向预应力对SWCNT固有频率和频率漂移的影响。研究表明，碳纳米管质量传感器的灵敏度可以达到10-21g。
建立了双壁碳纳米管（DWCNT）质量传感器振动特性分析的非局部Euler梁模型。利用带集中质量的两根相互嵌套的非局部Euler梁模拟含附加质量的DWCNT质量传感器，内外管之间的范德华力由线弹簧模拟，分别得到了含附加质量的内外管长度不同的悬臂DWCNT、以及两端固支DWCNT振动特性分析的传递函数解。此外，对内外管受不同轴向预应力的两端固支DWCNT进行了稳定性分析，在此基础上，讨论了内外管不同预应力对两端固支DWCNT固有频率和频率漂移的影响。
建立了SWCNT质量传感器振动特性分析的非局部Timoshenko梁模型。考虑横截面剪切变形和转动惯量的影响，通过含集中质量的非局部Timoshenko梁模拟携带附加质量的SWCNT质量传感器，采用传递函数方法得到了悬臂和两端固支SWCNT的固有频率。分析比较了Timoshenko梁模型和Euler梁模型在SWCNT振动特性分析中的差异性。研究表明，Timoshenko梁模型能更精确地模拟SWCNT的振动，对短粗SWCNT或SWCNT高阶模态的影响尤为突出。
建立了多壁碳纳米管（MWCNT）质量传感器振动特性分析的非局部Timoshenko梁模型。基于非局部Timoshenko理论，推导了含附加质量的内外管长度不同悬臂DWCNT以及两端固支DWCNT振动特性分析的传递函数解。考虑端部附加质量转动惯量的影响，利用传递函数方法，得到了悬臂MWCNT传感器的固有频率，结果可应用于生物传感器的病毒辨识。比较分析了单壁和双壁Timoshenko模型，以及Timoshenko梁模型和Euler梁模型在DWCNT振动特性分析中的差异性。研究表明，附加质量转动惯量降低了悬臂MWCNT的基频。
建立了单层石墨烯（SLGS）质量传感器振动特性分析的非局部Kirchhoff板模型。基于非局部Kirchhoff板理论，分别在直角坐标系和柱坐标系下推导了含附加质量的矩形和圆形SLGS质量传感器的控制方程，考虑简支和固支两种支撑方式，利用Galerkin近似方法进行求解。研究了非局部参数、附加质量以及SLGS形状对系统基频和频率漂移的影响。
由于纳米尺度下结构和材料的弹性模量具有尺度效应，经典理论不能诠释这一现象。本文将非局部弹性理论成功用于碳纳米管/石墨烯微纳米质量传感器的振动特性分析，并将传递函数方法和Galerkin近似方法的应用范围拓展到了微纳米力学领域。相关结论可为微纳米质量传感器的设计提供理论依据和参考价值。
Materials and structures at nano scale such as carbon nanotubes and graphenesheets, have small scale, low weight, extremely high rigidity as well as high intensity,and are highly sensitive to their environment change. Therefore, they are very promisingcandidates as atomic-resolution mass sensors that are used to detect the micro-masssuch as bacteria, virus or molecule of some gas. The principle of mass detection usingthe nanosized mass sensor from a vibration analysis is in quantifying the change in itsresonant frequency due to added mass. Therefore, the study of the vibrationcharacteristic of carbon nanotubes/graphenes based mass sensor is significant both intheory and in practical application. The emphasis of this dissertation is to establishtheoretical models suiting for dynamic behaviors of micro/nano-mass sensor. Themodels are based on nonlocal elasticity theory, which captures scale-dependent elasticmodulus of carbon nanotubes/graphenes. The transfer function method(TFM) orGalerkin method is used to solve the governing equations. The main achievements aresummarized as follows:
The nonlocal Euler beam model is established to analyze the vibrationcharacteristics of single-walled carbon nanotube(SWCNT) based mass sensor. ASWCNT mass sensor carrying an attached micro-mass is modeled as a nonlocal Eulerbeam with a concentrated mass. Taking into account the support conditions ofcantilevers and clamped-clamped beams, the governing equations are solved by TFM.Furthermore, the critical buckling stress and natural frequencies of the SWCNT basedsensors under initial axial stress are computed using TFM. The effects of nonlocalparameter, attached mass, support condition and axial stress on the natural frequenciesand frequency shifts are discussed. The obtained results show that the mass sensitivityof the carbon nanotube based mass sensor can reach at least10-21g.
Nonlocal Euler beam model for vibration characteristic analysis of double-walledcarbon nanotube (DWCNT) based mass sensor is established. The DWCNT mass sensorcarrying a nanoparticle is modeled as two nonlocal Euler-Bernoulli beams, and theinteraction between two tubes is governed by van der Waals force. The nonlocal Eulerbeam theory and TFM are used to analyze the vibration characteristic of cantilever andclamped-clamped DWCNT with different inner and outer wall lengths. Moreover, theeffect of different initial axial stress in inner tubes and outer tubes on the naturalfrequencies and frequency shift of clamped DWCNT based sensor is discussed.
A nonlocal Timoshenko beam model is established to analyze the dynamicbehavior of SWCNT based mass sensor. Taking into account the effects of sheardeformation and rotary inertia, the nonlocal Timoshenko beam model with aconcentrated mass is used to model a SWCNT sensor with an attached micro-mass. The natural frequencies are computed by the TFM. The difference of the vibrationcharacteristic of SWCNT according to the theories of Timoshenko beams and of Eulerbeams is analyzed. The results show that Timoshenko beam model is more adequatethan Euler beam model, especially for short beams or higher-order vibration modes.
Nonlocal Timoshenko beam model for vibration characteristic of multiwall-walledcarbon nanotubes(MWCNT) based mass sensor is established. Based on the nonlocalTimoshenko beam model, TFM is used to obtain the natural frequencies of theclamped-free and clamped-clamped DWCNT with different inner and outer wall lengths.The natural frequencies of a MWCNT-based biosensor carrying a spherical nanoscalebio-object at the free end are calculated using the nonlocal Timoshenko beam theoryand TFM. The influence of the rotary inertia of the bio-object itself is considered. Thedifference between single-walled and double-walled Timoshenko model, as well as theTimoshenko and Euler beam theory in vibration of DWCNT based mass sensor isanalyzed. The results show that the rotary inertia decreases the fundamental frequenciesof MWCNT based biosensor.
The nonlocal Kirchhoff plate model is established for vibration characteristic ofsingle-layered graphene sheet (SLGS) based mass sensor. Using the nonlocal Kirchhoffplate theory, the dynamic governing equations of the rectangle and circular SLGS withan attached micro-mass is derived under Cartesian coordinate system and cylindricalcoordinate system, respectively. The Galerkin method is used to obtain the naturalfrequencies. In the analysis, the clamped and simply supported support conditions areboth considered. The effect of nonlocal parameter, attached mass and the shape ofSLGS on the natural frequency are discussed.
The elastic modulus of materials and structures at nano scale exhibits strongscale-dependent property, and this property cannot be interpreted by the classicalcontinuum mechanics approach. In the present dissertation, the nonlocal elasticitytheory is developed to study the vibration of CNT/grapheme based micro/nano-masssensor. Moreover, the classical TFM and Galerkin method are extended tomicro-/nano-mechanics. The obtained results provide a theoretical foundation and arehelpful to the design of CNT/SLGS based resonator as nanomechanical mass sensor.
引文
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