Dual resonance excitation system for the contact mode of atomic force microscopy

We propose an improved system that enables simultaneous excitation and measurements of at least two resonance frequency spectra of a vibrating atomic force microscopy (AFM) cantilever. With the dual resonance excitation system it is not only possible to excite the cantilever vibrations in different frequency ranges but also to control the excitation amplitude for the individual modes. This system can be used to excite the resonance frequencies of a cantilever that is either free of the tip-sample interactions or engaged in contact with the sample surface. The atomic force acoustic microscopy and principally similar methods utilize resonance frequencies of the AFM cantilever vibrating while in contact with the sample surface to determine its local elastic modulus. As such calculation demands values of at least two resonance frequencies, two or three subsequent measurements of the contact resonance spectra are necessary. Our approach shortens the measurement time by a factor of two and limits the influence of the AFM tip wear on the values of the tip-sample contact stiffness. In addition, it allows for in situ observation of processes transpiring within the AFM tip or the sample during non-elastic interaction, such as tip fracture.     

Mapping of lateral vibration of the tip in atomic force microscopy at the torsional resonance of the cantilever

Lateral vibration of the tip in atomic force microscopy was mapped at the torsional resonance of the cantilever by attaching a shear piezo element at the base of the cantilever or under the sample. Fixed frequency excitation and self-excitation of torsional motion were implemented. The lateral vibration utilized as measured by an optical lever was in the order of 10 pm to 3 nm, and its frequency approximately 450 kHz for a contact-mode silicon nitride cantilever. The amplitude and phase of the torsional motion of the cantilever was measured by a lock-in-amplifier or a rectifier and plotted in x and y as the sample was raster scanned. The imaging technique gave contrast between graphite terraces, self-assembled monolayer domains, silicon and silicon dioxide, graphite and mica. Changing contrast was observed as silicon islands oxidized in atmosphere, showing that the imaging technique can detect change in lateral tip mobility due to changes occurring near the surface. Torsional self-excitation showed nanometric features of self-assembled monolayer islands due to different lateral dissipation. Dependence of torsional resonance frequency on excitation amplitude, and contrast change due to driving frequency around resonance were observed.     

Coupled lateral bending-torsional vibration sensitivity of atomic force microscope cantilever

We study the influence of the contact stiffness and the ration between cantilever and tip lengths on the resonance frequencies and sensitivities of lateral cantilever modes. We derive expressions to determine both the effective resonance frequency and the mode sensitivity of an atomic force microscope (AFM) rectangular cantilever. Once the contact stiffness is given, the resonance frequency and the sensitivity of the vibration modes can be obtained from the expression. The results show that each mode has a different resonant frequency to variations in contact stiffness and each frequency increased until it eventually reached a constant value at very high contact stiffness. The low-order vibration modes are more sensitive to vibration than the high-order mode when the contact stiffness is low. However, the situation is reversed when the lateral contact stiffness became higher. Furthermore, increasing the ratio of tip length to cantilever length increases the vibration frequency and the sensitivity of AFM cantilever.     

A torsional resonance mode AFM for in-plane tip surface interactions

Changing the method of tip/sample interaction leads to contact, tapping and other dynamic imaging modes in atomic force microscopy (AFM) feedback controls. A common characteristic of these feedback controls is that the primary control signals are based on flexural deflection of the cantilever probes, statically or dynamically. We introduce a new AFM mode using the torsional resonance amplitude (or phase) to control the feedback loop and maintain the tip/surface relative position through lateral interaction. The torsional resonance mode (TRmode™) provides complementary information to tapping mode for surface imaging and studies. The nature of tip/surface interaction of the TRmode facilitates phase measurements to resolve the in-plane anisotropy of materials as well as measurements of dynamic friction at nanometer scale. TRmode can image surfaces interleaved with TappingMode™ with the same probe and in the same area. In this way we are able to probe samples dynamically in both vertical and lateral dimensions with high sensitivity to local mechanical and tribological properties. The benefit of TRmode has been proven in studies of water adsorption on HOPG surface steps. TR phase data yields approximately 20 times stronger contrast than tapping phase at step edges, revealing detailed structures that cannot be resolved in tapping mode imaging. The effect of sample rotation relative to the torsional oscillation axis of the cantilever on TR phase contrast has been observed. Tip wear studies of TRmode demonstrated that the interaction forces between tip and sample could be controlled for minimum tip damage by the feedback loop.     

Study of the tip mass and interaction force effects on the frequency response and mode shapes of the AFM cantilever

The vibrational characteristics of an atomic force microscope (AFM) cantilever beam play a key role in dynamic mode of the atomic force microscope. As the oscillating AFM cantilever tip approaches the sample, the tip–sample interaction force influences the cantilever dynamics. In this paper, we present a detailed theoretical analysis of the frequency response and mode shape behavior of a cantilever beam in the dynamic mode subject to changes in the tip mass and the interaction regime between the AFM cantilever system and the sample. We consider a distributed parameter model for AFM and use Euler–Bernoulli method to derive an expression for AFM characteristics equation contains tip mass and interaction force terms. We study the frequency response of AFM cantilever under variations of interaction force between AFM tip and sample. Also, we investigate the effect of tip mass on the frequency response and also the quality factor and spring constant of each eigenmodes of AFM micro-cantilever. In addition, the mode shape analysis of AFM cantilever under variations of tip mass and interaction force is investigated. This will incorporate the presentation of explicit analytical expressions and numerical analysis. The results show that by considering the tip mass, the resonance frequencies of the cantilever are decreased. Also, the tip mass has a significant effect on the mode shape of the higher eigenmodes of the AFM cantilever. Moreover, tip mass affects the quality factor and spring constant of each modes.     

Study of the sensitivity and resonant frequency of the torsional modes of an AFM cantilever with a sidewall probe based on a nonlocal elasticity theory

A relationship based on a nonlocal elasticity theory is developed to investigate the torsional sensitivity and resonant frequency of an atomic force microscope (AFM) with assembled cantilever probe (ACP). This ACP comprises a horizontal cantilever and a vertical extension, and a tip located at the free end of the extension, which makes the AFM capable of topography at sidewalls of microstructures. First, the governing differential equations of motion and boundary conditions for dynamic analysis are obtained by a combination of the basic equations of nonlocal elasticity theory and Hamilton's principle. Afterward, a closed‐form expression for the sensitivity of vibration modes has been obtained using the relationship between the resonant frequency and contact stiffness of cantilever and sample. These analysis accounts for a better representation of the torsional behavior of an AFM with sidewall probe where the small‐scale effect are significant. The results of the proposed model are compared with those of classical beam theory. The results show that the sensitivities and resonant frequencies of ACP predicted by the nonlocal elasticity theory are smaller than those obtained by the classical beam theory. Microsc. Res. Tech. 78:408–415, 2015. © 2015 Wiley Periodicals, Inc.     

Microthermogravimetry using a microcantilever hot plate with integrated temperature-compensated piezoresistive strain sensors

This paper reports thermogravimetric analysis of nanogram samples of paraffin using a microcantilever hot plate. The microcantilever hot plate has an integrated temperature-controlled heater and integrated temperature-compensated strain-sensing piezoresistors. The microcantilever vibration amplitude was measured using either a laser and a position sensitive photodiode, or using the piezoresistors. The cantilever resonance was measured as the cantilever was heated, such that the analyte mass could be measured as a function of temperature. Both optical and piezoresistive methods were employed to generate thermogravimetric curves for analytes in the range of 1-3 ng, and the results of the two methods compared well.     

基于特征模型的压电悬臂板振动控制

航天器上悬臂板型挠性附件在扰动作用下将引起包括弯曲和扭转模态的振动,这将影响系统的稳定性和控制精度,尤其是在平衡点附近低频模态频率上的小幅值残余振动很难快速抑制。为了快速抑制压电智能挠性悬臂板系统,包括弯曲和扭转模态的振动,提出采用基于特征模型的非线性黄金分割自适应控制,组合非线性切换逻辑积分阻尼器算法。首先,优化配置压电传感器和驱动器实现了悬臂板的弯曲和扭转模态在检测和驱动上的解耦;其次,设计并建立压电挠性悬臂板试验平台,进行试验模态辨识并获得了悬臂板系统弯曲和扭转振动的模态频率和频响特性;最后,进行压电智能悬臂板的弯曲和扭转振动模态主动振动几种方法试验的比较研究。试验结果表明,采用的控制方法能够快速地抑制压电智能挠性悬臂板的振动。     

Dynamic analysis of tapping mode atomic force microscope (AFM) for critical dimension measurement

Transient dynamics of tapping mode atomic force microscope (AFM) for critical dimension measurement are analyzed. A simplified nonlinear model of AFM is presented to describe the forced vibration of the micro cantilever-tip system with consideration of both contact and non-contact transient behavior for critical dimension measurement. The governing motion equations of the AFM cantilever system are derived from the developed model. Based on the established dynamic model, motion state of the AFM cantilever system is calculated utilizing the method of averaging with the form of slow flow equations. Further analytical solutions are obtained to reveal the effects of critical parameters on the system dynamic performance. In addition, features of dynamic response of tapping mode AFM in critical dimension measurement are studied, where the effects of equivalent contact stiffness, quality factor and resonance frequency of cantilever on the system dynamic behavior are investigated. Contact behavior between the tip and sample is also analyzed and the frequency drift in contact phase is further explored. Influence of the interaction between the tip and sample on the subsequent non-contact phase is studied with regard to different parameters. The dependence of the minimum amplitude of tip displacement and maximum phase difference on the equivalent contact stiffness, quality factor and resonance frequency are investigated. This study brings further insights into the dynamic characteristics of tapping mode AFM for critical dimension measurement, and thus provides guidelines for the high fidelity tapping mode AFM scanning.     

Dynamic analysis of an axially moving robot manipulator supported by bearings

In this study, a robot manipulator is modelled as a cantilever beam, which moves in an axial direction, has a lumped mass at the end, and is supported by intermediate springs. Considering the tip mass and intermediate springs in the modeling, we derive the equations of motion in which the rigid-body motion is coupled with the flexible motions, and then analyze the transverse vibrations of the beam. Furthermore, we study the tip mass effects on the natural frequencies and the corresponding mode shapes. The natural frequency loci veering is analyzed for variations in the tip mass and the spring position/stiffness. In addition, we investigate the exchange and localization of modes around these veering regions as well as the parameter effects on the mode shapes. Using a Short-time Fourier transform (STFT), the relationship between the dynamic characteristics and dynamic responses are described. It is found that the dynamic characteristics of the beam are dependent on the veering distance. It is also shown via dynamic responses that the mode exchanges occur when a veering distance is close.     

三圆盘非线性扭振系统的内共振

建立了具有平方非线性三圆盘扭振系统的数学模型,应用非线性振动平均法,求得了系统满足双重共振条件的一次近似及对应的稳态解。求得了用椭圆函数表示的系统内共振条件下无阻尼自由振动的解析解,指出系统有能量交换现象,存在周期运动、概周期运动以及调幅调相运动。     

Optical microcantilever consisting of channel waveguide for scanning near-field optical microscopy controlled by atomic force

We develop a novel optical microcantilever for scanning near-field optical microscopy controlled by atomic force mode (SNOM/AFM). The optical microcantilever has the bent channel waveguide, the corner of which acts as aperture with a large tip angle. The resonance frequency of the optical microcantilever is 9 kHz, and the spring constant is estimated to be 0.59 N/m. The optical microcantilever can be operated in contact mode of SNOM/AFM and we obtain the optical resolution of about 200 nm, which is as same size as the diameter of aperture. We confirm that the throughput of optical microcantilever with an aperture of 170 nm diameter would be improved to be more than 10(-5).     

齿式联接不对中转子的弯扭耦合振动特性分析

以齿式联轴器的不对中故障为研究对象,推导不对中激励力表达式,并从动力学的角度,建立不对中弯扭耦合振动微分方程。基于该微分方程,文中先从理论分析角度,定性分析弯振对扭振的影响和扭振对弯振的影响,接着从变转速过程、质量偏心、不对中量以及阻尼系数对弯扭耦合特性影响几方面进行数值仿真分析。计算表明,弯振与扭振是相互耦合的,质量偏心是耦合的前提。弯振频率包含工频和二倍频成分等,不对中越严重,二倍频所占比例越大,其轴心轨迹为双环椭圆。弯扭耦合将激发各阶谐波扭振,包括工频、二倍频和三倍频,其中工频扭振占主导地位。     

Optical microcantilever consisting of channel waveguide for scanning near-field optical microscopy controlled by atomic force

We develop a novel optical microcantilever for scanning near-field optical microscopy controlled by atomic force mode (SNOM/AFM). The optical microcantilever has the bent channel waveguide, the corner of which acts as aperture with a large tip angle. The resonance frequency of the optical microcantilever is 9 kHz, and the spring constant is estimated to be 0.59 N/m. The optical microcantilever can be operated in contact mode of SNOM/AFM and we obtain the optical resolution of about 200 nm, which is as same size as the diameter of aperture. We confirm that the throughput of optical microcantilever with an aperture of 170 nm diameter would be improved to be more than 10⁻⁵.     

Geometric parameters effect of the atomic force microscopy smart piezoelectric cantilever on the different rough surface topography quality by considering the capillary force

Nowadays, the atomic force microscopy (AFM) is widely used in the nanotechnology as a powerful nano‐robot. The surface topography in Nanoscale is by far one of the most important usages of the AFM device. Hence, in this article, the vibration motion of a piezoelectric rectangular cross‐section micro‐cantilever (MC) which oscillates in the moist environment has been examined based on the Timoshenko beam theory. After extracting the MC governing equations according to Hamilton's principle, the finite element method has been used to discretize the motion equations. The surface topography has been simulated for various roughness forms in the tapping and non‐contact modes by considering the effects of the Van der Waals, capillary and contact forces. Also, the experimental results obtained from the glass surface topography have been simulated. The results illustrate that the time delay in higher natural frequencies in the tapping mode is shorter in comparison with the non‐contact mode, especially, for the lower natural frequencies. The sensitivity analysis of the natural frequencies, topography depth and time delay have been simulated. Results indicate that the most effective parameter is the MC length. In the first mode, the first section length has the highest effect on the surface topography time delay, also, in the second vibration mode; the most effective parameter on the time delay is the MC tip length based on the simulation results.     

Amplitude and frequency modulation torsional resonance mode atomic force microscopy of a mineral surface

Scanning probe imaging in a shear force mode allows for the characterization of in-plane surface properties. In a standard AFM, shear force imaging can be realized by the torsional resonance mode. In order to investigate the imaging conditions on mineral surfaces, a torsional resonance mode atomic force microscope was operated in amplitude (AM) and frequency modulation (FM) feedback. Freshly cleaved chlorite was investigated, which showed brucite-like and talc-like surface areas. In constant amplitude FM mode, a slight variation in energy dissipation was observed between both surfaces. Amplitude and frequency vs. distance curves revealed that the tip was in repulsive contact with the specimen during imaging.     

2K-H行星传动的修正扭转模型建立与固有特性分析

建立了2K-H均布行星传动的修正扭转模型,以揭示系统的固有频率与振动模态特性。该模型在纯扭转模型的基础上,进一步计入了行星轮轴心沿系杆切向的位移与行星轮轴的轴承支承刚度对系统动力学性能的影响。无阻尼固有特性分析表明,2K-H均布行星传动中存在三种典型的振动模式,即中心构件扭转振动模式、行星轮扭转振动模式与行星轮平移振动模式。求解相应的子特征值问题,获得了两种行星轮振动模式所对应的系统固有频率的封闭解析表达式。固有频率的参数灵敏度分析表明,行星轮轴的轴承支承刚度是影响2K-H均布行星传动的一阶固有频率的关键因素。固有频率敲击试验表明,与纯扭转模型相比,修正扭转模型更具工程应用价值,可用以指导2K-H均布行星传动的动态设计。     

Nonlinear dynamic analysis of coupled gear-rotor-bearing system with the effect of internal and external excitations

Extensive studies on nonlinear dynamics of gear systems with internal excitation or external excitation respectively have been carried out. However, the nonlinear characteristics of gear systems under combined internal and external excitations are scarcely investigated. An eight-degree-of-freedom(8-DOF) nonlinear spur gear-rotor-bearing model, which contains backlash, transmission error, eccentricity, gravity and input/output torque, is established, and the coupled lateral-torsional vibration characteristics are studied. Based on the equations of motion, the coupled spur gear-rotor-bearing system(SGRBS) is investigated using the Runge-Kutta numerical method, and the effects of rotational speed, error fluctuation and load fluctuation on the dynamic responses are explored. The results show that a diverse range of nonlinear dynamic characteristics such as periodic motion, quasi-periodic motion, chaotic behaviors and impacts exhibited in the system are strongly attributed to the interaction between internal and external excitations. Significantly, the changing rotational speed could effectively control the vibration of the system. Vibration level increases with the increasing error fluctuation. Whereas the load fluctuation has an influence on the nonlinear dynamic characteristics and the increasing excitation force amplitude makes the vibration amplitude increase, the chaotic motion may be restricted. The proposed model and numerical results can be used for diagnosis of faults and vibration control of practical SGRBS.     

一种新的转子碰摩故障诊断特征的研究

建立Jeffcott碰摩转子的弯扭耦合振动非线性微分方程，通过数值仿真手段，着重分析转子碰摩故障的扭振时频域特点及其非线性振动特性，并描述动静间隙对碰摩转子扭振非线性特性的影响。研究表明，转子碰摩激起丰富的扭振频谱信息，主要包括N/2X、N/3X等分频成分以及1X、2X等倍频成分，且分频幅值往往比倍频大许多。在一定转速范围内，周期运动与复杂的非线性运动交替出现。而动静间隙对转子扭振特性影响明显，动静间隙越小，转子动静碰摩越严重，频谱成分越丰富，运动形式更趋复杂，相应的各复杂运动区的范围也越宽。文中揭示的扭振特征为转子碰摩的状态识别与诊断提供了一种新方法。     

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.