Experimental Analysis of Tip Vibrations at Higher Eigenmodes of QPlus Sensors for Atomic Force Microscopy

QPlus sensors are non-contact atomic force microscope probes constructed from a quartz tuning fork and a tungsten wire with an electrochemically etched tip. These probes are self-sensing and offer an atomic-scale spatial resolution. Therefore, qPlus sensors are routinely used to visualize the chemical structure of adsorbed organic molecules via the so-called bond imaging technique. This is achieved by functionalizing the AFM tip with a single CO molecule and exciting the sensor at the first vertical cantilever resonance mode. Recent work using higher-order resonance modes has also resolved the chemical structure of single organic molecules. However, in these experiments, the image contrast can differ significantly from the conventional bond imaging contrast, which was suspected to be caused by unknown vibrations of the tip. This work investigates the source of these artefacts by using a combination of mechanical simulation and laser vibrometry to characterize a range of sensors with different tip wire geometries. The results show that increased tip mass and length cause increased torsional rotation of the tuning fork beam due to the off-center mounting of the tip wire, and increased flexural vibration of the tip. These undesirable motions cause lateral deflection of the probe tip as it approaches the sample, which is rationalized to be the cause of the different image contrast. The results also provide a guide for future probe development to reduce these issues.

Nonlinear Dynamic Modeling and Analysis of an L-Shaped Multi-Beam Jointed Structure with Tip Mass

A dynamic model of an L-shaped multi-beam joint structure is presented to investigate the nonlinear dynamic behavior of the system. Firstly, the nonlinear partial differential equations (PDEs) of motion for the beams, the governing equations of the tip mass, and their matching conditions and boundary conditions are obtained. The natural frequencies and the global mode shapes of the linearized model of the system are determined, and the orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to derive a set of nonlinear ordinary differential equations (ODEs) that govern the motion of the L-shaped multi-beam jointed structure. The accuracy of the model is verified by the comparison of the natural frequencies solved by the frequency equation and the ANSYS. Based on the nonlinear ODEs obtained in this model, the dynamic responses are worked out to investigate the effect of the tip mass and the joint on the nonlinear dynamic characteristic of the system. The results show that the inertia of the tip mass and the nonlinear stiffness of the joints have a great influence on the nonlinear response of the system.

Amplitude resonance response and feedback control of cantileverbeams with tip-mass under aerodynamic load

The dynamic of a cantilever beam with tip mass is studied under an aerodynamic loading. The effects of coupling is investigated by tacking into account the fluid flow. Using the multiple time scale method, the approximative solutions are found and the study of their stability is made by the Routh-Hurwitz stability criterion. The influence of parameters on the system is studied at the harmonic and subharmonic resonances. The results show that, the effects of tip mass can be neglected in harmonic resonance case ,while they are more important in subharmonic resonance cases. The results equally demonstrate that an increase of the stable state fluid velocity reduces the amplitude of vibrations. In addition, the hysteresis phenomenon studies show that it is principally induced by nonlinearity coefficients. Finally, time-delay feedback control is applied and the effects of controlling are observed on amplitude response curve at the harmonic resonance, from where we note that optimized choice of control parameters can be useful in controlling vibrations.

Experimentally Verified Analytical Models of Piezoelectric Cantilevers in Different Design Configurations

This paper deals with analytical modelling of piezoelectric energy harvesting systems for generating useful electricity from ambient vibrations and comparing the usefulness of materials commonly used in designing such harvesters for energy harvesting applications. The kinetic energy harvesters have the potential to be used as an autonomous source of energy for wireless applications. Here in this paper, the considered energy harvesting device is designed as a piezoelectric cantilever beam with different piezoelectric materials in both bimorph and unimorph configurations. For both these configurations a single degree-of-freedom model of a kinematically excited cantilever with a full and partial electrode length respecting the dimensions of added tip mass is derived. The analytical model is based on Euler-Bernoulli beam theory and its output is successfully verified with available experimental results of piezoelectric energy harvesters in three different configurations. The electrical output of the derived model for the three different materials (PZT-5A, PZZN-PLZT and PVDF) and design configurations is in accordance with lab measurements which are presented in the paper. Therefore, this model can be used for predicting the amount of harvested power in a particular vibratory environment. Finally, the derived analytical model was used to compare the energy harvesting effectiveness of the three considered materials for both simple harmonic excitation and random vibrations of the corresponding harvesters. The comparison revealed that both PZT-5A and PZZN-PLZT are an excellent choice for energy harvesting purposes thanks to high electrical power output, whereas PVDF should be used only for sensing applications due to low harvested electrical power output.

Revisited on the Free Vibration of a Cantilever Beam with an Asymmetrically Attached Tip Mass

Cantilever with an asymmetrically attached tip mass arises in many engineering applications. Both the traditional method of separation of variables and the method of Laplace transform are employed in the present paper to solve the eigenvalue problem of the free vibration of such structures, and the effect of the eccentric distance along the vertical direction and the length direction of the tip mass is considered here. For the traditional method of separation of variables, tip mass only affects to the boundary conditions, and the eigenvalue problem of the free vibration is solved based on the nonhomogeneous boundary conditions. For the method of Laplace transform, the effect of the tip mass is introduced in the governing equation with the Dirac function, and the eigenvalue problem then can be solved through Laplace transform with homogeneous boundary conditions. The computed results with these two methods are compared well with the numerical solution obtained by finite element method and approximate analytical solutions, and the effect of tip mass dimensions on the natural frequencies and corresponding mode shapes is also given.

Split Cantilever Multi-Resonant Piezoelectric Energy Harvester for Low-Frequency Application

This paper presents a new way to design a broadband harvester for harvesting high energy over a low-frequency range of 10–15 Hz. The design comprises a cantilever beam with two parallel grooves to form three dissimilar length parallel branches, each with an unequal concentrated tip mass. The piezoelectric material covers the whole length on both sides of the beam to form a bimorph. Appropriate geometry and mass magnitudes are obtained by a parametric study using the Finite Element Method. The design was simulated in COMSOL Multiphysics to study its response. The first three bending modes were utilized in energy harvesting, resulting in three power peaks at their respective fundamental frequencies. The adequate load resistance determined was 5.62 kΩ, at which maximum power can be harvested. The proposed harvester was compared to two other harvesters presented in the literature for validation: First, an optimized conventional harvester while the proposed harvester is operating at adequate load resistance. Second, a multimodal harvester, while the proposed harvester is operating at a 10 kΩ load. The suggested harvester proved to be more efficient by harvesting sufficiently higher broadband energy and is applicable in a wide range of vibration environments because of its adaptability in design.

A Parametric Study on the Aeroelasticity of Flared Hinge Folding Wingtips

This paper presents a parametric study on the aeroelasticity of cantilever wings equipped with Flared Hinge Folding Wingtips (FHFWTs). The finite element method is utilized to develop a computational, low-fidelity aeroelastic model. The wing structure is modelled using Euler–Bernoulli beam elements, and unsteady Theodorsen’s aerodynamic strip Theory is used for aerodynamic load predictions. The PK method is used to estimate the aeroelastic boundaries. The model is validated using three rectangular, cantilever wings whose properties are available in literature. Then, a rectangular, cantilever wing is used to study the effect of folding wingtips on the aeroelastic response and stability boundaries. Two scenarios are considered for the aeroelastic analysis. In the first scenario, the baseline, rectangular wing is split into inboard and outboard segments connected by a flared hinge that allows the outboard segment to fold. In the second scenario, a folding wingtip is added to the baseline wing. For both scenarios, the influence of fold angle, hinge-line angle (flare angle), hinge stiffness, tip mass and geometry are assessed. In addition, the load alleviation capability of FHFWT is evaluated when the wing encounters discrete (1-cosine) gusts. Finally, the hinge is assumed to exhibit cubic nonlinear behavior in torsion, and the effect of nonlinearity on the aeroelastic response is assessed and analyzed for three different cases.

Analytical model and energy harvesting analysis of a vibrating slender rod with added tip mass in three-dimensional space

In the paper, a new 3D energy harvesting system is provided. This work discussed the Lagrange approach to derive the differential equations of motion in the case of energy harvesting systems. An electromechanical system consists of a mechanical resonator, a piezoelectric transducer and electrical circuit with the load resistor. A flexible slender rod clamped at the bottom and loaded by the tip mass is proposed as the resonator. Moving in the 3D space, it enables the system to avoid the gravitational potential barrier of the straight vertical shape in case of buckling. This paper investigates the response of the rod deflection and the root mean square power output of selected vibration mode shapes with an attached tip mass.