Effect of Tip Mass on Modal Flexural Sensitivity of Rectangular AFM Cantilevers to Surface Stiffness Variations

Effect of tip mass on frequency response and sensitivity of AFM cantilever in liquid

Approximate solutions for vibrations of flexible beam-type appendages subjected to tip mass are studied while uniform and exponential profiles for arm deployment are simulated. Applying an equivalent dynamical system and following Lagrangian approach, the equations of motion of the system are derived as nonlinear ordinary differential equations (ODEs) (with time-varying coefficients), in which the effect of the tip mass can be considered as some nonlinearity added to the ‘no tip mass’ case dynamics. The approximate closed-form solutions are obtained through a novel methodology using a computer algorithm, in which the solutions of the ‘no tip mass’ case are expanded by imposing quadratic perturbations on the independent variable. The mean square of errors (MSEs) for the obtained approximate analytical solution is computed. Using this method, the amplitude and frequency of the arm response are presented by the algebraic equations, which help the parametric design of such systems. In addition, effects of tip mass as an indicator of nonlinearities added to the system dynamics, on the amplitude and frequency of the beam response, are investigated during arm deployment.

The nonlinear dynamics of a resonating carbon nanotube (CNT) cantilever having an attached mass at the tip (“tip mass”) were investigated by incorporating electrostatic forces and intermolecular interactions between the CNT and a conducting plane surface. This work enables applications of CNT resonating sensors for tiny mass detection and provides a better understanding of the dynamics of CNT cantilevers. The effect of tip mass on a resonating CNT cantilever is normally characterized by the fundamental frequency shift in the linear resonance regime. However, there are more complex dynamics in the nonlinear resonance regime, such as secondary resonances with parametric excitation. The latter have been limited to nano-cantilevers without tip mass or to axially excited micro-beams. To analyze the nonlinear dynamics, we developed a differential equation model that includes both geometric and inertial nonlinear terms for the large vibration amplitudes at increasing drive forces. In our approach, we used Galerkin discretization techniques and numerical integration methods. The CNT cantilever exhibited complex nonlinear responses due to the applied AC and DC voltages and various tip masses. The nonlinear model had a softer response for increasing tip mass than those of the linear model with the same driving conditions. At low applied voltages, the cantilever had linear amplitude and phase responses at primary and secondary superharmonic resonance frequencies. The response branches were softened at the primary resonance through saddle-node (SN) bifurcation from harmonic electrostatic excitation at higher applied voltages. After SN bifurcation, the lower branch of the solution near resonance became unstable. In addition, theoretical analyses were performed on more complex nonlinear responses and stability changes with tip mass variations, such as period-doubling (PD) bifurcation at subharmonic resonance frequencies.

Based on the Hamilton’s principle, a nonlinear mathematical model of the cantilever-type piezoelectric energy harvester with a tip mass is systematically derived under parametric and external excitations. The proposed model accounts for geometric and electro-mechanical coupling nonlinearities, damping nonlinearity and the inextensibility condition of beam. Using the Galerkin approach, the proposed model is converted into the electro-mechanical coupling Mathieu–Duffing equations. Analytical solutions of the frequency–response curves are presented by the multiple scales method. Nonlinear characteristics of the energy harvesters are explored under parametric excitation and hybrid parametric and external excitations. Analytical results provided new insights into the effects of tip mass and nonlinear damping on the performance of the energy harvester. The results show that with the tip mass increasing, the frequency–response curves of the energy harvester change from the nonlinear hardening type to the nonlinear softening type and the operating bandwidth and the output voltages of the energy harvester enlarge. For parametrical excitation, variation of the quadratic damping does not alter the initial threshold of the harvesters and the position of two transcritical bifurcation points of the frequency–response curves. The initiation threshold decreases with the tip mass increasing. Hybrid parametric and external excitations enhance the bandwidth and output voltage of the energy harvester, which will probably be used as an ideal way to improve the performance of the energy harvesting system.

Broadband and tunable PZT energy harvesting utilizing local nonlinearity and tip mass effects

Coupled bending and torsional vibration of a beam with in-span and tip attachments

There are some important parameters that must be taken into consideration in the design process of the actuator for a flexible single link robot. Among these important factors are the hub inertia and the proper tip mass (including the payload) for optimum performance of the flexible arm system. The main objective of the paper is to study the effect of the actuator inertia as well as the tip mass in the flexible hub torque through the solution of the inverse dynamics problem. For comparison, two reference trajectories; Gaussian velocity, and a polynomial profile are applied. The equations of motion are derived with the help of the extended Hamilton principle using the virtual link coordinate system and the pinned-pinned boundary conditions.

The vibration analysis of a rotating cantilever beam with tip mass was studied by using DTM(differential transformation method). DTM is one of the numerical methods, for finding series solutions by transforming differential equations to algebraic ones similar with Laplace transform. The advantages of the DTM are that it is easy to understand and is effective in finding numerical solutions. Applying DTM, the natural frequencies of a rotating cantilever beam were obtained taking into consideration the effects of tip mass. Also, convergence study of DTM was performed to decide the number of terms used in eigenvalue problems. Numerical results obtained by DTM show good agreement with those by other methods. As a result, it is expected that DTM can be a useful method in vibration analysis such as that of a rotating cantilever beam with tip mass.

This paper presents a dynamic model of a flexible hub-beam system with a tip mass. With the consideration of the second-order term of the coupling deformation field, dynamic equations of the flexible hub-beam system are developed by using assumed mode method (AMM) and Lagrangian principle. The corresponding dynamics model of the tip mass is developed in a consistent manner. Numerical simulations and Theoretical analysis show that the second-order term has a significant effect on the dynamic characteristics of the system and the dynamic stiffening is accounted for. The effect of tip mass on dynamic behavior of the multibody system is also discussed.

Dynamics modelling of a flexible hub–beam system with a tip mass

Influences of the tip mass, excitation mode of Frequency Modulated Atomic Force Microscope (FM-AFM) on the resonance frequency shift in force modulation (FM) mode are studied. Governing equations of motion are determined based on Timoshenko beam model with concentrated end mass. Approach point and base amplitude are set such that the FM-AFM remains just in FM mode. Either the linearized and nonlinear Derjaguin-Muller-Toporov (DMT) model are investigated. Then frequency shifts are determined for various interaction force regimes. It is showed the effect of tip mass on frequency shift is significant even for small tips. Nonlinear model shows lower frequency shifts in comparison with linearized model. It is showed that the amplitude of response is increased by increasing the tip mass and order of base excitation. Deviation of frequency shift between linearized and nonlinear solution are studied. It is declared that the error between linearized and nonlinear model is complicated. A deviation index is used for explaining behavior of error while tip mass and excitation mode are changed. It is showed, this index predicts the trend of error in all excitation modes and force cases. Behavior of system is linearizing by increasing the order of excitation, generally.

This work treats the problem of dynamic modeling and state space approximation for robotic manipulators with flexibility. Case studies are planar manipulators with a single flexible link together with clamped-free ends and tip mass conditions. In this paper, complete dynamic modeling of the flexible beam without premature linearization in the formulation of the dynamics equations is developed, whereby this model is capable of reproducing nonlinear dynamic effects, such as the beam stiffening due to the centrifugal and the Coriolis forces induced by rotation of the joints, giving it the capability to predict reliable dynamic behavior. On the other hand, in order to show the joint flexibility effects on the model dynamic behaviors, manipulator with structural and joint flexibility is considered. Thus, a reliable model for flexible beam is then presented. The model is founded on two basic assumptions: inextensibility of the neutral fiber and moderate rotations of the cross sections in order to account for the foreshortening of the beam due to bending. To achieve flexible manipulator control, the standard form of state space equations for a flexible manipulator system (flexible link and actuator) is very important. In this study, finite difference method for discretization of the dynamic equations is used and the state space equations of the flexible link with tip mass considering complete dynamic of the system are obtained. Simulation results indicated substantial improvements on dynamic behavior and it is shown that the joint flexibility has a considerable effect on the dynamic behavior of rotating flexible arm that should not be simply neglected. The effects of tip mass is proved to be increasing the elastic deformations’ amplitudes and increasing stability.

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.

Thermal flutter of a space beam is devastating for the safety of whole spacecraft. A coupled thermal-structural dynamic model of a space beam and its approximate solution are derived for the stability analysis. A new analytical criterion for thermal flutter of a beam considering the effects of tip mass and damping is established according to the first Lyapunov method. The exact limits on beam length, tip mass, damping and incident angle of heat flux are obtained from the presented criterion and are verified by numerical results. The presented criterion provides a clear design guideline to prevent the thermal flutter of a space beam from happening in space engineering.

Galloping beams were exposed to the wind free stream and is used for sustainable wind-power harnessing. In this paper, the effect of tip mass on the performance of a galloping energy harvester is investigated by simple modeling of the system, which is useful for broad engineering applications of these systems. Here, the piezoelectric layer attached to a cantilever beam with a tip mass exposed to the wind free stream is used as an energy harvester. A fluid–solid interaction model is used to simulate the problem. The fluid–solid interaction model is composed of the experimental data for aerodynamic loads and one-dimensional structural model of piezoelectric and beam material with Euler–Bernoulli beam theory. The governing partial differential equations of the system are solved analytically by use of the approximation method. The resulting model is confirmed by preceding experimental results. The effects of the tip mass length ratio on the onset of galloping, the level of the produced voltage, and the harvested power are determined analytically. As shown by increase of the length of tip mass for the constant beam and piezoelectric length, the inertia of the system increases while the tip displacement and onset of galloping decrease.