Effect Of Tip Mass Research Articles

Amplitude resonance response and feedback control of cantilever beams 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.

Improving Energy Harvesting Efficiency by Vibration-Induced Stresses of Piezoelectric Patch Glued Tapered Beams

In this study, the effects of taper ratio and boundary conditions on the energy harvest performance of a beam element were examined. For these purpose, different taper ratio beams were analyzed numerically. The energy harvesting process was achieved by gluing a piezoelectric patch onto the cantilever tapered beam. Different taper ratio beams were designed and the effects of these taper ratio on stress change were investigated. In piezoelectric materials, when mechanical stress or strain is applied to the material, they generate electrical potential energy as a response. In order to increase the stresses on the tapered beam, the boundary condition was applied to be the thin edge of the tapered beam element in this study. In this way, the effect of tip mass was created and it was aimed to increase the stress magnitude due to vibration on the beam. Stress changes and displacement magnitudes of beams examined by applying load on beams having different taper ratio. The effect of these alterations on energy harvest efficiency was analyzed.

Performance analysis of piezoelectric energy harvesters with a tip mass and nonlinearities of geometry and damping under parametric and external excitations

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.

Effect of Tip Mass Length Ratio on Low Amplitude Galloping Piezoelectric Energy Harvesting

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.

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

In this paper, a broadband piezoelectric energy harvester with an applied restoring force is studied. The restoring force is modeled as a spring in the boundary conditions of the system. The system is composed of a clamped-free beam with a tip mass which is supported by a spring at the free end. A piezoelectric layer is bounded to the upper surface of the beam and the system is exposed to an external harmonic base excitation. A nonlinear distributed parameter model of the harvester is derived using the Hamilton's principle utilizing the Euler–Bernoulli beam theory assumption. The nonlinear strain-displacement field is used to take the large transverse deflections effects into account. The reduced-order model is derived by implementing the Galerkin discretization method and the dimensionless form of the equations is used to analyze the system. The effect of the piezoelectric layer on the free vibrations of the structure is determined. Effects of tip mass and base acceleration on the frequency response of the system are investigated. The results show that, the tip mass can amplify the scavenged voltage and tune the resonance frequency. To study the influence of different types of the restoring force, linear, hardening, and pure nonlinear behavior are considered for the applied spring. The results demonstrate that by applying a pure nonlinear restoring force, the resonance bandwidth of the harvester increases which causes the harvester to generate energy in a larger frequency bandwidth and the output voltage increased remarkably in comparison to clamped-free energy harvester. This phenomenon enhances the harvester efficiency in the cases of the excitations with the time-varying frequency or the random excitations and the results in a broadband energy harvester. Finally, studying the frequency response curves for the different values of the spring location declares that, as the location of the attached spring changes, the nonlinear resonance frequency alters. The maximum increase in the bandwidth is accompanied by the case where spring is attached at the free end of the beam.

Effects of Tip Mass and Interaction Force on Nonlinear Behavior of Force Modulation FM-AFM Cantilever

AbstractInfluences 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.

Stability Criterion for Thermal Flutter of a Space Beam

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.

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

The effect of tip mass on the frequency response and sensitivity of atomic force microscope (AFM) cantilever in the liquid environment is investigated. For this purpose, using Euler–Bernoulli beam theory and considering tip mass and hydrodynamic functions in a liquid environment, an expression for the resonance frequencies of AFM cantilever in liquid is derived. Then, based on this expression, the effect of the surface contact stiffness on the flexural mode of a rectangular AFM cantilever in fluid is investigated and compared with the case where the AFM cantilever operates in the air. The results show that in contrast with an air environment, the tip mass has no significant impact on the resonance frequency and sensitivity of the AFM cantilever in the liquid. Hence, analysis of AFM behaviour in liquid environment by neglecting the tip mass is logical.

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

In this paper, the effect of the tip mass on the sensitivity of flexural vibration modes of an AFM rectangular cantilever has been studied and a characteristics equation for the AFM cantilever has been derived. Then, based on the characteristics equation, a closed form expression for the flexural modal sensitivity of the rectangular cantilever to the surface stiffness variations has been derived. Using numerical experiments, the effect of various tip masses on the sensitivity has been investigated. The results show that for low stiffness materials, the sensitivity of AFM cantilevers with smaller tip masses is more than the cantilever with greater tip masses. While for the materials with high stiffness, the difference in sensitivities is not significant. However, for the cantilevers with bigger tip masses sensitivity is higher.

Theoretical investigation of nonlinear resonances in a carbon nanotube cantilever with a tip-mass under electrostatic excitation

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.

Approximate analytical solutions of an axially moving spacecraft appendage subjected to tip mass

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.

VIBRATION ANALYSIS OF A ROTATING TAPERED COMPOSITE BEAM WITH TIP MASS

In this study, free vibration responses of a rotating tapered composite beam with tip mass are investigated. The energy expressions for the kinetic and potential energies of a rotating composite beam with tip mass have been formulated. The energy expressions are then applied in Lagrange’s equations to develop the equation of motion of a composite beam with tip mass. The stiffness and mass matrices for a standard composite beam element with two end nodes with two degrees of freedom at each node are derived. Various parametric studies are performed to investigate the effect of tip mass and the rotational speed on the variation of natural frequencies of the composite beam. The investigations are also done to study the effect of hub radius on the natural frequencies. It is shown that the addition of tip mass increases the stiffness of the structure and consequently increases the natural frequencies.

Complete dynamic modeling and approximate state space equations of the flexible link manipulator

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.

Vibration of a tapered cantilever of constant thickness and linearly tapered width

The vibration of a cantilever beam with constant thickness and linearly tapered sides is solved using a novel accurate, efficient initial value numerical method. The effects of tip mass, base fixity, and taper on the natural frequencies are determined. This geometrically anisotropic beam vibrates in a mixture of modes in two perpendicular directions.

끝단 집중 질량을 갖는 회전 외팔보의 DTM을 이용한 진동 해석

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.

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

In this paper, the coupled flexural–torsional free and forced vibrations of a beam with tip and/or in-span attachments are studied. First, a mathematical model is established, which consists of a beam with several tip attachments, i.e, a tip mass of non-negligible dimensions, a linear spring grounding the tip mass, and a torsional spring connected at the end of the beam. The modal functions of this model and the orthogonality condition among them are derived. For the purpose of verification the properties of the tip attachments are changed, and the numerical results obtained are compared with those given in the relevant literature. Effects of tip mass and distributed mass in-span on natural frequencies and modes are investigated for two cantilever beams with different cross sections. An application of the orthogonality condition in the case of a beam with tip mass is also presented for a forced vibration example.

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

This paper presents a finite-element model for a flexible hub–beam system with a tip mass. Both viscous damping and air drag force are introduced into this model. The complete coupling between the system rigid and flexible degrees of freedom is allowed since the start of the formulation and developing the system kinematic variables. Based on deformation theory and geometric constraints, a second order approximation for the displacement field is proposed and the dynamic stiffening is accounted for. Hamilton's principle is utilized in deriving the equations of motion. The corresponding dynamics models of the tip mass and damping forces are developed in a consistent manner through formulating their energy expressions and applying Hamilton's principle. The finite element method is employed for spatial discretization due to its versatility, high accuracy and convergence. Numerical simulations show that the second order term in deformation field can have significant effect on dynamics behavior of flexible multibody systems. It is also shown that the traditional linear model cannot account for dynamic stiffening and may lead to erroneous result in some high-speed systems because the deformation field commonly used in structural dynamics is straight employed in this model. In contrast, the developed model (CCM) based on the second order deformation field can predict valid results. The effects of tip mass and damping on dynamics behavior of the hub–beam system are also discussed.

Self-adaptive vibration control of smart composite beams using recurrent neural architecture

A self-adapting vibration control system is developed for damping augmentation in smart composite beams. The conventional vibration control approaches are limited by the requirement of an explicit and often accurate identification of the system dynamics and subsequent offline design of an optimal controller. In the present study a self-adapting vibration control system is developed. A hybrid system comprised of a dynamic diagonal recurrent neural network (DRNN) and an adaptable feed forward neural network is used to control the beam vibrations. Sensing and actuation are achieved using piezoelectric sensors and actuators. A finite element model based on a higher-order shear deformation theory is used to simulate the vibration response of laminated composite beams with integrated piezoelectric sensors and actuators. The dynamic effects of mass and stiffness of the piezoelectric patches are considered in the model. The performance of the DRNN controller is verified for arbitrary initial conditions and loadings. A robustness study including the effects of tip mass, structural parameter variation and partial loss of sensor output is performed. The performance with partial failure of control actuation is also examined. It is seen that the robustness and control capabilities of the hybrid control system are excellent.

DYNAMIC ANALYSIS OF ROTATING CURVED BEAM WITH A TIP MASS

The dynamic characteristics of a rotating curved beam are investigated. The equations of motion include all dynamic effects such as Coriolis force, centrifugal force and acceleration. The analysis of the rotating beam takes into account the coupling between rigid-body motion and elastic deformation, such that geometrically non-linear effects are included in the model. For dynamic analysis, the time responses for accelerating motion and torque-driven motion are calculated. The natural frequencies for curved beams of various radii of curvature are then calculated as the rotating speed increases. This study mainly discussed the effect of curvature that can change the characteristics of the beam. The effects of tip mass on the dynamic response of the beam are also studied.

Causal inverse dynamics of a flexible hub-arm system through Liapunov's second method

SUMMARYThe main objective of this work is to study the performance of a flexible single hub-arm system. The equations of motion are derived using the extended Hamilton principle. The Liapunov functional is used as a condition for the stability analysis. The Liapunov functional is considered as the sum of the internal energy of the flexible beam. The required drive torque was obtained directly through the solution of the inverse dynamic problem. Although the flexible link is nonminimum phase in nature, the use of Liapunov and the PD controller guarantee the causality for the stable case. The effects of tip mass as well as its inertia in the case of stable and asymptotic stable systems were investigated to ensure the validity of this procedure.