On the eigenfrequencies of cantilevered beams carrying a tip mass and spring-mass in-span

The present study is concerned with the derivation of the eigenfrequencies and their sensitivity of a cantilevered Bernoulli-Euler beam carrying a tip mass (primary system) to which a spring-mass (secondary system) is attached in-span. After establishing the exact frequency equation of the combined system, a Dunkerley-based approximate formula is given for the fundamental frequency. Using the normal mode method, a second approximate frequency equation is established which is then used for the derivation of a sensitivity formula for the eigenfrequencies. The frequency equations of some simpler systems are obtained from the general equation as special cases. These frequency equations are then numerically solved for various combinations of physical parameters. The comparison of the numerical results with those from exact frequency equations indicate clearly that the eigenfrequencies of the combined system described above can be accurately determined by the present method.     

Active disturbance rejection boundary control of Timoshenko beam with tip mass

In this paper, the active disturbance rejection control (ADRC) is utilized to stabilize the vibration of perturbed Timoshenko beam model with tip mass. The boundary control design is based on a hybrid PDE–ODE model, and is accompanied with designing a high-gain extended state observer (ESO) that is used to estimate the boundary disturbances. By transforming the model into the appropriate state space, the semigroup theory is employed to prove the well-posedness of the closed-loop system. To this end, it is proved by a frequency domain method that the semigroup generated by the system operator is exponentially stable, which allows to conclude the boundedness of perturbed closed-loop system response. The stability of the closed-loop system is further analyzed using the Lyapunov approach. Simulation results are presented to illustrate the efficacy of the suggested method.     

Application of a LIPCA for the structural vibration suppression of an aluminum cantilever beam with a tip mass

Use of bare PZT as an actuator in the field of active vibration suppression may cause some drawbacks such as critical breaks in the installation process, short circuits in the host material and low fatigue performance. To alleviate these problems, we developed a new actuator called a lightweight piezocomposite actuator (LIPCA). The LIPCA has five layers: three glass-epoxy layers, a carbon-epoxy layer and a PZT layer. We implemented a LIPCA as an actuator to suppress the vibration of an aluminum cantilever beam with a tip mass. For the control algorithm in our test, we used positive position feedback. The filter frequency for this type of feedback should be tuned to the frequency of the target mode. The first three experimental natural frequencies of the aluminum cantilever beam agree well with the results of finite element methods. The effectiveness of using a LIPCA as an actuator in active vibration suppression was investigated with respect to the time and frequency domains, and the experimental results show that LIPCAs can significantly reduce the amplitude of forced vibrations as well as the settling time of free vibrations.     

On the dynamics of tapered vibro-impacting cantilever with tip mass

This paper explores nonlinear dynamic behavior of vibro-impacting tapered cantilever with tip mass with regard to frequency response analysis. A typical frequency response curve of vibro-impacting beams displays well-known resonance frequency shift along with a hysteric jump and drop phenomena. We did a comprehensive parametric analysis capturing the effects of taper, tip-mass, stop location, and gap on the non-smooth frequency response. Analysis is presented in a non-dimensional form useful for other similar cases. Simulation results are further validated with corresponding experimental results for a few cases. Illustrative comparison of simulation results for varying parameters brings out several interesting aspects of variation in the nonlinear behavior.     

The mechanical analysis of a mass-spring load supported on a beam system

The static deflection and dynamic characteristics of a mass-spring system supported on beam systems are investigated in this paper. In statics, it shows that the maximum deflection is reduced considerably when a clamped-free beam is replaced by a beam system which consists of a primary beam one end of which is clamped and the other end is supported by a subsidiary beam. The addition of a subsidiary beam leads to axial forces in both beams, the primary one in tension and the subsidiary in compression. The dynamic characteristic shows that the natural frequency of the mass-spring system decreases. In some cases it becomes imaginary, because buckling occurs in the subsidiary beam. This means that the effects of the addition of a subsidiary beam are not always of a positive nature, with respect to the stiffness of the whole system. At low frequencies, the response of the mass is larger than that of the mass supported on a motionless foundation. At high frequencies the dynamic characteristics of the foundation influence the vibration of the mass only a little; i.e. it moves as if the foundation were motionless.     

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.     

Dynamic response analysis of a cantilevered beam due to an elastic impact

The beam structure models with impact or contact parts under impact forces have been applied to the design of mechanical and electronic accessories. Switches, pick-ups and sensors are the typical examples of the structure to be designed to colliding with other parts. The damping characteristics and stiffness of an impacter have a great effect on dynamic responses of structures subjected to impact forces. Since the response characteristics of structures have an effect on the generation of impact forces, both theoretical and experimental investigations for the dynamic relationships between the impacter and impact mechanism are required for the structural design to reduce impact forces or eliminate unwanted vibration modes. In this paper, in order to examine the relationship between the changes of the stiffness and damping of an impacter and the variations of the dynamic characteristics of the impact model of a cantilevered beam with an impacter, the impact force of the impacter and response characteristics of the cantilevered beam were analyzed by both numerical simulation and experiments. Since the stiffness and damping of the impacter are highly nonlinear, the contact model using revised Hertz-model was established by experiments. Also, the results of numerical analyses for the dynamic response and impact force of a cantilevered beam with an impacter have a good agreement with experimental results. It is believed that the mathematical model and the analytical method presented here can be used for the design analysis and the improvement of performances of the beam-shape switches and pickup devices.     

The effect of off-end tip distance on the nanomanipulation based on rectangular and V-shape cantilevered AFMs

The AFM system, which is used as a nanomanipulator, includes a probe consistent of a cantilever and a tapered tip. In cantilevers, the tip can be located in different distances from the cantilever free end. This causes to change in stiffness of the cantilever and therefore changing in pushing force of the nanomanipulation. In this paper, the effect of the tip distance on the cantilever stiffness is studied using the equations of Hazel, and Neumeister and Ducker (ND), and a new equation to correct the torsional stiffness of V-shaped cantilevers (VSC) is proposed, which is based on the ND equation. Then, the effect of distance on pushing force of AFM-based nanomanipulations with rectangular cantilevered (RC) and VSC AFMs is simulated. The obtained results using proposed equation show that increasing of distance causes to non-linear increment of torsional stiffness of VSC. Error of the proposed equation is achieved less than 3% in comparison with result of torsional stiffness equation of ND. Moreover, it is observed that the torsional stiffness of VSC predicted by Hazel’s equation is considerably inaccurate. In nanomanipulation studies, the necessary pushing forces of nanoparticle motion are increased by increment of distance, for both types of cantilevers (RC and VSC). Moreover, critical time for RC AFM increases, but in the case of VSC AFM, the critical time decreases at first, then it is almost constant at a limited range of d, and finally it starts to increase by increasing the distance.     

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.     

考虑附加质量的中心刚体-柔性悬臂梁系统的动力特性研究

对考虑附加质量的中心刚体-柔性悬臂梁系统的动力特性进行研究.首先采用Hamilton原理和有限元离散化方法,在计入柔性梁由于横向变形而引起的轴向变形的二阶耦合量的条件下,给出该系统的刚柔耦合动力学方程(即一次近似耦合模型),以及相应的非惯性系下的动力学模型,然后通过数值仿真对系统的动力特性进行研究.仿真结果显示,即使是小的附加质量也会对系统动力特性产生重要影响,附加质量使得梁的响应幅值变大和响应频率降低,且会影响柔性梁和中心刚体的终点位置.附加质量的影响随系统大范围运动的角速度的增大而变大.当系统大范围运动为低速时,传统的混合坐标模型仍然会导致较大误差;当系统大范围运动为高速时,传统的混合坐标模型存在失效的可能.     

Dynamic stability of a cantilevered Timoshenko beam on partial elastic foundations subjected to a follower force

This paper presents the dynamic stability of a cantilevered Timoshenko beam with a concentrated mass, partially attached to elastic foundations, and subjected to a follower force. Governing equations are derived from the extended Hamilton’s principle, and FEM is applied to solve the discretized equation. The influence of some parameters such as the elastic foundation parameter, the positions of partial elastic foundations, shear deformations, the rotary inertia of the beam, and the mass and the rotary inertia of the concentrated mass on the critical flutter load is investigated. Finally, the optimal attachment ratio of partial elastic foundation that maximizes the critical flutter load is presented.     

An assumed mode method and finite element method investigation of the coupled vibration in a flexible-disk rotor system with lacing wires

The Assumed mode method (AMM) and Finite element method (FEM) were used. Their results were compared to investigate the coupled shaft-torsion, disk-transverse, and blade-bending vibrations in a flexible-disk rotor system. The blades were grouped with a spring. The flexible-disk rotor system was divided into three modes of coupled vibrations: Shaft-disk-blade, disk-blade, and blade-blade. Two new modes of coupled vibrations were introduced, namely, lacing wires-blade and lacing wires-disk-blade. The patterns of change of the natural frequencies and mode shapes of the system were discussed. The results showed the following: first, mode shapes and natural frequencies varied, and the results of the AMM and FEM differed; second, numerical calculation results showed three influencing factors on natural frequencies, namely, the lacing wire constant, the lacing wire location, and the flexible disk; lastly, the flexible disk could affect the stability of the system as reflected in the effect of the rotational speed.     

Defining the parameters of a cantilever tip AFM by reference structure

A method of measurement and control of atomic force microscope (AFM) probe parameters is offered. The AFM real cantilever parameters are defined.     

Analysis of rotating nonuniform pretwisted beams with an elastically restrained root and a tip mass

Using Hamilton's principle derives the governing differential equations for the coupled bending–bending vibration of a rotating pretwisted beam with an elastically restrained root and a tip mass, subjected to the external transverse forces and rotating at a constant angular velocity. Using the mode expansion method derives the closed-form solutions of the dynamic and static systems. The orthogonal condition for the eigenfunctions of the system with elastic boundary conditions is discovered. The self-adjointness of the system is proved. Moreover, the Green functions of the system are obtained. The symmetric properties of the Green functions are revealed. The frequency response on the steady response of the beam is also investigated.     

Accurate low DOF modeling of a planar compliant mechanism with flexure hinges: the equivalent beam methodology

A methodology for accurate and efficient finite elements method (FEM) simulations of planar compliant mechanisms with flexure hinges is presented. First, using symmetry/antisymmetry boundary conditions and 3D elements, one-eighth of a single hinge is simulated to determine its true stress/stiffness characteristics. A set of fictitious beams is derived, which have the identical characteristics. This set is used in conjunction with other beams that model relatively stiff links to generate an equivalent model of an entire mechanism consisting of the beam elements only. The model has a low number of degrees-of-freedom (DOF) and appears to be more accurate than any 2D FEM models, even those with very large number of DOF. The methodology has been developed specifically for the right circular flexure hinge; however, it can be applied to all types of revolute flexure hinges.     

Parametric vibrations of a horizontal beam with a concentrated mass at one end

This investigation treats the steady state response of parametric vibration of a simply supported horizontal beam, carrying a concentrated mass at one end and subjected to a periodic axial displacement excitation at the other end under the influence of gravity. Non-linear terms arising from longitudinal inertia of a concentrated end mass and beam elements are included in the equation of motion. By using the one mode approximation and applying Galerkin's method, the governing equation of motion is reduced to a non-linear ordinary differential equation with periodic coefficient. The harmonic balance method is applied to solve the equation and the dynamic response is derived. Experimentally determined amplitude-frequency curves are presented, and are found to be in good agreement with the theory.     

On the stability of non-conservative plane mass loaded beam structures (static approach with the load-distribution method)

Stability problems can generally be regarded as kinetic problems since it is a matter of investigating the behaviour of a disturbance-induced motion away from the equilibrium position. Loading, mass distribution, frictional forces and the nature of the disturbance are thereby of importance. For most conservative problems a simpler calculation can be applied using the static instead of the kinematic equilibrium. For the static stability criterion, the methods of energetic and force equilibrium are available. In the absence of non-conservative problems the kinematic stability criterion is commonly applied.

This work shows that, when applying the load-distribution method a number of solutions for non-conservative mass loaded systems can be found. Thereby, the static stability criterion is used under the prerequisite of infinitesimal deformations.     

Stabilization of the phase-space portrait parameters of a hydrogen ion beam

This investigation carried out variations in the transverse phase-space portrait of an ion beam during a 200-μs pulse of a proton injector in the linear accelerator at the Institute for Nuclear Research. The parameters of the accelerated 400-keV beam at the entrance into the beam transport channel and the beam from the ion source were measured. Numerical simulation of ion-beam acceleration in the proton injector was performed. The noiseless mode of beam generation in the duoplasmatron was obtained. The generator of the duoplasmatron discharge current with a discharge current instability of ±1% or less was developed and put into operation. A compensated resistive-capacitive voltage divider was used in the accelerating tube. The beam emittance measurements showed that the variations in the phase-space portrait of the beam from the upgraded injector during the 200-μs pulse were absent within the measurement accuracy. The normalized emittance for 90% of the beam current was 0.15π × cm × mrad at an ion current of 65 mA.     

Influence of tip mass on dynamic behavior of cracked cantilever pipe conveying fluid with moving mass

In this paper, we studied about the effect of the open crack and a tip mass on the dynamic behavior of a cantilever pipe conveying fluid with a moving mass. The equation of motion is derived by using Lagrange’s equation and analyzed by numerical method. The cantilever pipe is modelled by the Euler-Bernoulli beam theory. The crack section is represented by a local flexibility matrix connecting two undamaged pipe segments. The influences of the crack, the moving mass, the tip mass and its moment of inertia, the velocity of fluid, and the coupling of these factors on the vibration mode, the frequency, and the tip-displacement of the cantilever pipe are analytically clarified.