Concentrated Tip Mass Research Articles

In-plane pipe whip: Post-failure dynamic response

Pressurised pipes are ubiquitous in chemical, nuclear and power plants. A sudden failure and release of high-speed fluid cause large inelastic displacements characterised by a whipping-type motion, which can ultimately hinder the surrounding structural and functional systems. In this paper, a beam user element has been developed, implemented and applied to analyse the in-plane flexural dynamic response of pipe whip. The two-dimensional Euler–Bernoulli beam element is based on the corotational kinematic formulation and elastoplastic constitutive models that include metal plasticity and moment–curvature relationships obtained from numerical bending tests, which highlighted the existence of two new dimensionless groups that govern the flexural response of slender pipes and enable the creation of moment–curvature master curves for thick and thin pipes. The corotational beam element formulation is compared against an analytical rigid-perfectly plastic model, numerical simulations employing shell elements and available experimental results, showing very good accuracy in the prediction of the inelastic deformation response of pipe whip and its hazardous area of influence. Furthermore, parametric studies are performed to investigate the effect of load intensity, cross-sectional geometry and concentrated tip mass on the post-failure deformation modes, plastic hinge formation and extension of the hazard zone. The presented results represent valid tools to assess the safety of industrial piping systems undergoing failure, and to optimally design pipe whip restraints.

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.

Nonlinear dynamic response of a multilayer piezoelectric nanocomposite microbeam with tip mass

The nonlinear forced response of multilayer piezoelectric cantilever microbeams reinforced by carbon nanotubes with a concentrated tip mass is investigated. The geometrically exact formulation based on the Cosserat theory of rods is employed to study large amplitude vibrations. The constitutive law of the nanocomposite layers is based on the Eshelby-Mori-Tanaka homogenization approach while the piezoelectric layers are modeled according to the standard piezoelectric constitutive formulation. The enforced inextensibility and unshearability constraints lead to a partial differential equation (PDE) governing the flexural motion of the multilayer microbeams. The obtained PDE is coupled to the ordinary differential equation for the motion of the shuttle mass. The Faedo-Galerkin approach is implemented to discretize the problem. The method of multiple scales is employed to obtain the frequency response of the system under a primary resonance base excitation. The frequency response highlights the effects of the carbon nanotubes volume fraction, tip mass, and force amplitude. An interesting result is obtained showing the feasibility of the second mode exploitation rather than the first mode for mass sensing purposes.

Nonlinear micromechanically analysis of forced vibration of the rectangular-shaped atomic force microscopes incorporating contact model and thermal influences

The linear and nonlinear forced vibrations of microprobe in contact-mode atomic force microscopes are presented based on a rectangular beam model of modified couple stress theory. The Hamilton principle is employed to derive governing PDEs for the first and second modes transformed into nonlinear ODEs by the Galerkin procedure coupled with time multiple scales method. The important effect of linear tip inertia is regarded despite of others. The impact of damping, thermal loading, and constant contact stiffness on the amplitude and phase angle-frequencies are investigated. Location of a concentrated tip mass with tip-sample interaction are modeled by delta Dirac functions. Communicated by Francesco Tornabene.

Solar Panels as Tip Masses in Low Frequency Vibration Harvesters

Tip masses are used in cantilevered piezoelectric energy harvesters to shift device resonance towards the required frequency for harvesting and to improve the electric power generation. Tip masses are typically in the form of concentrated passive weights. The aim of this study is to assess the inclusion of solar panels as active tip masses on the dynamics and power generation performance of cantilevered PVDF (polyvinylidene fluoride)-based vibration energy harvesters. Four different harvester geometries with and without solar panels are realized using off-the-shelf components. Our experimental results show that the flexible solar panels considered in this study are capable of reducing resonance frequency by up to 14% and increasing the PVDF power generated by up to 54%. Two analytical models are developed to investigate this concept; employing both an equivalent concentrated tip mass to represent the case of flexible solar panels and a distributed tip mass to represent rigid panels. Good prediction agreement with experimental results is achieved with an average error in peak power of less than 5% for the cases considered. The models are also used to identify optimum tip mass configurations. For the flexible solar panel model, it is found that the highest PVDF power output is produced when the length of solar panels is two thirds of the total length. On the other hand, results from the rigid solar panel model show that the optimum length of solar panels increases with the relative tip mass ratio, approaching an asymptotic value of half of the total length as the relative tip mass ratio increases significantly.

Fundamental frequencies of a torsional cantilever nano beam for dynamic atomic force microscopy (dAFM) in tapping mode

The aim of this investigation is to study the motion of an elastically restrained beam used in tapping mode atomic force microscopy (TM-AFM), which is to be utilized in manufacturing at nano-scale. TM-AFM uses high frequency oscillations to remove material or shape nano structures. Euler–Bernoulli theory and Eringen’s theory of non-local continuum are used to model the nano machining structure composed of a nanobeam and a single degree of freedom, spring-mass system. The system is modelled as a beam with a torsional spring boundary condition that is rigidly restrained in the transverse direction at one end; and at the free end is a transverse linear spring attached to the tip. The other end of the spring is attached to a mass, resulting in a single degree of freedom spring-mass system. When the linear spring constant is infinite, the free end behaves as a beam with a concentrated tip mass. When the mass is infinite, the boundary condition is that of a linear spring. When the tip mass is zero, the configuration is that of a torsionally restrained-free beam. The motion of the tip of the beam and tip mass can be investigated to observe the tip frequency response, displacement and force. The tip displacement frequency contains information about the maximum displacement amplitude and therefore the sample penetration depth.

집중 질량을 갖는 폭 변단면 외팔보의 굽힘 진동 해석

A transfer matrix method has been developed to determine the more accurate natural frequencies for the bending vibration of Bernoulli-Euler beam with linearly reduced width and a concentrated tip mass. The proposed method can be computed an infinite number of the natural frequencies using a single element. Using the differential equation, shear force, and bending moment in which can be deduced by the diverse variational principles, a transfer matrix is formulated. The roots of the differential equation are computed by the Frobenius method. The effect of the concentrated mass for the natural frequencies of width-tapered beams is examined through a parametric study, and to show the accuracy of the proposed method, the computed results compared with those obtained from commercial finite element analysis program(ANSYS).

Dropping Impact Characteristics Analysis of a Cubic Nonlinear Packaging System with a Cantilever Beam Type Elastic Critical Component with Concentrated Tip Mass

A mathematical model for a cubic nonlinear packaging system with a cantilever beam type critical component with concentrated tip mass is proposed. The finite element method and the implicit finite difference method together with the Rung-Kutta method are applied to study the dropping impact dynamics of the critical component and the effect of system parameters, such as the value of the concentrated tip mass and the frequency of the main component, is discussed. The results show that the relative displacement and acceleration change remarkably with the length of the cantilever beam, and the maximum internal stress occurs at the joint end of the critical component. With the increase of the value of the concentrated tip mass and/or a higher frequency of the main component, the amplitudes of the responses increase obviously.

Analytical solutions to the free vibration of a double-walled carbon nanotube carrying a bacterium at its tip

We calculate the natural frequencies and mode shapes of a cantilevered double-walled carbon nanotube carrying a rigid body—representative of a bacterium or virus—at the tip of the outer nanotube. By idealizing the nanotubes as Bernoulli-Euler beams, we are able to obtain exact expressions for both the mode shapes and characteristic frequency equation. Separate analyses are performed for the special case of a concentrated tip mass and the more complicated situation where the tip body also exhibits inertia and mass center offset from the beam tip.

Spherical tuned liquid damper for vibration control in wind turbines

A tuned liquid damper (TLD), which consisted of two-layer hemispherical containers, partially filled with water, was investigated as a cost-effective method to reduce the wind-induced vibration of wind turbines. A 1/20 scaled test model was designed to investigate its performance on the shaking table. Three groups of equivalent ground accelerations were inputted to simulate the wind-induced dynamic response under different load cases. The influence of rotors and nacelle was assumed to be a concentrated tip mass. A series of free and forced vibration experiments were performed on the shaking table. The experimental results indicated that the spherical TLD could effectively improve the damping capacity of the test model. The standard deviation of the dynamic response could be effectively reduced when the excitation frequency was approximately equal to its fundamental frequency. For “overspeed” and “extreme operating gust” load cases, the standard deviations of the dynamic responses were reduced more than 40% when the liquid mass was about 2% of the generalized mass; for “parking” load cases, the corresponding standard deviation was reduced more than 50% when the liquid mass was only 1% of the generalized mass. That is to say, the spherical TLD can effectively improve the anti-fatigue performance of the wind turbine tower.

Finite element vibration analysis of a partially covered cantilever beam with concentrated tip mass

The work presented in this paper is the theoretical investigation of the dynamical behavior of a cantilever beam, partially covered by damping and constraining layers, with concentrated mass at the free end. A finite element method is used in order to obtain the resonant frequencies and loss factors. The resonant frequencies and loss factors for different physical and geometrical parameters are determined. The variations of these two parameters are found to be strongly dependent on the geometrical and physical properties of the constraining layers and the mass ratio.

Eigenvalue sensitivity in the stability analysis of Beck's column with a concentrated mass at the free end

The stability of a uniform cantilever column carrying a concentrated tip mass and subjected to a follower force at the free end is investigated by using the finite element method and the technique of eigenvalue sensitivity. In the present finite element formulation, the shape functions, which contain a shear deformation parameter, are developed from the static deflection of a Timoshenko beam, and the effects of both transverse shear deformation and rotatory inertia are included. The advantage of such a formulation is that if a suitable value of the shear deformation parameter is given, a specific beam model is then simulated without too much additional work to modify the original formulation. In order to obtain promptly the critical flutter load, a simple and direct method which utilizes the eigenvalue sensitivity with respect to the follower force is proposed instead of the conventional trial-and-error technique. Numerical results demonstrate the excellent agreement of the present solutions and the rapid rate of convergence of the iterative procedure. They also show that the critical flutter load is more affected by the effect of transverse shear deformation than by that of rotatory inertia.

Impact of a finite elastic-viscoplastic bar

A method for predicting the response of strain-rate sensitive structures under dynamic loading is developed. It is based on a finite difference method, the incremental theory of plasticity, and an elastic work-hardening viscoplastic material idealization. The strain-rate effect, loading and unloading conditions, and wave interactions are automatically accounted for, and adjusted if necessary, as the deformation proceeds. No iteration is required even if the field equations are nonlinear (e.g. non-linear constitutive equations, large deformation, or complicated geometry). We solve as an example the small deflection of a finite bar with a concentrated tip mass. The accuracy is comparable to that obtained by the well-known method of characteristics, a powerful tool for solving elastic-viscoplastic wave problems but which is restricted to small deflections and simple geometry. Because of the form of the constitutive relation selected (elastic work-hardening visco-plastic), several important new features of the dynamics response are brought out. These features are not revealed when simpler, computationally-convenient constitutive relations, such as rigid ideal-viscoplastic, rigid work-hardening viscoplastic and elastic ideal-viscoplastic are used.

Hanging Cable as Rotational Vibration Absorber

Many basic textbooks present an analysis of free vibration of a uniform cable hanging in a gravitational field. An extension to illustrate the effects of a concentrated tip mass is given in Ref. 3. The purpose of this note is twofold: 1. To investigate the free vibration of the rigid body-cable system. 2. to show how a uniform hanging cable can be used as a dynamic absorber for a rigid body constrained to rotate about a fixed axis containing its mass center and subjected to sinusoidal moment disturbances.

Rotating Beam with Tip Mass Analyzed by a Variational Method

Hamilton's principle is used in the determination of a variational principle, which is stated in forms making it suitable for the case of: (1) moderately large displacements of a beam which is initially stressfree, or (2) small displacements of a beam having initial stresses. In the latter form, the principle is applied to the problem of determining the effect of tip mass and rotational speed on the natural frequencies of a rotating beam carrying a concentrated tip mass. The variational principle stated here allows for an independent selection of displacement and moment coordinate functions in the manner suggested by Reissner. It thus imposes fewer constraints than would be imposed by the Rayleigh-Ritz method of assuming displacement coordinate functions only. The effect of tip mass is shown to be much greater on the frequency of the second mode than on the fundamental mode.