Linear Vibration Research Articles

Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam

Abstract In this present work, linear and non-linear vibration of multi-supported nanobeams, which are a fundamental part of the nano-electromechanical systems, is examined. To the best of the researchers’ knowledge, there is no study performed into multi-supported nanobeam in the literature. The governing equations of the system are obtained by dint of the Hamilton principle and solved via the perturbation technique which is divided linear and non-linear parts of the main equations. The natural frequencies and mode shapes are calculated from the linear problem. The non-linear natural frequencies and amplitude-phase modulation graphs are obtained from the non-linear equation. All equations are written in generalized form, and 3, 4 and 5 supported nanobeams are investigated in detail. The nonlocal coefficient, support number and position and end condition types are focused on. The three to one internal resonance cases are also investigated. It is occurred that the clamped-end conditions shift right in the hardening behaviour graphs more than the simply supported condition. Moreover, it is shown that the supported numbers play a significant role in natural frequency.

Comparison of the Natural Vibration Frequencies of Timoshenko and Bernoulli Periodic Beams.

This paper deals with the linear natural vibrations analysis of beams where the geometric and material properties vary periodically along the beam axis. In contrast with homogeneous prismatic beams, the frequency spectra of such beams are irregular as there exist enlarged intervals between some adjacent frequencies. Presented here are two averaged models of beams based on the tolerance modelling approach. The assumptions of classical Euler–Bernoulli and Timoshenko–Ehrenfest beam theories are adopted as the foundations. The resulting mathematical models are systems of differential equations with constant, weight-averaged coefficients. This makes it possible to apply any classical method of solution suitable for homogeneous beams, such as Galerkin orthogonalization. Here, emphasis is placed on the comparison of natural frequencies neighbouring the frequency band-gaps that are obtained from these two theories. Two basic cases of material and geometric property distribution in a periodicity cell are studied, and the natural frequencies and mode shapes are obtained for a simply supported beam. The results are supported by a comparison with the finite element method and partially exact solutions.

Pressure of Electromagnetic Radiation on a Linear Vibrator

Nowadays the pressure of electromagnetic radiation in the optical range is widely used in laser traps (so called optical tweezers or single-beam gradient force trap) to control the position of microparticles, biological cells and other microscopic objects. This is possible by focusing the laser radiation into the area of several micrometers in size. The intensity of the radiation in the area is sufficient to hold particles in the beam and manipulate them. We are interested to research similar possibility in the microwave range of wavelengths. However we had faced a number of difficulties in this range: the size of the focal region is much larger, the radiation intensity is less, and to control microscopic objects by means of radiation pressure very high powers are required. And we decided to consider the known effect of a very strong interaction of thin conducting fibers (metal, semiconductor, graphite) with microwave radiation. The efficiency factor of radiation pressure on such objects reaches values of several hundreds and thousands. This can be used to control objects in the form of electrically thin metal conductors by means of radiation pressure. Methods for calculating the pressure of electromagnetic radiation on an infinitely long circular cylinder are known. In this paper we propose a method for calculating the radiation pressure on a circular cylinder (vibrator), the length of which is comparable to the radiation wavelength. We have found out that when the vibrator length is close to half the wavelength, the radiation pressure efficiency factor is much larger than for an infinite cylinder. We have obtained the dependence of the radiation pressure efficiency factor on the length and diameter of an absolutely reflecting and impedance vibrator. It decreases with decreasing conductivity. An infinite cylinder at a certain value of conductivity has a maximum of the radiation pressure efficiency factor.

Nonlinear finite element analysis of vibration of multi-walled carbon nanotubes with geometric imperfection resting on elastic foundations in a thermal–magnetic environment

This paper presents nonlinear finite element analysis of the systems of partial differential equations (PDEs) that govern the vibration dynamics and stability responses of slightly curved Multi-walled carbon nanotubes (MWCNTs) by means of nonlocal elasticity principle. The MWCNTs under investigation rest on Pasternak and Winkler foundations and operate in a magneto–thermal environment. With the aid of the nonlinear finite element analysis (FEA) of the developed governing equations, the effects of primary geometric imperfection, nonlocal parameter, magnetic field, temperature, elastic foundation parameters, number of walls, and boundary conditions on the nonlinear vibration of the CNTs are investigated. The simulated and presented results show how frequency ratio reduces with an increase in the number of elastic foundation parameters, strength of magnetic field term, temperature, tube wall numbers and ratio of curvature radius to length of the slightly curved nanotubes under investigation. Similar trend of results is recorded for the different boundary conditions considered. However, frequency ratio is highest for MWCNTs with clamped-pinned boundary conditions and lowest for the case of clamped supports at both ends. Furthermore, it is established that the vibration of quadruple MWCNTs approximates that of linear vibration even as the foundations parameters are augmented. Such result can be used to restrains chaos in vibration of CNTs. It is hoped that the results from this study will help in the design of MWCNTs for various applications.

Probability Density Evolution Algorithm for Stochastic Dynamical Systems Based on Fractional Calculus

With the increasing load and speed of trains, the problems caused by various random excitations (such as safety and passenger comfort) have become more prominent and thus arises the necessity to analyze stochastic dynamical systems, which is important in both academic and engineering circles. The existing analysis methods are inadequate in terms of computational accuracy, computational efficiency, and applicability in solving complex problems. For that, a new efficient and accurate method is used in this paper, suitable for linear and nonlinear random vibration analysis of large structures as well as static and dynamic reliability assessment. It is the direct probability integration method, which is extended and applied to the random vibration reliability analysis of dynamical systems. Dynamical models of the dynamic system and coupled system “three-car vehicle-rail-bridge” are established, the time-varying differential equations of motion are derived in detail, and the dynamic response of the system is calculated using the explicit Newmark algorithm. The simulation results show the influence of the number of representative points on the smoothness of the image of the probability density function and the accuracy of the calculation results.

Using the discrete model for the processing of natural vibrations, tapered beams made of AFG materials carrying masses at different spots

A discrete physical model based on modal analysis theory is used for the first time to investigate the linear free vibration of tapered Euler-Bernoulli beams constructed of axial functional gradient (AFG) materials and carrying masses concentrated at different locations, applied for CC, CF and SS end conditions. The considered beam will be modeled by the present model composed of N masses (N-DOF) connected by (N+1) small bars, the mass of the beam is totally distributed only on the set of N masses, this distribution is governed by the law of variation of the geometric and material properties of the beam. The components modeling the bending stiffness of the considered beam are (N+2) coil springs whose stiffness depends essentially on the law describing the variation of the squared moment and Young's modulus along the beam axis. After the construction of the stiffness matrix and the mass matrix, the Lagrange formalization is used to present the problem in mathematical form. Thus, the results of dimensional frequencies using several types of tapered beams carrying masses at different places have been discussed in this work. It is noted in the following that the discrete system is composed of (N+n)masses (taking into account the concentric masses carried by the beam).

Nonlinear vibration suppression of S-S and C-C laminated composite cylindrical shells with magnetostrictive strips

The focus of this study is on the control of nonlinear vibration of laminated composite cylindrical shells under S–S and C–C boundary conditions through magnetostrictive strips. The shells are modeled considering the effects of shear deformation and rotary inertia while geometrical nonlinearity is modeled using von Karman nonlinear theory. In order to obtain ordinary differential equation of the system, Ritz method and beam model are hired. The combination of multiple scales method and modal analysis is used for obtain of the vibration characteristics of the shells. Some results of literature are employed to investigate the validity of the results of this study. Lyapunov's indirect method is used to study asymptotical stability of the results of systems around zero equilibrium point. Finally, the effects of several parameters on the linear and nonlinear vibration characteristics are reported.

Coupled vibration analysis of fluid-filled baffled tank equipped with Kirchhoff plate

In the present paper, free coupled vibration of partially fluid-filled baffled tank equipped with elastic thin plate is studied by a variational fluid-domain-decomposition scheme, focusing on obtaining the analytically approximate coupled plate bulging and fluid sloshing modes. Different types of fluid-coupled elastic thin plates are considered: partially submerged elastic baffle, elastic baffle partially covering the free fluid surface, immersed elastic baffle, and elastic thin plates forming the tank wall. The elastic thin plate is modeled by the Kirchhoff plate, and the lateral plate deflection is approximated using the summation of its dry (in vacuum) modes multiplied by unknown weight coefficients. The baffled tank is partially filled with a fluid having multi free surface and assumed to be inviscid, incompressible, and irrotational. The present scheme is based on a variational principle related to free linear vibration of the thin elastic plates coupled with bounded multi-layer immiscible fluid. The eigenfrequency equation of the coupled plate–fluid system obtained by the present scheme has a much more unified and compact form when compared with the previous domain-decomposition scheme employing the Rayleigh–Ritz method or the Galerkin method. This is realized by constructing the variational formulation of the coupled plate–fluid problem expressed in differential form. The convergence and accuracy of the present scheme are studied for both convex and non-convex liquid domains. The results from the present scheme agree well with published numerical or experimental results.

Simplified bar-system model for tubular structures by considering local joint flexibility

Local joint flexibility (LJF) may produce significant effect on performance of tubular structures. According to the local deformation characteristics at tubular joints, a simplified bar-system model is presented by using tie-strut members to simulate the local flexibility at joints. Based on proposed equations for calculating the LJFs of different types of tubular joints in literature, analytical equations for determining the dimensions and material properties of the presented tie-strut members are deduced. Additionally, nonlinear static pushover test was conducted for a small-scale tubular structure specimen and linear vibration test was carried out for another specimen. Load-displacement curves at some specified positions in the static test and displacement-time curves at some specified positions in the vibration test are obtained. Three different finite element models are then presented, which include using 3-D shell elements to simulate the entire tubular structures (3-D model), using beam elements to simulate the entire structures with rigid connection assumed at tubular joints (rigid frame model), and using beam elements to simulate the tubular members and using the presented tie-strut model to simulate the tubular joints (presented bar-system model). Finite element results obtained from the three different methods are compared with the experimental results, and the comparison indicates that both 3-D model and the presented bar-system model can produce accurate predictions compared to the experimental results while the rigid frame model overestimates the stiffness and load-carrying capacity and underestimates the vibrating amplitude of the specimens. As the presented bar-system model has much higher efficiency compared to 3-D model, it is more challenging for analyzing both linear and nonlinear behavior of tubular structures in practical design.

Investigation of delamination effect on nonlinear vibration behaviors of a composite plate resting on nonlinear elastic foundation

The aim of the present work is to investigate the delamination effect on nonlinear vibration behaviors of composite plate resting on nonlinear elastic foundation. The composite plate is assumed to be pre-delaminated with various delamination locations and modeled with an improved layerwise theory and finite element method. The delamination is simulated by the Heaviside unit step function allowing for the discontinuity of displacement fields. By means of the variational principle, the nonlinear governing equations can be obtained and further solved by the direct iterative method. The model takes into account three foundation parameters so as to investigate their effect on the nonlinear vibration characteristics. Three longitudinal locations and three interfacial locations are considered, and both clamped-free and clamped–clamped boundary conditions are considered to investigate the nonlinear vibration behaviors. The results show that different from the linear vibration, with the existence of delamination, the nonlinear fundamental frequencies may exceed the fundamental frequency of the healthy plate when the vibration amplitude ratio reaches a certain value. This finding may help engineers understand the nonlinear vibration behaviors and further detect the delamination in composite plate resting on nonlinear foundations in presence of delamination.

Online double-sided identification and eliminating system of unclosed-glumes rice seed based on machine vision

An online double-sided identification and eliminating system based on machine vision was developed to identify the unclosed-glumes rice seed by analyzing its double-sided image simultaneously. The vibrating plate and linear vibration conveyor achieved the transfer of rice seed from disordering to spacing, and the identified unclosed-glumes rice seed was eliminated by the blowing jet. Hough linear detection and extracted feature were used to identify the unclosed-glumes rice seeds. The algorithm achieved the accuracy of 88.1% for normal seeds and 87.7% for unclosed-glumes seeds when using double-sided image acquired online. Multi-thread processing was used for double-sided images to shorten the code execution time. The online double-sided identification and eliminating system achieved the average accuracy of 83.7% for normal seeds and 83.3% for unclosed-glumes seeds. Results show that the system has a good adaptation of different rice seed varieties and achieved desired accuracy for online identification and eliminating of unclosed-glumes rice seed.

Design, Modeling, and Characterization of a Tubular Linear Vibration Energy Harvester for Integrated Active Wheel System

Design, Modeling, and Characterization of a Tubular Linear Vibration Energy Harvester for Integrated Active Wheel System

Analysis of Axisymmetric Vibration of Functionally-Graded Circular Nano-Plate Based on the Integral Form of the Strain Gradient Model

In this paper, it is aimed to analyze the linear vibrational behavior of functionally-graded (FG) size-dependent circular nano-plates using the integral form of the non-local strain gradient (NSG) model. The linear axisymmetric vibration of the circular FG nano-plates based on the non-local strain gradient (NSG) model is the focal point of this study. In this regard, the non-local elasticity theory (NET) and strain gradient (SG) models are used in conjunction with Hamilton's principle to obtain the governing equations. Discretization of the obtained governing equations is performed with the help of generalized differential quadrature rule (GDQR) and Galerkin weighted residual method (GWRM). The analysis is focused on the effect of non-local and material parameters, as well as the aspect ratio, heterogeneity index of FG material, different boundary conditions, and frequency number on the overall behavior of nano-plate. On using the Galerkin method, a system of linear differential equations is obtained and solved to determine the natural linear frequencies and mode shapes. The obtained results are then compared with the existing results in the literature. On using the proposed procedure in this paper, the dynamic behavior of nano-plate under different boundary conditions can be well described. In addition, the existing deficiencies in other non-local theories can be eliminated. The results of this investigation can be considered as a turning point in the improvement of theoretical results for achieving a better prediction of vibrational behavior in nanostructures.

Resource Allocation in Vibration Energy Harvesters

This paper studies the optimization of mechanics and resource allocation in linear vibration energy harvesters operating under an overall size constraint. In such harvesters, the volume aspect ratio and the stroke direction should be carefully selected to maximize output power and power density. Further, because the proof mass and spring stroke must share the available size along the stroke direction, a mass-stroke design trade-off results; the trade-off is optimized here. The optimization further leads to a universal metric the enables the comparison of linear vibration energy harvesters designed to operate in a limited volume. Examination of this metric results in guidelines and best practices for selecting harvester dimensions and shape for a given volume constraint, all based on the dimensional harvester optimization for maximum size-constrained output power. This paper offers several key contributions. First, it provides an evaluation and optimization of the impact that volume shaping and volumetric resource allocation have on the output power of linear vibration energy harvesters. Second, it develops a “universal normalized surface” relating output power to dimensional harvester design. The surface expresses the performance limit of space-constrained linear vibration energy harvesters in terms of their output power density. This leads to universal guidelines for harvester design. Finally, benchmarking harvesters reported in the literature against the universal surface demonstrates that their output performances lie on or below the surface, despite large variations in their exogenous design variables such as vibration amplitude, frequency, volume etc. Thus, this paper offers an effective method to evaluate mechanical design and optimization in state-of-the-art linear vibration energy harvesters under volume constraints, a scenario suited for IoT applications. [2021-0074]

Design and validation of the tubular linear vibration generator applied to energy harvester based on VIV

AbstractA tubular linear vibration generator applied to the energy harvester based on vortex‐induced is proposed and analyzed in this paper. First, the permanent magnet of the generator is placed on the inner side of the winding, and the outer side of the winding is improved by adding magnetically permeable housing to increase the generated power. Then, the optimal magnet magnetization method, size of the winding coil, wire diameter, and housing material are evaluated by finite element analysis. Finally, the prototype using radial magnets is fabricated and fixed on the energy harvester. The vibration displacement, the induced voltage of the coil, and the generated power at different water velocities are measured. The generators with different permeable housings are fabricated to verify the accuracy of the theory. The experiment results show the maximum average power harvested is 5.5 mW of the TVLG at the water speed of 0.7 m/s, for the coil of 0.2 mm wire diameter, 8 cm height, and 2,000 turns. Under the effective water speeds, the power obtained by the generator with magnetically permeable housing is more than 70% higher than that obtained by the generator without housing.

Video Image Correlation-Based Method Used for the Study of the Torsional Vibrations of an Adder Gearbox

In this paper, a study of the vibrations that appear in the transmission shafts of an adder gearbox used for a heavy truck is made. The truck has two engines on only one chassis and the power offered by these engines is summated and transmitted to the truck or the working machine. This type of transmission is used for oil production installation for the army. During the transportation of the installation to the workplace, only one engine is running, after mounting installation, both engines are running. This paper studies the vibrations of the adder gearbox, a complex construction, subjected to multiple operating requirements. In this regard, the authors first performed accuracy (accuracy) tests of the VIC 3D system on an original experimental mini-stand. Measurements performed on a mini disc demonstrated the validity and accuracy of the method, even if the cameras used were not high resolution. The authors applied the same principle in the case of the adder box from the truck, obtaining useful results for those in the field. The experimental method uses the facilities of a contactless optical measurement method (VIC-3D), which provides a high-accuracy quantitative linear and angular vibration analysis. The VIC measurement method offers, based on a frontal viewing of the disk during the resonance phenomenon and by simple calculus on the monitored linear displacements, the corresponding angular amplitude.

Nonlocal thermoelasticity: Transient heat conduction effects on the linear and nonlinear vibration of single-walled carbon nanotubes

This novel study presents a framework to investigate the effects of transient heat conduction on the nonlinear deflection and vibration of single-walled carbon nanotubes (SWCNTs) based on Eringen’s nonlocal elasticity and nonlocal heat conduction theories. The first phase of this research involves modifying the hyperbolic heat conduction theory by including a nonlocality term, in which the scale of energy transport is taken into account. The next phase includes a thermoelasticity solution for carbon nanotubes (CNTs), which is developed by Eringen’s nonlocal theory and implementing the obtained temperature. The differential quadrature method (DQM) is used to solve the transversely induced nonlocal heat conduction. We have shown the effect of thermal waves on the mode shapes and nonlinear vibration of a SWCNT by considering phase lag parameters. All boundaries are considered under a temperature-jump at the nanometer scale to consider the boundary phonon scattering. The effects of several parameters, including time-dependent boundary conditions, time lag, nonlocal parameter, length and radius, and end supports on the frequency response of SWCNTs are investigated.

Free nonlinear oscillations of cantilever circular cylindrical shells

AbstractFree nonlinear oscillations of cantilever cylindrical shell are studied. The Sanders‐Koiter shell theory and Donnell's nonlinear shell theory are used to obtain the mathematical model of the structure vibrations. Using the assumed‐mode technique, the finite‐DOFs nonlinear dynamical system, which describes the free nonlinear oscillations of the cylindrical shell, is obtained. The harmonic balance technique is applied to analyze free nonlinear vibrations. The Rayleigh‐ Ritz technique is implemented to analyze the structure linear vibrations. The properties of the system linear vibrations are analyzed using the results, which are obtained by the Rayleigh‐ Ritz technique and the commercial software ANSYS. The soft backbone curve is obtained to describe free nonlinear vibrations.

Computational analysis of the nonlinear vibrational behavior of perforated plates with initial imperfection using NURBS-based isogeometric approach

Abstract In this article, an isogeometric analysis through NURBS basis functions is presented to study the nonlinear vibrational behavior of perforated plates with initial imperfection. In this regard, the governing equations of plate dynamics, as well as the displacement–strain relations, are derived using the Mindlin–Reissner plate theory by considering von Karman nonlinearity. The geometry of the structure is formed by selecting the order of NURBS basis functions and the number of control points according to the physics of the problem. Since similar basis functions are utilized to estimate the accurate geometry and displacement field of the domain, the order of the basic functions and the number of control points are optimized for the proper approximation of the unknown field variables. By utilizing the energy approach and Hamilton principle and discretizing the equations of motion, the vibrational response of the perforated imperfect plate is extracted through an eigenvalue problem. The results of linear vibrations, geometrically nonlinear vibrations, and nonlinear vibrations of imperfect plates are separately validated by considering the previously reported findings, which shows a satisfactory agreement. Thereafter, a coefficient of the first mode shape is considered as the initial imperfection and the vibrational analysis is reexamined. Furthermore, the nonlinear vibrations of the perforated plate with initial imperfection are analysed using an iterative approach. The effects of the perforated hole, initial imperfection, and geometric nonlinearity are also addressed and discussed.

Non-linear normal modes of plane cable trusses

A model for the non-linear dynamic behaviour of pre-stressed cable trusses is presented. The model provides the basis for a simplified description of the (in-plane) non-linear dynamical behaviour of biconcave and bi-convex cable structures (as well as of single cables) subject to different loading (and mass distribution) conditions. The simplicity of the proposed approach is coupled with its ability to schematically represent a wide range of actual structures, including light pedestrian suspended cable bridges and cable roofs with vertical harnesses, under different vertical loading conditions. Although aiming at the needs of the practitioner, in search for a simplified and intuitive understanding of the effects of nonlinearities on the dynamic response of cable structures (to confirm and explain results from more detailed FEM models), the proposed approach also paves the way for the investigation in the extensive taxonomy of the non-linear dynamic behaviour of cable structures. The present paper focuses on non-linear normal modes (NNMs) arising as a continuation of linear vibration modes of the structure when energy is increased. The identification of non-linear normal modes is achieved through harmonic balance; a time-integration analysis is then performed on the same model, with initial conditions derived from the harmonic balance approach, demonstrating consistent results. To further test the model, its behaviour is compared to simulations on a 66-degrees of freedom finite element truss model, with satisfactory results.