ZERO MODE NATURAL FREQUENCY AND NONLINEAR VIBRATION OF COUPLED LATERAL AND TORSION OF A LARGE TURBINE GENERATOR

Zero mode natural frequency (ZMNF) is found during experiments. The ZMNF andvibrations resulted by it are studied. First, calculating method of the ZMNF excited byelectromagnetic in vibrational system of coupled mechanics and electrics are given from the view ofmagnetic energy. Laws that the ZMNF varies with active power and exciting current are obtained andare verified by experiments. Then, coupled lateral and torsional vibration of rotor shaft system isstudied by considering rest eccentricity, rotating eccentricity and swing eccentricity. UsingLargrange-Maxwell equation when three phases are asymmetric derives differential equation of thecoupled vibration. With energy method of nonlinear vibration, amplitude-frequency characteristics ofresonance are studied when rotating speed of rotor equals to ZMNF. The results show that ZMNF willoccur in turbine generators by the action of electromagnetic. Because ZMNF varies withelectromagnetic parameters, resonance can occur when exciting frequency of the rotor speed is fixedwhereas exciting current change. And also find that a generator is in the state of large amplitudein rated exciting current.

Natural Frequency and Main Vibration Mode Analysis of the Braking System of Metallurgical Crane

The safety brake system of metallurgy bridge crane is generally composed of two separated block brake, brake disc, and torsion shaft. The analysis of natural frequency and main vibration mode on this two-degree torsion vibration system is the basement to study the vibration model and vibration performance. In this work, we investigated natural frequency of the braking system of metallurgical crane with analytic method. This provides a systematic guidance towards a successful brake system design

Effect of vehicle weight on natural frequencies of bridges measured from traffic-induced vibration

Recently,ambient vibration test(AVT)is widely used tu estimate dynamic characteristics of large civil struc- tures.Dynamic characteristics ean be affected by various envirnnmental factors such as humidity,intensity of wind,and temperature.Besides these environmental conditions,tire mass of vehicles may change the measured valnes when traffic-in- duced vibration is used as a source of AVT tor bridges.The effect of vehicle mass on dynamic characteristics is investigated through traffic-induced vibration tests on three bridges;(1)three-span suspension bridge(128m+404m+128m),(2) five-span continuous steel box girder bridge(59m+3@ 95m+59m),(3)simply supported plate girder bridge(46m). Acceleration histories of each measurement location under normal traffic are recorded for 30 minutes at field.These recor- ded histories are divided into individual vibrations and are combined into two groups aceording to the level of vibration;one by heavy vehicles such as trucks and buses and the other by light vehicles such as passenger cars.Separate processing of the two groups of signals shows that,for the middle and long-span bridges,the difference can be hardly detected,but,for the short span bridges whose mass is relatively small,the measured natural frequencies can change up to 5.4%.

Investigation of Microstructure, Natural Frequencies and Vibration Modes of Dragonfly Wing

In the present work, a thorough investigation on the microstructural and morphological aspects of dragonfly wings was carried out using scanning electron microscope. Then, based on this study and the previous reports, a precise three-dimensional numerical model was developed and natural frequencies and vibration modes of dragonfly forewing were determined by finite element method. The results shown that dragonfly wings are made of a series of adaptive materials, which form a very complex composite structure. This bio-composite fabrication has some unique features and potential benefits. Furthermore, the numerical results show that the first natural frequency of dragonfly wings is about 168 Hz and bending is the predominant deformation mode in this stage. The accuracy of the present analysis is verified by comparison of calculated results with experimental data. This paper may be helpful for micro aerial vehicle design concerning dynamic response.

ON THE REPEATED NATURAL FREQUENCIES FOR TORSIONAL VIBRATION OF SHAFTS

To investigate the repeated frequency condition (RFC) for torsional vibration of shafts' system, the transfer matrix method was adopted. Firstly, the transfer relationship from the boundary to engaging disks of double shafts' system (DSS) was constructed. Secondly, the RFC of DSS was deduced out and the methods to select mode shape were presented. Finally, the relationship was extended to multilevel transmission system (MTS), and the RFC of this system was explored. The conclusions is this: 1) the necessary RFC requires the existence of joint engaging couple (JEC); 2) for DSS, the sufficient is the number of boundary transfer factors (f(B)) larger than 2; 3) the whale system can be split into independent groups, the total multiplicity is the sum of independent solution number of every group, the latter is the number of independent f(B), = 0 inside the group minus 1.

Natural vibration of cantilever porous twisted plate with variable thickness in different directions

In this paper,the blade is assumed to be a rotating variable thickness cantilever twisted plate structure,and the natural vibrations of variable thickness cantilever twisted plate made of metal porous material are studied.It is assumed that the thickness of the plate changes along spanwise direction and chordwise direction,respectively,and it changes in both directions.The classical thin shell theory,the first and second fundamental forms of surface and von Karman geometric relationship are employed to derive the total potential energy and kinetic energy of the cantilever twisted plate,in which the centrifugal force potential due to high rotational speed is included.Then,according to the Rayleigh-Ritz procedure and applying the polynomial functions which satisfy the cantilever boundary conditions,the dynamic system expressed by equations of motion is reduced to an eigenvalue problem.By numerical simulation,the frequency curves and the mode shapes of the twisted plate can be obtained to reveal the internal connection between natural vibration and the parameters.A series of comparison studies are performed to verify the accuracy of the present formulation and calculations,in which compared data come from experimental,finite element method and theoretical calculation,respectively.The influence of pre-twist angle,three different forms of thickness taper ratio and rotational speed on natural vibration,mode exchange and frequency veering phenomenon of the system is discussed in detail.In addition,the approach proposed here can efficiently extract analytical expressions of mode functions for rotating variable thickness cantilever twisted plate structures.

Natural frequency analysis of laminated piezoelectric beams with arbitrary polarization directions

Piezoelectric devices exhibit unique properties that vary with different vibration modes,closely influenced by their polarization direction.This paper presents an analysis on the free vibration of laminated piezoelectric beams with varying polarization directions,using a state-space-based differential quadrature method.First,based on the electro-elasticity theory,the state-space method is extended to anisotropic piezoelectric materials,establishing state-space equations for arbitrary polarized piezoelectric beams.A semi-analytical solution for the natural frequency is then obtained via the differential quadrature method.The study commences by examining the impact of a uniform polarization direction,and then proceeds to analyze six polarization schemes relevant to the current research and applications.Additionally,the effects of geometric dimensions and gradient index on the natural frequencies are explored.The numerical results demonstrate that varying the polarization direction can significantly influence the natural frequencies,offering distinct advantages for piezoelectric elements with different polarizations.This research provides both theoretical insights and numerical methods for the structural design of piezoelectric devices.

Research on Automatic Test System of Engine Blade Natural Frequency

Blades are one of the important components on aircraft engines.If they break due to vibration failure,the normal operation of the entire engine will be offected.Therefore,it is necessary to measure their natural frequency before installing them on the engine to avoid resonance.At present,most blade vibration testing systems require manual operation by operators,which has high requirements for operators and the testing process is also very cumbersome.Therefore,the testing efficiency is low and cannot meet the needs of efficient testing.To solve the current problems of low testing efficiency and high operational requirements,a high-precision and high-efficiency automatic test system is designed.The testing accuracy of this system can reach ±1%,and the testing efficiency is improved by 37% compared to manual testing.Firstly,the influence of compression force and vibration exciter position on natural frequency test is analyzed by amplitude-frequency curve,so as to calibrate servo cylinder and fourdimensional motion platform.Secondly,the sine wave signal is used as the excitation to sweep the blade linearly,and the natural frequency is determined by the amplitude peak in the frequency domain.Finally,the accuracy experiment and efficiency experiment are carried out on the developed test system,whose results verify its high efficiency and high precision.

Fatigue behavior of aluminum stiffened plate subjected to random vibration loading

Vibration tests were carried out on three types of stiffened aluminum plates with fully clamped boundaries under random base excitation. During the test, the response of the specimens was monitored using strain gauges. Based on the strain history, the accumulation of fatigue damage of the stiffened plates was estimated by means of the rainflow cycle counting technique and the Miner linear damage accumulation model in the time domain. Utilizing the change of natural frequencies, a nonlinear model was fitted for predicting the fatigue damage of plate and then the foregone failure criterion of 5% reduction in natural frequency is improved. The influence of section and spacing of the stiffeners on the vibration fatigue behavior of the aluminum plate was investigated. The results show that the fatigue life of aluminum plate increases with adding either T or L section riveted stiffeners. With the same cross-sectional area of stiffener, the T section stiffened plate shows longer fatigue life than L section stiffened plate. Meanwhile, the vibration fatigue life also shows great sensitivity to the spacing between the stiffeners.

Analysis of vibration reduction characteristics and applicability of steel-spring floating-slab track

A coupled dynamics computation model for metro vehicles, along with a steel-spring floating-slab track, is developed based on the theory of vehicle-track coupled dynamics. Using the developed model, the influences of the thickness, length and mass of floating-slab, spring rate and its arrangement space, running speed, etc. on the time and frequency domain characteristics of steel-spring fulcrum force are analyzed. The applicability of steel-spring floatingslab track is discussed through two integrated example cases of metro and buildings possessing distinct natural vibra- tion characteristics. It is concluded that, it is quite significant, in the optimization modular design of the parameters of steel-spring floating-slab track, to take the matching relationship of both the amplitude-frequency characteristics of steel-spring fulcrum force and natural vibration characteristics of integrated structures into comprehensive consideration. In this way the expensive steel-spring floating-slab track can be economically and efficiently utilized according to the site condition, and at the same time, the economic losses and bad social impact resulted from the resonance during usage of steel-spring floating-slab track can be avoided.

Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment

Free vibration of statically thermal postbuckled functionally graded material （FGM） beams with surface-bonded piezoelectric layers subject to both temperature rise and voltage is studied. By accurately considering the axial extension and based on the Euler-Bernoulli beam theory, geometrically nonlinear dynamic governing equations for FGM beams with surface-bonded piezoelectric layers subject to thermo-electro- mechanical loadings are formulated. It is assumed that the material properties of the middle FGM layer vary continuously as a power law function of the thickness coordinate, and the piezoelectric layers are isotropic and homogenous. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above mentioned non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations. One is for the postbuckling, and the other is for the linear vibration of the beam superimposed upon the postbuckled configuration. Using a shooting method to solve the two sets of ordinary differential equations with fixed-fixed boundary conditions numerically, the response of postbuckling and free vibration in the vicinity of the postbuckled configuration of the beam with fixed-fixed ends and subject to transversely nonuniform heating and uniform electric field is obtained. Thermo-electric postbuckling equilibrium paths and characteristic curves of the first three natural frequencies versus the temperature, the electricity, and the material gradient parameters are plotted. It is found that the three lowest frequencies of the prebuckled beam decrease with the increase of the temperature, but those of a buckled beam increase monotonically with the temperature rise. The results also show that the tensional force produced in the piezoelectric layers by the voltage can efficiently increase the critical buckling temperature and the natural frequency.

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material （FGM） beams is studied based on the Levinson beam theory （LBT）. Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.

Vibration of composite beams with two overlapping delaminations

Delaminations in composite laminates may de-velop from small cracks due to fabrication and impact load-ing,or from places of high stress concentration.The locationsof the delaminations are not determinate.In this research,ananalytical solution for the free vibration of a composite beamwith two overlapping delaminations is presented.The dela-minated beam is analyzed as seven interconnected beamsusing the delaminations as their boundaries.The continuityand equilibrium conditions are satisfied between the adjoin-ing regions of the beams.Classical beam theory is applied toeach of the beams.Complex vibration behaviors emerge fordifferent sizes and locations of the delaminations.Compar-ison with analytical results reported in the literature verifiesthe validity of the present solution.

Vibration of axially moving beam supported by viscoelastic foundation

In this paper, transverse vibration of an axially moving beam supported by a viscoelastic foundation is analyzed by a complex modal analysis method. The equation of motion is developed based on the generalized Hamilton＇s principle. Eigenvalues and eigenfunctions are semi-analytically obtained. The governing equation is represented in a canonical state space form, which is defined by two matrix differential operators. The orthogonality of the eigenfunctions and the adjoint eigenfunctions is used to decouple the system in the state space. The responses of the system to arbitrary external excitation and initial conditions are expressed in the modal expansion. Numerical examples are presented to illustrate the proposed approach. The effects of the foundation parameters on free and forced vibration are examined.

Flapwise Bending Vibration Analysis of Rotating Tapered Rayleigh Beams for the Application of Offshore Wind Turbine Blades

The flapwise bending vibrational equations of tapered Rayleigh beam are derived based on Hamilton’s principle.The corresponding vibrational characteristics of rotating tapered Rayleigh beams are investigated via variational iteration method(VIM).Natural frequencies and corresponding mode shapes are examined under various rotation speed,taper ratio and slenderness ratio focusing on two types of tapered beam.The convergence of VIM is examined as part of the paper.Validation of VIM solution is made by referring to results available in other literature and corresponding results show that VIM is capable of yielding precise results in a very efficient way.

Analysis of the flexural vibration of ship's tail shaft by transfer matrix method

A ship's tail shaft has serious flexural vibration due to the cantilevered nature of the propeller's blades.Analysis of the nature frequency of flexural vibration is vital to be able to provide effective shock absorption for a ship's tail shaft.A mathematic model of tail shaft flexural vibrations was built using the transfer matrix method.The nature frequency of flexural vibration for an electrically propelled ship's tail shaft was then analyzed,and an effective method for calculating it was proposed:a genetic algorithm(GA),which calculates the nature frequency of vibration of a system.Sample calculations,with comparisons by the Prohl method under conditions bearing isotropic support,showed this method to be practical.It should have significant impact on engineering design theory.

Free Vibration Analysis of Functionally Graded Beams with General Elastically End Constraints by DTM

The differential transformation method (DTM) is applied to investigate free vibration of functionally graded beams supported by arbitrary boundary conditions, including various types of elastically end constraints. The material properties of functionally graded beams are assumed to obey the power law distribution. The main advantages of this method are known for its excellence in high accuracy with small computational expensiveness. The DTM also provides all natural frequencies and mode shapes without any frequency missing. Fundamental frequencies as well as their higher frequencies and mode shapes are presented. The significant aspects such as boundary conditions, values of translational and rotational spring constants and the material volume fraction index on the natural frequencies and mode shapes are discussed. For elastically end constraints, some available results of special cases for isotropic beams are used to validate the present results. The new frequency results and mode shapes of functionally graded beams resting on elastically end constraints are presented.

Method for Detecting Multi-Modal Vibration Characteristics of Landmines

The acoustic vibration characteristics of landmines are investigated by means of modal analysis. According to the mechanical structure of landmines, a certain number of points are marked on the landmine shell to analyze its multi-modal vibration characteristics, based on laser self-mixing interferometer and taking 69 plastic landmine as an example, the vibration detection experiment system is built to show the results of analytical method of multi-modal testing. The first and second order natural frequencies of the bricks are 38 HZ and 106 HZ, 112 HZ and 232 HZ for plastic landmines, and 74 HZ and 290 HZ for metal landmines. The first and second order natural frequencies of the bricks are far smaller than those of plastic landmines and metal landmines. This indicates that landmines show multi-modal vibration characteristics under external excitation, which are significantly different from those of bricks. The findings can be used for further research on acoustic landmines detection technology.

Research on Horizontal Vibration of Heavy-Weight Drill Pipes in Directional Drilling

The mode of load and deformation of directional drilling string and the expression of trigonometric series of deflection equation are established by means of elastic deformation energy and of the vertical and horizontal bending. A calculation formula for natural frequency of horizontal resonance and rotational speed is derived based on the calculation method by Ritz, with which analysis is made for the cause and affecting factors of the excessive abrasion of heavy-weight drill pipe in high-angle holes so as to provide reference and basis for rational selection of drilling parameters and drilling tools in the future high-angle directional drilling.

Analytical and Experimental Study of Free Vibration of Beams Carrying Multiple Masses and Springs

The structures in engineering can be simplified into elastic beams with concentrated masses and elastic spring supports. Studying the law of vibration of the beams can be a help in guiding its design and avoiding resonance. Based on the Laplace transform method, the mode shape functions and the frequency equations of the beams in the typical boundary conditions are derived. A cantilever beam with a lumped mass and a spring is selected to obtain its natural frequencies and mode shape functions. An experiment was conducted in order to get the modal parameters of the beam based on the NExT-ERA method. By comparing the analytical and experimental results, the effects of the locations of the mass and spring on the modal parameter are discussed. The variation of the natural frequencies was obtained with the changing stiffness coefficient and mass coefficient, respectively. The findings provide a reference for the vibration analysis methods and the lumped parameters layout design of elastic beams used in engineering.