Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

Analysis of Dynamic Characteristics of Forced and Free Vibrations of Mill Roll System Based on Fractional Order Theory

In this paper,the main researches are focused on the horizontal nonlinear vibration characteristics of roll systems for rolling mill,mainly including the study of forced vibration and free vibration of the roller.Firstly,the nonlinear damping parameters and nonlinear stiffness parameters within interface of the rolling mill are both considered,and a fractional-order differential term is also introduced to model the horizontal nonlinear vibration.Secondly,the averaging method is introduced to solve the forced vibration system of the mill roll system,and the amplitude-frequency characteristic curves of the system are obtained for different orders,external excitation amplitudes,stiffness and fractional order coefficients.Thirdly,the amplitude-frequency and phasefrequency characteristics of the free vibration of the mill roll system are investigated at different fractional orders.Then,the accuracy of the averaging method for solving the dynamic characteristics of the system is verified by numerical analysis,and the effect of the fractional differential term coefficients and order on the dynamic characteristics of the roll system are investigated.Finally,the time-frequency characteristics and phase-frequency characteristics of free vibration systems at different fractional orders are studied.The validity of the theoretical study is also verified through experiments.

Closed-form steady-state solutions for forced vibration of second-order axially moving systems

Second-order axially moving systems are common models in the field of dynamics, such as axially moving strings, cables, and belts. In the traditional research work, it is difficult to obtain closed-form solutions for the forced vibration when the damping effect and the coupling effect of multiple second-order models are considered.In this paper, Green's function method based on the Laplace transform is used to obtain closed-form solutions for the forced vibration of second-order axially moving systems. By taking the axially moving damping string system and multi-string system connected by springs as examples, the detailed solution methods and the analytical Green's functions of these second-order systems are given. The mode functions and frequency equations are also obtained by the obtained Green's functions. The reliability and convenience of the results are verified by several examples. This paper provides a systematic analytical method for the dynamic analysis of second-order axially moving systems, and the obtained Green's functions are applicable to different second-order systems rather than just string systems. In addition, the work of this paper also has positive significance for the study on the forced vibration of high-order systems.

Computations of Cross-Link Forces and Vibrations for the Test Bed of the Full Channel Gas

In order to control the cross? link forces and the vibration frequencies of the test bed of the full channel gas within the allowable ranges, the analyses of forces and deformation of the test bed was done, for the variously restrained elastic movable frame and the rigid one, the vibration frequencies were computed respectively by means of the methods of mechanics of materials, elasticity and vibration mechanics, the cross link forces and the vibration frequencies of the test bed were tested. The results of theoretical computation comparatively approach the experimental results. The computational methods could be used to availably estimate the design parameters relevant to the test bed of the full channel gas.

Study on Electromagnetic Force and Vibration of Turbogenerator End Windings under Impact Load(Ⅰ): Analysis of Electromagnetic Force of End Windings under Impact Load

In this paper, the boundary value problem (BVP) of 3 D transient eddy current field in the end region in the case that the generator is affected by impact load is specified. Besides, ways to implement discrete methods in both time domain and space domain during the solution of the problem are investigated. The Crank Nicolson scheme is utilized to attain the iterative format of time differential, after taking factors that can ensure both computation precision and stability into consideration. In this paper, the magnetic distribution in the end region of a turbogenerator in the case that the generator is affected by impact load is specified. As a result, it provides foundation for further study of electromagnetic force and electromagnetic vibration in the end region of the turbogenerator.

The Behavior of a Gas Bubble in a Square Cavity Filled with a Viscous LiquidUndergoing Vibrations

External vibrations are known to be one of the promising ways to control the behavior of multiphase systems. Thecomputational modeling of the behavior of a gas bubble in a viscous liquid in a horizontal cylinder of squarecross-section, which undergoes linearly polarized translational oscillations in weightless conditions, has been carried out. Under vibrations, the bubble moves towards the wall of the vessel with acceleration determined by theamplitudes and frequency of vibrations. Near the wall, at a distance of the order of the thickness of the viscousStokes boundary layer, the effects of viscosity become more important and, as a result, the bubble is repelled fromthe wall. After some oscillations, equilibrium conditions are attained where the attractive force balances the repulsive force;accordingly, the average position of the bubble ceases to change. The numerical modelling shows thatthe average behaviors of a deformable bubble near a wall under normal and tangential vibration are similar.

Vibration analysis of maglev three-span rigid frame bridge considering magnetic force

The dynamic interaction between the maglev vehicle and the three-span rigid frame bridge is discussed. With the consideration of magnetic force, the interaction model is developed. Numerical simulations are performed to study the dynamic characteristics of the bridge during vehicle movement along the bridge. The results show that a reasonable value of the linear stiffness ratio of columns to beams is between 2. 0 and 3.0. The dynamic responses of the bridge are aggravated with the decrease in bending rigidity and the increase in vehicle speed and the span ratio of the bridge. It is suggested that a definite way is to control impact coefficients and acceleration in the dynamic design of the bridge. It is unsuitable to adopt the moving load model and the moving mass model in the design. The proposed results can serve in the design of high-speed maglev three-span rigid frame bridges.

NON-LINEAR FORCED VIBRATION OF AXIALLY MOVING VISCOELASTIC BEAMS

The non-linear forced vibration of axially moving viscoelastic beams excited by the vibration of the supporting foundation is investigated. A non-linear partial-differential equation governing the transverse motion is derived from the dynamical, constitutive equations and geometrical relations. By referring to the quasi-static stretch assumption, the partial-differential non-linearity is reduced to an integro-partial-differential one. The method of multiple scales is directly applied to the governing equations with the two types of non-linearity, respectively. The amplitude of near- and exact-resonant steady state is analyzed by use of the solvability condition of eliminating secular terms. Numerical results are presented to show the contributions of foundation vibration amplitude, viscoelastic damping, and nonlinearity to the response amplitude for the first and the second mode.

In-plane forced vibration of curved pipe conveying fluid by Green function method

The Green function method （GFM） is utilized to analyze the in-plane forced vibration of curved pipe conveying fluid, where the randomicity and distribution of the external excitation and the added mass and damping ratio are considered. The Laplace transform is used, and the Green functions with various boundary conditions are obtained subsequently. Numerical calculations are performed to validate the present solutions, and the effects of some key parameters on both tangential and radial displacements are further investigated. The forced vibration problems with linear and nonlinear motion constraints are also discussed briefly. The method can be radiated to study other forms of forced vibration problems related with pipes or more extensive issues.

Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates

This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.

A Simplified Nonlinear Model of Vertical Vortex-Induced Force on Box Decks for Predicting Stable Amplitudes of Vortex-Induced Vibrations

Wind-tunnel tests of a large-scale sectional model with synchronous measurements of force and vibration responses were carried out to investigate the nonlinear behaviors of vertical vortex-induced forces （VIFs） on three typical box decks （i.e., fully closed box, centrally slotted box, and semi-closed box）. The mechanisms of the onset, development, and self-limiting phenomenon of the vertical vortex-induced vibration （VlV） were also explored by analyzing the energy evolution of different vertical VIF components and their contributions to the vertical VIV responses. The results show that the nonlinear components of the vertical VIF often differ from deck to deck; the most important components of the vertical VIF, governing the stable amplitudes of the vertical VIV responses, are the linear and cubic components of velocity contained in the self-excited aerodynamic damping forces. The former provides a constant negative damping ratio to the vibration system and is thus the essential power driving the development of the VIV amplitude, while the latter provides a positive damping ratio proportional to the square of the vibration velocity and is actually the inherent factor making the VIV amplitude self-limiting. On these bases, a universal simplified nonlinear mathematical model of the vertical VIF on box decks of bridges is presented and verified in this paper; it can be used to predict the stable amplitudes of the vertical VIV of long-span bridges with satisfactory accuracy.

Numerical simulation of a short flexible pipe subject to forced motion and vortex-induced vibration

A series of numerical sinmlations about a small scale （aspect ratio： 63.2） flexible pipe undergoing forced harmonious oscillation and vortex-induced vibration （VIV） have been taken into account. The wake hydrodynamics and pipe deformation were accomplished by ANSYS MFX solution strat- egy designed for fluid-structure interaction （FSI） problem with well-performed LES model. The configuration of structured mesh, multi-domain design, different mesh stiffness admeasured by User Fortran ensured that the numerical task was competent to deal with large deformation related to this case. The introduction of instantaneous amplitude definition and modeless component decom- position method （Chen and Kim, 2008） was helpful to reveal much more information from modal analysis. Most results from numerical simulation are generally consistent with those from model test （Choi and Hong, 2000） via the comparison between them. As supplementary to model test, visualization of the vortex wake was also provided. It has been proved that the forced oscillation doesn＇t only excite a complicated dumbbell-like wake pattern around the outer thimble, but also results in inner flow inside the PVC pipe. The velocity of the inner flow increases with the frequency of forced oscillation.

Prediction Model for Vortex-Induced Vibration of Circular Cylinder with Data of Forced Vibration

A model based on the data from forced vibration experiments is developed for predicting the vortex-induced vibra- tions （VIV） of elastically mounted circular cylinders in flow. The assumptions for free and forced vibration tests are explored briefly. Energy equilibrium is taken into account to set up the relationship between the dynamic response of selfexcited oscillations and the force coemcients from forced vibration experiments. The gap between these two cases is bridged straightforwardly with careful treatment of key parameters. Given reduced mass m^# and material damping ratio of an elastically mounted circular cylinder in flow, the response characteristics such as amplitude, frequency, lock-in range, added mass coefficient, cross-flow fluid force and the corresponding phase angle can be predicted all at once. In- stances with different combination of reduced mass and material damping ratio are compared to investigate their effects on VIV. The hysteresis phenomenon can be interpreted reasonably. The predictions and the results from recent experiments carried out by Wifliamson＇ s group are in rather good agreement.

Wind-induced vibration control of bridges using liquid column damper

The potential application of tuned liquid column damper (TLCD) for suppressing wind-induced vibration of long span bridges is explored in this paper.By installing the TLCD in the bridge deck,a mathematical model for the bridge-TLCD system is established.The governing equations of the system are developed by considering all three displacement components of the deck in vertical,lateral,and torsional vibrations,in which the interactions between the bridge deck,the TLCD,the aeroelastic forces,and the aerodynamic forces are fully reflected.Both buffeting and flutter analyses are carried out.The buffeting analysis is performed through random vibration approach,and a critical flutter condition is identified from flutter analysis.A numerical example is presented to demonstrate the control effectiveness of the damper and it is shown that the TLCD can be an effective device for suppressing wind-induced vibration of long span bridges,either for reducing the buffeting response or increasing the critical flutter wind velocity of the bridge.

Experimental Research on the Mechanism of Lift Amplification and Vibration Suppression of Lorentz Force

The Lorentz force generated by electromagnetic field on the surface of the cylinder in the electrolyte solution may modify the structure of the flow boundary layer effectively. The transient control process of Lorentz force is investigated experimentally for lift amplification and vibration suppression. The experiments are conducted in a rotating annular tank filled with a low-conducting electrolyte. A cylinder with an electro-magnetic actuator is placed into the electrolyte. The lift force of cylinder is measured using the strain gages attached to a fixed beam, and the flow fields are visualized by the dye markers. The results show that the upper vortex on the cylinder is suppressed, and the wake becomes a line and leans to the lower side under the action of upside Lorentz force while the lower vortex on the cylinder is suppressed and limited in a small region. Therefore, the value of lift increases with the variation of flow field. However, the vortexes on the cylinder are suppressed fully under the action of symmetrical Lorentz force which leads to the suppression of lift oscillation and then the vibration of cylinder are suppressed fully.

Numerical Simulation of Self-Excited and Forced Vibration of Circular Cylinders in Current

Numerical simulations of a low-mass-damping circular cylinder which can oscillate freely at transverse and stream- wise directions are presented in this work. The Navier-Stokes equations are solved with finite volume method, and large eddy simulation of vortex is also performed in the calculation. In order to implement dynamic mesh, overlapping grids are generated to lessen the computation for mesh field itself. Self-excited vibrations are firstly calculated to obtain the average amplitudes and frequencies of the target circular cylinder in the current flow situation, and then forced oscillations are implemented with parameters obtained in vortex-induced vibrations previously. With slight amplitude modulation, time series of displacements in vortex-induced vibrations are essentially harmonic. Regarding the fluid force, which are larger in forced oscillations than those in corresponding self-excited cases because the fluid subtracts energy from the forced cylinders. The phase angles between forces and displacements are 0° and 180° for self-excited ease and forced case respectively. In vortex-induced vibrations, the interactions between fluid and structure produce some weakly energetic vortices which induce the modulations of amplitude and frequency.

PI force feedback control for large flexible structure vibration with active tendons

Active tendon, consisting of a displacement actuator and a collocated force sensor, was first presented by Preumont and his co-workers to attenuate the vibration of large flexible space structures, and the control algorithm adopted by them was integral force feedback. This paper presents a new proportional-integral （PI） force feedback algorithm to achieve larger damping ratios for the structure without the requirement of structure model. Stability of the control system is shown, and simulations of a structure similar to JPL-MPI demonstrate the effectiveness of the proposed algorithm for vibration control of space structures.

The Interface Stress Field in the Elastic System Consisting of the Hollow Cylinder and Surrounding Elastic Medium Under 3D Non-Axisymmetric Forced Vibration

The paper develops and employs analytical-numerical solution method for the study of the time-harmonic dynamic stress field in the system consisting of the hollow cylinder and surrounding elastic medium under the non-axisymmetric forced vibration of this system.It is assumed that in the interior of the hollow cylinder the point-located with respect to the cylinder axis,non-axisymmetric with respect to the circumferential direction and uniformly distributed time-harmonic forces act.Corresponding boundary value problem is solved by employing of the exponential Fourier transformation with respect to the axial coordinate and by employing of the Fourier series expansion of these transformations.Numerical results on the frequency response of the interface normal stresses are presented and discussed.

Wind-Induced Vibration Control for Substation Frame on Viscous Damper

In order to study the wind-induced vibration control effect of the viscous damper on the large-scale substation frame,this paper takes the large-scale 1000 kV substation frame of western Inner Mongolia as an example.The time-history sample of pulsating wind load is simulated by harmonic superposition method based on Matlab software.6 kinds of viscous damper arrangement schemes have been designed,and SAP2000 finite element software is used for fine modeling and input wind speed time history load for nonlinear time history analysis.The displacement and acceleration of a typical node are the indicators of wind vibration control.The wind-induced vibration control effects of different schemes under different damping parameters have compared,and the damping parameters are analyzed for the optimal layout scheme.The results show that a viscous damper has installed in the lower layers of the substation;a viscous damper is placed between the ground column and the lattice beam.It is an integrated optimal solution.The wind-induced vibration control effect of the optimal scheme is sensitive to the viscous damper parameters,and the control effect does not increase linearly with the increase of the damping index and the damping coefficient.Corresponding to different damping indexes,the damping coefficient has a better range of values.

A flexible multi-body model of a surface miner for analyzing the interaction between rock-cutting forces and chassis vibrations

The discontinuous nature of rock cutting can easily cause unwanted vibrations in the structure of a surface miner.If these vibrations are not properly addressed,the related stress cycles can gradually damage the chassis resulting in fatigue failures.These events can seriously undermine the safety of operators and digging operations may be stopped for days,with an obvious economic impact.This work presents an analysis of the dynamics of a surface miner,focusing on the interaction between cutting machine dynamics and cutting forces,which is a new approach for this type of machine.For this purpose,the authors developed a numerical model of the cutting process made up of(1)a multi-body model of the cutting machine,which takes into account the chassis's flexibility;(2)a model of the rotating cutting head;and(3)a model of the interaction between the cutting head and rock,based on Shao's model.The model was compared with experimental results and then used to investigate the effects of cutting speed and cutting depth on the machine dynamics.