Nonlinear Vibration Equation Research Articles

Research on Lateral Nonlinear Vibration Behavior of Composite Shaft-Disk Rotor System

The characteristics of the lateral nonlinear vibration in composite shaft-disk rotor system with nonlinear deformation are studied. Firstly, the equations of the kinetic energy of the composite shaft, the disk and the eccentric mass as well as the equations of the strain energy of the composite shaft are derived. Based on these equations, the nonlinear vibration equations are deduced by using the Lagrange equation. Then, the frequency response curves and time response curves of the system are obtained by using the IHB method and verified by the fourth order Runge-Kutta method. Experimental results show that the external damping coefficient, the size of eccentric mass only influences the nonlinear amplitude. Moreover, the ply-angle, thickness to diameter(T/D) ratio, length to diameter (L/D) ratio, and the position of disk in the shaft not only produce an effect on the nonlinear amplitude, but also influence the nonlinear vibration frequency.

Vertical Vibration Control of Cold Rolling Mill’s Rolls Based on Time-delay Feedback Control Method

In this paper, a two degree of freedom nonlinear vertical vibration equation of the cold rolling mill with the dynamic rolling force was established, then the delay feedback control method was introduced into the equation to controlled the vertical vibration of the system. The amplitude-frequency equations of primary resonance of system was carried out by using the multi-scale method, and the resonance characteristics of different parameters of delay feedback control method were obtained by adopting the actual parameters of rolling mill. It is found that the size of the resonance amplitude value was effectively controlled and the resonance region and jumping phenomenon of the system were eliminated by selecting the appropriate time-delay parameters combination, which provides an effective theoretical reference for solving mill vibration problems.

Nonlinear vibration of oscillatory systems using semi-analytical approach

In this paper, He\'s Variational Approach (VA) is used to solve high nonlinear vibration equations. The proposed approach leads us to high accurate solution compared with other numerical methods. It has been established that this method works very well for whole range of initial amplitudes. The method is sufficient for both linear and nonlinear engineering problems. The accuracy of this method is shown graphically and the results tabulated and results compared with numerical solutions.

Nonlinear Dynamic Analysis of High‐Voltage Overhead Transmission Lines

According to a generalized Hamilton’s principle, three‐dimensional (3D) nonlinear vibration equations for overhead transmission lines that consider geometric nonlinearity are established. Based on the characteristics of an actual transmission line, the 3D equations are simplified to two‐dimensional equations, and the nonlinear vibration behavior of transmission lines is investigated by combining theoretical analysis with numerical simulation. The results show that transmission lines have inherently nonlinear vibration characteristics. When in free vibration, a transmission line can undergo nonlinear internal resonance, even when its initial out‐of‐plane energy is relatively low; as its initial out‐of‐plane energy increases, the coupling of in‐plane and out‐of‐plane vibration becomes stronger. When forced to vibrate by an external excitation, due to the combined action of internal and primary resonance, the vibration energy of a transmission line transfers from the out‐of‐plane direction to the in‐plane direction that is not directly under the excitation, resulting in an increase in the dynamic tension and the displacement amplitude of the transmission line. Increasing damping can consume the vibration energy of a transmission line but cannot prevent the occurrence of internal resonance.

Effects of temperature variations on nonlinear planar free and forced oscillations at primary resonances of suspended cables

This paper is concerned with the analysis of the free and forced nonlinear vibrations of the suspended cable with thermal effects. By introducing the new thermal stressed configuration, the effect of temperature variation on the nonlinear vibration equations of motion is reflected by a non-dimensional cable tension variation factor. The partial differential equations of the planar motion are discretized to the ordinary equations via the Galerkin method, and the single-mode discretization is investigated. The Lindstedt–Poincare method and multiple scales method are applied to obtain the higher-order approximate solutions of the nonlinear free vibrations and primary resonances, respectively. Parametric investigations of temperature effects on the linear and nonlinear vibration characteristics of the suspended cable with different sag-to-span ratios and excitation amplitudes are performed. The results of the perturbation analysis show that the nonlinear free and forced vibration characteristics would be changed by the temperature variations qualitatively and quantitatively, depending on the sag-to-span ratio and the excitation amplitude. The asymmetric phenomena between the effects of warming and cooling conditions on the vibration characteristics can be observed. The crossover points between next two mode frequencies are shifted under the temperature variation, and these would significantly influence the internal resonances of the suspended cable. Finally, temperature effects on the time-history diagram of the cable axial total tension force are investigated.

时滞影响下受控斜拉索的参数振动稳定性

The effects of time delays on the primary parametric vibration of controlled stay cables under axial excitation were studied. In view of cable sag and geometric nonlinearity, the nonlinear parametric vibration equation for the controlled stay cable system under axial excitation was built based on the Hamiltonian principle. Then the dynamic system with time delay was formulated by means of the Galerkin method. The multiscale method was used to analyze the primary parametric resonance of the controlled stay cable system and obtain the effects of different time delays and control gains on the time histories of the parametric vibration and the stability region of the controlled stay cable. The study shows that time delay weakens the vibration controlling effects on the stay cable, and the stability region of the parametric vibration is shifted. The larger the time delay is, the worse the controlling effects will be. The work plays a guiding role in the parametric design of the control system for stay cables.

Static and dynamic responses of a microcantilever with a T-shaped tip mass to an electrostatic actuation

In this study, nonlinear static and dynamic responses of a microcantilever with a T-shaped tip mass excited by electrostatic actuations are investigated. The electrostatic force is generated by applying an electric voltage between the horizontal part of T-shaped tip mass and an opposite electrode plate. The cantilever microbeam is modeled as an Euler–Bernoulli beam. The T-shaped tip mass is assumed to be a rigid body and the nonlinear effect of electrostatic force is considered. An equation of motion and its associated boundary conditions are derived by the aid of combining the Hamilton principle and Newton’s method. An exact solution is obtained for static deflection and mode shape of vibration around the static position. The differential equation of nonlinear vibration around the static position is discretized using the Galerkin method. The system mode shapes are used as its related comparison functions. The discretized equations are solved by the perturbation theory in the neighborhood of primary and subharmonic resonances. In addition, effects of mass inertia, mass moment of inertia as well as rotation of the T-shaped mass, which were ignored in previous works, are considered in the analysis. It is shown that by increasing the length of the horizontal part of the T-shaped mass, the amount of static deflection increases, natural frequency decreases and nonlinear shift of the resonance frequency increases. It is concluded that attaching an electrode plate with a T-shaped configuration to the end of the cantilever microbeam results in a configuration with larger pull-in voltage and smaller nonlinear shift of the resonance frequency compared to the configuration in which the electrode plate is directly attached to it.

High conservative nonlinear vibration equations by means of energy balance method

This paper presents He`s Energy Balance Method (EBM) for solving nonlinear oscillatory differential equations. Three strong nonlinear cases have been studied analytically. Analytical results of the EBM are compared with numerical solutions using Runge-Kutta`s algorithm. The effects of different important parameters on the nonlinear response of the systems are studied. The results show the presented method is potentially to solve high nonlinear vibration equations.

Nonlinear vibration mechanism for fabrication of crimped nanofibers with bubble electrospinning and stuffer box crimping method

This paper aims to produce nanoscale crimped fibers using stuffer box crimping and bubble electrospinning. Nanofibers are originally obtained via a ruptured bubble and then crimped with the stuffer box crimping method. During this spinning process, the governing equations for nonlinear transverse vibration of an axially moving viscoelastic beam with finite deformation are established using the Hamiltonian principle. The crimp frequency is affected by many factors, including spinning conditions and mechanical properties of fibers. The obtained governing equations can be further used for numerical or analytical study of the crimping mechanism.

Nonlinear free vibration of a rotating circular plate under the static load in magnetic field

The nonlinear free vibration of a conductive rotating thin circular plate subjected to static loads in magnetic field was investigated. The nonlinear vibration equations of the motion on a spinning circular thin plate were derived. According to the set of a displacement function, the approximate solution of the static problem and the magneto elastic axisymmetric vibration differential equation of the round plate were obtained through the application of Galerkin integral method. The characteristic formula between nonlinear frequency and amplitude using the static load as a parameter was obtained by the method of multiple scales. According to the numerical calculation, the characteristic curves of the relationship between frequency and the parameter such as the static load, the magnetic induction intensity and the speed of rotation were obtained. The nonlinear vibration characteristics of a spinning plate with clamped boundary condition were described according to the curves drawn by Matlab. We also showed that different parameters (including magnetic induction intensity, static load, time and spinning velocity.) have effect on the frequency of the plate.

Thermal vibration of a simply supported single-walled carbon nanotube with thermal stress

The thermally induced vibration of a simply supported single-walled carbon nanotube (SWCNT) subject to thermal stress is investigated by using the models of planar and non-planar nonlinear beams with initial stress, respectively. The dynamic equations of nonlinear stochastic vibration of the SWCNT are established, with the geometric nonlinearity of the large deformation taken into account. The thermal vibration of SWCNT is predicted by numerically integrating both the dynamic equations of the nonlinear beam models via the Runge–Kutta algorithm of fourth order. The root-mean-square (RMS) amplitude and the stationary probability density of the thermal vibration of the SWCNT are obtained via the planar and non-planar nonlinear beam models with simply supported boundary conditions for both pre-buckling case and post-buckling case. The RMS amplitude of the thermal vibration of the SWCNT is given by using the numerical integration of probability density function. The RMS amplitude of thermal vibration of a SWCNT predicted via the non-planar nonlinear model with thermal stress is lower than that predicted via the planar nonlinear beam model, but higher than that predicted via the planar linear model.

Nonlinear bending and vibration of functionally graded tubes resting on elastic foundations in thermal environment based on a refined beam model

This paper studies the nonlinear bending and vibration problems of functionally graded tubes with temperature-dependent material properties based on a refined beam model. The tubes are exposed to a uniform distributed temperature field and are placed on elastic foundation. The refined beam model for tubes can satisfy the stress boundary conditions on inner and outer surfaces. The governing equations of nonlinear bending and vibration for the functionally graded tubes are derived by using Hamilton's principle and are solved by introducing a two-step perturbation technique. Some comparisons for bending and vibration are presented to valid the correctness of present beam model and solution method. In numerical results, the effects of transverse shear deformation, the volume fraction, inner radius and elastic foundation stiffness as well as the temperature on the natural frequency, amplitude–frequency responses and nonlinear bending responses are discussed.

Bifurcation Observation of Combining Spiral Gear Transmission Based on Parameter Domain Structure Analysis

This study considers the bifurcation evolutions for a combining spiral gear transmission through parameter domain structure analysis. The system nonlinear vibration equations are created with piecewise backlash and general errors. Gill’s numerical integration algorithm is implemented in calculating the vibration equation sets. Based on cell-mapping method (CMM), two-dimensional dynamic domain planes have been developed and primarily focused on the parameters of backlash, transmission error, mesh frequency and damping ratio, and so forth. Solution demonstrates thatPeriod-doubling bifurcation happens as the mesh frequency increases; moreover nonlinear discontinuous jump breaks the periodic orbit and also turns the periodic state into chaos suddenly. In transmission error planes, three cell groups which arePeriod-1,Period-4, andChaoshave been observed, and the boundary cells are the sensitive areas to dynamic response. Considering the parameter planes which consist of damping ratio associated with backlash, transmission error, mesh stiffness, and external load, the solution domain structure reveals that the system step into chaos undergoesPeriod-doubling cascade withPeriod-2m(m: integer) periodic regions. Direct simulations to obtain the bifurcation diagram and largest Lyapunov exponent (LE) match satisfactorily with the parameter domain solutions.

Nonlinear Vibration Signal Tracking of Large Offshore Bridge Stayed Cable Based on Particle Filter

Abstract The stayed cables are key stress components of large offshore bridge. The fault detection of stayed cable is very important for safe of large offshore bridge. A particle filter model and algorithm of nonlinear vibration signal are used in this paper. Firstly, the particle filter model of stayed cable of large offshore bridge is created. Nonlinear dynamic model of the stayed-cable and beam coupling system is dispersed in temporal dimension by using the finite difference method. The discrete nonlinear vibration equations of any cable element are worked out. Secondly, a state equation of particle filter is fitted by least square algorithm from the discrete nonlinear vibration equations. So the particle filter algorithm can use the accurate state equations. Finally, the particle filter algorithm is used to filter the vibration signal of bridge stayed cable. According to the particle filter, the de-noised vibration signal can be tracked and be predicted for a short time accurately. Many experiments are done at some actual bridges. The simulation experiments and the actual experiments on the bridge stayed cables are all indicating that the particle filter algorithm in this paper has good performance and works stably.

Nonlinear resonance of the rotating circular plate under static loads in magnetic field

The rotating circular plate is widely used in mechanical engineering, meanwhile the plates are often in the electromagnetic field in modern industry with complex loads. In order to study the resonance of a rotating circular plate under static loads in magnetic field, the nonlinear vibration equation about the spinning circular plate is derived according to Hamilton principle. The algebraic expression of the initial deflection and the magneto elastic forced disturbance differential equation are obtained through the application of Galerkin integral method. By mean of modified Multiple scale method, the strongly nonlinear amplitude-frequency response equation in steady state is established. The amplitude frequency characteristic curve and the relationship curve of amplitude changing with the static loads and the excitation force of the plate are obtained according to the numerical calculation. The influence of magnetic induction intensity, the speed of rotation and the static loads on the amplitude and the nonlinear characteristics of the spinning plate are analyzed. The proposed research provides the theory reference for the research of nonlinear resonance of rotating plates in engineering.

Internal resonance of an axially moving unidirectional plate partially immersed in fluid under foundation displacement excitation

This paper studies characteristics of 1:3 internal resonances and its bifurcations for an axially moving unidirectional plate partially immersed in a fluid under foundation displacement excitation. The fluid is assumed to be inviscid and incompressible. Based on Von Kármán large deflection equations of thin plates, and consider the influences of axial movement of the plate, axial tension, fluid‐structure interaction and foundation displacement, nonlinear vibration equations of an axially moving plate are established. By applying the Galerkin technique, the vibration equations are discretized, and thus achieve nonlinear differential equations on mode coordinates. Due to the influence of foundation displacement, there are more than two excitation items in the mode equations. Moreover, excitation amplitude is not a constant value, but a function of excitation frequency. Adopting numerical method and approximate analysis method, the equations are solved, moreover frequency‐response curves and time histories under internal resonance system are also obtained, the stability of periodic solution is also discussed. Global bifurcation phenomenon of averaging equations is studied for 1:3 internal resonance system. Changing course of periodic solution of the system and its complex nonlinear dynamic characteristics are revealed through bifurcation diagrams and phase trajectory diagrams.

Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions

In this paper, parametric and 3:1 internal resonance of axially moving viscoelastic beams on elastic foundation are analytically and numerically investigated. The beam is restricted by viscous damping force. The beam’s material obeys the Kelvin model in which the material time derivative is used. The governing equations of coupled planar vibration and the associated boundary conditions are derived from the generalized Hamilton principle. The effects of the nonhomogeneous boundary conditions due to the viscoelasticity are highlighted, while the boundary conditions are assumed to be homogeneous in previous studies. In small but finite stretching problems, the equation is simplified into a governing equation of transverse nonlinear vibration. It is a nonlinear integro-partial differential equation with time-dependent and space-dependent coefficients. The dependence of the tension on the finite axial support rigidity is also modeled. The method of multiple scales is directly applied to establish the solvability conditions. The nonlinear steady-state oscillating response along with the stability and bifurcation of the beam is investigated. A detailed study is carried out to determine the influence of the viscoelastic coefficient and the viscous damping coefficient on dynamic behavior of the system. The numerical calculations by the differential quadrature scheme confirm the approximate analytical results.

Calculation of Combination Coefficient and Drag Coefficient for the Throwing Materials of Inertial Rotary Vibrating Screen

The structural characteristics and the working principle of inertial rotary vibrating screen are introduced in this paper. The inertial rotary vibrating screen is a kind of high efficient equipment for the screening classification of dry and wet fine materials. It has the features of simple and compact structure, convenient installation and disassembly, stable work operation, no sieve pore blocking, high screening efficiency, and the whole machine closure without polluting environment. The nonlinear vibration equations of inertial rotary vibrating screen are established, considering the nonlinear forces of materials. An approximate solution and the material combination coefficient and drag coefficient for the vibration equations are solved by equivalent linearization method of nonlinear theory. A calculation example is also given in this paper.

Nonlinear vibration of fluid-conveying single-walled carbon nanotubes under harmonic excitation

In this paper, the nonlinear vibration of a single-walled carbon nanotube conveying fluid is investigated utilizing a multidimensional Lindstedt–Poincaré method. Considering the geometric large deformation of the single-walled carbon nanotube and external harmonic excitation force, based on nonlocal elastic theory and Euler–Bernoulli beam theory, the nonlinear vibration equation of a fluid-conveying single-walled carbon nanotube is established. Analyzing the equation through the multidimensional Lindstedt–Poincaré method, and from the solvability condition of the nonlinear vibration equation, the cubic algebraic equation which indicates the amplitude–frequency relation is obtained. Based on the root discriminant of the cubic equation, the first order primary response of the pinned–pinned carbon nanotube is discussed. The relations among internal resonance, the amplitude and frequency of the external excitation force are analyzed in detail. When the external excite force frequency is around the first mode natural frequency, the first mode primary resonance occurs. If simultaneously the first two modes natural frequency ratio is around 3, internal resonance occurs and the internal resonance region depends on the amplitude of external excitation force.

Mathematical solution for nonlinear vibration equations using variational approach

In this paper, we have applied a new class of approximate analytical methods called Variational Approach (VA) for high nonlinear vibration equations. Three examples have been introduced and discussed. The effects of important parameters on the response of the problems have been considered. Runge-Kutta's algorithm has been used to prepare numerical solutions. The results of variational approach are compared with energy balance method and numerical and exact solutions. It has been established that the method is an easy mathematical tool for solving conservative nonlinear problems. The method doesn't need small perturbation and with only one iteration achieve us to a high accurate solution.