Nonlinear Vibration Differential Equations Research Articles

Study on Nonlinear Vibration of Vertical Lifting Section of Bulk Grain Entrainment Ship Unloader

In view of the prominent problem that nonlinear vibration of a belt conveyor can easily occur during the vertical grain-conveyance process due to the coupling effect of airflow clamping and traction of the conveyor belt, which seriously affects the efficiency and stability of conveying materials by the belt conveyor, a method of solving the vibration analysis of the vertical lifting section of the bulk grain belt unloader by using nonlinear vibration is proposed. Firstly, based on the laminated plate theory, the vertical lifting belt and the grain material clamped by the belt are laminated. The nonlinear vibration differential equation of the vertical lifting section of the bulk grain-carrying ship unloader is established by elastic–plastic mechanics, and solved by perturbation theory and Galerkin discrete analysis. The vibration response curve and structural natural frequency of the vertical lifting section of the bulk grain-carrying machine are obtained by numerical solution, and the influence of the volume content of the clamped material on the vibration response and structural natural frequency of the lifting section is analyzed. This study provides theoretical support for the design of pressure-supply parameters, overall structure and operation parameters of the subsequent entrainment ship unloader, promotes the rapid development of the entrainment ship unloader, provides theoretical support for the design, manufacture, later operation, and maintenance of the entrainment ship unloader, and thus provides equipment and technical support for building an efficient and intelligent port.

Magnetic-Solid Coupling Nonlinear Vibration of an Axially Moving Thin Plate Under Air-Gap Magnetic Field

The nonlinear inherent vibration of an axially moving ferromagnetic thin plate under the action of the armature air-gap magnetic field is investigated. Based on the nonlinear elasticity theory, the energy relationship equations of the thin plate are given. Based on the electromagnetic theory and the solution of Laplace’s magnetic potential equation, magnetic force on the ferromagnetic rectangular plate under air-gap magnetic field environment is deduced. The Hamilton variational principle is applied to derive the magneto-solid coupled nonlinear vibration equation of the axially moving ferromagnetic thin plate. The two-degree-of-freedom nonlinear vibration differential equations containing static load terms with boundary conditions of SSSS and SSCC are obtained by discretizing through the Galerkin method. The multi-scale method is applied to solve the second-order approximation for deriving the first two orders’ intrinsic frequency of the nonlinear system. The variation laws of first two orders’ nonlinear inherent vibration with axial velocity, magnetic potential, initial air-gap thickness, and initial value are given through numerical examples, and the comparative analysis is performed. The results show that the first-order and second-order inherent vibration frequencies decrease with the increase of axial velocity and magnetic potential and increase with the increase of initial air-gap magnetic field thickness. Different materials and different boundary conditions have greater influence on the first-order and second-order inherent frequencies, and show obvious nonlinear characteristics. The results can provide references for analyzing and controlling the vibration behavior of moving structures in electromagnetic environment.

Magneto-Aero-Elastic Superharmonic and Subharmonic Resonances, Bifurcations and Chaos of Conductive Spinning Circular Plates

This paper focuses on the magneto-aero-elastic nonlinear resonances, bifurcations and chaos for superharmonic and subharmonic responses of conductive circular plates spinning in air and magnetic fields. The electromagnetic forces acting on in-plane are presented by using electromagnetic field theory, and a rotational dissipative damping aerodynamic model is employed to characterize the aerodynamic forces acting on the plates. Considering the effects of the geometric nonlinearity and centrifugal tensioning, the magneto-aero-elastic governing equations of spinning conductive circular plates are derived by the Hamiltonian principle. Based on the mode shapes with Bessel functions, the transverse nonlinear vibration differential equations of spinning plates with clamped-edge are derived by the Galerkin method. Amplitude-frequency equations for subharmonic and superharmonic responses are obtained by using the approach of multiple scales, respectively. Numerical calculations indicate the influences of important parameters (e.g. magnetic field intensity, excitation amplitude and rotational speed) on amplitude frequency characteristics. Moreover, it is demonstrated by the numerical results that system responses are sensitive to the variation of these important parameters, where the complex intermittency, period-doubling and chaotic dynamical behaviors exist.

Bifurcation and chaos analysis of a pretensioned moving printed electronic laminated membrane considering aerothermoelasticity

The bifurcation and chaotic motion characteristics are studied for a roll-to-roll flexible printed electronic laminated membrane subjected to initial tension in a steady temperature field. First, an aerodynamic model of the moving membrane is established by using an improved Newton's law for internal friction. Based on von Karman's large deflection theory, a nonlinear vibration differential equation is derived for the laminated membrane under multiple physical fields using Hamilton's principle and solved numerically using the fourth-order Runge-Kutta method. The effects of the temperature, air resistance and other parameters on the nonlinear vibration of the membrane are analyzed by using the bifurcation diagram, Lyapunov exponent diagram and corresponding phase diagram, Poincaré diagram and time history diagram. The results show that the membrane exhibits abundant chaos and bifurcation phenomena under a coupling field. The parameters of the membrane can be varied to control the nonlinear vibration characteristics. This study provides theoretical guidance for improving the transmission stability of electronic membranes and realizing high-precision overprinting.

The Approximate Solution of Nonlinear Vibration of Tennis Based on Nonlinear Vibration Differential Equation

Abstract In this paper, a nonlinear vibration differential equation is established for the moment when a tennis racket hits the ball. At the same time, we deduce the relationship between the tennis racket string tension and the separation time and distance of the hitting speed. Then the article analyzes the vibration equation of the tennis racket damping ball. We use ANSYS finite element analysis software for solid modeling and modal analysis of tennis rackets. The natural frequency of the tennis racket was obtained through the verification results of the experimental modal analysis. A qualitative relationship between the natural frequency of the vibration-absorbing ball system and some parameters was found.

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells

The nonlinear vibration responses of functionally graded shells with different cone angles under external load are studied in this paper. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the first shear deformation theory, von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations was simplified into a single degree of freedom nonlinear vibration differential equation based on the Volmir’s assumption. Finally, the nonlinear motion equation was solved by the harmonic balance method and Runge-Kutta method, and the amplitude frequency response characteristic curves of FGMs conical shell were obtained. The effects of different material distribution function and the ceramic volume fraction exponents on the amplitude frequency response curve of conical shell were discussed. The bifurcation diagrams of conical shells with different cone angles and time process diagrams and phase diagrams with different excitation amplitudes were described. The motion characteristic was characterized by Poincaré map. The results show that FGMs conical shell presents the nonlinear characteristics of "hardening" spring. The chaotic motion of FGMs conical shell is restrained and FGMs conical shell is not prone to produce motion instability with the increase of cone angle. The motion of FGMs conical shell presents a process from periodic motion to multi-periodic motion and then to chaos with the increase of excitation amplitude.

Combination Resonances of Traveling Unidirectional Plates Partially Immersed in Fluid With Time-Dependent Axial Velocity and Axially Varying Tension

Abstract Nonlinear vibrations of axially moving plates partially immersed in fluid are investigated in this paper. The system has time dependency in velocity and tension in axial direction. The Galerkin method is used to solve the nonlinear vibration differential equation. The method of multiple scales and Runge–Kutta method are applied to solve the nonlinear vibration response of the system. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. The effects of mean velocity, amplitude of pulsating velocity, mean tension, amplitude of pulsating tension, and pulsating frequency on the complex dynamics of the system are obtained. The study results reveal rich dynamic behaviors of fluid–structure coupling system.

Thermal buckling and free vibration of composite sandwich curved panels

For this article, the thermal buckling and free vibration of Composite Sandwich Curved Panels (CSCP) in a thermal environment are analyzed. To achieve the purposes, Hamilton’s principle is used to deduce the non-linear vibration differential equations of CSCP based on the first-order discrete layer model. In these equations, non-linear change of lateral and longitudinal displacements are taken into account because of the temperature change. Later, these equations are solved by the Navier method and are authenticated by comparison with finite element simulation results. At last, the influence rules of structural parameters on Critical Buckling Temperature (CBT) of CSCP are studied and exhibited vividly. Meanwhile, some new results about thermal buckling and free vibration of CSCP are presented for the first time.

Nonlinear natural vibration of a circular plate in the non-uniform induced magnetic field

The magnetoelastic nonlinear natural vibration of the conductive thin circular plate in non-uniform induced magnetic field in the current-carrying circular coil is presented. Based on electromagnetic theory, elliptic integral expression of magnetic field and electromagnetic force is deduced. The magnetoelastic vibration equation of plate is derived through the Hamilton principle. Then through Galerkin method, nonlinear vibration differential equation is developed. By the multiple-scale method, the natural frequency expression is achieved. Through numerical calculation, the magnetoelastic vibration characteristics diagrams are drawn. The influences of coil current, plate geometric parameters and time on natural frequency, electromagnetic torque and singular point are discussed. The results show that natural frequency increases when plate thickness becomes larger, but decreases as current or plate radius increases. Along radial direction, magnetic field increases nonlinearly and electromagnetic torque manifests different variations under different boundary conditions. Additionally, when the current is off and on, equilibrium point is center and stable focus, respectively. The obtained results are validated by comparing with the previously reported work and those from numerical method.

Double Resonance of Magnetism-Solid Coupling of in-Plane Moving Thin Plates With Linear Loads and Elastic Supports

For the in-plane moving thin plates with linear loads and elastic supports in magnetic field, the potential energy, the kinetic energy and the electromagnetic force expressions of the system were given. Based on the Hamiltonian variational principle, the magnetism-solid coupling nonlinear vibration equation for the in-plane moving strip plate was deduced. For the clamped-hinged boundary condition, the variable separation method and the Galerkin method were employed to obtain the 2DOF nonlinear vibration differential equations containing the simple harmonic linear load and the electromagnetic damping force terms. The multiscale method was used to analytically solve the principal-internal resonance problem, and the 1st-order state equation and the resonance response characteristic equation for the system under the double joint resonance were obtained. Through numerical examples, the 1st- and 2nd-order resonance amplitude curves of the in-plane moving thin plate were obtained. The effects of different parameters and load positions on vibration characteristics of the system were analyzed. The results show that, for the principal-internal resonance occurring in the system, the multivaluedness and jumping phenomenon of the solution are obvious, and the effects of the elastic support and the linear load position on the resonance are significant. Additionally, the 1st- and 2nd-order resonance multivalued solution areas appear and disappear simultaneously, which reflects obvious internal resonance characteristics.

Analysis of nonlinear twisting vibration characteristics of orthogonal curve-face gear drive

Considering input shaft and output shaft excitation, the time-varying meshing stiffness, static transmission error and gear backlash, a nonlinear dynamic model of the twisting vibration of the orthogonal curve-face gear transmission was established, and the nonlinear vibration differential equation was listed. Using the forth-order Runge–Kutta numerical integral method to solve the equation with MATLAB, the dynamic response of orthogonal curve-face gear transmission was obtained. By analyzing the response images, the impact of different meshing frequency was summarized. Different meshing frequencies lead to different dynamic responses, including single-cycle response, 9-cycle response, quasi-periodic and chaotic response. When the gear backlash changes, the vibration amplitude of the system also changes.

Investigation on Nonlinear Vibration of the Cable in Cable-Stayed Bridge in 3D Space

Cable-stayed bridge is main kind of modern large-span bridges. Wind load and vehicle load are main dynamic force on cable stayed-bridge, which can cause large amplitude vibration of cables. In the paper, the nonlinear vibration differential equations of cable with dynamic displacement at its end point in 3D space was derived. The differential equations were analysed by the method of multiple scales and the vibration characteristic was discussed. The differential equations were solved by numerical method. The Trajectory of the cable middle point was plotted. The result of this paper indicated that the vibration amplitude of cable in 3D space is relate to the frequency and amplitude of the dynamic displacement of the cable end point. The results in the paper could be a reference for the research of cable vibration in cable-stayed bridge.

Dynamic Nonlinearity and Nonlinear Single-Degree-of-Freedom Model for Cable Net Glazing

AbstractThe nonlinear vibration differential equation and vibration frequency of cable net glazing subject to earthquake loading was determined, and a geometrically nonlinear single-degree-of-freedom model for cable net glazing was developed. The nonlinear response spectra were established, and nonlinear time history analysis with finite element (FE) models was conducted to verify them. The nonlinear vibration differential equation and frequency obtained as described in this paper provide a basis for the nonlinear single-degree-of-freedom model for cable net glazing. The analytical formula for the nonlinear frequency with a simplified expression is highly precise and convenient for use in engineering practice. For larger-amplitude seismic waves, the difference between the linear and nonlinear response spectra are more obvious. As the natural period of cable net glazing is always less than 2 s, the linear response spectra in the Chinese code for the seismic design of buildings can be used in the seismic de...

Nonlinear response spectra of cable net facades

Cable net facades are widely used as a new kind of glazing system and have many advantages, such as pleasing esthetics and easy constructability. With the characteristics of low stiffness, light weight, and small damping, cable net facades are obviously geometrically nonlinear, and the response spectrum method may have some limitations in the seismic design of cable net façades. Therefore, the nonlinear vibration differential equation and natural frequency of the cable net subject to earthquakes was deduced, and then the geometrically nonlinear single-degree of freedom model was founded. Second, with the geometrically nonlinear single-degree of freedom model, the nonlinear seismic response of the cable net was calculated with the time history method subject to four groups of site conditions, and the nonlinear response spectra were founded. Third, the linear and nonlinear acceleration and displacement spectra were compared thoroughly. Lastly, shaking table tests were conducted to verify the accuracy of the nonlinear response spectra.

Nonlinear Dynamics Analysis on Super Low Noise Mining System

in this paper, super low noise mining system is analyzed with dynamics model, every constructional element is studied with analysis of kinematics and dynamics, the path of the mass center, angular velocity and angular acceleration of every constructional element is obtained. Nonlinear vibration differential equations of the system are obtained according to Alembert principle. Through numerical analysis, the enforced situation and dynamo load torque of the joint between every constructional element is gained, the influence of crank angle on system response is acquired, and the result has a certain reference value to optimization of the system.

Precise Hsu’s method for analyzing the stability of periodic solutions of multi-degrees-of-freedom systems with cubic nonlinearity

This paper presents a new precise Hsu’s method for investigating the stability regions of the periodic motions of an undamped two-degrees-of-freedom system with cubic nonlinearity. Firstly, the incremental harmonic balance (IHB) method is used to obtain the solution of nonlinear vibration differential equations. Hsu’s method is then adopted for computing the transition matrix at the end of one period, and the precise time integration algorithm is adjusted to improve the computational precision. The stability regions of the system obtained from the precise Hsu’s, Hsu’s and improved numerical integration methods are compared and discussed.

Magneto-elastic combination resonances analysis of current-conducting thin plate

Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electromagnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed. Some complex dynamic performances such as period-doubling motion and quasi-period motion are discussed.

A simplified calculating method of nonlinear frequency of cable net under mean wind load

The cable net supported glass curtain wallas the most advanced technique in dot point supported glass curtain wall, is widely used in China. Because of its large deflection and high nonlinearity under wind load, the dynamic performance of the cable net is greatly different from that of the conventional linear structures. The continuous membrane theory is used to construct the nonlinear vibration differential equation of the cable net, and the harmonic balance method is used to solve the analytic formula of the nonlinear frequency. In order to verify the accuracy of the above analytic formula, the results of the formula and the nonlinear FEM time-history method are compared and found to be in good agreement. Furthermore, the nonlinear vibration differential equation and the nonlinear frequency obtained in this paper are the basis for the wind-induced response analysis of a cable net under fluctuating wind load.

Nonlinear Vibration Systems with Two Parallel Random Excitations

Systems of nonlinear vibration differential equations are investigated where the non-linearities are given by polynomials of any degree. The random excitations are induced by two parallel processes. These random excitations of an often applied type are expressed by linear functionals of weakly correlated processes with correlation length \epsilon . The moments of the solutions and their first and second derivatives are expanded with respect to \epsilon where all terms up to order \epsilon^2 are included. Approximations of the correlation functions are given explicitely. Only the quadratic and cubic non-linearities have an influence on the correlation functions in this approximation order.