Dynamic Behavior Analysis of an Axially Loaded Beam Supported by a Nonlinear Spring-Mass System

Dynamic analysis of an Euler–Bernoulli beam with nonlinear supports is receiving greater research interest in recent years. Current studies usually consider the boundary and internal nonlinear supports separately, and the system rotational restraint is usually ignored. However, there is little study considering the simultaneous existence of axial load, lumped mass and internal supports for such nonlinear problem. Motivated by this limitation, the dynamic behavior of an axially loaded beam supported by a nonlinear spring-mass system is solved and investigated in this paper. Modal functions of an axially loaded Euler–Bernoulli beam with linear elastic supports are taken as trail functions in Galerkin discretization of the nonlinear governing differential equation. Stable steady-state response of such axially loaded beam supported by a nonlinear spring-mass system is solved via Galerkin truncation method, which is also validated by finite difference method. Results show that parameters of nonlinear spring-mass system and boundary condition have a significant influence on system dynamic behavior. Moreover, appropriate nonlinear parameters can switch the system behavior between the single-periodic state and quasi-periodic state effectively.

Nonlinear Phenomena in Axially Moving Beams with Speed-Dependent Tension and Tension-Dependent Speed

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

On general convergence behaviours of finite-dimensional approximants for abstract linear inverse problems

In the framework of abstract linear inverse problems in infinite-dimensional Hilbert space we discuss generic convergence behaviours of approximate solutions determined by means of general projection methods, namely outside the standard assumptions of Petrov–Galerkin truncation schemes. This includes a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.

Design of a Magnetic Interaction-Based Vibration Absorber for Continuous Beam

Classical absorber for vibration suppression of a continuous structure is constructed as a spring-mass oscillator, which only provides coupling force to suppress the vibration of primary structure. In this study, absorber beam is introduced and coupled on the continuous primary beam with magnetic interaction. Thus, the magnetic interaction and coupling bending moment affect the responses of primary beam. Based on the model of the system and Galerkin truncation, the natural frequencies for different magnetic parameters are obtained, which demonstrates that the fundamental frequency can be reduced to zero and the vibration of primary beam can be suppressed in a wide frequency band. Considering the vibration suppression on frequency band, we propose two criteria to evaluate the vibration suppression effect: one is the width of band for vibration suppression and the other is the width for vibration absorption. The two criteria not only show the vibration reduction effect but also correspond to different vibration suppression mechanism. Due to the advantages of zero fundamental frequency induced by the proposed magnetic interaction coupling and wide vibration suppression frequency band, utilizing absorber beam in vibration suppression of continuous structure has potential applications for flexible aim in the fields of manufacturing and aerospace.

Study on the dynamic vibration characteristics of viscoelastic yarn bundles

In order to study the influence of yarn bundle vibration characteristics on the vibration and noise of tufted carpet looms, a yarn bundle vibration model was proposed in this paper, which was based on the viscoelasticity of the yarn bundle, and the correctness of the transverse vibration equation of the yarn bundle was verified by experiments. Different creep models of the yarn bundle were fitted with the experimental data, and the transverse vibration equation of the axial motion viscoelastic yarn bundle was established by using Burgers four-element constitutive model. Then, the Galerkin truncation method was used to solve the partial differential vibration equation of the yarn bundle and solve the equation. Finally, the correctness of the vibration equation is verified by comparison between the experimental results and the numerical simulation results. The results show that the vibration equation is suitable for studying the transverse dynamic vibration characteristics of the yarn bundle.

Vibration characteristics of yarn in the tufted carpet machine

To control the yarn vibration in a reasonable range and to improve the quality of tufted carpet, it is very important to understand the vibration characteristics of yarn correctly. The transverse vibration equation of yarn is established using Newton’s second law in different paths, and then the yarn vibration characteristic curves in different regions are obtained. Firstly, the yarn path is divided and the optimal constitutive model of tufted carpet yarn is determined. Secondly, the transverse vibration is built by analyzing the force of yarn means. Then, the partial differential equation of yarn vibration is simplified as an ordinary differential equation by the Galerkin truncation method. The equation is solved numerically by using the Runge–Kutta method, obtaining the yarn amplitude in different regions. The vibration characteristics of the yarns after the jacquard parts are emphatically analyzed, and the effects of the speed, tension and damping coefficient on the vibration characteristics of yarns are also discussed. Finally, the results are verified by experiments.

Transmissibility of Bending Vibration of an Elastic Beam

This paper proposes an isolation transmissibility for the bending vibration of elastic beams. At both ends, the elastic beam is considered with vertical spring support and free to rotate. The geometric nonlinearity is considered. In order to implement the Galerkin method, the natural modes and frequencies of the bending vibration of the beam are analyzed. In addition, for the first time, the elastic continuum supported by boundary springs is solved by direct numerical method, such as the finite difference method (FDM). Moreover, the detailed procedure of FDM processing boundary conditions and initial conditions is presented. Two numerical approaches are compared to illustrate the correctness of the results. By demonstrating the significant impact, the necessity of elastic support at the boundaries to the vibration isolation of elastic continua is explained. Compared with the vibration transmission with one-term Galerkin truncation, it is proved that it is necessary to consider the high-order bending vibration modes when studying the force transmission of the elastic continua. Furthermore, the numerical examples illustrate that the influences of the system parameters on the bending vibration isolation. This study opens up the research on the vibration isolation of elastic continua, which is of profound significance to the analysis and design of vibration isolation for a wide range of practical engineering applications.

Stability under Galerkin truncation of A-stable Runge–Kutta discretizations in time

We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup, and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semi-flow by an implicit A-stable Runge–Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semi-flow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. We then estimate the Galerkin truncation error for the semi-flow of the evolution equation, its Runge–Kutta discretization and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrodinger equation.

Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam in the Supercritical Regime

This paper focuses on the bifurcation and chaos of an axially accelerating viscoelastic beam in the supercritical regime. For the first time, the nonlinear dynamics of the system under consideration are studied via the high-order Galerkin truncation as well as the differential and integral quadrature method (DQM & IQM). The speed of the axially moving beam is assumed to be comprised of a constant mean value along with harmonic fluctuations. The transverse vibrations of the beam are governed by a nonlinear integro-partial-differential equation, which includes the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation and the DQM & IQM are, respectively, applied to reduce the equation into a set of ordinary differential equations. Furthermore, the time history of the axially moving beam is numerically solved based on the fourth-order Runge–Kutta time discretization. Based on the numerical solutions, the phase portrait, the bifurcation diagrams and the initial value sensitivity are presented to identify the dynamical behaviors. Based on the nonlinear dynamics, the effects of the truncation terms of the Galerkin method, such as 2-term, 4-term, and 6-term, are studied by comparison with DQM & IQM.