Nonlinear dynamics of an inclined beam subjected to a moving load

In this paper, the nonlinear dynamic response of an inclined pinned-pinned beam with a constant cross section, finite length subjected to a concentrated vertical force traveling with a constant velocity is investigated. The study is focused on the mode summation method and also on frequency analysis of the governing PDEs equations of motion. Furthermore, the steady-state response is studied by applying the multiple scales method. The nonlinear response of the beam is obtained by solving two coupled nonlinear PDEs governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-pint of the beam are obtained for various load velocity ratios and the outcome results have been illustrated and compared to the results with those obtained from traditional linear solution. The appropriate parametric study considering the effects of the linear viscous damping, the velocity of the traveling load, beam inclination angle under zero or nonzero axial load are carried out to capture the influence of the effect of large deflections caused by stretching effects due to the beam’s immovable ends. It was seen that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also in the case where the object leaves the inclined beam, its planar motion path is derived and the targeting accuracy is investigated and compared with those from the rigid solution assumption. Moreover, the stability analysis of steady-state response for the modes equations having quadratic nonlinearity was carried out and it was observed from the frequency response curves that for the considered parameters in the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon can be predicted for the longitudinal excitation.     

Numerical algorithm for natural frequencies computation of an axially moving beam model

An algorithm for numerical computation of natural frequencies of the axially moving Euler-Bernoulli beam is presented. It is tested against data found in the literature and against known analytical expressions of its limiting models—axially moving string and stationary beam—where good agreements were found. The numerical algorithm always stays within real algebra. Roots of the polynomial can be computed out of only three real numbers and the expressions for determinant evaluations are deduced in a numerically stable way.     

The influences of both offset and mass moment of inertia of a tip mass on the dynamics of a centrifugally stiffened visco-elastic beam

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end of the beam and is located along its extended axis. This system can be thought of as an extremely simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem is solved by using Frobenius method of solution in power series. The characteristic equation is then solved numerically. The simulation results are tabulated for a variety of the nondimensional rotational speeds, tip mass, tip mass offset, mass moment of inertia and internal damping parameters. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained. Some numerical results are given in graphical form. The numerical results obtained, indicate clearly that the tip mass offset and mass moment of inertia are important parameters on the eigencharacteristics of rotating beams so that they have to be included in the modeling process.     

Nonlinear dynamics of a planar beam–spring system: analytical and numerical approaches

The multiple timescales method is applied to the exact partial differential equations of the planar motion of a hinged–simply supported beam with a linear axial spring of arbitrary stiffness. The forced-damped and free oscillations of the system around frequencies corresponding to nth natural bending mode are examined thoroughly and compared with numerical simulations as well as with already published results obtained by Lindstedt–Poincaré method. A special numerical technique using explicit finite element method to draw the frequency–response curves is appositely developed. The well-known jump phenomena between resonant and non-resonant branches, as well as superharmonic resonances, have been detected numerically.     

Delayed saturation controller for vibration suppression in a stainless-steel beam

This paper considers the effect of time delays on the saturation control of first-mode vibration of a stainless-steel beam. Time delay is commonly caused by measurements of the system states, transport delay, on-line computation, filtering and processing of data, calculating and executing of control forces as required in control processing. The method of multiple scales is employed to obtain the analytical solutions of limit cycles and their stability and to investigate the bifurcations of the system under consideration. All the predictions from analytical solutions are in agreement with the numerical simulation. The analytical results show that a delay can change the range of the saturation control, either widening or shrinking the effective frequency bandwidth. Thus, vibration control of a beam can be achieved using an appropriate choice of the delay in a self-feedback signal. From the examples illustrated, this paper provides a positive example that time delay can also be utilized to suppress vibration in systems when time delay cannot be neglected.     

Nonlinear response and nonsmooth bifurcations of an unbalanced machine-tool spindle-bearing system

In this effort, a six-degree-of-freedom (DOF) model is presented for the study of a machine-tool spindle-bearing system. The dynamics of machine-tool spindle system supported by ball bearings can be described by a set of second order nonlinear differential equations with piecewise stiffness and damping due to the bearing clearance. To investigate the effect of bearing clearance, bifurcations and routes to chaos of this nonsmooth system, numerical simulation is carried out. Numerical results show when the inner race touches the bearing ball with a low speed, grazing bifurcation occurs. The solutions of this system evolve from quasi-periodic to chaotic orbit, from period doubled orbit to periodic orbit, and from periodic orbit to quasi-periodic orbit through grazing bifurcations. In addition, the tori doubling process to chaos which usually occurs in the impact system is also observed in this spindle-bearing system.     

Bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances

The bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances are investigated. The combined effects of the nonlinear term due to the cable’s geometric and coupled behavior between the modes of the beam and the cable are considered. The nonlinear partial-differential equations are derived by the Hamiltonian principle. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. The bifurcation diagrams in three separate loading cases, namely, excitation acting on the cable, on the beam and simultaneously on the beam and cable, are analyzed with changing forcing amplitude. Based on careful numerical simulations, bifurcations and possible chaotic motions are represented to reveal the combined effects of nonlinearities on the dynamics of the beam and the cable when they act as an overall structure.     

Dynamics of firing patterns, synchronization and resonances in neuronal electrical activities： experiments and analysis

Recent advances in the experimental and theoretical study of dynamics of neuronal electrical firing activities are reviewed. Firstly, some experimental phenomena of neuronal irregular firing patterns, especially chaotic and stochastic firing patterns, are presented, and practical nonlinear time analysis methods are introduced to distinguish deterministic and stochastic mechanism in time series. Secondly, the dynamics of electrical firing activities in a single neuron is concerned, namely, fast-slow dynamics analysis for classification and mechanism of various bursting patterns, one- or two-parameter bifurcation analysis for transitions of firing patterns, and stochastic dynamics of firing activities （stochastic and coherence resonances, integer multiple and other firing patterns induced by noise, etc.）. Thirdly, different types of synchronization of coupled neurons with electrical and chemical synapses are discussed. As noise and time delay are inevitable in nervous systems, it is found that noise and time delay may induce or enhance synchronization and change firing patterns of coupled neurons. Noise-induced resonance and spatiotemporal patterns in coupled neuronal networks are also demonstrated. Finally, some prospects are presented for future research. In consequence, the idea and methods of nonlinear dynamics are of great significance in exploration of dynamic processes and physiological functions of nervous systems.     

Flow and heat transfer over an unsteady stretching sheet in a micropolar fluid

The unsteady laminar flow of an incompressible micropolar fluid over a stretching sheet is investigated. The unsteadiness in the flow and temperature fields is caused by the time-dependence of the stretching velocity and the surface temperature. Effects of the unsteadiness parameter, material parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined.     

Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of buoyancy force and thermal radiation

In this paper an analysis has been made to study heat and mass transfer in two-dimensional stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet in the presence of buoyancy force and thermal radiation. The similarity solution is used to transform the problem under consideration into a boundary value problem of nonlinear coupled ordinary differential equations containing Prandtl number, Schmidt number and Sherwood number which are solved numerically with appropriate boundary conditions for various values of the dimensionless parameters. Comparison of the present numerical results are found to be in excellent with the earlier published results under limiting cases. The effects of various physical parameters on the boundary layer velocity, temperature and concentration profiles are discussed in detail for both the cases of assisting and opposing flows. The computed values of the skin friction coefficient, local Nusselt number and Sherwood number are discussed for various values of physical parameters. The tabulated results show that the effect of radiation is to increase skin friction coefficient, local Nusselt number and Sherwood number.     

Chemical reaction and variable viscosity effects on flow and mass transfer of a non-Newtonian visco-elastic fluid past a stretching surface embedded in a porous medium

This work deals with the study of the boundary layer flow and mass transfer of a visco-elastic fluid immersed in a porous medium over a stretching surface in the presence of surface slip, chemical reaction and variable viscosity. The partial differential equations governing the flow have been transformed by similarity transformation into a system of coupled nonlinear ordinary differential equations which is solved numerically by means of the fourth order Runge-Kutta integration scheme coupled with the shooting technique. The effects of various involved interesting parameters on the velocity fields and concentration fields are shown graphically and investigated. In addition, tabulated results for the local skin-friction coefficient and the local Sherwood number are presented and discussed.     

Nonlinear instabilities in a vibratory gyroscope subjected to angular speed fluctuations

Nonlinear instabilities in a single-axis vibrating MEMS gyroscope that is subjected to periodic fluctuations in input angular rates are investigated. For the purpose of characterizing the bifurcation behavior associated with the steady-state, when the angular rate input is subject to small intensity periodic fluctuations, dynamic behavior of periodically perturbed nonlinear gyroscopic systems is studied in detail. An asymptotic approach based on the method of averaging has been employed for this purpose, and closed-form conditions for the frequency response due to parametric resonances have been obtained. This behavior has been illustrated via amplitude-frequency plots.     

Utilizing nonlinear phenomena to locate grazing in the constrained motion of a cantilever beam

Grazing behavior in soft impact dynamics of a harmonically based excited flexible cantilever beam is investigated. Numerical and experimental methods are employed to study the dynamic behavior of macro- and micro-scale cantilever beam–impactor systems. For off-resonance excitation at two and a half times the fundamental frequency, the response of the oscillating cantilever experiences period doubling as the separation distance or clearance between the beam axis and the contact surface is decreased. The nonlinear phenomenon is studied by using phase portraits, Poincaré sections, and spectral analysis. Motivated by atomic force microscopy, this general dynamic behavior is studied as a means to locating the separation distance corresponding to grazing where the contact force is minimized.     

A consistent equilibrium in a cross-section of an elastic–plastic beam

A phenomenon of inequality of equilibrium and constitutive internal forces in a cross-section of elastic–plastic beams is common to many finite element formulations. It is here discussed in a rate-independent, elastic–plastic beam context, and a possible treatment is presented. The starting point of our discussion is Reissners finite-strain beam theory, and its finite element implementation. The questions of the consistency of interpolations for displacements and rotations, and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem. Approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method. The novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross-section. An advantage of the present approach is that the locations, where the balance of equilibrium and constitutive moments should be satisfied, can be prescribed in advance. In order to minimize the error, explicit analytical expressions are used for the constitutive forces; for a rectangular cross-section and bilinear constitutive law, they are given in Appendix A. The comparison between the results of the two finite element formulations, the one using consistent, and the other inconsistent equilibrium in a cross-section, is shown for a cantilever beam subjected to a point load. The problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method, in which the coordinate system is transformed such to follow high gradients of curvature.     

Autoparametric interaction of a liquid surface in a rectangular tank with an elastic support structure under 1:1 internal resonance

Autoparametric interaction of a liquid free surface in a rectangular tank with an elastic support structure, which is subjected to vertical excitation, is investigated. When the natural frequency of the structure is equal to the lowest natural frequency of liquid sloshing, this system is categorized as an autoparametric system with an internal resonance ratio 1:1. The structure is elastically supported so there is a higher possibility that the 1:1 internal resonance can be observed. The nonlinear theoretical analysis is conducted for a fluid assumed to be perfect in a tank with a finite liquid depth. The equations of motion for the first three sloshing modes are derived employing Galerkin’s technique and considering both the nonlinearity of the fluid motion, and the viscous damping effect. Then the theoretical frequency response curves for the harmonic oscillations of the structure and sloshing are determined using van der Pol’s method. The frequency response curves show that high amplitudes of the structure’s vibrations facilitate the liquid sloshing. Furthermore, the influence of the internal detuning parameter is investigated by showing the frequency response curves and bifurcation sets. Hopf bifurcations may occur followed by amplitude-modulated motions. The theoretical results are in quantitative agreement with the experimental data.     

Implementation and efficiency analysis of parallel computation using OpenACC: a case study using flow field simulations

The Open Accelerator (OpenACC) application programming interface is a relatively new parallel computing standard. In this paper, particle-based flow field simulations are examined as a case study of OpenACC parallel computation. The parallel conversion process of the OpenACC standard is explained, and further, the performance of the flow field parallel model is analysed using different directive configurations and grid schemes. With careful implementation and optimisation of the data transportation in the parallel algorithm, a speedup factor of 18.26× is possible. In contrast, a speedup factor of just 11.77× was achieved with the conventional Open Multi-Processing (OpenMP) parallel mode on a 20-kernel computer. These results demonstrate that optimised feature settings greatly influence the degree of speedup, and models involving larger numbers of calculations exhibit greater efficiency and higher speedup factors. In addition, the OpenACC parallel mode is found to have good portability, making it easy to implement parallel computation from the original serial model.     

Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.     

Nonlinear response of a flexible Cartesian manipulator with payload and pulsating axial force

In this present work, the nonlinear response of a single-link flexible Cartesian manipulator with payload subjected to a pulsating axial load is determined. The nonlinear temporal equation of motion is derived using D’Alembert’s principle and generalised Galerkin’s method. Due to large transverse deflection of the manipulator, the equation of motion contains cubic geometric and inertial types of nonlinearities along with linear and nonlinear parametric and forced excitation terms. Method of normal forms is used to determine the approximate solution and to study the dynamic stability and bifurcations of the system. These results are found to be in good agreement with those obtained by numerically solving the temporal equation of motion. Influences of amplitude of the base excitation, mass ratio, and amplitude of static and dynamic axial load on the steady state responses of the system are investigated for three different resonance conditions. For some specific conditions, the results obtained in this work are found to be in good agreement with the previously published experimental work. The results obtained in this work will find applications in the design of flexible Cartesian manipulators with payload.