Dynamics of 1D mass–spring system with a negative stiffness spring realized by magnets: Theoretical and experimental study

Realization of negative stiffness(NS)in damping low frequency acoustic and mechanical vibration is relevant in engineering applications.In this work,assemblage of two repelling magnets was used to produce negative magnetic spring(NMS).A mass–spring system with NMS is experimented where the free and forced vibrations of the system are examined.The anti-phase movement is observed due to the presence of proposed NMS,confirming the analytical solution.We further showed the dynamics of the system containing NS spring could also be derived from Hamilton’s principle.     

Autoparametric amplification of two nonlinear coupled mass–spring systems

The gearboxes of machines generally operate under a time-varying state rather than under steady-state conditions. However, it is difficult to investigate the nonlinear dynamics of a time-varying gear system. A gear system model of a railway vehicle was proposed in consideration of its time-varying mesh stiffness, nonlinear backlash, transmission error, time-varying external excitation, and rail irregularity. To obtain the nonlinear behaviors of a time-varying stochastic gear system, a quasi-static analysis was performed to observe its doubling-periodic bifurcation, chaotic motion, and transition from a lower to a higher power periodic motion. Based on the energy comparison results, the time-varying stochastic gear system was degraded to a time-varying system to simplify the calculation. Furthermore, the nonlinear response of the time-varying system was computed using the Runge–Kutta method and was compared with the results of a quasi-static analysis that employed a short-time Fourier transform method. The results of the quasi-static analysis were consistent with the results of the time–frequency analysis for the time-varying gear system except for the result at 3180 r/min, which represented a short period wherein the process transitioned to chaos. Hence, the comparison demonstrates the applicability of the quasi-static analysis for the nonlinear behavior analysis of a time-varying stochastic system.     

Wave-based control of planar motion of beam-like mass–spring arrays

Wave-based control (WBC) is a simple and relatively new technique for motion control of under-actuated flexible systems. To date it has been mainly applied to rectilinear lumped flexible systems. The current work focuses on a development of WBC to control two-dimensional beam-like structures in which an actuator, attached to one end, acts to translate and rotate the structure through an arbitrary path in the plane. In this work, first a lumped model of a beam is developed using mass–spring arrays. The lumped beam model is of interest here as a benchmark control challenge. It can also be considered as a model of various lumped or distributed mass structures. To check the latter, the mode shapes and frequencies are first compared with those of classical beam theory. This involved a new technique to find mode shapes and frequencies for arrays. The control strategy is then presented and tested for a range of manoeuvres. As a system to be controlled, the mass–spring array presents many challenges. It has many degrees of freedom, many undamped vibration modes, is highly under-actuated, and sensing of system states is difficult. Despite these challenges, WBC performs well, combining a fairly rapid response with active vibration damping and zero steady-state error. The controller is simple to implement and of low order. It does not need or use any system model and is very robust to system changes.     

Critical oscillations of mass–spring systems due to nonsmooth friction

Critical values of the parameters governing the dynamics of simple systems appear when Coulomb friction is not regularized. We explore such systems using a method based on the fact that under constant or analytical data the trajectory exists, is unique and is also sufficiently regular. In fact these properties justify elementary analytical computations on successive time intervals where the condition used to connect the solution from one interval to the other is due to the regularity. Although the systems are simple the dynamics turn out to be quite complex and thus furnish an interesting benchmark for contact dynamics numerical codes. Among other possible applications we choose to present here how to use a mass–spring chain with Coulomb friction to slow down in a progressive and regular manner an oncoming mass with a given initial velocity.     

Dynamics of a mass–spring–pendulum system with vastly different frequencies

We investigate the dynamics of a simple pendulum coupled to a horizontal mass?Cspring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass?Cspring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and ??/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.     

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.     

On the “bead,hoop and spring” (BHS) dynamical system

Here, we make the theoretical and numerical analysis of the non-linear equation describing the evolution of the “bead, hoop and spring” (BHS) dynamical system derived by Ochoa and Clavijo in (Eur. J. Phys. 27:1277–1288, 2006). In particular, we solve by standard techniques of non-linear physics an approximation of their equation neglecting the centrifugal effect before giving a more mathematical and exact treatment. The analogy with phase transitions is underlined. We point out the existence of finite-time singularities in the phase-space and we derive a criterion for possible oscillations.     

High-order synchronization of stick–slip process: experiments on spring–slider system

In the last years it has been shown that the synchronization and triggering of dynamic events by weak external forcing is ubiquitous and is observed in biological systems, lasers, electronic networks, etc. In the present paper, new experimental data on the phase synchronization in frictional system induced by a weak electromagnetic or mechanical periodic forcing are analyzed. For quantitative analysis of stick–slip time series, modern tools of nonlinear dynamics were used. Stick–slip events were identified by recording acoustic emissions, which accompany slip displacements. The spring–slider system in stick–slip regime is considered as a proxy of active tectonic fault, generating earthquakes. The effect of high-order synchronization of stick–slip events by weak electromagnetic or mechanical periodic forcing, as well as the phenomenon of phase time delay of the synchronized slip events behind the forcing phase, was discovered. These findings can help to find new regularities in seismic time series.     

New ANCF solid-beam element: relationship with Bézier volume and application on leaf spring modeling

Acta Mechanica Sinica - An eight-node solid-beam element based on absolute nodal coordinate formulation (ANCF) which uses cubic interpolation at the longitudinal direction and linear at the...     

Solution and asymptotic analysis of a boundary value problem in the spring–mass model of running

We consider the classic spring–mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires us to choose the stiffness to ascertain that after a complete step, the spring returns to its equilibrium position. Motivated by numerical calculations and real data, we conduct a rigorous asymptotic analysis in terms of the Poicaré–Lindstedt series. The perturbation expansion is furnished by an interplay of two time scales what has an significant impact on the order of convergence. Further, we use these asymptotic estimates to prove that there exists a unique solution to the aforementioned boundary value problem and provide an approximation to the sought stiffness. Our results rigorously explain several observations made by other researchers concerning the dependence of stiffness on the initial angle of the stride and its velocity. The theory is illustrated with a number of numerical calculations.

     

Characteristics of elemental carbon and organic carbon in PM10 during spring and autumn in Chongqing, China

PM10 （particulate matter with aerodynamic diameter less than 10 μm） samples were collected simultaneously at nine urban sites and one urban background site during two intensive observation campaigns in 2006. Concentrations of elemental carbon （EC） and organic carbon （OC） in PM10 were analyzed using an element analyzer. The characteristics regarding spatial and seasonal distribution patterns of OC and EC concentrations and their contributions to PM10 mass, as well as correlation between OC and EC, were investigated in detail. The average OC and EC concentrations for urban sites were 57.5 ± 20.8 and 8.3 ± 3.9 μg/m^3, respectively, both being around three times higher than those for urban background site. As a whole, EC concentrations did not show distinct seasonal variations, though OC concentrations were generally higher in autumn than in spring. For urban sites, total carbonaceous aerosol （TCA） accounted for 33.2% in spring and 35.0% in autumn of PM10 mass. The OC and EC concentrations were found significantly correlated to each other both in spring and in autumn, implying the existence of similar emission sources such as coal combustion. The OC/EC ratios generally exceeded 2.0, indicating the presence of secondary organic carbon （SOC）, whose estimated concentration for urban Chongqing was 26.7 and 39.4μg/m^3, accounting for 48.9 and 61.9% of the total OC observed in the samples, in spring and in autumn, respectively.