PID control of second-order systems with hysteresis

The efficacy of proportional, integral and derivative (PID) control for set point regulation and disturbance rejection is investigated in a context of second-order systems with hysteretic components. Two basic structures are studied: in the first, the hysteretic component resides (internally) in the restoring force action of the system ('hysteretic spring' effects); in the second, the hysteretic component resides (externally) in the input channel (e.g. piezo-electric actuators). In each case, robust conditions on the PID gains, explicitly formulated in terms of the system data, are determined under which asymptotic tracking of constant reference signals and rejection of constant disturbance signals is guaranteed. Keywords: hysteresis; non-linear systems; PID control; tuning regulator

Vibrational Power Flow Analysis of Rods and Beams

A new method to model vibrational power flow and predict the resulting energy density levels in uniform rods and beams is investigated. This method models the flow of vibrational power in a manner analogous to the flow of thermal power in a heat conduction problem. The classical displacement solutions for harmonically excited, hysteretically damped rods and beams are used to derive expressions for the vibrational power flow and energy density in the rod and beam. Under certain conditions, the power flow in these two structural elements will be shown to be proportional to the energy density gradient. Using the relationship between power flow and energy density, an energy balance on differential control volumes in the rod and beam leads to a Poisson's equation which models the energy density distribution in the rod and beam. Coupling the energy density and power flow solutions for rods and beams is also discussed. It is shown that the resonant behavior of finite structures complicates the coupling of solutions, especially when the excitations are single frequency inputs. Two coupling formulations are discussed, the first based on the receptance method, and the second on the travelling wave approach used in Statistical Energy Analysis. The receptance method is the more computationally intensive but is capable of analyzing single frequency excitation cases. The traveling wave approach gives a good approximation of the frequency average of energy density and power flow in coupled systems, and thus, is an efficient technique for use with broadband frequency excitation

Equation of State of Wet Granular Matter

A theory is derived for the nonequilibrium probability currents of the capillary interaction which determines the pair correlation function near contact. This yields an analytic expression for the equation of state, P = P(N/V,T), of wet granular matter for D=2 dimensions, valid in the complete density range from gas to jamming. Driven wet granular matter exhibits a van-der-Waals-like unstable branch at granular temperatures T<T_c corresponding to a first order segregation transition of clusters. For the realistic rupture length of the liquid bridge, s_crit=0.07 d, the critical point is located at T_c = 0.274 E_cb. While the critical temperature weakly depends on the rupture length, the critical density phi_c is shown to scale with s_crit according to s_crit = 4d (sqrt(phi_J / phi_c) -1). The segregation transition is closely related to the precipitation of granular droplets reported for the free cooling of one-dimensional wet granular matter [Phys. Rev. Lett. 97, 078001 (2006)], and extends the effect to higher dimensional systems. Since the limiting case of sticky bonds, E_cb >> T, is of relevance for aggregation in general, simulations have been performed which show very good agreement with the theoretically predicted coordination K of capillary bonds as a function of the bond length s_crit. This result implies that particles that stick at the surface, s_crit=0, form isostatic clusters.Comment: 29 pages, 20 figure

Periodically Forced Nonlinear Oscillators With Hysteretic Damping

We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as $t\rightarrow \infty$. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence of the amplitude of the periodic solution on the different parameters of the model and discuss the differences with the linear model. We point out certain undesirable features of the solutions, which have also been alluded to in the literature for the linear Bishop's model, but persist in the nonlinear case as well. Finally, we discuss an alternative hysteretic damping oscillator model first proposed by Reid [2], which appears to be free from these difficulties and exhibits remarkably rich dynamical properties when extended in the nonlinear regime.Comment: Accepted for publication in the Journal of Computational and Nonlinear Dynamic

Explosive synchronization in weighted complex networks

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. Given a set of phase oscillators networking with a generic wiring of connections and displaying a generic frequency distribution, we show how combining dynamical local information on frequency mismatches and global information on the graph topology suggests a judicious and yet practical weighting procedure which is able to induce and enhance explosive, irreversible, transitions to synchronization. We report extensive numerical and analytical evidence of the validity and scalability of such a procedure for different initial frequency distributions, for both homogeneous and heterogeneous networks, as well as for both linear and non linear weighting functions. We furthermore report on the possibility of parametrically controlling the width and extent of the hysteretic region of coexistence of the unsynchronized and synchronized states

Hysteretic and chaotic dynamics of viscous drops in creeping flows with rotation

It has been shown in our previous publication (Blawzdziewicz,Cristini,Loewenberg,2003) that high-viscosity drops in two dimensional linear creeping flows with a nonzero vorticity component may have two stable stationary states. One state corresponds to a nearly spherical, compact drop stabilized primarily by rotation, and the other to an elongated drop stabilized primarily by capillary forces. Here we explore consequences of the drop bistability for the dynamics of highly viscous drops. Using both boundary-integral simulations and small-deformation theory we show that a quasi-static change of the flow vorticity gives rise to a hysteretic response of the drop shape, with rapid changes between the compact and elongated solutions at critical values of the vorticity. In flows with sinusoidal temporal variation of the vorticity we find chaotic drop dynamics in response to the periodic forcing. A cascade of period-doubling bifurcations is found to be directly responsible for the transition to chaos. In random flows we obtain a bimodal drop-length distribution. Some analogies with the dynamics of macromolecules and vesicles are pointed out.Comment: 22 pages, 13 figures. submitted to Journal of Fluid Mechanic