Forecasting sales of slow and fast moving inventories

Traditional computerised inventory control systems usually rely on exponential smoothing to forecast the demand for fast moving inventories. Practices in relation to slow moving inventories are more varied, but the Croston method is often used. It is an adaptation of exponential smoothing that (1) incorporates a Bernoulli process to capture the sporadic nature of demand and (2) allows the average variability to change over time. The Croston approach is critically appraised in this paper. Corrections are made to underlying theory and modifications are proposed to overcome certain implementation difficulties. A parametric bootstrap approach is outlined that integrates demand forecasting with inventory control. The approach is illustrated on real demand data for car parts.     

Characterization of ZnGeP2 for Parametric Generation with Near-Infrared Femtosecond Pumping

Values for the nonlinear refractive index (n₂     

Two models for the parametric forcing of a nonlinear oscillator

Instabilities associated with 2:1 and 4:1 resonances of two models for the parametric forcing of a strictly nonlinear oscillator are analyzed. The first model involves a nonlinear Mathieu equation and the second one is described by a 2 degree of freedom Hamiltonian system in which the forcing is introduced by the coupling. Using averaging with elliptic functions, the threshold of the overlapping phenomenon between the resonance bands 2:1 and 4:1 (Chirikov’s overlap criterion) is determined for both models, offering an approximation for the transition from local to global chaos. The analytical results are compared to numerical simulations obtained by examining the Poincaré section of the two systems.     

Dynamic stability of rotating cylindrical shells subjected to periodic axial loads

In this paper, the dynamic stability of rotating cylindrical shells under static and periodic axial forces is investigated using a combination of the Ritz method and Bolotin’s first approximation. The kernel particle estimate is employed in hybridized form with harmonic functions, to approximate the 2-D transverse displacement field. A system of Mathieu–Hill equations is obtained through the application of the Ritz energy minimization procedure. The principal instability regions are then obtained via Bolotin’s first approximation. In this formulation, both the hoop tension and Coriolis effects due to the rotation are accounted for. Various boundary conditions are considered, and the present results represent the first instance in which, the effects of boundary conditions for this class of problems, have been reported in open literature. Effects of rotational speeds on the instability regions for different modes are also examined in detail.     

Parametric random excitation of nonlinear coupled oscillators

The stochastic bifurcation and response statistics of nonlinear modal interaction under parametric random excitation are studied analytically, numerically and experimentally. Two basic definitions of stochastic bifurcation are first introduced. These are bifurcation in distribution and bifurcation in moments. bifurcation in moments is examined for the case of a coupled oscillator subjected to parametric filtered white noise. The center frequency of the excitation is selected to be close to either twice the first mode or second mode natural frequencies or the sum of the two. The stochastic bifurcation in moments is predicted using the Fokker-Planck equation together with gaussian and non-Gaussian closures and numerically using the Monte Carlo simulation. When one mode is parametrically excited it transfers energy to the other mode due to nonlinear modal interaction. The Gaussian closure solution gives close results to those predicted numerically only in regions well remote from bifurcation points. However, bifurcation points predicted by the non-Gaussian closure are in good agreement with those estimated by numerical simulation. Depending on the excitation level, the probability density of the excited mode is strongly non-Gaussian and exhibits multi-maxima as predicted by Monte Carlo simulation. Experimental tests are carried out at relatively low excitation levels. In the neighborhood of stochastic bifurcation in mean square the measured results exhibit different regimes of response characteristics including zero motion and occasional small random motion regimes. These two regimes are characterized by the phenomenon of on-off intermittency. Both regimes overlap and thus it is difficult to locate experimentally the bifurcation point.     

Non-linear systems under parametric white noise input: Digital simulation and response

Monte Carlo technique is constituted of three steps. Therefore, improving such technique in practice means, improving the procedure used in one of the three following steps: (i) sample paths of the stochastic input process, (ii) calculation of the outputs corresponding to the generated input samples by using methods of classical dynamics and (iii) estimating statistics of the output process from sample outputs related to the previous step. For linear and non-linear systems driven by parametric impulsive inputs such as normal or non-normal white noises, a general integration method requires a considerable reduction of the integration step when the impulse occurs, treating the impulse as a physical one, by means of a window function of finite duration. This makes Monte Carlo simulation very prohibitive from a computational time point of view. While knowing the exact jump value of the response at impulse occurring that is expressed by a numerical series, the aforementioned problem is overcome because there is no need to reduce the integration step saving computational time, reliability being equal as shown by means of a numerical example.     

On the Need for Control of Nonlinear Oscillations in Machine Systems

Oscillations in machines are invariably nonlinear. This is either because of inertial coupling effects between different motions of the moving components, material and constitutive phenomena giving rise to stiffness modifications, nonlinear dissipation mechanisms, large deflections, or, as is most likely, some sort of combination of all of these. The net effect of nonlinear vibrations is that at best the machine may well behave a little differently from the way the designer intended, or at worst, in a manner which renders it completely unsuitable for the job. The extent of such problems depends on the nature and the scale of the nonlinearities that are present but it is safe to say that nonlinear oscillations can rarely be completely overlooked in precision machinery analysis and design. The unifying theme in this paper is pendulum motion, firstly in the case of a mobile gantry crane for container stacking where we wish to minimise such motion and converge on a target, and then secondly in the case of a vibration absorber in which we choose to initiate pendulum motion within a special absorber, for the purposes of vibration minimisation. The third example involves the potential for pendulum motion at a very much larger scale and summarises the main control problem that is likely to be encountered in a fully deployed momentum exchange propulsion tether operating in space. The paper discusses the general mathematical issues that pertain to pendulum motion in each of the three cases. This motion is investigated initially in the context of the mobile gantry crane, in the form of a basic three dimensional dynamical model. A feedback linearised controller is shown to offer some advantages for the control of such a system and then a simulation based on data from a practical implementation of this within a real-time control system on a 1/10 laboratory scale model is discussed. It is recalled that the real-time effectiveness of the controller can be compromised by relatively slow sensing and data logging hardware but that despite this some useful performance gains can still be obtainable using this sort of control strategy. The second example comprises an autoparametric vibration absorber and here it is shown how even a simple hunting controller can exploit the mode-locking and wide detuning region effects inherent in autoparametric systems. Further experimental results are discussed in the case of a hunting controller for detuning a vertically oriented parametrically excited pendulum in order to exploit and enhance the powerful and persistent absorption available during autoparametric interaction. The paper concludes with a summary review of the third problem in which the theoretical attitude dynamics of a motorised momentum exchange space propulsion tether are summarised and it is shown that they need to be controlled for reliable and optimal payload velocity boost from both circular parking orbits and elliptical transfer orbits about the Earth.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

参量阵电控波束偏转方法研究与实验测试*

为了使参量阵获得更快的波束转动能力,提出了一种基于相控阵技术的电控波束偏转方法。该方法通过同步改变原波波束的空间分布,间接实现了参量阵差频波束的电控偏转。通过仿真发现,较低的原波频率、较大的原波频率差异、较小的阵元间距和原波波长比值d/λ,是在波束偏转时减小差频波主瓣宽度和栅瓣强度的有利条件。随后采用所提方法,对频率为2 kHz的差频波,完成了偏转角为5°、10°与20°时的波束偏转实验。实验生成的差频波主瓣分布在预设方向,栅瓣同时也得到了有效的抑制,达到了波束偏转的目的。     

Mappings Which Have a Parametric Representation in Several Complex Variables

In this paper, the sharp distortion theorems of the Fr\''echet-derivative type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball of complex Banach spaces are established, and the corresponding results of the above generalized mappings on the unit polydisk in $\mathbb{C}^n$ are also given. Meanwhile, the sharp distortion theorems of the Jacobi determinant type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball with an arbitrary norm in $\mathbb{C}^n$ are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in $\mathbb{C}^n$ are got as well. Thus, some known results in prior literatures are generalized.