Additive resonances of a controlled van der Pol-Duffing oscillator

The trivial equilibrium of a controlled van der Pol-Duffing oscillator with nonlinear feedback control may lose its stability via a non-resonant interaction of two Hopf bifurcations when two critical time delays corresponding to two Hopf bifurcations have the same value. Such an interaction results in a non-resonant bifurcation of co-dimension two. In the vicinity of the non-resonant Hopf bifurcations, the presence of a periodic excitation in the controlled oscillator can induce three types of additive resonances in the forced response, when the frequency of the external excitation and the frequencies of the two Hopf bifurcations satisfy a certain relationship. With the aid of centre manifold theorem and the method of multiple scales, three types of additive resonance responses of the controlled system are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. The amplitudes of the free-oscillation terms are found to admit three solutions; two non-trivial solutions and the trivial solution. Of two non-trivial solutions, one is stable and the trivial solution is unstable. A stable non-trivial solution corresponds to a quasi-periodic motion of the original system. It is also found that the frequency-response curves for three cases of additive resonances are an isolated closed curve. It is shown that the forced response of the oscillator may exhibit quasi-periodic motions on a three-dimensional torus consisting of three frequencies; the frequencies of two bifurcating solutions and the frequency of the excitation. Illustrative examples are given to show the quasi-periodic motions. © 2008 Elsevier Ltd. All rights reserved

Nonlinear response of a forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two

Non-resonant bifurcations of codimension two may appear in the controlled van der Pol-Duffing oscillator when two critical time delays corresponding to a double Hopf bifurcation have the same value. With the aid of centre manifold theorem and the method of multiple scales, the non-resonant response and two types of primary resonances of the forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. It is shown that the non-resonant response of the forced oscillator may exhibit quasi-periodic motions on a two- or three-dimensional (2D or 3D) torus. The primary resonant responses admit single and mixed solutions and may exhibit periodic motions or quasi-periodic motions on a 2D torus. Illustrative examples are presented to interpret the dynamics of the controlled system in terms of two dummy unfolding parameters and exemplify the periodic and quasi-periodic motions. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation. © 2008 Elsevier Ltd. All rights reserved

On the critical forcing amplitude of forced nonlinear oscillators

© Versita sp. z o.o. The steady-state response of forced single degree-of-freedom weakly nonlinear oscillators under primary resonance conditions can exhibit saddle-node bifurcations, jump and hysteresis phenomena, if the amplitude of the excitation exceeds a certain value. This critical value of excitation amplitude or critical forcing amplitude plays an important role in determining the occurrence of saddle-node bifurcations in the frequency-response curve. This work develops an alternative method to determine the critical forcing amplitude for single degree-of-freedom nonlinear oscillators. Based on Lagrange multipliers approach, the proposed method considers the calculation of the critical forcing amplitude as an optimization problem with constraints that are imposed by the existence of locations of vertical tangency. In comparison with the Gröbner basis method, the proposed approach is more straightforward and thus easy to apply for finding the critical forcing amplitude both analytically and numerically. Three examples are given to confirm the validity of the theoretical predictions. The first two present the analytical form for the critical forcing amplitude and the third one is an example of a numerically computed solution

Modeling of Evaporation Losses in Sewage Sludge Drying Bed

A model for evaporation losses in sewage sludge drying bed was derived from first principles. This model was developed based on the reasoning that the rate at which evaporation is taking place is directly proportional to the instantaneous quantity of water in the sludge. The aim of this work was to develop a model to assist in the design of drying beds for effective treatment of wastes derived from households’ septic systems.  Data generated after evaporation experiment in sewage sludge drying bed over a period of 15 days at 24 hours intervals was used for the modeling. The model was calibrated and regressed using data for days 1 to 8 by employing the Pearson’s correlation coefficient, “ ” was found to be -0.96124, after which verification was done and regressed using data for days 9 to 15 and  was found to be 0.9969 that these fits validates the model. Materials balance show that evaporation reduces with increase in seepage losses which can be enhanced with use of conditioners and coarser sands because of their larger pore spaces. It was discovered that evaporation from the sludge follows first order kinetic in agreement with the fundamental concept. The model will help in the prediction of total water losses due to evaporation over a given period which would aid in the design of drying beds.http://dx.doi.org/10.4314/njt.v34i4.3

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation. © 2009 Elsevier Ltd. All rights reserved

Modeling of Seepage Losses in Sewage Sludge Drying Bed

This research was carried out to develop a model governing seepage losses in sewage sludge drying bed. The model will assist in the design of sludge drying beds for effective management of wastes derived from households’ septic systems. In the experiment conducted this study, 125kg of sewage sludge, 90.7% moisture content was thoroughly mixed and intermittently into a sand drying bed of dimensions 1.0m length, 0.3m width, 0.8m depth including overboard and having a 50mm diameter drain pipe. Seepage were measured at 24 hours intervals for 15 days, after which seepage model was derived from first principles based on the concept that seepage is inversely proportional to time. The model is in the form and for this model, two cases exist, cases I and II respectively. For case I, as time, tends to infinity (i.e. t →∞) at which point seepage has completely stopped (i.e qs=0) and the seepage curve intercepts the x-axis so that the intercept a = 0, data generated was modeled first by calibration using odd number values of seepage corresponding to 1 to 15 days. Coefficient of correlation after calibration was found to be r=-0.8474, and after verification using even number values of seepage corresponding to 2 to 14days, r =0.8474 was the coefficient of correlation between measured and calculated quantities of seepage which validates the model. For seepage model case II, this was at the initial stage of application of sludge into the drying bed at which point seepage was still taking place so that the intercept , then ‘a’ was determined by trial and error. Again, calibration and verification was done as in case I and correlation of measured and calculated seepage gave r=0.972, this high value of ‘r’ validates the model.http://dx.doi.org/10.4314/njt.v34i1.

Effect of Conditioner on Dewaterability of Sewage Sludge

A model for dewaterability of conditioned sludge was derived based on data generated after dewaterability experiments using six sand drying beds. In these experiments, Six beds were used, the first bed contain no conditioner and served as control experiment. Ferric Chloride was added to the remaining five beds in the neighborhood of 1%, 2%, 3%, 4%, and 5% respectively, corresponding to 30g, 60g, 90g, 120g, and 150g of FeCl3 Seepage (sg) was derived as a linear function of time. The coefficient of correlation obtained from the linear regression of seepage against time corresponding to 30g, 60g, 90g, 120g and 150g are 0.860, 0.844, 0.781, 0.722 and 0.535 respectively. Moisture content of sludge was 86.69% when no coagulant was added, it was 69.47%, 57.32%, 54.01%, 52.79% and 47.81% when 30g, 60g, 90g, 120g and 150g doses were used with a range of 38.88%. Dewaterability enhancement ratio was 1:1 for control experiment, 1:27.72, 1:46.59, 1:51.72, 1:53.62 and 1:61.34 for 30g, 60g, 90g, 120g, and 150g doses respectively. Statistical test parameter, bp-valueb was computed for each coagulant dose and hypothesis tested at 95% level of significance. The bp-valueb results corresponding to 30g, 60g, 90g, 120g and 150g are 0.000, 0.001, 0.003, 0.008 and 0.073. Based on the decision rule, it was concluded that the model is adequate. This modelB is good, but its deficiency lies in the fact that the coefficient of correlation between measured and calculated quantities of seepage at specified coagulant doses decrease with increase in coagulant dosage. Though, the higher the quantity of coagulant used, the higher the dewaterability ofB sewage sludge which is a big plus. The disadvantage of using coagulants in dewatering sludge lies in the fact that the effluent cannot be recycled for consumption because of the toxic Ferric Chloride that has been used as conditioner. It is recommended that further research be carried out to improve on this already existing model.http://dx.doi.org/10.4314/njt.v33i3.2

Steady-state response of fluid-structure interactions in hydraulic piping system of passive interconnected suspensions

Copyright © 2016 Inderscience Enterprises Ltd. Pressure changes in the liquid-filled fluid circuit of a hydraulically interconnected suspension (HIS) system can induce vibrations of the whole pipeline and the associated structure, and hence become a source of structural noise which degrades ride comfort. This paper presents a numerical and experimental investigation into the vibration of the hydraulic piping system of a passive interconnected suspension. The transfer matrix method (TMM) is used to develop a mathematical model, which consists of various pipe sections, hose sections, concentrated masses, spring supports, elbows, damper valves, and accumulators. Laboratory experiments are performed on two liquid-filled piping systems. The measured steady-state responses of the hydraulic circuits are compared with those obtained from numerical simulations of the developed model. It is found that the developed model of the hydraulic system has a reasonable accuracy in the frequency range of interest, and thus can be employed to optimise the design of the hydraulic system

A partial skeleton of an enantiornithine bird from the early Cretaceous of northwestern China

Although recent discoveries from Lower Cretaceous sediments in northeastern China have greatly improved our understanding of the initial stages of avian diversification in eastern Asia, the early evolution of Aves elsewhere on the continent remains poorly understood. In 2004, a collaborative field effort directed by personnel from the Chinese Academy of Geological Sciences and Carnegie Museum of Natural History recovered multiple partial to nearly complete avian skeletons from outcrops of the Lower Cretaceous Xiagou Formation exposed in the Changma Basin of northwestern Gansu Province, China. Here we describe a thrush-sized partial skeleton comprised of a fragmentary pelvic girdle and largely complete hind limbs. A phylogenetic analysis of 20 avian ingroup taxa and 169 anatomical characters places the specimen in Enantiomithes, and within that clade, in Enenantiomithes. When coupled with additional recent discoveries from the Changma Basin, the new skeleton improves our understanding of early avian evolution and diversification in central Asia