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The primary and subharmonic resonances of a nonlinear single-degree-of-freedom system under feedback control with a time delay are studied by means of an asymptotic perturbation technique. Both external (forcing) and parametric excitations are included. By means of the averaging method and multiple scales method, two slow-flow equations for the amplitude and phase of the primary and subharmonic resonances and all other parameters are obtained. The steady state (fixed points) corresponding to a periodic motion of the starting system is investigated and frequency-response curves are shown. The stability of the fixed points is examined using the variational method. The effect of the feedback gains, the time-delay, the coefficient of cubic term, and the coefficients of external and parametric excitations on the steady-state responses are investigated and the results are presented as plots of the steady-state response amplitude versus the detuning parameter. The results obtained by two methods are in excellent agreement

The primary and subharmonic resonances of a nonlinear single-degree-of-freedom (SDOF) system under feedback control with a time delay have been studied by means of an asymptotic perturbation technique. Both external (forcing) and parametric excitations have been included. By means of the averaging method and multiple scales method, two slow-flow equations for the amplitude and phase of the primary and subharmonic resonances and all other parameters are obtained, respectively. The steady state solutions (fixed points) for the original system are investigated. The stability of the fixed points is examined by using the variational method. The effect of the feedback gains, time-delay, the coefficient of cubic term, the coefficients of external and parametric excitations on the steady state responses are investigated and the results are presented as plots of the steady state response amplitude versus the detuning parameter. The results obtained by the two methods are in excellent agreement. There exist saddle node bifurcations for the case of primary resonance and the solutions lose stability for the case of resonance subharmonic.

Article informationBackground: Water pollution by heavy metals is a dangerous health problem causing multiple system diseases. Natural materials, such as nigella sativa and propolis, appear to offer a good preventive of pollution in comparison to more costly technologies currently in use. The Aim of The Work:This study aimed to evaluate and compare the potential protective effects of propolis and nigella sativa against the Cadmium and Lead toxicity harmful effects on the kidney structure and functions of adult male rats. Materials and Methods:Seventy adult male albino rats were chosen as an animal model for this study, divided into seven equal groups [each 10 rats]: Group I, the control group received the standard diet and normal saline [1 ml/kg body weight [BW]/day]; Group II for cadmium [Cd]; Group III for cadmium plus nigella sativa; Group IV for cadmium plus propolis; Group V for lead [Pb]; Group VI for lead plus nigella sativa and Group VII for lead plus Propolis. Each rat received [0.5 ml/rat] of its prepared solution orally every day for 15 days. At the end of the experiment, rats were sacrificed and blood samples were collected for the assessment of kidney functions. Then, the kidney was removed and prepared for histopathological and immunohistochemical examination. Finally, the kidney tissue homogenate was prepared for assessments of renal malondialdehyde [MDA].Results: Exposure of rats to Cd. chloride and Pb. acetate resulted in a significant increase in serum creatinine, urea, uric acid, and renal MDA levels and induced histopathological alterations in kidney tissue. But concomitant administration of lead or cadmium with nigella sativa or propolis were associated with amelioration of the kidney impairment induced by lead or cadmium. Conclusion:The natural antioxidants, nigella sativa and propolis, are capable of minimizing the hazardous effects of cadmium chloride or lead acetate on the kidney.