M. Saifur Rahman

An analytical technique for solving a class of strongly nonlinear conservative systems

Abstract Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.

Design of a surface plasmon resonance refractive index sensor with high sensitivity

This paper presents a highly sensitive photonic crystal fiber (PCF) refractive index sensor based on the surface plasmon resonance (SPR) effect operating in the telecommunication wavelengths. Gold is used as the plasmonic material due to its chemical stability and titanium dioxide (TiO2) is used to shift the resonance wavelength in the telecommunication bands. Both materials are deposited sequentially on the PCF surface, which is comparatively easy to fabricate. Numerical investigations show that the proposed sensor exhibits very high wavelength sensitivity of 10,800  nm/RIU and amplitude sensitivity of 514  RIU−1 in the sensing range between 1.46 and 1.48. Moreover, it exhibits maximum sensor resolution of 9.25×10−6  RIU and high linearity over a wide sensing range. The proposed sensor can be practically realized due to its simple and straightforward structure.

An analytical technique to find approximate solutions of nonlinear damped oscillatory systems

Abstract Combining Krylov–Bogoliubov–Mitropolskii (KBM) and harmonic balance methods, an analytical technique is presented to determine approximate solutions of nonlinear oscillatory systems with damping. The first approximate perturbation solutions in which the unperturbed solutions contain two harmonic terms agree with numerical solutions nicely even if the damping force is significant. With suitable examples it has been shown that the combination of classical KBM and harmonic balance methods sometimes fails to measure satisfactory results; but the combination of extended KBM method (by Popov) and harmonic balance method always give the desired results. The method is illustrated by several examples and the solutions are compared to some existing solutions.

A GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-TH ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING

Based on the multiple-time-scale (MTS) method, a general formula has been presented for solving an n-th, n = 2, 3, , order ordinary differential equation with strong linear damping forces. Like the solution of the unified Krylov-Bogoliubov-Mitropolskii (KBM) method or the general Strubles method, the new solution covers the un-damped, under-damped and over-damped cases. The solutions are identical to those obtained by the unified KBM method and the general Strubles method. The technique is a new form of the classical MTS method. The formulation as well as the determination of the solution from the derived formula is very simple. The method is illustrated by several examples. The general MTS solution reduces to its classical form when the real parts of eigen-values of the unperturbed equation vanish.

Multi-level residue harmonic balance solution for the nonlinear natural frequency of axially loaded beams with an internal hinge

ABSTRACT In this article, an analytical approach, namely, multi-level residue harmonic balance is introduced and developed for the nonlinear free vibration analysis of axially loaded beams with an internal hinge. The main advantage of this method is that only one set of nonlinear algebraic equations is required to be solved for obtaining the zero level solution while the high accuracy of the higher level solutions can be obtained by solving a set of linear equations. The new approximate analytical solution method is developed for solving the governing differential equations. The accuracy and efficiency of the proposed method are verified by a numerical method. In the comparison, the results obtained from the proposed method well agree with those from other methods. The effects of vibration amplitude, axial force, and hinge location on the fundamental frequencies of various beam cases are investigated. The optimum and worst hinge locations are also studied.

Sensitivity enhancement of porous silicon based SPR sensor using graphene-M 0 S 2 hybrid structure

In this paper a theoretical porous silicon surface based plasmon resonance (SPR) sensor has been presented consisting graphene-Melybdenum sulphide (MoS2) hybrid structure for enhancing sensor detection sensitivity. The biosensor uses perfectly matched layer (PML) boundary condition incorporating on its computational domain to improve its surface plasmon resonance characteristics. Here, graphene-MoS2 hybrid sheet is used to detect the refractive index change of the sensor surface, which is cause of the reaction of biomolecules. Our calculations show that the graphene-MoS2 hybrid structure on silicon porous sensor has 25% more sensitivity than the conventional silicon resonant sensor. The enhanced sensitivity is for increasing SPR angle about 25% by adding graphene-MoS2 hybrid structure.