the harmonic index h(g) , of a graph g is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in e(g), where deg (u) denotes the degree of a vertex u in v(g). in this paper we define the harmonic polynomial of g. we present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in caterpill...

The radial responses of free or encapsulated micro-bubbles excited by a plane wave of large wavelength are governed by NonLinear Ordinary Differential Equations (NL-ODEs). The nonlinear frequency response details the harmonic content of the time response and constitutes the expected outcome of a high order harmonic analysis. In this paper, a high order harmonic balance analysis of the ”RNNP” (b...

A unique feature of oscillators is that small but sustained external perturbations lead to unboundedly large changes in phase, thereby making standard harmonic balance (HB) inapplicable to realistic oscillator phase macromodels. In this paper, we rectify this situation by presenting a novel extension of HB that is capable of handling oscillator phase macromodels. Key to the new method, termed P...

In this paper, based on the nonlocal Timoshenko beam theory, a nonlinear model is presented for the vibrational behavior of carbon nanotubes (CNTs) embedded in elastic medium in thermal environment. Using the Timoshenko beam theory and nonlocal elasticity of Eringen, the influences of rotary inertia, transverse shear deformation and small scale effect are taken into account. To model the intera...

this paper presents a new control methodology for active power filters that provide an adaptive online harmonic estimation with partial and selective harmonic reduction schemes, which has been implemented within an integrated controller. the proposed approach is to provide partial and selective reduction of those individual harmonics which exceed the recommended levels as set by regulatory bodi...