K. M. Khalil

Harmonic and Sub-Harmonic Periodic Solutions (1/2, 1/3) of Generalized Mathieu-Van der Pol-Duffing Equations

A. M. El-Naggar, K. M. Khalil & A. M. Omran · Journal of Scientific Research and Reports · 2017

The frequency-locking area of harmonic and subharmonic ( 1/2, 1/3 ) solutions in a fast harmonic excitation Mathieu-Van der PolDuffing equation is studied. A perturbation technique is then performed on the slow dynamic near the harmonic and subharmonic ( 1/2, 1/3 ) solutions, to...

Open access Research Article 10.9734/JSRR/2017/32122

Superharmonic Solutions of Order (2,3) to a MEMS Governed the Motion of a TM-AFM Cantilever

A. M. Elnaggar, K. M. Khalil & A. S. Rahby · Journal of Scientific Research and Reports · 2016

The superharmonic solutions of a weakly nonlinear second order differential equation governed the dynamic behavior of a microcantilever based on TM (Tapping mode) AFM (Atomic force microscopy) are studied analytically by applying the method of multiple scales (MMS). The modulatio...

Open access Research Article 10.9734/JSRR/2016/28140

Periodic Solutions of a Modified Duffing Equation Subjected to a Bi-Harmonic Parametric and External Excitations

A. M. Elnaggar, A. F. El-Bassiouny, K. M. Khalil & A. M. Omran · Journal of Advances in Mathematics and Computer Science · 2016

In this paper, we investigated the periodic solutions of type superharmonic and subsuperharmonic of modified Duffing equation subjected to a bi-harmonic parametric and external excitations. The method of multiple scales is used to construct a first order uniform expansion of appr...

Open access Research Article 10.9734/BJMCS/2016/25684