^{1}Dept. of Mechanical Engineering, Babol University of Technology, Babol, Iran

^{2}Assistant Professor, Department of Mechanical Engineering, Babol University of Technology, , Babol, Iran

^{3}Department of Mechanical Engineering, Babol University of Technology, Semnan, Iran

Abstract

In this paper, Newton Harmonic Balance Method (NHBM) is applied to obtain the analytical solution for an electron beam injected into a plasma tube where the magnetic field is cylindrical and increases towards the axis in inverse proportion to the radius. Periodic solution is analytically verified and consequently the relation between the Natural Frequency and the amplitude is obtained in an analytical form. A comparison of the period of the oscillation and obtained solution with the exact result illustrates that the NHBM is a powerful and efficient tool for solving nonlinear vibration equations.

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