Approximation Technique for Solving High-Order Non-Oscillatory Vibration Systems with Slowly Changing Coefficients Represented by Strong Non-Linearity and Multiple Integrated Roots

A formula for the asymptotic solution of an over-damped strongly nonlinear vibration system combining the extended KBM method and the harmonic balance (HB) method with slowly varying coefficients is proposed. This article aims to establish a slowly time-varying solution of an over-damped nonlinear vibration system where one eigenvalue is an integer multiple (greater than two hundred times) of the other eigenvalues. The integrated multiple eigenvalues can provide a better result than other eigenvalues for strong linearity (even if). We found solutions by considering initial conditions and comparing the percentage error of present solutions with unified solutions by using this procedure in examples. Finally, the findings are addressed, especially to improve the physical prospects and shown graphically by using Excel, Dev C++, MATHEMATICA, and MATLAB software.