Abstract
•Analytical solution and numerical simulation of non-linear differential equations.•Multiple scale perturbation technique is used to obtained the first order approximation.•PPF, IRC and NIPPF controllers to reduce the vibration.•Stability study subjected to external excitation.•Effect of different parameters in the system.
This paper presents a comparison between three different controller methods added to a cracked beam under the action of a harmonic excitation. Those three controllers are Positive Position Feedback (PPF), Integral Resonant Control (IRC) and Nonlinear Integral Positive Position Feedback (NIPPF) which be added to the measured system. The multiple scales method (MSM) is applied for getting the approximate solution on behalf of measured design. This method is effective to solve the major equations of measured system. Stability and effect of different coefficients of the system are demonstrated. The approximate solution response is established via numerical simulation outcome. NIPPF controller is the best one gives better results compared to the other two controllers in decreasing the high amplitude of the system. Comparison between mathematical solution and numerical simulation are considered. Relationship of formerly available papers is considered.