Criteria of vibration control in delayed third-order critically damped Duffing oscillation.

Control forces of delayed third-order critically damped Duffing equation is proposed in this study. The analytical solution is based on the modified HPM. The solution is demonstrated by introducing a proper function used to suppress the vibration of the nonlinear oscillations. The outcome of the modified homotopy expansion by the exponential negative delay parameter reveals that approximations using this technique are more accurate. To test the accuracy of the recommended modified technique, the obtained results are compared with the numerical solution. Excellent agreement with numerical testing is observed. Further, the suggested control function is eligible for establishing highly accurate approximate analytical frequency and periodic solutions. Based on He's frequency formula a quasi-exact solution for the nonlinear oscillation has been derived. It is effective and simple to utilize analytic tools for nonlinear problems. These results reveal the accuracy and quality of the suggested technique. [ABSTRACT FROM AUTHOR]