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

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Abstract

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.

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Abbreviations

y :

Displacement variable

t :

Independent variable

P :

First damping coefficient

\(\mu\) :

Second damping coefficient

\(\sigma\) :

Linear frequency

\(Q\) :

Duffing coefficient

\(\eta\) :

Coefficient of velocity control force

\(\lambda\) :

Coefficient of displacement control force

\(\tau\) :

Time-delay parameter

\(\Omega\) :

Total frequency(non-conservative frequency)

A :

Amplitude of the oscillation

\(U(t)\) :

Transformed variable

\(\phi\) :

Proper decay parameter

\(\alpha\) :

First damped coefficient of the linearized equation

\(\beta\) :

Second damped coefficient of the linearized equation

\(\omega\) :

He’s frequency (conservative frequency)

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  1. Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

    Yusry O. El-Dib

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El-Dib, Y.O. Criteria of vibration control in delayed third-order critically damped Duffing oscillation. Arch Appl Mech 92, 1–19 (2022). https://doi.org/10.1007/s00419-021-02039-4

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