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Article History

Received: 2 September 2021

Revised: 5 April 2022

Accepted: 18 April 2022

First Online: 26 May 2022

Declarations

:

: None.

: The Volterra series response representation for a general nonlinear system under multi-tone harmonic excitation is given byEquation removedEquation removedThen the response series in velocity InlineEquation removed becomesEquation removedwhere InlineEquation removedEquation removedInlineEquation removed, where InlineEquation removedNow, for a general polynomial nonlinearity up to cubic term for multi-tone excitation, equation of motion becomesEquation removedSubstituting Eqs. (InternalRef removed–InternalRef removed) in Eq. (InternalRef removed), one obtainsEquation removedEquation removedEquation removedEquation removedEquating coefficients of InlineEquation removed both sides in Eq. (InternalRef removed), <i>n</i> = 1,2,3…., one obtainsEquation removedEquation removedFor <i>n</i> > 1,Coefficient of InlineEquation removed in first line of Eq. (InternalRef removed) isEquation removedCoefficient of InlineEquation removed in second line of Eq. (InternalRef removed) isEquation removedsuch that, InlineEquation removed and InlineEquation removedCoefficient of InlineEquation removed in third line of Eq. (InternalRef removed) isEquation removedsuch that, InlineEquation removed and InlineEquation removed.Coefficient of InlineEquation removed in fourth line of Eq. (InternalRef removed) isEquation removedsuch that, InlineEquation removed and InlineEquation removedEquation removedCoefficient of InlineEquation removed in fifth line of Eq. (InternalRef removed) isEquation removedsuch that, InlineEquation removed and InlineEquation removedEquation removedSum of all these terms coming from LHS of Eq. (InternalRef removed) will be zero as there is no such term on the RHS for <i>n</i> > 1. Therefore,Equation removedThis gives,Equation removedEquation removedSynthesis of InlineEquation removed and InlineEquation removed for damping nonlinearity with square and cubic terms.If coefficients of nonlinear stiffness InlineEquation removed then, Eq. (InternalRef removed) becomesEquation removed

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