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Article History

Received: 2 September 2021

Revised: 5 April 2022

Accepted: 18 April 2022

First Online: 26 May 2022

Declarations

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: None.

: The Volterra series response representation for a general nonlinear system under multi-tone harmonic excitation is given byThen the response series in velocity becomeswhere , where Now, for a general polynomial nonlinearity up to cubic term for multi-tone excitation, equation of motion becomesSubstituting Eqs. (–) in Eq. (), one obtainsEquating coefficients of both sides in Eq. (), <i>n</i> = 1,2,3…., one obtainsFor <i>n</i> > 1,Coefficient of in first line of Eq. () isCoefficient of in second line of Eq. () issuch that, and Coefficient of in third line of Eq. () issuch that, and .Coefficient of in fourth line of Eq. () issuch that, and Coefficient of in fifth line of Eq. () issuch that, and Sum of all these terms coming from LHS of Eq. () will be zero as there is no such term on the RHS for <i>n</i> > 1. Therefore,This gives,Synthesis of and for damping nonlinearity with square and cubic terms.If coefficients of nonlinear stiffness then, Eq. () becomes

: The symbols and description listed in “List of symbols”.