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Identification and Parameter Estimation of Nonlinear Damping Using Volterra Series and Multi-Tone Harmonic Excitation

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Abstract

Purpose

Response characteristics of nonlinear systems have been extensively studied for system identification. But all these studies mainly employ single tone harmonic excitation. In contrast, there are very few research literatures on the use of multi-tone harmonic excitation, obviously due to the challenges in more complicated formulation of response characteristics. This research intends to identify a polynomial type of damping nonlinearity using Higher-order Frequency Response Functions (HoFRFs) and harmonic amplitude measurement data under multi-tone harmonic excitation.

Methods

In the present study, the Volterra series is employed to demonstrate benefits of using multi- tone harmonic excitation for identification of damping nonlinearity. It is shown a large number of combination tones of higher harmonics are formed in the response spectrum. Response harmonic amplitude series are formulated for these harmonics using higher order Volterra kernel synthesis for both symmetric and asymmetric forms of damping nonlinearity.

Results and conclusion

A novel parameter estimation algorithm is presented to first estimate the nonlinear parameter and then the linear modal parameters of the system using two experiments only, whereas, for single-tone harmonic excitation, one would require at least six to eight experiments. The signal strength of higher harmonics is studied for selection of most effective frequency combinations in the multi-tone excitation. Numerical simulations with a typical two-tone excitation demonstrate that fairly accurate estimates of nonlinear damping parameters and linear modal parameters can be obtained with proper selection of frequency pair and excitation level.

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Abbreviations

A, B:

Excitation force amplitudes

\({\beta }_{2}\) :

Second order nonlinear damping parameter

\({\beta }_{3}\) :

Third order nonlinear damping parameter

\({c}_{1}\) :

Linear damping coefficient

\({c}_{2}\) :

Square damping coefficient

\({c}_{3}\) :

Cubic damping coefficient

\(f(t)\) :

Excitation force

ξ:

Damping ratio

\({h}_{1}\left({\tau }_{1}\right)\) :

First order Volterra kernel

\({h}_{2}\left({\tau }_{1},{\tau }_{2}\right)\) :

Second order Volterra kernel

\({h}_{n}\left({\tau }_{1},{\tau }_{2},\dots ,{\tau }_{n}\right)\) :

nth order Volterra kernel

\({H}_{1}(\omega )\) :

First order Volterra kernel transform

\({H}_{2}(\omega ,\omega )\) :

Second order Volterra kernel transform

\({H}_{3}\left(\omega ,\omega ,\omega \right)\) :

Third order Volterra kernel transform

\({H}_{n}\left({\omega }_{1},{\omega }_{2},\dots ,{\omega }_{n}\right)\) :

nth Order Volterra kernel transform or Frequency Response Function

\({k}_{1}\) :

Linear stiffness coefficient

\(m\) :

Mass of the system

t :

Time

\(\tau\) :

Non-dimensional time

\({\omega }_{1}\) :

Two-tone first driving frequency

\({\omega }_{2}\) :

Two-tone second driving frequency

\({\omega }_{n}\) :

Natural frequency

\({\omega }_{p,q,s,u}\) :

General higher harmonic of \(\omega\)

\({\Omega }_{E}\) :

Non-dimensional excited frequency

\({\Omega }_{1}=\frac{{\omega }_{1}}{{\omega }_{n}}\) :

Non-dimensional two-tone first driving frequency

\({\Omega }_{2}=\frac{{\omega }_{2}}{{\omega }_{n}}\) :

Non-dimensional two-tone secnd driving frequency

x(t):

Response function

\(\dot{x}(t)\) :

Velocity of system

\(\ddot{x}\left(t\right)\) :

Acceleration of system

\(X\) :

Amplitude of system

\(X\left(m\omega \right)\) :

mth harmonic response amplitude

\(X\left(\omega \right)\) :

First harmonic amplitude

\(X\left(2\omega \right)\) :

Second harmonic amplitude

\(X\left(3\omega \right)\) :

Third harmonic amplitude

\(\eta \left(\tau \right)\) :

Non-dimensional response

\({\eta }^{\prime}\left(\tau \right)\) :

Non-dimensional Velocity

\(\eta^{\prime\prime} \left(\tau \right)\) :

Non-dimensional acceleration

\(\overline{\eta }\left(\Omega \right)\) :

Non-dimensional first harmonic amplitude

\(\overline{\eta }\left(2\Omega \right)\) :

Non-dimensional second harmonic amplitude

\(\overline{\eta }\left(3\Omega \right)\) :

Non-dimensional third harmonic amplitude

\({x}_{1}\left(t\right)\) :

First response component

\({x}_{2}\left(t\right)\) :

Second response component

\({x}_{3}\left(t\right)\) :

Third response component

\({x}\left(t\right)\) :

Total response

\(X\left({m}_{1}{\omega }_{1}+{m}_{2}{\omega }_{2}\right)\) :

Response harmonic amplitude for a Combination tone \(\left({m}_{1}{\omega }_{1}+{m}_{2}{\omega }_{2}\right)\)

\(X\left({\omega }_{1}\right)\) :

First harmonic amplitude for a driving frequency \({\omega }_{1}\)

\(X\left({\omega }_{2}\right)\) :

First harmonic amplitude for a driving frequency \({\omega }_{2}\)

\(X\left({2\omega }_{1}\right)\) :

Second harmonic amplitude for a frequency \({2\omega }_{1}\)

\(X\left(2{\omega }_{2}\right)\) :

Second harmonic amplitude for a frequency \({2\omega }_{2}\)

\(X\left({\omega }_{1}+{\omega }_{2}\right)\) :

Second harmonic amplitude for a combination tone \({(\omega }_{1}+{\omega }_{2})\)

\(X\left({\omega }_{1}-{\omega }_{2}\right)\) :

Second harmonic amplitude for a combination tone \(({\omega }_{1}-{\omega }_{2})\)

\(X\left({3\omega }_{1}\right)\) :

Third harmonic amplitude for a frequency \(3{\omega }_{1}\)

\(X\left(3{\omega }_{2}\right)\) :

Third harmonic amplitude for a frequency \(3{\omega }_{2}\)

\(X\left(2{\omega }_{1}+{\omega }_{2}\right)\) :

Third harmonic amplitude for a combination tone \(\left(2{\omega }_{1}+{\omega }_{2}\right)\)

\(X\left(2{\omega }_{1}-{\omega }_{2}\right)\) :

Third harmonic amplitude for a combination tone \(\left(2{\omega }_{1}-{\omega }_{2}\right)\)

\(X\left(2{\omega }_{2}+{\omega }_{1}\right)\) :

Third harmonic amplitude for a combination tone \(\left(2{\omega }_{2}+{\omega }_{1}\right)\)

\(X\left(2{\omega }_{2}-{\omega }_{1}\right)\) :

Third harmonic amplitude for a combination tone \(\left(2{\omega }_{2}-{\omega }_{1}\right)\)

\(\overline{\eta }\left({m}_{1}{\Omega }_{1}+{m}_{2}{\Omega }_{2}\right)\) :

Non-dimensional response harmonic amplitude for a combination tone \(\left({m}_{1}{\Omega }_{1}+{m}_{2}{\Omega }_{2}\right)\)

\(\overline{\eta }\left({\Omega }_{1}\right)\) :

Non-dimensional first harmonic amplitude for a driving frequency \({\Omega }_{1}\)

\(\overline{\eta }\left({\Omega }_{2}\right)\) :

Non-dimensional first harmonic amplitude for a driving frequency \(\left({\Omega }_{2}\right)\)

\(\overline{\eta }\left(2{\Omega }_{1}\right)\) :

Non-dimensional harmonic amplitude for a frequency \(2{\Omega }_{1}\)

\(\overline{\eta }\left(2{\Omega }_{2}\right)\) :

Non-dimensional harmonic amplitude for a frequency \(2{\Omega }_{2}\)

\(\overline{\eta }\left({\Omega }_{1}+{\Omega }_{2}\right)\) :

Non-dimensional second harmonic amplitude for a combination tone \(\left({\Omega }_{1}+{\Omega }_{2}\right)\)

\(\overline{\eta }\left({\Omega }_{1}-{\Omega }_{2}\right)\) :

Non-dimensional second harmonic amplitude for a combination tone \(\left({\Omega }_{1}-{\Omega }_{2}\right)\)

\(\overline{\eta }\left(3{\Omega }_{1}\right)\) :

Non-dimensional third harmonic amplitude for a frequency \(3{\Omega }_{1}\)

\(\overline{\eta }\left(3{\Omega }_{2}\right)\) :

Non-dimensional third harmonic amplitude for a frequency \(\left(3{\Omega }_{2}\right)\)

\(\overline{\eta }\left(2{\Omega }_{1}+{\Omega }_{2}\right)\) :

Non-dimensional third harmonic amplitude for a combination tone \(\left(2{\Omega }_{1}+{\Omega }_{2}\right)\)

\(\overline{\eta }\left(2{\Omega }_{1}-{\Omega }_{2}\right)\) :

Non-dimensional third harmonic amplitude for a combination tone \(\left(2{\Omega }_{1}-{\Omega }_{2}\right)\)

\(\overline{\eta }\left(2{\Omega }_{2}+{\Omega }_{1}\right)\) :

Non-dimensional third harmonic amplitude for a combination tone \(\left(2{\Omega }_{2}+{\Omega }_{1}\right)\)

\(\overline{\eta }\left(2{\Omega }_{2}-{\Omega }_{1}\right)\) :

Non-dimensional third harmonic amplitude for a combination tone \(\left(2{\Omega }_{2}-{\Omega }_{1}\right)\)

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Authors and Affiliations

  1. Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, 440010, India

    Hari Prasad Chintha & Animesh Chatterjee

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  1. Hari Prasad Chintha
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Correspondence to Hari Prasad Chintha.

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Appendix-A: Synthesis of Higher Order FRFs

The Volterra series response representation for a general nonlinear system under multi-tone harmonic excitation is given by

$${x}_{n}\left(t\right) ={x}_{1}\left(t\right)+{x}_{2}\left(t\right)+\dots = \quad \sum_{n=1}^{\infty }\frac{1}{{2}^{n}}\sum {A}^{p+q}{B}^{s+u}{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right){e}^{j{\omega }_{p,q,s,u}t}$$
(A.1)
$$x\left(t\right)=\sum_{n=1}^{\infty }\frac{1}{{2}^{n}}\sum {A}^{p+q}{B}^{s+u}{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right){e}^{j{\omega }_{p,q,s,u}t}$$
(A.2)

Then the response series in velocity \(\dot{x}(t)\) becomes

$$\dot{x}\left(t\right)=\sum_{n=1}^{\infty }\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}\sum_{p+q+s+u=n}^{j}{\omega }_{p,q,s,u}{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right){e}^{j{\omega }_{p,q,s,u}t}$$
(A.3)

where \(H_n^{p,q,s,u}\left( \omega \right) = {H_n}\left( {\underbrace {{\omega _1} \ldots }_{p\;times},\underbrace { - {\omega _1} \ldots }_{q\;times},\underbrace {{\omega _2} \ldots }_{s\;times},\underbrace { - {\omega _2} \ldots }_{u\;times},} \right)\)

$${\omega }_{p,q,s,u}=\left(p-q\right){\omega }_{1}+\left(s-u\right){\omega }_{2}$$

\({C}_{p,q,s,u}=\frac{n!}{p!q!s!u!}\), where \(n=p+q+s+u\)

Now, for a general polynomial nonlinearity up to cubic term for multi-tone excitation, equation of motion becomes

$$m\ddot{x}\left(t\right)+{c}_{1}\dot{x}\left(t\right)+{c}_{2}{\dot{x}}^{2}\left(t\right)+{c}_{3}{\dot{x}}^{3}\left(t\right){k}_{1}x\left(t\right)+{k}_{2}{x}^{2}\left(t\right)+{k}_{3}{x}^{3}(t)=A\,\,\mathrm{cos}\,\,\left({\omega }_{1}t\right)+B\,\,\mathrm{cos}\,\,\left({\omega }_{2}t\right)$$
(A.4)

Substituting Eqs. (A.1–A.3) in Eq. (A.4), one obtains

$$\sum_{n=1}^{\infty }\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}\sum_{p+q+s+u=n}^{j}{ C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right){e}^{j{\omega }_{p,q,s,u}t}\left[-m{\omega }_{p,q,s,q}^{2}+{k}_{1}+j{c}_{1}{\omega }_{p,q,s,u}\right]$$
$$+{k}_{2}{\left[\sum_{n=1}^{\infty }\frac{1}{{2}^{n}}\sum {A}^{p+q}{B}^{s+u}{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right){e}^{j{\omega }_{p,q,s,u}t}\right]}^{2}$$
$$+{k}_{3}{\left[\sum_{n=1}^{\infty }\frac{1}{{2}^{n}}\sum {A}^{p+q}{B}^{s+u}{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right){e}^{j{\omega }_{p,q,s,u}t}\right]}^{3}+{c}_{2}{\left[\sum_{n=1}^{\infty }\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}\sum_{p+q+s+u=n}^{j}{\omega }_{p,q,s,u}{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right){e}^{j{\omega }_{p,q,s,u}t}\right]}^{2}$$
$$+{c}_{3}{\left[\sum_{n=1}^{\infty }\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}\sum_{p+q+s+u=n}^{j}{\omega }_{p,q,s,u}{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right){e}^{j{\omega }_{p,q,s,u}t}\right]}^{3}=\frac{A}{2} \left({e}^{j{\omega }_{1}t}+{e}^{-j{\omega }_{1}t}\right)+\frac{B}{2}\left({e}^{j{\omega }_{2}t}+{e}^{-j{\omega }_{2}t}\right)$$
(A.5)

Equating coefficients of \(\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}{e}^{j{\omega }_{p,q,s,u}t}\) both sides in Eq. (A.5), n = 1,2,3…., one obtains

$$H_{1} \left( {\omega_{1} } \right) = \frac{1}{{\left( { - m\omega_{1}^{2} + k_{1} + jc_{1} \omega_{1} } \right)}},{\text{for}}\,\, n = { 1}$$
(A.6)
$$H_{1} \left( {\omega_{2} } \right) = \frac{1}{{\left( { - m\omega_{2}^{2} + k_{1} + jc_{1} \omega_{2} } \right)}},{\text{for}}\,\, n = { 1}$$
(A.7)

For n > 1,

Coefficient of \(\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}{e}^{j{\omega }_{p,q,s,u}t}\) in first line of Eq. (A.5) is

$${C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right)\left[-m{\omega }_{p,q,s,q}^{2}+{k}_{1}+j{c}_{1}{\omega }_{p,q,s,u}\right]=\frac{{ C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right)}{{H}_{1}({\omega }_{p,q,s,u})}$$

Coefficient of \(\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}{e}^{j{\omega }_{p,q,s,u}t}\) in second line of Eq. (A.5) is

$${k}_{2}\sum \left\{{C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}$$

such that, \({p}_{1}+{q}_{1}+{s}_{1}+{u}_{1}={n}_{1},\; {p}_{2}+{q}_{2}+{s}_{2}+{u}_{2}={n}_{2}\) and \({n}_{1}+{n}_{2}=n\)

Coefficient of \(\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}{e}^{j{\omega }_{p,q,s,u}t}\) in third line of Eq. (A.5) is

$${k}_{3}\sum \left\{{C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\left\{{C}_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{H}_{{n}_{3}}^{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}\left(\omega \right)\right\}$$

such that, \({p}_{1}+{q}_{1}+{s}_{1}+{u}_{1}={n}_{1}, {p}_{2}+{q}_{2}+{s}_{2}+{u}_{2}={n}_{2}, {p}_{3}+{q}_{3}+{s}_{3}+{u}_{3}={n}_{3}\) and \({n}_{1}+{n}_{2}+{n}_{3}=n\).

Coefficient of \(\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}{e}^{j{\omega }_{p,q,s,u}t}\) in fourth line of Eq. (A.5) is

$${c}_{2}\sum \left\{{j{\omega }_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{j{\omega }_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}$$

such that, \({p}_{1}+{q}_{1}+{s}_{1}+{u}_{1}={n}_{1}, {p}_{2}+{q}_{2}+{s}_{2}+{u}_{2}={n}_{2}\) and \({n}_{1}+{n}_{2}=n\)

$${p}_{1}+{p}_{2}=p, {q}_{1}+{q}_{2}=q, {s}_{1}+{s}_{2}=s, and\,\, {u}_{1}+{u}_{2}=u$$

Coefficient of \(\frac{1}{{2}^{n}}{A}^{p+q}{B}^{s+u}{e}^{j{\omega }_{p,q,s,u}t}\) in fifth line of Eq. (A.5) is

$${c}_{3}\sum \left[\begin{array}{c}\left\{{j{\omega }_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{j{\omega }_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\\ \left\{j{\omega }_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{C}_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{H}_{{n}_{3}}^{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}\left(\omega \right)\right\}\end{array}\right]$$

such that, \({p}_{1}+{q}_{1}={n}_{1}, {p}_{2}+{q}_{2}={n}_{2}, {p}_{3}+{q}_{3}={n}_{3}\) and \({n}_{1}+{n}_{2}+{n}_{3}=n\)

$${p}_{1}+{p}_{2}+{p}_{3}=p, {q}_{1}+{q}_{2}+{q}_{3}=q, {s}_{1}+{s}_{2}+{s}_{3}=s, \mathrm{and} {u}_{1}+{u}_{2}+{u}_{3}=u$$

Sum of all these terms coming from LHS of Eq. (A.5) will be zero as there is no such term on the RHS for n > 1. Therefore,

$$\begin{aligned}\frac{{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right)}{{H}_{1}({\omega }_{p,q,s,u})}+{k}_{2}\sum \left\{{C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\\+{k}_{3}\sum \left\{{C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\left\{{C}_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{H}_{{n}_{3}}^{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}\left(\omega \right)\right\}\\+{c}_{2}\sum \left\{{j{\omega }_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{j{\omega }_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\\+{c}_{3}\sum \left[\begin{array}{c}\left\{{j{\omega }_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{j{\omega }_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\\ \left\{j{\omega }_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{C}_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{H}_{{n}_{3}}^{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}\left(\omega \right)\right\}\end{array}\right]=0 \end{aligned}$$
(A.8)

This gives,

$$\frac{{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right)}{{H}_{1}({\omega }_{p,q,s,u})}=-\left[{k}_{2}\sum_{\begin{array}{c}{p}_{i}+{q}_{i}+{s}_{i}+{u}_{i}={n}_{i}\\ {n}_{1}+{n}_{2}=n\end{array}}\sum \left\{{C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}+{k}_{3}\sum_{\begin{array}{c}{p}_{i}+{q}_{i}+{s}_{i}+{u}_{i}={n}_{i}\\ {n}_{1}+{n}_{2}+{n}_{3}=n\end{array}}\left\{{C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\left\{{C}_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{H}_{{n}_{3}}^{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}\left(\omega \right)\right\}\right]$$
$$-\left[{c}_{2}\sum_{\begin{array}{c}{p}_{i}+{q}_{i}+{s}_{i}+{u}_{i}={n}_{i}\\ {n}_{1}+{n}_{2}=n\end{array}}\left\{{j{\omega }_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{j{\omega }_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}+{c}_{3}\sum_{\begin{array}{c}{p}_{i}+{q}_{i}+{s}_{i}+{u}_{i}={n}_{i}\\ {n}_{1}+{n}_{2}+{n}_{3}=n\end{array}}\left[\begin{array}{c}\left\{{j{\omega }_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{j{\omega }_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\\ \left\{j{\omega }_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{C}_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{H}_{{n}_{3}}^{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}\left(\omega \right)\right\}\end{array}\right]\right]$$
(A.9)

Synthesis of \({H}_{2}\left(\omega ,\omega \right)\) and \({H}_{3}\left(\omega ,\omega ,\omega \right)\) for damping nonlinearity with square and cubic terms.

If coefficients of nonlinear stiffness \({k}_{2}={k}_{3}=0\) then, Eq. (A.9) becomes

$$\frac{{C}_{p,q,s,u}{H}_{n}^{p,q,s,u}\left(\omega \right)}{{H}_{1}({\omega }_{p,q,s,u})}=-\left[{c}_{2}\sum_{\begin{array}{c}{p}_{i}+{q}_{i}+{s}_{i}+{u}_{i}={n}_{i}\\ {n}_{1}+{n}_{2}=n\end{array}}\left\{{j{\omega }_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{j{\omega }_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}+{c}_{3}\sum_{\begin{array}{c}{p}_{i}+{q}_{i}+{s}_{i}+{u}_{i}={n}_{i}\\ {n}_{1}+{n}_{2}+{n}_{3}=n\end{array}}\left[\begin{array}{c}\left\{{j{\omega }_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}C}_{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}{H}_{{n}_{1}}^{{p}_{1},{q}_{1},{s}_{1},{u}_{1}}\left(\omega \right)\right\}\left\{j{\omega }_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{C}_{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}{H}_{{n}_{2}}^{{p}_{2},{q}_{2},{s}_{2},{u}_{2}}\left(\omega \right)\right\}\\ \left\{j{\omega }_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{C}_{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}{H}_{{n}_{3}}^{{p}_{3},{q}_{3},{s}_{3},{u}_{3}}\left(\omega \right)\right\}\end{array}\right]\right]$$
(A.10)

Appendix-B: List of Symbols

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Chintha, H.P., Chatterjee, A. Identification and Parameter Estimation of Nonlinear Damping Using Volterra Series and Multi-Tone Harmonic Excitation. J. Vib. Eng. Technol. 10, 2217–2239 (2022). https://doi.org/10.1007/s42417-022-00535-7

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