Transmissibility function analysis for the detection and localization of damage with nonlinear features in MDOF structural systems

Abstract

Transmissibility function analysis refers to the evaluation of the ratio between the spectra of two different outputs of a system and has been widely used for the detection and localization of damage in linear structural systems. In this article, a new transmissibility function analysis methodology is proposed to detect and localize damage with nonlinear features in multidegree-of-freedom (MDOF) structural systems. The methodology assumes the MDOF systems are subject to a general input with only the frequency range known a priori so is applicable to address damage detection and location issues for a wide range of engineering structures and systems. This methodology is derived based on the concept of nonlinear output frequency response functions (NOFRFs) and the concept of transmissibility of nonlinear systems introduced based on the NOFRFs. The transmissibility function analysis over output frequencies contributed by system nonlinearities is exploited to detect and localize damage with nonlinear features in MDOF structural systems. Both numerical simulations and experimental studies on a cracked beam structure in the laboratory have been conducted. The results verify the effectiveness and demonstrate the potential applications of the proposed methodology in the detection and localization of damage in practical engineering structures.

Keywords

Nonlinear output frequency response functions damage detection nonlinearity transmissibility MDOF system finite element method dynamic system

Introduction

Infrastructures and mechanical systems inevitably suffer damage along with aging, which has drawn significant research interest in localization and quantification of damage in structural systems. Vibration-based damage identification has been developed by many researchers.^1–6 This is because changes in physical structures naturally induce changes in vibration-related system properties such as for example, natural frequency, modal damping, and mode shape. Vibration-based damage identification methods include methods that evaluate changes in system natural frequency, changes in mode shape, changes in frequency response function, and changes in transmissibility functions. Among these, the evaluation of changes in transmissibility functions is preferable since transmissibility functions not only have better sensitivity to damage but also require no prior information on loadings on linear dynamic systems due to the removal of input loading information by definition.⁷

Transmissibility function is traditionally defined as the ratio between the output spectra of two degrees of freedom in an MDOF system and was originally applied for structural damage identification in Chen et al.⁸ Recently, many transmissibility function-based damage indicators have been proposed,^9–12 showing that the transmissibility-based methodologies outperform the classical frequency response function (FRF) in terms of the sensitivity of damage identification. In Johnson and Adams⁷ the significance of poles and zeros for the localization of damage in dynamic systems has been investigated and then its sensitivity analysis has been studied on a 3 DOF system. More recently, it has been proved that the transmissibility function converges to the mode shape ratio when frequency bandwidth is restricted to the system’s poles, and damage could be identified according to the changes in the transmissibility functions around these poles.⁹ Based on this principle, a method to detect the existence and quantify the severity of damage has been proposed¹⁰ but the method cannot be used to localize damage. In Cheng et al.,¹¹ the strain transmissibility measured using distributed fiber optics is used for damage localization, showing that the transmissibility of strain is more sensitive to local damage than the transmissibility of displacement and the transmissibility of acceleration. A transmissibility-based analytical methodology has been developed to localize damage induced by mass and stiffness changes in MDOF mass–spring–damper systems.¹²

However, existing transmissibility function analysis mainly focuses on the identification of linear damage including, for example, changes in mass and stiffness, and is unable to detect changes induced by nonlinear damage.¹³ Damage makes engineering structures behave nonlinearly in many scenarios, for example, breathing cracks (bilinear stiffness), postbuckled structures (duffing nonlinearity), and rattling joints (systems with discontinuity characteristics) are typical nonlinear damage in MDOF structural systems.^14–17 Therefore, the study of nonlinear damage identification is of great practical importance.

Recently, the concept of transmissibility of nonlinear systems was proposed by Lang et al.,¹⁸ which is defined as the ratio of nonlinear output frequency response functions (NOFRFs) between two measurement points. The NOFRFs are a relatively new concept and a powerful and effective tool for the analysis of nonlinear systems in the frequency domain.¹⁹ On the basis of the concept of transmissibility of nonlinear systems, Zhao et al.²⁰ developed a technique for the identification of nonlinear damage in MDOF structural systems. In this technique, an MDOF system is assumed to be subject to a harmonic input and the damage detection is performed by evaluating the transmissibility functions of the system responses at super-harmonics. However, harmonic input excitations can only be implemented in purposely designed test campaigns. Consequently, the harmonic input-based technique is not applicable to the identification of damage in real infrastructures such as, for example, long-span bridges where only structural responses to more complicated ambient loading excitations are available for analysis.²¹

In the present study, a new methodology for the detection and localization of damage with nonlinear features is developed for MDOF structural systems. The two key novelties of this article are (1) extending the existing NOFRF-based transmissibility to more general input cases. General input excitations can cover various ambient loading inputs, including, for example, loadings from cars running on a long-span bridge and impacts of winds on tower structures. The new methodology is derived based on the properties of the transmissibility of nonlinear systems within and outside the frequency ranges of system inputs, and its implementation can be straightforwardly achieved by evaluating conventional transmissibility functions over output frequencies contributed by system nonlinearities; (2) formulating a new damage indicator to detect and locate the nonlinear components. Both numerical simulations and experimental studies are conducted to verify the effectiveness and demonstrate potential applications of the proposed methodology in the detection and localization of damage in practical engineering structures.

This article is structured as follows: In section “Transmissibility of Nonlinear Systems,” the concept of transmissibility of nonlinear systems is introduced. Section “Transmissibility function of nonlinear MDOF structural systems under general inputs,” is concerned with the revelation of significant properties of the transmissibility of nonlinear systems. These properties lead to some valuable features with the transmissibility functions of MDOF systems under general inputs and enable the derivation of a novel methodology for detecting and localizing nonlinear components in MDOF structural systems. The procedure of the new methodology is summarized in section “Damage detection and localization methodology.” Numerical simulations and experimental studies are conducted in sections “Simulation study,” and “Experimental study,” respectively. Conclusions are finally reached in section “Conclusion.”

Transmissibility of nonlinear systems

The Volterra series representation of nonlinear systems

Nonlinear systems that are stable at zero equilibrium can be expressed using a Volterra series as²²

y (t) = \sum_{\bar{n} = 1}^{\bar{N}} \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} h_{\bar{n}} (τ_{1}, \dots, τ_{\bar{n}}) Π_{i = 1}^{\bar{n}} u (t - τ_{i}) d τ_{i}

(1)

where $u (t)$ and $y (t)$ are the system input and output, respectively, $h_{\bar{n}} (τ_{1}, \dots, τ_{\bar{n}}), \bar{n} = 1, \dots, \bar{N}$ is the Volterra kernel, and $\bar{N}$ is the maximum order of the system nonlinearity. The frequency-domain representation of system Equation (1) can be given as²²

{\begin{matrix} Y (j ω) = \sum_{\bar{n} = 1}^{\bar{N}} Y_{\bar{n}} (j ω) \forall ω \\ Y_{\bar{n}} (j ω) = \frac{1 / \sqrt{n}}{{(2 π)}^{\bar{n} - 1}} \int_{ω_{1} + \dots + ω_{\bar{n}} = ω} H_{\bar{n}} (j ω_{1}, \dots, j ω_{\bar{n}}) Π_{i = 1}^{\bar{n}} U (j ω_{i}) d σ_{\bar{n} ω} . \end{matrix}

(2)

$σ_{\bar{n} ω}$ represents the hyperplane $ω_{1} + \dots + ω_{\bar{n}} = ω$ . Besides $U (j ω)$ and $Y (j ω)$ are the system input and output spectrum, respectively, $Y_{\bar{n}} (j ω)$ is the $\bar{n}$ th order output spectrum, and

\begin{matrix} H_{\bar{n}} (j ω_{1}, \dots, j ω_{\bar{n}}) = \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} h_{\bar{n}} (τ_{1}, \dots, τ_{\bar{n}}) \times \\ e^{- (ω_{1} τ_{1} + \dots + ω_{\bar{n}} τ_{\bar{n}}) j} d τ_{1} \dots d τ_{\bar{n}} . \end{matrix}

(3)

$H_{\bar{n}} (j ω_{1}, \dots, j ω_{\bar{n}})$ is known as the $\bar{n}$ th order Generalized Frequency Response Function (GFRF), which is the $\bar{n}$ -dimensional Fourier transform of $h_{\bar{n}} (τ_{1}, \dots, τ_{\bar{n}})$ .

The NOFRFs of nonlinear systems

The nonlinear output frequency response functions (NOFRFs) are a concept introduced in 2005 (Lang and Billings²³) as a new extension of linear FRF to nonlinear cases, and defined as

\begin{matrix} G_{\bar{n}} (j ω) = \frac{\int_{ω_{1 + \dots + ω_{\bar{n}} = ω}} H_{\bar{n}} (j ω_{1}, \dots, j ω_{n}) Π_{i = 1}^{\bar{n}} U (j ω_{i}) d σ_{\bar{n} ω}}{\int_{ω_{1 + \dots + ω_{\bar{n}} = ω}} Π_{i = 1}^{\bar{n}} U (j ω_{i}) d σ_{\bar{n} ω}} \\ \bar{n} = 1, . . ., \bar{N} \end{matrix}

(4)

under the condition of

U_{\bar{n}} (j ω) = \int_{ω_{1 + \dots + ω_{\bar{n}} = ω}} Π_{i = 1}^{\bar{n}} U (j ω_{i}) d σ_{\bar{n} ω} \neq 0 .

(5)

As stated in Equation (4), one of the main advantages of NOFRFs is their one-dimensional nature. This characteristic has led to suggestions of replacing the multidimensional GFRFs to simplify the frequency analysis of nonlinear systems.²³ Equations (2) and (4) reveal that the output spectrum of system Equation (1) can be effectively characterized using NOFRFs, as demonstrated in Equation (6). Notably, this representation is analogous to the output spectrum description in linear systems,

Y (j ω) = \sum_{\bar{n} = 1}^{\bar{N}} Y_{\bar{n}} (j ω) = \sum_{\bar{n} = 1}^{\bar{N}} G_{\bar{n}} (j ω) U_{\bar{n}} (j ω) .

(6)

The NOFRFs of single input multiple outputs nonlinear systems

For single input multiple outputs (SIMO) nonlinear systems with $n$ outputs, the Volterra series representation is

y_{i} (t) = \sum_{\bar{n} = 1}^{\bar{N}} \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} h_{(i, \bar{n})} (τ_{1}, \dots, τ_{\bar{n}}) Π_{i = 1}^{\bar{n}} u (t - τ_{i}) d τ_{i} .

(7)

In Equation (7), $y_{i} (t)$ and $u (t)$ are the ith output and the input of the system, respectively, $h_{(i, \bar{n})} (τ_{1}, \dots, τ_{\bar{n}})$ is the $\bar{n}$ th order Volterra kernel associated with the ith system output.

The NOFRFs associated with the ith output of the system can be defined as

G_{i, \bar{n}} (j ω) = \frac{\int_{ω_{1 + \dots + ω_{\bar{n}} = ω}} H_{i, \bar{n}} (j ω_{1}, \dots, j ω_{\bar{n}}) Π_{i = 1}^{\bar{n}} U (j ω_{i}) d σ_{\bar{n} ω}}{\int_{ω_{1 + \dots + ω_{\bar{n}} = ω}} Π_{i = 1}^{\bar{n}} U (j ω_{i}) d σ_{\bar{n} ω}}

i = 1, 2, \dots, n; \bar{n} = 1, \dots, \bar{N} .

(8)

The spectrum of the ith output can then be represented using the NOFRFs as

Y_{i} (j ω) = \sum_{\bar{n} = 1}^{N} Y_{i, \bar{n}} (j ω) = \sum_{\bar{n} = 1}^{N} G_{i, \bar{n}} (j ω) U_{\bar{n}} (j ω) i = 1, 2, \dots, n .

(9)

The transmissibility of nonlinear systems

Transmissibility in SIMO linear systems refers to the ratio between the spectra of two distinct outputs, which is equivalent to the ratio of two system FRFs evaluated at the corresponding output points.^7,10 The concept of transmissibility in SIMO nonlinear systems was initially presented in Peng et al.,¹⁹ where it is described as the ratio of system NOFRFs between two different outputs, given by

T_{i, k}^{\bar{n} L} (j ω) = \frac{G_{(i, \bar{n})} (j ω)}{G_{(k, \bar{n})} (j ω)} = \frac{G_{(i, \bar{n})} (j ω) U_{\bar{n}} (j ω)}{G_{(k, \bar{n})} (j ω) U_{\bar{n}} (j ω)} = \frac{Y_{(i, \bar{n})} (j ω)}{Y_{(k, \bar{n})} (j ω)}

(10)

where $i, k \in {1, 2, \dots, m},$ $\bar{n} = 1, \dots, \bar{N}$ and $i < k .$

It is obvious the transmissibility of nonlinear systems Equation (10) reduces to the traditional transmissibility of linear systems when $\bar{N} = 1$ as

T_{i, k}^{L} (j ω) = \frac{H_{(i, 1)} (j ω)}{H_{(k, 1)} (j ω)} = \frac{G_{(i, 1)} (j ω)}{G_{(k, 1)} (j ω)} i, k \in {1, 2, \dots, n} .

(11)

An important feature of NOFRFs is their independence from changes in the system input by a constant, as noted in Peng et al.¹⁹ This property ensures that the transmissibility of nonlinear systems, as defined in Equation (11), remains insensitive to such changes. This attribute is analogous to the property of transmissibility in linear systems.

Transmissibility function of nonlinear MDOF structural systems undergeneral inputs

The properties of transmissibility in nonlinear MDOF structural systems under harmonic inputs were investigated by Peng et al.,¹⁹ this does not account for interactions between input components at different frequencies in nonlinear dynamic systems. When the loading inputs are multitone, band-limited, or random signals, the system dynamics become more complicated compared to the single harmonic case due to the interactions between input components. Furthermore, as harmonic inputs are not representative of ambient loadings in most practical operational conditions, the applicability of transmissibility analysis for damage detection and localization must be expanded. Therefore, in this section, we broaden the scope of our investigation to encompass the transmissibility function of nonlinear MDOF structural systems under general inputs, with the goal of enhancing the practicality of our findings.

Nonlinear MDOF structural systems

A wide class of linear MDOF structural systems can be described by

M \overset{\cdot}{y} (t) + C \overset{\cdot}{y} (t) + Ky (t) = F (t)

(12)

where $y (t) = {[y_{1} (t) \dots y_{n} (t)]}^{T}$ and $F (t) = {[\overset{J - 1}{\overset{︷}{0 \dots 0}} u (t) \overset{n - J}{\overset{︷}{0 \dots 0}}]}^{T}$ are the system output displacements and input force vector, respectively. $u (t)$ is the input force applied on the $J$ th degree of freedom of the system with a broadband frequency range $[a, b]$ with $b > a \geq 0$ .

\begin{matrix} M = [\begin{matrix} m_{1} & 0 & \begin{matrix} \dots & 0 \end{matrix} \\ 0 & m_{2} & \begin{matrix} \dots & 0 \end{matrix} \\ \begin{matrix} ⋮ \\ 0 \end{matrix} & \begin{matrix} ⋮ \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ 0 \end{matrix} & \begin{matrix} ⋮ \\ m_{n} \end{matrix} \end{matrix} \end{matrix}] \\ C = [\begin{matrix} c_{1} + c_{2} & - c_{2} & \begin{matrix} 0 & \begin{matrix} \dots & 0 \end{matrix} \end{matrix} \\ - c_{2} & c_{2} + c_{3} & \begin{matrix} - c_{3} & \begin{matrix} ⋱ & ⋮ \end{matrix} \end{matrix} \\ \begin{matrix} 0 \\ ⋮ \\ 0 \end{matrix} & \begin{matrix} ⋱ \\ ⋱ \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ - c_{n - 1} \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} ⋱ & ⋮ \end{matrix} \\ \begin{matrix} c_{n - 1} + c_{n} & - c_{n} \end{matrix} \\ \begin{matrix} - c_{n} & c_{n} + c_{n + 1} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] \\ K = [\begin{matrix} k_{1} + k_{2} & - k_{2} & \begin{matrix} 0 & \begin{matrix} \dots & 0 \end{matrix} \end{matrix} \\ - k_{2} & k_{2} + k_{3} & \begin{matrix} - k_{3} & \begin{matrix} ⋱ & ⋮ \end{matrix} \end{matrix} \\ \begin{matrix} 0 \\ ⋮ \\ 0 \end{matrix} & \begin{matrix} ⋱ \\ ⋱ \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} ⋱ \\ - k_{n - 1} \\ 0 \end{matrix} & \begin{matrix} \begin{matrix} ⋱ & ⋮ \end{matrix} \\ \begin{matrix} k_{n - 1} + k_{n} & - k_{n} \end{matrix} \\ \begin{matrix} - k_{n} & k_{n} + k_{n + 1} \end{matrix} \end{matrix} \end{matrix} \end{matrix}] . \end{matrix}

$M$ , $C$ , and $K$ are the mass, damping, and stiffness matrices of the system, respectively. The nonlinear MDOF structural system considered in the present study are as shown in Figure 1, sharing the same structure as system Equation (12) but contains $\bar{L} (\bar{L} \geq 1)$ nonlinear components (nonlinear springs or damping) located between $L_{r} - 1$ and $L_{r}$ masses with $r = 1, 2, \dots, \bar{L}$ . Assume that the restoring force $Non F_{L_{r}}$ produced by the $r$ th ( $1 \leq r \leq \bar{L}$ ) nonlinear component can be approximated by a polynomial function.

\begin{matrix} Non F_{L_{r}} = \sum_{l = 1}^{\bar{N}} α_{(L_{r}, l)} {(y_{L_{r}} (t) - y_{L_{r} - 1} (t))}^{l} \\ + \sum_{l = 1}^{\bar{N}} β_{(L_{r}, l)} {({\overset{\cdot}{y}}_{L_{r}} (t) - {\overset{\cdot}{y}}_{L_{r} - 1} (t))}^{l} \end{matrix}

(13)

where $α_{(L_{r}, l)}$ and $β_{(L_{r}, l)}$ are the coefficients of the polynomial. Thus, the nonlinear MDOF structural systems can be described as

M \overset{\cdot\cdot}{y} (t) + C \overset{\cdot}{y} (t) + Ky (t) = F (t) + NF (t)

(14)

where

NF (t) = \sum_{r = 1}^{\bar{L}} n f_{r} (t)

(15)

and

n f_{r} (t) = {[\overset{L_{r} - 2}{\overset{︷}{0 \dots 0}} - Non F_{L_{r}} Non F_{L_{r}} \overset{n - L_{r}}{\overset{︷}{0 \dots 0}}]}^{T} r = 1, 2, \dots, \bar{L .}

(16)

Figure 1.

MDOF nonlinear system.

Output frequency range of Nonlinear MDOF structural systems

In linear systems, the output frequency range corresponds to the input frequency range. However, this is not the case in nonlinear systems. For instance, if the frequency range of input u(t) of the system Equation (14) is $[a, b] with b > a \geq 0$ , then it is known from Lang and Billings²² that the frequency ranges of each of the $n$ system outputs $y (t) = {[y_{1} (t) \dots y_{n} (t)]}^{T}$ can be determined as follows:

f_{Y} = f_{Y_{\bar{N}}} \cup f_{Y_{\bar{N} - (2 p^{*} - 1)}} f_{Y_{\bar{n}}} = {\begin{matrix} ⋃_{k = 0}^{i^{*} - 1} I_{k} when \frac{\bar{n} b}{(a + b)} - [\frac{\bar{n} a}{(a + b)}] < 1 \\ ⋃_{k = 0}^{i^{*}} I_{k} when \frac{\bar{n} b}{(a + b)} - [\frac{\bar{n} a}{(a + b)}] \geq 1 \end{matrix}

i^{*} = [\frac{\bar{n} a}{(a + b)}] + 1, I_{i^{*}} = (0, \bar{n} b - i^{*} (a + b))

I_{k} = (\bar{n} a - k (a + b), \bar{n} b - k (a + b)) for k = 0, \dots, i^{*} - 1

(17)

where $f_{Y}$ represents the nonnegative frequency range of the system outputs and $f_{Y_{\bar{n}}}$ represents the nonnegative frequency range produced by the $\bar{n}$ th order system nonlinearity. $[\cdot]$ means to take the integer part. This is much more complicated than the linear system case. Depending on the orders of nonlinearities that exist in the system, the particular value of $p^{*}$ in Equation (17) can be taken as 1, 2, …, [ $\bar{N} / 2$ ]. If the GFRFs of the system $H_{N - (2 i - 1)} (\cdot) = 0$ , for i = 1,2,…,q-1, and $H_{N - (2 q - 1)} (\cdot) \neq 0$ , then $p^{*} = q$ . For example, if the system has nonlinearities up to the third order, and the input frequency range is [7, 10] Hz, then N = 3, $p^{*} = q = 1$ , and from Equation (17), $f_{Y_{2}}$ and $f_{Y_{3}}$ can be obtained as $f_{Y_{2}} = [0, 3] \cup [14, 20] Hz$ and $f_{Y_{3}} = [4, 13] \cup [21, 30] Hz$ . Therefore, the output frequency range $f_{Y} = f_{Y_{\bar{N}}} \cup f_{Y_{\bar{N} - (2 p^{*} - 1)}} = [0, 3] \cup [4, 13] \cup$ $[14, 20] \cup [21, 30] Hz$ . In brief, due to its nonlinear nature, the system produces an output signal that contains a more extensive and diverse range of frequency components compared to the input signal.

Transmissibility function and associated properties of nonlinear MDOF structural systems under general inputs

The transmissibility function of nonlinear output frequency response functions between two consecutive masses i and i+1 at the frequency $\bar{ω}$ produced by the $\bar{n}$ th order system nonlinearity is defined in Lang et al.,¹⁸ as shown below

α_{i, i + 1}^{\bar{n}} (\bar{ω}) = \frac{G_{(i, \bar{n})} (j \bar{ω})}{G_{(i + 1, \bar{n})} (j \bar{ω})} (i = 1, \dots, n - 1; \bar{n} = 1, \dots, \bar{N}) .

(18)

Equation (18) for system Equation (14) provides the basis for a series of fundamental propositions concerning the transmissibility of the NOFRF in response to general inputs. These propositions can be summarized as follows:

Proposition 1: Outside the DOFs connected to nonlinear components ( $1 \leq i < L_{1} - 1 and L_{\bar{L}} \leq i < n; \bar{L} \geq 2$ ), if the system output response at the frequency $\bar{ω}$ is produced by the $(\bar{n} = 2, \dots, n_{i}$ )th order system nonlinearities where $n_{i}$ is the nonlinear order associated with the ith system output with $2 \leq n_{i} < \bar{N}$ , then

α_{i, i + 1} (\bar{ω}) = \frac{G_{(i, 2)} (j \bar{ω})}{G_{(i + 1, 2)} (j \bar{ω})} = \frac{G_{(i, 3)} (j \bar{ω})}{G_{(i + 1, 3)} (j \bar{ω})} = \dots = \frac{G_{(i, n_{i})} (j \bar{ω})}{G_{(i + 1, n_{i})} (j \bar{ω})}

(19)

where

\begin{matrix} α_{i, i + 1} (\bar{ω}) = \\ {\frac{k_{i + 1} + j \bar{ω} c_{i + 1}}{- m_{i} {\bar{ω}}^{2} + j \bar{ω} (c_{i} + c_{i + 1}) + k_{i} + k_{i + 1} - (j \bar{ω} c_{i} + k_{i}) α_{i - 1, i}^{\bar{n}} (\bar{ω})} |}_{\bar{ω} \in f_{Y_{\bar{n}}}} . \end{matrix}

Proof 1 of Proposition 1: see Appendix 1.

This proposition implies that the transmissibility function of the NOFRF between two consecutive masses i and i+1 remains constant for the frequency $\bar{ω}$ generated by the nonlinearities of the $(\bar{n} = 2, \dots, n_{i}$ )th order system. Furthermore, this proposition holds true regardless of the number or position of the input loads. The proof of this statement is also available in Appendix 1.

Proposition 2: Outside the DOFs connected to nonlinear components ( $1 \leq i < L_{1} - 1 and L_{\bar{L}} \leq i < n; \bar{L} \geq 2$ ), if the system output response at the frequency $\bar{ω}$ is produced by the $(\bar{n} = 2, \dots, n_{i}$ )th order system nonlinearities with $1 < n_{i} < \bar{N}$ , then

α_{i, i + 1}^{F_{1}} (\bar{ω}) = \frac{Y_{i}^{F_{1}} (j \bar{ω})}{Y_{i + 1}^{F_{1}} (j \bar{ω})} = \frac{Y_{i}^{F_{2}} (j \bar{ω})}{Y_{i + 1}^{F_{2}} (j \bar{ω})} = α_{i, i + 1}^{F_{2}} (\bar{ω}) .

(20)

Equation (20) uses the superscripts $F_{1}$ and $F_{2}$ to denote two different loading conditions that have the same frequency range for the input signal, but differ in the level of input excitation.

Proof of Proposition 2: According to Proposition 1, the following relationship holds

\begin{matrix} α_{i, i + 1} (\bar{ω}) = \frac{G_{(i, 2)} (j \bar{ω}) U_{2} (j \bar{ω})}{G_{(i + 1, 2)} (j \bar{ω}) U_{2} (j \bar{ω})} = \frac{G_{(i, 3)} (j \bar{ω}) U_{3} (j \bar{ω})}{G_{(i + 1, 3)} (j \bar{ω}) U_{3} (j \bar{ω})} \\ = \dots = \frac{G_{(i, n_{i})} (j \bar{ω}) U_{n_{i}} (j \bar{ω})}{G_{(i + 1, n_{i})} (j \bar{ω}) U_{n_{i}} (j \bar{ω})} (1 < n_{i} < \bar{N}) . \end{matrix}

(21)

Therefore,

\begin{matrix} α_{i, i + 1} (\bar{ω}) = \frac{Y_{(i, 2)} (\bar{ω})}{Y_{(i + 1, 2)} (\bar{ω})} = \frac{Y_{(i, 3)} (\bar{ω})}{Y_{(i + 1, 3)} (\bar{ω})} = \dots = \\ \frac{Y_{(i, n_{i})} (\bar{ω})}{Y_{(i + 1, n_{i})} (\bar{ω})} (1 < n_{i} < \bar{N}) \end{matrix}

(22)

and

\begin{matrix} \frac{Y_{i} (\bar{ω})}{Y_{i + 1} (\bar{ω})} = \frac{Y_{(i, 2)} (j \bar{ω}) + \dots + Y_{(i, n_{i})} (j \bar{ω})}{Y_{(i + 1, 2)} (j \bar{ω}) + \dots + Y_{(i + 1, n_{i})} (j \bar{ω})} \\ = α_{i, i + 1} (\bar{ω}) (1 < n_{i} < \bar{N}) . \end{matrix}

(23)

The validity of Proposition 2 is ascertained by the fact that $α_{i, i + 1} (\bar{ω})$ is solely dependent on the linear parameters of the system proven in Proposition 1, which leads to the invariance of the transmissibility functions under the two loading scenarios $F_{1}$ and $F_{2}$ . It should be noted that the applicability of Propositions 1 and 2 is limited to MDOF systems with at least two nonlinear components $\bar{L} \geq 2$ . Additional details are available in Appendix 2.

Proposition 3: Outside the DOFs connected to only single nonlinear component ( $J < L_{1}; 1 \leq i \leq J - 1 or L_{1} \leq i < n$ ), or $(J \geq L_{1}; 1 \leq i < L_{1} - 1 or J \leq i < n),$ if the system output response at the linear output frequency $ω \in [a, b], 0 \leq a < b$ , then

α_{i, i + 1}^{F_{1}} (ω) = \frac{Y_{i}^{F_{1}} (j ω)}{Y_{i + 1}^{F_{1}} (j ω)} = \frac{Y_{i}^{F_{2}} (j ω)}{Y_{i + 1}^{F_{2}} (j ω)} = α_{i, i + 1}^{F_{2}} (ω) .

(24)

Proof 3 of Proposition 3: see Appendix 3.

The proposition asserts that the output-based transmissibility function remains invariant in the presence of two loading scenarios characterized by identical input frequency bands $[a, b]$ but differing input excitation levels.

Damage detection and localization methodology

Based on Proposition 2, a new damage indicator, as presented in Equation (26), can be proposed to detect the existence and localize nonlinear components in MDOF systems. This methodology involves a five-step procedure that can be summarized as the following diagram.

Simulation study

In order to validate the effectiveness of the proposed methodology, a 10 DOFs Mass-spring-damper system has been simulated and coded in commercial software Matlab^® shown in Figure 2. Damping is assumed to be proportional damping, for example, $C = μ K$ .

Figure 2.

Ten DOFs mass-spring-damper system.

The detailed parameters are assigned as follows:

m_{1} = m_{2} = \dots = m_{10} = 1

(25)

k_{1} = k_{2} = k_{3} = k_{4} = k_{5} = k_{10} = k_{11} = 3.6 \times 10^{4}

(26)

k_{6} = k_{7} = k_{8} = 0.8 k_{1} k_{9} = 0.9 k_{1}

(27)

μ = 0.01; C = μ K .

(28)

Equation (13) takes into account a level of nonlinearity of up to three, $\bar{N} = 3$

α_{(L_{r}, 2)} = 0.8 \cdot k_{1}^{2} α_{(L_{r}, 3)} = 0.4 \cdot k_{1}^{3} β_{(L_{r}, 2)} = 0 β_{(L_{r}, 3)} = 0

(29)

where $r = 1, 2, \dots, \bar{L}$ , $\bar{L}$ is the number of nonlinear components in the above system.

The general input¹⁷ is given here:

u (t) = A \cdot \frac{3 (\sin (2 π \cdot 10 t) - \sin (2 π \cdot 7 t))}{2 π t} A = 1; 10

t = - 5.115 : 0.01 : 5.125 s .

(30)

Equation (30) serves to generate an input signal with a bandwidth of [7, 10] Hz at a sampling frequency of 100 Hz. $A$ in the equation represents the excitation level. When $A$ is assigned a value of 1, the time-domain and frequency-domain representations of the input signal are illustrated in Figures 3 and 4, respectively.

Figure 3.

Time history of the general input.

Figure 4.

The spectrum of the general input.

Identification of single nonlinear component for MDOF system

Case study 1 : $\bar{L} = 1$ considering that two inputs $F_{1}$ and $F_{2}$ are applied to the same mass.

Input $F_{1}$ with an amplitude of $A = 1$ and Input $F_{2}$ with an amplitude of $A = 10$ are applied on the 3rd mass. The nonlinearity is due to the nonlinear spring connecting mass 5 and mass 6, that is $L_{1} = 6$ .

Assuming that the system is linear and that the input is applied to the 3^rd mass with an amplitude of $A = 1$ , the response spectrum is obtained by analyzing the dynamic response of the system for all masses using the fourth-order Runge–Kutta method.²⁴

Given that the system under consideration is linear, the output frequency range matches that of the input, spanning from 7 to 10 Hz, as illustrated in Figure 5. However, incorporating two nonlinear springs into the system results in a remarkable change in the output spectrum, as shown in Figure 6. The nonlinearity causes a partial transfer of output energy from the original frequency range of [7, 10] Hz to both lower-frequency regions ranging from [0, 6] Hz and higher-frequency regions (11, 50] Hz. This redistribution of energy across the frequency spectrum is a significant demonstration of the impact of nonlinearities on the behavior of a system. Equation (25) is employed to calculate the transmissibility function for each of the two successive masses at linear frequency range $ω_{1} \leq ω_{2} \in [7, 10]$ Hz, which serves to formulate the damage indicator in Equation (26). In this case, $ω_{1} = 8$ and $ω_{2} = 10$ Hz are selected.

Figure 5.

Output spectra of the 10-DOF linear system (in logarithms).

Figure 6.

Output spectra of the 10-DOF nonlinear system (in logarithms).

From Equation (28), any damage indicator $D I_{i}$ ,for which $J \leq i < L_{1}$ , would be estimated as a positive value. As demonstrated in Figure 7, the damage region is identified as from $D I_{3}$ to $D I_{5}$ due to their nonzero values. Consequently, it can be easily inferred that the two nonlinear components are positioned between the 5th and 6th masses.

Figure 7.

Damage indicator based on [8, 10] Hz.

Identification of multiple nonlinear components for MDOF system

Case study 2 : $\bar{L} = 2$ considering that two inputs $F_{1}$ and $F_{2}$ are applied to two different masses.

The 3rd and the 7th masses are subjected to input forces $F_{1}$ and $F_{2}$ , with amplitudes $A = 1$ and $A = 10$ , respectively. The analysis considers $\bar{L} = 2$ nonlinear components, where the associated nonlinear springs are placed between the 5th and 6th mass, and the 8th and 9th mass, respectively, such that $L_{1} = 6$ , $L_{2} = 9$ . As evidenced by the substantial nonlinear spectral energy within the range shown in Figure 8, the frequency range of interest for the analysis is chosen to be [24, 26] Hz.

Figure 8.

Damage indicator based on [24, 26] Hz.

According to Equation (27), any damage indicator $D I_{i}$ , for which $L_{1} - 1 \leq i < L_{\bar{L}}$ , would be estimated as a positive value. As indicated in Figure 7, the damage region is identified as the interval between $D I_{5}$ to $D I_{8}$ due to their nonzero values. Therefore, it can be inferred that the nonlinear components are located between the 5th and 9th masses.

Case study 3 : $\bar{L} = 3$ Considering the input $F_{1}$ as multipoints loading and $F_{2}$ as single point loading.

In this case, $F_{1}$ acts on the 2nd mass, 4th mass, and 8th mass at the same time, whereas $F_{2}$ acts only on the 5th mass. The number of nonlinear components is set to 3, $\bar{L} = 3$ and the nonlinear springs are applied between the 3rd mass and the 4th mass ( $L_{1} = 4$ ), the 4th mass and the 5th mass ( $L_{2} = 5$ ) and the 5th mass and the 6th mass ( $L_{3} = 6$ ), respectively. The selected frequency range remains [24, 26] Hz.

The only region with nonzero damage indicators, as depicted in Figure 9, spans from $D I_{3}$ to $D I_{5}$ . Based on Equation (28), it can be inferred that $L_{1} = 4$ , $L_{3} = 6$ , indicating that the nonlinear components are situated between the 3rd mass and the 6th mass. This finding is consistent with the information provided regarding the damage scenario.

Figure 9.

Damage indicator based on [24, 26] Hz

The output-based transmissibility function at frequency $\bar{ω}$ produced by the $\bar{n}$ th order system nonlinearities is able to detect and localize the nonlinear components in nonlinear MDOF systems. This has been demonstrated by the above simulation studies, confirming its effectiveness. One of the key advantages of this methodology is its independence from the location and number of inputs applied to the system if $\bar{L} \geq 2$ , which has a great potential impact on the practical application of engineering infrastructures. One limitation of the proposed methodology is that it necessitates prior knowledge of the number of nonlinear components in the system to determine the suitable frequency range for analysis, as reflected in the policies between Equations (27) and (29). In practice, however, MDOF systems cannot serve as a substitute for complex structures such as continuous beam structures. The finite element method (FEM) is a commonly used alternative for simulating continuous structures. In addition, nonlinearity present in a damaged element can propagate to its neighboring elements, making the analysis of complex continuous systems challenging. On the contrary, this results in a favorable aspect of the proposed methodology, which avoids a scenario characterized by a sole nonlinear component.

Identification of a single crack on a beam structure in FEM

Consider a Euler–Bernoulli beam element shown in Figure 10. The displacement along the $x$ direction is ignored.

Figure 10.

Two nodes beam element.

There are four DOFs in the beam element, define them as $δ^{e} = {[v_{i}, θ_{i}, v_{i + 1}, θ_{i + 1}]}^{T}$ in which $v_{i}$ and e the bending and rotation displacement at the ith node, respectively.

The element mass matrix $M^{e}$ , the element damping matrix $C^{e}$ , and the element stiffness matrix $K^{e}$ have the following form

\begin{matrix} M^{e} = \frac{ρ Al}{420} [\begin{matrix} 156 & 22 l & 54 & - 13 l \\ 22 l & 4 l^{2} & 13 l & - 3 l^{2} \\ 54 & 13 l & 156 & - 22 l \\ - 13 l & - 3 l^{2} & - 22 l & 4 l^{2} \end{matrix}] \\ M^{e} = \frac{ρ Al}{420} [\begin{matrix} 156 & 22 l & 54 & - 13 l \\ 22 l & 4 l^{2} & 13 l & - 3 l^{2} \\ 54 & 13 l & 156 & - 22 l \\ - 13 l & - 3 l^{2} & - 22 l & 4 l^{2} \end{matrix}], \\ C^{e} = \frac{cl}{420} [\begin{matrix} 156 & 22 l & 54 & - 13 l \\ 22 l & 4 l^{2} & 13 l & - 3 l^{2} \\ 54 & 13 l & 156 & - 22 l \\ - 13 l & - 3 l^{2} & - 22 l & 4 l^{2} \end{matrix}] \end{matrix}

\begin{matrix} C^{e} = \frac{cl}{420} [\begin{matrix} 156 & 22 l & 54 & - 13 l \\ 22 l & 4 l^{2} & 13 l & - 3 l^{2} \\ 54 & 13 l & 156 & - 22 l \\ - 13 l & - 3 l^{2} & - 22 l & 4 l^{2} \end{matrix}], \\ K^{e} = \frac{EI}{l^{3}} [\begin{matrix} 12 & 6 l & - 12 & 6 l \\ 6 l & 4 l^{2} & - 6 l & 2 l^{2} \\ - 12 & - 6 l & 12 & - 6 l \\ 6 l & 2 l^{2} & - 6 l & 4 l^{2} \end{matrix}] \\ K^{e} = \frac{EI}{l^{3}} [\begin{matrix} 12 & 6 l & - 12 & 6 l \\ 6 l & 4 l^{2} & - 6 l & 2 l^{2} \\ - 12 & - 6 l & 12 & - 6 l \\ 6 l & 2 l^{2} & - 6 l & 4 l^{2} \end{matrix}] . \end{matrix}

where $ρ$ is the density of the beam, $A$ is the area of the cross-section, $c$ is the damping coefficient, $E$ is the young’s modulus, and $I$ is the moment of inertia of the beam.

If a beam contains a crack, there will be a nonlinear phenomenon in its vibration. A simplified model of open crack was proposed by Sinha et al.,²⁵ which can introduce the damage for the beam elements.

The flexural rigidity close to the crack, $E I_{e} (ξ)$ , is given by

E I_{e} (ξ) = {\begin{matrix} E I_{0} - E (I_{0} - I_{c}) \frac{ξ - ξ_{1}}{ξ_{c} - ξ_{1}} & ξ_{1} \leq ξ \leq ξ_{c} \\ E I_{0} - E (I_{0} - I_{c}) \frac{ξ_{2} - ξ}{ξ_{2} - ξ_{c}} & ξ_{c} \leq ξ \leq ξ_{2} \end{matrix}

(31)

where $I_{0} = ω d^{3} / 12$ and $I_{c} = ω {(d - d_{c})}^{3} / 12$ are the second moment of the areas of the undamaged beam and at the cracked element. $l$ is the length of the $e$ th element, $ξ_{c}$ is the location of the crack within the $e$ th element, $ξ_{1} = ξ_{c} - l_{c}$ and $ξ_{2} = ξ_{c} + l_{c}$ are the position on either sides of crack where the stiffness reduction begins. It has been estimated that $I_{c} = 1.5 d_{c}$ where $d_{c}$ is the depth of the crack. Assume the stiffness reduction caused by an open crack falls within a single element, and then the stiffness matrix $K_{c}^{e}$ of the cracked element can be written as:

K_{c}^{e} = K^{e} - K_{e, c}

(32)

where $K_{e, c}$ is the reduction in the stiffness matrix.

The governing equation in finite element form can be written as:

{\begin{matrix} M \overset{\cdot\cdot}{δ} + C \overset{\cdot}{δ} + K δ = F & crack is close \\ M \overset{\cdot\cdot}{δ} + C \overset{\cdot}{δ} + (K - K_{c}) δ = F & crack is open \end{matrix}

(33)

A 10-element (11 nodes) Euler–Bernoulli cantilever beam is employed and simulated in the commercial software MATLAB, shown in Figure 11. The dimensions of the beam are set as 1.00 × 0.015 × 0.015 m³ (Length × Width × Depth). It has a mass density of 7680 kg/m³, an elastic modulus of 208 GPa, and a damping ratio of 0.001. The beam is subjected to a breathing crack of half the beam’s depth, precisely located at the midpoint of the 6th element. The broadband input, given by Equation (34), is considered under two scenarios. Figure 11(a) illustrates a single load acting on DOF $v_{9}$ , and Figure 11(b) depicts two loads acting on DOF $v_{3}$ and DOF $v_{10}$ .

Figure 11.

The FEM model of a cantilever beam. (a) Single load at node 10. (b) Two loads at node 4 and node 11.

f_{1} (t) = A \cdot \frac{3 (\sin (2 π \cdot 10 t) - \sin (2 π \cdot 5 t))}{2 π t} A = 1; 10

f_{2} (t) = A \cdot \frac{3 (\sin (2 π \cdot 10 t) - \sin (2 π \cdot 5 t))}{2 π t} A = 1; 10

t = - 5.115 : 0.01 : 5.125 s .

(34)

Utilizing the fourth-order Runge–Kutta method, the dynamic response of the cracked beam model is obtained. To assess the existence of the crack, the transmissibility outside the driving frequency range, [12, 14] Hz, is analyzed. Figure 12 demonstrates the identification outcomes. The position of the crack, located between the DOFs $v_{5}, v_{6}$ or $θ_{5}, θ_{6}$ within element 6, is predefined, see Figure 11. The damage indicators for elements 6 and 7 are recognized as high-risk region by applying a simple statistical threshold mean(DI values) + std(DI values), denoted by red dash-dot lines, as shown in Figure 12(a). Similarly, the damage indicator obtained from the rotational displacement, as shown in Figure 12(b), also identifies the crack located at the 6th element. Furthermore, the aforementioned consistent outcome is also confirmed for the two-load scenario, as presented in Figure 12(c) and (d).

Figure 12.

Damage indicator of the FEM model. (a) Damage indicator of [12, 14] Hz based on bending displacement under a single load. (b) Damage indicator of [12, 14] Hz based on rotation displacement under a single load. (c) Damage indicator of [12, 14] Hz based on bending displacement under two loads. (d) Damage indicator of [12, 14] Hz based on rotation displacement under two loads.

The effectiveness of the proposed methodology in identifying a single nonlinear component in a beam FEM model is evident. This can be attributed to the beam element’s higher number of degrees of freedom compared to MDOF systems, which enables the propagation of nonlinearity from the crack to other elements. This observation confirms our conclusion that the number and placement of loading inputs have no effect on NOFRF transmissibility ( $\bar{n} \geq 2$ ) within the nonlinear frequency range in MDOF systems.

Experimental study

The focus of this section is to localize the breathing crack behaving nonlinearly in a steel-beam structure by the proposed damage indicator based on the NOFRF transmissibility function through the use of an innovative distributed fiber optical sensing-based technique. In contrast to the conventional transducers that suffer from single point sensing, making it challenging to cover most vibration data and impossible to precisely localize the damages such as cracks. Instead, distributed fiber optics, together with LUNA ODiSI-B Optical interrogator,²⁶ is adopted to acquire vibrations in terms of dynamic strain, which aids in offering a dense measurement solution, readers are referred to References^27–29 for more details.

The ODiSI-B Series integrator is designed by American company Luna Innovations, based on the Rayleigh scattering, and specifically engineered to embrace the testing challenges of advanced materials and systems. It offers high-speed, fully distributed strain and temperature measurements with high spatial resolution, with hundreds of sensing locations per meter on a single fiber, all of which are interrogated at frequencies up to 250 Hz.

The most effective methodology for generating breathing cracks is through fatigue-inducing experiments. However, in this case study, a more efficient and time-saving method for generating “breathing crack” has been employed. A rectangular block notch measuring 40 × 3 × 2 mm (length × width × thickness) was removed, and a steel block of the same dimensions was filled in with adhesive, as illustrated in Figure 13.

Figure 13.

Configuration of beam structure with crack and distributed fiber optics.

The investigation is performed on a clamped-clamped steel beam with a dimension of 1.5 × 0.04 × 0.015 m. The installation configuration of distributed fiber optics is denoted by the red dashed line in the diagram above. A distributed fiber-optic sensor with an effective sensing area of 1.28 m is attached to the surface of the beam. The crack is located 700 mm to the left of the initial effective section, as illustrated in Figure 13. With a sampling frequency of 250 Hz and a sensing spacing of 2.6125 mm, the effective sensing region encompasses 497 sensors. The crack, as shown in Figure 14, is approximately at the 267th sensor.

Figure 14.

Crack in the beam.

Figures 15 and 16 demonstrate the configuration of the corresponding experimental system. A broadband chip input signal is generated by NI-MyDAQ, which is transmitted to the power amplifier and reaches the mini-Shake to generate vibrations into the beam. The generated dynamic responses of the beam are then collected via the attached distributed fiber optics, demodulated by LUNA OSiDI-B integrator, and saved in PC2. A zoom-in view of glued fiber-optics is presented in Figure 17.

Figure 15.

System configuration.

Figure 16.

Configuration of the experimental system.

Figure 17.

Zoom-in view of fiber optics.

To illustrate the feasibility and effectiveness of the proposed methodology, the operational scenario (the input on the left side of the beam, and its configuration is shown in Figure 18) has been considered in the experimental verification test.

Figure 18.

Configuration of scenario 1.

Chirp signals with a bandwidth of [25, 45] Hz with two different excitation levels A and B (B/A = 5) are considered inputs, as displayed in Figure 19 and 20 in the time domain and frequency domain, respectively, with unit amplitude.

Figure 19.

A chirp input signal in time domain.

Figure 20.

A chirp input signal in frequency domain.

The input signal is generated by the minishaker positioned 570 mm to the left of the fiber optics, which corresponds to the 219th sensor. The crack is located 700 mm to the left of the fiber optics, corresponding to the 267th sensor. Under excitation level A, the strain responses of all 497 sensors, including the distributed fiber optics, in the time domain are presented in Figure 21 for the 20-s duration of the chirp input.

Figure 21.

Strain responses of 497 sensors in time domain.

To minimize the effects of noise, the test is repeated five times. Subsequently, the output-based transmissibility function is computed based on the spatially dense dynamic responses of the cracked beam. The frequency range of interest for the damage indicator is selected as [120, 140] Hz, with higher values indicating a greater likelihood of structural damage. Damage indicators are computed and presented in Figure 22, which identifies the high-risk region within the beam element range of 240–270 due to the presence of nonlinearities by setting a simple threshold. This finding is in agreement with the observed damage scenario, where the crack is located near the 267th beam element within the aforementioned high-risk area. Moreover, it is noteworthy that the employment of the dense measurement technique allows for enhanced localization of the crack.

Figure 22.

Damage indicator.

Conclusion

This article presents a novel methodology for detecting and localizing damage with nonlinear features in MDOF systems subject to general inputs. The methodology leverages the intrinsic characteristics of the transmissibility of nonlinear systems defined based on the NOFRF concept, and exploits the transmissibility between adjacent masses over the frequencies generated by the system nonlinearities. An output-based transmissibility function is then formulated and utilized to identify single and/or multiple nonlinearly damaged components in the system. The proposed methodology has been validated through relevant simulation studies, demonstrating its feasibility and effectiveness. The premilitary experimental investigation on a steel beam structure with a breathing crack using distributed fiber-optic sensors has generated a damage indicator with acceptable fluctuations, demonstrating the potential of the proposed methodology in the identification of damage on practical engineering structures.

Yet, the proposed methodology has certain limitations: (1) The propositions are based on chain-like MDOF systems that do not account for coupling effects between DOFs and the complexity of degrees of freedom in a real continuous system. In reality, DOFs are often internally linked, which can lead to the propagation of nonlinearity from one specific location to surrounding elements. Therefore, Proposition 3 is not very applicable to complex engineering structures when considering only a single nonlinear component. Although the presented FEM and experimental studies verify the feasibility of detecting single nonlinearity based on Proposition 2 rather than Proposition 3, a detailed and rigorous theoretical study will be continued; (2) This experimental case study demonstrates the process of identifying a single nonlinear component, such as a breathing crack, using distributed fiber optics. This supports our proposed methodology based on the concept of the NOFRF-based transmissibility function under general inputs. However, determining a suitable and reliable threshold, such as statistical thresholding based on noise level, for identifying high-risk regions requires further research and discussion. (3) The proposed methodology is currently only able to localize the entire region covered by nonlinear components. Precisely locating the individual nonlinear components remains challenging.

This study has preliminarily established a framework for nonlinear damage identification and localization of MDOF systems based on NOFRFs under general inputs. Further investigation will focus on addressing the summarized limitations to develop a robust and more generalized solution applicable to complex engineering structures.

Footnotes

Appendix 1

The first row of governing equations of the system Equation (14) can be written as

(A1.1)

\begin{matrix} m_{1} {\overset{\cdot\cdot}{y}}_{1} (t) + (c_{1} + c_{2}) {\overset{\cdot}{y}}_{1} (t) - c_{2} {\overset{\cdot}{y}}_{2} (t) + (k_{1} + k_{2}) y_{1} (t) \\ - k_{2} y_{2} (t) = 0 . \end{matrix}

According to the Volterra series theory of nonlinear systems and the definition of the NOFRFs, the ith output nonlinear SIMO system Equation (14) can be represented as

(A1.2)

\begin{matrix} y_{i} (t) = \sum_{\bar{n} = 1}^{n_{i}} y_{i, \bar{n}} (t) = \frac{1}{2 π} \sum_{\bar{n} = 1}^{n_{i}} \int_{- \infty}^{\infty} G_{(i, \bar{n})} (j ω) U_{\bar{n}} (j ω) e^{j ω t} d ω \end{matrix}

where $n_{i}$ is the nonlinear order associated with the ith system output with $2 \leq n_{i} < \bar{N}$ . Assume that the considered nonlinear order is identical for all system outputs, that is $n_{1} = \dots = n_{n} .$

Substituting Equations (A1.2) into (A1.1) yields

\begin{matrix} m_{1} \frac{1}{2 π} \sum_{\bar{n} = 1}^{n_{1}} \int_{- \infty}^{\infty} {(j ω)}^{2} G_{(1, \bar{n})} (j ω) U_{\bar{n}} (j ω) e^{j ω t} d ω + (c_{1} + c_{2}) \frac{1}{2 π} \\ \sum_{\bar{n} = 1}^{n_{1}} \int_{- \infty}^{\infty} (j ω) G_{(1, \bar{n})} (j ω) U_{\bar{n}} (j ω) e^{j ω t} d ω \end{matrix}

\begin{matrix} - c_{2} \frac{1}{2 π} \sum_{\bar{n} = 1}^{n_{1}} \int_{- \infty}^{\infty} (j ω) G_{(2, \bar{n})} (j ω) U_{\bar{n}} (j ω) e^{j ω t} d ω + (k_{1} + k_{2}) \frac{1}{2 π} \\ \sum_{\bar{n} = 1}^{n_{1}} \int_{- \infty}^{\infty} G_{(1, \bar{n})} (j ω) U_{\bar{n}} (j ω) e^{j ω t} d ω \end{matrix}

(A1.3)

- k_{2} \frac{1}{2 π} \sum_{\bar{n} = 1}^{n_{1}} \int_{- \infty}^{\infty} G_{(2, \bar{n})} (j ω) U_{\bar{n}} (j ω) e^{j ω t} d ω = 0

which implies that

\begin{matrix} m_{1} \frac{1}{2 π} \int_{\bar{ω} \in f_{Y_{\bar{n}}}} {(j \bar{ω})}^{2} G_{(1, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω}) e^{j \bar{ω} t} d \bar{ω} + (c_{1} + c_{2}) \frac{1}{2 π} \\ \int_{\bar{ω} \in f_{Y_{\bar{n}}}} (j \bar{ω}) G_{(1, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω}) e^{j \bar{ω} t} d \bar{ω} \end{matrix}

\begin{matrix} - c_{2} \frac{1}{2 π} \int_{\bar{ω} \in f_{Y_{\bar{n}}}} (j \bar{ω}) G_{(2, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω}) e^{j \bar{ω} t} d \bar{ω} + (k_{1} + k_{2}) \frac{1}{2 π} \\ \int_{\bar{ω} \in f_{Y_{\bar{n}}}} G_{(1, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω}) e^{j \bar{ω} t} d \bar{ω} \end{matrix}

(A1.4)

- k_{2} \frac{1}{2 π} \int_{\bar{ω} \in f_{Y_{\bar{n}}}} G_{(2, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω}) e^{j \bar{ω} t} d \bar{ω} = 0 (\bar{n} = 1, \dots, n_{1})

where $f_{Y_{\bar{n}}}$ represents the output frequency range contributed by the $\bar{n}$ th ( $\bar{n} = 2, \dots, n_{1}$ ) order system nonlinearity, while $f_{Y_{\bar{n}}}$ ( $\bar{n} = 1$ ) belongs to linear output frequency range.

Considering the general nature of $U_{\bar{n}} (j \bar{ω})$ , Equation (A1.4) implies that

\begin{matrix} {- m_{1} {\bar{ω}}^{2} G_{(1, \bar{n})} (j \bar{ω}) |}_{\bar{ω} \in f_{Y_{\bar{n}}}} + {j \bar{ω} (c_{1} + c_{2}) G_{(1, \bar{n})} (j \bar{ω}) |}_{\bar{ω} \in f_{Y_{\bar{n}}}} \\ - {j \bar{ω} c_{2} G_{(2, \bar{n})} (j \bar{ω}) |}_{\bar{ω} \in f_{Y_{\bar{n}}}} + {(k_{1} + k_{2}) G_{(1, \bar{n})} (j \bar{ω}) |}_{\bar{ω} \in f_{Y_{\bar{n}}}} \end{matrix}

(A1.5)

- {k_{2} G_{(2, \bar{n})} (j \bar{ω}) |}_{\bar{ω} \in f_{Y_{\bar{n}}}} = 0 (\bar{n} = 1, \dots, n_{1}) .

Substitute NOFRF transmissibility Equation (18) into Equation (A1.5). Therefore,

(A1.6)

α_{1, 2}^{\bar{n}} (\bar{ω}) = {\frac{k_{2} + j \bar{ω} c_{2}}{- m_{1} {\bar{ω}}^{2} + j \bar{ω} (c_{1} + c_{2}) + k_{1} + k_{2}} |}_{\bar{ω} \in f_{Y_{\bar{n}}}} (\bar{n} = 1, \dots, n_{1}) .

Note that here $\bar{ω}$ is belonging to the both linear ( $\bar{n} = 1$ ) and nonlinear frequency range ( $\bar{n} = 2, \dots, n_{1}$ ).

Similarly, the NOFRF transmissibility for the $i$ th row ( $i < min (L_{r} - 1, L_{r}, J) or i > max (L_{r} - 1, L_{r}, J), r = 1, 2, \dots, \bar{L}$ ) of the system matrix can be yielded as below:

α_{i, i + 1}^{\bar{n}} (\bar{ω}) = {\frac{k_{i + 1} + j \bar{ω} c_{i + 1}}{- m_{i} {\bar{ω}}^{2} + j \bar{ω} (c_{i} + c_{i + 1}) + k_{i} + k_{i + 1} - (j \bar{ω} c_{i} + k_{i}) α_{i - 1, i}^{\bar{n}} (\bar{ω})} |}_{\bar{ω} \in f_{Y_{\bar{n}}}}

(A1.7)

(i < min (L_{r} - 1, L_{r}, J) or i > max (L_{r} - 1, L_{r}, J); \bar{n} = 1, \dots, n_{i}) .

From the general expression for the NOFRF transmissibility Equation (A1.7) where the masses don’t connect with nonlinear components and are not subject to the input, it can be known that NOFRF transmissibility only depends on its linear physical parameters, independent of nonlinearities and input.

The NOFRF transmissibility between mass $L_{1} - 1$ and mass $L_{1}$ can be obtained as Equation (A1.8)

\begin{matrix} α_{L_{1} - 1, L_{1}}^{\bar{n}} (\bar{ω}) = {\frac{k_{L_{1}} + j \bar{ω} c_{L_{1}} - \frac{P_{\bar{n}} (Non F_{L_{1}} (j \bar{ω}))}{G_{(L_{1}, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω})}}{- m_{L_{1} - 1} {\bar{ω}}^{2} + j \bar{ω} (c_{L_{1} - 1} + c_{L_{1}}) + k_{L_{1} - 1} + k_{L_{1}} - (j \bar{ω} c_{L_{1} - 1} + k_{L_{1} - 1}) α_{L_{1} - 2, L_{1} - 1}^{\bar{n}}} |}_{\bar{ω} \in f_{Y_{\bar{n}}}} \end{matrix}

(A1.8)

(\bar{n} = 1, \dots, n_{L_{1} - 1})

where

(A1.9)

P_{1} (Non F_{L_{1}}) + P_{2} (Non F_{L_{1}}) + \dots + P_{n_{L_{1} - 1}} (Non F_{L_{1}}) = Non F_{L_{1}} .

Using $P_{1} (Non F_{L_{1}}), P_{2} (Non F_{L_{1}}), \dots, P_{n_{L_{1} - 1}} (Non F_{L_{1}})$ to denote the corresponding nonlinear restoring force for each nonlinear order. And they satisfy the given condition of Equation (A1.9).

The following relationship holds for the NOFRF transmissibility where the mass is subjected to the input,

(A1.10)

\begin{matrix} α_{J, J + 1}^{\bar{n}} (\bar{ω}) \\ = {\begin{matrix} {\frac{k_{J + 1} + j \bar{ω} c_{J + 1}}{- m_{J} {\bar{ω}}^{2} + j \bar{ω} (c_{J} + c_{J + 1}) + k_{J} + k_{J + 1} - (j \bar{ω} c_{J} + k_{J}) α_{J - 1, J}^{\bar{n}} - \frac{U (j \bar{ω})}{y_{J} (j \bar{ω})}} |}_{\bar{ω} \in f_{Y_{\bar{n}}}} (\bar{n} = 1) \\ {\frac{k_{J + 1} + j \bar{ω} c_{J + 1}}{- m_{J} {\bar{ω}}^{2} + j \bar{ω} (c_{J} + c_{J + 1}) + k_{J} + k_{J + 1} - (j \bar{ω} c_{J} + k_{J}) α_{J - 1, J}^{\bar{n}}} |}_{\bar{ω} \in f_{Y_{\bar{n}}}} (\bar{n} = 2, \dots, n_{J}) \end{matrix} . \end{matrix}

One interesting thing needs to be mentioned that the establishment of $U (j \bar{ω}) = 0$ when $\bar{n} = 2, \dots, n_{J} .$ , thus leading to the vanishing of the term $- \frac{U (j \bar{ω})}{y_{J} (j \bar{ω})}$ linked to the loading input at a nonlinear frequency range $\bar{ω} \notin [a, b]$ . As a result, the relationship in Equation (A1.10) draws a significant conclusion that NOFRF transmissibility at a nonlinear frequency range is irrelevant to the number and locations of loading input that is when $\bar{n} \geq 2$ .

It can be known that Equation (A1.10) conforms to the general expression (A1.7) of NOFRF transmissibility.

Since the calculation of the NOFRF transmissibility is a forward iterative process, each NOFRF transmissibility includes the effect of the previous NOFRF transmissibility, thus the NOFRF transmissibility $α_{i, i + 1}^{\bar{n}} (j \bar{ω})$ when $i > L_{1} - 1$ is affected by the nonlinear component $Non F_{L_{1}}$ . Therefore, the NOFRF transmissibility $α_{i, i + 1}^{\bar{n}}$ at $\bar{ω}$ is produced by system nonlinearities outside the DOFs connected to nonlinear components ( $1 < i < L_{1} - 1$ ) only depends on linear system parameters.

Starting from the last row of the moving equation system (14) between mass $n$ and mass $n$ -1, as a reverse iterative process, the following conclusion can be drawn: The NOFRF transmissibility $α_{i, i + 1}^{\bar{n}}$ at $\bar{ω}$ is produced by system nonlinearities outside the DOFs connected to nonlinear components ( $L_{\bar{L}} < i < n$ ) only depends on linear system parameters.

Appendix 2

All terms associated with $U (j \bar{ω})$ become zero in the nonlinear MDOF system Equation (14) if the system output response at the frequency $\bar{ω}$ is produced by $(\bar{n} = 2, \dots, n_{i}$ )th order system nonlinearities with 1 < $n_{i}$ < $\bar{N}$ , then the system can be expressed as

(A2.1)

Γ (j \bar{ω}) Y (j \bar{ω}) = Δ (j \bar{ω})

where

Δ (j \bar{ω}) = {[Δ_{1} (j \bar{ω}), Δ_{2} (j \bar{ω}), \dots, Δ_{n} (j \bar{ω})]}^{T}

Δ_{i} (j \bar{ω}) = {\begin{matrix} \begin{matrix} 0 \end{matrix} & if \begin{matrix} i \neq L_{r} - 1, L_{r} \end{matrix} \\ - Non F_{L_{r}} & if i = L_{r} - 1 \\ Non F_{L_{r}} & if i = L_{r} \end{matrix} (1 \leq r \leq L_{\bar{L}}; 1 \leq i \leq n)

Y (j \bar{ω}) = {[Y_{1} (j \bar{ω}), Y_{2} (j \bar{ω}), \dots, Y_{n} (j \bar{ω})]}^{T}

Γ (j \bar{ω}) = Θ [M, K, C] (j \bar{ω}) .

Note: $Θ [M, K, C]$ is a matrix only containing the physical parameters of the system.

Then the output spectra can be rewritten as

(A2.2)

Y (j \bar{ω}) = Γ^{- 1} (j \bar{ω}) Δ (j \bar{ω}) .

Therefore, the output spectrum at nonlinear frequency components of mass i can be obtained

(A2.3)

Y_{i} (j \bar{ω}) = \sum_{r = 1}^{\bar{L}} {(\begin{matrix} {Γ^{- 1}}_{i, L_{r} - 1} (j \bar{ω}) \\ {Γ^{- 1}}_{i, L_{r}} (j \bar{ω}) \end{matrix})}^{T} (\begin{matrix} - Non F_{L_{r}} (j \bar{ω}) \\ Non F_{L_{r}} (j \bar{ω}) \end{matrix}) .

${Γ^{- 1}}_{i, L_{r} - 1} (j \bar{ω})$ is the element at the $i$ th row, $L_{r} - 1$ th column of an inversed matrix of $Γ (j \bar{ω})$ that only contains the physical parameters of the system. Consider if there is only one nonlinear component in the system, that is, $\bar{L} = 1$ , the output spectrum of mass i can be calculated as the following expression

(A2.4)

Y_{i} (j \bar{ω}) = - {Γ^{- 1}}_{i, L_{r} - 1} (j \bar{ω}) \cdot Non F_{L_{1}} (j \bar{ω}) + {Γ^{- 1}}_{i, L_{r} - 1} (j \bar{ω}) \cdot Non F_{L_{r}} (j \bar{ω}) (1 \leq i \leq n) .

Then the output-based transmissibility function for each pair of consecutive masses can be derived as

(A2.5)

α_{i, i + 1} (\bar{ω}) = \frac{Y_{i} (j \bar{ω})}{Y_{i + 1} (j \bar{ω})} = \frac{- {Γ^{- 1}}_{i, L_{1} - 1} (j \bar{ω}) + {Γ^{- 1}}_{i, L_{1}} (j \bar{ω})}{- {Γ^{- 1}}_{i + 1, L_{1} - 1} (j \bar{ω}) + {Γ^{- 1}}_{i + 1, L_{1}} (j \bar{ω})} (1 \leq i \leq n) .

If there is more than one nonlinear component, that is, $\bar{L} > 1$

(A2.6)

α_{i, i + 1} (\bar{ω}) = \frac{Y_{i} (j \bar{ω})}{Y_{i + 1} (j \bar{ω})} = \frac{\sum_{r = 1}^{\bar{L}} {(\begin{matrix} {Γ^{- 1}}_{i, L_{r} - 1} (j \bar{ω}) \\ {Γ^{- 1}}_{i, L_{r}} (j \bar{ω}) \end{matrix})}^{T} (\begin{matrix} - Non F_{L_{r}} (j \bar{ω}) \\ Non F_{L_{r}} (j \bar{ω}) \end{matrix})}{\sum_{r = 1}^{\bar{L}} {(\begin{matrix} {Γ^{- 1}}_{i + 1, L_{r} - 1} (j \bar{ω}) \\ {Γ^{- 1}}_{i + 1, L_{r}} (j \bar{ω}) \end{matrix})}^{T} (\begin{matrix} - Non F_{L_{r}} (j \bar{ω}) \\ Non F_{L_{r}} (j \bar{ω}) \end{matrix})} (1 \leq i \leq n) .

Equation (A2.5) states that the output-based transmissibility function at nonlinear frequencies for each consecutive pair of masses is insensitive to the presence of a nonliar component and solely relies on the physical properties of the system. However, the existence of multiple nonlinear components results in a dependence of the output-based transmissibility function at nonlinear frequencies not only on the physical parameters of the system but also on the nonlinear restoring force, as shown in Equation (A2.6). Consequently, the presence of a solitary nonlinear component does not impact the output-based transmissibility function at nonlinear frequencies.

Appendix 3

Considering when $J < L_{1}$ , NOFRF transmissibility for the $i$ th row ( $1 \leq i < J$ ) of the system matrix can be yielded as below:

α_{i, i + 1}^{\bar{n}} (j \bar{ω}) = {\frac{k_{i + 1} + j \bar{ω} c_{i + 1}}{- m_{i} {\bar{ω}}^{2} + j \bar{ω} (c_{i} + c_{i + 1}) + k_{i} + k_{i + 1} - (j \bar{ω} c_{i} + k_{i}) α_{i - 1, i}^{\bar{n}}} |}_{\bar{ω} \in f_{Y_{\bar{n}}}}

(A3.1)

(i < J; \bar{n} = 1, \dots, n_{i}) .

The NOFRF transmissibility where the masses don’t connect with nonlinear components and are not subject to the input only depends on its linear physical parameters, independent of nonlinearities and input. Therefore, the output-based transmissibility function remains unchanged under two different loading inputs at the same mass J for the masses i $(1 \leq i < J$ )

(A3.2)

α_{i, i + 1}^{F_{1}} (\bar{ω}) = \frac{Y_{i}^{F_{1}} (j \bar{ω})}{Y_{i + 1}^{F_{1}} (j \bar{ω})} = \frac{Y_{i}^{F_{2}} (j \bar{ω})}{Y_{i + 1}^{F_{2}} (j \bar{ω})} = α_{i, i + 1}^{F_{2}} (\bar{ω}) \bar{ω} \in [a, b] .

Equation (A3.2) still holds valid via a forward iterative process in the governing equations of the system (14) until $i = J$ . The NOFRF transmissibility between mass $J$ and mass $J + 1$ is shown in equation (A3.3),

(A3.3)

α_{J, J + 1}^{\bar{n}} (j \bar{ω}) = {\frac{k_{J + 1} + j \bar{ω} c_{J + 1} + \frac{δ (U (j \bar{ω}))}{G_{(J, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω})}}{- m_{J} {\bar{ω}}^{2} + j \bar{ω} (c_{J} + c_{J + 1}) + k_{J} + k_{J + 1} - (j \bar{ω} c_{J} + k_{J}) α_{J - 1, J}^{\bar{n}}} |}_{\bar{ω} \in f_{Y_{\bar{n}}}}

where $δ (U (j \bar{ω})) = {\begin{matrix} U (j \bar{ω}) \bar{n} = 1 \\ 0 \bar{n} = 2, . . ., n_{J} \end{matrix} .$ From (A3.3), the NOFRF transmissibility $α_{J, J + 1}^{\bar{n}} (j \bar{ω})$ $(\bar{n} = 1)$ at the linear output frequency range between the mass $J$ and mass $J + 1$ not only depends on system physical parameters but also on the term $of \frac{U (j \bar{ω})}{G_{(J, 1)} (j \bar{ω}) U (j \bar{ω})}$ , which bears the impact of the NOFRF associated with the $J$ th output of the system at the linear frequency range. Therefore, the output-based transmissibility function between the mass $J$ and mass $J + 1$ varies under two different loading inputs. Then,

(A3.4)

α_{i, i + 1}^{F_{1}} (\bar{ω}) = \frac{Y_{i}^{F_{1}} (j \bar{ω})}{Y_{i + 1}^{F_{1}} (j \bar{ω})} \neq \frac{Y_{i}^{F_{2}} (j \bar{ω})}{Y_{i + 1}^{F_{2}} (j \bar{ω})} = α_{i, i + 1}^{F_{2}} (\bar{ω}) \bar{ω} \in [a, b] .

From an alternative viewpoint of the backward iterative process in the governing Equation (14), the NOFRF transmissibility for the $i$ th row ( $n \geq i \geq L_{1}$ ) of the system matrix can be yielded as below:

(A3.5)

α_{i, i - 1}^{\bar{n}} (j \bar{ω}) = {\frac{k_{i} + j \bar{ω} c_{i}}{- m_{i} {\bar{ω}}^{2} + j \bar{ω} (c_{i} + c_{i + 1}) + k_{i} + k_{i + 1} - (j \bar{ω} c_{i + 1} + k_{i + 1}) α_{i + 1, i}^{\bar{n}}} |}_{\bar{ω} \in f_{Y_{\bar{n}}}} .

As seen from Equation (A3.5), the same conclusion as Equation (A3.3) can be drawn for the output-based transmissibility function between mass $i$ and mass $i - 1$ ( $n \geq i \geq L_{1}$ ), which only relies on the linear physical parameters of the system.

(A3.6)

α_{i, i - 1}^{F_{1}} (\bar{ω}) = \frac{Y_{i}^{F_{1}} (j \bar{ω})}{Y_{i - 1}^{F_{1}} (j \bar{ω})} = \frac{Y_{i}^{F_{2}} (j \bar{ω})}{Y_{i - 1}^{F_{2}} (j \bar{ω})} = α_{i, i - 1}^{F_{2}} (\bar{ω}) \bar{ω} \in [a, b] .

The relationship Equation (A3.6) holds true until $i = L_{1}$ . For the relationship among the masses connected to the right side of the nonlinear component can be rewritten as

α_{L_{1}, L_{1} - 1}^{\bar{n}} (j \bar{ω}) = {\frac{k_{L_{1}} + j \bar{ω} c_{L_{1}} + \frac{P_{L_{1}} (Non F_{L_{1}} (j \bar{ω}))}{G_{(L_{1}, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω})}}{- m_{L_{1}} {\bar{ω}}^{2} + j \bar{ω} (c_{L_{1}} + c_{L_{1} + 1}) + k_{L_{1}} + k_{L_{1} + 1} - (j \bar{ω} c_{L_{1} + 1} + k_{L_{1} + 1}) α_{L_{1} + 1, L_{1}}^{\bar{n}}} |}_{\bar{ω} \in f_{Y_{\bar{n}}}}

(A3.7)

(\bar{n} = 1, \dots, n_{L_{1}})

where

P_{1} (Non F_{L_{1}}) + P_{2} (Non F_{L_{1}}) + \dots + P_{n_{L_{1}}} (Non F_{L_{1}}) = Non F_{L_{1}} .

Using $P_{1} (Non F_{L_{1}}), P_{2} (Non F_{L_{1}}), \dots, P_{n_{L_{1}}} (Non F_{L_{1}})$ to denote the corresponding nonlinear restoring force for each nonlinear order. And they satisfy the given condition of (A3-8). When $\bar{n} = 1$ , $α_{L_{1}, L_{1} - 1}^{1} (j \bar{ω})$ is associated with the nonlinear feature $\frac{P_{L_{1}} (Non F_{L_{1}} (j \bar{ω}))}{G_{(L_{1}, \bar{n})} (j \bar{ω}) U_{\bar{n}} (j \bar{ω})}$ , and this leads to the following relationship:

(A3.9)

α_{L_{1}, L_{1} - 1}^{F_{1}} (\bar{ω}) = \frac{Y_{i}^{F_{1}} (j \bar{ω})}{Y_{i - 1}^{F_{1}} (j \bar{ω})} \neq \frac{Y_{i}^{F_{2}} (j \bar{ω})}{Y_{i - 1}^{F_{2}} (j \bar{ω})} = α_{L_{1}, L_{1} - 1}^{F_{2}} (\bar{ω}) \bar{ω} \in [a, b] .

Therefore, outside the DOFs connected to one single nonlinear component ( $1 \leq i \leq J - 1 or L_{1} \leq i < n$ ), if the system output response at the linear output frequency $ω \in [a, b], 0 \leq a < b$ , then

(A3.10)

α_{i, i + 1}^{F_{1}} (ω) = \frac{Y_{i}^{F_{1}} (j ω)}{Y_{i + 1}^{F_{1}} (j ω)} = \frac{Y_{i}^{F_{2}} (j ω)}{Y_{i + 1}^{F_{2}} (j ω)} = α_{i, i + 1}^{F_{2}} (ω) .

The derivation for the scenario of $L_{1} \leq J$ is similar to the case of $J < L_{1}$ , therefore, it is omitted here.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Liangliang Cheng

Alfredo Cigada

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