Short remarks on the paper: A novel methodology for a time-delayed controller to prevent nonlinear system oscillations

Abstract

In this brief communication, we highlight several critical inaccuracies in the paper entitled “A novel methodology for a time-delayed controller to prevent nonlinear system oscillations,” identified by the DOI: 10.1177/14613484231195276. The core of the non-perturbative approach discussed in this paper is the introduction of a trial solution with an indeterminate nonlinear frequency. We elucidate the technical missteps and provide necessary corrections. The scrutinized article presents ample opportunities for enhancement, and we propose several amendments. Additionally, we reveal that the numerical outcomes showcased in the mentioned paper expose deficiencies in the proposed approximate solution. With the amendments implemented herein, a more precise numerical solution is achieved.

Keywords

Duffing type oscillation equivalent linearized equation non-perturbative approach El-Dib’s frequency and damping coefficient formulas time-delay

Introduction

In the study outlined in Ref 1, the focus is on a Duffing-type equation representing a forced delayed nonlinear oscillator. The approach taken is a non-perturbative method aimed at approximating a solution for equation (1). This method's cornerstone is the transformation of the nonlinear oscillator equation into an equivalent linearized version. Typically, this results in a second-order linear equation with forced damping, where the key task is to identify a new damping coefficient and natural frequency. The authors of this study leverage El-Dib's frequency formula and damping coefficient, as detailed in References ^2–4, to guide their approach. The non-perturbative technique involves introducing a trial solution characterized by an unknown nonlinear frequency. This approach is reminiscent of the strategies employed in Galerkin’s method. Additionally, the study references He's publications, which are cited in Ref. 1. The essence of the method lies in the selection of an appropriate trial solution, which is crucial for achieving accurate and meaningful results.

In the realm of oscillator frequency analysis, there are four distinct levels of frequency consideration, each building upon the previous to incorporate additional factors and complexities.

1. Zero-Level Frequency (Natural Frequency): This is the most fundamental level, often used in the multiple scales method. At this level, the frequency is considered without accounting for nonlinear contributions.⁵

2. First-Level Frequency (Extended Frequency): Introduced by Professor He,⁶ Anjum, and He⁷ in the context of the homotopy perturbation method, this level, known as the extended frequency, includes nonlinear contributions to the frequency calculation. He’s Frequency Formula or Duffing’s Frequency developed by Professor He,^8–10 this level expands upon the first level by incorporating both the natural frequency and nonlinear contributions. It’s a more comprehensive approach, often referred to as He’s frequency formula or the Duffing’s frequency.

3. Second-Level Frequency: This level further extends the frequency analysis by including the effects of damping forces, building upon the first and zero level frequencies.^11–13

4. Third-Level Frequency: It represents the most advanced stage of frequency analysis, encompassing all previous levels and potentially additional factors. Recently, El-Dib et al.^14,15 have applied these concepts to the study of forced nonlinear oscillations in fractal space, utilizing the annihilator operator. Notably, El-Dib's frequency formula has been effectively used to solve the nonlinear Mathieu equation without resorting to a perturbative approach, showcasing its utility in complex oscillator analysis.¹⁶

Technical comments

In the mentioned paper, the researchers formulated a trial solution based on the natural frequency, denoted as ω, as outlined in equation (3). By employing the natural frequency ω in their trial solution, the authors are aligning their analysis with the intrinsic oscillatory properties of the system. The natural frequency is a crucial parameter, as it defines the frequency at which a system naturally oscillates in the absence of nonlinear forces or damping. It is widely recognized that the solution presented in this equation corresponds to a solution for a specific harmonic equation:

\ddot{y} + ω^{2} y = 0 .

(1)

Inserting equations (3)–(5) of Ref. 1 into the original fundamental equation (1), one gets

\begin{array}{l} (\frac{1}{4} A^{2} (3 a_{1} + a_{2} - 3 ω^{2} a_{3}) + β_{1} \cos ω τ + ω β_{2} \sin ω τ) A \cos ω t + \frac{1}{4} (a_{1} - a_{2} - a_{3}) A^{3} \cos 3 ω t \\ + (- 2 μ ω^{2} + β_{1} A \sin ω τ - ω β_{2} \cos ω τ) A \sin ω t = F \cos Ω t . \end{array}

(2)

Removing the secular terms assuming that Ω not approaching the natural frequency ω yields

ω^{2} - ω \frac{4 β_{2}}{3 A^{2} a_{3}} \sin ω τ - \frac{1}{3 A^{2} a_{3}} ((3 a_{1} + a_{2}) A^{2} + 4 β_{1} \cos ω τ) = 0,

(3)

ω^{2} + ω \frac{β_{2}}{2 μ} \cos ω τ - \frac{β_{1} A}{2 μ} \sin ω τ = 0 .

(4)

If $ω, β_{1}, β_{2}, a_{1}, a_{2}$ , and $a_{3}$ are known coefficients then we obtain some constrain on the problem. These constraints confine the problem to a highly specific scenario. Essentially, if we assume that the frequency ω is not predetermined, then the aforementioned equations result in a redundancy in the determination of ω. Such redundancy suggests the presence of a flawed or inadequate approximation in the solution.

If the natural frequency ω is used in the trial solution then what we are looking for?

A scrutiny of the aforementioned harmonic equation to the original equation (1) from Ref. 1 indicates the elimination of all nonlinear terms. In essence, undertaking this comparison results in the disappearance of nonlinear effects. This can be exemplified by considering the most basic form of the Duffing oscillator

\ddot{y} + ω^{2} y + Q y^{3} = 0 .

(5)

Professor He^8–10 has suggested the trial solution with an unknown frequency Θ such that

y (t) = A \cos Θ t .

(6)

Subsequently, He dedicated considerable effort across numerous publications to approximate the unknown frequency Θ, which is expressed as follows:

Θ^{2} = ω^{2} + \frac{3}{4} Q A^{2} .

(7)

Thus, if we posit that Θ is substituted by ω, it follows that nonlinear contributions are absent. Numerous researchers have adopted He’s methodology as evidenced in various scholarly works. However, none have utilized the natural frequency in place of the total frequency, even though they introduced a trial solution. This suboptimal selection of the proposed solution has led to unsatisfactory outcomes, culminating in a deficient approximation.

Considering the above He’s proposed trial solution and the adjusted period (T = 2π/Θ), we arrive at the accurate formulations of El-Dib’s frequency equation (10)^2–4 and El-Dib’s damping coefficient equation (12) as referenced in Ref ¹. This approach allows for the precise estimation of the equivalent frequency ω_eq and the equivalent damping coefficient μ_eq, ensuring a more accurate representation of the system's dynamics.

ω_{e q}^{2} = ω^{2} + \frac{3}{4} A^{2} a_{1} + \frac{1}{4} A^{2} (a_{2} - 3 a_{3}) Θ^{2} + β_{1} \cos Θ τ + Θ β_{2} \sin Θ τ .

(8)

μ_{e q} = 2 μ ω + β_{2} \cos Θ τ - \frac{β_{1}}{Θ} \sin Θ τ,

(9)

The analytical solution of equation (14) is missing in Ref. 1. The correction regarding to the solution of equation (14) is performed in the form

\begin{array}{l} y (t) = (A - \frac{F (ω_{e q}^{2} - Ω^{2})}{{(ω_{e q}^{2} - Ω^{2})}^{2} + {μ_{e q}}^{2} Ω^{2}}) e^{- \frac{1}{2} μ_{e q} t} \cos Θ t + \frac{1}{2 Θ} μ_{e q} (A - \frac{F (ω_{e q}^{2} - 3 Ω^{2})}{{(ω_{e q}^{2} - Ω^{2})}^{2} + {μ_{e q}}^{2} Ω^{2}}) e^{- \frac{1}{2} μ_{e q} t} \sin Θ t \\ + \frac{F ((ω_{e q}^{2} - Ω^{2}) \cos Ω t + Ω μ_{e q} \sin Ω t)}{{(ω_{e q}^{2} - Ω^{2})}^{2} + {μ_{e q}}^{2} Ω^{2}} . \end{array}

(10)

where the total frequency Θ is estimated to be

\begin{array}{l} Θ^{4} (1 - \frac{1}{4} A^{2} (a_{2} - 3 a_{3})) - Θ^{3} β_{2} \sin Θ τ - \frac{1}{2} Θ (2 μ ω + β_{2} \cos Θ τ) β_{1} \sin Θ τ + \frac{1}{4} β_{1}^{2} \sin^{2} Θ τ \\ - (ω^{2} + \frac{3}{4} A^{2} a_{1} - μ^{2} ω^{2} - (\frac{1}{4} β_{2}^{2} \cos Θ τ + μ ω β_{2} - β_{1}) \cos Θ τ) Θ^{2} = 0 . \end{array}

(11)

The relationship between frequency and amplitude in this context exhibits a transcendental nature. As a result, the crucial frequency formula was overlooked in Ref. 1. This oversight renders the numerical results presented therein both invalid and unreliable. Additionally, the incorporation of a periodic force in the stability analysis has been notably omitted.

To accurately ascertain the stability conditions for the unforced problem, one can normalize the delayed parameter τ by the frequency Θ, such that τ = $\hat{τ}$ / Θ. Consequently, equation (11) is transformed as follows:

C_{4} Θ^{4} + C_{3} Θ^{3} + C_{2} Θ^{2} + C_{1} Θ + C_{0} = 0,

(12)

The stability conditions are performed in the form¹⁷

3 C_{1}^{2} - 8 C_{0} C_{2} > 0, 4 C_{2}^{2} - 9 C_{1} C_{3} > 0, 3 C_{3}^{2} - 8 C_{2} C_{4} > 0,

(13)

where

\begin{array}{l} C_{4} = \frac{1}{4} A^{2} (a_{2} - 2 a_{3}) - 1, \\ C_{3} = β_{2} \sin \hat{τ}, \\ C_{2} = (ω^{2} + \frac{3}{4} A^{2} a_{1} - μ^{2} ω^{2}) - (\frac{1}{4} β_{2}^{2} \cos \hat{τ} + μ ω β_{2} - β_{1}) \cos \hat{τ}, \\ C_{1} = \frac{1}{2} (2 μ ω + β_{2} \cos \hat{τ}) β_{1} \sin \hat{τ} \\ C_{0} = - \frac{1}{4} β_{1}^{2} \sin^{2} \hat{τ} . \end{array}

(14)

Supplementary comments

Regarding the multiple-scale solution

In Ref. 1, the authors utilized the multiple scales method on the derived linear equation (14), while maintaining the initial conditions (2) as they were. This approach involved applying a transformation (21) exclusively to equation (14), neglecting its application to the initial conditions (2). Nonetheless, the appropriate corrections for equations. (22) and (23) should be as follows¹⁸:

D_{0}^{2} y_{0} + ω_{e q}^{2} y_{0} = 0; y_{0} (0; T_{1}) = 0, D_{0} y_{0} (0; T_{1}) = A (T_{1}),

(15)

\begin{array}{l} D_{0}^{2} y_{1} + ω_{e q}^{2} y_{1} = - 2 D_{0} D_{1} y_{0} - μ_{e q} D_{0} y_{0} + F \cos Ω T_{0}; \\ y_{1} (0; T_{1}) = 0, D_{0} y_{1} (0; T_{1}) + D_{1} y_{0} (0; T_{1}) = 0 . \end{array}

(16)

It is noted that equation (20) becomes redundant, as equation (14) in Ref. 1 does not explicitly incorporate delayed terms, a fact that becomes evident upon closer examination.

There exists a discrepancy and inconsistency in the analysis of resonance, stemming from the definition of the detuning factor. The authors delineate the detuning parameter σ as (Ω = ω + ρ σ), which fails to align with equations (24) and (25) in Ref. 1. Consequently, if equation (24) represents the zero-order solution, then equation (25) is erroneous. The accurate formulation should be as follows:

D_{0}^{2} y_{1} + ω_{e q}^{2} y_{1} = (- 2 i ω_{e q} D_{1} A - i μ_{e q} ω_{e q} A) e^{i ω_{e q} T_{0}} + \frac{1}{2} F e^{i σ T_{1} + i ω T_{0}} + c c .

(17)

The secular terms are, only, the coefficients of $e^{i ω_{e q} T_{0}}$ not $e^{i ω T_{0}}$ . Therefore, equation (26) is incorrect.

To address the ambiguity and inconsistency present, a reevaluation of the resonance definition is imperative. Given that the methodology is applied to equation (14), the relevant natural frequency should be ω_eq, not the original natural frequency ω. Hence, the accurate definition of the detuning parameter should be established as follows:

Ω = ω_{e q} + ρ σ

(18)

As a result, the proper form for equation (25) is

D_{0}^{2} y_{1} + ω_{e q}^{2} y_{1} = (- 2 i ω_{e q} D_{1} A - i μ_{e q} ω_{e q} A + \frac{1}{2} F e^{i σ T_{1}}) e^{i ω_{e q} T_{0}} + c c .

(19)

At this stage, the correct secular terms are done.

It's important to note that while the resonance concept as defined in (18) is apt for the multiple scales method, it does not align with the exact solution presented in (10). This discrepancy highlights the need for careful consideration when applying different analytical methods to complex dynamical systems.

Regarding the numerical calculations

The concordance between the theoretical and numerical solutions, as observed in the comparison, is limited to scenarios with very small amplitudes. This limitation suggests a deficiency in the approximate solutions when dealing with relatively large amplitudes. A robust analysis should encompass a wide range of oscillation amplitudes, rather than being confined to a narrow interval. This is evident from the various figures presented in Ref. 1, (Figures 2(b) and (c), and 7(a) and (b)), which illustrate the discrepancies across different amplitude ranges.

Figures 2(b) and (c) are plotted for a non-delayed system. While Figures 7(a) and (b) are for the delay system.

Figure 1.

The numerical comparison for the analytical solution of equation (14) and the numerical solution of equation (1). For the numerical values: $ω = 3; μ = 0.1; a_{1} = a_{2} = a_{3} = 0.1; f = . 01; β_{1} = 0; β_{2} = 0; A = 1; Ω = 3 .$

Figure 2.

The numerical comparison for the analytical solution of equation (14) and the numerical solution of equation (1). For the numerical values: $ω = 3; μ = 0.1; a_{1} = 0.1; a_{2} = 0.1; a_{3} = 0.1; f = 1.2; β_{1} = 0; β_{2} = 0; A = 1; Ω = 0.5$ .

Figure 3.

Figure 4.

The correct stability behavior for the unforced system is.

Figure 5.

The stable region has been distinguished by the shaded area for the variation of the delay parameter. For the numerical parameter given as: $μ = 0.01; a_{1} = 0.1; a_{2} = 0.1; a_{3} = 0.1; β_{1} = 0.3; β_{2} = 0.5 .$

Discussions

Several nonlinear oscillation models are formulated without incorporating a natural frequency. In such scenarios, the methodology proposed by the authors becomes inapplicable. An effective methodology should be versatile enough to be applicable under all conditions, including those where the natural frequency is not a factor.

The stability condition outlined in (32) is flawed, as it erroneously suggests that the frequency formula solely comprises the power Θ². An expansion of this condition, as delineated in equation (11), reveals a fourth-order polynomial frequency formula, which is more apt for establishing the required stability criteria, as demonstrated in equation (13).

An estimated method was used to convert the original nonlinear equation to the linear equation (14). With this approximate approach, there is an exact solution for linear equation (14). However, the multiple scales perturbation method is an additional approximate method. The application of multi-approximate methods degrades the finding's accuracy.

It raises a pertinent question: why do authors pursue approximate solutions when an exact solution, as presented in equation (10), is readily available? Typically, this approach is adopted to evaluate a new approximation method by contrasting it with the exact solution. However, in this instance, there is a noticeable absence of comparison between the exact solution in equation (10) and the outcomes derived from applying the approximate solution via the multiple scales method.

The accurate numerical representations for Figure 2(b) and (c) are provided herein. The substitution Ω = ω+σ, contrary to what one might assume, does not signify a resonance scenario. On the other hand, the correction proposed earlier in the perturbation analysis, Ω = ω_eq +σ, does indeed denote primary resonance within the context of the perturbation method. However, it’s important to note that this does not apply to the non-perturbative approach.

Turning our attention back to the non-perturbative solution equation (10), it’s clear that there is no risk of division by zero, meaning the denominator of the fraction will never approach zero. Consequently, the solution presented in equation (10) points to a non-resonance case. This highlights the need for a more sophisticated analysis to explore the behavior of the periodic force in resonance scenarios, especially when employing a non-perturbative approach.

Researchers consistently strive for precision, particularly at small amplitude values $A \geq 1$ . However, the limited accuracy observed at very low amplitudes A highlights a fundamental weakness in their analytical approach. Notably, solutions derived from the correctly formulated trial solution equation (6) demonstrate enhanced accuracy when juxtaposed with numerical solutions, underscoring the importance of precise initial assumptions in theoretical analyses.

Conclusion

In this concise note, we elucidate a fundamental error present in the results published in1, arising from the use of the natural frequency instead of the linear frequency in the proposed trial solution. We provide a corrective solution, as detailed in (6), and address the necessary adjustments throughout the entire article to rectify this mistake. Furthermore, we introduce the frequency-amplitude formula, which was notably absent from the discussed article.¹ Additionally, we offer suggestions on how to simplify the frequency-amplitude equation, which can become intricate due to the presence of transcendental formulations. Moreover, we establish and illustrate the stability criteria, a crucial aspect that was overlooked in Ref. 1, and we present these corrected stability conditions in the absence of a periodic force. It is worth noting that while the non-perturbation method transforms the nonlinear equation into its linear equivalent, for which an exact solution exists, this exact solution was omitted in Ref. 1 but is provided comprehensively in this note. The application of the perturbation method to this exact solution is found to be unnecessary. Finally, we emphasize the significance of understanding the concept of resonance and its occurrence.

Footnotes

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 iD

Yusry O El-Dib

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