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Abstract

This paper introduces a novel methodology to determine the frequency-amplitude relationship of fractal-order viscoelastic polymer materials using the two-scale fractal dimension transform, the equivalent power-form representation of the conservative restoring forces, and a simple coordinate transformation to eliminate viscoelastic effects. Then, the ancient Chinese algorithm Ying Bu Zu Shu and He’s formulation are used for obtaining the frequency-amplitude relationship. Simulation results obtained from the derived expressions exhibit good agreement when compared to numerical integration solutions. This article elucidates how the molecular structure of polymer chains influences the relaxation oscillations as a function of the fractal parameter values.

Keywords

Two-scale fractal dimension transform power-form equivalent transformation ancient Chinese algorithm He’s frequency formulation fractal and damped polymer chains oscillator

Introduction

The material damping properties serve to modify the oscillations’ amplitude through energy dissipation. In this sense, polymers are used for energy absorption and vibration damping attenuation in several engineering devices since the damping forces are small compared to the elastic and inertia ones.^1,2 The oscillation’s decay in free vibrations can be explained considering damping effects that prevent oscillations by energy dissipation due to the material molecule movement. Of course, factors such as viscoelastic material properties and glass-transition phenomena could influence damping effects. It is also known that when a rubber material vibrates, its molecules adopt new conformations releasing the vibration energy. Recently, Sarkheil³ identified that the mobility of polymer molecules is an indication of the molecular structure fractal behavior. Schiessel and Blumen⁴ found that dynamic processes in polymers such as mechanical relaxation dynamics and cross-linking at the solgel transition exhibit fractal behavior. Based on these findings, Elías-Zúñiga et al.⁵ investigated how the fractal molecular structure of polymer materials changes as the oscillation amplitude varies and how this polymer material behavior can be used to identify molecular structure defects and intermolecular cohesion.

It is evident that including damping effects in the mathematical models provides qualitative and quantitative material response behavior that could agree with experimental observations. Furthermore, products such as vibration dampers, bridge bearings, seismic absorbers, and building/engine mounts, to name a few, are designed considering a material’s high damping value to mitigate undesirable vibration effects.⁶ As expected, the addition of damping effects in the governing equation of motion in conjunction with strongly nonlinear restoring forces increases the complexity in deriving approximate analytical solutions valid for the whole solution domain. Therefore, one needs to identify which solution technique must be used to get the desired solution. A good overview of some available techniques to get the approximate solution of strongly nonlinear and damped equations such as variational approaches, homotopy techniques, He’s frequency formulation, Taylor series, iteration methods, equivalent power-form approach, energy techniques, simple frequency formulations, and ancient Chinese mathematical methods are given in Refs. 7–34 and references cited therein.

The goal of this article is to introduce a straightforward methodology to solve the fractal and damped differential equation of motion that models the dynamic response of polymer materials using the two-scale fractal dimension transform,^35–38 the equivalent power-form representation of the restoring force, and a coordinate transformation to eliminate the damping term. Then, the frequency-amplitude relationship is determined by using the Ying Bu Zu Shu or ancient Chinese algorithm of the Chinese method.^39–42

Mathematical model

Before introducing the fractal mathematical model that describes the dynamics of a viscoelastic Langevin polymer chain first, we recall the simple definition of the two-scale dimension transform and its relevance in modeling multiscale phenomena that describe behavior at a given length or time scales, playing a critical role in bridging qualitative and quantitative methods for engineering problem analysis. In fact, the two-scale theory models each phenomenon with a large scale to characterize continuous media with the classic calculus, and on the smaller scale, to elucidate system molecular effects and to reflect discontinuous phenomena. The two-scale dimension transform has been used to study the N/MEMS in a fractal space (porous medium),⁴³ the Fangzhu water collector from air,^44–48 the vibration attenuation and vibration absorption of the porous concrete,⁴⁹ the smart receptor system for exact printing of nano/micro devices modeled by a fractal Duffing equation,⁵⁰ the rheological properties of SiC-based print paste using a fractal model for predicting the material viscosity,⁵¹ the fractal model of current generation in porous electrodes,⁵² the fractal Toda oscillator that describes the intensity fluctuation of Nd:Yag lasers,⁵³ the nonlinear vibration of the carbon nanotube embedded in fractal medium,⁵⁴ the shallow water waves traveling along an unsmooth boundary,⁵⁵ to describe the flow of the shallow water of harbor considering an unsmooth boundary,⁵⁶ and packaging cushioning systems⁵⁷ among others. Furthermore, modeling polymer molecules’ dynamics using fractal derivatives and the two-scale dimension transform aids in identifying molecular structure defects, intermolecular cohesion, and the evolution of the polymer storage modulus as a function of frequency and of the fractal parameter.⁵

Considering the relevance that polymers have in engineering applications, there is a need for a better understanding of their network structure behavior that affects physical and mechanical properties; therefore, a mathematical model that takes into account fractal patterns that are formed in the preparation of polymers and their composites is highly desirable.⁵⁸ In this article, we use a dynamics mathematical model that can be obtained from the fractal Lagrange equation

\frac{d}{d t^{α}} (\frac{\partial (T - V)}{\partial {\dot{q}}_{i}}) - \frac{\partial (T - V)}{\partial q_{i}} = Q_{i}

(1)

that considers the Langevin polymer chain’s viscoelastic effects. In equation (1), the kinetic and Langevin potential energies T and V can be computed from the following expressions

\begin{array}{l} T & = & \frac{1}{2} M {(\frac{d q_{i}}{d t^{α}})}^{2} \end{array}

(2)

\begin{array}{l} V & = & k N T \int_{0}^{x / L_{1}} ℒ^{- 1} (q_{i}) d q_{i} \end{array}

(3)

Here,

ℒ^{- 1} (x)

is known as the inverse Langevin function defined as

x = ℒ (β) = coth (β) - 1 / β

with β =

ℒ^{- 1} (x)

,^59,60 q_i are the generalized coordinates, the overdot denotes derivative with respect to time, Q_i are the generalized nonconservative forces, M is the mass, and α is the fractal dimension.^61–63 Thus, for a viscoelastic material, the governing equation of motion becomes

\frac{d}{d t^{α}} (\frac{d λ}{d t^{α}}) + 2 ν \frac{d λ}{d t^{α}} + K_{1} ℒ^{- 1} (λ) = 0, λ (0) = A, \frac{d λ}{d t^{α}} = 0

(4)

where K₁ = kNT/(ML₁), 2ν = ν₀/(ML₁), λ is the chain stretch, L₁ is the undeformed chain length, and ν₀ is the damping constant.

Recalling the fractal derivative definition with respect to time

\frac{d λ}{d t^{α}} (t_{0}^{α}) = Γ (1 + α) lim_{\frac{(t - t_{0}) \to Δ t}{Δ t \neq 0}} \frac{λ (t) - λ (t_{0})}{{(t - t_{0})}^{α}}

(5)

that agrees with the classical differential derivative when the two-scale fractal dimension takes the value of one. Thus, using the fractal dimension transform^64–66

τ = t^{α}

(6)

Equation (4) can be written in the form

\frac{d^{2} x}{d τ^{2}} + 2 ν ω_{n} \frac{d λ}{d τ} + K_{1} ℒ^{- 1} (λ) = 0, λ (0) = A, \frac{d}{d τ^{1 / α}} λ (0) = 0

(7)

Next, the approximate representation form of

ℒ^{- 1} (λ)

is introduced

ℒ^{- 1} (λ) ≃ \frac{λ (30 - 26 λ + 7 λ^{2})}{3 (1 - a λ) (10 + a λ)}

(8)

where

ω_{n} = \sqrt{K_{1}}

, a = 1/x₀, and x₀ describes the maximum possible chain extension⁶¹; therefore, equation (7) simplifies to

\frac{d^{2} λ}{d τ^{2}} + 2 ν ω_{n} \frac{d λ}{d τ} + \frac{K_{1}}{3} \frac{λ (30 - 26 λ + 7 λ^{2})}{(1 - a λ) (10 + a λ)} = 0

(9)

To find the frequency-amplitude expression for equation (9) using the ancient Chinese algorithm Ying Bu Zu Shu, we first use the approach proposed in Refs.^67–74 to find its power-form equivalent representation, and then, we introduce a coordinate transformation to eliminate the damping term.

Power-form equivalent representation

Following the transformation approach introduced in Refs.^67–74, equation (7) can be written in the following form

\frac{d^{2} λ}{d τ^{2}} + 2 ν ω_{n} \frac{d λ}{d τ} + a_{1} λ + a_{2} sgn (λ) | λ |^{m} = 0

(10)

where a₁, a₂, and m are parameters that can be determined using the weighted mean square error U, which is defined as

U = \int_{0}^{σ} {(\frac{K_{1}}{3} \frac{λ (30 - 26 λ + 7 λ^{2})}{(1 - a λ) (10 + a λ)} - a_{1} λ - a_{2} sgn (λ) {| λ |}^{m})}^{2} w (λ) d λ

(11)

with

\partial U / \partial a_{1} = 0, \partial U / \partial a_{2} = 0

(12)

where the weighted function w(λ) is assumed to have the form

w (λ) = λ^{N_{1}}

. Using equations (11) and (12), we get the following expressions for a₁ and a₂

\begin{array}{l} a_{1} & = & 1 / (33 a^{4} {(- 1 + m)}^{2}) K_{1} (11 (- 7 a^{2} {(m - 1)}^{2} - ((637 + 6 a (39 + 5 a)) (3 + N_{1}) \\ {(2 + m + N_{1})}^{2}) / ((1 + N_{1}) σ^{2}) + a (63 + 26 a) (3 + N_{1}) {(2 + m + N_{1})}^{2} / (2 + \\ N_{1}) σ) - a^{2} (70 + a (26 + 3 a)) (3 + N_{1}) {(1 + 2 m + N_{1})}_{2} F_{1} [1,2 + m + N_{1}, \\ 3 + m + N_{1}, - a σ / 10] - a^{2} (7 - 26 a + 30 a^{2}) (3 + N_{1}) {(1 + 2 m + N_{1})}_{2} \\ F_{1} [1,2 + m + N_{1}, 3 + m + N_{1}, a σ] + 1 / σ^{2} (3 + N_{1}) {(2 + m + N_{1})}^{2} \\ Γ [1 + N_{1}] (100 {(70 + a (26 + 3 a))}_{2} F_{1} [1,1 + N_{1}, 2 + N_{1}, - a σ / 10] + \\ {(7 - 26 a + 30 a^{2})}_{2} {\bar{F}}_{1} [1,1 + N_{1}, 2 + N_{1}, a σ])) \end{array}

(13)

\begin{array}{l} a_{2} & = & 1 / (33 a^{4} {(m - 1)}^{2} (1 + N_{1}) (2 + N_{1})) K_{1} (2 + m + N_{1}) (1 + 2 m + N_{1}) \\ σ^{(- 1 - m)} (11 (3 + N_{1}) ((637 + 6 a (39 + 5 a)) (2 + N_{1}) - a (63 + 26 a) (1 + \\ N_{1}) σ) + a^{2} (70 + a (26 + 3 a)) (1 + N_{1}) (2 + N_{1}) σ^{2}_{2} F_{1} [1,2 + m + N_{1}, \\ 3 + m + N_{1}, - a σ / 10] + a^{2} (7 - 26 a + 30 a^{2}) (1 + N_{1}) (2 + N_{1}) σ_{2}^{2} \\ _{2} F_{1} [1,2 + m + N_{1}, 3 + m + N_{1}, a σ] + Γ [4 + N_{1}] (- 100 (70 + a (26 + \\ 3 {a))}_{2} {\bar{F}}_{1} [1,1 + N_{1}, 2 + N_{1}, - a σ / 10] \\ + {(26 a - 7 - 30 a^{2})}_{2} {\bar{F}}_{1} [1,1 + N_{1}, 2 + N_{1}, a σ])) \end{array}

(14)

where ₂F₁[a, b; c; z] is the hypergeometric function ₂F₁(a, b; c; z), and ₂

{\bar{F}}_{1} [a, b, c, z]

is the regularized hypergeometric function ₂F₁(a, b; c; z)/Γ(c).⁷⁵ Substitution of equations (13) and (14) into equations (11) and minimization of the resulting expression, yield the value of σ, N₁, and m.

Frequency-amplitude formulation

Before we start with the derivation of the frequency-amplitude formulation of equation (10) using the Chinese algorithm and He’s formulation, it is important to point out some of its advantages over traditional analytical methods. This method does not depend on small or large physical parameters. It has a high convergence rate, and its accuracy is higher when compared to Newton’s iteration method.⁷⁶ This ancient algorithm has been also used to derive, by modifying Chun-Hui He’s algorithm, an efficient iterative algorithm that converges to the optimal solution with only a few number of iterations.⁷⁷ Furthermore, it has excellent flexibility in choosing the functions needed to identify the trial solutions; it can be used to obtain the frequency-amplitude formulation of nonlinear oscillators without a linear term that makes it harder to find an approximate solution when using perturbation methods.³⁴ This ancient Chinese algorithm can be also combined with other techniques such as the power-form equivalent representation, Padé approximants, Chebyshev polynomials, Fourier series, to name a few. One possible limitation of this algorithm might be related to the incapacity of getting the transient response of nonlinear forced oscillators. However, the extent of this limitation certainly requires further investigation.

In what follows, we focus our efforts on finding the frequency-amplitude expression for equation (10). One can see that the determination of the frequency-amplitude expression for equation (10) using He’s formulation and the ancient Chinese algorithm is not a straightforward process because of the presence of the damping term. Therefore, to avoid potential complications in the derivation of the frequency-amplitude relationship using this ancient algorithm, the following coordinate transformation⁷⁸

λ = e^{- ν ω_{n} τ} y (τ)

(15)

is introduced to eliminate the damping term in equation (10). This coordinate transformation yields the following equation of motion

\frac{d^{2} y}{d τ^{2}} + (a_{1} - ν^{2} ω_{n}^{2}) y + a_{2} e^{- ν τ ω_{n} (m + 1)} sgn (y) {| y |}^{m} = 0

(16)

Thus, the frequency-amplitude expression of equation (16) will be determined using He’s formulation based on the ancient Chinese algorithm Ying Bu Zu Shu originated from an ancient Chinese mathematics monograph—The Nine Chapters on the Mathematical Art (see Refs.^39,79–81 and references cited therein). In this formulation, it is first assumed that the approximate trial solutions x₁(t) and x₂(t) of equation (16) are given as

y_{1} = A_{1} \cos (ω_{1} τ), y_{2} = A_{2} \cos (ω_{2} τ)

(17)

where A₁ and A₂ need to be defined. Then, the substitution of equation (17) into equation (16) yields the trial residual functions

R_{i} = A_{i} \cos (ω_{i} τ) (a_{1} - ω_{i}^{2} - ν^{2} ω_{n}^{2}) + a_{2} \exp (- (m + 1) ν τ ω_{n}) A_{i}^{m} \cos {(ω_{i} τ)}^{m}

(18)

where i = 1, 2 (no sum).^39–42 Next, the average trail residuals

{\tilde{R}}_{i}

defined by the following equation

{\tilde{R}}_{i} = \frac{4}{T_{i}} \int_{0}^{T_{i} / 4} R_{i} d τ

(19)

are determined with T_i = 2π/ω_i. To simplify the integration of equation (19), we use Fourier series to write the term

\cos {(S ω_{i})}^{m}

in equivalent form as⁸²

\cos {(τ ω_{i})}^{m} ≃ b_{1 m} \cos (τ ω_{i}) + b_{3 m} \cos (3 τ ω_{i})

(20)

with

b_{1 m} = \frac{2 Γ (1 + m / 2]}{π^{1 / 2} Γ [(3 + m) / 2]}, b_{3 m} = \frac{(m - 1) Γ (1 + m / 2]}{π^{1 / 2} Γ [(5 + m) / 2]}

(21)

where Γ[…] is the Gamma function. Thus, substitution of equation (18) into equation (19), yields

\begin{array}{l} {\tilde{R}}_{i} & = & 4 / T_{i} (A_{i} a_{1} / ω_{i} - A_{i} ω_{i} - A_{i} ν^{2} ω_{n}^{2} / ω_{i} + \\ (A_{i}^{m} b_{1 m} a_{2} e^{- (1 / 4) (1 + m) T_{1} ν ω_{n}} (ω_{i} + (1 + m) ν (e^{1 / 4 (1 + m) T_{i} ν ω_{n}}) ω_{n})) / (ω_{i}^{2} + \\ {(1 + m)}^{2} ν^{2} ω_{n}^{2}) + (A_{i}^{m} b_{3 m} a_{2} e^{- 1 / 4 (1 + m) T_{i} ν ω_{n}}) (- 3 ω_{1} + \\ (1 + m) ν e^{1 / 4 (1 + m) T_{i} ν ω_{n}} ω_{n})) / (9 ω_{i}^{2} + {(1 + m)}^{2} ν^{2} ω_{n}^{2})) \end{array}

(22)

By using He’s formula

ω_{A E}^{2} = \frac{ω_{1}^{2} {\tilde{R}}_{2} - ω_{2}^{2} {\tilde{R}}_{1}}{{\tilde{R}}_{2} - {\tilde{R}}_{1}}

(23)

the approximate frequency-amplitude expression for equation (9) is found. Here, it is assumed that ω₁ = 1, ω₂ = 2, A₁ = A₂ = A as discussed in Ref.^39–42

The following section will address the applicability of equation (23) to obtain the fractal frequency-amplitude response curves of equation (9).

Numerical results

To illustrate the applicability of our proposed approach to obtain the frequency-amplitude response curve of the fractal and damped polymer chain oscillator, let us consider the case for which the parameter values are a = 0.5; ν = 0.1; K₁ = 1; and α = 0.9, 1, and 1.1 with y(0) = A = 0.5 and $\dot{y} (0) = 0$ . Figure 1 shows the amplitude-time response curves plotted using the values of a₁ = 1.0032, a₂ = −0.3113, σ = 1, N₁ = −2.9, m = 1.75, and ω_AE = 0.9265. Small discrepancies are observed in Figure 1 between the numerical integration solution (solid lines) of equation (9) and the approximate solution (dashed lines) obtained from

λ (τ) = A e^{- ν τ^{1 / α_{ω_{n}}}} \cos (ω_{A E} τ^{1 / α})

(24)

As a second example, let us consider the system parameter values of a = 0.85; ν = 0.1; K₁ = 0.25; and α = 0.9, 1, and 1.1 with y(0) = A = 0.9 and

\dot{y} (0) = 0

. Figure 2 shows that our proposed solution (dashed lines) given by equation (24) follows the numerical integration solution of equation (9) well. Furthermore, our derived solution is able to capture the variation of the wave length propagation as well as the decaying system oscillations as a function of the fractal exponent values. In this case, the parameter values of m, N₁, and σ used to plot the amplitude-time curves shown in Figure 2 were 2.5, −2.9, and 0.5, respectively, with a₁ = 0.2495, a₂ = 0.0097, and ω_AE = 0.5025.

Figure 1.

Amplitude-time response curves for a fractal viscoelastic polymer chain obtained by considering the parameter values of a = 0.5, ν = 0.1, K₁ = 1, and A = 0.5. Here, the solid lines represent the numerical integration solution of equation (9), while the dashed lines describe the approximate solutions obtained by using equation (24) with a₁ = 1.0032, a₂ = −0.3113, σ = 1, N₁ = −2.9, m = 1.75, and ω_AE = 0.9265. The discrepancies observed in these plots are mostly due to the replacement of the inverse of the Langevin function by an equivalent power-form representation, and the use of Fourier series to simplify the integration of equation (19). However, the fractal approximate amplitude-time curves given by equation (24) follow numerical integration solutions well.

Figure 2.

Amplitude-time response curves for a fractal viscoelastic polymer chain obtained by considering the parameter values of a = 0.85, ν = 0.1, K₁ = 0.25, and A = 0.9. Here, the solid lines represent the numerical integration solution of equation (9), while the dashed lines describe the approximate solutions obtained by using equation (24) with a₁ = 0.2495, a₂ = 0.0097, σ = 0.5, N₁ = −2.9, m = 2.5, and ω_AE = 0.5025. These curves indicate a modification of the molecular chain stretching magnitude during motion influenced by the polymer fractal molecular structure.

The proposed methodology capture fractal phenomena that occur at the polymer molecular level. In fact, the computed amplitude-time curves indicate that the chain’s mobility is less localized when α < 1 since the polymer molecular structure exhibits fluid-like behavior with steady flow viscosity and low vibration frequency.^83,84 Although there is evidence that in polymerfluidgels (PFGs), high energy-dissipation property can be precisely tailored at desired frequencies,⁸⁵ one needs to bear in mind that the localized strain accumulation process, that occurs at lower frequencies, could lead to the crack initiation, followed by its propagation, and failure of the polymer part. In other words, the polymer material macroscopic behavior strongly depends on the material fractal structure.⁸⁶

When considering α > 1, the polymer material exhibits solid-like behavior due to geometrical changes in the chemical network during gelation because the dynamic properties depend strongly on the material composition and structure.⁸⁷ Therefore, understanding the fractal nature of the material molecular structure can help to develop a polymer material with the desired dynamic response that fulfills engineering specifications.

Conclusions

This article elucidates how our proposed solution methodology can be extended to study the relaxation oscillations of a fractal viscoelastic polymer chain oscillator. We have used this methodology to obtain the frequency-amplitude relationship of a mathematical model that describes the dynamics of fractal viscoelastic polymer Langevin chains at the material molecular level. The accuracy of the derived solution is confirmed by plotting the amplitude-time curves that indicate less localized chains mobility when α < 1 because the polymer exhibits fluid-like behavior. For fractal values bigger than one, the material exhibits solid-like behavior which induces the largest decay in the oscillations amplitude of the polymer chain oscillator.

The derived approximate frequency-amplitude relationship sheds new light on understanding how the fractal parameter value correlates with relaxation oscillations of polymer chains. Therefore, understanding the fractal nature of the material molecular structure can help to have polymer materials with the desired dynamic response.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Tecnológico de Monterrey funded this research through the Research Group of Nanotechnology for Devices Design, and by the Consejo Nacional de Ciencia y Tecnología de México (Conacyt), Project Numbers 242269, 255837, and 296176.

ORCID iDs

Alex Elías-Zúñiga

Luis M Palacios-Pineda

Daniel Olvera Trejo

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