Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms

Abstract

In this work we consider the nonlinear equivalent representation form of oscillators that exhibit nonlinearities in both the elastic and the damping terms. The nonlinear damping effects are considered to be described by fractional power velocity terms which provide better predictions of the dissipative effects observed in some physical systems. It is shown that their effects on the system dynamics response are equivalent to a shift in the coefficient of the linear damping term of a Duffing oscillator. Then, its numerical integration predictions, based on its equivalent representation form given by the well-known forced, damped Duffing equation, are compared to the numerical integration values of its original equations of motion. The applicability of the proposed procedure is evaluated by studying the dynamics response of four nonlinear oscillators that arise in some engineering applications such as nanoresonators, microresonators, human wrist movements, structural engineering design, and chain dynamics of polymeric materials at high extensibility, among others.

1. Introduction

The aim of this paper focuses on using a nonlinear approach to transform the forced nonlinear equation: 𝑥̈+𝑓(𝑥)+𝑓₁𝜈𝑥̇,𝜅𝑥̇^𝑝=𝑄₀cos𝜔_𝑓𝑡,𝑥(0)=𝑥₀,𝑥̇(0)=0, (1) with nonlinear damping terms into an equivalent forced, linearly damped Duffing’s equation. Here we assume that 𝑓(𝑥) is the system restoring force which could have rational or irrational conservative force terms, 𝑓₁(𝜈𝑥̇,𝜅𝑥̇^𝑝) represents the system nonlinear dissipative effects, 𝜈 and 𝜅 are damping constants, 𝑝 is the exponent of the velocity, 𝑄₀ is the driving force magnitude, 𝜔_𝑓 is the system driving frequency, 𝑡 is the current time, and 𝑥₀ is the initial amplitude. The main motivation in studying the equivalent representation form of (1) with nonlinear damping terms for which 𝑝>0 comes from the fact that the addition of nonlinear damping to the system could remove undesirable effects over the nonresonant regions that can help to improve the overall performance of Duffing-type vibration isolators [1, 2]. Furthermore, during the study of the dynamics response of resonators made from carbon nanotubes and graphene, Lifshitz and Cross [3] and Eichler et al. [4] concluded that damping is strongly dependent on the amplitude of motion and that its effects are better described by nonlinear damping forces. They also concluded that the nonlinearities could be associated with a dissipation channel exterior to the resonator, such as the manner in which the resonator is clamped by its boundaries to the surrounding material, friction effects associated with the sliding between the nanotube/graphene and the metal electrode, and the phonon-phonon interactions, among others. To quantify the nonlinear dissipative effects observed during the performance of micromechanical oscillators, Zaitzev and coworkers designed a doubly clamped beam oscillator and performed several experimental studies to understand the phenomenon of nonlinear damping. They found that nonlinear damping plays an important role in the dynamics response of the micromechanical beam oscillator [5].

On the other hand, it is known that structural engineering design utilizes nonlinear damper devices to reduce the forces exerted in the dampers that could exceed the device force capacity during the structure dynamics response to earthquakes. In this case, the exponent of the velocity is selected on the interval of 0.3<𝑝<1.95 [6]. Martínez-Rodrigo and Romero [7] found that when the nonlinear dampers velocity exponent 𝑝 is slightly less that 1, the forces in the dampers can be reduced more than 35% during the retrofitting of a multistory that leads to a similar structural performance when compared to the usage of linear dampers. Similar results were reported in [8] in which the utilization of nonlinear viscous dampers reduces the displacement response of existing girder bridges and arch bridge structures.

Since nonlinear dampers with fractional powers in the velocity terms are commonly used to model the rhythmic movement of the wrist, the inclusion of a nonlinear damping term for which 𝑝=1/5 is considered in (1). This dynamics model is known as the one-fifth power law model [9]. Of course, there are other models that consider different values of 𝑝 to characterize the damping and the elastic nonlinearities observed in different experimental measurements [10–12]. In an attempt to cover the gap between viscous, dry friction, and drag forces in turbulent fluids, Litak and coworkers considered a nonlinear damping term with a fractional exponent to model the behavior of double well oscillators. They identified the critical values of 𝑄₀ that induced chaotic vibrations in the double well system with a nonlinear damping term [13]. Later, Borowiec and coworkers applied the Melnikov criterion to examine the global homoclinic bifurcation and the transition to chaos in a forced Duffing oscillator with nonlinear fractional damping term. They found, by using perturbation methods, the critical forcing amplitude above from which the system can behave chaotically [14].

It is clear that the inclusion of nonlinear damping terms in (1) allows a better prediction of the system dynamics response behavior [15–23]. Therefore, the aim of this paper focuses on modifying the nonlinear transformation approach introduced in [24] to find the equivalent representation form of (1) in which nonlinear damping with velocity terms to the power 𝑝 is included.

2. A Nonlinear Transformation Procedure

To transform (1) into a driving, linearly damped differential equation of the Duffing type, we will follow the procedure introduced in [24] and consider first the transformation of the conservative force terms 𝑓(𝑥), which do not have a cubic-like representation form, by using the classical Chebyshev polynomials of the first kind [25–29], since this provides a uniform approximation and requires a smaller number of expansion terms in comparison, for instance, to Taylor series to obtain good accuracy [30]: 𝑓(𝑥)=^𝑁_𝑛=0𝑏_2𝑛+1(𝑥)𝑇_2𝑛+1(𝑥), (2) where 𝑏_2𝑛+1=2𝜋⁺¹₋₁1√1−𝑥²𝑓(𝑥)𝑇_2𝑛+1(𝑥)𝑑𝑥, (3) and 𝑇_2𝑛−1 are the Chebyshev polynomials of the first kind defined as 𝑇_2𝑛+1(𝑥)=cos(2𝑛+1)cos⁻¹(𝑥,[])𝑥∈−1,1,𝑛=0,1,2,…. (4)

The first three terms are given by 𝑇₁(𝑥)=𝑥,𝑇₃(𝑥)=4𝑥³𝑇−3𝑥,₅(𝑥)=16𝑥⁵−20𝑥³+5𝑥. (5)

Thus, the equivalent restoring force 𝑓(𝑥) can be written as 𝑓(𝑥)≡𝑏₁𝑞𝑇₁𝑦+𝑏₃𝑞𝑇₃𝑦≈𝛼(𝐴)𝑥+𝛽(𝐴)𝑥³+𝛾(𝐴)𝑥⁵. (6)

Substituting (2) into (3), by using (5), provides the following equivalent representation form of 𝑓(𝑥): 𝑓(𝑥)≈𝛼(𝑥)𝑥+𝛽(𝑥)𝑥³+𝛾(𝑥)𝑥⁵. (7)

Then, (1) can be equivalently written as 𝑑²𝑥𝑑𝑡²+𝛼(𝑥)𝑥+𝛽(𝑥)𝑥³+𝛾(𝑥)𝑥⁵+𝑓₁𝜈𝑥̇,𝜅𝑥̇^𝑝−𝑄₀cos𝜔_𝑓𝑡≈0. (8)

Since nonlinear damping terms are involved in the dissipative force expression 𝑓₁(𝜈𝑥̇,𝜅𝑥̇^𝑝), we modify our cubication nonlinear transformation approach introduced in [24] by adding a constant term 𝑐(𝑥) to the resulting Duffing equation. This term will take into account the asymmetric dynamics behavior described by (1) due to the presence of the nonlinear dissipative effects. For instance, let us consider the forced Duffing equation with nonlinear damping terms: 𝑥̈+𝜈𝑥̇+𝜅₀𝑥̇|𝑥̇|^𝑝−1+𝛼𝑥+𝛽𝑥³=𝑄₀cos𝜔_𝑓𝑡,with𝑥(0)=𝑥₀,𝑥̇(0)=0, (9) where 𝑦 denotes the displacement of the system, 𝜈 and 𝜅₀ are damping coefficients, 𝛼, 𝛽, and 𝑄₀ are system constant parameters, and 𝜔_𝑓 represents the driving frequency. Figure 1 shows the phase portrait of (1) by considering the values of 𝑝=1 and 2.5. Notice that when 𝑝≠1, the corresponding phase portrait exhibits asymmetric dynamic behavior about the equilibrium point (0,0). Therefore, we write the restoring force term of (8) in the form 𝐹(𝑥,𝑥̇)=𝛼(𝑥)𝑥+𝛽(𝑥)𝑥³+𝛾(𝑥)𝑥⁵+𝑓₁𝜈𝑥̇,𝜅𝑥̇^𝑝≡𝛿(𝑥)𝑥+𝜖(𝑥)𝑥³+𝜅𝜈₁+𝜈𝑥̇+𝑐(𝑥), (10) where 𝜈₁, 𝑐(𝑥), 𝛿(𝑥), and 𝜖(𝑥) can be found from 𝐹₁𝛿,𝜖,𝑐,𝜈₁=^𝜎₀𝑓₁𝜈𝑥̇,𝜅𝑥̇^𝑝+𝛼𝑥+𝛽𝑥³+𝛾𝑥⁵+𝑄₀cos𝜔_𝑓𝑡−𝜅𝜈₁+𝜈𝑥̇−𝛿𝑥−𝜖𝑥³−𝑐−𝑄₀cos𝜔_𝑓𝑡²𝐹𝑑𝑥⟶min,₂𝛿,𝜖,𝑐,𝜈₁=^𝑣₀𝑓₁𝜈𝑥̇,𝜅𝑥̇^𝑝+𝛼𝑥+𝛽𝑥³+𝛾𝑥⁵+𝑄₀cos𝜔_𝑓𝑡−𝜅𝜈₁+𝜈𝑥̇−𝛿𝑥−𝜖𝑥³−𝑐−𝑄₀cos𝜔_𝑓𝑡²𝑑𝑥̇⟶min,𝜕𝐹₁𝛿,𝜖,𝑐,𝜈₁𝜕𝛿=0,𝜕𝐹₁𝛿,𝜖,𝑐,𝜈₁𝜕𝜖=0,𝜕𝐹₁𝛿,𝜖,𝑐,𝜈₁𝜕𝑐=0,𝜕𝐹₂𝛿,𝜖,𝑐,𝜈₁𝜕𝜈₁=0. (11)(12)(13)(14)

Figure 1 

Steady-state phase portraits of 𝑥̈+𝜈𝑥̇+𝜅0𝑥̇|𝑥̇|𝑝−1+𝛼𝑥+𝛽𝑥3 = 𝑄0cos(𝜔𝑓𝑡) when 𝛼=1, 𝛽=1, 𝜈=0.1, 𝜅=0.9, 𝑄=1, 𝜔𝑓=1, and 𝑝=1 and 2.5.

In (11) and (12), 𝜎 and 𝜐 are fitting parameters whose values must be fixed to ensure that the equivalent restoring forces are qualitatively and quantitatively similar to those of the original equations of motion. During the transformation process, the values of 𝑄₀ and 𝜔_𝑓 are assumed to remain constant. Once the expressions to determine 𝜈₁, 𝑐(𝑥), 𝛿(𝑥), and 𝜖(𝑥) are found, then (1) can be written as the following Duffing-type equation of motion: 𝑑²𝑥𝑑𝑡²+𝜅𝜈₁̇(𝑥)𝑥(𝑥)𝑥+𝜈𝑥+𝛿+𝜖³+𝑐(𝑥)≈𝑄₀cos𝜔_𝑓𝑡. (15) Notice that (1) only contains linear damping terms since the nonlinear damped effects are now contained in the (𝜅|𝜈₁|+𝜈)𝑥̇ term [18].

To assess the accuracy of the proposed solution procedure, we will next examine the dynamics response of forced oscillators with nonlinear damping terms such as the Duffing equation, the cubic-quintic Duffing equation, the rational form elastic term oscillator, and the finite extensibility nonlinear oscillator (FENO).

3. Forced Duffing Equation with Nonlinear Damping

Here we consider the forced, damped Duffing equation given by (9) and introduce the following change of variable 𝑥=𝐴/𝑦 which transforms it into an equation of the form 𝑥̈+𝜈𝑥̇+𝜅𝑥̇|𝑥̇|^𝑝−1+𝐴𝑥+𝐵𝑥³=𝑄cos𝜔_𝑓𝑡,with𝑥(0)=1,𝑥̇(0)=0, (16) where 𝜅=𝜅₀𝑥₀^(𝑝−1) and 𝑄=𝑄₀/𝑥₀. We next use (12) and (14) to derive the following forced, damped Duffing equation of the form 𝑥̈+𝜅𝜈₁+𝜈𝑥̇+𝐴𝑥+𝐵𝑥³+𝑐=𝑄cos𝜔_𝑓𝑡, (17) where 𝑐 and 𝜈₁ are given by 𝜐𝑐=−2𝜅𝑝−1^𝑝,𝜈𝑝+2₁=3𝑝𝜐^(𝑝−1)4+4𝑝+𝑝². (18)

Here the values of 𝜐 must satisfy (12). To assess the accuracy of (17) when compared to the numerical integration of (16), let us consider the parameter values of 𝐴=1, 𝐵=1, 𝜈=0.1, 𝜅=0.9, 𝑄=1, and 𝜔_𝑓=1 with 𝑝= 0.1, 0.387, 1.5, and 2.5 with the initial conditions of 𝑥(0)=1 and 𝑥̇(0)=0. Figure 2 shows the comparison of the amplitude-time curves obtained from the numerical integration solutions of (16) and (17). As we can see from these amplitude-time plots, the predicted curves obtained from (17) follow well the numerical integration solutions of (16). In fact, the computed root-mean-square error (RMSE) values are 0.1196, 0.0874, 0.1530, and 0.1351 for the 𝑝 values of 0.1, 0.387, 1.5, and 2.5, respectively. In Figure 2, the solid lines represent the numerical integration solutions of (16), while the dashed lines represent the numerical integration solutions of (17). Therefore, we can conclude that our solution procedure provides an equivalent equation of motion that describes the dynamics behavior of the oscillatory system (16) well even when nonlinear damping terms are considered.

(a)

(b)

(c)

(d)

Figure 2 

Amplitude-time response curves of (16) and (17). Here, the solid lines represent the numerical integration solutions of (16), while the dashed lines represent the numerical integration solutions of (17). The parameter values are 𝐴=1, 𝐵=1, 𝜈=0.1, 𝜅=0.9, 𝑄=1, and 𝜔𝑓=1, with 𝑥(0)=1 and 𝑥̇(0)=0. (a) 𝑝=0.1, 𝑐=0.8535, 𝜈1=0.0574, and 𝜐=2.75; (b) 𝑝=0.387, 𝑐=0.5544, 𝜈1=0.3646, and 𝜐=1.6; (c) 𝑝=1.5, 𝑐=−0.0221, 𝜈1=0.5677, and 𝜐=0.195; (d) 𝑝=2.5, 𝑐= −0.0434, 𝜈1=0.3451, and 𝜐=0.35.

To further assess the accuracy of our proposed approach, we will next study the forced cubic-quintic Duffing oscillator with linear and nonlinear damping terms.

4. Cubic-Quintic Duffing Equation with Nonlinear Damping Terms

In this case, we derive the equivalent representation form of the cubic-quintic oscillator in which nonlinear damping terms are considered. Here, we assume that the equation of motion has the form 𝑦̈+2𝜈𝑦̇+𝜂₀𝑦²𝑦̇+𝜅₀𝑦̇𝑦̇^𝑝−1+𝛼𝑦+𝛽𝑦³+𝛾𝑦⁵=𝑄₀cos𝜔_𝑓𝑡, (19) with initial conditions given by 𝑦(0)=𝑦₀ and 𝑦(̇0)=0. In (19) 𝑦 denotes the displacement of the system, 𝜈, 𝜂₀, and 𝜅₀ are damping coefficients, 𝛼, 𝛽, and 𝛾 are system constant parameters, and 𝑄₀ and 𝜔_𝑓 are the driving force and frequency, respectively. Notice that the nonlinear damping terms 𝜂₀𝑦²𝑦̇+𝜅₀𝑦|̇𝑦|̇^𝑝−1 of (19) are commonly used to model the dynamics of nanoresonators and microresonators. See [3–5] and references cited therein. If we let 𝑥=𝐴/𝑦, thus, (19) becomes 𝑥̈+2𝜈𝑥̇+𝜂𝑥²𝑥̇+𝜅𝑥̇|𝑥̇|^𝑝−1+𝐴𝑥+𝐵𝑥³+𝐺𝑥⁵=𝑄cos𝜔_𝑓𝑡, (20) with 𝑥(0)=1, 𝑥̇(0)=0, 𝐴=𝛼, 𝐵=𝛽𝑦²₀, 𝐺=𝛾𝑦⁴₀, 𝜂=𝜂₀𝑦₀, 𝜅=𝜅₀𝑦₀^(𝑝−1), and 𝑄=𝑄₀/𝑦₀. In accordance with our proposed nonlinear transformation approach, the system terms 2𝜈𝑥̇+𝜂𝑥²𝑥̇+𝜅𝑥̇^𝑝+𝐴𝑥+𝐵𝑥³+𝐺𝑥⁵ are replaced by an equivalent cubic-like polynomial expression of the form 2𝜈𝑥̇+𝜂𝑥²𝑥̇+𝜅𝑥̇|𝑥̇|^𝑝−1+𝐴𝑥+𝐵𝑥³+𝐺𝑥⁵≡𝜅𝜈₁+𝜈𝑥̇+𝛿𝑥+𝜖𝑥³+𝑐. (21)

Thus, the equivalent nonlinear transformation form of (20) is given as 𝑥̈+𝜅𝜈₁+𝜈𝑥̇+𝛿𝑥+𝜖𝑥³+𝑐≈𝑄cos𝜔_𝑓𝑡, (22) where 𝛿=𝐴+5𝜂𝜐𝜎−1225𝐺𝜎⁴,563𝜖=𝐵+21𝜂𝜐+40𝐺𝜎³,𝜈162𝜎₁=3𝑝𝜐^𝑝−1+2+𝑝252𝜐𝜈+273𝜂𝜐𝜎²−80𝐺𝜎⁵,𝜐252𝜅𝜐𝑐=−2𝜅𝑝−1^𝑝+2+𝑝848𝐺𝜎⁵−2541𝜂𝜐𝜎².2268 (23)

To evaluate the accuracy of (22), we use the parameter values of 𝛼=1, 𝛽=5, 𝛾=1, 𝜈=0.1, 𝜂₀=0.15, 𝜅₀=0.25, 𝑄=1, and 𝜔_𝑓= 0.5, 1.5, 2, and 2.75 with the initial conditions of 𝑥(0)=1 and 𝑥̇(0)=0. Figure 3 illustrates the amplitude versus time plots by considering a nonlinear damping term of order 𝑝=3/4. The solid black lines represent the numerical integration of (19) while the red dots represent the numerical integration of (22). Notice that in all cases, the predicted solutions obtained from (22) follow well the numerical integration solution of the original equation of motion. The computed parameter values are 𝛿=0.9147, 𝜖=5.5863, 𝜈₁=1.395, 𝑐=0.0572, 𝜎=0.685, and 𝜐=0.05. In all cases, the maximum RMSE value does not exceed 0.0666. As a second example, let us consider the following system parameter values: 𝛼=1, 𝛽=1, 𝛾=0.1, 𝜈₀=0.05, 𝜂₀=0.15, 𝜅₀=0.1, 𝜔_𝑓=2.75, and 𝑄=1. Here the computed values for 𝛿, 𝜖, 𝜈₁, and 𝑐 are, respectively, 0.9975, 1.0679, 1.2806, and −0.0045 with 𝜎=0.645 and 𝜐=0.11. The corresponding RMSE value computed on 0⩽𝑡⩽800 is 0.0188. Figure 4 illustrates the amplitude-time response curves, the phase space, and the Morlet continuum wavelet transform plots. As usual, the solid black lines represent the numerical integration of (20) while the red dots describe the numerical integration solution of (22). From these plots, it is clear that the equivalent representation form (22) of (20) describes well the system dynamics response. Of course, the equivalent expression (22) can be used if different system parameter values are considered.

(a)

(b)

(c)

(d)

Figure 3 

Amplitude-time response curves of (19) and (22). The parameter values are 𝛼=1, 𝛽=5, 𝛾=1, 𝜈=0.1, 𝜂0=0.15, 𝜅0=0.25, 𝑄=1, 𝑝=3/4, and 𝜔𝑓= 0.5, 1.5, 2, and 2.75 with 𝑥(0)=1 and 𝑥̇(0)=0. The computed parameter values are 𝛿=0.9147, 𝜖=5.5863, 𝜈1=1.395, 𝑐=0.0572, 𝜎=0.685, and 𝜐=0.05. (a) Amplitude-time response curves for 𝜔𝑓=0.5; (b) amplitude-time response curves for 𝜔𝑓=1.5; (c) amplitude time-response curves for 𝜔𝑓=2; (d) amplitude-time response curves for 𝜔𝑓=2.75. Here the solid black lines represent the numerical integration solution of (19) while the red dots represent the prediction obtained by using the derived equivalent equation of motion (22).

Figure 4 

Amplitude-time response curves of (19) and (22). The parameter values are 𝛼=1, 𝛽=1, 𝛾=0.1, 𝜐0=0.05, 𝜂0=0.15, 𝜅0=0.1, 𝜔𝑓=2.75, and 𝑄=1, with 𝑥0=1 and 𝑥̇(0)=0. Here 𝛿=0.9975, 𝜖=1.0679, 𝜐1=1.2806, and 𝑐=−0.0045, with 𝜎=0.645 and 𝜐=0.11. (a) Morlet continuum wavelet transform plot; (b) transient response phase portrait on 0⩽𝑡⩽50; (c) the steady-state system response phase portrait on 750⩽𝑡⩽800; (d) amplitude-time response curves. Here the solid lines represent the numerical integration solution of (19), while the red dots represent predicted values obtained by using the derived equivalent equation of motion (22).

We will next derive the equivalent representation form of the forced power-form elastic term oscillator with linear and nonlinear damping terms.

5. The Forced Power-Form Elastic Term Oscillator with Nonlinear Damping

The equation of motion of this oscillator is assumed to be given as 𝑑²𝑦𝑑𝑡²+2𝜈𝑑𝑦𝑑𝑡+𝜅₀𝑦̇𝑦̇^𝑝−1+𝜔²_𝑛𝑦+ℎsgn𝑦𝑦^𝑚=𝑄₀cos𝜔_𝑓𝑡,𝑦(0)=𝐴,𝑦̇(0)=0, (24) where 𝜔_𝑛, 𝜈, 𝜅₀, and ℎ are constant parameters and 𝑚 can take any nonnegative real value on 0≤𝑚<∞ [31, 32]. By introducing the coordinate transformation 𝑥=𝑦/𝐴, (24) can be written as 𝑑²𝑥𝑑𝑡²+2𝜈𝑑𝑥𝑥̇|𝑥̇|𝑑𝑡+𝜅^𝑝−1+𝜔²_𝑛𝑥+𝑐₁sgn(𝑥)|𝑥|^𝑚=𝑄cos𝜔_𝑓𝑡,with𝑐₁=ℎ𝐴^(𝑚−1),𝜅=𝜅₀𝐴^(𝑝−1),𝑄𝑄=⁰𝐴𝑥(0)=1,𝑥̇(0)=0. (25)

The restoring conservative forces 𝜔²_𝑛𝑥+𝑐₁sgn(𝑥)|𝑥|^𝑚 can be replaced by a cubic-quintic polynomial by using the Chebyshev polynomial expansion (2) which provides 𝜔²_𝑛𝑥+𝑐₁sgn(𝑥)|𝑥|^𝑚≈𝛼₂𝑥+𝛽𝑥³+𝛾𝑥⁵+Δ𝑥⁷+𝜀𝑥⁹, (26) where 𝛼₂=5𝑐₁(𝑚−9)(𝑚−7)(𝑚−5)(𝑚−3)Γ[𝑚/2+1]8√𝜋Γ[(11+𝑚)/2]+𝜔²_𝑛,𝛽=−10𝑐₁(𝑚−9)(𝑚−7)(𝑚−5)(𝑚−1)Γ[𝑚/2+1]√)],𝜋Γ[(11+𝑚/2𝛾=42𝑐₁(𝑚−9)(𝑚−7)(𝑚−3)(𝑚−1)Γ[𝑚/2+1]√)],𝜋Γ[(11+𝑚/2Δ=−64𝑐₁(𝑚−9)(𝑚−5)(𝑚−3)(𝑚−1)Γ[𝑚/2+1]√)],𝜋Γ[(11+𝑚/2𝜀=32𝑐₁(𝑚−7)(𝑚−5)(𝑚−3)(𝑚−1)Γ[𝑚/2+1]√)].𝜋Γ[(11+𝑚/2 (27)

Here Γ[•] represents the Euler gamma function. We next use our solution procedure previously described and determine the equivalent representation form of 𝑑²𝑥𝑑𝑡²+2𝜈𝑑𝑥𝑥̇|𝑥̇|𝑑𝑡+𝜅^𝑝−1+𝛼₂𝑥+𝛽𝑥³+𝛾𝑥⁵+Δ𝑥⁷+𝜀𝑥⁹=𝑄cos𝜔_𝑓𝑡. (28) This gives the following equivalent equation of motion: 𝑑²𝑥𝑑𝑡²+𝜅𝜈₁+𝜈𝑑𝑥𝑑𝑡+𝛿𝑥+𝜖𝑥³+𝑐=𝑄cos𝜔_𝑓𝑡, (29) where 𝛿=𝛼₂−25𝛾𝜎⁴−6335Δ𝜎⁶−6681𝜀𝜎⁸,143𝜖=𝛽+100𝛾𝜎²+781455Δ𝜎⁴+432𝜀𝜎⁶,𝜐2574𝑐=−2𝜅𝑝−1^𝑝+𝑝+2212𝛾𝜎⁵+56771Δ𝜎⁷+99712𝜀𝜎⁹,𝜈715₁=3𝑝𝜐^𝑝−1+𝜈𝑝+2𝜅−14300𝛾𝜎⁵+28665Δ𝜎⁷+40824𝜀𝜎⁹.45045𝜐 (30)

To study the influence in the system dynamics of the nonlinear damping term of (24), the parameter values of 𝑚=3/2, 𝐴=1, 𝜔_𝑛=1, ℎ=1, 𝑄₀=1, 𝜈=0.1, 𝜅=0.9, 𝜔_𝑓=0.75, and 𝑝= 1/4, 3/4, 1.5, and 2.75 are considered. Figure 5 illustrates the amplitude-time response curves obtained by numerically integrating (25) and (29). As we can see from Figure 5, the numerical integration of (29) closely follows the amplitude-time response curve obtained from (25). Here, the black solid and the red dots represent, respectively, the numerical integration solution of (25) and (29). Table 1 shows the estimated RMSE values and the computed parameter values of 𝛿, 𝜖, 𝑐, 𝜈₁, 𝜎, and 𝜐. Based on these results, it is concluded that our equivalent representation form (29) describes well the numerical integration solution of (25).

Table 1 

Computed values of the forced power-form elastic term oscillator with nonlinear damping term. The assumed parameter values are 𝑚=3/2, 𝐴=1, 𝜔_𝑛=1, ℎ=1, 𝑄₀=1, 𝜈=0.1, 𝜅=0.9, and 𝜔_𝑓=3/4.

(a)

(b)

(c)

(d)

Figure 5 

Amplitude-time response curves of (25) and (29). The parameter values are 𝑚=3/2, 𝐴=1, 𝜔𝑛=1, ℎ=1, 𝑄0=1, 𝜈=0.1, 𝜅=0.9, and 𝜔𝑓=0.75, with 𝑥0=1 and 𝑥̇(0)=0. (a) Amplitude-time response curves for 𝑝=1/4; (b) amplitude-time response curves for 𝑝=3/4; (c) amplitude time-response curves for 𝑝=1.5; (d) amplitude-time response curves for 𝑝=2.75. Here the solid black lines represent the numerical integration solution of (25), while the red dots represent the prediction obtained by using the derived equivalent equation of motion (29).

We will next develop the equivalent representation form of the finite extensibility nonlinear oscillator with nonlinear damping terms that models the chain dynamics of polymeric materials at high extensibility.

6. The Finite Extensibility Nonlinear Oscillator (FENO)

As a final example, we now focus our attention on determining the equivalent representation form of the FENO dynamical system: 𝑑²𝑦𝑑𝑡²+2𝜈𝑑𝑦𝑦̇𝑦̇𝑑𝑡+𝜅^𝑝−1+𝑦1−𝐴²𝑦²=𝑄cos𝜔_𝑓𝑡,𝑦(0)=1,𝑦̇(0)=0, (31) where 𝜈 and 𝜅=𝜅₀𝐴^(𝑝−1) are the damping coefficients, 𝑄=𝑄₀/𝐴 is the driving force, 𝜔_𝑓 represents the driving frequency, and 𝐴 is the initial oscillation amplitude on 0<𝐴<1 [33]. By using Chebyshev polynomials, the conservative force of (31) can be written in equivalent form as follows 𝑦1−𝐴²𝑦²≡𝛼(𝐴)𝑦+𝛽(𝐴)𝑦³+𝛾(𝐴)𝑦⁵+Δ(𝐴)𝑦⁷+𝜀(𝐴)𝑦⁹, (32) in which 𝛼(𝐴)=1𝐴¹⁰1−𝐴²𝐴⁴9504−56001−𝐴²+𝐴⁸330−501−𝐴²−46081−𝐴²−1+48𝐴⁶251−𝐴²−66+256𝐴²351−𝐴²,𝛽(𝐴)=−4416𝐴¹⁰1−𝐴²𝐴⁶2244−8001−𝐴²+128𝐴²71−561−𝐴²+38401−𝐴²−1+𝐴⁸251−𝐴²−198+8𝐴⁴5251−𝐴²,𝛾(𝐴)−913=−32𝐴¹⁰1−𝐴²4𝐴⁶913−3151−𝐴²+128𝐴²125−981−𝐴²+69121−𝐴²−1+𝐴⁸351−𝐴²−297+16𝐴⁴4411−𝐴²,Δ(𝐴)=−779256𝐴¹⁰1−𝐴²64𝐴²41−321−𝐴²+8𝐴⁶71−241−𝐴²+11521−𝐴²−1+𝐴⁸51−𝐴²−44+80𝐴⁴141−𝐴²,𝜀(𝐴)−25=−512𝐴¹⁰1−𝐴²64𝐴²9−71−𝐴²+𝐴⁸1−𝐴²−9−40𝐴⁶1−𝐴²−3+2561−𝐴²−1+48𝐴⁴51−𝐴²−9. (33) Thus, (31) becomes 𝑑²𝑦𝑑𝑡²+2𝜈𝑑𝑦𝑦̇𝑦̇𝑑𝑡+𝜅^𝑝−1+𝛼(𝐴)𝑦+𝛽(𝐴)𝑦³+𝛾(𝐴)𝑦⁵+Δ(𝐴)𝑦⁷+𝜀(𝐴)𝑦⁹=𝑄cos𝜔_𝑓𝑡. (34)

We next determine the Duffing-like representation form of (34) by using (11)–(14). This yields 𝑑²𝑦𝑑𝑡²+𝜅𝜈₁+𝜈𝑑𝑦𝑑𝑡+𝛿𝑦+𝜖𝑦³+𝑐=𝑄cos𝜔_𝑓𝑡, (35) where 𝛿=𝛼−25𝛾𝜎⁴−6335Δ𝜎⁶−6681𝜀𝜎⁸,143𝜖=𝛽+100𝛾𝜎²+781455Δ𝜎⁴+432𝜀𝜎⁶,𝜐2574𝑐=−2𝜅𝑝−1^𝑝+𝑝+2212𝛾𝜎⁵+56771Δ𝜎⁷+99712𝜀𝜎⁹,𝜈715₁=3𝑝𝜐^𝑝−1+𝜈𝑝+2𝜅−14300𝛾𝜎⁵+28665Δ𝜎⁷+40824𝜀𝜎⁹.45045𝜐 (36) Notice that equations in (36) are similar to those derived for the forced power-form elastic term oscillator with nonlinear damping term but with different expressions to determine the values of 𝛼, 𝛽, 𝛾, Δ, and 𝜀.

To illustrate the accuracy attained from the derived nonlinear equivalent equation of motion of the FENO oscillator, let us consider the parameter values of 𝜈, 𝜅₀, 𝑄₀, 𝜔_𝑓, and 𝐴 to be, respectively, 0.1, 0.9, 1, 0.75, and 0.9 with 𝑝= 0.25, 0.5, 1.25, and 2.5. In this case 𝛼=1.224, 𝛽=−3.1263, 𝛾=19.0743, Δ=−31.445, and 𝜀=19.4869. The computed values of 𝛿, 𝜖, 𝑐, 𝜈₁, 𝜎, and 𝜐 are summarized in Table 2. Figure 6 exhibits the amplitude-time plots computed from the numerical integration solutions of (31) and (35). Figure 6 shows that the amplitude-time response curves obtained form (35) describes well the qualitative and quantitative behavior of (31). In fact, the RMSE does not exceed the value of 0.1984. Here the black solid lines represent the numerical integration solution of (31), while the red dots represent the numerical integration results obtained from (35).

Table 2 

Computed values of the finite extensibility nonlinear oscillator (FENO) with nonlinear damping term. The assumed parameter values are 𝐴=0.9, 𝜈=0.1, 𝑄₀=1, and 𝜅₀=0.9, with 𝜔_𝑓=0.75.

(a)

(b)

(c)

(d)

Figure 6 

Amplitude-time response curves of (31) and (35). The parameter values are 𝐴=0.9, 𝜈=0.1, 𝑄0=1, and 𝜅0=0.9, with 𝜔𝑓=0.75. (a) Amplitude-time response curves for 𝑝=1/4; (b) amplitude-time response curves for 𝑝=1/2; (c) amplitude time-response curves for 𝑝=1.25; (d) amplitude-time response curves for 𝑝=2.5. Here the solid black lines represent the numerical integration solution of (31) while the red dots represent the prediction obtained by using the derived equivalent equation of motion (35).

7. Conclusions

In this paper, we have introduced a nonlinear procedure to obtain the equivalent representation form of forced nonlinear oscillators that involve nonlinear damping that depends on velocity terms to the power 𝑝 in which 𝑝 is a positive number that can be even odd or fractional; that is, 𝑝>0. Then, we have obtained the equivalent equation of motion of oscillators that are commonly used to study, among others, nanoresonators, microresonators, vibration isolators, the dynamics of the human movement, the retrofitting of existing girder bridges and arch bridge structures, and the finite extensibility nonlinear oscillator that is used to characterize chain dynamics of polymeric-like materials. The comparison among the numerical integration solutions of the original equations of motion and the equivalent ones exhibits good agreement in all the cases considered here even at larger nonlinear stiffness and damping parameter values. In fact, the computed RMSE values do not exceed of 0.1984 for all the oscillators studied here. Therefore, we can conclude that our nonlinear solution procedure which is developed to determine the equivalent representation form of forced oscillators with nonlinear stiffness and damping effects describes well the qualitative and quantitative dynamics behavior of the original equations of motion. Furthermore, the equivalent equations of the original equations of motion derived by using our nonlinear transformation approach can be used to study the system bifurcations and basins of attraction, since it provides equivalent representation forms that are close to those provided by Melnikov analysis [18].

Conflict of Interests

The authors declare that they have no conflict of interests with any mentioned identities in their paper.

Acknowledgments

This work was funded by Tecnológico de Monterrey-Campus Monterrey, through the Research Chair in Nanomaterials for Medical Devices and Research Chair in Intelligent Machines. Additional support was provided by the European Union Seventh Framework Programme (FP7 PEOPLE-2009) under the grant agreement IRSES no. 247476 and from Consejo Nacional de Ciencia y Tecnología (Conacyt), México.