MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 383947 10.1155/2013/383947 383947 Research Article Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms Elías-Zúñiga Alex 1 http://orcid.org/0000-0002-4385-6269 Olvera Daniel 1 Ferrer Real Inés 2 Martínez-Romero Oscar 1 Sanjuán Miguel A. F. 1 Centro de Innovación en Diseño y Tecnología Tecnológico de Monterrey, Campus Monterrey E. Garza Sada 2501 Sur 64849 Monterrey NL Mexico itesm.mx 2 Department of Mechanical Engineering and Industrial Construction University of Girona, Maria Aurelia Capmany 61, 17071 Girona Spain udg.edu 2013 13 11 2013 2013 22 07 2013 17 09 2013 17 09 2013 2013 Copyright © 2013 Alex Elías-Zúñiga et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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: (1)x¨+f(x)+f1(νx˙,κx˙p)=Q0cos(ωft),hhhhhhhhhhhhhhhx(0)=x0,x˙(0)=0, 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(νx˙,κx˙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, ωf 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  and Eichler et al.  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 .

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 . Martínez-Rodrigo and Romero  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  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 . 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 . 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 . 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 .

It is clear that the inclusion of nonlinear damping terms in (1) allows a better prediction of the system dynamics response behavior . Therefore, the aim of this paper focuses on modifying the nonlinear transformation approach introduced in  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  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 , 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 : (2)f(x)=n=0Nb2n+1(x)T2n+1(x), where (3)b2n+1=2π-1+111-x2f(x)T2n+1(x)dx, and T2n-1 are the Chebyshev polynomials of the first kind defined as (4)T2n+1(x)=cos[(2n+1)cos-1(x)],hhhlhhhx[-1,1],n=0,1,2,.

The first three terms are given by (5)T1(x)=x,T3(x)=4x3-3x,T5(x)=16x5-20x3+5x.

Thus, the equivalent restoring force f(x) can be written as (6)f(x)b1(q)T1(y)+b3(q)T3(y)α(A)x+β(A)x3+γ(A)x5.

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

Then, (1) can be equivalently written as (8)d2xdt2+α(x)x+β(x)x3+γ(x)x5d2xdt2+f1(νx˙,κx˙p)-Q0cos(ωft)0.

Since nonlinear damping terms are involved in the dissipative force expression f1(νx˙,κx˙p), we modify our cubication nonlinear transformation approach introduced in  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: (9)x¨+νx˙+κ0x˙|x˙|p-1+αx+βx3=Q0cos(ωft),hhhhhhhhhhhhhhhwith  x(0)=x0,x˙(0)=0, where y denotes the displacement of the system, ν and κ0 are damping coefficients, α, β, and Q0 are system constant parameters, and ωf 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 p1, 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 (10)F(x,x˙)=α(x)x+β(x)x3+γ(x)x5+f1(νx˙,κx˙p)δ(x)x+ϵ(x)x3+(κ|ν1|+ν)x˙+c(x), where ν1, c(x), δ(x), and ϵ(x) can be found from (11)F1(δ,ϵ,c,ν1)=0σ((ωft)f1(νx˙,κx˙p)+αx+βx3hhhh+γx5+Q0cos(ωft)hhhh-(κ|ν1|+ν)x˙-δx-ϵx3hhhh-c-Q0cos(ωft))2dxmin,(12)F2(δ,ϵ,c,ν1)=0v((ωft)(ωft)(ωft)f1(νx˙,κx˙p)+αxhhhh+βx3+γx5+Q0cos(ωft)hhhh-(κ|ν1|+ν)x˙-δx-ϵx3hhhh-c-Q0cos(ωft))2dx˙min,(13)F1(δ,ϵ,c,ν1)δ=0,F1(δ,ϵ,c,ν1)ϵ=0,(14)F1(δ,ϵ,c,ν1)c=0,F2(δ,ϵ,c,ν1)ν1=0.

Steady-state phase portraits of x¨+νx˙+κ0x˙|x˙|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 ωf 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: (15)d2xdt2+(κ|ν1|+ν)x˙+δ(x)xd2xdt2+ϵ(x)x3+c(x)Q0cos(ωft). Notice that (1) only contains linear damping terms since the nonlinear damped effects are now contained in the (κ|ν1|+ν)x˙ term .

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 (16)x¨+νx˙+κx˙|x˙|p-1+Ax+Bx3=Qcos(ωft),hhhhhhhhhhhhhhhwith  x(0)=1,x˙(0)=0, 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 (17)x¨+(κ|ν1|+ν)x˙+Ax+Bx3+c=Qcos(ωft), where c and ν1 are given by (18)c=-2κ(p-1)υpp+2,ν1=3pυ(p-1)4+4p+p2.

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 ωf=1 with p= 0.1, 0.387, 1.5, and 2.5 with the initial conditions of x(0)=1 and x˙(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.

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 x˙(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 (19)y¨+2νy˙+η0y2y˙+κ0y˙|y˙|p-1y¨+αy+βy3+γy5=Q0cos(ωft), 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 ωf are the driving force and frequency, respectively. Notice that the nonlinear damping terms η0y2y˙+κ0y˙|y˙|p-1 of (19) are commonly used to model the dynamics of nanoresonators and microresonators. See  and references cited therein. If we let x=A/y, thus, (19) becomes (20)x¨+2νx˙+ηx2x˙+κx˙|x˙|p-1+Axx¨+Bx3+Gx5=Qcos(ωft), with x(0)=1, x˙(0)=0, A=α, B=βy02, G=γy04, η=η0y0, κ=κ0y0(p-1), and Q=Q0/y0. In accordance with our proposed nonlinear transformation approach, the system terms 2νx˙+ηx2x˙+κx˙p+Ax+Bx3+Gx5 are replaced by an equivalent cubic-like polynomial expression of the form (21)2νx˙+ηx2x˙+κx˙|x˙|p-1+Ax+Bx3+Gx52νx˙(κ|ν1|+ν)x˙+δx+ϵx3+c.

Thus, the equivalent nonlinear transformation form of (20) is given as (22)x¨+(κ|ν1|+ν)x˙+δx+ϵx3+cQcos(ωft), where (23)δ=A+5ηυσ12-25Gσ463,ϵ=B+5(21ηυ+40Gσ3)162σ,ν1=3pυp-12+p+252υν+273ηυσ2-80Gσ5252κυ,c=-2κ(p-1)υp2+p+848Gσ5-2541ηυσ22268.

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 ωf= 0.5, 1.5, 2, and 2.75 with the initial conditions of x(0)=1 and x˙(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, ωf=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 0t800 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.

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 x˙(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).

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 x˙(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 0t50; (c) the steady-state system response phase portrait on 750t800; (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 (24)d2ydt2+2νdydt+κ0y˙|y˙|p-1+ωn2yd2ydt2+hsgn(y)|y|m=Q0cos(ωft),hhhhhhhhhhhhhy(0)=A,y˙(0)=0, where ωn, ν, κ0, and h are constant parameters and m can take any nonnegative real value on 0m< [31, 32]. By introducing the coordinate transformation x=y/A, (24) can be written as (25)d2xdt2+2νdxdt+κx˙|x˙|p-1+ωn2xd2xdt2+c1sgn(x)|x|m=Qcos(ωft),hhhwith  c1=hA(m-1),κ=κ0A(p-1),hhhhhQ=Q0Ax(0)=1,x˙(0)=0.

The restoring conservative forces ωn2x+c1sgn(x)|x|m can be replaced by a cubic-quintic polynomial by using the Chebyshev polynomial expansion (2) which provides (26)ωn2x+c1sgn(x)|x|mα2x+βx3+γx5+Δx7+εx9, where (27)α2=5c1(m-9)(m-7)(m-5)(m-3)Γ[m/2+1]8πΓ[(11+m)/2]+ωn2,β=-10c1(m-9)(m-7)(m-5)(m-1)Γ[m/2+1]πΓ[(11+m)/2],γ=42c1(m-9)(m-7)(m-3)(m-1)Γ[m/2+1]πΓ[(11+m)/2],Δ=-64c1(m-9)(m-5)(m-3)(m-1)Γ[m/2+1]πΓ[(11+m)/2],ε=32c1(m-7)(m-5)(m-3)(m-1)Γ[m/2+1]πΓ[(11+m)/2].

Here Γ[] represents the Euler gamma function. We next use our solution procedure previously described and determine the equivalent representation form of (28)d2xdt2+2νdxdt+κx˙|x˙|p-1+α2x+βx3d2xdt2+γx5+Δx7+εx9=Qcos(ωft). This gives the following equivalent equation of motion: (29)d2xdt2+(κ|ν1|+ν)dxdt+δx+ϵx3+c=Qcos(ωft), where (30)δ=α2-25γσ463-35Δσ666-81εσ8143,  ϵ=β+100γσ281+7(455Δσ4+432εσ6)2574,c=-2κ(p-1)υpp+2+212γσ5567+71Δσ799+712εσ9715,    ν1=3pυp-1p+2+νκ-14300γσ5+28665Δσ7+40824εσ945045υ.

To study the influence in the system dynamics of the nonlinear damping term of (24), the parameter values of m=3/2, A=1, ωn=1, h=1, Q0=1, ν=0.1, κ=0.9, ωf=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).

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

p δ ϵ c ν 1 σ υ RMSE
1/4 2.2003 −0.3094 0.5011 0.3964 −1 1.85 0.1293
3/4 2.2003 −0.3094 0.0609 0.8878 −1 1.85 0.0730
3/2 2.2260 −0.3513 0.0320 0.9539 0.95 0.6 0.0937
11/4 2.2260 −0.3513 −0.0111 0.6684 0.96 0.6 0.0759

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 x˙(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: (31)d2ydt2+2νdydt+κy˙|y˙|p-1d2ydt2+y(1-A2y2)=Qcos(ωft),hhhhhhhhhhhy(0)=1,y˙(0)=0, where ν and κ=κ0A(p-1) are the damping coefficients, Q=Q0/A is the driving force, ωf represents the driving frequency, and A is the initial oscillation amplitude on 0<A<1 . By using Chebyshev polynomials, the conservative force of (31) can be written in equivalent form as follows (32)y(1-A2y2)α(A)y+β(A)y3+γ(A)y5+Δ(A)y7+ε(A)y9, in which (33)α(A)=1A101-A2(A4(9504-56001-A2)hhhhhhhhhhh+A8(330-501-A2)hhhhhhhhhhh-4608(1-A2-1)hhhhhhhhhhh+48A6(251-A2-66)hhhhhhhhhhh+256A2(351-A2-44)),β(A)=16A101-A2(A6(2244-8001-A2)hhhhhhhhhhh+128A2(71-561-A2)hhhhhhhhhhh+3840(1-A2-1)hhhhhhhhhhh+A8(251-A2-198)hhhhhhhhhhh+8A4(5251-A2-913)),γ(A)=-32A101-A2(4A6(913-3151-A2)hh-h4hhhhhhhh+128A2(125-981-A2)hhhh-4hhhhhhh+6912(1-A2-1)hhhhhh-4hhhhh+A8(351-A2-297)hhhhhhhh-4hhh+16A4(4411-A2-779)),Δ(A)=256A101-A2(64A2(41-321-A2)hhhhhhhhhhh+8A6(71-241-A2)hhhhhhhhhhh+1152(1-A2-1)hhhhhhhhhhh+A8(51-A2-44)hhhhhhhhhhh+80A4(141-A2-25)),ε(A)=-512A101-A2(64A2(9-71-A2)hhhhhhhhh6hhh+A8(1-A2-9)hhhhhhhhhh6hh-40A6(1-A2-3)hhhhhhhhhh6hh+256(1-A2-1)hhhhhhhhhhh6h+48A4(51-A2-9)). Thus, (31) becomes (34)d2ydt2+2νdydt+κy˙|y˙|p-1+α(A)y+β(A)y3d2ydt2+γ(A)y5+Δ(A)y7+ε(A)y9=Qcos(ωft).

We next determine the Duffing-like representation form of (34) by using (11)–(14). This yields (35)d2ydt2+(κ|ν1|+ν)dydt+δy+ϵy3+c=Qcos(ωft), where (36)δ=α-25γσ463-35Δσ666-81εσ8143,ϵ=β+100γσ281+7(455Δσ4+432εσ6)2574,c=-2κ(p-1)υpp+2+212γσ5567+71Δσ799+712εσ9715,ν1=3pυp-1p+2+νκ-14300γσ5+28665Δσ7+40824εσ945045υ. 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, ωf, 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).

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 ωf=0.75.

p δ ϵ c ν 1 σ υ RMSE
1/4 0.6432 2.0089 0.7490 0.2392 -0.800 4.8 0.1984
1/2 0.4782 2.3696 0.1949 0.5017 -0.875 4.8 0.1270
5/4 0.6919 1.8831 0.0371 1.0742 0.750 0.8 0.0819
3/2 0.6601 1.9668 0.0894 0.3851 0.785 0.5 0.0927

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 .

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.

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