Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms

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.


Introduction
The aim of this paper focuses on using a nonlinear approach to transform the forced nonlinear equation: ẍ +  () +  1 (] ẋ ,  ẋ  ) =  0 cos (  ) ,  (0) =  0 , ẋ (0) = 0, (1) with nonlinear damping terms into an equivalent forced, linearly damped Duffing's equation.Here we assume that () is the system restoring force which could have rational or irrational conservative force terms,  1 (] ẋ ,  ẋ  ) represents the system nonlinear dissipative effects, ] and  are damping constants,  is the exponent of the velocity,  0 is the driving force magnitude,   is the system driving frequency,  is the current time, and  0 is the initial amplitude.The main motivation in studying the equivalent representation form of (1) with nonlinear damping terms for which  > 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 <  < 1.95 [6].Martínez-Rodrigo and Romero [7] found that when the nonlinear dampers velocity exponent  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  = 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  to characterize the damping and the elastic nonlinearities observed in different experimental measurements [10][11][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  0 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][16][17][18][19][20][21][22][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  is included.

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 (), which do not have a cubiclike representation form, by using the classical Chebyshev polynomials of the first kind [25][26][27][28][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]: where and  2−1 are the Chebyshev polynomials of the first kind defined as The first three terms are given by Thus, the equivalent restoring force () can be written as Substituting (2) into (3), by using (5), provides the following equivalent representation form of ():  () ≈  ()  +  ()  3 +  ()  5 .
Then, (1) can be equivalently written as Since nonlinear damping terms are involved in the dissipative force expression  1 (] ẋ ,  ẋ  ), we modify our cubication nonlinear transformation approach introduced in [24] by adding a constant term () 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: with  (0) =  0 , ẋ (0) = 0, (9) where  denotes the displacement of the system, ] and  0 are damping coefficients, , , and  0 are system constant parameters, and   represents the driving frequency.Figure 1 shows the phase portrait of (1) by considering the values of  = 1 and 2.5.Notice that when  ̸ = 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 where ] 1 , (), (), and () can be found from In ( 11) and ( 12),  and  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  0 and   are assumed to remain constant.Once the expressions to determine ] 1 , (), (), and () are found, then (1) can be written as the following Duffing-type equation of motion: Notice that (1) only contains linear damping terms since the nonlinear damped effects are now contained in the (|] 1 |+ ]) 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).

Forced Duffing Equation with
Nonlinear Damping Here we consider the forced, damped Duffing equation given by ( 9) and introduce the following change of variable  = / which transforms it into an equation of the form where  =  0  (−1) 0 and  =  0 / 0 .We next use ( 12) and ( 14) to derive the following forced, damped Duffing equation of the form where  and ] 1 are given by Here the values of  must satisfy (12).To assess the accuracy of ( 17) when compared to the numerical integration of ( 16), let us consider the parameter values of  = 1,  = 1, ] = 0.1,  = 0.9,  = 1, and   = 1 with  = 0.1, 0.387, 1.5, and 2.5 with the initial conditions of (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 rootmean-square error (RMSE) values are 0.1196, 0.0874, 0.1530, and 0.1351 for the  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 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  = 1,  = 1, ] = 0.1,  = 0.9,  = 1, and equation of motion that describes the dynamics behavior of the oscillatory system (16) well even when nonlinear damping terms are considered.
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.

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 with initial conditions given by (0) =  0 and ẏ (0) = 0.In (19)  denotes the displacement of the system, ],  0 , and  0 are damping coefficients, , , and  are system constant parameters, and  0 and   are the driving force and frequency, respectively.Notice that the nonlinear damping terms (19) are commonly used to model the dynamics of nanoresonators and microresonators.See [3][4][5] and references cited therein.If we let  = /, thus, (19) becomes with (0) = 1, ẋ (0) = 0,  = ,  =  2 0 ,  =  4 0 ,  =  0  0 ,  =  0  (−1) 0 , and  =  0 / 0 .In accordance with our proposed nonlinear transformation approach, the system terms (c) amplitude time-response curves for   = 2; (d) amplitude-time response curves for   = 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).
2] ẋ + 2 ẋ + ẋ  ++ 3 + 5 are replaced by an equivalent cubic-like polynomial expression of the form Thus, the equivalent nonlinear transformation form of (20) is given as where To evaluate the accuracy of ( 22), we use the parameter values of  = 1,  = 5,  = 1, ] = 0.1,  0 = 0.15,  0 = 0.25,  = 1, and   = 0.5, 1.5, 2, and 2.75 with the initial conditions of (0) = 1 and ẋ (0) = 0. Figure 3 illustrates the amplitude versus time plots by considering a nonlinear damping term of order  = 3/4.The solid black lines represent the numerical integration of ( 19) while the red dots represent the numerical 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).
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  = 0.9147,  = 5.5863, ] 1 = 1.395,  = 0.0572,  = 0.685, and  = 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:  = 1,  = 1,  = 0.1, ] 0 = 0.05,  0 = 0.15,  0 = 0.1,   = 2.75, and  = 1.Here the computed values for , , ] 1 , and  are, respectively, 0.9975, 1.0679, 1.2806, and −0.0045 with  = 0.645 and  = 0.11.The corresponding RMSE value computed on 0 ⩽  ⩽ 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.
We will next derive the equivalent representation form of the forced power-form elastic term oscillator with linear and nonlinear damping terms.

The Forced Power-Form Elastic Term Oscillator with Nonlinear Damping
The equation of motion of this oscillator is assumed to be given as where   , ],  0 , and ℎ are constant parameters and  can take any nonnegative real value on 0 ≤  < ∞ [31,32].By 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).
introducing the coordinate transformation  = /, (24) can be written as The restoring conservative forces  2   +  1 sgn()||  can be replaced by a cubic-quintic polynomial by using the Chebyshev polynomial expansion (2) which provides where Here Γ[•] represents the Euler gamma function.We next use our solution procedure previously described and determine the equivalent representation form of This gives the following equivalent equation of motion: where To study the influence in the system dynamics of the nonlinear damping term of (24), the parameter values of  = 3/2,  = 1,   = 1, ℎ = 1,  0 = 1, ] = 0.1,  = 0.9,   = 0.75, and  = 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 amplitudetime 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 , , , ] 1 , , and .Based on these results, it is concluded that our equivalent representation form (29) describes well the numerical integration solution of (25).
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.

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: where ] and  =  0  (−1) are the damping coefficients,  =  0 / is the driving force,   represents the driving frequency, and  is the initial oscillation amplitude on 0 <  < 1 [33].By using Chebyshev polynomials, the conservative force of (31) can be written in equivalent form as follows  (1 −  2  2 ) ≡  ()  +  ()  3 +  ()  5 + Δ ()  7 +  ()  9 , in which 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 , , , Δ, and .
To illustrate the accuracy attained from the derived nonlinear equivalent equation of motion of the FENO oscillator, let us consider the parameter values of ],  0 ,  0 ,   , and  to be, respectively, 0.1, 0.9, 1, 0.75, and 0.9 with  = 0.25, 0.  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).

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  in which  is a positive number that can be even odd or fractional; that is,  > 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].