We applied an approach to obtain the natural frequency of the generalized Duffing oscillator u¨ + u + α3u3 + α5u5 + α7u7 + ⋯ + αnun=0 and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflection u¨+αu|u|n−1=0. This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power of n, the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain.
1. Introduction
Although a large amount of the efforts on dynamical systems are related to second-order differential equations, some dynamical systems can be described by nonlinear (second-order) differential equations. Attention in nonlinear oscillator equations involving the second temporal derivative of displacement has recently been focused on the existence of periodic solutions. The study of nonlinear periodic oscillator is of interest to many researchers and various methods of solution have been suggested. Several approaches have been proposed to deal with different kinds of oscillator equations, for example, [1–7]. He in [8] used Hamiltonian method to calculate the analytical approximate periodic solutions of nonlinear oscillator equations. The approximations to the periodic solution and the angular frequency obtained by He were not accurate enough.
Yildirim et al. [9] and Khan et al. [10], respectively, applied a higher order Hamiltonian formulation combined with parameters for nonlinear oscillators. Our concern in this work is the derivation of amplitude-frequency relationship for the nonlinear oscillator equations u¨+u+α3u3+α5u5+α7u7+⋯+αnun=0 and u¨+αu|u|n-1=0. The attention here has been restricted primarily to odd positive integer power for the first equation and rational powers greater than unity for the second equation. There are examples of systems, however, for which these exponents can be of noninteger order, for instance, the flexible elements of vibration isolators made of wire-mesh and felt materials, cable isolators, and radially loaded rubber cylinder.
In the present work, the mentioned parameters are the undetermined values in the assumed solution. In the parameters technique, the motion has been assumed as u=∑k=0NA2k+1cos(2k+1)ωt where ω, Ak, and k=0,1,2,… are the angular frequency of motion and Fourier coefficients, respectively. The method in this approach to obtain the parameters is quite different from the method in He’s Hamiltonian technique. Hence, the present technique is not similar to He’s Hamiltonian technique. Finally, the paper provides some accurate results for the angular frequency ω of the motion.
2. Analysis of the Method2.1. Generalized Duffing Oscillator
First, we consider a general form of nonlinear oscillator
(1)u¨+u+α3u3+α5u5+α7u7+⋯+αnun=0
with initial conditions
(2)u(0)=A,u˙(0)=0,
where A, α2k+1, and k=1,2,3,… are constants. Multiplying both sides of (1) by 2u˙ and integrating, with initial conditions, we get
(3)u˙2+u2+α3u42+α5u63+⋯+2αnun+1n+1=A2+α3A42+α5A63+⋯+2αnAn+1n+1.
In this approach the solution of the problem is assumed to be
(4)u=∑k=0NA2k+1cos(2k+1)ωt.
Differentiating (4) leads to the results
(5)u˙=-ω∑k=0NA2k+1(2k+1)sin(2k+1)ωt(6)u¨=-ω2∑k=0NA2k+1(2k+1)2cos(2k+1)ωt.
From the initial condition equations (2) and (4), we have
(7)A=∑i=0NA2k+1.
Substituting (4) and (5) into (3) at different values of ωt is mentioned in Table 1.
Collocation points for different ωt.
N=1
π/2
—
—
—
N=2
π/4
π/2
—
—
For N=1 the following equation is obtained:
(8)ω2(∑k=0NA2k+1(2k+1))2=A2+α3A42+α5A63+⋯+2αnAn+1n+1.
For N=2, corresponding number of equations will be obtained, respectively. Considering the acceleration at ωt=0, from (1), (2), and (6), we get the following equation:
(9)ω2∑k=0NA2k+1(2k+1)2=(A+α3A3+α5A5+α7A7+⋯+αnAn).
The numerical solution of these algebraic equations can be obtained by symbolic software or an iterative scheme. From these system of equations, the approximate frequency-amplitude relationship of a nonlinear oscillator up to higher order will be attained. From (3), the period of the motion is given by(10)T=∫-AA2duA2+α3(A4/2)+α5(A6/3)+⋯+2αn(An+1/(n+1))-(u2+α3(u4/2)+α5(u6/3)+⋯+2αn(un+1/(n+1))).The angular frequency of the motion can be expressed by the relation
(11)ω=2πT.
2.2. Noninteger Order Force-Deflection Oscillator
Now, consider a nonlinear oscillator of the form
(12)u¨+αu|u|n-1=0,n>1
with initial conditions
(13)u(0)=A,u˙(0)=0.
Multiply both sides of (12) by 2u˙ and integrate, with initial conditions
(14)u˙2+2αu2|u|n-1n+1=2αA2|A|n-1n+1.
Proceeding in the same way as in the previous section, the obtained algebraic equations can be solved by symbolic software. From these system of equations, one can obtain approximate frequency-amplitude relationship of a nonlinear oscillator up to higher order. From (14), the period of the motion is obtained as
(15)T=∫-AA2du2α(A2|A|n-1/(n+1))-2α(u2|u|n-1/(n+1)).
The angular frequency of the motion can be expressed by the relation
(16)ω=2π(∫-AA2du×(2αA2|A|n-1n+1-2αu2|u|n-1n+1)-1/2)-1.
It should be pointed out that the square root under the integral sign in (16) should be positive.
Consider the following nonlinear Duffing oscillator:
(17)u¨+u+α3u3=0
with initial conditions u(0)=A and u˙(0)=0. Assume that the solution can be expressed as
(18)u=A1cosωt+A3cos3ωt.
According to the initial conditions,
(19)A1+A3=A.
Substituting (18) and its derivative into (3) at ωt=π/2, the following equation is obtained:
(20)ω2(A1-3A3)2=A2+α3A42.
From (17) the acceleration at ωt=0 has been obtained as follows:
(21)ω2(A1+9A3)=A+α3A3.
Using MATHEMATICA the simplified values will be in the following form:
(22)A1=5A8-A8(1+α3A2)+A16+22α3A2+7α32A48(1+α3A2),A3=3A8+A8(1+α3A2)-A16+22α3A2+7α32A48(1+α3A2),ω(A)=13(1+α32A2+4+72α3A2).
The frequency-amplitude relationship of nonlinear oscillator for α3=1, A=1, has been obtained as
(23)ω2(app)=1.32111921.
From (3), the frequency of the motion for A=1 is
(24)ω(Exact)=(3/2)π2EllipticK[-1/3]≈1.31777606,
where EllipticK[m] is the complete elliptic integral of the first kind. The three-parameter approach provides good approximations to the exact frequency and the relative error lower than 0.2536%.
Since the accuracy of the obtained results in three-parameter technique is not so high, the four-parameter technique has been introduced as follows:
(25)u=A1cosωt+A3cos3ωt+A5cos5ωt,
where A1, A3, A5, and ω are four undetermined parameters. Four equations can be formulated for the solution of four parameters. According to the initial conditions,
(26)A=A1+A3+A5.
Substituting (25) and its derivative into (3) at the times ωt=π/2 and ωt=π/4 as mentioned in Table 1, the following equations have been obtained:
(27)ω22(A1+3A3-5A5)2+u12+α3u142=A2+α3A42ω2(A1-3A3+5A5)2=A2+α3A42,
where u1=(1/2)(A1-A3-A5).
The acceleration at ωt=0, from (17), resulted with the following equation:
(28)ω2(A1+9A3+25A5)2=A+α3A3.
It is difficult to obtain the analytical expression for the unknown values; we derived the numerical results for the unknown parameters ω, A1, A3, and A5 by using (26)–(28). After some mathematical simplification by taking A=1,α3=1, the following values have been obtained:
(29)A1=0.981716,A3=0.0179609,A5=0.000322698,ω=1.31777939
which is very close to the exact solution and highly accurate in comparison with [11, 12, 17, 18].
The following nonlinear Duffing oscillator is considered:
(30)u¨+u+α3u3+α5u5=0
with initial conditions u(0)=A and u˙(0)=0. Let the solution be of the form
(31)u=A1cosωt+A3cos3ωt.
According to the initial conditions
(32)A1+A3=A.
Substituting (31) and its derivatives into (3) at ωt=π/2, the following equation is obtained:
(33)ω2(A1-3A3)2=A2+α3A42+α5A63.
The acceleration will be procured from (30) at ωt=0 as follows:
(34)ω2(A1+9A3)=A+α3A3+α5A5.
After some mathematical simplification using MATHEMATICA, the following values were achieved:(35)A1=5A8+-3A+α5A524(1+α3A2+α5A4)+A144+198α3A2+63α32A4+168α5A4+102α3α5A6+40α52A824(1+α3A2+α5A4),A3=3A8+3A-α5A524(1+α3A2+α5A4)-A144+198α3A2+63α32A4+168α5A4+102α3α5A6+40α52A824(1+α3A2+α5A4),ω=59+4A29α3+11A427α5+127144+198α3A2+63α32A4+168α5A4+102α3α5A6+40α52A8.
From (3), the exact period of the motion for A=1 has been obtained as
(36)ω(Exact)=1.52358602.
The three-parameter approach provides good approximations to the exact frequency and the relative error lower than 1.6332%.
Since the accuracy of the obtained results in three-parameter technique is not so high, the four-parameter technique has been introduced as follows:
(37)u=A1cosωt+A3cos3ωt+A5cos5ωt,
where A1, A3, A5, and ω are four undetermined parameters. Four equations can be formulated for the solution of four parameters. According to the initial conditions,
(38)A=A1+A3+A5.
Substituting (37) and its derivative into (3) at the times ωt=π/2 and ωt=π/4 as mentioned in Table 1, the following equations have been obtained:
(39)ω2(A1+3A3-5A5)2+u12+α3u142+α5u163=A2+α3A42+α5A63ω2(A1-3A3+5A5)2=A2+α3A42+α5A63,
where u1=(1/2)(A1-A3-A5).
By the acceleration at ωt=0, from (30), the following equation will get
(40)ω2(A1+9A3+25A5)2=A+α3A3+α5A5.
Similarly, from (38)–(40), four unknowns ω, A1, A3, and A5 can be solved numerically by using MATHEMATICA taking A=1 to get
(41)A1=0.968372,A3=0.0296105,A5=0.00201743,ω=1.52376242
which is very close to the exact solutions. The four-parameter approach provides good approximations to the exact frequency and the relative error lower than 0.0115%.
Consider the following nonlinear non-integer force-deflection oscillator:
(42)u¨+u|u|n-1=0,n>1
with initial conditions
(43)u(0)=A,u˙(0)=0.
The differential (42) with (43) has an exact analytical solution in the form of the Ateb cam function [13, 14]:
(44)u=Acam(t|A|n-1(n+1)2)
which is the inverse incomplete Euler Beta function. The exact period of the oscillation is
(45)T=42|A|n-1(n+1)B(1n+1,12),
where B(m,n) is the complete Beta function. Consider that the solution can be written as
(46)u=A1cosωt+A3cos3ωt.
According to the initial conditions,
(47)A1+A3=A.
Substituting (46) into (14) at ωt=π/2, the following equation has been obtained:
(48)ω2(A1-3A3)2=A2|A|n-1n+1.
By the acceleration at ωt=0, from (46) and (42), the following equation will be achieved:
(49)ω2(A1+9A3)2=An.
An analytical frequency-amplitude relation has been heeded from (47)–(49), by MATHEMATICA 8 built-in utilities, as
(50)A1=3A8-An+1-(7+3n)A8(n+1)-22A|A|1-n(5+3n)|A|2n+2(n+1)|A|2,A3=5A8+An+1-(7+3n)A8(n+1)-A|A|1-n(5+3n)|A|2n+222(n+1)|A|2,ω=13|A|n-1(n+1)(1+4+3n).
The four-parameter approach provides good approximations to the exact frequency. The computed results and its comparison with exact frequency, Lindstedt-Poincaré (LP) method “ωLP”, and modified Lindstedt-Poincaré (MLP) method “ωMLP” of second order have been tabulated in Table 2.
The frequencies ωLP and ωMLP given by [15], ωPresent* and ωPresent** given by three-parameter and four-parameter approaches, ωExact given by (16) as a function of various values of the parameters n and A=1.
n
ωLP
ωMLP
ωPresent*
ωPresent**
ωExact
1
1
1
1
1
1
4/3
0.97013
0.95452
0.83543
0.96326
0.96916
3/2
0.95671
0.93298
0.82546
0.94793
0.95469
5/3
0.94418
0.91212
0.81649
0.93417
0.94081
2
0.92130
0.87214
0.80103
0.91045
0.91468
4. Conclusions
The parameter method [16] gives the approximate solution for the generalized Duffing and non-integer order oscillator equations. Accuracy and validity of the obtained results have been examined by comparing it with the exact ones in time histories and table. Figures 1 and 2 are depicted for cubic and quanta oscillator equations and Figure 3 for non-integer oscillator equation. The frequency of vibration depends on the initial amplitude, the coefficient of nonlinearity, and the value of the fractional power. The main results of the paper obtained by this method can be summarized as follows.
It has been observed that if numbers of parameters are increased, then the method will give the better results.
The nonlinear oscillator equation has converted to nonlinear algebraic equation by this approach which can be solved by numerical methods that leads to a better result.
It has also been observed that presented results for oscillator equations (as in numerical experiments Sections 3.1 and 3.2) are accurate in comparison with [11, 12, 17, 18].
The parameters method may also provide for large values of amplitude of the motion. The method is also valid for any arbitrary values of α3, α5, α, and n. It will provide liberty to solve any kind of nonlinear oscillator problem which is not suitable for established methods.
Figures 4–6 have been plotted for comparison of the numerical results with presented and existing methods in the literature. The LP and MLP methods give the same results for motion in the first approximation.
The higher order MLP gives minor corrections in LP method. It has been observed that four-parameter approximations give the better results as shown in Figure 5.
(a) Comparison for the u versus u˙ trajectory for the case of A=1, (b) t versus u (displacement A=1), (c) t versus u (displacement A=100), (d) t versus u (displacement A=1000).
(a) Comparison for the u versus u˙ trajectory for the case of A=1 and (b) t versus u (displacement A=1).
(a) Comparison for the u versus u˙ trajectory for the case of A=1 and (b) t versus u (displacement A=1).
Comparison of the present method with numerical values and [18] for ω versus A of Duffing Oscillator (a) with n=3, α3=1 (b) with n=3, α3=1, α5=1.
Comparison of various methods for ω versus A of force-deflection oscillator.
Comparison of LP method and numerical values for ω versus A of force-deflection oscillator.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to express their sincere gratitude to the referees and Professor Mehdi Ahmadian for their careful assessment, fruitful remarks, and suggestions regarding the initial version of the paper. The author Najeeb Alam Khan is highly thankful and grateful to the Dean of Faculty of Sciences, University of Karachi, Karachi, Pakistan, for facilitating this research work.
HeJ. H.KhanN. A.JamilM.AliS. A.Nadeem KhanA.Solutions of the force-free duffing-van der pol oscillator equationBarariA.KimiaeifarA.NejadM. G.MotevalliM.SfahaniM. G.A closed form solution for nonlinear oscillators frequencies using amplitude-frequency formulationMickensR. E.Generalization of the method of harmonic balanceTiwariS. B.RaoB. N.SwamyN. S.SaiK. S.NatarajaH. R.Analytical study on a Duffing-harmonic oscillatorBayatM.PakarI.On the approximate analytical solution to non-linear oscillation systemsYounesianD.AskariH.SaadatniaZ.YazdiM. K.Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance methodHeJ.-H.Hamiltonian approach to nonlinear oscillatorsYildirimA.SaadatniaZ.AskariH.KhanY.YazdiM. K.Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approachKhanN. A.JamilM.AraA.Multiple-parameter Hamiltonian approach for higher accurate approximations of a nonlinear oscillator with discontinuityChenY. M.LiuJ. K.A new method based on the harmonic balance method for nonlinear oscillatorsRenZ.-F.LiuG.-Q.KangY.-X.FanH.-Y.LiH.-M.RenX.-D.GuiW.-K.Application of He's amplitude-frequency formulation to nonlinear oscillators with discontinuitiesCveticaninL.KovacicI.RakaricZ.Asymptotic methods for vibrations of the pure non-integer order oscillatorRosenbergR. M.The Ateb(h)-functions and their propertiesCveticaninL.Oscillator with fraction order restoring forceChenY. Z.Multiple-parameters technique for higher accurate numerical solution of Duffing-harmonic oscillationHeJ.-H.An improved amplitude-frequency formulation for nonlinear oscillatorsGanjiS. S.GanjiD. D.BabazadehH.SadoughiN.Application of amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back