Higher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational Approach

Abstract

He's variational approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). Three levels of approximation have been used. We obtained 1.6% relative error for the first approximate period, 0.3% relative error for the second-order approximate period. The third approximate solution has the accuracy as high as 0.1%.

1. Introduction

Considerableattention has been directed toward the solution of nonlinear equations since they play crucial role in applied mathematics, physics, and engineering problems. In general, the analytical approximationto solution of a given nonlinear problem ismoredifficult than the numerical solution approximation. During the past decades, several types of methods are proposed to obtain approximate solution of nonlinear equations of various types. Among them are variational iteration methods [1–7], homotopy perturbation method [8–15], modified Lindstedt-Poincare method [16], parameter expansion method [17, 18], and variational methods [19–21]. The variational method is different from any other variational methods in open literature, and it is only valid for nonlinear oscillators [22]. Paper [23] is an example of use of variational approach method in nonlinear oscillator problem.

When we examine the frequency amplitude relations of some nonlinear oscillators, it is seen that paper [24] focuses on only the first-order solutions.

Variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions [20].

In the present study, we have investigated the application of variational approach to nonlinear oscillator with discontinuity.

2. A Variational Method

Let us consider a general nonlinear oscillator in the form

𝑢^′′+𝑓(𝑢)=0. (2.1)

He proposed a variational principle for (2.1) as follows [20]:

𝐽(𝑢)=₀^𝑇/41(−2𝑢^′2+𝐹(𝑢))𝑑𝑡, (2.2) where 𝑇 is period of the nonlinear oscillator, 𝜕𝐹/𝜕𝑢=𝑓. Actually, the upper limit is originally 𝑇 instead of 𝑇/4. Normally, it works in most of the cases. Let us suppose that 𝑓(𝑢)=sgn(𝑢) such

sgn(𝑢)=−1,𝑢<0,+1,𝑢≥0, (2.3)

therefore 𝐹(𝑢)=−𝑢,𝑢<0,𝑢,𝑢≥0. (2.4)

But this form is not suitable for discontinuity equation. Therefore, we propose the equation in the form of

𝐽(𝑢)=₀^𝑇/21(−2𝑢^′2+|𝑢|)𝑑𝑡. (2.5)

Assume that its solution can be expressed as

𝑢(𝑡)=𝐴cos𝜔𝑡+𝐵(cos𝜔𝑡−cos3𝜔𝑡)+𝐶(cos3𝜔𝑡−cos5𝜔𝑡)+⋯, (2.6) where 𝜔 is the frequency of the oscillator.

Inserting (2.6) into (2.5) yields

=𝐽(𝐴,𝐵,𝐶,…,𝜔)_0𝑇/21{−2𝜔²[𝐴sin𝜔𝑡+𝐵(sin𝜔𝑡−3sin3𝜔𝑡)+𝐶(3sin3𝜔𝑡−5sin5𝜔𝑡)+⋯]²+|𝐴cos𝜔𝑡+𝐵(cos𝜔𝑡−cos3𝜔𝑡)+𝐶(cos3𝜔𝑡−cos5𝜔𝑡)+⋯|} (2.7)

Let us define 𝜏=𝜔𝑡. Then (2.7) becomes

𝐽(𝐴,𝐵,𝐶,..,𝜔)=^𝜋₀1{−2𝜔[𝐴sin𝜏+𝐵(sin𝜏−3sin3𝜏)+𝐶(3sin3𝜏−5sin5𝜏)+⋯]²+1𝜔|𝐴cos𝜏+𝐵(cos𝜏−cos3𝜏)+𝐶(cos3𝜏−cos5𝜏)+⋯|}𝑑𝜏. (2.8)

Using the Ritz method, we require

𝜕𝐽𝜕𝜔=0,𝜕𝐽𝜕𝐴=0,𝜕𝐽𝜕𝐵=0,𝜕𝐽𝜕𝐶=0,…,𝜕𝐽𝜕𝜔=−^𝜋₀{12[𝐴sin𝜏+𝐵(sin𝜏−3sin3𝜏)+𝐶(3sin3𝜏−5sin5𝜏)+⋯]²+1𝜔²|𝐴cos𝜏+𝐵(cos𝜏−cos3𝜏)+𝐶(cos3𝜏−cos5𝜏)+⋯|}𝑑𝜏. (2.9)(2.10)

By a careful inspectation, we find that

𝜕𝐽𝜕𝜔<0. (2.11)

Thus, the conditions in (2.9) reduce to

𝜕𝐽𝜕𝐴=0,𝜕𝐽𝜕𝐵=0,𝜕𝐽𝜕𝐶=0,…. (2.12)

3. Application

Consider the following nonlinear oscillator with discontinuity:

𝑢^′′+sgn(𝑢)=0, (3.1) with initial conditions

𝑢(0)=𝐴,𝑑𝑢(0)𝑑𝑡=0. (3.2)

Its variational formulation can be written as follows:

𝐽(𝑢)=₀^𝜋/21(−2𝑢^′2+𝑢)𝑑𝑡+^𝜋_𝜋/21(−2𝑢^′2−𝑢)𝑑𝑡. (3.3)

For the first approximation assume that 𝑢(𝑡) is in the following form:

𝑢(𝑡)=𝐴cos𝜔𝑡. (3.4)

Substitute this first approximation into  (3.3):

1𝐽(𝐴,𝜔)=2𝐴−4𝜋𝐴²𝜔². (3.5)

The stationary condition with respect to A reads

𝜕𝐽𝜕𝐴=2−𝜋𝐴2𝜔²=0, (3.6) which leads to the result

2𝜔=√=𝐴𝜋1.128379√𝐴, (3.7) and the approximate period can be obtained as follows:

𝑇_1app(𝐴)=𝜋√𝜋𝐴=5.568328√𝐴,relativeerror=1.6%. (3.8)

This solution agrees with Liu’s solution obtained by He’s modified Lindsted-Poincaré method [16], Rafei et al.’s solution obtained by He’s variational iteration method [2], Wu et al.’s solution obtained by the low-order harmonic balance method [25], and A. Belendéz et al.’s solution obtained by He’s homotopy perturbation method [9].

Secondly, to obtain a more accurate result, define 𝑢 as follows:

𝑢=𝐴cos𝜔𝑡+𝐵(cos𝜔𝑡−cos3𝜔𝑡). (3.9)

Notice that (3.9) satisfies the initial conditions (3.2).

Substituting (3.9) into (3.3), we obtain

𝐽(𝐴,𝐵,𝜔)=₀^𝑇/41(−2(𝐴+𝐵)²𝜔²sin²𝜔𝑡+3𝐵(𝐴+𝐵)𝜔²sin𝜔𝑡sin3𝜔𝑡)𝑑𝑡+₀^𝑇/49(−2𝐵²𝜔²sin²+3𝜔𝑡+(𝐴+𝐵)cos𝜔𝑡−𝐵cos3𝜔𝑡)𝑑𝑡_𝑇/2𝑇/41(−2(𝐴+𝐵)²𝜔²sin²𝜔𝑡+3𝐵(𝐴+𝐵)𝜔²+sin𝜔𝑡sin3𝜔𝑡)𝑑𝑡_𝑇/2𝑇/49(−2𝐵²𝜔²sin²3𝜔𝑡−(𝐴+𝐵)cos𝜔𝑡+𝐵cos3𝜔𝑡)𝑑𝑡. (3.10) The 𝐽(𝐴,𝐵,𝜔) in (3.10) can be obtained as follows:

8𝐽(𝐴,𝐵,𝜔)=2𝐴+31𝐵−4𝜋𝐴²𝜔²−12𝜋𝐴𝐵𝜔²−52𝜋𝐵²𝜔². (3.11)

The stationary condition with respect to A and B reads

𝜕𝐽1𝜕𝐴=2−2𝐴𝜋𝜔²−12𝐵𝜋𝜔²=0,𝐴=10427𝜋𝜔²,𝜕𝐽=8𝜕𝐵3−12𝐴𝜋𝜔²−5𝐵𝜋𝜔²4=0,𝐵=27𝜋𝜔², (3.12)(3.13) from which the relationship between oscillator frequency and amplitude can be determined.

From (3.12) we have

2𝜔=326=3𝐴𝜋1.10729√𝐴, (3.14) and the approximate period can be obtained as follows:

𝑇_2app(𝐴)=27𝐴𝜋³√26=5.67440𝐴,relativeerror=0.31%. (3.15)

In this study, we obtained the relative error as 1.6% for the first-order approximation while the other researchers [2, 16] obtained the relative error as 1.8%. The reason for the difference in the relative error is that the other researchers take less precision in the decimal numbers during calculations. In [9], the frequency 𝜔=1.107452/√𝐴 and the period 𝑇_2app(𝐴)=5.67440√𝐴 were found for the same problem by second-order approximation and the relative error was calculated as %0.30.

Equation (3.1) was approximately solved in [25] using an improved harmonic balance method that incorporates salient features of both Newtons’s method and the harmonic balance method. In [25], the following results for the first and second-order approximations were obtained:

𝑇_WSL1(𝐴)=5.568328√𝑇𝐴,relativeerror=1.6%_WSL2(𝐴)=5.67440√,𝐴,relativeerror=0.31%. (3.16)

To obtain a more accurate result, define 𝑢 as follows:

𝑢=𝐴cos𝜔𝑡+𝐵(cos𝜔𝑡−cos3𝜔𝑡)+𝐶(cos3𝜔𝑡−cos5𝜔𝑡). (3.17)

Notice that (3.17) satisfies the initial conditions (3.2).

Substituting (3.17) into (3.3) gives

8𝐽(𝐴,𝐵,𝐶,𝜔)=2𝐴+3𝐵−16115𝐶−4𝜋𝜔²(𝐴²+2𝐴𝐵+10𝐵²−18𝐵𝐶+34𝐶²). (3.18)

The stationary condition with respect to A, B, and C reads

𝜕𝐽1𝜕𝐴=2−2𝐴𝜋𝜔²−12𝐵𝜋𝜔²=0,𝐴=131083375𝜋𝜔²,𝜕𝐽=8𝜕𝐵3−12𝐴𝜋𝜔²−5𝐵𝜋𝜔²+92𝐶𝜋𝜔²=0,𝐵=3923375𝜋𝜔²,𝜕𝐽𝜕𝐶=−1615−17𝐶𝜋𝜔²+92𝐵𝜋𝜔²=0,𝐶=−4125𝜋𝜔². (3.19)

Hence the approximate frequency is

𝜔=13108=3375𝐴𝜋1.11188√𝐴. (3.20)

Therefore, approximate period of the nonlinear oscillator can be obtained as follows:

𝑇_3app(𝐴)=23375𝜋³𝐴√13108=5.65098𝐴,relativeerror=0.1%. (3.21)

For this nonlinear problem in (3.1), the exact period is given as follows [25]:

𝑇_𝑒(𝐴)=4√2𝐴=5.656854√𝐴. (3.22)

The period values and these relative errors obtained in this method for nonlinear oscillator with discontinuity are the following:

𝑇_1app(𝐴)=5.568328√𝑇𝐴,relativeerror=1.6%,_2app(𝐴)=5.67440√𝑇𝐴,relativeerror=0.31%,_3app(𝐴)=5.65098√𝐴,relativeerror=0.10%. (3.23)

Equation (3.1) was approximately solved in [25] using an improved harmonic balance method that incorporates salient features of both Newtons’s method and the harmonic balance method. In [25], the following result for the third-order approximations was obtained:

𝑇_WSL3(𝐴)=5.650976√𝐴,relativeerror=0.10%. (3.24)

Equation (3.1) was approximately solved in [9] using a homotopy perturbation method. In [9], the following result for the third-order approximations was obtained:

𝑇₃(𝐴)=5.653609√𝐴,relativeerror=0.057%. (3.25)

By using above values, the periodic function 𝑢(𝑡) can be written for three levels of approximation as follows:

𝑢_𝑎1(𝑡)=𝐴cos(𝜔₁𝑢𝑡),_𝑎2(𝑡)=1.03846𝐴cos(𝜔₂𝑡)−0.0384615𝐴cos(3𝜔₂𝑢𝑡),_𝑎3(𝑡)=1.02991𝐴cos(𝜔₃𝑡)−0.0381446𝐴cos(3𝜔₃𝑡)+0.00823924𝐴cos(5𝜔₃𝑡). (3.26)

The normalized exact periodic solution 𝑢_ex/𝐴 has been obtained by numerically integrating (3.1) and (3.2) and compared with approximate solutions (3.26) in Figure 1. Here nondimensional time h is defined as follows: 𝑡ℎ=𝑇_𝑒. (3.27)

(a)

(b)

(c)

Figure 1 

Comparison of approximate normalized functions (dashed line) with exact normalized functions (continuous line) for (a) first approximation (b) second approximation (c) third approximation.

4. Conclusions

He’s variational approach is modified for nonlinear oscillator with discontinuities. The method has been applied to obtain three levels of approximation of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn(u). We reached 1.6%, 0.31%, and 0.1% relative errors for the first, second, and third approximate periods, respectively. One can obtain higher-order accuracy by extending the idea given in this paper.