A complement to period/frequency estimation of a nonlinear oscillator

Introduction

Recently Tian and Liu gave a complete review on frequency–amplitude formulae for nonlinear oscillators, ¹ which give a very fast estimation of the periodic property with relative accuracy; this review article gave an immediate response in academic community. Ren gave an extension by replacing ω 2 in He’s frequency–amplitude formula by ω 4 ^,2 and Wang and An applied the formula to fractional oscillators ³ and many more applications will be appeared in future. However, all formulae cannot obtain a more accurate solution. Some famous analytical methods, e.g., the homotopy perturbation method^4–13 and the variational iteration method,^14–16 can continue their solution processes until a needed accuracy is obtained. This paper tries to give an effective modification so that any accurate solutions can be obtained.

He’s frequency–amplitude formulation

Consider a general nonlinear oscillator

R ( t ) = x ″ + f ( x ) = 0

(1)

x ( 0 ) = A , x ′ ( 0 ) = 0

(2)

He’s frequency–amplitude formulation ¹⁷ is to choose two arbitrary solutions in the forms

x 1 ( t ) = A cos ⁡ ω 1 t

(3)

x 2 ( t ) = A cos ⁡ ω 2 t

(4)

ω 2 = ω 2 2 R 1 ( t 1 ) − ω 1 2 R 2 ( t 2 ) R 1 ( t 1 ) − R 2 ( t 2 )

(5)

ω 1 t 1 = ω 2 t 2

(6)

There are other modifications of He’s frequency–amplitude formulation as summarized by Tian and Liu, 1 and the last modification was given by Ren, ² which is

ω 4 = ω 2 4 R 1 ( t 1 ) − ω 1 4 R 2 ( t 2 ) R 1 ( t 1 ) − R 2 ( t 2 )

(7)

This modification did not improve the accuracy, and the calculation becomes complex.

A modification

The shortcoming of He’s frequency–amplitude formulation is that the solution process cannot be continued if a higher accurate solution is needed. To overcome this shortcoming, we can choose the following two trial solutions like that in the homotopy perturbation method ¹⁰

x 1 ( t ) = a cos ⁡ ω 1 t + b cos ⁡ 3 ω 1 t

(8)

x 2 ( t ) = a cos ⁡ ω 2 t + b cos ⁡ 3 ω 2 t

(9)

We use the Duffing equation as an example

x ″ + x + ε x 3 = 0

(10)

The residuals for two trial solutions are, respectively, as

R 1 ( t ) = − a ω 1 2 cos ⁡ ω 1 t − 9 b ω 1 2 cos ⁡ 3 ω 1 t + a cos ⁡ ω 1 t + b cos ⁡ 3 ω 1 t + ε ( a cos ⁡ ω 1 t + b cos ⁡ 3 ω 1 t ) 3

(11)

R 2 ( t ) = − a ω 2 2 cos ⁡ ω 2 t − 9 b ω 2 2 cos ⁡ ω 2 t + a cos ⁡ 3 ω 2 t + b cos ⁡ 3 ω 2 t + ε ( a cos ⁡ ω 2 t + b cos ⁡ 3 ω 2 t ) 3

(12)

a + b = A

(13)

For simplicity we set t = 0 in equations (11) and (12)

R 1 ( 0 ) = − a − 9 b + A + ε A 3

(14)

R 2 ( 0 ) = − 4 a − 36 b + A + ε A 3

(15)

By He’s frequency–amplitude formulation, we have

ω 2 = ω 2 2 R 1 ( 0 ) − ω 1 2 R 2 ( 0 ) R 1 ( 0 ) − R 2 ( 0 ) = 4 ( − a − 9 b + A + ε A 3 ) − ( − 4 a − 36 b + A + ε A 3 ) ( − a − 9 b + A + ε A 3 ) − ( − 4 a − 36 b + A + ε A 3 ) = 3 A ( 1 + ε A 2 ) 3 a + 27 b = A ( 1 + ε A 2 ) a + 9 b = A ( 1 + ε A 2 ) 9 A − 8 a

(16)

In case when b = 0, we have the same result as that by equation (5). In equation (16) we have a free parameter to improve the accuracy of the frequency.

The approximate solution is

x ( t ) = a cos ⁡ ω t + b cos ⁡ 3 ω t

(17)

In order to obtain a closed solution, we need an additional equation to determine a, b and ω . Equation (17) results in the following residual equation

R ( t ) = − a ω 2 cos ⁡ ω t − 9 b ω 2 cos ⁡ 3 ω t + a cos ⁡ ω t + b cos ⁡ 3 ω t + ε ( a cos ⁡ ω t + b cos ⁡ 3 ω t ) 3

(18)

We set

R ( t = 2 π ω N ) = 0

(19)

R ( t = π 6 ω ) = − a 3 ω 2 2 + a 3 2 + ε ( a 3 2 ) 3 = 0

(20)

= ω 2 1 + 3 4 ε a 2

(21)

Solving equations (13), (16) and (21) simultaneously, a closed solution can be obtained. It is obvious that a ≈ A , the result is more accurate than that by the homotopy perturbation method.

Discussion and conclusions

Hereby we show that the modification of He’s frequency–amplitude formulation can lead to any accuracy of the frequency of a nonlinear oscillator, and the present modification is also valid for other modifications of He’s frequency–amplitude formulation summarized in Tian and Liu. ¹ A general trial solution can be written as

x ( t ) = ∑ i = 1 M a i cos ⁡ ( ( 2 i − 1 ) ω t )

(22)

According to the initial conditions, we have

∑ i = 1 M a i = A

(23)

We need additional M-2 equations to have a closed solution, we set

R ( t = 2 π ω N i ) = 0 , ( i = 1 ∼ M − 2 )

(24)

Solving equations (16), (23) and (24) simultaneously, we can obtain a closed solution.

He’s frequency–amplitude formulation was widely used to a fast insight into the periodic property of a nonlinear oscillator, so it has been attracting much attention in engineering. Now this simple formulation is modified to solve a nonlinear oscillator for any accuracy, making it much attractive for both mathematics and engineering applications.