He’s frequency–amplitude formulation for nonlinear oscillator with damping

Introduction

In this study, we intend to seek approximate frequency of the following nonlinear oscillator with damping ¹ through He’s frequency–amplitude formulation

u ″ + ku 1 + c u ′ 2 + bu = 0 , u ( 0 ) = A , u ′ ( 0 ) = 0

(1)

It was hardly possible for scientists all over the world to solve nonlinear oscillators accurately in early stage of nonlinear science. Now the condition is changed, some effective methods have been developed for the analyses of nonlinear oscillators, such as the variational approach,^2–6 the homotopy perturbation methods,^7–13 the variational iteration method,^14–17 He’s frequency–amplitude formulation,^18–21 the parameter expansion method, ²² the Hamiltonian approach, ²³ the stability analysis, ²⁴ the max-min approach, ²⁵ the energy balance method, ²⁶ and so on. Although most of the methods are suitable for conservative oscillators, there is some literature focusing on damping cases. Gheorghe and Cristian reported the stability of solutions to damped nonlinear oscillators.^27,28 Ene et al. employed the optimal homotopy perturbation method for nonlinear oscillators with non-viscous exponential damping. ²⁹ Tian et al. studied the effects of fractional-order intrinsic damping, fractional-order external damping and nonlinear noise on resonant behaviors.^30,31 There are other references^32–34 on the investigation of properties of nonlinear oscillators with random damping. This paper will demonstrate the effectiveness and simplicity of He’s frequency–amplitude formulation for approximate frequency of nonlinear oscillator with damping.

He’s frequency–amplitude formulation

He’s frequency–amplitude formulation was originated from an ancient Chinese mathematics method. Below is an algebraic equation

F ( x ) = 0

(2)

Let x₁ and x₂ be its two trial solutions, which produce two residuals F(x₁) and F(x₂), respectively; according to the ancient Chinese mathematics method, an approximate solution of the algebraic equation can be written as

x = x 2 F ( x 1 ) − x 1 F ( x 2 ) F ( x 1 ) − F ( x 2 )

(3)

Equation (3) generally gives an accurate solution, although with the extremely simple calculating procedure.

He’s frequency–amplitude formulation was proposed by Prof. Ji-Huan He for nonlinear oscillators^35,36 as below

u ″ + F ( u ) = 0 , u ( 0 ) = A , u ′ ( 0 ) = 0

(4)

R 1 ( t ) = F ( u 1 ) − ω 1 2 A cos ω 1 t

(5)

R 2 ( t ) = F ( u 2 ) − ω 2 2 A cos ω 2 t

(6)

Meanwhile, u 1 = A cos ⁡ ω 1 t and u 2 = A cos ⁡ ω 2 t are, respectively, the exact solutions to the following two linear differential equations

u ″ + ω 1 2 u = 0

(7)

u ″ + ω 2 2 u = 0

(8)

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

(9)

Solving process

We rewrite equation (1) in the form

u ″ + ( b + k ) u + c u ′ 2 ( u ″ + bu ) = 0

(10)

Here, we substitute a trial function u = A cos ⁡ ω t , which is the solution to linear equation u ″ + ω 2 u = 0 , into equation (10), we get a residual

R ( t ) = ( b + k − ω 2 ) A cos ⁡ ω t + ω 2 c A 3 ( b − ω 2 ) sin ⁡ 2 ω t cos ⁡ ω t

(11)

When ω = 1, it becomes

R 1 ( t ) = ( b + k − 1 ) A cos ⁡ t + c A 3 ( b − ω 2 ) sin ⁡ 2 t cos ⁡ t

(12)

According to He’s frequency–amplitude formulation in equation (9), an approximate frequency of equation (10) can be calculated by

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

(13)

Geng and Cai set the phase of residuals by t = T 12 , ³⁷ where T ( = 2 π ω ) is the period, that is to say that the phases of equations (11) and (12) are both π 6 .

R ( T 12 ) = 3 2 ( b + k − ω 2 ) A + 3 8 ω 2 c A 3 ( b − ω 2 )

(14)

R 1 ( T 12 ) = 3 2 ( b + k − 1 ) A + 3 8 c A 3 ( b − 1 )

(15)

Substituting equations (14) and (15) into equation (13), we get

ω 2 = 1 4 c A 2 ω 2 + b + k − 1 4 c A 2 ( b − ω 2 − 1 ) + 1

(16)

Equation (16) is an algebraic equation, we rewrite it as

c A 2 ω 4 + ( 4 − bc A 2 ) ω 2 − 4 ( b + k ) = 0

(17)

ω = bc A 2 − 4 + ( bc A 2 − 4 ) 2 + 16 c A 2 ( b + k ) 2 c A 2

(18)

Summary

After the brief solving procedure shown in the context above, we obtained the reliable relationship between the frequency and the amplitude of damped nonlinear vibration, which includes the main parameters affecting the frequency property. He’s frequency–amplitude formulation provides an efficient method for the analysis of nonlinear oscillators with damping.