Introduction

Nonlinear vibration arises everywhere in engineering, it is of utter importance to have a fast insight into its frequency or period property, and a simple mathematical method is very much appreciated for practical applications. Though there are many analytical methods for nonlinear vibrations, among which the amplitude-frequency formulation [1] and the max-min approach [2] are widely adopted for this purpose due to shorter calculation with relatively higher accuracy. Other analytical methods for nonlinear oscillators are summarized in Refs. [3, 4]. In this paper we will suggest a remarkably simple way with a relatively acceptable accuracy to conservative nonlinear oscillators with odd nonlinearities.

Amplitude-Frequency Relationship

To illustrate the basic solution process of the new method, we first consider a linear oscillator in the form

$$\begin{aligned} {u}''+ku=0 \end{aligned}$$
(1)

where k is a constant.

The square of its frequency can be easily obtained, which reads

$$\begin{aligned} \omega ^{2}=\frac{dg(u)}{du}={g}'(u)=k \end{aligned}$$
(2)

where g(u) is the restoring force, \(g(u)=ku\).

Now we consider a nonlinear oscillator in the form

$$\begin{aligned} {u}''+f(u)=0 \end{aligned}$$
(3)

where f(u) is a nonlinear restoring force, it requires \(f(u)/u>0\) and f(0) \(=\) 0. We extend Eq. (2) to nonlinear cases, that is

$$\begin{aligned} \omega ^{2}=\frac{df(u)}{du} \end{aligned}$$
(4)

Equation (4) is valid only for the linear case, df(u) / du is a function of u for nonlinear oscillators. Locating at \(u=\frac{i}{N}A(i=1,2,3,\ldots ,N-1)\), where A is the amplitude, we have

$$\begin{aligned} \omega _{i}^{2} =\frac{df}{du}(u=iA/N), \quad i=1,2,3,\ldots ,N-1 \end{aligned}$$
(5)

The square of its frequency is approximately written as

$$\begin{aligned} \omega ^{2} =\frac{{\mathop \sum \nolimits _{i=1}^{N-1}} {\omega _{i}^{2} } }{N-1} \end{aligned}$$
(6)

When N=2, we have

$$\begin{aligned} \omega ^{2}\left. {={f}'(u)} \right| _{u=A/2} \end{aligned}$$
(7)

Equation (7) can be used for a fast qualitative analysis of a nonlinear oscillator.

Consider the Duffing equation, which is

$$\begin{aligned} {u}''+u+\varepsilon u^{3}=0, \;u(0)=A,{u}'(0)=0 \end{aligned}$$
(8)

Hereby \(f(u)=u+\varepsilon u^{3}\). By Eq. (7), the square of its frequency can be immediately obtained, which is

$$\begin{aligned} \omega ^{2}\left. {={f}'(u)} \right| _{u=A/2} =1+3\varepsilon \left( \frac{A}{2}\right) ^{2}=1+\frac{3}{4}\varepsilon A^{2} \end{aligned}$$
(9)

When \(\varepsilon \) is small, i.e., \(\varepsilon <<1\), Eq. (9) is equivalent to that obtained by the perturbation method; when \(\varepsilon \) tends to infinite, the exact frequency reads [3, 4]

$$\begin{aligned} \omega _{ex} =0.9318\sqrt{\varepsilon A^{2}} \end{aligned}$$
(10)

The accuracy of the obtained frequency by Eq. (9) reaches 7 % even when \(\varepsilon \rightarrow \infty \).

If we set N \(=\) 3 and N \(=\) 4, respectively, in Eq. (6), we have

$$\begin{aligned} \omega ^{2}=\frac{1+3\varepsilon \left( \frac{A}{3}\right) ^{2}\hbox {+}1+3\varepsilon \left( \frac{2A}{3}\right) ^{2}}{2}=1+\frac{8}{9}\varepsilon A^{2}, \quad \hbox {N}=3 \end{aligned}$$
(11)

and

$$\begin{aligned} \omega ^{2}=\frac{1+3\varepsilon \left( \frac{A}{4}\right) ^{2}\hbox {+}1+3\varepsilon \left( \frac{2A}{4}\right) ^{2}\hbox {+}1+3\varepsilon \left( \frac{3A}{4}\right) ^{2}}{3}=1+\frac{7}{8}\varepsilon A^{2},\quad \hbox {N}=4 \end{aligned}$$
(12)

The accuracy of the frequency improves to 1.18 and 0.38 %, respectively, when \(\varepsilon \rightarrow \infty \).

Discussion and conclusions

Equation (6) is such constructed only for simple calculation, there are many alternative determinations of the square of frequency, for example

$$\begin{aligned} \omega ^{2} =\frac{{\mathop \sum \nolimits _{i=1}^{N}} {\omega _{i}^{2} } }{N} \end{aligned}$$
(13)

where \(\omega _{i}\) is defined by Eq. (5), or in a more general form

$$\begin{aligned} \omega ^{2} =\frac{{\mathop \sum \nolimits _{i=1}^{N}} {{f}'(u_i )} }{N} \end{aligned}$$
(14)

where \(u_{i} (i=1\sim N)\) are location points, \(0<u_{i} <A\). For Duffing equation, we set N \(=\) 2 and locate two points:\(u_{1} = 0.5A\) and \(u_{2} =0.6A\), from Eq. (14) we have

$$\begin{aligned} \omega ^{2} =\frac{{f}'(0.5A)+{f}'(0.6A)}{2}=1+0.915\varepsilon A^{2} \end{aligned}$$
(15)

The accuracy of the frequency is 2.65 %.

The most simple calculation is

$$\begin{aligned} \omega ^{2} ={f}'(u_i ), \quad 0<u_{i} <A \end{aligned}$$
(16)

The accuracy, however, depends greatly upon the location point. Hereby we give a criterion for choosing a suitable location point, see Table 1.

Table 1 Criterion for choosing a location point
Full size table

For Duffing equation, we have \(u{f}''(u)=6\varepsilon u^{2}>0\), we choose \(u_{i} =0.51A\):

$$\begin{aligned} \omega ^{2} =\frac{df}{du}(u=0.51A)=1+0.7803\varepsilon A^{2} \end{aligned}$$
(17)

The accuracy of the obtained frequency improves from 7 % for \(u=0.5A\) to 5.2 % for \(u=0.51A\).

Consider another example in the form

$$\begin{aligned} {u}''+u^{1/3}=0,\;u(0)=A,{u}'(0)=0 \end{aligned}$$
(18)

Hereby \(f(u)=u^{1/3}\), which satisfies the condition: \(u{f}''(u)<0\), therefore the location point should be \(0<u_{i} <A/2\). We choose two location points \(u=0.5A\) and \(u=0.2A\) for comparison.

By Eq. (16), we have

$$\begin{aligned} \omega ^{2}= & {} \frac{1}{3}(0.5A)^{-2/3}=0.5291A^{-2/3}\end{aligned}$$
(19)
$$\begin{aligned} \omega ^{2}= & {} \frac{1}{3}(0.2A)^{-2/3}=0.9746A^{-2/3} \end{aligned}$$
(20)

The exact frequency for Eq. (18) is

$$\begin{aligned} \omega _{ex} =1.070451A^{-1/3} \end{aligned}$$
(21)

It is obvious that the accuracy improves from 32.05 % for \(u=0.5A\) to 7.77 % for \(u=0.2A\), showing that the criterion given in Table 1 is practicable.

We conclude that this paper might give the most simple and direct way to outline the general solution property of a nonlinear oscillator, while the accuracy is always remarkable contrast to those obtained by the perturbation method. The error by the perturbation method tends to infinity when \(\varepsilon \rightarrow \infty \) for Duffing equation [4], while all predictions in this paper are relatively acceptable even when \(\varepsilon \rightarrow \infty \). The utmost simplicity of the solution process makes the method much attractive for practical applications. The examples given in this paper can be used as a paradigm for many other applications.