Introduction

Nonlinear vibration arises everywhere in science, engineering and other disciplines, since most phenomena in our world today, are essentially nonlinear and are described by nonlinear equations. It is very important in applications to have a version of the frequency (or period) to have a better understanding of the phenomena modeled through differential equations that contain terms with high nonlinearities, and a simple mathematical method is very useful for practical applications.

Recently many analytical methods have appeared to obtain the approximate solutions of nonlinear systems, such as the parameter-expansion method [30], the harmonic balance method [3, 22, 28, 31], the energy balance method [23, 36], the Hamiltonian approach [13, 37], the use of special functions [6, 7], the max–min approach [14, 39], the Adomian decomposition technique [10], the variational iteration method [15, 16, 21, 33, 35] and homotopy perturbation [2, 4, 9, 11, 17,18,19] are some examples. An excellent study, in which many of these techniques can be found in detail to solve nonlinear problems of oscillatory type can be seen in [20].

Recently, In [12] an analytical approximate technique for large and small amplitudes oscillations of a class of conservative single degree-of-freedom systems with odd non-linearity is proposed. In this study, we have applied new method to find the approximate solutions of nonlinear differential equation governing strongly nonlinear oscillators and have made a comparison with the exact solution. The most interesting features of the used method are its simplicity and its excellent accuracy of both period and corresponding periodic solution for the entire range of oscillation amplitude. Finally, four examples are presented to describe the solution methodology and to illustrate the usefulness and effectiveness of the proposed technique.

He’s New Amplitude–Frequency Relationship

Consider a one-dimensional, free nonlinear oscillator (undamped and unforced) governed by

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

with the initial conditions

$$\begin{aligned} u(0)=A,\quad u'(0)=0 \end{aligned}$$
(2)

where a prime denotes differentiation with respect to t, u is the displacement, and the nonlinear restoring force f(u) is an odd function of u, i.e. \(f (-u)=-f (u)\) and satisfies \(f (u)/u>0\) for \(u\in [-A,A]\), \(u\ne 0\). It is obvious that \(u = 0\) is the equilibrium position. The system oscillates between the symmetric bounds \(-A\) and A. If f(u) is a nonlinear function, both period T and frequency \(\omega =2\pi /T\) of the corresponding oscillation are dependent upon the amplitude of oscillation A. The relationship between the frequency and amplitude is the main property of a nonlinear oscillator; see [5, 8] and references therein.

A simple realization of the harmonic oscillator in classical mechanics is a particle which is acted upon by a restoring force proportional to its displacement from its equilibrium position. Considering motion in one dimension, this means

$$\begin{aligned} f=-ku \end{aligned}$$
(3)

Such a force might originate from a spring which obeys Hooke’s law. The force constant k is a measure of the stiffness of the spring. Now applying Newton’s second law to the force from Eq. (3), we obtain

$$\begin{aligned} f=mu''=-ku \end{aligned}$$
(4)

where m is the mass of the body attached to the spring, which is itself assumed massless. This leads to a linear differential equation of familiar form

$$\begin{aligned} u''+\omega ^2 u=0, \quad \omega ^2 =k/m. \end{aligned}$$
(5)

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

$$\begin{aligned} \omega ^2=F'(u) \end{aligned}$$
(6)

where F(u) is the restoring force, \(F(u) = \omega ^{2} u\).

Let us now consider a general nonlinear oscillator of the form given by Eq. (1). In this case, the restoring force is given by f(u). We extend Eq. (6) to nonlinear cases, that is

$$\begin{aligned} \omega ^2=f'(u). \end{aligned}$$
(7)

According to He’s new amplitude–frequency formulation, the approximate frequency as a function of A can be obtained as follows [12]:

$$\begin{aligned} \omega ^{2}(A)=\frac{\sum _{i=1}^{N}\omega _{i}^{2}(A)}{N} \end{aligned}$$
(8)

with each \(\omega _{i}^{2}(A)\) defined by

$$\begin{aligned} \omega _{i}^{2}(A)=f'(u_{i}) \end{aligned}$$
(9)

where \(u_i\) are location points, \(0<u_{i}<A\). Explicitly, \(u_{i}=iA/N\) for every \(i=1,2,\ldots , N-1\).

The simplest way to calculate the frequency is given by

$$\begin{aligned} \omega ^{2}(A)=f'(u_{i}), \end{aligned}$$
(10)

for some \(0<u_{i}<A\). The accuracy, however, depends greatly upon the location point.

In Table 1 we present the criteria suggested by He [12] for choosing a suitable location point \(u_i\).

Table 1 Criterion for choosing a location point
Full size table

Therefore, according to Eq. (10) the analytical approximate frequency \(\omega \) as a function of A is

$$\begin{aligned} \omega _{{ app}}(A)=\sqrt{f'(u_{i})}. \end{aligned}$$
(11)

For conservative oscillations with an odd restoring force f(u), there exists a periodic motion around the equilibrium point \(u = 0\) with frequency \(\omega \) and amplitude A. Therefore, a reasonable and the simplest initial approximation of u(t) which satisfies the initial condition given by Eq. (2) is [25]:

$$\begin{aligned} u(t)=A\cos (\omega t). \end{aligned}$$
(12)

From Eq. (11) we obtain the following approximate periodic solution to (1)

$$\begin{aligned} u_{{ app}}(t)=A\cos \left( \sqrt{f'(u_{i})}\cdot t\right) . \end{aligned}$$
(13)

Numerical Examples

In this section, we will give four examples to illustrate the use and the efectiveness of the present approach.

Nonlinear oscillators in physics, engineering, biology, mathematical and related fields have been the focus of attention for many years. The Duffing equation is a well-known nonlinear differential equation which is related to many practical engineering systems such as the classical nonlinear spring system with odd nonlinear restoring characteristics and more recently in different physical phenomena. There have been many variations of Duffing equation, one of them is the cubic–quintic Duffing equation. The systems modelled by the cubic–quintic Duffing equation include the nonlinear dynamics of a slender elastica, the compound Korteweg–de Vries (KdV) equation in nonlinear wave systems, or the propagation of a short electromagnetic pulse in a nonlinear medium, structural dynamics, among others [31, 32]. In Examples 1 and 2, we will illustrate that the proposed method is remarkably effective and applicable in solving the generalized Duffing equation even with strong nonlinearities. Finally, in the Examples 3 and 4 He’s new amplitude–frequency relationship will be used to solve the nonlinear differential equation related to plasma physics and oscillations of a mass attached to a stretched elastic wire, respectively.

Example 1

Consider the cubic–quintic Duffing nonlinear oscillator, which is modelled by the following second-order differential equation

$$\begin{aligned} u''+u+u^{3}+u^{5}=0, \end{aligned}$$
(14)

with initial conditions

$$\begin{aligned} u(0)=A,\;\; u'(0)=0. \end{aligned}$$
(15)
Fig. 1
figure 1

Comparison of analytical approximation (dashed) and exact solution (black) for \(A=1/10\) in Example 1

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Fig. 2
figure 2

Comparison of analytical approximation (dashed) and exact solution (black) for \(A=50\) in Example 1

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In the present example we have \(f(u)=u+u^{3}+u^{5}\), it is clear that f is an odd function and satisfies \(f (u)/u>0\).

Calculating we have \(f'(u)=1+3u^{2}+5u^{4}\) and \(f''(u)=6u+20u^{3}\), hence \(uf''(u)>0\). Now, considering the criterion given in Table 1 we must take the location points \(A/2\le u_{i}<A\). If we take \(u_{i}=0.5772A\) and consider the proposed approach in Eq. (11), one can assume for the frequency–amplitude formulation

$$\begin{aligned} \omega _{{ app}}(A)=\sqrt{1+3(0.5772)^{2}A^{2}+5(0.5772)^{4}A^{4}}. \end{aligned}$$
(16)

We, therefore, obtain the following periodic solution:

$$\begin{aligned} u_{{ app}}(t)=A\cos \left( \sqrt{1+3(0.5772)^{2}A^{2}+5(0.5772)^{4}A^{4}}\cdot t\right) \end{aligned}$$
(17)

which has a high accuracy (see Figs. 1, 2).

The exact frequency for the present example is given by [38]:

$$\begin{aligned} \omega _{{ ex}}(A) = \dfrac{2\pi }{\displaystyle \int _{0}^{\pi /2}\frac{4d\theta }{\sqrt{1+\frac{1}{2}\left( 1 +\sin ^{2}\theta \right) A^{2}+\frac{1}{3}\left( 1+\sin ^{2}\theta +\sin ^{4}\theta \right) A^{4}}}}. \end{aligned}$$
(18)

From Table 2, it can be observed that Eq. (16) yield excellent analytical approximate periods for both small and large values of oscillation amplitude A.

Table 2 Comparison between frequencies \(\omega _{{ app}}(A)\) and \(\omega _{{ ex}}(A)\) for different values of A
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The obtained results in this example reveals that the presented method is very effective, simple and exact and is valid for small and large amplitudes.

Example 2

In this example, we consider the following nonlinear Duffing oscillator:

$$\begin{aligned} u''+u+u^{5}=0, \end{aligned}$$
(19)

subject to the initial conditions

$$\begin{aligned} u(0)=A,\;\; u'(0)=0. \end{aligned}$$
(20)

For this problem,

$$\begin{aligned} f(u)=u+u^{5}, \end{aligned}$$

it is clear that f is an odd function and satisfies \(f (u)/u>0\).

Derivating we have, \(f'(u)=1+5u^{4}\) and \(f''(u)=20u^{3}\), hence \(uf''(u)=20u^{4}>0\). Therefore, considering the criterion given in Table 1 we must take the location points \(A/2\le u_{i}<A\). If we take \(u_{i}=0.5779A\) and consider the proposed approach in Eq. (11), one can assume for the frequency–amplitude formulation

$$\begin{aligned} \omega _{{ app}}(A)=\sqrt{1+5(0.5779)^{4}A^{4}}. \end{aligned}$$
(21)

The exact frequency for the present problem was established in [16] and is given by

$$\begin{aligned} \omega _{{ ex}}(A) = \frac{\pi \sqrt{A^{4}+3}}{2\sqrt{3}}\left( \int _{0}^{\pi /2}\frac{1}{\sqrt{1+\Big (\frac{ A^{4}}{A^{4}+3}\Big )(\sin ^{2}\theta +\sin ^{4}\theta )}}\;d\theta \right) ^{-1}. \end{aligned}$$
(22)

To illustrate and verify accuracy of these approximate analytical approach, a comparison of approximate frequencies \(\omega _{{ app}}(A)\) for different values of amplitude A and the exact frequencies \(\omega _{{ ex}}(A)\) is presented in Table 3. Note that the approximation is very accurate for small values and large values of A. From Table 3 we can see that

$$\begin{aligned} \lim \limits _{A\rightarrow 0^{+}}\frac{\omega _{{ app}}(A)}{\omega _{{ ex}}(A)}=1 \quad \text{ and }\quad \lim \limits _{A\rightarrow \infty }\frac{\omega _{{ app}}(A)}{\omega _{{ ex}}(A)}=0.999922. \end{aligned}$$
(23)
Table 3 Comparison between frequencies \(\omega _{{ app}}(A)\) and \(\omega _{{ ex}}(A)\) for different values of A
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Considering the approximation for the frequency obtained in Eq. (21) the approximate solution of Eq. (19) becomes

$$\begin{aligned} u_{{ app}}(t)=A\cos \left( \sqrt{1+5(0.5779)^{4}A^{4}}\cdot t\right) . \end{aligned}$$
(24)

For this example we will not show graphs as we did in the previous example, because the high precision would not allow the distinction between them.

Example 3

A problem of some importance in plasma physics concerns an electron beam injected into a plasma tube where the magnetic field is cylindrical and increases towards the axis in inverse proportion to the radius. The governing equation for the path u of the electrons is modelled by the following second-order differential equation [1, 24]:

$$\begin{aligned} u''+\frac{1}{u}=0, \end{aligned}$$
(25)

with initial conditions

$$\begin{aligned} u(0)=A,\;\; u'(0)=0. \end{aligned}$$
(26)

The exact solution for Eq. (25) as a function of A was obtained in [29] and this is

$$\begin{aligned} \omega _{{ ex}}(A) = 2\pi \Bigg [ 2\sqrt{2}A \int _{0}^{1}\frac{d s}{\sqrt{\ln (1/s)}}\Bigg ]^{-1}. \end{aligned}$$
(27)

To use the method presented in the “He’s New Amplitude–Frequency Relationship” section, we will consider \(f(u)=\frac{1}{u}\), it is clear that f is an odd function and satisfies \(f (u)/u>0\).

Calculating, we get \(f'(u)=-\frac{1}{u^2}\) and \(f''(u)=\frac{2}{u^3}\), hence \(uf''(u)>0\). Now, considering again the criterion given in Table 1 we must take the location points \(A/2\le u_{i}<A\). If we take \(u_{i}=0.799A\) and consider the proposed approach in Eq. (11), one can assume for the frequency–amplitude formulation

$$\begin{aligned} \omega _{{ app}}(A)= & {} \sqrt{\frac{1}{\left( \frac{799}{1000}\right) ^{2}A^{2}}} =\frac{1000}{799A}. \end{aligned}$$
(28)
$$\begin{aligned} \lim \limits _{A\rightarrow 0^{+}}\frac{\omega _{{ app}}(A)}{\omega _{{ ex}}(A)}= & {} \lim \limits _{A\rightarrow \infty }\frac{\omega _{{ app}}(A)}{\omega _{{ ex}}(A)}=0.9986. \end{aligned}$$
(29)
Table 4 Comparison between frequencies \(\omega _{{ app}}(A)\) and \(\omega _{{ ex}}(A)\) for different values of A
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Finally, considering the approximation (28), we have obtain the following periodic solution of the Eq. (25)

$$\begin{aligned} u_{{ app}}(t)=A\cos \left( \frac{1000}{799A}\cdot t\right) . \end{aligned}$$
(30)

The obtained solution is of remarkable accuracy, as shown in Table 4 and Fig. 3.

Fig. 3
figure 3

Comparison of analytical approximation (dashed) and exact solution (black) for \(A=100\) in Example 3

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Example 4

The governing nonlinear differential equation of motion and the associated initial conditions for a mass attached to a stretched elastic wire are [28]:

$$\begin{aligned} u''+u+\frac{u}{\sqrt{1+u^2}}=0, \quad u(0)=A,\;\; u'(0)=0. \end{aligned}$$
(31)

Which, \(f(u)=u+\frac{u}{\sqrt{1+u^2}}\). Its derivatives are:

$$\begin{aligned} f'(u)=1+\frac{1}{\sqrt{\left( 1+u^2\right) ^3}}, \quad f''(u)= -\frac{3u}{\sqrt{\left( 1+u^2\right) ^5}}. \end{aligned}$$
(32)

From Eq. (32) we have \(uf''(u)<0\). Considering the criterion given in Table 1 we must take the location points \(A<u_{i}<A/2\). If we take \(u_{i}=0.48A\) and consider the proposed approach in Eq. (11), one can assume for the frequency–amplitude formulation

$$\begin{aligned} \omega _{{ app}}(A)=\sqrt{1+\frac{1}{\left( 1 +\left( \frac{48}{100}\right) ^2A^2\right) ^{3/2}}}. \end{aligned}$$
(33)

The nonlinear oscillator described in Eq. (31) is a conservative system. By integrating Eq. (31) and using the initial conditions, we arrive at

$$\begin{aligned} \omega _{{ ex}}(A)=\frac{1}{2} \pi \left( \int _0^{\frac{\pi }{2}} \frac{A\cos \theta }{\sqrt{A^{2}\cos ^{2}\theta -2\big (\sqrt{1+A^2\sin ^{2}\theta }-\sqrt{1+A^2}\big )}} d\theta \right) ^{-1} \end{aligned}$$
(34)

By taking into account our approximation made through He’s frequency–amplitude formulation Eq. (33) and \(\omega _{{ ex}}(A)\) from Eq. (34) we can calculate the Table 5 for small and large values of A.

Table 5 Comparison between frequencies \(\omega _{{ app}}(A)\) and \(\omega _{{ ex}}(A)\) for different values of A
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Also, considering the approximation given by Eq. (33), we have obtain the following periodic solution of the Eq. (31)

$$\begin{aligned} u_{{ app}}(t)=A\cos \left( \sqrt{1+\frac{1}{\left( 1+\left( \frac{48}{100}\right) ^2A^2\right) ^{3/2}}}\cdot t\right) . \end{aligned}$$
(35)

The obtained solution is very acceptable accuracy, as shown in Figs. 4 and 5.

We can conclude that formula (33) is valid for the whole range of values of amplitude of oscillation and its maximum relative error is 5.3% and this is obtained when \(A = 10\). We can also see that, for very large or very small values of A, we have

$$\begin{aligned} \lim \limits _{A\rightarrow 0^{+}}\frac{\omega _{{ app}}(A)}{\omega _{{ ex}}(A)}= \lim \limits _{A\rightarrow \infty }\frac{\omega _{{ app}}(A)}{\omega _{{ ex}}(A)}=1. \end{aligned}$$
(36)
Fig. 4
figure 4

Comparison of analytical approximation (dashed) and exact solution (black) for \(A=10\) in Example 4

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Fig. 5
figure 5

Comparison of analytical approximation (dashed) and exact solution (black) for \(A=100\) in Example 4

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The study of nonlinear problems arisen in many areas of physics and also engineering is very complex issue for scientists. Since most phenomena in our world are essentially nonlinear and are described by nonlinear equations, it is very difficult to solve nonlinear problems and in general, it is often more difficult to get an analytic approximation than a numerical one for given nonlinear problems. One of the critical problems in materials science field is the behavior of elastic materials. The extensive literature on the topic is now available and we can only mention a few recent interesting investigations in [26, 27, 34].

Conclusions

He’s new amplitude–frequency relationship recently established by He [12] is proved to be a powerful mathematical tool for use in the search for periodic solutions of nonlinear oscillators. It is simple, straightforward and effective. Moreover the approximate analytical solutions are valid for small as well as large amplitudes of oscillation.

The new method applied in this paper is of potential and can be applied to other strongly nonlinear oscillators with more general restoring forces provided that they meet the requirements established in “He’s New Amplitude–Frequency Relationship” section. Finally, four examples have been presented to illustrate excellent accuracy of the analytical approximate periods and the corresponding periodic solutions; being our main contribution in the present study to find the location points involved in Eq. (10). All numerical work and graphics were performed with the Mathematica software package.