He's frequency–amplitude formulation for nonlinear oscillators using Jacobi elliptic functions

Abstract

In this work, the Duffing’s type analytical frequency–amplitude relationship for nonlinear oscillators is derived by using Hés formulation and Jacobi elliptic functions. Comparison of the numerical results obtained from the derived analytical expression using Jacobi elliptic functions with respect to the exact ones is performed by considering weak and strong Duffing’s nonlinear oscillators.

Introduction

Many dynamic systems that arise in physics and engineering applications are known to be modeled by a homogeneous nonlinear differential equation whose closed-form solution is unknown, Then, numerical or approximate methods have to be used to predict its dynamic response behavior and determine the frequency–amplitude relationship needed to understand its qualitative and quantitative dynamic response.^1–18 However, in a few cases, the closed-form solution of some nonlinear oscillators is known. In fact, the cubic, the quadratic–cubic, and the cubic–quintic Duffing-type oscillators are those whose analytical closed form solutions are described exactly in terms of Jacobi elliptic functions.^19–22 It is also known that when the rational or irrational functions describe the restoring forces of nonlinear oscillators, the original nonlinear differential equations can be written in an equivalent form using methods that transform the original equations into Duffing-type oscillators and thus, the approximate frequency–amplitude relationship can be determined by the analytical closed-form solutions of the corresponding Duffing-type oscillator.

However, to find a simple and accurate frequency–amplitude expression, Professor He has used an ancient Chinese mathematical algorithm in which the frequency–amplitude of various nonlinear oscillators was found.^23–25

Therefore, the article aims to enhance the accuracy of He’s formulation using trial residuals based on Jacobi elliptic functions, which are a more general class of periodic functions that include the trigonometric functions as a particular case. To demonstrate the accuracy achieved by using Jacobi functions to find the approximate frequency–amplitude expression by He’s algorithm, two cases are examined. The first case focuses on determining the approximate frequency of the Fangzhu singular equation of motion, while the second case focuses on obtaining the frequency–amplitude equation for Duffing-type oscillators considering even and odd nonlinearities.

Preliminaries: He’s frequency–amplitude approach

In this section, we provide a brief summary of He’s formulation to derive frequency–amplitude expressions of nonlinear oscillators of the form

$$\ddot{u}+f^{}\left(u\right)=0,\mathrm{\hspace{1em}}u\left(0\right)=A_0,\mathrm{\hspace{1em}}\dot{\mathit{u}}\left(0\right)=v_0$$

(1)using the ancient Chinese algorithm^23–25

$${\omega }_{He}^2=\frac{{\omega }_1^2{R}_2\left(t\right)-{\omega }_2^2{R}_2\left(t\right)}{R_2\left(t\right)-R_1\left(t\right)}$$

(2)where ω $t_1$ and ω $t_2$ are location points, ${\mathrm{\omega }}_i$ are arbitrary chosen frequencies with ${\mathrm{\omega }}_2>{\mathrm{\omega }}_1$ , and $R_i\left(t\right)$ are trial residual functions defined as $R_i\left(t\right)=\ddot{u_i}+f\left(u_i\right)$ , where

$$R_1\left(t\right)=-A{\omega }_1^2\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{1}t+f\left(A\mathrm{co}\mathrm{s}{\omega }_{1}t\right)$$

(3)

$$R_2\left(t\right)=-A{\mathrm{\omega }}_2^2\cos {\mathrm{\omega }}_{2}t+f\left(A\cos {\mathrm{\omega }}_{2}t\right)$$

(4)and $u_i\left(t\right)=A_i\cos {\mathrm{\omega }}_{i}t$ , $i=1, 2$ . $R_i\left(t\right)$ are assumed to be time-dependent and therefore, their average values can be computed from the following expression

$$R_i=\frac{4}{T}{\int }_0^{T/4}{R}_i\left(t\right)w\left(t\right)\mathrm{d}t$$

(5)where $T$ is the motion period, and $w\left(t\right)$ is a weighted function that can take any simple form.

We shall next modify He’s approach by introducing Jacobi elliptic functions instead of trigonometric ones.

Frequency–amplitude formulation based on elliptic functions

It is well known that most of the nonlinear oscillators can be expressed in the form

$$\ddot{u}+f^{\mathrm{*}}\left(u\right)=0,\mathrm{\hspace{1em}}u\left(0\right)=A_0,\mathrm{\hspace{1em}}\dot{u}\left(0\right)=v_0$$

(6)where $A_0$ is the initial amplitude of oscillation and $f^{\mathrm{*}}\left(u\right)$ are the conservative system restoring forces. If the following transformation $y=u/A_0$ is introduced, then equation (6) can be written as

$$\ddot{y}+f\left(y\right)=0,\mathrm{\hspace{1em}}y\left(0\right)=1,\mathrm{\hspace{1em}}\dot{y}\left(0\right)=v$$

(7)

Based on the ancient Chinese method, two trial functions are needed to find the approximate solution of equation (7) that are assumed to be $y_1=A_{1}cn\left({\mathrm{\omega }}_{1}t,k_1^2\right)$ , and $y_2=A_{2}cn\left({\mathrm{\omega }}_{2}t,k_2^2\right)$ that satisfy the following Duffing differential equations

$$\ddot{y_1}+{\mathrm{\omega }}_1^2{y}_1+{\mathrm{\alpha }}_1{y}_1^3=0,\text{ and }\ddot{y_2}+{\mathrm{\omega }}_2^2{y}_2+{\mathrm{\alpha }}_2{y}_2^3=0$$

(8)where $A_i$ , ${\mathrm{\omega }}_i$ , and $k_i$ are trial amplitudes, frequencies, and modulus of the Jacobi elliptic functions $cn\left({\mathrm{\omega }}_{1}t,k_1^2\right)$ and $cn\left({\mathrm{\omega }}_{2}t,k_2^2\right)$ that need to be found. Thus, the residuals to find the approximate solution of equation (7) are

$$R_1\left(t\right)=f\left(y_1\right)-A_1\left(1-2k_1^2\right){\mathrm{\omega }}_1^{2}cn\left({\mathrm{\omega }}_{1}t,k_1^2\right)-2A_1{k}_1^2{\mathrm{\omega }}_1^{2}c{n}^3\left({\mathrm{\omega }}_{1}t,k_1^2\right)$$

(9)

$$R_2\left(t\right)=f\left(y_2\right)-A_2\left(1-2k_2^2\right){\mathrm{\omega }}_2^{2}cn\left({\mathrm{\omega }}_{2}t,k_2^2\right)-2A_2{k}_2^2{\mathrm{\omega }}_2^{2}c{n}^3\left({\mathrm{\omega }}_{2}t,k_2^2\right)$$

(10)

Since the restoring forces of nonlinear oscillators can be expressed in equivalent form as polynomial expressions of the form

$$f\left(y_i\right)={\mathrm{\alpha }}_0{y}_i+{\mathrm{\alpha }}_1{y}_i^3+{\mathrm{\alpha }}_2{y}_i^5+{\mathrm{\alpha }}_3{y}_i^7+{\mathrm{\alpha }}_{22}{y}_i^2$$

(11)then, equations (9) and (10) can be written as

$$R_1\left(t\right)=\frac{1}{64}(32A_1^2{\alpha }_{22}+\left(64A_1{\alpha }_0+48A_1^3{\alpha }_1+40A_1^5{\alpha }_2+35A_1^7{\alpha }_3+32A_1\left(-2+k_1^2\right){\omega }_1^2\right)\cos \left({\varphi }_1\right)+32A_1^2{\alpha }_{22}\cos \left(2{\varphi }_1\right)+\left(16A_1^3{\alpha }_1+20A_1^5{\alpha }_2+21A_1^7{\alpha }_3-32A_1{k}_1^2{\omega }_1^2\right)\cos \left(3{\varphi }_1\right)+A_1^5\left(4{\alpha }_2+7A_1^2{\alpha }_3\right)\cos \left(5{\varphi }_1\right)+A_1^7{\alpha }_3\cos \left(7{\varphi }_1\right))$$

(12)

$$R_2\left(t\right)=\frac{1}{64}(32A_2^2{\alpha }_{22}+\left(64A_2{\alpha }_0+48A_2^3{\alpha }_1+40A_2^5{\alpha }_2+35A_2^7{\alpha }_3+32A_2\left(-2+k_2^2\right){\omega }_2^2\right)\cos \left({\varphi }_2\right)+32A_2^2{\alpha }_{22}\cos \left(2{\varphi }_2\right)+\left(16A_2^3{\alpha }_1+20A_2^5{\alpha }_2+21A_2^7{\alpha }_3-32A{k}_2^2{\omega }_2^2\right)\cos \left(3{\varphi }_2\right)+A_2^5\left(4{\alpha }_2+7A_2^2{\alpha }_3\right)\cos \left(5{\varphi }_2\right)+A_2^7{\alpha }_3\cos \left(7{\varphi }_2\right))$$

(13)where the identities $\cos {\varphi }_i=cn\left({\omega }_{i}t,k_i^2\right)$ and $\sin {\varphi }_i=sn\left({\omega }_{i}t,k_i^2\right)$ have been used. ²⁶ Here ${\varphi }_i$ represents the Jacobi amplitude given by the expression

$${\mathrm{\varphi }}_i=am\left({\mathrm{\omega }}_{i}t,k_i^2\right)={\int }_0^{{\omega }_{i}t}\mathrm{d}n\left({\omega }_{i}t,k_i^2\right)\mathrm{d}\left({\omega }_{i}t\right)$$

(14)where $dn\left({\omega }_{i}t,k_i^2\right)$ is a Jacobi elliptic function.

From He’s formulation, the residual equations (12) and (13) are found using the following relations

$${\overline{R}}_i=\frac{4}{T}{\int }_0^{T/4}{R}_i\left(t\right)\mathrm{d}t$$

(15)that yield after integration the expressions of ${\overline{R}}_i$

$${\overline{R}}_i=1/\left(1680T{\omega }_i\right)(840A_i^{2}T{\alpha }_{22}{\omega }_i+(6720A_i{\alpha }_0+5040A_i^3{\alpha }_1+4200A_i^5{\alpha }_2+3675A_i^7{\alpha }_3-6720A{\omega }_i^2+3360A{k}_i^2{\omega }_i^2)sn\left(T{\omega }_i/4,k_i^2\right)+1680A_i^2{\alpha }_{22}sn\left(T{\omega }_i/2,k_i^2\right)+\left(560A_i^3{\alpha }_1+700A_i^5{\alpha }_2+735A_i^7{\alpha }_3-1120A{k}_i^2{\omega }_i^2\right)sn\left(3T{\omega }_i/4,k_i^2\right)+\left(84A_i^5{\alpha }_2+147A_i^7{\alpha }_3\right)sn\left(5T{\omega }_i/4,k_i^2\right)+15A_i^7{\alpha }_{3}sn\left(7T{\omega }_i/4,k_i^2\right))$$

(16)

Recalling that $cn$ and $sn$ are periodic Jacobi elliptic functions with period $T=4K\left(k_i^2\right)/{\omega }_i$ , where $K\left(k_i^2\right)$ are the complete elliptic integral of the first kind with modulus $k_i^2,$ then equation (16) reduces to

$${\overline{R}}_i=\left(210A_i{\alpha }_0+140A_i^3{\alpha }_1+112A_i^5{\alpha }_2+96A_i^7{\alpha }_2-70\left(3A_i-2A_i{k}_i^2\right){\omega }_i^2+105A_i^2{\alpha }_{22}K\left(k_i^2\right)\right)/\left(210K\left(k_1^2\right)\right)$$

(17)

Thus from equation (2), the approximate frequency–amplitude relationship of nonlinear oscillators modeled by an equation of the form

$$\ddot{y_i}+{\alpha }_0{y}_i+{\alpha }_1{y}_i^3+{\alpha }_2{y}_i^5+{\alpha }_3{y}_i^7+{\alpha }_{22}{y}_i^2=0,\mathrm{ }{y}_i\left(0\right)=1,\mathrm{ }\dot{y_i}\left(0\right)=v$$

(18)is given as

$${\omega }_{\mathrm{AE}}^2=(-2{\omega }_1^2\left(105A_2{\alpha }_0+70A_2^3{\alpha }_1+56A_2^5{\alpha }_2+48A_2^7{\alpha }_3+35\left(-3A_2+2A_2{k}_2^2\right){\omega }_2^2\right)K\left(k_1^2\right)+(2\left(105A_1{\alpha }_0+70A_1^3{\alpha }_1+56A_1^5{\alpha }_2+48A_1^7{\alpha }_3+35\left(-3A_1+2A_1{k}_1^2\right){\omega }_1^2\right){\omega }_2^2+105{\alpha }_{22}\left(-A_2{\omega }_1+A_1{\omega }_2\right)\left(A_2{\omega }_1+A_1{\omega }_2\right)K\left(k_1^2\right))K\left(k_2^2\right))/\times (-2\left(105A_2{\alpha }_0+70A_2^3{\alpha }_1+56A_2^5{\alpha }_2+48A_2^7{\alpha }_3+35\left(-3A_2+2A_2{k}_2^2\right){\omega }_2^2\right)K\left(k_1^2\right)+(\left(210A_1{\alpha }_0+140A_1^3{\alpha }_1+112A_1^5{\alpha }_2+96A_1^7{\alpha }_3\right)+70\left(-3A_1+2A_1{k}_1^2\right){\omega }_1^2+105\left(A_1-A_2\right)\left(A_1+A_2\right){\alpha }_{22}K\left(k_1^2\right))K\left(k_2^2\right))$$

(19)

In the next section, we shall next evaluate the accuracy of using Jacobi elliptic functions by examining the approximate frequency–amplitude solution of the Fangzhu singular oscillator, and the accuracy of equation (19) in providing the frequency–amplitude value of nonlinear Duffing-type oscillators considering weak and strong nonlinearities.

Results

The accuracy of the proposed approach is evaluated by examining two cases.

Case (a): The Fangzhu singular oscillator

First, the frequency–amplitude of the Fangzhu singular oscillator is determined. This singular oscillator arises during the mathematical modeling of the Fangzhu water harvester device considering the influence of its nanoscale surface morphology. ²⁷ Its equation of motion is given as

$$\ddot{y}+{\omega }_0^{2}y+\frac{Q}{y^{\alpha }}=g\left(t\right)\text{ with }y\left(0\right)=A\text{, }\dot{y}\left(0\right)=0$$

(20)where $y$ is the distance of the attracted molecule from its equilibrium position, $\alpha$ is a positive fractional or integer number, $g\left(t\right)$ is a driving force, and ${\omega }_0^{}$ represents a positive number related to the water harvester device surface morphology.^27–29 Introducing the transformation $t=\tau /Q$ equation (20) becomes

$$\frac{d^{2}y}{d{\tau }^2}+{\omega }^{2}y+\frac{1}{y^a}=G\left(\tau \right)$$

(21)

Since we are interested in determining the frequency–amplitude relation of equation (21), it is assumed that $G\left(\tau \right)=0$ . Thus, equation (21) becomes

$$\frac{d^{2}y}{d{\tau }^2}+{\omega }^{2}y+\frac{1}{y^{\alpha }}=0,\text{ with }y\left(0\right)=A\text{, }\dot{y}\left(0\right)=0$$

(22)

Notice that equation (22) does not have critical points. Furthermore, equation (22) has exact solution when $\omega =0$ and $\alpha =1$ and a pseudo-period of the weak periodic solution given by $T=2\sqrt{2\pi }A$ .^30,31 However, the closed-form solution of equation (22) is unknown. Therefore, in order to determine the approximate frequency–amplitude expression of the weak periodic solution of the Fangzhu equation, it is assumed that $y_i\left(t\right)=A_{i}cn\left({\omega }_{i}t,k_1^2\right)$ , $i=1\text{,}2$ . Thus, from equation (22) the following trial residuals are obtained

$$R_i\left(t\right)=A_i\left(-{\omega }_i^{2}cn\left({\omega }_{i}t,k_1^2\right)d{n}^2\left({\omega }_{i}t,k_1^2\right)+k_i^2{\omega }_i^{2}cn\left({\omega }_{i}t,k_1^2\right)sn\left({\omega }_{i}t,k_1^2\right)+f\left(y_i\left(t\right)\right)\right)$$

(23)where $f\left(y_i\left(t\right)\right)={\omega }^2{y}_i+1/y_i^{\alpha }$ . Expanding equation (23) yields

$$R_i\left(t\right)=1+A_i{\omega }_0^{2}cn\left(\tau {\omega }_i,k_i^2\right){\left(A_{i}cn\left(\tau {\omega }_i,k_i^2\right)\right)}^{\alpha }-A_i{\omega }_i^{2}cn\left(\tau {\omega }_i,k_i^2\right){\left(A_{i}cn\left(\tau {\omega }_i,k_i^2\right)\right)}^{\alpha }dn{\left(\tau {\omega }_i,k_i^2\right)}^2+A_i{k}_i^2{\omega }_i^{2}cn\left(\tau {\omega }_i,k_i^2\right){\left(A_{i}cn\left(\tau {\omega }_i,k_i^2\right)\right)}^{\alpha }sn{\left(\tau {\omega }_i,k_i^2\right)}^2$$

(24)

If the Jacobi elliptic identities $\cos {\varphi }_i=cn\left({\omega }_{i}t,k_i^2\right)$ and $\sin {\varphi }_i=sn\left({\omega }_{i}t,k_i^2\right)$ are applied then, equation (24) becomes

$$R_i\left(t\right)=1+\cos {\varphi }_i{\left(A_i\cos {\varphi }_i\right)}^{\alpha }\left(A_i{\omega }_0^2+\left(k_i^2-1\right)A_i{\omega }_i^2-A_i{k}_i^2{\omega }_i^2\cos 2{\varphi }_i\right)$$

(25)

The trial residual functions $R_i\left(t\right)$ are computed using equation (15). This yields, after considering the limits of integration $\tau \to T/4$ and $\tau \to 0$ , and the fact that the period of the Jacobi elliptic functions $sn\left({\omega }_{i}t,k_1^2\right)$ and $cn\left({\omega }_{i}t,k_1^2\right)$ is $4K\left(k_i^2\right)/{\omega }_i$ , the expression

$${\overline{R}}_i=1+\frac{A_i^{\alpha \sqrt{\pi }}\left(A_i\left(3+\alpha \right){\omega }_0^2-A_i\left(3-2k_i^2+\alpha \right){\omega }_i^2\right)\mathrm{\Gamma }\left(1+\alpha /2\right)}{4K\left(k_i^2\right)\mathrm{\Gamma }\left(\left(5+\alpha \right)/2\right)}$$

(26)

Using equation (2) gives the approximate frequency–amplitude relation

$${\omega }_{AE}^2=(\left(\left(3+\alpha \right){\omega }_0^2-\left(3-2k_1^2+\alpha \right){\omega }_1^2\right){\omega }_2^{2}K\left(k_2^2\right)+K\left(k_1^2\right)({\omega }_1^2\left(-\left(3+\alpha \right){\omega }_0^2+\left(3-2k_2^2+\alpha \right){\omega }_2^2\right)+\left(4\left({\omega }_2^2-{\omega }_1^2\right)K\left(k_2^2\right)\mathrm{\Gamma }\left(\left(5+\alpha \right)/2\right)\right)/\times \left(A^{1+\alpha \sqrt{\pi }}\mathrm{\Gamma }\left(1+\alpha /2\right)\right)))/\left(-\left(\left(3+\alpha \right){\omega }_0^2+\left(3-2k_2^2+\alpha \right){\omega }_2^2\right)K\left(k_1^2\right)+\left(\left(3+\alpha \right){\omega }_0^2-\left(3-2k_1^2+\alpha \right){\omega }_1^2\right)K\left(k_2^2\right)\right)$$

(27)where $A_1=A_2=A$ .^23–25 Equation (27) illustrates that the frequency varies inversely with the oscillatory amplitude. This qualitative behavior of ${\omega }_{AE}$ as a function of the oscillation amplitude agrees with the results obtained by He et al. ²⁹ Furthermore, when $\alpha =1$ and ${\omega }_0=0$ , equation (27) provides the frequency value of 1.2533 if $A=1$ , ${\omega }_1=1$ , ${\omega }_2=2$ , and $k_1=k_2=0.5968i$ , where $i=\sqrt{-1}$ . This result agrees with the frequency value obtained from the exact solution derived by Gadella and Lara, García and Gasull.^30,31 If now $\alpha =8/53$ and ${\omega }_0=1$ with $k_1=k_2=0.4254i$ , using equation (27) gives ${\omega }_{AE}=1.5196$ while the exact numerical frequency value of equation (22) is ${\omega }_{\mathit{Exact}}=1.5167$ . This implies a relative error value of 0.188%. For values of $0.01\le \alpha \le 1$ and $0<{\omega }_0\le 100$ with $k_1=k_2=0.4254i$ , the relative error does not exceed the value of 5.8%, which corroborates the accuracy of our proposed approach.

On the other hand, it is important to point out that our solution procedure can be extended to study the fractal form of equation (22) given as

$$\frac{d^{2\gamma }y}{d{\tau }^{2\gamma }}+{\omega }^{2}y+\frac{1}{y^{\alpha }}=0,\text{ with }y\left(0\right)=A\text{, }\dot{y}\left(0\right)=0$$

(28)where $\gamma$ is a fractal parameter. Using the two-scale transform method $T_1={\tau }^{\gamma }$ , equation (28) becomes ¹³

$$\frac{d^{2}y}{d{T}_1^2}+{\omega }^{2}y+\frac{1}{y^{\alpha }}=0\text{ with }y\left(0\right)=A\text{, }\dot{y}\left(0\right)=0$$

(29)for which the solution $y\left(\tau \right)=\mathit{Acn}\left({\omega }_1{\tau }^{\gamma },k_1^2\right)$ holds. ²⁸

Case (b): Duffing-type oscillators

The second case focuses on determining the frequency–amplitude relationship for Duffing-type cubic, cubic–quintic, cubic–quintic–heptic oscillators, and for the quadratic Duffing–Helmhotz oscillator, since these arise in many physical and engineering systems and their closed-form solutions are known.^1,2 Furthermore, these Duffing-type oscillators can be used to transform nonlinear oscillators that have rational or irrational restoring forces into polynomial ones.^32–40

Table 1 illustrates the errors attained by considering different values of the nonlinear terms ${\alpha }_1$ , ${\alpha }_2$ , ${\alpha }_{22}$ , and ${\alpha }_3$ in equation (18). For comparison purposes, Table 1 lists the frequency values obtained by using the approach introduced by Ren and Hu to derive the frequency–amplitude equation ⁴¹

$${\omega }_{RE}^2={\alpha }_0+\frac{2}{3}{\alpha }_{\mathit{1}}{A}^2+\frac{8}{15}{\alpha }_2{A}^4+\frac{16}{35}{\alpha }_3{A}^6+\frac{\pi }{4}{\alpha }_{22}A$$

(30)where the term related to the quadratic nonlinearity of equation (18) has been considered. In all cases examined here, the values of ${\alpha }_0=1$ , $A=1$ , $A_1=A_2=1,k_1^2=k_2^2=0.06$ with ${\omega }_1=1,$ and ${\omega }_2=2{\omega }_1$ were selected to minimized the error. Notice from Table 1 that in general, the error attained from equation (19) is smaller than that of ${\omega }_{RE}$ . This reduced error is expected since the Duffing-type equations’ exact solution is based on Jacobi elliptic functions. Table 1 shows the values of ${\omega }_{AE}$ and ${\omega }_{RE}$ are similar and compare well with respect to the exact frequency, ${\omega }_{\mathit{Exact}}$ , whose value can be obtained by numerical integration of equation (18). It is also seen from Table 1 that for increasing values of the nonlinear parameters ${\alpha }_i$ , the errors attained tend to decrease.

Table 1.

Exact and estimated frequency values for Duffing-like nonlinear oscillators.

${\mathit{\alpha }}_{22}$	${\mathit{\alpha }}_1$	${\mathit{\alpha }}_2$	${\mathit{\alpha }}_3$	${\mathit{\omega }}_{\mathit{Exact}}$	${\mathit{\omega }}_{\mathit{AE}}$	Relative error (%)	${\mathit{\omega }}_{\mathit{RE}}$	Relative error (%)
1	1	1	1	1.9099	1.8970	0.6799	1.8554	2.9378
10	10	10	10	5.2040	5.1586	0.8804	5.0423	3.2073
100	100	100	100	16.1641	16.0232	0.8793	15.6606	3.2152
1000	1000	1000	1000	51.0218	50.5773	0.8788	49.4322	3.2156
1	1	0	0	1.6043	1.6021	0.1349	1.5659	2.4536
10	10	0	0	4.0744	4.0366	0.9374	3.9396	3.4228
100	100	0	0	12.5214	12.3923	1.0420	12.0916	3.5548
1000	1000	0	0	39.4794	39.0681	1.0527	38.1191	3.5684
0	1	0	0	1.3177	1.3176	0.0121	1.2909	2.0745
0	10	0	0	2.8667	2.8259	1.4391	2.7688	3.5308
0	100	0	0	8.5335	8.3956	1.6435	8.2259	3.7395
0	1000	0	0	26.804	26.3721	1.6633	25.8392	3.7597
0	0	1	0	1.2647	1.2638	0.0708	1.2382	2.1343
0	0	10	0	2.5836	2.5685	0.5893	2.5166	2.6635
0	0	100	0	7.5425	7.5231	0.2579	7.3711	2.3253
0	0	1000	0	23.6406	23.5923	0.2045	23.1157	2.2708
0	0	0	1	1.2305	1.2320	0.1183	1.2071	1.9412
0	0	0	10	2.3879	2.4090	0.8759	2.3603	1.1680
0	0	0	100	6.8362	6.9757	2.0001	6.8347	0.0206
0	0	0	1000	21.3722	21.8456	2.1671	21.4043	0.1497
0	1	1	0	1.5235	1.5131	0.6447	1.4832	2.72015
0	1	1	1	1.6753	1.6636	0.7001	1.6300	2.7766
0	10	10	10	4.3059	4.2782	0.6477	4.1918	2.7232
0	100	100	100	13.2511	13.1780	0.5546	12.9118	2.6281
0	1000	1000	1000	41.7858	41.5600	0.5433	40.7203	2.6166
0	0	1	1	1.4450	1.4399	0.3541	1.4108	2.4235
0	0	10	10	3.3589	3.3703	0.3366	3.3022	1.7184
0	0	100	100	10.1327	10.2086	0.7439	10.0024	1.3028
0	0	1000	1000	31.8817	32.1370	0.7943	31.4877	1.2513

Conclusion

He's formulation for getting the frequency–amplitude relationship for nonlinear oscillators provides accurate results when the two trial residuals are based on Jacobi elliptic functions. The simplicity of Professor He’s formulation can now be expanded to obtain analytical expressions for the frequency of nonlinear oscillators, since most of these can be written as a Duffing-type equation using transformation techniques. Furthermore, using the two-scale transform, our results can be applied to obtain approximate frequency–amplitude expressions for the fractal Fangzhu singular oscillator and for the fractal Duffing-type equations. Therefore, this paper provides evidence of the applicability of He’s formulation for all homogeneous single-degree-of-freedom nonlinear oscillators found in physics, aerospace, and engineering applications.