Rapidly Convergent Solution of Nonlinear Oscillators with General Non-rational Restoring Force

Abstract

Purpose

A rapidly convergent solution for nonlinear oscillatory system with non-rational restoring force has been presented.

Method

The energy balance method has been extended and applied to solve said nonlinear oscillator.

Result

It has been verified that the second approximate solution is better than that obtained by the usual energy balance method and it is close to corresponding third approximate solution obtained by harmonic balance method.

Appendices

Appendix 1

For “Example 3.1” (Harmonic Balance Method)

In general, a third approximate solution of Eq. (12) is chosen in a form [37]

$$x(t) = A((1 - u - v)\cos \varphi + u\cos 3\varphi + v\cos 5\varphi ).$$

(37)

Substituting Eq. (37) into Eq. (12) and equating, respectively, the coefficient of $\cos \varphi$, $\cos 3\varphi$, and $\cos 5\varphi$, the following algebraic equations are obtained

$$- A(1 - u - v)\,\omega^{2} + A^{q - 1} I_{0} (8 + 6q + q^{2} + 16u - 12qu - 4q^{2} u + 12qv - 12q^{2} v)/(q + 4) = 0 ,$$

(38)

$$\begin{aligned} - 9Au\omega^{2} + A^{q - 1} I_{0} ( - 48 + 4q + 8q^{2} + q^{3} - 192u + 184qu + 12q^{2} u \hfill \\ - 4q^{3} u + 96v - 168qv + 84q^{2} v - 12q^{3} v)/(q^{2} + 10q + 24) = 0 \hfill \\ \end{aligned} ,$$

(39)

$$\begin{aligned} - 25Av\omega^{2} + A^{q - 1} I_{0} (384 - 176q - 28q^{2} + 8q^{3} + q^{4} + 1536u - 2176qu + 568q^{2} u + 76q^{3} u \hfill \\ \quad - 4q^{4} u - 1536v + 1824qv - 552q^{2} v + 276q^{3} v - 12q^{4} v)/(q^{3} + 18q^{2} + 104q + 192) = 0 \hfill \\ \end{aligned} .$$

(40)

Eliminating $\omega^{2}$ between Eqs.(38) and (39) and then Eqs.(38) and (40), ignoring $u^{3} ,u^{4} ,u^{5} , \cdots$ and $v^{2} ,v^{3} , \ldots$ terms, the following equations are obtained

$$\begin{aligned} - 48 + 4q + 8q^{2} + q^{3} - (576 + 216q + 104q^{2} + 14q^{3} )u \hfill \\ + ( - 672 + 320q + 312q^{2} + 40q^{3} )u^{2} + (144 - 172q + 76q^{2} - 13q^{3} )v = 0, \hfill \\ \end{aligned}$$

(41)

$$\begin{aligned} 384 - 176q - 28q^{2} + 8q^{3} + q^{4} + (1152 - 2000q + 596q^{2} 68q^{3} - 5q^{4} )u \hfill \\ + ( - 1536 + 2176q - 568q^{2} - 76q^{3} + 4q^{4} )u^{2} \hfill \\ - (11520 + 8000q + 4024q^{2} + 232q^{3} + 38q^{4} )v = 0. \hfill \\ \end{aligned}$$

(42)

Solving Eqs. (41) and (42) simultaneously, we get the values of $u$ and $v$. Then substituting these values of $u$, $v$ in Eq. (38) and simplifying, we obtain third approximation of the frequency $\omega = \omega_{3}$ as well as period $T_{3} = 2\pi /\omega_{3}$.

Appendix 2

For “Example 3.2” (Harmonic Balance Method)

Generally, a second approximate solution of Eq. (17) is chosen in a form [37]

$$x(t) = A((1 - u)\,\cos \varphi + u\cos 3\varphi ).$$

(43)

Substituting Eq. (43) into Eq. (17) and equating the coefficient of $\cos \varphi$ and $\cos 3\varphi$, the following algebraic equations are obtained

$$A(9 - 9u) + A^{1/3} (320m_{0} - 128m_{0} u - 96m_{0} u^{2} ) - A(9 - 9u)\omega^{2} = 0,$$

(44)

$$99Au - A^{1/3} (704m_{0} - 1760m_{0} u - 1824m_{0} u^{2} ) - 891Au\omega^{2} = 0,$$

(45)

where $m_{0} = \frac{\varGamma (7/6)}{4\sqrt \pi \varGamma (11/3)}.$

Eliminating $\omega^{2}$ between Eqs.(44) and (45) and then ignoring higher order terms of $u$ (more than two), we obtain

$$792m_{0} + (891A^{2/3} + 32868m_{0} )u - (891A^{2/3} + 14328m_{0} )u^{2} = 0 .$$

(46)

This quadratic equation has two roots. The smallest of $\left| u \right|$ is the required value of $u$. Substituting the value of $u$ in Eq. (44) and simplifying, then the second approximate frequency $\omega_{2}$ is obtained.

Similarly, a third approximate solution of Eq. (17) is chosen in a form [37]

$$x(t) = A((1 - u - v)\cos \varphi + u\cos 3\varphi + v\cos 5\varphi ).$$

(47)

Substituting Eq. (47) into Eq. (17) and the coefficient of $\cos \varphi$, $\cos 3\varphi$, and $\cos 5\varphi$ are equating, the following algebraic equations are obtained

$$A(9 - 9u - 9v) + A^{1/3} (320m_{0} - 128m_{0} u - 96m_{0} v) - A(9 - 9u - 9v)\,\omega^{2} = 0 ,$$

(48)

$$99Au - A^{1/3} (704m_{0} - 1760m_{0} u + 192m_{0} v) - 891Au\,\omega^{2} = 0 ,$$

(49)

$$99Av + A^{1/3} (352m_{0} - 544m_{0} u + 1536m_{0} v) - 2475Av\,\omega^{2} = 0 ,$$

(50)

where $m_{0} = \frac{\varGamma (7/6)}{4\sqrt \pi \,\varGamma (11/3)}.$

Eliminating $\omega^{2}$ between Eqs.(48) and (49) and between (48) and (50) then ignoring $u^{3} ,u^{4} ,u^{5} , \ldots$ and $v^{2} ,v^{3} , \ldots$ terms, we obtain

$$792m_{0} + (891A^{2/3} + 32868m_{0} )u - (891A^{2/3} + 12276m_{0} )u^{2} - 576m_{0} v = 0 ,$$

(51)

$$( - 396 + 1008\,u - 612u^{2} )m_{0} + (2673A^{2/3} + 97668m_{0} )v = 0 .$$

(52)

Solving Eqs. (51) and (52) simultaneously, obtained value of $u$ and $v$ are used in Eq. (48), then the frequency $\omega = \omega_{3}$ is obtained.

Appendix 3

The expression $x^{q} - A^{q}$ can be written as $\sin^{2} \varphi \times \frac{{\,(x^{q} - A^{q} )}}{{\sin^{2} \varphi }}$ and substituting the value of $x$ from Eq. (5) then $\frac{{x^{q} - A^{q} }}{{\sin^{2} \varphi }}$ can be easily expanded in a Fourier series $A^{q} (C_{0} + C_{2} \cos 2\varphi + \cdots )$, thus

$$x^{q} - A^{q} = A^{q} \sin^{2} \varphi \,(C_{0} + C_{2} \cos 2\varphi + \cdots ),$$

(53)

where $C_{0}$, $C_{2}$ are Fourier coefficients which are expressed (with help of Mathematica software) as

$$C_{0} = ( - 2 - q - 4(2 + q)u + 8( - 1 + q)u^{2} )I_{0} ,$$

(54)

$$C_{2} = 8/q + \frac{{( - (2 + q)^{2} (4 + q) - 4q^{2} (4 + q)u + 8( - 2 + q)( - 1 + q)qu^{2} I_{0} }}{q\,(2 + q/2)},$$

(55)

where $I_{0} = \frac{\varGamma ((1 + q)/2)}{\sqrt \pi \,\varGamma (2 + q/2)}.$

Appendix 4

We modified the term $x^{{{\raise0.7ex\hbox{$8$} \!\mathord{\left/ {\vphantom {8 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}}$ as following way:

$x$ can be written as $x = x_{0} + u_{1}$where $x_{0} = A\cos \omega t$ and $u_{1} = A( - {\kern 1pt} u\,\cos \omega t + u\,\cos 3\omega t)$ which satisfy Eq. (5) so

$$\begin{aligned} x^{{{\raise0.7ex\hbox{$8$} \!\mathord{\left/ {\vphantom {8 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} & = (x_{0} + u_{1} )^{{{\raise0.7ex\hbox{$8$} \!\mathord{\left/ {\vphantom {8 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} \\ & \cong x_{0}^{{{\raise0.7ex\hbox{$8$} \!\mathord{\left/ {\vphantom {8 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} + \frac{8}{3}x_{0}^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} u_{1} + \frac{20}{9}x_{0}^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} u_{1}^{2} , \\ \end{aligned}$$

(56)

where ignoring more than $u_{1}^{2}$ terms.