Analytical approximate solutions for asymmetric conservative oscillators

Abstract

This paper focuses on the construction of analytical approximate solutions for an asymmetric conservative single-degree-of-freedom oscillator. First, based on the asymmetric oscillator, two symmetric ones are introduced; then, the second-order Newton iteration method and the harmonic balance method are applied to the two oscillators, respectively; finally, the analytical approximate solutions of the asymmetric oscillator are constructed and expressed by the oscillation amplitudes. Each iterative step needs the Fourier series representations of the restoring force functions and their first and second derivatives, of the two symmetric oscillators. Using only one iterative step can obtain accurate analytical approximate solution valid for a large range of oscillation amplitudes. Two examples are presented to illustrate use and high accuracy of the proposed approach.

Appendices

Appendix A: Analytical approximate solutions to strongly nonlinear conservative symmetric oscillators

Consider the nonlinear oscillator

$$\begin{aligned} \frac{\hbox {d}^{2}y}{\hbox {d}t^{2}}+n\left( y \right) =0,\hbox { y}\left( 0 \right) =B,\hbox { }\frac{\hbox {d}y}{\hbox {d}t}\left( 0 \right) =0 \end{aligned}$$

(A1)

where $n\left( y \right)$ satisfies $n\left( {-y} \right) =-n\left( y \right)$ and $yn\left( y \right) >0$ for $y\in [-B,B],\hbox { y}\ne 0$. Three analytical approximate solutions to Eq. (A1) by using the second-order Newton iteration approach and the HB method [17] are shown as follows. The first approximate period and periodic solution are

$$\begin{aligned} T_1 \left( B \right) ={2\pi }/{\sqrt{\Omega _1 \left( B \right) }} \end{aligned}$$

(A2)

and

$$\begin{aligned} y_1 \left( t \right) =B\cos \left[ {\sqrt{\Omega _1 \left( B \right) }\hbox { }t} \right] , \end{aligned}$$

(A3)

respectively, where

$$\begin{aligned} \Omega _1 \left( B \right) ={\lambda _1 \left( B \right) }/B \end{aligned}$$

(A4)

in which

$$\begin{aligned} \lambda _1 \left( B \right) =\frac{4}{\pi }\int _0^{\pi /2} {n\left( {B\cos \tau } \right) } \cos \tau \hbox {d}\tau \end{aligned}$$

(A5)

The second approximate period and periodic solution are

$$\begin{aligned} T_2 \left( B \right) ={2\pi }/{\sqrt{\Omega _2 \left( B \right) }} \end{aligned}$$

(A6)

and

$$\begin{aligned} y_2 \left( t \right) =\left[ {B+c_1 \left( B \right) } \right] \cos \left[ {\sqrt{\Omega _2 \left( B \right) }t} \right] -c_1 \left( B \right) \cos \left[ {3\sqrt{\Omega _2 \left( B \right) }t} \right] , \end{aligned}$$

(A7)

respectively, where

$$\begin{aligned} \Omega _2 \left( B \right) =\frac{\lambda _1 }{B}-\frac{\lambda _3 \left[ {\left( {\mu _0 -\mu _4 } \right) B-2\lambda _1 } \right] }{B\left[ {\left( {\mu _2 +\mu _4 -\mu _0 -\mu _6 } \right) B+18\lambda _1 } \right] }, \quad c_1 (B)=-\frac{2\lambda _3 B}{\left( {\mu _2 +\mu _4 -\mu _0 -\mu _6 } \right) B+18\lambda _1 }\nonumber \\ \end{aligned}$$

(A8)

in which

$$\begin{aligned}&\lambda _3 \left( B \right) =\frac{4}{\pi }\int _0^{\pi /2} {n\left( {B\cos \tau } \right) } \cos 3\tau \hbox {d}\tau ,\nonumber \\&\mu _{2(n-1)} \left( B \right) =\frac{4}{\pi }\int _0^{\pi /2} {n_y \left( {B\cos \tau } \right) } \cos [2(n-1)\tau ]\hbox {d}\tau ,\hbox { }n=1,\hbox { }2,\hbox { }\ldots \end{aligned}$$

(A9)

The third approximate period and periodic solution are

$$\begin{aligned} T_3 \left( B \right) ={2\pi }/{\sqrt{\Omega _3 \left( B \right) }} \end{aligned}$$

(A10)

and

$$\begin{aligned} y_3 \left( t \right)= & {} \left[ {B+d_1 \left( B \right) } \right] \cos \left[ {\sqrt{\Omega _3 \left( B \right) }t} \right] +\left[ {-d_1 \left( B \right) +d_2 \left( B \right) } \right] \cos \left[ {3\sqrt{\Omega _3 \left( B \right) }t} \right] \nonumber \\&-d_2 \left( B \right) \cos \left[ {5\sqrt{\Omega _3 \left( B \right) }t} \right] , \end{aligned}$$

(A11)

respectively, where

$$\begin{aligned} \Omega _3 \left( B \right)= & {} \frac{\lambda _1 }{B}+\frac{\lambda _3 \left( {Q_3 P_1 -Q_1 P_3 } \right) +\lambda _5 \left( {Q_1 P_2 -Q_2 P_1 } \right) }{B\left( {Q_2 P_3 -Q_3 P_2 } \right) },\hbox { }d_1 \left( B \right) =\frac{-P_3 \lambda _3 +P_2 \lambda _5 }{Q_2 P_3 -Q_3 P_2 },\nonumber \\ d_2 \left( B \right)= & {} \frac{Q_3 \lambda _3 -Q_2 \lambda _5 }{Q_2 P_3 -Q_3 P_2 } \end{aligned}$$

(A12)

in which

$$\begin{aligned} P_1 \left( B \right)= & {} \frac{1}{8}\left[ {4\mu _2 -4\mu _6 +c_1 \left( {2\nu _3 -2\nu _5 -\nu _7 +\nu _9 } \right) } \right] ,\nonumber \\ Q_1 \left( B \right)= & {} \frac{1}{8}\left[ {-8\Omega _2 +4\mu _0 -4\mu _4 +c_1 \left( {3\nu _1 -3\nu _3 -\nu _5 +\nu _7 } \right) } \right] ,\nonumber \\ P_2 \left( B \right)= & {} \frac{1}{8}\left[ {-72\Omega _2 +4\mu _0 -4\mu _2 +4\mu _6 -4\mu _8 +c_1 \left( {2\nu _1 -4\nu _3 +3\nu _5 -2\nu _9 +\nu _{11} } \right) } \right] ,\nonumber \\ Q_2 \left( B \right)= & {} \frac{1}{8}\left[ {72\Omega _2 -4\mu _0 +4\mu _2 +4\mu _4 -4\mu _6 +c_1 \left( {-3\nu _1 +5\nu _3 -\nu _5 -2\nu _7 +\nu _9 } \right) } \right] ,\nonumber \\ P_3 \left( B \right)= & {} \frac{1}{8}\left[ {200\Omega _2 -4\mu _0 +4\mu _2 +4\mu _8 -4\mu _{10} +c_1 \left( {-2\nu _1 +3\nu _3 -2\nu _5 +2\nu _7 -2\nu _{11} +\nu _{13} } \right) } \right] ,\nonumber \\ Q_3 \left( B \right)= & {} \frac{1}{8}\left[ {-4\mu _2 +4\mu _4 +4\mu _6 -4\mu _8 -c_1 \left( {\nu _1 +\nu _3 -4\nu _5 +\nu _7 +2\nu _9 -\nu _{11} } \right) } \right] ,\nonumber \\ \lambda _5 \left( B \right)= & {} \frac{4}{\pi }\int _0^{\pi /2} {n\left( {A\cos \tau } \right) } \cos 5\tau \hbox {d}\tau ,\nonumber \\ \nu _{2n-1} \left( B \right)= & {} \frac{4}{\pi }\int _0^{\pi /2} {n_{yy} \left( {B\cos \tau } \right) } \cos [(2n-1)\tau ]\hbox {d}\tau ,\hbox { }n=1,\hbox { }2,\hbox { }\ldots \end{aligned}$$

(A13)

In Eqs. (A9) and (A13), the subscript y indicates the derivative of n(y) with respect to y. It is obvious that the coefficients $\lambda _{2n-1} \left( B \right) ,\hbox { }\mu _{2(n-1)} \left( B \right) ,\hbox { }\nu _{2n-1} \left( B \right) \hbox { }\left( n=1,\hbox { }2,\hbox { }\ldots \right)$ should be calculated to obtain the approximate solutions above.

Appendix B

The coefficients $d_1^i$, $d_2^i$, $E_1^i$ and $D_1^i$ in Eq. (13) are given by

$$\begin{aligned} d_1^i= & {} \frac{2310\left( {-1} \right) ^{i}A_i ^{2}\left[ {55\left( {-1} \right) ^{i-1}A_i +21\pi } \right] \left[ {46438822A_i ^{2}+35691045\left( {-1} \right) ^{i-1}A_i \pi +6756750\pi ^{2}} \right] }{D_1^i },\\ d_2^i= & {} \frac{6435(-1)^{i-1}A_i ^{2}\left[ {55(-1)^{i-1}A_i +21\pi } \right] \left[ {426834A_i ^{2}+484330(-1)^{i-1}A_i \pi +121275\pi ^{2}} \right] }{D_1^i }\\ E_1^i= & {} 12966921466026784(-1)^{i-1}A_i ^{5}+25114501127000964A_i ^{4}\pi +19374272110822860(-1)^{i-1}A_i ^{3}\pi ^{2} \\&+7441413396832800A_i ^{2}\pi ^{3}+1423084718805750(-1)^{i-1}A_i \pi ^{4}+108409908481875\pi ^{5}, \\ D_1^i= & {} 234906113646548A_i ^{4}+364260716787660(-1)^{i-1}A_i ^{3}\pi +210858535364400A_i ^{2}\pi ^{2} \\&+53999601405750(-1)^{i-1}A_i \pi ^{3}+5162376594375\pi ^{4}. \end{aligned}$$

The coefficients $E_2^i$, $E_3^i$ and $D_2^i$ in Eq. (31) are given by

$$\begin{aligned} E_2^i= & {} 2.94613\times 10^{-13}(-1)^{i-1}A_i -7.93131\times 10^{-13}A_i ^{2}-777.231(-1)^{i-1}A_i ^{3}+772.927A_i ^{4},\\ E_3^i= & {} -9.81257\times 10^{-29}+1.66583\times 10^{-28}(-1)^{i-1}A_i +3.62553\times 10^{-12}A_i ^{2}\\&-9.46558\times 10^{-12}(-1)^{i-1}A_i ^{3} \\&-8881.75A_i ^{4}+9893.47(-1)^{i-1}A_i ^{5}, \\ D_2^i= & {} 8.48703\times 10^{-12}(-1)^{i-1}-150948A_i +446559(-1)^{i-1}A_i ^{2}-427380A_i ^{3}+131072(-1)^{i-1}A_i ^{4}. \end{aligned}$$

The coefficients $E_k^i \left( {k=4,5,6,7} \right)$ and $D_3^i$ in Eq. (36) are given by

$$\begin{aligned} E_4^i= & {} -3.71253\times 10^{-36}(-1)^{i-1}+5.07015\times 10^{-35}(-1)^{i-1}A_i^2 -0.0192766A_i^3\\&+2.46728\times 10^{-13}(-1)^{i-1}A_i^4 \\&-1.69077\times 10^{14}A_i^5 +4.92658\times 10^{14}(-1)^{i-1}A_i^6 -7.91152\times 10^{14}A_i^7\\&+7.56128\times 10^{14}(-1)^{i-1}A_i ^{8} \\&-4.30023\times 10^{14}A_i ^{9}+1.34733\times 10^{14}(-1)^{i-1}A_i ^{10}-1.79394\times 10^{13}A_i ^{11}, \\ E_5^i= & {} -7.42507\times 10^{-36}(-1)^{i-1}A_i^2 -1.23364\times 10^{12}(-1)^{i-1}A_i^4 +1.0452\times 10^{13}A_i^5 \\&-3.70693\times 10^{13}(-1)^{i-1}A_i^6 +7.15405\times 10^{13}A_i^7 -8.13040\times 10^{13}(-1)^{i-1}A_i ^{8} \\&+5.44889\times 10^{13}A_i ^{9}-1.99578\times 10^{13}(-1)^{i-1}A_i ^{10}+3.08332\times 10^{12}A_i ^{11}, \\ E_6^i= & {} -5.73292\times 10^{-60}(-1)^{i-1}+4.81592\times 10^{-43}A_i -2.05164\\&\times 10^{-26}(-1)^{i-1}A_i^2 +2.50484\times 10^{-10}A_i^3 \\&-1.35661\times 10^{6}(-1)^{i-1}A_i^4 -5.40672\times 10^{6}A_i^5 +8.45414\times 10^{7}(-1)^{i-1}A_i^6 +8.40628\times 10^{20}A_i^7 \\&-4.02267\times 10^{21}(-1)^{i-1}A_i ^{8}+2.01162\times 10^{22}A_i ^{9}-6.01020\times 10^{22}(-1)^{i-1}A_i ^{10} \\&+1.19199\times 10^{23}A_i ^{11}-1.64759\times 10^{23}(-1)^{i-1}A_i ^{12}+1.61944\times 10^{23}A_i ^{13} \\&-1.13183\times 10^{23}(-1)^{i-1}A_i ^{14}+5.51174\times 10^{22}A_i ^{15}-1.78097\times 10^{22}(-1)^{i-1}A_i ^{16} \\&+3.43628\times 10^{21}A_i ^{17}-2.99898\times 10^{20}(-1)^{i-1}A_i ^{18},\\ E_7^i= & {} 8.48703\times 10^{-12}(-1)^{i-1}-150948A_i +446559(-1)^{i-1}A_i^2 -427380A_i^3 +131072(-1)^{i-1}A_i^4 ,\\ D_3^i= & {} 2.25773\times 10^{-44}-2.16890\times 10^{-27}(-1)^{i-1}A_i +7.55671\times 10^{-11}A_i^2 -1.12808\times 10^{6}(-1)^{i-1}A_i^3 \\&+6.10963\times 10^{21}A_i^4 -7.82821\times 10^{22}(-1)^{i-1}A_i^5 +4.60599\times 10^{23}A_i^6 -1.64753\times 10^{24}(-1)^{i-1}A_i^7 \\&+3.99631\times 10^{24}A_i^8 -6.94016\times 10^{24}(-1)^{i-1}A_i^9 +8.87557\times 10^{24}A_i^{10} \\&-8.46095\times 10^{24}(-1)^{i-1}A_i^{11} +6.01046\times 10^{24}A_i^{12} -3.14140\times 10^{24}(-1)^{i-1}A_i^{13} \\&+1.17374\times 10^{24}A_i^{14} -2.96755\times 10^{23}(-1)^{i-1}A_i^{15} +4.54833\times 10^{22}A_i^{16} \\&-3.18991\times 10^{21}(-1)^{i-1}A_i^{17}. \end{aligned}$$