A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms

Abstract

In this article, we present a straightforward methodology to obtain the normal forms of nonlinear systems subjected to external periodic excitation. Moreover, the nonlinear systems are not mandated to be minimally excited and may be parametrically excited or possess constant coefficients. This methodology applies an intuitive state augmentation scheme which serves to liberate the analysis from traditional approaches that require special strategies such as ‘book-keeping’ parameters, detuning parameters, ad hoc new unsolved differential equations and variables. Because our technique affiliates the excitation frequency terms with the augmented states in a direct, consistent and explicit manner, this approach is applicable to a broad range of nonlinear systems with single or multiple periodic excitations. We performed analysis of the forced Duffing and the Mathieu–Duffing equations via normal forms computed by our methodology. The analysis scrutinized the systems amplitude variations in time and frequency domains. Observed conformity between the normal forms results and the numerically integrated results validated the reliability of our unified approach to accurately construct normal forms of nonlinear systems with external periodic excitation.

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Acknowledgements

Funding to support this research was partially provided by the Interplanetary Initiative of Arizona State University.

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Correspondence to Peter M. B. Waswa.

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Appendix A: Sample of periodic TDNF terms in Eq. (56)

Appendix A: Sample of periodic TDNF terms in Eq. (56)

$$\begin{aligned} \begin{array}{llll} \begin{bmatrix} {\dot{v}}_{1} \\ {\dot{v}}_{2} \\ {\dot{v}}_{3} \\ {\dot{v}}_{4} \\ {\dot{v}}_{5}\\ {\dot{v}}_{6} \\ \end{bmatrix}=&{} \begin{bmatrix} {\mathcal {V}}_{11} \\ {\mathcal {V}}_{21}\\ -i\pi v_{3} + i(2.2204 \times 10 ^{-16})v_{4}\\ - i(2.2204 \times 10 ^{-16})v_{3}+i\pi v_{4}\\ -i4\pi v_{5}\\ i4\pi v_{6}\\ \end{bmatrix} \end{array} \end{aligned}$$

where

$$\begin{aligned}&{\mathcal {V}}_{11}= -i1.73732v_{1} +i(3.33067\times 10^{-16} ) v_{2} \\&\quad +i(1.85037\times 10^{-17}) v_{1}^3 v_{3}\\&\quad +i(1.11022\times 10^-{16}) v_{1}^2 v_{2} v_{3} \\&\quad -i(2.31296\times 10^{-18}) v_{2}^3 v_{3} \\&\quad +i(3.70074\times 10^{-17}) v_{1}^3 v_{4} \\&\quad -i(9.25186\times 10^{-18}) v_{1}^2 v_{2} v_{4} \\&\quad +i(9.25186\times 10^{-18}) v_{1} v_{2}^2 v_{4} \\&\quad +i(2.31296\times 10^{-18}) v_{2}^3 v_{4} \\&\quad +i(6.93889\times 10^{-18}) v_{5} \\&\quad +i(9.03502\times 10^{-21}) v_{1}^2 v_{3} v_{5} \\&\quad +i(1.807\times 10^{-20}) v_{1} v_{2} v_{3} v_{5}\\&\quad +i(9.03502\times 10^{-21}) v_{2}^2 v_{3} v_{5} \\&\quad +i(1.807\times 10^{-20}) v_{1} v_{2} v_{4} v_{5} \\&\quad -i(9.03502\times 10^{-21}) v_{2}^2 v_{4} v_{5} \\&\quad -i(8.82326\times 10^{-24}) v_{2} v_{3} v_{5}^2 \\&\quad -i(4.41163\times 10^{-24}) v_{1} v_{4} v_{5}^2 \\&\quad -i(4.41163\times 10^{-24}) v_{2} v_{4} v_{5}^2 \\&\quad -i(2.15412\times 10^{-27}) v_{3} v_{5}^3 \\&\quad +i(2.15412\times 10^{-27}) v_{4} v_{5}^3 \\&\quad +i(6.93889\times 10^{-18}) v_{6} \\&\quad -i(1.807\times 10^{-20}) v_{1} v_{2} v_{4} v_{6} \\&\quad +i(5.29396\times 10^{-23}) v_{1} v_{3} v_{5} v_{6} \\&\quad +i(1.14702\times 10^{-22}) v_{2} v_{3} v_{5} v_{6}\\&\quad -i(1.76465\times 10^{-23}) v_{1} v_{4} v_{5} v_{6} \\&\quad +i(1.76465\times 10^{-23}) v_{2} v_{4} v_{5} v_{6} \\&\quad +i(1.29247\times 10^{-26}) v_{3} v_{5}^2 v_{6} \\&\quad +i(8.61646\times 10^{-27}) v_{4} v_{5}^2 v_{6} \\&\quad +i(4.41163\times 10^{-24}) v_{1} v_{3} v_{6}^2 \\&\quad +i(4.41163\times 10^{-24}) v_{2} v_{3} v_{6}^2 \\&\quad -i(4.30823\times 10^{-27}) v_{4} v_{5} v_{6}^2 \\&\quad +(5.78241\times 10^{-19}-i9.03502\times 10^{-21} ) \\&\quad v_{1}^3 v_{3} \cos (2\pi t) \\&\quad +i(3.75857\times 10^{-18}-6.32451\times 10^{-20} ) \\&\quad v_{1}^2 v_{2} v_{3} \cos (2\pi t) \\&\quad +(8.67362\times 10^{-19}+i4.51751\times 10^{-21} ) \\&\quad v_{1} v_{2}^2 v_{3} \cos (2\pi t) \\&\quad +i(2.25875\times 10^{-21}) v_{2}^3 v_{3} \cos (2\pi t) \\ \end{aligned}$$
$$\begin{aligned}&\quad +(2.74665\times 10^{-18}-i6.77626\times 10^{-21} ) \\&\quad v_{1}^3 v_{4} \cos (2\pi t) \\&\quad -(2.89121\times 10^{-19}-i4.51751\times 10^{-21} ) \\&\quad v_{1}^2 v_{2} v_{4} \cos (2\pi t) \\&\quad +(5.64689\times 10^{-22}-i3.5293\times 10^{-23} ) \\&\quad v_{1}^2 v_{3} v_{5} \cos (2\pi t)\\&\quad +(1.12938\times 10^{-21}-i1.41172\times 10^{-22} ) \\&\quad v_{1} v_{2} v_{3} v_{5} \cos (2\pi t)\\&\quad +i(5.64689\times 10^{-22}) v_{2}^2 v_{3} v_{5} \cos (2\pi t) \\&\quad +i(3.5293\times 10^{-23} ) v_{1}^2 v_{4} v_{5} \cos (2\pi t) \\&\quad +(2.25875\times 10^{-21}-i7.05861\times 10^{-23} ) \\&\quad v_{1} v_{2} v_{4} v_{5} \cos (2\pi t)\\&\quad -(5.64689\times 10^{-22}+i3.5293\times 10^{-23} ) \\&\quad v_{2}^2 v_{4} v_{5} \cos (2\pi t) \\&\quad -(5.51454\times 10^{-25}-i6.89317\times 10^{-26} ) \\&\quad v_{1} v_{3} v_{5}^2 \cos (2\pi t) \\&\quad -i(1.10291\times 10^{-24}) v_{2} v_{3} v_{5}^2 \cos (2\pi t) \\&\quad -(5.51454\times 10^{-25}-i1.37863\times 10^{-25} ) \\&\quad v_{1} v_{4} v_{5}^2 \cos (2\pi t) \\&\quad +i(1.10291\times 10^{-24}) v_{2} v_{4} v_{5}^2 \cos (2\pi t) \\&\quad +(2.69265\times 10^{-28}+i1.6829\times 10^{-29}) \\&\quad v_{3} v_{5}^3 \cos (2\pi t) \\&\quad -(2.69265\times 10^{-28}+i3.36581\times 10^{-29}) \\&\quad v_{4} v_{5}^3 \cos (2\pi t) \\&\quad +i(4.33681\times 10^{-19} ) v_{6} \cos (2\pi t) \\&\quad -i(5.64689\times 10^{-22}) v_{2}^2 v_{3} v_{6} \cos (2\pi t) \\&\quad -(5.64689\times 10^{-22}+i3.5293\times 10^{-23}) \\&\quad v_{1}^2 v_{4} v_{6} \cos (2\pi t) \\&\quad +(6.61744\times 10^{-24}-i3.44659\times 10^{-26}) \\&\quad v_{1} v_{3} v_{5} v_{6} \cos (2\pi t) \\&\quad +(1.43378\times 10^{-23}-i6.89317\times 10^{-26}) \\&\quad v_{2} v_{3} v_{5} v_{6} \cos (2\pi t) \\&\quad -i(2.20581\times 10^{-24}) v_{1} v_{4} v_{5} v_{6} \cos (2\pi t) \\&\quad +i(1.10291\times 10^{-24}) v_{2} v_{4} v_{5} v_{6} \cos (2\pi t) \\&\quad +(1.07706\times 10^{-27}-i3.36581\times 10^{-29}) \\&\quad v_{3} v_{5}^2 v_{6} \cos (2\pi t) \\&\quad -i(3.36581\times 10^{-29}) v_{4} v_{5}^2 v_{6} \cos (2\pi t) \\&\quad +(1.10291\times 10^{-24}+i6.89317\times 10^{-26}) \\&\quad v_{1} v_{3} v_{6}^2 \cos (2\pi t) \\ \end{aligned}$$
$$\begin{aligned}&\quad +(5.51454\times 10^{-25}+i1.03398\times 10^{-25}) \\&\quad v_{2} v_{3} v_{6}^2 \cos (2\pi t) \\&\quad +(5.51454\times 10^{-25}+i6.89317\times 10^{-26}) \\&\quad v_{1} v_{4} v_{6}^2 \cos (2\pi t) \\&\quad +i(1.6829\times 10^{-29}) v_{3} v_{5} v_{6}^2 \cos (2\pi t) \\&\quad -(5.38529\times 10^{-28}+i3.36581\times 10^{-29}) \\&\quad v_{4} v_{5} v_{6}^2 \cos (2\pi t) \\&\quad -(2.69265\times 10^{-28}-i1.6829\times 10^{-29}) \\&\quad v_{3} v_{6}^3 \cos (2\pi t) \\&\quad -i(1.6829\times 10^{-29}) v_{4} v_{6}^3 \cos (2\pi t) \\&\quad +i(8.82326\times 10^{-24}) v_{1}^2 v_{3} v_{5} \cos (2\pi t)^2 \\&\quad +(1.76465\times 10^{-23}-i4.41163\times 10^{-24}) \\&\quad v_{1} v_{2} v_{3} v_{5} \cos (2\pi t)^2 \\&\quad + \cdots \\&\quad + i(1.8684\times 10^{-45}+i7.47359\times 10^{-45} ) \\&\quad v_{4} v_{6}^3 \cos (4\pi t) \sin (4\pi t)^3 \sin (2 \pi t)^3 \\&\quad -(2.98944\times 10^{-43}-i1.8684\times 10^{-45}) \\&\quad v_{4} v_{5}^3 \sin (2\pi t) \sin (4\pi t)^3 \sin (2 \pi t)^3 \\&\quad -(5.97887\times 10^{-44}-i1.79366\times 10^{-43} ) \\&\quad v_{3} v_{5}^2 v_{6} \sin (2\pi t) \sin (4\pi t)^3 \sin (2 \pi t)^3\\&\quad +i(5.97887\times 10^{-44}) v_{4} v_{5}^2 v_{6} \\&\quad \sin (2\pi t) \sin (4\pi t)^3 \sin (2 \pi t)^3\\&\quad +i(1.19577\times 10^{-43}) v_{3} v_{5} v_{6}^2 \\&\quad \sin (2\pi t) \sin (4\pi t)^3 \sin (2 \pi t)^3\\&\quad +i(9.34199\times 10^{-46}) v_{4} v_{6}^3 \\&\quad \sin (2\pi t) \sin (4\pi t)^3 \sin (2 \pi t)^3 \\&\quad -(2.3355\times 10^-i46+2.91937\times 10^{-47}) \\&\quad v_{3} v_{5}^3 \sin (4\pi t)^4 \sin (2 \pi t)^3 \\&\quad -i(4.67099\times 10^{-46}) v_{3} v_{5}^2 v_{6} \sin (4\pi t)^4 \sin (2 \pi t)^3\\&\quad +(4.67099\times 10^-i46+2.3355\times 10^{-46} ) \\&\quad v_{4} v_{5}^2 v_{6} \sin (4\pi t)^4 \sin (2 \pi t)^3 \\&\quad -(2.3355\times 10^{-46}+i4.67099\times 10^{-46} ) \\&\quad v_{3} v_{5} v_{6}^2 \sin (4\pi t)^4 \sin (2 \pi t)^3\\&\quad +(4.67099\times 10^{-46}+i2.3355\times 10^{-46} ) \\&\quad v_{4} v_{5} v_{6}^2 \sin (4\pi t)^4 \sin (2 \pi t)^3 \\&\quad -(1.16775\times 10^{-46}-i2.91937\times 10^{-47} ) \\&\quad v_{3} v_{6}^3 \sin (4\pi t)^4 \sin (2 \pi t)^3. \end{aligned}$$
$$\begin{aligned}&{\mathcal {V}}_{21} = -i(3.33067\times 10^-16) v_{1} + i(1.73732) v_{2} \\&\quad + i(4.62593\times 10^{-18} ) v_{1}^3 v_{3} \\&\quad - i(4.62593\times 10^{-17} ) v_{1}^2 v_{2} v_{3} \\&\quad - i(9.25186\times 10^{-17} ) v_{1} v_{2}^2 v_{3} \\&\quad + i(3.70074\times 10^{-17} ) v_{2}^3 v_{3} \\&\quad - i(7.17019\times 10^{-17} ) v_{1}^3 v_{4} \\&\quad - i(9.25186\times 10^{-18} ) v_{1}^2 v_{2} v_{4} \\&\quad + i(9.25186\times 10^{-18} ) v_{1} v_{2}^2 v_{4} \\&\quad + i(4.62593\times 10^{-18} ) v_{2}^3 v_{4} \\&\quad - i( 6.93889\times 10^{-18}) v_{5} \\&\quad - i(1.807\times 10^{-20} ) v_{1} v_{2} v_{3} v_{5} \\&\quad - i(9.03502\times 10^{-21} ) v_{2}^2 v_{3} v_{5} \\&\quad + i(1.807\times 10^{-20} ) v_{1} v_{2} v_{4} v_{5} \\&\quad + i(9.03502\times 10^{-21} ) v_{2}^2 v_{4} v_{5} \\&\quad + i(4.41163\times 10^{-24} ) v_{2} v_{3} v_{5}^2 \\&\quad + i(4.41163\times 10^{-24} ) v_{1} v_{4} v_{5}^2 \\&\quad + i(8.82326\times 10^{-24} ) v_{2} v_{4} v_{5}^2 \\&\quad - i(6.93889\times 10^{-18} ) v_{6} \\&\quad + i(1.807\times 10^{-20} ) v_{1}^2 v_{3} v_{6} \\&\quad + i(9.03502\times 10^{-21} ) v_{1}^2 v_{4} v_{6} \\&\quad + i(1.807\times 10^{-20} ) v_{1} v_{2} v_{4} v_{6} \\&\quad + i(9.03502\times 10^{-21} ) v_{2}^2 v_{4} v_{6} \\&\quad - i(7.05861\times 10^{-23} ) v_{1} v_{3} v_{5} v_{6} \\&\quad - i(9.70559\times 10^{-23} ) v_{2} v_{3} v_{5} v_{6} \\&\quad + i(1.14702\times 10^{-22} ) v_{1} v_{4} v_{5} v_{6} \\&\quad - i(8.61646\times 10^{-27} ) v_{3} v_{5}^2 v_{6} \\&\quad + i(1.29247\times 10^{-26} ) v_{4} v_{5}^2 v_{6} \\&\quad - i(4.41163\times 10^{-24} ) v_{2} v_{3} v_{6}^2 \\&\quad - i(8.82326\times 10^{-24} ) v_{1} v_{4} v_{6}^2 \\&\quad + i(4.30823\times 10^{-27} ) v_{3} v_{5} v_{6}^2 \\&\quad + i(8.61646\times 10^{-27} ) v_{4} v_{5} v_{6}^2 \\&\quad + i(2.15412\times 10^{-27} ) v_{3} v_{6}^3 \\&\quad - i(2.15412\times 10^{-27} ) v_{4} v_{6}^3 \\&\quad + (1.4456\times 10^{-19} + i2.25875\times 10^{-21} ) \\&\quad v_{1}^3 v_{3} \cos ( 2\pi t) \\ \end{aligned}$$
$$\begin{aligned}&\quad - (1.4456\times 10^{-18} + i1.807\times 10^{-20} ) \\&\quad v_{1}^2 v_{2} v_{3} \cos ( 2\pi t) \\&\quad - (3.46945\times 10^{-18} - i4.51751\times 10^{-21} ) \\&\quad v_{1} v_{2}^2 v_{3} \cos ( 2\pi t) \\&\quad + (1.59016\times 10^{-18} + i2.48463\times 10^{-20} ) \\&\quad v_{2}^3 v_{3} \cos ( 2\pi t) \\&\quad - (2.38524\times 10^{-18} + i2.03288\times 10^{-20} ) \\&\quad v_{1}^3 v_{4} \cos ( 2\pi t) \\&\quad - (1.4456\times 10^{-18} + i4.51751\times 10^{-21} ) \\&\quad v_{1}^2 v_{2} v_{4} \cos ( 2\pi t) \\&\quad - i( 2.25875\times 10^{-21} ) v_{2}^3 v_{4} \cos ( 2\pi t) \\&\quad - i( 4.33681\times 10^{-19} ) v_{5} \cos ( 2\pi t) \\&\quad + i( 7.05861\times 10^{-23} ) v_{1} v_{2} v_{3} v_{5} \cos ( 2\pi t) \\&\quad - (5.64689\times 10^{-22} - i3.5293\times 10^{-23} ) \\&\quad v_{2}^2 v_{3} v_{5} \cos ( 2\pi t) \\&\quad - i(5.64689\times 10^{-22} ) v_{1}^2 v_{4} v_{5} \cos (2\pi t)\\&\quad + i(1.12938\times 10^{-21} ) v_{1} v_{2} v_{4} v_{5} \cos ( 2\pi t)\\&\quad - (5.64689\times 10^{-22} - i3.5293\times 10^{-23} ) \\&\quad v_{2}^2 v_{4} v_{5} \cos ( 2\pi t) \\&\quad + i(1.10291\times 10^{-24} ) v_{2} v_{3} v_{5}^2 \cos ( 2\pi t) \\&\quad + i(5.51454\times 10^{-25} ) v_{1} v_{4} v_{5}^2 \cos ( 2\pi t) \\&\quad + (1.10291\times 10^{-24} - i3.44659\times 10^{-26} ) \\&\quad v_{2} v_{4} v_{5}^2 \cos ( 2\pi t) \\&\quad + i( 1.6829\times 10^{-29} ) v_{3} v_{5}^3 \cos ( 2\pi t) \\&\quad - i( 1.6829\times 10^{-29} ) v_{4} v_{5}^3 \cos ( 2\pi t) \\&\quad - (5.64689\times 10^{-22} -i3.5293\times 10^{-23} ) \\&\quad v_{1}^2 v_{3} v_{6} \cos ( 2\pi t) \\&\quad - i( 3.5293\times 10^{-23} ) v_{2}^2 v_{3} v_{6} \cos ( 2\pi t) \\&\quad + i(5.64689\times 10^{-22} ) v_{1}^2 v_{4} v_{6} \cos ( 2\pi t) \\ \end{aligned}$$
$$\begin{aligned}&\quad + (2.25875\times 10^{-21} + i1.41172\times 10^{-22} ) \\&\quad v_{1} v_{2} v_{4} v_{6} \cos ( 2\pi t) \\&\quad + (5.64689\times 10^{-22} + i3.5293\times 10^{-23} ) \\&\quad v_{2}^2 v_{4} v_{6} \cos ( 2\pi t) \\&\quad - (6.61744\times 10^{-24} + i4.30823\times 10^{-26} ) \\&\quad v_{1} v_{3} v_{5} v_{6} \cos ( 2\pi t) \\&\quad - (1.10291\times 10^{-23} - i3.44659\times 10^{-26} ) \\&\quad v_{2} v_{3} v_{5} v_{6} \cos ( 2\pi t) \\&\quad + (4.41163\times 10^{-24} + i6.89317\times 10^{-26} ) \\&\quad v_{1} v_{4} v_{5} v_{6} \cos ( 2\pi t) \\&\quad - (5.38529\times 10^{-28} - i1.6829\times 10^{-29} ) \\&\quad v_{3} v_{5}^2 v_{6} \cos ( 2\pi t) \\&\quad + (5.38529\times 10^{-28} - i1.6829\times 10^{-29} ) \\&\quad v_{4} v_{5}^2 v_{6} \cos ( 2\pi t) \\&\quad + i(5.51454\times 10^{-25} ) v_{1} v_{3} v_{6}^2 \cos ( 2\pi t) \\&\quad - (5.51454\times 10^{-25} + i6.89317\times 10^{-26} ) \\&\quad v_{2} v_{3} v_{6}^2 \cos ( 2\pi t) \\&\quad - (1.65436\times 10^{-24} +i 6.89317\times 10^{-26} ) \\&\quad v_{1} v_{4} v_{6}^2 \cos ( 2\pi t) \\&\quad - i(5.51454\times 10^{-25} ) v_{2} v_{4} v_{6}^2 \cos ( 2\pi t) \\&\quad + (5.38529\times 10^{-28} + i3.36581\times 10^{-29} ) \\&\quad v_{3} v_{5} v_{6}^2 \cos (2\pi t)+ \cdots \\&\quad -(5.97887\times 10^{-43} + i1.19577\times 10^{-43}) \\&\quad v_{4} v_{5} v_{6}^2 \sin (2\pi t) \sin ( 4\pi t)^3 \sin ( 2 \pi t)^3 \\&\quad - (2.98944\times 10^{-43} + i1.8684\times 10^{-45} ) \\&\quad v_{3} v_{6}^3 \sin (2\pi t) \sin (4\pi t)^3 \sin (2 \pi t)^3\\&\quad - i(2.91937\times 10^{-47} ) v_{4} v_{5}^3 \sin ( 4\pi t)^4 \sin ( 2 \pi t)^3\\&\quad - (2.3355\times 10^{-46} + i4.67099\times 10^{-46} ) \\&\quad v_{3} v_{5}^2 v_{6} \sin ( 4\pi t)^4 \sin ( 2 \pi t)^3 \\&\quad - (2.3355\times 10^{-46} - i4.67099\times 10^{-46}) \\&\quad v_{4} v_{5}^2 v_{6} \sin ( 4\pi t)^4 \sin ( 2 \pi t)^3 \\&\quad +i ( 4.67099\times 10^{-46} ) v_{4} v_{5} v_{6}^2 \\&\quad \sin ( 4\pi t)^4 \sin ( 2 \pi t)^3 \\&\quad - (2.3355\times 10^{-46} - i2.91937\times 10^{-47} ) \\&\quad v_{4} v_{6}^3 \sin ( 4\pi t)^4 \sin (2 \pi t)^3. \end{aligned}$$

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Waswa, P.M.B., Redkar, S. A direct approach for simplifying nonlinear systems with external periodic excitation using normal forms. Nonlinear Dyn 99, 1065–1088 (2020). https://doi.org/10.1007/s11071-019-05334-6

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Keywords

  • Normal forms
  • Forced nonlinear dynamics
  • Lyapunov–Floquet
  • Mathieu–Duffing
  • Parametric excitation