Comparison of instantaneous phase differences between externally and parametrically excited suspended cable

Many investigations on the response phase of cables mainly focus on the phase shift value in the linear solution, while the effect of the higher-order approximate terms (HOAT) is often omitted. To ascertain the effect of the HOAT on response phases, instantaneous phase-frequency characteristics of a classical externally- and parametrically-excited suspended cable are investigated. The Galerkin method is used to discretize the motion equations into ordinary differential equations, and the Multiple Scales Method (MSM) is used to solve these equations. Afterward, cable responses under these two types of excitation with different Irvine parameters λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{2}$$\end{document} and excitation frequency Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} are numerically solved, and instantaneous phase differences between the responses and excitations are obtained by using the Hilbert transform. Then, variation characteristics of the instantaneous phase differences and corresponding amplitudes are analyzed in the (λ2,Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{2},\Omega $$\end{document}) plane. It is shown that if the HOAT are not considered, the phase shifts of cable response would be constants. On the contrary, if they are included, the drift term (DT) and the doubling-frequency term (DFT) in the HOAT would vary periodically with time. Due to the difference in the frequency-response equation’s right-hand terms between these two excitations, response amplitudes are different, affecting the phase-frequency characteristics through the DT and the DFT. The response-excitation instantaneous phase difference amplitude pmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\max }$$\end{document} under external and parametric excitation are both suddenly increase in the local region centered on λ2≈3.0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{2}\approx 3.0$$\end{document} and Ω≈1.125\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \approx 1.125$$\end{document} and present a near-antisymmetrical distribution. However, the sudden-change-region of the former is a long and narrow band along the axis of λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{2}$$\end{document} in the (λ2,Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^{2},\Omega $$\end{document}) plane, while that of the latter is a point field. Besides, values of the former are significantly larger than of the latter.