Nonlinear planar vibrations of a cable with a linear damper

Abstract

This paper minutely studies the effects of the damper on nonlinear behaviors of a cable–damper system. By modeling the damper as a combination of a viscous damper and a linear spring, the primary resonance and subharmonic resonance (1/2 order and 1/3 order) of the cable are explored. The equation of motion of the cable is treated by using Galerkin’s method, and the ordinary differential equation (ODE) is obtained subsequently. To solve the ODE, the method of multiple timescales is applied, and the modulation equations corresponding to different types of resonance are derived. Then, the frequency–/force–response curves are acquired by utilizing Newton–Raphson method and pseudo-arclength algorithm, so as to explore the vibration suppression effect from a nonlinear point of view. Meanwhile, the time histories, phase portraits and Poincaré sections are also provided to discuss the influences of the damping and spring stiffness on the nonlinear characteristics of the cable. The results show that the damper has a significant effect on nonlinear resonances of the cable, and the increase in damping (spring stiffness) makes the dual characteristic gradually convert to the softening (hardening) characteristic.

Appendices

Appendix A

The expressions of the Galerkin integral coefficients in Eq. (10) are given as follows:

$b_{11} = \omega_{c}^{2} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} (x)} {\text{d}}x}}\left[K_{d} \phi_{c}^{2} (l_{1} ) - \frac{{\lambda_{c} \int_{0}^{1} {y^{\prime}_{c} (x)\phi^{\prime}_{c} (x)dx} \int_{0}^{1} {y^{\prime\prime}_{c} (x)\phi_{c} (x)dx} }}{{\beta_{c}^{2} }} - \int_{0}^{1} {\frac{{\phi_{c} (x)\phi^{\prime\prime}_{c} (x)}}{{\beta_{c}^{2} }}} {\text{d}}x\right]$;

$b_{12} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} (x)} {\text{d}}x}}\left[\mu_{c} \int_{0}^{1} {\phi_{c}^{2} (x)} {\text{d}}x + C_{d} \phi_{c}^{2} (l_{1} )\right]$;

$b_{13} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} (x)} {\text{d}}x}}\left[ - \frac{{\lambda_{c} \int_{0}^{1} {\phi^{\prime}_{c} (x)\phi^{\prime}_{c} (x){\text{d}}x} \int_{0}^{1} {y^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{2\beta_{c}^{2} }} - \frac{{\lambda_{c} \int_{0}^{1} {y^{\prime}_{c} (x)\phi^{\prime}_{c} (x){\text{d}}x} \int_{0}^{1} {\phi^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{\beta_{c}^{2} }}\right]$;

$b_{14} = - \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} (x)} {\text{d}}x}}\frac{{\lambda_{c} \int_{0}^{1} {\phi^{\prime}_{c} (x)\phi^{\prime}_{c} (x){\text{d}}x} \int_{0}^{1} {\phi^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{2\beta_{c}^{2} }}$; $F = \frac{{\int_{0}^{1} {F_{1} (x)\phi_{c} (x){\text{d}}x} }}{{\int_{0}^{1} {\phi_{c} (x){\text{d}}x} }}$; $b_{15} = \frac{{F\int_{0}^{1} {\phi_{c} (x){\text{d}}x} }}{{\int_{0}^{1} {\phi_{c}^{2} (x)} {\text{d}}x}}$.

Appendix B

The expressions of the coefficients in Eqs. (22), (25), (42), (54), and (55) are given as follows:

$\Gamma_{11} = \left( - 3b_{14} + \frac{{10b_{13}^{2} }}{{3\omega_{c}^{2} }}\right)$; $\Gamma_{12} = \left(b_{14} + \frac{{2b_{13}^{2} }}{{3\omega_{c}^{2} }}\right)$; $\overline{\Gamma }_{11} = \frac{{b_{12} b_{15} }}{{8\omega_{c} }}$; $\overline{\Gamma }_{12} = - \frac{{b_{15} \sigma }}{{4\omega_{c} }}$; $\overline{\Gamma }_{13} = - \frac{{5b_{13}^{2} b_{15} }}{{12\omega_{c}^{4} }} + \frac{{3b_{14} b_{15} }}{{8\omega_{c}^{2} }}$; $\overline{\Gamma }_{14} = \frac{{7b_{13}^{2} b_{15} }}{{18\omega_{c}^{4} }} - \frac{{3b_{14} b_{15} }}{{4\omega_{c}^{2} }}$; $\overline{\Gamma }_{15} = - \frac{{11b_{12} b_{13}^{2} }}{{9\omega_{c}^{3} }} + \frac{{3b_{12} b_{14} }}{{2\omega_{c} }}$; $\overline{\Gamma }_{16} = \frac{{485b_{13}^{4} }}{{54\omega_{c}^{6} }} - \frac{{173b_{13}^{2} b_{14} }}{{6\omega_{c}^{4} }} + \frac{{15b_{14}^{2} }}{{8\omega_{c}^{2} }}$;

$\Gamma_{21} = \frac{{b_{12}^{2} }}{4} - 6b_{14} \Lambda^{2} + \frac{{3b_{13}^{2} \Lambda^{2} }}{{\omega_{c}^{2} }} - \frac{{4b_{13}^{2} \Lambda^{2} }}{{\Omega (\Omega + 2\omega_{c} )}}$; $\Gamma_{22} = \frac{{b_{13} \Lambda \sigma }}{{\omega_{c} }}$; $\Gamma_{23} = - \frac{{2b_{12} b_{13} \Lambda \Omega }}{{\Omega^{2} - \omega_{c}^{2} }}$; $\Gamma_{24} = - 3b_{14} + \frac{{10b_{13}^{2} }}{{3\omega_{c}^{2} }}$.

$\Gamma_{31} = - 3b_{14} \Lambda - \frac{{4b_{13}^{2} \Lambda }}{{\Omega (\Omega - 2\omega_{c} )}} - \frac{{2b_{13}^{2} \Lambda }}{{3\omega_{c}^{2} }}$; $\Gamma_{32} = - b_{12} \omega_{c}$; $\Gamma_{33} = - 6b_{14} \Lambda^{2} - \frac{{4b_{13}^{2} \Lambda^{2} }}{{\Omega (\Omega - 2\omega_{c} )}} + \frac{{4b_{13}^{2} \Lambda^{2} }}{{\omega_{c}^{2} }} - \frac{{4b_{13}^{2} \Lambda^{2} }}{{\Omega (\Omega + 2\omega_{c} )}}$; $\Gamma_{34} = - 3b_{14} + \frac{{10b_{13}^{2} }}{{3\omega_{c}^{2} }}$.

Appendix C

To seek a second-order approximation, the solution of q_c is uniformly expanded in power series of ε, namely

$$q_{c} = \sum\limits_{i = 1}^{3} {\varepsilon^{i - 1} q_{ci} (T_{0} ,T_{2} )} + O(\varepsilon^{3} ).$$

(C.1)

Here, it should be noted that q_c is found to be independent of T₁ by eliminating the secular terms in the equation of order $\varepsilon^{1}$. Therefore, T₁ is dropped in Eq. (C.1). Following similar treatment to that of higher-order expansion, the following autonomous modulation equations can be derived:

$$D_{2}^{1} a = - \frac{{b_{12} a}}{2} + \frac{{b_{15} \sin \alpha }}{{2\omega_{c} }},$$

(C.2)

$$aD_{2}^{1} \alpha = a\sigma + \frac{{\Gamma_{11} a^{3} }}{{8\omega_{c} }} + \frac{{b_{15} \cos \alpha }}{{2\omega_{c} }}.$$

(C.3)

Letting $D_{2}^{1} a = D_{2}^{1} \alpha = 0$ and eliminating $\alpha$, Eqs. (C.2) and (C.3) can be simplified as

$$\frac{{b_{12}^{2} a^{2} }}{4} + \left(a\sigma + \frac{{\Gamma_{11} a^{3} }}{{8\omega_{c} }}\right)^{2} = \frac{{b_{15}^{2} }}{{4\omega_{c}^{2} }}.$$

(C.4)

The stable solutions of the system can be obtained according to Eq. (C.4).