Revisited dynamic modeling and eigenvalue analysis of the cable-stayed beam

Abstract

In this study, the Hamilton’s principle is applied to revisit the dynamic modeling of the cable-stayed beam, and the motion equations governing the nonlinear response of the cable-stayed beam are derived. The corresponding boundary terms are transformed to the dynamic equilibrium conditions through the continuity of the displacement at the anchoring point. Following the standard condensation procedure, the condensed model of the cable-stayed beam is determined. The eigenvalue analysis is performed to determine the closed-form eigenvalue solution of the linear problems, and two types of eigenvalue solution are obtained. It is shown that the frequency spectrum of the cable-stayed beam exhibits the curve veering and crossover phenomena. Corresponding to these phenomena, the mode shapes of the cable-stayed beam may exhibit the coupling characteristic. Finally, the discrete model of the cable-stayed beam is determined, and the possible nonlinear interactions are discussed.

Graphical abstract

Appendix

The other coefficients in discrete model of the cable-stayed beam are:

$$\begin{aligned} \Gamma _{kijh}= & {} \frac{\alpha }{2}\int \limits ^1_0\phi ^c_{k}\phi ^c_{i,x_cx_c}\mathrm {d}x_c \int \limits ^1_0\phi ^c_{j,x_c}\phi ^c_{h,x_c}\mathrm {d}x_c\nonumber \\&+\frac{\Xi }{2m}\cos ^3\theta \int \limits ^1_0\phi ^b_k\phi ^b_{i,x_bx_b}\mathrm {d}x_b\int \limits ^1_0\phi ^b_{j,x_b}\phi ^b_{h,x_b} \mathrm {d} x_b\nonumber \\&-\frac{\alpha }{2}\cos ^2\theta \phi ^b_{k}(1)\phi ^c_{i,x_c}(1) \int \limits ^1_0\phi ^c_{j,x_c}\phi ^c_{h,x_c}\mathrm {d}x_c\nonumber \\&-\frac{\alpha \kappa }{2}\cos \theta \phi ^b_{k}(1) \phi ^b_{i,x_b}(1)\int \limits ^1_0\phi ^b_{j,x_b}\phi ^b_{h,x_b}\mathrm {d}x_b, \end{aligned}$$

(39)

$$\begin{aligned} \Upsilon _{kijh}= & {} \frac{\alpha }{2}\int \limits ^1_0\phi ^c_k\phi ^c_{i,x_cx_c}\mathrm {d}x_c \int \limits ^1_0\psi ^c_{j,x_c}\psi ^c_{h,x_c}\mathrm {d}x_c\nonumber \\&+\frac{\Xi }{2m}\cos ^3\theta \int \limits ^1_0\phi ^b_k\phi ^b_{i,x_bx_b}\mathrm {d}x_b \int \limits ^1_0\psi ^b_{j,x_b}\psi ^b_{h,x_b}{\mathrm {d}} x_b\nonumber \\&-\frac{\alpha }{2}\cos ^2\theta \phi ^b_{k}(1)\phi ^c_{i,x_c}(1) \int \limits ^1_0\psi ^c_{j,x_c}\psi ^c_{h,x_c}\mathrm {d}x_c\nonumber \\&-\frac{\alpha \kappa }{2}\cos \theta \phi ^b_{k}(1)\phi ^b_{i,x_b}(1) \int \limits ^1_0\psi ^b_{j,x_b}\psi ^b_{h,x_b}\mathrm {d}x_b, \end{aligned}$$

(40)

$$\begin{aligned} \Theta _{kijh}= & {} \frac{\alpha }{2}\int \limits ^1_0\psi ^c_k\psi ^c_{i,x_cx_c} \mathrm {d}x_c\int \limits ^1_0\psi ^c_{j,x_c}\psi ^c_{h,x_c}\mathrm {d}x_c\nonumber \\&+\frac{\Xi }{2m}\cos ^3\theta \int \limits ^1_0\psi ^b_k\psi ^b_{i,x_bx_b} \mathrm {d}x_b\int \limits ^1_0\psi ^b_{j,x_b}\psi ^b_{h,x_b}\mathrm {d}x_b\nonumber \\&-\frac{\alpha }{2}\psi ^b_{k}(1)\psi ^c_{i,x_c}(1) \int \limits ^1_0\psi ^b_{j,x_b}\psi ^b_{h,x_b}\mathrm {d}x_b\nonumber \\&-\frac{\alpha \kappa }{2}\cos \theta \psi ^b_{k}(1)\psi ^b_{i,x_b}(1) \int \limits ^1_0\psi ^b_{j,x_b}\psi ^b_{h,x_b}\mathrm {d}x_b, \end{aligned}$$

(41)

$$\begin{aligned} \Delta _{kijh}= & {} \frac{\alpha }{2}\int \limits ^1_0\psi ^c_k\psi ^c_{i,x_cx_c} \mathrm {d}x_c\int \limits ^1_0\phi ^c_{j,x_c}\phi ^c_{h,x_c}\mathrm {d}x_c\nonumber \\&+\frac{\Xi }{2m}\cos ^3\theta \int \limits ^1_0\psi ^b_k\psi ^b_{i,x_bx_b}\mathrm {d}x_b \int \limits ^1_0\phi ^b_{j,x_b}\phi ^b_{h,x_b}\mathrm {d}x_b\nonumber \\&-\frac{\alpha }{2} \psi ^b_{k}(1)\psi ^b_{i,x_b}(1)\int \limits ^1_0 \phi ^c_{j,x_c}\phi ^c_{h,x_c}\mathrm {d}x_c\nonumber \\&-\frac{\alpha \kappa }{2}\cos \theta \psi ^b_{k}(1)\psi ^b_{i,x_b}(1) \int \limits ^1_0\phi ^b_{j,x_b}\phi ^b_{h,x_b}\mathrm {d}x_b, \end{aligned}$$

(42)

$$\begin{aligned} \mu ^v_k= & {} \frac{1}{2}\int \limits ^1_0c^v_c(\phi ^c_k)^2\mathrm {d}x_c +\frac{\cos ^3\theta }{2m}\int \limits ^1_0c^v_b(\phi ^b_k)^2\mathrm {d}x_b,\nonumber \\ f_k= & {} \int \limits ^1_0p^c_v\phi ^c_k\mathrm {d}x_c+\frac{\cos ^3\theta }{m} \int \limits ^1_0p^b_v\phi ^b_k\mathrm {d}x_b, \end{aligned}$$

(43)

$$\begin{aligned} \mu ^w_k= & {} \frac{1}{2}\int \limits ^1_0c^w_c(\psi ^c_{k})^2\mathrm {d}x_c +\frac{\cos ^3\theta }{2m}\int \limits ^1_0c^w_b(\psi ^b_k)^2\mathrm {d}x_b,\nonumber \\ p_k= & {} \int \limits ^1_0p^c_w \psi ^c_k\mathrm {d}x_c+\frac{\cos ^3\theta }{m}\int \limits ^1_0p^b_w\psi ^b_k\mathrm {d}x_b. \end{aligned}$$

(44)