Vertical Vibrations of Suspension Bridges: A Review and a New Method

Abstract

Suspension bridges offer an elegant and economical solution for bridging over long spans with resultant low material content and ease of construction. Classical analytical methods for the linear vertical vibrations of a suspension bridge usually ignores the flexibility of hangers and bending stiffness of the main cable, and the bending stiffness and mass of the stiffening girder are uniformly distributed to the main cable. However, results show that the flexibility of the hangers has a significant effect on higher-order modal frequencies and may loss some modal information when using the inextensible hanger assumption. In view of this, this paper developed a succinct and universal analytical method for vertical flexural vibration analysis of the suspension bridge, which truly considers the vertical support stiffness of each hanger for the first time. The comparison with finite element solutions and field measurement results shows that the calculation accuracy of the proposed method is significantly improved compared with the classical analytical method. The calculation error of this paper is basically below 5%, besides, the mode-missing problem existing in the classical analytical methods is well solved.

Appendices

Appendix 1

1.1 The Explicit Expression of the Additional Cable Force $h_{j}^{{}}$

According to existing studies, the additional cable force of the jth segment can be expressed as [52]

$$h_{j}^{{}} \left( t \right) = EA\varepsilon_{j} \left( t \right) = \frac{EA}{{l_{j}^{e} }}\int_{0}^{{l_{j}^{e} }} {\left[ {\frac{{\partial u_{j} }}{{\partial x_{j} }}\frac{{{\text{d}}y_{j} }}{{{\text{d}}x_{j} }}} \right]{\text{d}}x_{j} }$$

(22)

For convenience, the following derivation still ignore the effect of lateral support on static configurations of the cable, at this time, the static profile of the cable can be described by quadratic parabola [1] (valid for sag-to-span ratio $e < {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-0pt} 8}$). Thus, combined with Eq. (8) the additional cable force $h_{j}^{{}}$ can be obtained by

$$\begin{aligned} h_{j} & = EA\varepsilon_{j}^{{}} \\ & = \frac{EA}{{l_{j}^{e} }}\int_{0}^{{l_{j} }} {\frac{{\partial u_{j} \left( {x_{j} ,t} \right)}}{{\partial x_{j} }}\frac{{{\text{d}}y\left( {x_{j} } \right)}}{{{\text{d}}x_{j} }}dx_{j} } \\ & = - \frac{4EAe}{{l_{0} l_{j}^{e} }}\int_{0}^{{l_{j} }} {\frac{{\partial u_{j} \left( {x_{j} ,t} \right)}}{{\partial x_{j} }}\left[ {2x_{j} + \left( {2l_{sj - 1} - l_{0} } \right)} \right]dx_{j} } \\ & = - \frac{4EAe}{{l_{0} l_{j}^{e} }}\left\{ {2\left[ {u_{j} \left( {x_{j} ,t} \right)x_{j} } \right]_{0}^{{l_{j} }} - 2\int_{0}^{{l_{j} }} {u_{j} \left( {x_{j} ,t} \right)dx_{j} } + \left( {2l_{sj - 1} - l_{0} } \right)\left[ {u_{j} \left( {x_{j} ,t} \right)} \right]_{0}^{{l_{j} }} } \right\} \\ & = \frac{8EAe}{{l_{0} l_{j}^{e} }}\left\{ {\int_{0}^{{l_{j} }} {u_{j} \left( {x_{j} ,t} \right)dx_{j} } + \left( {0.5l_{0} - l_{sj} } \right)u_{j} \left( {l_{j} ,t} \right) + \left( {l_{sj - 1} - 0.5l_{0} } \right)u_{j} \left( {0,t} \right)} \right\} \\ \end{aligned}$$

(23)

The additional cable force $h_{j}^{{}}$ of a multi-segment cable system considering the effect of the flexural stiffness, sag, inclination, etc. can be expressed as

$$h_{j}^{{}} = \frac{8EAe}{{l_{0} l_{j}^{e} }}\left\{ {\int_{0}^{{l_{j} }} {u_{j} \left( {x_{j} ,t} \right)dx_{j} } + \left( {0.5l_{0} - l_{sj} } \right)u_{j} \left( {l_{j} ,t} \right) + \left( {l_{sj - 1} - 0.5l_{0} } \right)u_{j} \left( {0,t} \right)} \right\}$$

(24)

where e is the sag-to-span ratio defined by $e = {{mgl_{0} \cos \theta } \mathord{\left/ {\vphantom {{mgl_{0} \cos \theta } {8H}}} \right. \kern-0pt} {8H}}$, $\mu_{j} = {{l_{j} } \mathord{\left/ {\vphantom {{l_{j} } {l_{0} }}} \right. \kern-0pt} {l_{0} }}$ is the relative length of jth cable segment, $l_{j}^{e}$ is the arch length of the jth cable segment

$$l_{j}^{e} = l_{0} \left[ {\mu_{sj} + 8e^{2} \mu_{sj}^{3} \left( {\frac{{1 + 4e\left( {1 - \mu_{sj}^{{}} } \right)\tan \theta }}{{\sqrt {1 + 16e^{2} \left( {1 - \mu_{sj}^{{}} } \right)^{2} } }}} \right)^{2} } \right] - l_{0} \left[ {\mu_{sj - 1} + 8e^{2} \mu_{sj - 1}^{3} \left( {\frac{{1 + 4e\left( {1 - \mu_{sj - 1}^{{}} } \right)\tan \theta }}{{\sqrt {1 + 16e^{2} \left( {1 - \mu_{sj - 1}^{{}} } \right)^{2} } }}} \right)^{2} } \right]$$

(25)

Equation (24) is the generalized expression of the additional cable force suitable for arbitrary boundary conditions. In particular, when the transverse displacement of the cable at the endpoint is constrained, then $u_{1} \left( {x_{1} \left| {_{ = 0} } \right.,t} \right) = u_{n} \left( {x_{n} \left| {_{{ = l_{n} }} } \right.,t} \right) = 0$ and we have

$$h_{1}^{{}} = \frac{8EAe}{{l_{0} l_{j}^{e} }}\left\{ {\int_{0}^{{l_{1} }} {u_{1} \left( {x_{1} ,t} \right)dx_{1} } + (0.5l_{0} - l_{1} )u_{1} \left( {x_{1} \left| {_{{ = l_{1} }} } \right.,t} \right)} \right\}$$

(26)

$$h_{n}^{{}} = \frac{8EAe}{{l_{0} l_{j}^{e} }}\left\{ {\int_{0}^{{l_{n} }} {u_{n} \left( {x_{n} ,t} \right)dx_{n} } + \left( {l_{sn - 1} - 0.5l_{0} } \right)u_{n} \left( {x_{n} \left| {_{ = 0} } \right.,t} \right)} \right\}$$

(27)

For a two-segment cable system, it can be seen that the Eqs. (26) and (27) will degenerate into the expression given in [52].

1.2 The Explicit Expression of ${\text{B}}^{(j)}$

Assume the particular solution ${{\hat{h}_{j} } \mathord{\left/ {\vphantom {{\hat{h}_{j} } {\tilde{\omega }^{2} }}} \right. \kern-0pt} {\tilde{\omega }^{2} }}$ in Eq. (12) has the following form

$$\frac{{\hat{h}_{j} }}{{\tilde{\omega }^{2} }} = {\mathbf{B}}^{(j)} \cdot \left\{ {\begin{array}{*{20}c} {A_{1}^{(j)} } & {A_{2}^{(j)} } & {A_{3}^{(j)} } & {A_{4}^{(j)} } \\ \end{array} } \right\}^{T}$$

(28)

then ${\mathbf{B}}^{(j)} { = }\left[ {\begin{array}{*{20}c} {b_{1}^{\left( j \right)} } & {b_{2}^{\left( j \right)} } & {b_{3}^{\left( j \right)} } & {b_{4}^{\left( j \right)} } \\ \end{array} } \right]$, ($i = 1,2,3,4$) can be determined as

• When $j = 1$

$$\begin{aligned} b_{0}^{\left( 1 \right)} & = \frac{{\eta_{1} }}{{\left( {\tilde{\omega }^{2} - 0.5\eta_{1} } \right)}}, \\ b_{1}^{\left( 1 \right)} & = b_{0}^{\left( 1 \right)} \left( {\frac{{1 - e^{{ - p\mu_{1} }} }}{p} + \left( {0.5 - \mu_{1} } \right)e^{{ - \mu_{1} p}} } \right), \\ b_{2}^{\left( 1 \right)} & = b_{0}^{\left( 1 \right)} \left( {\frac{{1 - e^{{ - p\mu_{1} }} }}{p} + 0.5 - \mu_{1} } \right), \\ b_{3}^{\left( 1 \right)} & = b_{0}^{\left( 1 \right)} \left( {\frac{{\sin (q\mu_{1} )}}{q} + \left( {0.5 - \mu_{1} } \right)\cos \left( {q\mu_{1} } \right)} \right), \\ b_{4}^{\left( 1 \right)} & = b_{0}^{\left( 1 \right)} \left( {\frac{{1 - \cos (q\mu_{1} )}}{q} + \left( {0.5 - \mu_{1} } \right)\sin \left( {q\mu_{1} } \right)} \right) \\ \end{aligned}$$

• When $j = n$

$$\begin{aligned} b_{0}^{\left( n \right)} & = \frac{{\eta_{n} }}{{\left( {\tilde{\omega }^{2} - 0.5\eta_{n} } \right)}}, \\ b_{1}^{\left( n \right)} & = b_{0}^{\left( n \right)} \left( {\frac{{1 - e^{{ - p\mu_{n} }} }}{p} + \left( {0.5 - \mu_{n} } \right)e^{{ - \mu_{n} p}} } \right), \\ b_{2}^{\left( n \right)} & = b_{0}^{\left( n \right)} \left( {\frac{{1 - e^{{ - p\mu_{n} }} }}{p} + 0.5 - \mu_{n} } \right), \\ b_{3}^{\left( n \right)} & = b_{0}^{\left( n \right)} \left( {\frac{{\sin \left( {q\mu_{n} } \right)}}{q} + \left( {0.5 - \mu_{n} } \right)\cos \left( {q\mu_{n} } \right)} \right), \\ b_{4}^{\left( n \right)} & = b_{0}^{\left( n \right)} \left( {\frac{{1 - \cos \left( {q\mu_{n} } \right)}}{q} + \left( {0.5 - \mu_{n} } \right)\sin \left( {q\mu_{n} } \right)} \right) \\ \end{aligned}$$

• Others

$$\begin{aligned} b_{0}^{\left( j \right)} & = \frac{{\eta_{j} }}{{\tilde{\omega }^{2} }}, \\ b_{1}^{(j)} & = b_{0}^{(j)} \left( {\frac{{1 - e^{{ - p\mu_{j} }} }}{p} + \left( {\mu_{sj - 1} - 0.5} \right) + \left( {0.5 - \mu_{sj} } \right)e^{{ - p\mu_{j} }} } \right), \\ b_{2}^{(j)} & = b_{0}^{\left( j \right)} \left( {\frac{{1 - e^{{ - p\mu_{j} }} }}{p} + \left( {0.5 - \mu_{sj} } \right) + \left( {\mu_{sj - 1} - 0.5} \right)e^{{ - p\mu_{j} }} } \right), \\ b_{3}^{(j)} & = b_{0}^{\left( j \right)} \left( {\frac{{\sin \left( {q\mu_{j} } \right)}}{q} + \left( {\mu_{sj - 1} - 0.5} \right) + \left( {0.5 - \mu_{sj} } \right)\cos \left( {q\mu_{j} } \right)} \right), \\ b_{4}^{(j)} & = b_{0}^{\left( j \right)} \left( {\frac{{1 - \cos \left( {q\mu_{j} } \right)}}{q} + \left( {0.5 - \mu_{sj} } \right)\sin \left( {q\mu_{j} } \right)} \right) \\ \end{aligned}$$

where $\begin{array}{*{20}l} {\eta_{i} } \hfill \\ \end{array} = 64\frac{{Al_{0}^{3} }}{{Il_{i}^{e} }}e^{2}$.

Appendix 2

2.1 Element Dynamic Stiffness Matrix ${\text{K}}^{\left( j \right)}$

$${\mathbf{K}}^{\left( j \right)} = \frac{EI}{{l^{3} }}\left[ {\begin{array}{*{20}c} {k_{11}^{\left( j \right)} } & {k_{12}^{\left( j \right)} } & {k_{13}^{\left( j \right)} } & {k_{14}^{\left( j \right)} } \\ {k_{21}^{\left( j \right)} } & {k_{22}^{\left( j \right)} } & {k_{23}^{\left( j \right)} } & {k_{24}^{\left( j \right)} } \\ {k_{31}^{\left( j \right)} } & {k_{32}^{\left( j \right)} } & {k_{33}^{\left( j \right)} } & {k_{34}^{\left( j \right)} } \\ {k_{41}^{\left( j \right)} } & {k_{42}^{\left( j \right)} } & {k_{43}^{\left( j \right)} } & {k_{44}^{\left( j \right)} } \\ \end{array} } \right]$$

(29)

Then the coefficients in matrix ${\mathbf{K}}^{\left( j \right)}$ can be determined by

$$\begin{aligned} k_{11}^{\left( j \right)} & = - p\left( {p^{2} - \gamma^{2} } \right)\left( {c_{11}^{\left( j \right)} - \varepsilon_{i} c_{21}^{\left( j \right)} } \right) - q\left( {q^{2} + \gamma^{2} } \right)c_{41}^{\left( j \right)} \\ k_{12}^{\left( j \right)} & = - p\left( {p^{2} - \gamma^{2} } \right)\left( {c_{12}^{\left( j \right)} - \varepsilon_{i} c_{22}^{\left( j \right)} } \right) - q\left( {q^{2} + \gamma^{2} } \right)c_{42}^{\left( j \right)} \\ k_{13}^{\left( j \right)} & = - p\left( {p^{2} - \gamma^{2} } \right)\left( {c_{13}^{\left( j \right)} - \varepsilon_{i} c_{23}^{\left( j \right)} } \right) - q\left( {q^{2} + \gamma^{2} } \right)c_{43}^{\left( j \right)} \\ k_{14}^{\left( j \right)} & = - p\left( {p^{2} - \gamma^{2} } \right)\left( {c_{14}^{\left( j \right)} - \varepsilon_{i} c_{24}^{\left( j \right)} } \right) - q\left( {q^{2} + \gamma^{2} } \right)c_{44}^{\left( j \right)} \\ k_{21}^{\left( j \right)} & = - p^{2} \left( {c_{11}^{\left( j \right)} + \varepsilon_{j} c_{21}^{\left( j \right)} } \right) + q^{2} c_{31}^{j} , \\ k_{22}^{\left( j \right)} & = - p^{2} \left( {c_{12}^{\left( j \right)} + \varepsilon_{i} c_{22}^{\left( j \right)} } \right) + q^{2} c_{32}^{j} \\ k_{23}^{\left( j \right)} & = - p^{2} \left( {c_{13}^{\left( j \right)} + \varepsilon_{i} c_{23}^{\left( j \right)} } \right) + q^{2} c_{33}^{j} , \\ k_{24}^{\left( j \right)} & = - p^{2} \left( {c_{14}^{\left( j \right)} + \varepsilon_{i} c_{24}^{\left( j \right)} } \right) + q^{2} c_{34}^{j} \\ k_{31}^{\left( j \right)} & = \left( {p^{2} - \gamma^{2} } \right)p\varepsilon_{i} c_{11}^{\left( j \right)} + p\left( {\gamma^{2} - p^{2} } \right)c_{21}^{\left( j \right)} \\ & \quad + \left( {q^{3} + q\gamma^{2} } \right)\left( { - S_{i} c_{31}^{\left( j \right)} + C_{i} c_{41}^{\left( j \right)} } \right) \\ k_{32}^{\left( j \right)} & = \left( {p^{2} - \gamma^{2} } \right)p\varepsilon_{j} c_{12}^{\left( j \right)} + p\left( {\gamma^{2} - p^{2} } \right)c_{22}^{\left( j \right)} \\ & \quad + \left( {q^{3} + q\gamma^{2} } \right)\left( { - S_{j} c_{32}^{\left( j \right)} + C_{i} c_{42}^{\left( j \right)} } \right) \\ k_{33}^{\left( j \right)} & = \left( {p^{2} - \gamma^{2} } \right)p\varepsilon_{j} c_{13}^{\left( j \right)} + p\left( {\gamma^{2} - p^{2} } \right)c_{23}^{\left( j \right)} \\ & \quad + \left( {q^{3} + q\gamma^{2} } \right)\left( { - S_{i} c_{33}^{\left( j \right)} + C_{i} c_{43}^{\left( j \right)} } \right) \\ k_{34}^{\left( j \right)} & = \left( {p^{2} - \gamma^{2} } \right)p\varepsilon_{j} c_{14}^{\left( j \right)} + p\left( {\gamma^{2} - p^{2} } \right)c_{24}^{\left( j \right)} \\ & \quad + \left( {q^{3} + q\gamma^{2} } \right)\left( { - S_{j} c_{34}^{\left( j \right)} + C_{i} c_{44}^{\left( j \right)} } \right) \\ k_{41}^{\left( j \right)} & = p^{2} \left( {\varepsilon_{j} c_{11}^{\left( j \right)} + c_{21}^{\left( j \right)} } \right) - q^{2} \left( {C_{j} c_{31}^{\left( j \right)} + S_{i} c_{41}^{\left( j \right)} } \right), \\ k_{42}^{\left( j \right)} & = p^{2} \left( {\varepsilon_{j} c_{12}^{\left( j \right)} + c_{22}^{\left( j \right)} } \right) - q^{2} \left( {C_{j} c_{32}^{\left( j \right)} + S_{j} c_{42}^{\left( j \right)} } \right) \\ k_{43}^{\left( j \right)} & = p^{2} \left( {\varepsilon_{j} c_{13}^{\left( j \right)} + c_{23}^{\left( j \right)} } \right) - q^{2} \left( {C_{j} c_{33}^{\left( j \right)} + S_{j} c_{43}^{\left( j \right)} } \right), \\ k_{44}^{\left( j \right)} & = p^{2} \left( {\varepsilon_{j} c_{14}^{\left( j \right)} + c_{24}^{\left( j \right)} } \right) - q^{2} \left( {C_{j} c_{34}^{\left( j \right)} + S_{j} c_{44}^{\left( j \right)} } \right) \\ \end{aligned}$$

where $\gamma^{2} = \frac{{Hl_{0}^{2} }}{EI}$, $\left. \begin{aligned} p \hfill \\ q \hfill \\ \end{aligned} \right\} = \sqrt {\sqrt {\left( {\frac{{Hl_{0}^{2} }}{2EI}} \right)^{2} + \omega^{2} \frac{{ml_{0}^{4} }}{EI}} \pm \frac{{Hl_{0}^{2} }}{2EI}}$, $\varepsilon_{j} = e^{{ - p\mu_{j} }}$, $C_{j} = \cos \left( {q\mu_{j} } \right)$, $S_{j} = \sin \left( {q\mu_{j} } \right)$ and

$$\begin{aligned} c_{11}^{\left( j \right)} & = \frac{{ - q\left( {p\left( {b_{3} + C_{j} } \right)\left( {C_{j} \varepsilon_{j} - 1} \right) - \left( {q - p\varepsilon_{j} b_{4} + b_{2} q} \right)S_{j} + \varepsilon_{j} pS_{j}^{2} } \right)}}{{y_{j} }} \\ c_{12}^{\left( j \right)} & = - \frac{{pb_{4}^{\left( j \right)} \left( {C_{j} - 1} \right) + C_{j} \left( {1 + b_{2}^{\left( j \right)} + b_{3}^{\left( j \right)} - C_{j} b_{2}^{\left( j \right)} - \left( {b_{3}^{\left( j \right)} + C_{j} } \right)\varepsilon_{j} } \right)pS_{j} - b_{4}^{\left( j \right)} \left( {\varepsilon_{j} - 1} \right)qS_{j} - \left( {b_{2}^{\left( j \right)} + \varepsilon_{j} } \right)qS_{2}^{\left( j \right)} }}{{y_{j} }} \\ c_{13}^{\left( j \right)} & = \frac{{q\left( {\left( {b_{2}^{\left( j \right)} + \varepsilon_{j} } \right)qS_{j} - p\left( {\left( {1 + b_{3}^{\left( j \right)} } \right)\left( { - 1 + C_{j} \varepsilon_{j} } \right) + b_{4}^{\left( j \right)} \varepsilon_{j} S_{j} } \right)} \right)}}{{y_{j} }} \\ c_{14}^{\left( j \right)} & = \frac{{ - b_{4}^{\left( j \right)} \left( {C_{j} - 1} \right)\varepsilon_{j} p - \left( {1 + b_{2}^{\left( j \right)} + b_{3}^{\left( j \right)} } \right)q + b_{2}^{\left( j \right)} C_{j} q + \left( {b_{3}^{\left( j \right)} + C_{j} } \right)\varepsilon_{j} q + \left( {1 + b_{3}^{\left( j \right)} } \right)\varepsilon_{j} pS_{j} }}{{y_{j} }} \\ c_{21}^{\left( j \right)} & = \frac{{q\left( {\left( {b_{3}^{\left( j \right)} + C_{j} } \right)\left( {C_{j} - \varepsilon_{j} } \right)p + \left( {b_{4}^{\left( j \right)} p + \left( {b_{1}^{\left( j \right)} + \varepsilon_{j} } \right)q} \right)S_{j} + pS_{j}^{2} } \right)}}{{y_{j} }} \\ c_{22}^{\left( j \right)} & = - \frac{{ - b_{4}^{\left( j \right)} \left( {C_{j} - 1} \right)\varepsilon_{j} p + C_{j} \left( {b_{1}^{\left( j \right)} - C_{j} - b_{1}^{\left( j \right)} C_{j} + b_{3}^{\left( j \right)} \left( {\varepsilon_{j} - 1} \right) + \varepsilon_{j} } \right)q + \left( {1 + b_{3}^{\left( j \right)} } \right)\varepsilon_{j} pS_{j} + b_{4}^{\left( j \right)} \left( {\varepsilon_{j} - 1} \right)qS_{j} - \left( {1 + b_{1}^{\left( j \right)} } \right)qS_{j}^{2} }}{{y_{j} }} \\ c_{23}^{\left( j \right)} & = - \frac{{\left( {1 + b_{3}^{\left( j \right)} } \right)\left( {C_{j} - \varepsilon_{j} } \right)pq + q\left( {b_{4}^{\left( j \right)} p + q + b_{4}^{\left( j \right)} q} \right)S_{j} }}{{y_{j} }} \\ c_{24}^{\left( j \right)} & = \frac{{b_{4}^{\left( j \right)} \left( {p - C_{j} p} \right) + \left( {b_{1}^{\left( j \right)} - C_{j} - b_{1}^{\left( j \right)} C_{j} + b_{3}^{\left( j \right)} \left( {\varepsilon_{j} - 1} \right) + \varepsilon_{j} } \right)q + \left( {1 + b_{3}^{\left( j \right)} } \right)pS_{j} }}{{y_{j} }} \\ c_{31}^{\left( j \right)} & = - \frac{{p\left( {b_{4}^{\left( j \right)} \left( {\varepsilon_{j}^{2} - 1} \right)p + \left( { - \left( {2 + b_{2}^{\left( j \right)} } \right)\varepsilon_{j} + b_{1}^{\left( j \right)} \left( {C_{j} \varepsilon_{j} - 1} \right) + C_{j} \left( {1 + b_{2}^{\left( j \right)} + \varepsilon_{j}^{2} } \right)} \right)q + \left( {\varepsilon_{j}^{2} - 1} \right)pS_{j} } \right)}}{{y_{j} }} \\ c_{32}^{\left( j \right)} & = \frac{{b_{4}^{\left( j \right)} \left( {\varepsilon_{j} - 1} \right)^{2} p + C_{j} \left( {\varepsilon_{j} - 1} \right)\left( {1 + b_{1}^{\left( j \right)} + b_{2}^{\left( j \right)} + \varepsilon_{j} } \right)q + \left( {1 + b_{1}^{\left( j \right)} + \varepsilon_{j} \left( {b_{2}^{\left( j \right)} + \varepsilon_{j} } \right)} \right)pS_{j} }}{{y_{j} }}, \\ c_{33}^{\left( j \right)} & = \frac{{p\left( {b_{4}^{\left( j \right)} \left( {\varepsilon_{j}^{2} - 1} \right)p - \left( {1 + b_{1}^{\left( j \right)} - C_{i} b_{2}^{\left( j \right)} + \left( {b_{2}^{\left( j \right)} - \left( {2 + b_{1}^{\left( j \right)} } \right)C_{j} } \right)\varepsilon_{j} + \varepsilon_{j}^{2} } \right)q} \right)}}{{y_{j} }}, \\ c_{34}^{\left( j \right)} & = \frac{{b_{4}^{\left( j \right)} \left( {\varepsilon_{j} - 1} \right)^{2} p - \left( {\varepsilon_{j} - 1} \right)\left( {1 + b_{1}^{\left( j \right)} + b_{2}^{\left( j \right)} + \varepsilon_{j} } \right)q - \left( {b_{2}^{\left( j \right)} + \left( {2 + b_{1}^{\left( j \right)} } \right)\varepsilon_{j} } \right)pS_{j} }}{{y_{j} }}, \\ c_{41}^{\left( j \right)} & = \frac{{p\left( { - \left( {b_{3}^{\left( j \right)} + C_{j} } \right)\left( {\varepsilon_{j}^{2} - 1} \right)p + \left( {1 + b_{2}^{\left( j \right)} + b_{1}^{\left( j \right)} \varepsilon_{j} + \varepsilon_{j}^{2} } \right)qS_{j} } \right)}}{{ - y_{j} }}, \\ c_{42}^{\left( j \right)} & = \frac{{ - \left( {b_{1}^{\left( j \right)} \left( {C_{j} - 1} \right) + C_{j} + b_{3}^{\left( j \right)} \left( {\varepsilon_{j} - 1} \right)^{2} - \left( {2 + b_{2}^{\left( j \right)} } \right)\varepsilon_{j} + C_{j} \varepsilon_{j} \left( {b_{2}^{\left( j \right)} + \varepsilon_{j} } \right)} \right)p + \left( {\varepsilon_{j} - 1} \right)\left( {1 + b_{1}^{\left( j \right)} + b_{2}^{\left( j \right)} + \varepsilon_{j} } \right)qS_{j} }}{{y_{j} }}, \\ c_{43}^{\left( j \right)} & = \frac{{p\left( {\left( {1 + b_{3}^{\left( j \right)} } \right)\left( { - 1 + \varepsilon_{j}^{2} } \right)p - \left( {b_{2}^{\left( j \right)} + \left( {2 + b_{1}^{\left( j \right)} } \right)\varepsilon_{j} } \right)qS_{j} } \right)}}{{ - y_{j} }}, \\ c_{44}^{\left( j \right)} & = \frac{{\left( { - 1 + b_{2}^{\left( j \right)} \left( { - 1 + C_{j} } \right) - b_{3}^{\left( j \right)} \left( { - 1 + \varepsilon_{j} } \right)^{2} + \left( {b_{1}^{\left( j \right)} \left( { - 1 + C_{j} } \right) + 2C_{j} - \varepsilon_{j} } \right)\varepsilon_{j} } \right)p}}{{y_{j} }} \\ \end{aligned}$$

$$\begin{aligned} y_{j} & = \left( { - b_{3}^{\left( j \right)} \left( {1 + C_{j} } \right)\left( { - 1 + \varepsilon_{j} } \right)^{2} + b_{1}^{\left( j \right)} \left( { - 1 + C_{j} } \right)\left( { - 1 + C_{j} \varepsilon_{j} } \right) + \left( {C_{j} - \varepsilon_{j} } \right)\left( { - 2 + b_{2}^{\left( j \right)} \left( { - 1 + C_{j} } \right) + 2C_{j} \varepsilon_{j} } \right)} \right)pq - \left( { - 1 + \varepsilon_{j} } \right) \\ & \quad \left( {\left( {1 + b_{3}^{\left( j \right)} } \right)\left( {1 + \varepsilon_{j} } \right)p^{2} - \left( {1 + b_{1}^{\left( j \right)} + b_{2}^{\left( j \right)} + \varepsilon_{j} } \right)q^{2} } \right)S_{j} + \left( {b_{2}^{\left( j \right)} + \left( {2 + b_{1}^{\left( j \right)} } \right)\varepsilon_{j} } \right)pqS_{j}^{2} + b_{4}^{\left( j \right)} p\left( {\left( { - 1 + C_{j} } \right)\left( { - 1 + \varepsilon_{j}^{2} } \right)p - \left( { - 1 + \varepsilon_{j} } \right)^{2} qS_{j} } \right) \\ \end{aligned}$$

The Global Dynamic Stiffness Matrix ${\mathbf{K}}^{\left( 0 \right)}$ for an n-Segment Cable System with the Clamped Boundary Condition

$$K^{\left( 0 \right)} = \left[ {\begin{array}{*{20}c} {k_{33}^{\left( 1 \right)} + k_{11}^{\left( 2 \right)} + k_{eq,1} } & {k_{34}^{\left( 1 \right)} + k_{12}^{\left( 2 \right)} } & {k_{13}^{\left( 2 \right)} } & {k_{14}^{\left( 2 \right)} } & 0 & 0 & \cdots & 0 & \cdots & 0 \\ {k_{43}^{\left( 1 \right)} + k_{21}^{\left( 2 \right)} } & {k_{44}^{\left( 1 \right)} + k_{22}^{\left( 2 \right)} } & {k_{23}^{\left( 2 \right)} } & {k_{24}^{\left( 2 \right)} } & 0 & 0 & \cdots & \cdots & \cdots & 0 \\ {k_{31}^{\left( 2 \right)} } & {k_{32}^{\left( 2 \right)} } & {k_{33}^{\left( 2 \right)} + k_{11}^{\left( 3 \right)} + k_{eq,2} } & {k_{34}^{\left( 2 \right)} + k_{12}^{\left( 3 \right)} } & {k_{13}^{\left( 3 \right)} } & {k_{14}^{\left( 3 \right)} } & 0 & \vdots & \vdots & \vdots \\ {k_{41}^{\left( 2 \right)} } & {k_{42}^{\left( 2 \right)} } & {k_{43}^{\left( 2 \right)} + k_{21}^{\left( 3 \right)} } & {k_{44}^{\left( 2 \right)} + k_{22}^{\left( 3 \right)} } & {k_{23}^{\left( 3 \right)} } & {k_{24}^{\left( 3 \right)} } & 0 & 0 & \cdots & 0 \\ 0 & 0 & {k_{31}^{\left( 3 \right)} } & {k_{32}^{\left( 3 \right)} } & \ddots & {} & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & {k_{41}^{\left( 3 \right)} } & {k_{42}^{\left( 3 \right)} } & {} & \ddots & 0 & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \cdots & 0 & {k_{33}^{{\left( {n - 2} \right)}} + k_{11}^{{\left( {n - 1} \right)}} + k_{eq,n - 2} } & {k_{34}^{{\left( {n - 2} \right)}} + k_{12}^{{\left( {n - 1} \right)}} } & {k_{13}^{{\left( {n - 1} \right)}} } & {k_{14}^{{\left( {n - 1} \right)}} } \\ 0 & 0 & \cdots & 0 & \cdots & 0 & {k_{43}^{{\left( {n - 2} \right)}} + k_{21}^{{\left( {n - 1} \right)}} } & {k_{44}^{{\left( {n - 2} \right)}} + k_{22}^{{\left( {n - 1} \right)}} } & {k_{23}^{{\left( {n - 1} \right)}} } & {k_{24}^{{\left( {n - 1} \right)}} } \\ \vdots & \vdots & \vdots & \vdots & \cdots & 0 & {k_{31}^{{\left( {n - 1} \right)}} } & {k_{32}^{{\left( {n - 1} \right)}} } & {k_{33}^{{\left( {n - 1} \right)}} + k_{11}^{\left( n \right)} + k_{eq,n - 1} } & {k_{34}^{{\left( {n - 1} \right)}} + k_{12}^{\left( n \right)} } \\ 0 & 0 & \cdots & 0 & \cdots & 0 & {k_{41}^{{\left( {n - 1} \right)}} } & {k_{42}^{{\left( {n - 1} \right)}} } & {k_{43}^{{\left( {n - 1} \right)}} + k_{21}^{\left( n \right)} } & {k_{44}^{{\left( {n - 1} \right)}} + k_{22}^{\left( n \right)} } \\ \end{array} } \right]$$