Parametric studies on the dynamic properties of stay cables interconnected with uniformly distributed cross-ties

Abstract

The cable-stayed bridge is one of the most popular super-long-span bridges. However, the stay cables are prone to large vibration because their inherent characteristics of low mass, small damping, and large flexibility. Keeping the stay cables calm is significant to ensure their safety. Over the past decade, the use of cross-tie has become a practical and effective method to suppress the cable vibrations. Extensive research has led to a better understanding of the mechanics of cable-cross-tie systems and revealed that the application of a single or a few cross-ties may yield a potential deficiency in local modal vibrations. Recently, the use of large numbers of small elastic cross-ties uniformly distributed along stay cables has been proposed to replace the single or multiple cross-ties traditionally used to suppress the large vibrations of stay cables and delay the “mode localization.” In a previous study, the closed-form solutions to the free vibration of two equal-length cables with uniformly distributed elastic cross-ties have been derived and a numerical procedure has been developed to calculate the resulting modal frequencies and mode shapes. The uniformly distributed cross-ties successfully delayed the “model localization.” In the present study, the effects of the stiffness, width, and location of the uniformly distributed cross-ties on the free vibration of the general cable-cross-tie system are further investigated through parametric analyses. The results indicate that the stiffness and width of the cross-ties significantly affect the modal frequencies and mode shapes of the cable cross-tie system, while the location of the cross-ties has a relatively small effect.

Keywords

Cable-stayed bridge stay cable vibration uniformly distributed cross-ties vibration control

Introduction

Stay cables are prone to large vibrations under external excitations, such as from the wind (Chen et al., 2015; Cheng et al., 2008; Kumarasena et al., 2007; Li et al., 2013b), rain and wind (Chen et al., 2013, 2016; He et al., 2012; Hikami and Shiraishi, 1988; Jing et al., 2015, 2017, 2016; Cheng et al., 2015; Matsumoto et al., 2003; Zuo et al., 2008; Zuo and Jones, 2010), and anchorage motion (Macdonald, 2016; Xia and Fujino, 2006). The large vibrations severely affect the bridge safety and extensive research has been conducted with the aim to suppress these undesirable vibrations. Several aerodynamic (Matsumoto et al., 1998) and mechanical methods (Chen et al., 2016; Tabatabai and Mehrabi, 2000) were proposed and successfully applied on stay cables. The use of cross-tie (Caracoglia and Jones, 2007; Caracoglia and Zuo, 2009; Yamaguchi and Alauddin, 2003; Yamaguchi and Nagahawatta, 1995) has become increasingly popular over the past decades. This strategy enhances the in-plane stiffness, increases the modal mass in comparison with the mass of an isolated-cable vibrating system, and redistributes energy, thus improving the cable stability.

Yamaguchi and Nagahawatta (1995) and Yamaguchi and Alauddin (2003) experimentally investigated the characteristics of the free vibration of a cable-cross-ties system and studied the effect of the nonlinearity of the cross-ties. They reported that cross-ties can increase the damping ratio of the stay cables, the flexible cross-ties are more effective to dissipate energy, and the nonlinearity of the cross-ties induces multi-harmonic and multi-modal vibrations. They also proposed an energy-based method to evaluate the damping ratio of the cable-cross-tie system. Sun et al. (2007) experimentally investigated the in-plane stiffness and damping of the stay cables connected by cross-ties. They concluded that stiffer cross-ties are more effective in enhancing the in-plane stiffness, whereas softer cross-ties contribute more in increasing the damping ratio.

Caracoglia and Jones (2005a, 2005b) derived the closed-form solutions for the free vibration of two cables interconnected with a single cross-tie and developed a numerical procedure to solve for the free vibration of the system with multiple cables interconnected with several cross-ties. They further numerically evaluated the effects of in-plane dampers on the effectiveness of the cross-ties (Caracoglia and Jones, 2007) and concluded that these dampers increase the damping ratio of the in-plane modes of the cable-cross-tie system. These numerical results were verified by Caracoglia and Zuo (2009). Zhou et al. (2014) also numerically investigated the effect of the in-plane dampers on the free vibration of the cable-cross-tie system and reported that the damping of the system depends on the cross-tie location. Ahmad et al. (2016) considered the inherent damping of the stay cables in the analytical procedure and further investigated the effects of the different parameters, including the length ratio, mass ratio, mass–tension ratio, and frequency ratio, and the number of cross-ties and stay cables, on the free vibration of the cable-cross-tie system (Ahmad and Cheng, 2013; Ahmad et al., 2016).

Through these studies, the effectiveness of the cross-ties in cable vibration suppression has been verified, the mechanics of the cable-cross-tie system were revealed, and the effects of different parameters on the effectiveness of the cross-tie have been investigated in detail. However, the fact that a single or a few cross-ties may have potential deficiencies in the local modal vibrations was reported by Caracoglia and Jones (2005b), Caracoglia and Zuo (2009), and Ahmad et al. (2016). Ahmad et al. (2016) proposed a new index, the degree of mode localization, to evaluate the global nature of the single mode of the cable network and suggested using more and softer cross-ties to avoid or delay the local mode.

In a recent paper (Jing et al., 2018), uniformly distributed small elastic cross-ties (UDSECs) were proposed to replace the single or a few cross-ties traditionally used to suppress stay cable vibration and to delay the “mode localization.” A numerical procedure was developed to calculate the modal frequencies and mode shapes of the two general stay cables interconnected by UDSECs by simplifying the UDSECs into a continuously distributed cross-tie (CDC). The results showed that this method is effective to calculate the modal frequencies and mode shapes of two general cables with UDSECs and the UDSECs successfully delay the “mode localization.” Although this strategy is more difficult in practical implementation compared with a single or several cross-ties, it would become much easier and applicable in view of the anticipated rapid development of robotic technology in the future.

In this article, parametric studies are conducted to investigate the effects of different parameters, including the stiffness, width, and location of the UDSECs, on the modal frequencies and mode shapes of the cable-cross-tie system. The results show that the free vibration of the cable-cross-tie system is obviously affected by the stiffness, width, and location of the UDSECs. However, the stiffness exerts more influence than that of either the width or location of the UDSECs. The increase of either the stiffness or width of the UDSECs results in higher modal frequencies and enhances the participation of the neighboring stay cables, which is helpful for the energy redistribution and improves the cable stability. In addition, the stiffness, width, and location of the UDSECs generally exert more influence on the lower modes, with the exception of the first mode.

General problem formulation

A network of two general stay cables with UDSECs is horizontally laid taut as shown in Figure 1 (Jing et al., 2018). The upper and lower stay cables are generally denoted as Cable 1 and Cable 2, respectively. Their characteristic parameters include the lengths $L_{1}$ and $L_{2}$ , masses per unit length $m_{1}$ and $m_{2}$ , and the pre-tensions in the horizontal direction, $T_{1}$ and $T_{2}$ , and $Δ L$ is the horizontal offset between Cables 1 and 2. The two stay cables are fixed at both ends and four nodes ( $P_{I}, I = 1, 2, 3, 4$ ), which separate them into six segments. These segments are referred to as Elements j-p, where j is the jth cable and p is the pth segment of each cable. The length of Element j-p is denoted as $L_{j, p}$ . The displacement of Element j-p is defined as $v_{j, p}$ according to the sub-coordinate $x_{j, p}$ , as shown in Figure 1. Numerous small elastic cross-ties orthogonally interconnect the two central segments. The interval of the cross-ties is $l$ and the stiffness of each elastic cross-tie is $k_{e}$ .

Figure 1.

Two general cables interconnected with uniformly distributed cross-ties (Jing et al., 2018).

A CDC method had been proposed to solve for the modal frequency and mode shape; in the method, the elastic cross-ties were equivalent to a CDC and the stiffness of the equivalent continuously distributed cross-tie was calculated as $k = k_{e} / l$ per unit length.

The kinematic equation of the in-plane free vibration of the new cable system is described as

m_{j, p} \frac{\partial^{2} v_{j, p} (x_{j, p}, t)}{\partial^{2} t} - T_{j, p} \frac{\partial^{2} v_{j, p} (x_{j, p}, t)}{\partial^{2} x_{j, p}} = 0 (j = 1, 2; p = 1, 3)

(1a)

{\begin{matrix} m_{1} {\overset{\cdot\cdot}{v}}_{1, 2} - T_{1} {v ″}_{1, 2} + k (v_{1, 2} - v_{2, 2}) = 0 \\ s_{m} m_{1} {\overset{\cdot\cdot}{v}}_{2, 2} - s_{T} T_{1} {v ″}_{2, 2} + k (v_{2, 2} - v_{1, 2}) = 0 \end{matrix}

(1b)

where $v_{j, p}$ is the vertical displacement of Element j-p, and the dimensionless parameters $s_{m} = m_{2} / m_{1}$ is the mass ratio and $s_{T} = T_{2} / T_{1}$ is the tension ratio.

For the elements without a distributed cross-tie (Elements 1-1, 1-3, 2-1, and 2-3 in Figure 1), the cable displacement is expressed as (Ahmad and Cheng, 2013; Ahmad et al., 2016; Caracoglia and Jones, 2005a, 2005b; Irvine and Caughey, 1974)

v_{j, p} = V_{j, p} (x_{j, p}) e^{i ω t} (j = 1, 2; p = 1, 3)

(2)

in which

V_{j, p} (x_{j, p}) = A_{j, p} \sin (\frac{α π}{L_{j}} f_{j} x_{j, p})

(3)

where $A_{j, p}$ is the unknown mode-shape coefficients determined from the boundary conditions, $α = ω / ω_{01} = ω (L_{1} / π) \sqrt{m_{1} / T_{1}}$ is the reduced frequency of the system, and $f_{j} = \sqrt{(T_{1} / T_{j}) (m_{j} / m_{1})}$ is the ratio of the fundamental frequency of the first to the jth cable.

For the elements with UDSECs (Elements 1-2 and 2-2 in Figure 1), the cable displacements are expressed as

{\begin{matrix} v_{1, 2} (x, t) = V_{1, 2} (x_{1, 2}) e^{i ω t} \\ v_{2, 2} (x, t) = V_{2, 2} (x_{1, 2}) e^{i ω t} \end{matrix}

(4)

in which $V_{1, 2} (x_{1, 2})$ and $V_{2, 2} (x_{2, 2})$ are the mode-shape functions of the two elements that are solved as

V_{1, 2} (x_{1, 2}) = C_{1} e^{r_{1} x_{1, 2}} + C_{2} e^{r_{2} x_{1, 2}} + C_{3} e^{r_{3} x_{1, 2}} + C_{4} e^{r_{4} x_{1, 2}}

(5)

V_{2, 2} (x_{2, 2}) = (1 - \frac{m_{1} ω^{2}}{k}) V_{1, 2} (x_{2, 2}) - \frac{T_{1}}{k} V_{1, 2}^{″} (x_{2, 2})

(6)

where $C_{i} (i = 1, 2, 3, 4)$ are the unknown coefficients denoting the mode shape of Element 1-2 that are determined from the boundary conditions.

The boundary condition functions of the cable network are written in the following formula

ϕ X = 0

(7)

where

X = {(\begin{matrix} A_{1, 1} & A_{1, 3} & A_{2, 1} & A_{2, 3} & C_{1} & C_{2} & C_{3} & C_{4} \end{matrix})}^{T}

(8)

Φ = (\begin{matrix} \sin (ϕ_{1} L_{1, 1}) & 0 & 0 & 0 & - 1 & - 1 & - 1 & - 1 \\ 0 & \sin (ϕ_{11} L_{1, 3}) & 0 & 0 & - e^{r_{1} L_{1, 2}} & - e^{r_{2} L_{1, 2}} & - e^{r_{3} L_{1, 2}} & - e^{r_{4} L_{1, 2}} \\ 0 & 0 & \sin (ϕ_{2} L_{2, 1}) & 0 & - K_{1} & - K_{2} & - K_{3} & - K_{4} \\ 0 & 0 & 0 & \sin (ϕ_{2} L_{2, 3}) & - K_{1} e^{r_{1} L_{2, 2}} & - K_{2} e^{r_{2} L_{2, 2}} & - K_{3} e^{r_{3} L_{2, 2}} & - K_{4} e^{r_{4} L_{2, 2}} \\ ϕ_{1} \cos (ϕ_{1} L_{1, 1}) & 0 & 0 & 0 & - r_{1} & - r_{2} & - r_{3} & - r_{4} \\ 0 & ϕ_{1} \cos (ϕ_{1} L_{1, 3}) & 0 & 0 & r_{1} e^{r_{1} L_{1, 2}} & r_{2} e^{r_{2} L_{1, 2}} & r_{3} e^{r_{3} L_{1, 2}} & r_{4} e^{r_{4} L_{1, 2}} \\ 0 & 0 & ϕ_{2} \cos (ϕ_{2} L_{2, 1}) & 0 & - r_{1} K_{1} & - r_{2} K_{2} & - r_{3} K_{3} & - r_{4} K_{4} \\ 0 & 0 & 0 & ϕ_{2} \cos (ϕ_{2} L_{2, 3}) & r_{1} K_{1} e^{r_{1} L_{2, 2}} & r_{2} K_{2} e^{r_{2} L_{2, 2}} & r_{3} K_{3} e^{r_{3} L_{2, 2}} & r_{4} K_{4} e^{r_{4} L_{2, 2}} \end{matrix})

(10)

in which $ϕ_{1} = (α π / L_{1}) f_{1}, ϕ_{2} = (α π / L_{1}) f_{2}$ , and $K_{s} = 1 - ((m_{1} ω^{2} + T_{1} r_{s}^{2}) / k)$ ( $s = 1, 2, 3, 4$ ).

The existence of the infinite set of nontrivial solutions ( $X$ ) to the homogeneous system (7) requires $det [ϕ] = 0$ , from which the nondimensional natural frequencies in terms of $α$ are calculated.

The set of $α$ solutions cannot generally be determined in the closed form for such a complex structure. Therefore, a numerical iterative procedure to calculate the modal frequencies and mode shapes of this system had been established as follows:

Step 1: Give the desired maximum natural circular frequency ( $ω_{max}$ ) and generate the vector of the circular frequency, $ω$ , which varies from 0 to $ω_{max}$ with an interval of 10⁻⁴, and the corresponding vector of reduced frequency, $α = ω / ω_{01}$ .

Step 2: Substitute $ω$ and $α$ into equation (9) and calculate the vector of $det [ϕ (ω)]$ .

Step 3: Define the frequency at the minimum $det [ϕ (ω)]$ as the suspected natural circular frequency.

Step 4: Substitute the suspected natural circular frequencies into equation (9) one by one, and calculate the corresponding eigenvalues and eigenvectors of the matrix $ϕ$ .

Step 5: If there is a zero eigenvalue in Step 4, the suspected natural circular frequency is confirmed to be the natural circular frequency of this system and the corresponding eigenvector is determined to be the mode-shape coefficient; otherwise, go back to Step 4 and move on to the next suspected natural circular frequency.

The vector of the mode-shape coefficients ( $X$ ) is normalized such that (Caracoglia and Jones, 2005)

\sum_{j = 1}^{2} \sum_{p = 1}^{3} \int_{0}^{L_{j, p}} m_{j} V_{j, p}^{2} (x_{j, p}) d x_{j, p} = 1

(10)

Two general stay cables interconnected with uniformly distributed cross-ties

Two stay cables of the Fred Hartman Bridge (AS20 and AS18 used by Caracoglia and Jones (2005b), Ahmad et al. (2016), Caracoglia and Zuo (2009), and Jing et al. (2018)) are applied to study the effects of stiffness, width, and location of the uniformly distributed cross-tie on the modal frequency and mode shapes of the cable network system. The parameters of the cable-cross-tie system are listed in Table 1. Cable AS20 is set as Cable 1 with a total length of 140 m, a mass of 70.1 kg/m, and a horizontal extension of 3351 kN. Cable AS18 is set as Cable 2, which has a total length of 112 m, a mass of 52.9 kg/m, and a horizontal extension of 2732 kN. The horizontal offset between the two cables ( $Δ L$ ) is set as 2.6 m. They are orthogonally connected by plenty of UDSECs, at the central part of Cable 2. The equivalent stiffness of the elastic cross-ties is defined to be 6.75 kN/m².

Table 1.

Parameters of the cable-cross-tie system (Caracoglia and Jones, 2005b; Caracoglia and Zuo, 2009).

$m_{1}$	70.1 kg	$m_{2}$	52.9 kg
$T_{1}$	3351 kN	$T_{2}$	2732 kN
$L_{1}$	140.0 m	$L_{2}$	112.0 m
$L_{1, 1}$	8.6 m	$L_{2, 1}$	6.0 m
$L_{1, 2}$ $L_{2, 2}$	100.0 m	$L_{2, 3}$	6.0 m
$L_{1, 3}$	31.4 m	$Δ L$	2.6 m
$k$	6.75 kN/m²

Parametric analyses

Parametric studies are conducted using the cable-cross-tie system described in section “Two general stay cables interconnected with uniformly distributed cross-ties” as the basic system. The effects of different parameters, including the stiffness, width, and location of the UDSECs, are investigated using the CDC method (Jing et al., 2018).

Effect of the stiffness of the UDSECs

The stiffness of the UDSECs is first nondimensionalized as follows

K = \frac{k}{m_{1} ω_{1, n}^{2}}

(11)

The effect of the nondimensional stiffness of the UDSECs on the modal frequencies and mode shapes of the cable-cross-tie system is investigated by changing the parameter k in equation (9). Both the width and location of the UDSECs are constant. The nondimensional stiffness varies in the range of 0.1–4.0, which is wide enough for practical applications. The modal frequencies and mode shapes of six cases with nondimensional stiffness of 0.1, 0.5, 1.0, 2.0, 3.0, and 4.0 (designated K-0.1, K-0.5, K-1.0, K-2.0, K-3.0, and K-4.0) are calculated and shown in Figures 2 to 4. Figure 2(a) shows the first 10 modal frequencies of the six cases. The modal frequencies generally increase as the stiffness of the UDSECs becomes higher, particularly the second to fifth, seventh and ninth modes. The modal frequencies of the first, sixth, and eighth modes are relatively less insensitive to the nondimensional stiffness. Figure 2(b) shows the rate of increase the modal frequencies with respect to those from Case K-0.1. The figure indicates that the nondimensional stiffness generally has more influence on the lower modes. When the nondimensional stiffness increases from 0.1 to 4.0, the modal frequencies of the second to fifth modes increase by 67.8%, 52.3%, 28.5%, and 28.3%, respectively, while the modal frequencies of the first, sixth, and eighth modes only increase by 14.5%, 5.6%, and 1.8%, respectively. In addition, the nondimensional stiffness shows different effects on different modes. Figure 3 shows that the modal frequencies of the second and third modes vary with the nondimensional stiffness. For the second mode, the modal frequency significantly increases when the nondimensional stiffness increases from 0.1 to 2.0 and then becomes constant when the nondimensional stiffness is above 2.0. However, for the third mode, the modal frequency is more sensitive when the nondimensional stiffness is in the range of 1.0–3.0.

Figure 2.

Effects of nondimensional stiffness on the modal frequencies: (a) modal frequencies and (b) increase rate of the modal frequency.

Figure 3.

The modal frequencies of the second and third modes vary with the nondimensional stiffness: (a) second modal frequencies and (b) third modal frequencies.

Figure 4.

Mode shapes vary with the nondimensional stiffness of the UDSECs.

Figure 4 shows the first to third, fifth and seventh mode shapes of Cases K-0.1, K-1.0, K-2.0, K-3.0, and K-4.0. A higher nondimensional stiffness always enhances the participation of the neighboring cable. When the nondimensional stiffness is 0.1 (Case K-0.1), the mode shapes of the cable network are very close to those of a single cable. For example, the first and third mode shapes of the cable network are very close to the first two mode shapes of Cable 1. This means that these mode vibrations are mainly concentrated on a single cable and are “mode localizations.” However, when the nondimensional stiffness is higher than 2.0, all mode shapes are the combination of both the target and neighboring cable vibrations. Since the participation of the neighboring cable redistributes the energy, the UDSECs with a higher nondimensional stiffness help avoid “mode localization” (Ahmad et al., 2016) and improve the stability of the stay cables.

In addition, the nondimensional stiffness also changes the combination pattern of some mode shapes. For example, when the nondimensional stiffness is 1.0, the second mode shape is the out-of-phase combination of the first basic modes of Cables 1 and 2 (Jing et al., 2018); however, when the nondimensional reduced stiffness reaches 3.0 (Case K-3.0), the second mode shape turns into the in-phase combination of the second basic modes of both Cables 1 and 2. Similarly, the pattern of the third mode shape changes at Case K-2.0.

Effect of the UDSECs width

The effect of the UDSECs width on the modal frequencies and mode shapes of the cable-cross-tie system is investigated by changing the parameters $L_{1, 1}$ , $L_{1, 2}$ , $L_{1, 3}$ , $L_{2, 1}$ , $L_{2, 2}$ , and $L_{2, 3}$ in equation (11). The UDSECs are located at the center of Cable 2 with a nondimensional stiffness of 1.0. The width of the UDSECs varies from 20 to 100 m and is nondimensionalized as follows

s = \frac{L_{2, 2}}{L_{2}} s

(12)

Five cases with widths of 100, 80, 60, 40, and 20 m, designated S-1 to S-5 with nondimensional widths of 89.3%, 71.4%, 53.6%, 35.7%, and 17.9%, respectively, are investigated. Their modal frequencies and mode shapes are calculated and shown in Figures 5 and 6, respectively. The results show that the nondimensional width has a relatively small effect on the modal frequencies. When the span ratio decreases from 71.4% to 17.9%, the largest decrement among the modal frequencies is approximately 18.9%, which occurs at the second mode. The decrements of the modal frequencies of the 1st and 4th to 10th modes are less than 5.0%. However, the mode shapes are significantly affected by the nondimensional width, as shown in Figure 6. When the nondimensional width of the UDSECs decreases, the participation of the neighboring cable decreases and results in “mode localization.” In particular, the third and fourth mode shapes become a single cable vibration as the nondimensional width decreases to 17.9%. This means that when the nondimensional width decreases, less energy is transferred from the target cable to the neighboring cable, reducing the effectiveness of the vibration suppression. In addition, the nondimensional width of the UDSECs also changes the combination pattern of some mode shapes, such as the second to fourth modes. However, the effect of the nondimensional width of the UDSECs on the first and seventh mode shapes is relatively smaller, similar to those for the modal frequencies.

Figure 5.

First 10 modal frequencies of Cases S-1 to S-5: (a) modal frequencies and (b) increase rate of the modal frequency.

Figure 6.

First to fourth, and seventh mode shapes of Cases S-1 to S-5.

Effect of the UDSECs location

The UDSEC location is indicated by the horizontal distance from the left end of Cable 2 to the center point of the UDSECs. The effect of the UDSECs location is investigated by changing the parameters $L_{1, 1}$ , $L_{1, 2}$ , $L_{1, 3}$ , $L_{2, 1}$ , $L_{2, 2}$ , and $L_{2, 3}$ in equation (11). The width of the UDSECs ( $L_{1, 2}$ ) is 60 m, which is approximately half of the length of Cable 2, and the nondimensional stiffness of the UDSECs is 1.0. The UDSEC location varies from 36 to 76 m. Five cases corresponding to UDSECs at 36, 46, 56, 66, and 76 m, named M-1 to M-5, respectively, are investigated. The modal frequencies and shapes are calculated and shown in Figures 7 and 8, respectively. The results show that the UDSEC location has a relatively small effect on the modal frequencies. The largest divergence of the modal frequencies among the five cases is approximately 10.0%. When the UDSECs are moved to the center of Cable 2, the modal frequency of the second mode increases by 10.3%. When the UDSECs are moved to the right side of Cable 2, the modal frequency of the third mode decreases by 3.0%.

Figure 7.

First, fourth, and seventh mode shapes when the UDSECs move from left side to right side: (a) modal frequencies and (b) increase rate of the modal frequency.

Figure 8.

First to fourth, and seventh mode shapes when the UDSECs move from left side to right side.

Figure 8 shows the first to fourth, and seventh mode shapes of the five cases. When the UDSECs are moved from the left side to right side, the second to fourth mode shapes showed obvious changes. However, the UDSEC location has little effect on the first and seventh mode shapes.

Coupling effects of both stiffness and width of the UDSECs on the modal frequencies

The above studies showed that the stiffness and width of the UDSECs have relatively large effects on the modal frequencies and mode shapes of the cable-cross-tie system. Therefore, the coupling effects of both the stiffness and width of the UDSECs on the modal frequencies of the cable-cross-tie system are further investigated, when they are located at the center of Cable 2. The nondimensional stiffness of the UDSECs varies from 0.1 to 9.0 for the purpose of the academic investigation and the nondimensional width of the UDSECs varies from 17.9% to 89.3%.

Figure 9 shows the variations of the modal frequencies of the first four modes. The figure agrees with the above results that the modal frequencies always increase with the nondimensional stiffness and width when the nondimensional stiffness is relatively small. However, when the nondimensional stiffness continues to increase, the modal frequencies reach their asymptotic values. The four modes have different ranges of sensitivities. When the nondimensional width equals 89.3%, the modal frequency of the first mode is sensitive when the nondimensional stiffness is lower than 0.7, that of the second mode is sensitive when the nondimensional stiffness is lower than 1.5, that of the third mode is more sensitive in the range from 1.0 to 3.0, and that of the fourth mode is more sensitive in the range from 3.0 to 6.0.

Figure 9.

Effects of both stiffness and width of the UDSECs on the modal frequencies of the first modes: (a) modal frequency of the first mode, (b) modal frequency of the second mode, (c) modal frequency of the third mode, and (d) modal frequency of the fourth mode.

The decrease of nondimensional width always results in the decrease of the asymptotic values and broadens the sensitive stiffness range of each model frequency. For example, when the nondimensional width is 89.3%, the modal frequency of the second mode becomes 1.75 Hz as the increase of the nondimensional stiffness, and the sensitive stiffness range is from 0 to 1.5. However, when the nondimensional width decreases to 17.9%, the asymptotic value becomes 1.5 Hz, and the sensitive stiffness range becomes 0 to 3.0. Figure 9(c) and (d) shows that the modal frequencies of the higher modes (third and fourth modes) are more sensitive to the nondimensional width of the UDSECs.

Conclusion

In the present study, parametric studies were conducted to investigate the effects of different parameters of the UDSECs, including the stiffness, width, and location, on the modal frequencies and mode shapes of the cable-cross-tie system using the CDC method developed by Jing et al. (2018). The main conclusions are obtained as follows:

Both the modal frequency and mode shape of the cable-cross-tie system are sensitive to the stiffness, width, and location of the UDSECs. The UDSECs stiffness exerts more influence than either the width or location.

The increase in the stiffness and width of the UDSECs results in higher modal frequencies and enhances the participation of the neighboring stay cables, which is helpful for the energy redistribution and improves the cable stability.

The stiffness, width, and location of the UDSECs generally exert more influence on the lower modes, with the expectation of the first mode.

The modal frequency of each mode becomes asymptotic when the stiffness of UDSECs continues to increase. Each modal frequency has its own sensitive stiffness range. The sensitive stiffness of higher modes is larger than those of the lower modes.

The decrease of UDSEC width results in the decrease of the asymptotic values and broadens the sensitive stiffness range of each modal frequency.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research described in this article was financially supported by the National Natural Science Foundations of China (nos 51708559, U1534206), Key Project of China Railway Corporation (2017T001-G), and the National Key R&D Program of China (no. 2017YFB1201204).

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