Using a Rigid Restraint with a Built-In Tuned Mass Damper to Control the Vibration of Cables

Abstract

Cables are widely utilized as load-carrying members due to their excellent mechanical properties. However, the inherent damping of cables is usually extremely low, thereby causing undesired vibrations to occur frequently under various external excitations. This study investigates the utilization of rigid restraints with a built-in tuned mass damper to mitigate the vibration of cables. First, the configuration of a rigid restraint with a built-in tuned mass damper is presented, followed by the development of a problem formulation for controlled cables using such a device. A discrete model is further established to describe the dynamic behavior of the system. Thereafter, a series of numerical simulations are conducted. The influence of the mass ratio of the tuned mass damper and installation position is analyzed. Then, examples are presented to verify the control effectiveness under sinusoidal excitations. As indicated by the numerical results, the proposed device can mitigate cable vibration exceptionally well. Taking aerodynamic effects into account, model cables and control devices are manufactured. Two installation positions, namely, quarter-span and mid-span, are considered. Wind tunnel tests are performed. As shown by the experimental tests, the proposed rigid restraint with a built-in tuned mass damper can suppress the first two modal vibrations. Overall, the rigid restraint with built-in tuned mass damper can mitigate cable vibration, though several issues should be further addressed.

1. Introduction

Due to superior mechanical properties, cables are widely utilized as load-bearing components in various structures. For example, in cable-stayed bridges, stay cables play crucial roles in carrying the weight of the entire bridge. In suspension bridges, suspender cables are essential for transferring loads. In photovoltaic applications, cables also work as primary elements to form large-span flexible support systems, addressing the limitations inherent in traditional rigid design [1].

However, cables often suffer from vibration issues in practical engineering scenarios. Sometimes, the mechanism of these vibrations is complex and not fully understood. In general, cable vibrations result from multiple factors, including the aerodynamic properties of the cable’s surface and corresponding structural parameters. Due to external excitations, such as wind, rain, and vehicular loads, cables can experience various types of vibrations [2]. Once these vibrations occur, it becomes difficult to control their amplitudes due to the extremely low inherent damping of cables. For instance, even in light wind and rain, cables are prone to vibrating violently and suddenly, which is known as wind–rain-induced vibration [3]. Other common wind-induced problems include buffeting, galloping, and vortex-induced vibrations. Undesired vibrations pose additional risks [4,5,6], including but not limited to, for example, the accelerating fatigue of cables, damage to the protective layer, concerns of pedestrians, and sometimes even the disturbance of normal operation.

To address unwanted vibrations, various control methods have been proposed in recent years. There are currently three control strategies based on the working mechanism—aerodynamic measurements, networking, and mechanical dampers—to mitigate cable vibrations. Aerodynamic measurements can alter the surface roughness of cables or disrupt waterline formation, thereby causing a mitigating effect [7]. To amplify airflow around a component using small attachments, spontaneous vibrations are transformed into controlled artificial vibrations [8]. Moreover, active jets have been introduced to improve the control of wind-induced vibrations [9,10]. The networking approach involves arranging auxiliary cables, which divide the main cable into several short ones to enhance rigidity. Varying the arrangement of auxiliary cables significantly alters the overall stiffness and other dynamic properties of cable networks [11]. According to research results, a uniform triangular grid is the most effective arrangement. However, the installed auxiliary cables cannot dissipate energy; thus, the vibration-reducing effect remains limited. In actual engineering, auxiliary cables can experience tension loss, leading to a loss of effectiveness or abnormal internal forces in cables [12,13]. Furthermore, auxiliary cables often affect the aesthetics of controlled structures, which is hard to accept most of the time.

Comparing the two methods above, attaching mechanical dampers is an effective and more attractive method. This popularity stems from the better technical completion, greater robustness, and easier installation of mechanical dampers. Using wire ropes, one type of damper was developed through cyclic loading experiments in three directions [14]. As shown by monitored data, attaching such wire rope dampers to actual bridges can reduce both in-plane and out-of-plane vibrations well. Using the Rayleigh–Ritz method, vibration modal functions were obtained. Then, the effectiveness of the wire damper used for composite beams under thermal conditions was confirmed [15]. Using a wide range of cyclic loading experiments, the effect of geometric parameters on the wire rope damper’s performance was investigated and optimized for cable vibration control [16]. A dual-damper arrangement is advantageous for improving damping ratios and cost-effectiveness. Through an analysis of vibration modes and parameter effects, a new vibration model was proposed to examine the control effects of double-tuned mass dampers [17]. To achieve multimode control, using spatially distributed tuned mass dampers along cables was suggested [18]. Equations for different spatial arrangements with tuned mass dampers in arbitrary directions were established. Using the finite element method, bridges with damped legs were modeled [19]. This approach addressed the problems of closely spaced natural frequencies, which might complicate the tracking modal order and damping changes during design. One study presented the connection of diagonal cables with negative stiffness dampers to increase overall damping [20]. To enhance damping efficiency, a practical negative stiffness device with adjustable stiffness was proposed to connect in parallel or series with a viscoelastic damper [21]. In addition to optimizing existing damper forms, researchers have provided new insights [22,23,24]. Scholars have investigated the magnetorheological phenomenon under varying magnetic field strengths. Such research shows different viscous coefficients that can be integrated into various dampers for vibration damping [25,26]. By precisely adjusting the state of the magnetorheological fluid [27], accurate damping control is achieved.

As discussed above, dampers are usually installed near the lower anchorage on the deck due to practical limitations. With the increase in cable length, attaching dampers has become a new challenge. In particular, the vertical arrangement of suspended cables creates unique structural characteristics, and attaching dampers becomes more difficult. To overcome the above drawback, this study proposes a novel device: a rigid restraint with a built-in tuned mass damper to control the vibration of suspended cables. This paper is arranged as follows: firstly, the formulation of cables connected by the proposed rigid restraint with a built-in tuned mass damper (TMD) is developed. The configuration of the proposed device is described in detail, followed by the establishment of a discrete model for the controlled cables. Then, numerical simulations are conducted to investigate the control performance under harmonic loads. After that, wind tunnel tests are further performed to verify the mitigation effect under wind loads. Lastly, conclusions are addressed based on the obtained results. The limitations of this study include the following aspects: firstly, the discrete numerical model developed in the manuscript does not account for factors such as the aero-elastic nature and fluid–structure interaction. Secondly, the choice of the dominant frequency, damping level, number, and location of the proposed device requires further refined study.

2. Formulation of Cables Linked by the Rigid Restraint with Built-In TMD

2.1. Configuration of Rigid Restraint with Built-In TMD for Cables

Figure 1 shows the details of a rigid restraint with a built-in TMD for suspended cables. In common cases, suspended cables consist of two cables near each other. The rigid restraint is made of lightweight materials with enough stiffness, thereby interconnecting two adjacent cables. The built-in TMD consists of a section of steel strand and a mass block. One end of the steel strand is attached to the rigid restraint. The other end is connected to the mass block. Through such a configuration, the steel strand acts as an elastic unit; therefore, a TMD is formed. By changing the length of the steel strand or the cross-sectional area, various stiffness levels can be achieved to tune the TMD. As the steel strand is axisymmetric, the formed TMD can work in multiple directions. The rigid restraint links two individual cables at the installed position. Due to the rigidity of the restraint, two separate cables maintain the same motion at the connection point all the time, thereby causing changes in the mode. The TMD part absorbs and dissipates the vibrating energy by adjusting its frequency to match the targeted mode of the cables.

A simplified model with a built-in TMD is established for the system of cables and rigid restraints. The main goal of the current study is to investigate the effectiveness of the proposed device. The nonlinearities, such as cable sag, are not considered. As shown in Figure 2, the adjacent cables are simplified as taut strings with two fixed ends, in which the flexural rigidity is not considered. Generally, the cable has a high ratio of tension to weight. Under such conditions, flexural rigidity only causes second-order effects, which can be neglected. Only one-direction vibration is considered. Compared with the large weight of cables, the weight of the rigid restraint is neglected. Meanwhile, the steel strand is replaced by the combination of a spring and a dashpot.

Within small deflection, the equations of motion for the system can be written as:

$$m_1{\ddot{v}}_1+c_1{\dot{v}}_1-T_1{v_1}^{″}=F\left(t\right)\delta \left(x-x_d\right)+f_{e1}\left(x,t\right)$$

(1)

$$m_2{\ddot{v}}_2+c_2{\dot{v}}_2-T_2{v_2}^{″}=F\left(t\right)\delta \left(x-x_d\right)+f_{e2}\left(x,t\right)$$

(2)

where $m_i$, $c_i$, $T_i$, and $f_{ei}\left(x,t\right)$ are the mass per unit length, damping per unit length, and tension force of cable #$i$($i=1,2$), respectively; the tension force can be measured by the load cell or calculated as $T=4m{f}^2{L}^2$, in which $m$ is the mass per unit length, $L$ is the length of the cable, and $f$ is the first natural frequency obtained from the acceleration responses of the cable; $v_i(i=1,2)$ is the transverse displacement of cable #i; ${\dot{v}}_i(i=1,2)$ is the corresponding velocity response; ${\ddot{v}}_i(i=1,2)$ is the corresponding acceleration response; and ${v^{″}}_i(i=1,2)$ is the second-order derivative of the displacement with respect to space. $F\left(t\right)$ is the control force provided by the control device. The stiffness of the rigid restraint is $k_c\to \infty$, so the displacement responses of two cables at location $x=x_d$ are always equal:

$$v_1\left(x_d,t\right)=v_2\left(x_d,t\right)$$

(3)

where $x_d$ is the attached position of the rigid restraint. Then, the control force of the TMD part has the following expression:

$$\begin{array}{l}F\left(t\right)=-m_d\left({\ddot{v}}_d-{\ddot{v}}_1\left(x_d\right)\right)\\ =c_d\left({\dot{v}}_d-{\dot{v}}_1\left(x_d\right)\right)+k_d\left(v_d-v_1\left(x_d\right)\right)\end{array}$$

(4)

where $m_d$ is the mass of the block; $c_d$ is the damping of the steel strand; $k_d$ is the stiffness of the steel strand; and $v_d$, ${\dot{v}}_d$, and ${\ddot{v}}_d$ are the displacement, velocity, and acceleration response of the mass block, respectively.

2.2. Discrete Model for the System

Due to the connection of the rigid restraint, the mode shape of the continuous model alters a lot. Thus, a discrete model with finite degrees is developed to describe the dynamic behavior of the cables using a rigid restraint with built-in TMD at an arbitrary position. As shown in Figure 3, one continuous cable is discretized into n + 1 masses. For the two ends, the mass is m/2. For other nodes, the mass is m. To link these masses, n springs with length l are utilized. The tension of the springs in horizontal conditions is equal to the cable tension force T. The damping of the cables themselves, $c_1$ and $c_2$, are not considered.

The rigid restraint with built-in TMD is attached at node n_d. According to the results [28], the motion equation is obtained in matrix form as:

$$\ddot{y}+Ky=L_{d}F\left(t\right)+L_e{F}_e\left(t\right)$$

(5)

$$\ddot{z}+Kz=L_{d}F\left(t\right)+L_e{F}_e\left(t\right)$$

(6)

where $y={\left[\begin{array}{cccc}{y}_1& y_2& \cdots & y_{n-1}\end{array}\right]}^T$ and $z={\left[\begin{array}{cccc}{z}_1& z_2& \cdots & z_{n-1}\end{array}\right]}^T$ are the displacement vectors for $n-1$ nodes of cables #1 and #2, respectively; $\ddot{y}$ and $\ddot{z}$ are the corresponding acceleration responses, $L_d={\left[\begin{array}{lllll}0\hfill & \cdots \hfill & 1\hfill & \cdots \hfill & 0\hfill \end{array}\right]}^T$ is the vector in which the $n_d$ element is unity, indicating the location of the control device; $L_e$ is the vector whose elements depend on the characteristics of external excitation; and $K$ is the stiffness matrix, as follows:

$$K={\displaystyle \frac{T}{lm}}\left[\begin{array}{ccccc}2& -1& & & \\ -1& 2& -1& & \\ & -1& \ddots & \ddots & \\ & & \ddots & \ddots & -1\\ & & & -1& 2\end{array}\right]$$

(7)

3. Numerical Study on Cable Vibration Control Using Rigid Restraint with Built-In TMD

3.1. Numerical Model and Its Verification

Two full-scale adjacent cables were adopted for a numerical simulation. The length of the cables was 265 m. The tension force was 1812 kN, with a mass per unit of 26 kg/m. In general, the inherent damping of cables is extremely low. According to on-site observation results, it is usually less than 0.3%. In addition, to ensure calculation convergence under uncontrolled conditions, the first modal damping for cables was assumed to be 0.1% herein. The external sinusoidal has the following form:

$$f_e\left(x,t\right)=\left\{\begin{array}{cc}\sin \left(\pi x/L\right)\cdot \sin \left({\omega }_{1}t\right)& t\le 50 \mathrm{s}\\ 0& t>50 \mathrm{s}\end{array}\right.$$

(8)

where ${\omega }_1$ is the first circular frequency of the cable.

Based on the above parameters, a Simulink model was established in MATLAB 2016a. The solving method was ode5 with a fixed time step of $1\times {10}^{-4}$ s. To verify the effectiveness of this model, one cable with an optimal linear viscous damping at location 3% was simulated. As shown in Figure 4, the cable enters into forced vibration for the first 50 s, after which it becomes free vibration due to the cut-off of external excitation. According to the decaying trend, the added damping by the linear viscous damping is 1.02%, which is very close to the analytical result of 1%.

3.2. Numerical Results Under Harmonic Excitation

A series of numerical simulations were conducted to investigate the control performance. For the rigid restraint with built-in TMD, two main parameters, the installation position and the mass ratio of the TMD relative to the cable, were considered.

In the beginning, the influence of installation position was analyzed for the first modal vibration of the cable. As shown in Figure 5a, the horizontal axis is the installation position relative to the cable length, while the vertical axis is the root mean square response of the cable at mid-span. The mass ratio was set to 0.01. With the installation position moving from the left end to the right end, the root mean square response decreases rapidly until the relative installation position, reaching about 0.2. After that, the curve enters a gentle stage and remains basically unchanged in the range of 0.2 to 0.8. When the mass ratio increases to 0.03, the variation pattern of the root mean square response is similar, as shown in Figure 5b. Within the range of 0.2 to 0.8, the root mean square response of the cable drops dramatically, which implies the control performance of the rigid restraint with built-in TMD.

Furthermore, the impact of the mass ratio was studied. As described in Figure 6, the horizontal axis is the mass ratio, and the vertical axis is the root mean square root response at the midspan. The value of the installation position relative to the cable length was set to 0.2. A rapid decline occurs when the mass ratio is less than 0.02, followed by a slow downward trend until the mass ratio reaches 0.1. Some fluctuations can be observed. Hence, when the mass ratio is larger than 0.02, continuing to increase the mass of the built-in TMD is not cost-effective.

Based on the aforementioned parameter analysis, for another case study, the installation location was selected as 0.2, and the mass ratio was chosen as 0.01. Figure 7 compares the time histories of displacement response at the mid-span of the cables. For the without-control condition, the cables exhibit forced vibration in the first 50 s and then enter free vibration in the later 50 s. For the controlled case, the cables’ displacement response presents the characteristics of a beat, with a significant reduction in amplitude due to the control effect of the rigid restraint with built-in TMD.

The root mean square responses along the cable are shown in Figure 8. As can be seen clearly, the maximum value under the uncontrolled case is about 0.075, while the peak value for the controlled case is only 0.02, achieving a reduction of over 70%.

By changing the exciting frequency in Equation (8), the frequency response curves of the root mean square response at the midspan were obtained, as shown in Figure 9. For the uncontrolled case, the frequency response curve has only one peak. When the excitation frequency is the same as the first-order natural frequency of the cable, the root mean square response reaches its maximum value. The cable enters a resonance state at this time. After the attachment of the rigid restraint with built-in TMD, the frequency response changes into a curve with two peaks, of which the amplitudes also decrease remarkably. This phenomenon is due to the additional degree of freedom provided by the TMD. Thus, the cable exhibits two resonant frequencies close to each other, which results in the beat in the time histories.

4. Experimental Investigation of Cable Vibration Control Using Rigid Restraint with Built-In TMD

Under wind loads, cable vibration is a complex, dynamic behavior. The problem of cable vibrations under wind load is of an aero-elastic nature. It is not only related to turbulence but also to a more complex mechanism of fluid–structure interaction. To simulate the dynamic behavior of cables under wind load, more factors need to be considered. For example, the spatial correlation of wind actions and fluid–structure interaction (which may be strongly nonlinear) should be considered. However, the discrete formulation provided in this study does not account for such factors. Thus, it is hard to simulate such an aerodynamic effect exactly using numerical methods, especially when considering the rigid restraint with a built-in TMD. Therefore, wind tunnel tests were adopted to investigate the proposed device.

4.1. Experimental Setup

Firstly, two flexible model cables were designed and manufactured. The length of the model cables adopted was 4.8 m due to the limitations of the wind tunnel. The cross-section was made of flexible strand wires wrapped around light materials with a diameter of 75 mm. At two ends, nuts were connected to adjust the tension force, thereby achieving the desired vibration frequency. The center spacing of the two cables was set to 414 mm. As shown in Figure 10, the two cables were placed horizontally in the wind tunnel.

The control device used in this experiment is shown in Figure 11. The rigid restraint part was made via 3D printing using lightweight materials. The TMD part consists of a section of steel strand, a mass block, and other necessary connections. The value of the mass block is fixed, while the length of the steel strand is adjustable. By changing the length of the steel strand, various resonant frequencies can be realized. The mass of the tuned mass damper in the experimental test was 600 g. The whole weight of the model cable was about 10 kg. The first natural frequency of the cable was 4.0 Hz. Thus, the mass ratio in the experimental test was about 6%.

Table 1. The parameters of the steel strand used for the TMD part.

Material Parameters	Values
Diameter of steel wire	1.80 mm
Number of steel wire	19
Diameter of steel strand	9.0 mm
Elastic modulus	$2\times {10}^{11}{\mathrm{N}/\mathrm{m}}^2$
Mass per unit length	0.38 kg/m

lists other parameters of the steel strand.

To obtain cable responses, a measurement system was established. For each cable, two B&K accelerometers were installed at a one-third span near the left end to collect the vibration response of the cross-wind and along-wind directions, respectively. Similarly, two accelerometers were located at a one-fifth span close to the right end. The tension force was measured by the load cell, which was connected in series with the cables at the left end. The wind velocity was obtained using a Pitot tube.

In experimental tests, the wind loads were adopted as follows: for uncontrolled cases, the wind speed increased from zero gradually until the first two modes of cables were excited. For controlled cases, the wind protocol was the same as in uncontrolled cases.

4.2. Experimental Results

Using flexible model cables, a series of wind tunnel tests were conducted. In total, five conditions were considered for experimental investigation. They were the uncontrolled case, where only the rigid restraint was installed at L/2; a case where both the rigid restraint and TMD were installed at L/2; a case where only the rigid restraint was installed at L/4; and a case where both the rigid restraint and TMD were installed at L/4. In Figure 12, the horizontal axis represents the reduced velocity $U/fD$, in which $U$ is the wind velocity, $f$ is the natural frequency of the cable, and $D$ is the diameter of the cable. The vertical axis represents the dimensionless displacement response $A/D$, in which $A$ is the displacement response integrated from measured acceleration. For the uncontrolled case, the vibration response exhibits two peaks, which indicates the first two modes of the cables. Compared with the upstream cable, the downstream cable exhibits more serious vibration. This is because the downstream cable is in the wake of the upstream cable, thereby experiencing more significant aerodynamic effects. After installing only the rigid restraint, a slight decrease in the vibration response occurred. Compared to install location L/4, a better control performance was achieved for install location L/2. When the TMD part was further added, the vibration response of the cables was significantly reduced. The first two modal vibrations basically disappeared from the cables. Thus, the proposed rigid restraint with built-in TMD exhibits a remarkable control effect on wind-induced vibration for both upstream and downstream cables.

4.3. Discussion

As a preliminary study, the present experiment verifies the mitigation effect of the rigid restraint with built-in TMD for wind-induced vibration of cables. Only one device is considered, and two installation positions, namely, the quarter-span and mid-span, are considered. According to the current results, the first two modal vibrations of the cables are effectively suppressed.

For the proposed device, the tuning strategies for controlling cable vibration are described as follows: firstly, the dimension of the rigid restraint is determined based on the diameter and center spacing of the two cables. Then, the parameters of the TMD part, including the mass and stiffness, are designed to match the targeted cable mode. Finally, different installations and numbers are compared to obtain the final control scheme. During the design process, environmental factors such as temperature may also affect the efficiency of the proposed device. Temperature changes can directly affect the length and tension, which might alter the natural frequencies of cables. Then, this offset of natural frequencies leads to a reduction in the control effect achieved by the TMD part. The material fatigue caused by energy dissipation can damage the steel strand used in the TMD part, which can lead to a decrease in stiffness. As a result, frequency deviation of the TMD part may occur, thereby reducing long-term control effectiveness.

Several other issues should be further addressed. For the TMD part, there is only one effective frequency, which theoretically can only target one certain modal vibration of the cable. However, on real bridges, cables often experience multi-modal vibrations, making it difficult to determine the dominant mode in advance. Moreover, the vibration energy is absorbed and transferred but not dissipated by the TMD. The steel strand used in the TMD can dissipate the energy of a part. However, the dissipation mechanism of the steel strand is not stable and reliable. As a next step, it is necessary to introduce an explicit damping mechanism in the TMD. Furthermore, the influence of the damping needs to be investigated, thereby obtaining optimized parameters for various conditions.

In actual engineering, several control devices may be required. If the rigid restraint is located at the node of some modes, there will be no control effect. Thus, it is necessary to clarify the impact of the number of restraints and the installation position.

5. Conclusions

This study investigated the use of rigid restraints with a built-in tuned mass damper for mitigating cable vibrations. In this study, the configuration of the proposed device was presented, and a formulation was established for the system, followed by a discrete model. Then, numerical simulation and experimental tests were conducted to study the control effectiveness. The following conclusions were obtained.

• The developed discrete model can describe the dynamic behavior of the controlled cables when the rigid restraint with a built-in tuned mass damper is located at an arbitrary location.
• The proposed device, with proper parameters, can reduce cable responses remarkably under sinusoidal excitations.
• The rigid restraint with a built-in tuned mass damper has a significant control effect on the wind-induced vibration of both upstream and downstream cables.

Compared with other existing methods, the proposed device achieves its control effect in two parts. One is the rigid restraint, which interconnects two separate cables. The other is the TMD part, which can effectively absorb vibrating energy with proper tuning. Moreover, according to the specific configuration, the proposed device can be easily installed at the required locations. As a next step, several issues, including choosing the dominant frequency, damping, and number and locations of the proposed device, can be further investigated in the near future.