Match of negative stiffness and viscous damping in a passive negative stiffness damper (NSD) is studied for the vibration control of stay cables in this paper. At first, a discrete model of the stay cable with an NSD attached perpendicularly near the support is established. Under sinusoidal excitations, forced responses of the system are derived theoretically, which results in an asymptotic form for the additional modal damping ratios. Then, experimental results are presented to verify the discrete model and the corresponding theoretical derivations. Subsequently, numerical analysis is performed further to show the optimal match of negative stiffness and viscous damping, which is a function of the attachment location. The energy dissipated by the NSD and the cable energy are analyzed, thereby demonstrating the change trend of the additional modal damping ratios. Moreover, the energy distribution along the cable is investigated to reveal the effect of the negative stiffness and viscous damping. This study demonstrates the control mechanism of negative stiffness and viscous damping in the passive damper and is of practical significance for designing the optimal match of the damper parameters for cable vibration control.

As critical components of cable-stayed bridges, stay cables are usually extremely flexible with low inherent damping, which makes them prone to various unwanted vibrations induced by kinds of external disturbances. For example, the deck motion might interact with the cables to result in nonlinear resonance [

To achieve such an aim, several countermeasures have been proposed, including the aerodynamic methods [

The negative stiffness provided by semiactive dampers depends on the real-time feedback and power supply, which makes the entire system complex and unreliable. To draw these drawbacks, passive NSDs were suggested to replace the semiactive ones. For example, prepressed springs were incorporated to produce negative stiffness behavior [

In the above studies, much attention was paid to the design and realization of passive NSDs. The control mechanism and the optimal match of negative stiffness and viscous damping in the passive damper are seldom mentioned. This paper investigates the optimal match of the negative stiffness and viscous damping, followed by revelation of their control mechanism for stay cables from the perspective of energy. Initially, a discrete model with infinite degrees is established to describe the stay cable with an NSD located close to the support. Steady-state forced responses of the system under harmonic loads are derived, and additional modal damping ratios are presented approximately. Then, experimental tests are presented to compare with the above derivations. After verifying the model, numerical simulation is conducted further to study the relationship between negative stiffness and viscous damping in a passive damper for the optimal control of stay cables. The energy dissipated by the passive NSD and the cable energy, which show the changing law of additional modal damping, are analyzed numerically. Furthermore, the spatial distribution of vibrational energy along the cable is formulated to reveal the effect of negative stiffness and viscous damping. Finally, conclusions from the current study are addressed.

A discrete model with

Discrete model of a stay cable with a passive NSD attached in the transverse direction.

For small deflections, three adjacent nodes

Force and motion diagram for the

With the damping force, the equation of motion for the node

Similarly, at the node

The nodes

Rewriting in the vector form, the modal shape for the

Sinusoidal excitation is assumed to act on the node

Then, the motion equation for the system is rewritten in the matrix form as

After reaching steady-state vibration, the system response has the following form:

Substitution of equation (

Hence, the response of each node in the discrete model can be obtained. In particular, the displacement of the node

Using the displacement on the node

Meanwhile, the potential energy of the stay cable in the

Then, the

Setting the derivative of

To suppress the cable vibration, Zhou and Li [

Making the dissipation energy per cycle the same, an equivalent viscous damping is adopted to replace the Coulomb friction part of the NSD when the cable-damper system vibrates in the

Figure

Equivalent hysteretic loops for the experimental NSD.

The length of the scaled stay cable used in the experiment [

For the discrete model, the parameters, such as the node mass

(a) Frequency ratios for the first three modes; (b) comparison of the natural frequencies with

The excitation force in the experiment is used to calculate the theoretical responses of the discrete model. The excitation position is

Displacements of the cable under the first modal vibration with the NSD: (a) damper location; (b) midspan.

Specially, the relationship of displacement amplitude to damping at the damper’s location is shown in Figure

Relationship of displacement amplitude to damping at the damper’s location.

The additional modal damping ratios provided by the NSD are demonstrated in Figure

Additional damping ratios provided by the damper: (a)

The absolute increment of the additional damping ratio reaches its maximum near the region of optimal damping values. To be different, the curves of the relative increment of the additional damping ratio and damping are plotted in Figure

Relative increment of the additional damping ratio with negative stiffness

The discrete model of the stay cable with an NSD has been deduced theoretically and validated experimentally in the former sections. In this section, numerical analysis will be further performed to reveal the optimal match of the negative stiffness and viscous damping based on the aforementioned derivations of the discrete model. Moreover, the influence and mechanism of the negative stiffness and viscous damping will be demonstrated in the perspective of energy.

The effect of the excitation position on the optimal negative stiffness for the cable is analyzed firstly. As shown in Figure

Change trend of the optimal negative stiffness to the excitation position.

Being different from the excitation position, the damper location mainly determines the value of the optimal negative stiffness. Figure

Change trend of the optimal negative stiffness to the damper location.

The relationship between the optimal damping and the negative stiffness is further described in Figure

Optimal damping corresponding to the negative stiffness.

The phase difference, as depicted in equation (

Phase difference at the damper location.

From the initial state, the excitation force maintains inputting energy until the cable-NSD system reaches its steady-state vibration. Under the steady state, the cable’s mechanical energy stays constant, while the excitation energy is always dissipated by the damper. Obviously, the cable mechanical energy or the energy dissipated by the NSD depends on the parameters of the damping and stiffness. The change tendencies of the first modal vibration are shown in Figure

Change trend of the cable mechanical energy or the energy dissipated by the NSD to the damping with different levels of stiffness: (a)

Figure

Change trend of the cable mechanical energy or the energy dissipated by the NSD to the stiffness with different levels of damping: (a)

The results of other modes are similar to those of the first mode, so the curves of other modes are not drawn here. Under the steady-state forced vibration, the cable mechanical energy means the vibration intensity of the cable. Only the energy dissipated by the NSD is not enough to evaluate the control performance. The cable mechanical energy has a minimum value of zero under the condition of optimal negative stiffness and zero damping. This zero mechanical energy means that the cable has the weakest vibration. In other words, the parameters for best control performance are the combination of the optimal negative stiffness and zero damping.

Figure

Additional modal damping ratio for the stay cable with various levels of stiffness.

Furthermore, Figure

Additional modal damping ratio for the stay cable with various levels of damping.

This section will analyze the energy distribution along the cable when the NSD is attached for vibration control.

The energy ratio of each node in the discrete model is demonstrated in Figure

Distribution of the cable mechanical energy under the first modal vibration: (a)

Under the second modal vibration, the energy ratio at the damper location is increased by the negative stiffness, as shown in Figure

Distribution of the cable mechanical energy under the second modal vibration: (a)

The match of negative stiffness and viscous damping for cable vibration mitigation is investigated in this paper. A discrete model for the stay cable with an NSD was firstly established, and the system dynamic characteristics, including the forced steady-state vibration and additional modal damping ratio, were performed theoretically. After that, experimental results were utilized to verify the discrete model and the corresponding derivations. Then, numerical analysis was carried out further to reveal the effect of negative stiffness and viscous damping in the cable vibration mitigation. Moreover, the energy dissipated by the NSD and the energy distribution along the cable were evaluated. Based on the results of this study, the following conclusions could be drawn.

A discrete model, which can effectively describe the system dynamic behavior, was established for the stay cable with an NSD attached near the support. The forced steady-state vibration response and the additional modal damping ratio based on this model agree with the experimental results well.

The cable energy and the energy dissipated by the NSD depend on the negative stiffness and viscous damping. With the optimal negative stiffness and zero damping, the cable mechanical energy declines to the weakest vibration condition.

In particular, the match of negative stiffness and viscous damping also has a significant impact on the energy distribution along the cable. More precisely, the negative stiffness concentrates the energy to the damper location. For the optimal negative stiffness and zero damping, the energy ratio at the damper location approaches its peak value.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This study was financially supported by Grants 51808175 and 51808172 from the National Natural Science Foundation of China and Projects 2018M640297 and 2018M641833 funded by the China Postdoctoral Science Foundation.