Viscous Damping Research Articles

H∞ optimum design and nonlinear seismic performance assessment of tuned-mass-damper-inerter (TMDI) supported by an auxiliary structure

Tuned Mass Damper Inerter (TMDI) and its simplified form, Tuned Inerter Damper (TID), are highly effective in mitigating seismic responses in buildings, particularly when rigidly connected to the ground. However, practical constraints often limit the feasibility of this ideal setup. This study introduces a novel configuration: the AS-type TMDI, which connects the TMDI to an auxiliary structure (AS) adjacent to the main building to simulate a ground connection. To design the AS-type TMDI, this research builds upon insights from studies on TMDIs in adjacent buildings and develops an analogous 3-DOF model representing the TMDI-linked adjacent-building system. This model incorporates two buildings of different heights, allowing the TMDI to be connected at any floor level. Utilizing fixed-point theory, the study derives an analytical solution for the optimal design of TMDI systems within this novel configuration. Additionally, a method is introduced to determine the optimal stiffness and design procedure for the AS itself. The performance of the proposed AS-type TMDI is evaluated through comprehensive nonlinear seismic analysis of a benchmark nine-story steel frame structure, accounting for material and geometric nonlinearities. Results demonstrate that the AS-type TMDI performs comparably to a virtually grounded TMDI in effectively reducing the seismic responses. Furthermore, this innovative configuration outperforms both a conventional retrofitting technique, where the building is rigidly connected to the AS, and a system where a viscous damper alone connects the main building to the AS. This highlights the AS-type TMDI's potential to significantly improve the building’s seismic performance.

A comparative study of low cyclic loading effects on plastic strain, nonlinear damping, and strength softening in fiber-reinforced recycled aggregate concrete

The incorporation of fibers significantly enhances the properties and structural performance of recycled aggregate concrete. This study introduces Fiber-Reinforced Recycled Aggregate Concrete (FRAC) by integrating steel fibers (SF) and polypropylene fibers (PPF) into the recycled aggregate concrete (RAC) matrix. A comprehensive series of cyclic loading tests, including reload and unload sequences, were conducted to meticulously analyze the deterioration of strength and energy dissipation capabilities of SF/PPF-reinforced RAC. The research examines the effects of fiber characteristics, volume ratios, lengths, and diameters on strength degradation after the peak stress along the stress-strain curve. Degradation models based on plastic strain were proposed for both SF-reinforced RAC and PPF-reinforced RAC. The study further investigates the fluctuation of strain energy in FRAC under low cyclic loading, revealing the correlation between plastic strain energy and equivalent viscous damping ratio. A nonlinear viscous damping model that accounts for the influence of the reinforcing index (RI) is proposed. This model effectively predicts the degradation of strength, the evolution of plastic strain, and the dynamics of nonlinear viscous damping in composite materials with varying fiber content. Ultimately, this research provides essential technical insights for the practical engineering application of RAC.

Inductive mode selective damping of structural vibrations

AbstractStructural vibrations pose significant challenges in various engineering applications and require effective damping strategies to improve the overall performance and longevity of systems. While some oscillations can significantly reduce the longevity of systems, others might contribute positively to the overall system performance. Thus, this contribution presents a novel method of an inductive damping concept, that targets only specific vibrational modes based on their mode shapes. By analyzing the used inductive damping element in the context of a single degree of freedom oscillator, it is shown, that it functions as an ideal viscous damper only if the dynamics of the electric part of the system are neglected. The concept of mode selective damping is presented, using two of these damping elements in an oscillator chain with three masses. By altering the wiring configurations connecting the two damping elements in the electrical domain, the damping behavior can be manipulated to selectively damp only specific modes according to their mode shapes.

A case study comparing seismic retrofitting techniques for a historically significant masonry building’s minaret

Historical masonry structures are extremely susceptible to earthquakes due to their characteristic features. Seismic performance and corresponding damage patterns vary between these buildings. Even though the main structure was undamaged, many minarets suffered damage or collapsed due to the transmission of large forces from the main mass to the minaret and the abrupt changes in cross-section due to the geometry of the minaret. This study uses an ancient masonry mosque as a case study, whose minaret and main building are constructed as a single structure. The mosque’s minaret under seismic excitation is the focus of this study. The adopted model is called Alaeddin Bey Mosque in Muş, Türkiye. The seismic performance assessment of the minaret, considering various retrofitting options, is mainly based on four critical parameters: base shear, acceleration, displacement, and maximum tensile forces in all three dimensions. The analyzed retrofitting methods include base isolation located in the basement of the mosque, viscous dampers placed only in the upper part of the minaret, Carbon Fiber-Reinforced Polymer fabric fitted to only the minaret, and steel plates applied to only the minaret. Representative structural models of the mosque have been modelled with SAP2000 software. The main novelty of this study is the use of viscous dampers in the minaret. It is the first time a design methodology has been introduced for viscous damper applications in minarets. This methodology aims to prevent local damage to the minaret due to the forces generated by the dampers, while also considering the constraints of limited internal space within the minaret. The finding of this study shows that viscous damper application yields significantly better results compared to the application of Carbon Fiber-Reinforced Polymer fabric and steel plates. However, although base isolation reduces the tensile stress values throughout the entire mosque to levels well below the material’s strength, viscous damper application in the minaret significantly reduces tensile stresses only in the minaret. As a result, viscous dampers are recommended for damage reduction in the minaret only. Otherwise, base isolation should be considered for reducing stress values throughout the entire mosque including the minaret. This study contributes towards the development of new seismic retrofitting methods for historic masonry buildings ‘minaret.

Fluid viscous dampers in tackle-damper configuration: An experimental study

This study presents dynamic test results on a recently developed damper placement known as the tackle-damper configuration. This innovative setup integrates a viscous damper alongside two inclined tackles, each comprising multiple turns of wire ropes threaded between upper and lower sheaves. This configuration has the ability to amplify damper deformations and forces based on the mechanical advantage of the tackle. Four specimens of tackle-damper configurations are tested under different dynamic harmonic and seismic loadings. Obtained results indicated that tackle-damper configuration changes pure viscous behaviour of the fluid viscous damper to an amplified viscoelastic behaviour. It is verified that energy dissipation in the tackle-damper configuration arises from two sources, i.e. fluid viscous damper and capstan friction between the wire ropes and the sheaves. Ideally, deformation and force amplification factors in the tackle-damper configuration should depend solely on the number of wire ropes and the inclination angle of the tackle. However, it has been found that true force amplification factor is much larger than the ideal value due to the capstan friction. Moreover, true deformation amplification factor can also be much smaller than the ideal value. This is attributed to the elastic stretching of individual wire ropes within the tackles. Therefore, both friction and wire rope elongation substantially affect the true amplification capabilities of the tackle-damper configuration.

Experimental study on fatigue damage mitigation in welded beam-to-column connections with passive vibration control

To investigate the effectiveness of a viscous fluid damper (VFD) in mitigating fatigue damage in welded beam-to-column connections, six specimens were manufactured with the full penetration groove weld process. These specimens were divided into three groups: S1, S2, and S3, each containing two specimens. One specimen in each group was equipped with VFD, while the other was not. All specimens underwent the same elastic cyclic loading stage, but a different plastic cyclic loading stage for each group. The study results indicate that the presence of VFD can significantly enhance the load-bearing capacity of the welded beam-to-column connection by 10%–15% when subjected to the same displacement amplitude at the beam end. The primary cause of failure in the welded beam-to-column connection is the fatigue damage of the weld toe on the beam flange, which requires more attention during the welding process. Additionally, the VFD can effectively reduce stress concentration at the weld seam area, leading to a maximum of 50% decrease in the maximum stress near the weld toe of the beam flange center, as observed in our study.

Effects of air damping on quality factors of different probes in tapping mode atomic force microscopy

Abstract The AFM probe in tapping mode is a continuous process of energy dissipation, from moving away from to intermittent contact with the sample surfaces. At present, studies regarding the energy dissipation mechanism of this continuous process have only been reported sporadically, and there are no systematic explanations or experimental verifications of the energy dissipation mechanism in each stage of the continuous process. The quality factors can be used to characterize the energy dissipation in TM-AFM systems. In this study, the vibration model of the microcantilever beam was established, coupling the vibration and damping effects of the microcantilever beam. The quality factor of the vibrating microcantilever beam under damping was derived, and the air viscous damping when the probe is away from the sample and the air squeeze film damping when the probe is close to the sample were calculated. In addition, the mechanism of the damping effects of different shapes of probes at different tip–sample distances was analyzed. The accuracy of the theoretical simplified model was verified using both experimental and simulation methods. A clearer understanding of the kinetic characteristics and damping mechanism of the TM-AFM was achieved by examining the air damping dissipation mechanism of AFM probes in the tapping mode, which was very important for improving both the quality factor and the imaging quality of the TM-AFM system. This study’s research findings also provided theoretical references and experimental methods for the future study of the energy dissipation mechanism of micro-nano-electromechanical systems.

Radial stiffness and damping of mountain bike tires subject to impact determined using the coefficient of restitution

Mountain bike tires are designed to encounter jumps, bumps, drops, and impacts. Despite their contribution to the vehicle suspension system, little has been published on their in-plane behaviour. In this work, a drop sled test bench is utilised to measure non-rolling dynamic radial stiffness and damping of numerous mountain bike tires across sizes, constructions, and usage categories. Each tire is treated as a classical, lumped Kelvin–Voigt model with a linear spring and viscous damper arranged in parallel. Identification of the system parameters is accomplished treating the tire as a bouncing ball. The coefficient of restitution from pre- and post-impact velocities is used to determine the dynamic stiffness and damping across two regimes of ‘impact’ and ‘in-contact’. Advantages of this approach over the classical logarithmic decrement are discussed. Measured first-bounce tire impact footprints exhibit a large ‘stadium’ shape compared to their elliptical static counterparts. A simple one degree-of-freedom, mass-spring-damper model with two pairs of dynamic stiffness and damping values is shown to faithfully reproduce the loaded, tire-only response for two of the inflation pressures tested. Measured dynamic radial stiffness and damping values are useful to compare tire behaviour and support future multibody simulation of the tire’s contribution to the suspension subsystem.

Evaluation of highly sensitive vibration states of nanomechanical resonators in liquid using a convolutional neural network

Nanomechanical resonators can detect various small physical quantities with high sensitivity using changes in resonant properties. However, viscous damping in liquids significantly reduces the measurement sensitivity. This study proposes convolutional neural network (CNN) vibration spectrum analysis to evaluate the highly sensitive vibration states of nanomechanical resonators, which are useful for in-liquid measurements. This research was carried out through the measurement of acetone concentration. First, we compared the concentration classification ability between the proposed and conventional methods and determined that the proposed method of analyzing vibration spectral changes using the CNN model can provide higher measurement sensitivity than the conventional measurement method of observing resonance properties changes and comparing the values for each measurement condition. This result shows that CNN-based spectral analysis is effective for the vibration spectra of in-liquid measurements. Next, gradient-weighted class activation mapping (Grad-CAM) was applied to verify which frequency bands are important for concentration classification in CNN model decision-making. The vibration states in these frequency bands were analyzed in terms of oscillation modes. This analysis revealed significant oscillation modes of the nanomechanical resonator in the liquid environment. Notably, in addition to the resonance states utilized in the conventional method, several other oscillation modes were found to be significant for measurements. This finding suggests that these oscillation modes may be highly sensitive for measurements in liquid environments. Among these oscillation modes, the mode with very small amplitude is highly promising for achieving unprecedented levels of sensitivity in sensing technologies.

A Study on the Amplification Effect and Optimum Control of the Intermediate Column–Lever Negative Stiffness Viscous Damper

A Study on the Amplification Effect and Optimum Control of the Intermediate Column–Lever Negative Stiffness Viscous Damper

Experimental study on seismic performance of replaceable shear links with low-yield-point steel

Based on the design concept of controllable seismic damage during earthquakes and rapid functional recovery after earthquakes, a replaceable shear link with low-yield-point steel was proposed. Quasi-static cyclic loading tests and numerical analysis were conducted on six specimens. The failure mode, seismic performance, energy dissipation behavior, and shear distribution of each specimen were carefully considered to investigate the effects of web steel, web stiffening configuration and stiffener-plate stiffness ratio. The results showed that the stiffened specimens showed full-sectional shear yielding behavior, while the non-stiffened specimens exhibited coupling behavior of web yielding and buckling. Specimens with reasonable design parameters had excellent ductility and energy dissipation capacity, with the plastic shear angle exceeding 0.157 rad and the maximum equivalent viscous damping coefficients greater than 0.51 (80% of the theoretical maximum value). Owing to the cyclic strengthening of the low-yield-point steel web and the contributions of stiffeners and flanges, the specimens showed obvious overstrength behavior, with the overstrength factor ranging from 1.99 to 2.39. The cross-stiffening configuration was conducive to enhancing the seismic performance of shear links. Smaller web width-to-thickness ratios were recommended to match with smaller stiffener-plate stiffness ratios. Extended cross stiffeners and diagonal stiffeners resulted in earlier cracking of the web welds, which adversely affected the ductility of the specimens.

Explainable AI model for predicting equivalent viscous damping in dual frame–wall resilient system

A prominent challenge in applying the direct displacement-based design (DDBD) method to the proposed dual frame–wall lateral force-resisting system lies in determining the equivalent viscous damping ratio (EVDR). However, the strong nonlinearity and complexity behind the equivalent procedure lead to limited choice, mostly trial and error based on experience, to explain and predict the EVDR in the context of traditional research. This study employs the XGBoost method to unravel intricate relationships of EVDR using over 5 million data points from nonlinear time-history (NLTH) analyses, encompassing various parameters including the fundamental period, ductility, subsystem stiffness ratios, post-yielding stiffness ratios of the subsystems and ground motion types. SHapley Additive exPlanations (SHAP) values consistently identify critical features relevant to the equivalent procedure. Comprehensive feature ablation tests further illuminate the robustness and susceptibility of each model. Additionally, the incorporation of Local Interpretable Model-agnostic Explanations (LIME) for local interpretability provides insights into the intricate decision-making mechanisms inherent in each model’s predictions. Both the predicting results from machine learning (ML) and traditional method are also compared. Findings highlight the relative importance of features for EVDR and present a refined prediction model. It underscores the pivotal role of model interpretability in reinforcing confidence in complex models and advocates for leveraging ML techniques to enhance the effectiveness and efficiency of the DDBD method in structural design.

Enhanced Energy Dissipation of Tuned Negative Stiffness Viscous Mass Damper (TNSVMD) for the Base-Isolated Structure (BIS)

Negative stiffness dampers (NSDs) have garnered significant attention in the research of base-isolated structures (BIS). This study proposes a tuned negative stiffness viscous mass damper (TNSVMD), which includes a tuning spring in series with the NSD and an inerter, to enhance the seismic control effect of the BIS. An analytical model of a BIS with the TNSVMD is initially established, followed by an investigation into frequency response analysis. The analytical expressions for the stochastic responses of the BIS with TNSVMD are derived, and further exploration is conducted on the influences of the negative stiffness ratio, dashpot damping ratio, and inerter mass ratio on the stochastic responses of the BIS. The mechanisms responsible for the damping enhancement and energy dissipation enhancement of the BIS with the TNSVMD are elucidated through the derivation of the analytical solution for the effective damping ratio and dissipated energy of the BIS with the TNSVMD. Two principles are utilized to determine the optimal parameters of the TNSVMD. These principles involve minimizing the earthquake input energy ratio of the BIS and minimizing the stochastic displacement response of the BIS. Finally, the effectiveness of the TNSVMD is validated through a design example using 100 ground motions. The results indicate that the displacement and acceleration responses of the BIS can be simultaneously controlled due to the weakening of stiffness and enhancement of the effective damping ratio, leading to higher energy dissipation efficiency of the TNSVMD. The increase in the inertial mass ratio enhances the seismic mitigation effect and reduces the optimal negative stiffness ratio and optimal dashpot damping ratio. Therefore, it raises the possibility of implementing the TNSVMD. The inherent damping ratio of the BIS has a significant impact on the control efficiency of the TNSVMD. For BIS systems with relatively small inherent damping ratios, the TNSVMD can achieve better control effectiveness.

Second Order Dynamics Featuring Tikhonov Regularization and Time Scaling

In a Hilbert setting we aim to study a second order in time differential equation, combining viscous and Hessian-driven damping, containing a time scaling parameter function and a Tikhonov regularization term. The dynamical system is related to the problem of minimization of a nonsmooth convex function. In the formulation of the problem as well as in our analysis we use the Moreau envelope of the objective function and its gradient and heavily rely on their properties. We show that there is a setting where the newly introduced system preserves and even improves the well-known fast convergence properties of the function and Moreau envelope along the trajectories and also of the gradient of Moreau envelope due to the presence of time scaling. Moreover, in a different setting we prove strong convergence of the trajectories to the element of minimal norm from the set of all minimizers of the objective. The manuscript concludes with various numerical results.

Seismic Performance of Bridge Expansion Joints with and without Viscous Dampers during the 6 February 2023 Kahramanmaraş Earthquakes

Expansion joints render bridge structures highly vulnerable to damage during strong ground motions. Failures of expansion joints triggered by earthquakes not only jeopardize the post-earthquake serviceability of the bridge but also have a significant impact on the bridgeâs overall seismic performance. Despite extensive investigations and efforts to integrate these measures into design specifications aimed at mitigating the consequences of relative movements between adjacent bridge spans, major earthquakes have still revealed instances of damage related to expansion joints. On 6 February 2023, strong earthquake sequences occurred in KahramanmaraÅ, Turkey, with magnitudes of M7.7 and M7.6. The fault lines and epicenters of these shallow earthquakes were near the city and town centers and caused severe structural damage to buildings and infrastructures. There are approximately 1000 railway and highway bridges in the earthquake-affected region. Although both highway and railway bridges have generally performed well, some bridges experienced structural damage during the KahramanmaraÅ earthquakes. A large number of damage on the bridges is due to pounding and opening relative movements in expansion joints. This paper presents a comprehensive seismic evaluation of expansion joint failure mechanisms on bridges without viscous dampers during the 2023 KahramanmaraÅ earthquake sequences and an in-depth investigation into the seismic performance of bridge expansion joints equipped with viscous dampers and shock transmission unit devices are implemented utilizing the strong ground motion data collected throughout the earthquake sequences. It can be stated that the near-fault induced significant directivity and fling effects, resulting in notable velocity pulses and permanent tectonic deformations, and that these effects contributed to the failures of expansion joints, viscous damper devices, pot bearings, and shear keys.

Comprehensive evaluation of seismic fragility in base isolated buildings enhanced with supplemental dampers considering pounding phenomenon

Base isolation is a proven technology effective in minimizing earthquake-induced damage to both the structural and non-structural elements (especially freestanding contents) of buildings. Nevertheless, it has been shown that allowing the flexible isolation layer to endure substantial demand displacements might subject the isolated structure to failure. This vulnerability arises from potential superstructure yielding due to increased deformations and reduced stiffness, such as those occurring through pounding against a moat wall during intense earthquakes. One alternative to controlling these large deformations is to use supplementary dampers at the isolation plane of the building, creating a hybrid base isolation system (HIS), which adds damping to the isolation system. In this research, the seismic performance of base isolated structures with lead-rubber bearing (LRB) are studied in conjunction with linear viscous dampers (LVD). Due to this aim, a base-isolated steel frame with 2 story X-type braces and a surrounding moat wall is modeled with two various configurations: LRB, and LRB with LVD. Fragility curves are developed for an ensemble of near-field (NF) pulse-like ground motions and for the maximum inter-story drift ratio (MIDR) as an engineering demand parameter (EDP). The incremental dynamic analysis (IDA) is conducted to calculate the damage probability of the structure by assuming different threshold values for damage states. It was found that adding viscous dampers to the isolation system balanced the behavior of the structure under near field pulse-type records; resulting in reduced isolation displacement while its adverse effects are insignificant and can be ignored. Moreover, the median spectral acceleration, namely the spectral acceleration associated to a probability of exceedance (POE) equal to 50 %, at the specific limit state, in the hybrid base isolation system is higher compared to the conventional one.

Forced-Vibration Characteristics of Bowtie-Shaped Honeycomb Composite Sandwich Panel with Viscoelastic Damping Layer.

The incorporation of viscoelastic layers in laminates can markedly enhance the damped dynamic characteristics. This study focuses on integrating viscoelastic layers into the composite facesheet of the bowtie-shaped honeycomb core composite sandwich panel (BHC-CSP). The homogenization of the damped BHC-CSP is performed by employing the variational asymptotic method. Based on the generalized total energy equation, the energy functional of the representative unit cell of the damped BHC-CSP is asymptotically analyzed. The warping function, derived following the principle of minimum potential energy, provides a basis for obtaining the corresponding Euler-Lagrange equation to ascertain the equivalent elastic properties of the damped BHC-CSP. Utilizing the developed two-dimensional equivalent model, the free-vibration characteristics of the damped BHC-CSP are examined across diverse boundary conditions while delving into the impact of an external viscous damping layer on the natural frequency of the damped BHC-CSP. The results reveal that intensified boundary constraints effectively diminish the effective vibration region of the damped BHC-CSP, thereby enhancing its overall stability. The introduction of a PMI foam layer proves effective in adjusting the stiffness and mass distribution of the damped BHC-CSP. Resonance characteristics are explored through frequency and time-domain analyses, highlighting the pivotal roles of the excitation position and receiver point in influencing the displacement and velocity responses. Although the stiffness is improved by incorporating a PMI foam layer, its effect on the damping performance of the damped BHC-CSP is minimal when compared to the T-SW308 foam layer.

Hydrodynamics of fluid resonance in a narrow gap between two boxes with different breadths

Violent fluid oscillations occur inside narrow gaps between closely side-by-side structures, threating the safety of the operations. In the current study, fluid resonance in the gap between two side-by-side different boxes is investigated by a two-dimensional viscous-flow numerical wave tank, which can simultaneously predict the viscous damping and free-surface nonlinearity. The accuracy of the established numerical model is fore validated by the published experimental data. The upstream box keeps fixed and has six varied breadths, the downstream box heaves freely under wave actions and remains a constant breadth. The current work aims to study the effects of the breadth ratio and heave motion of the lee-side box on the hydrodynamics of gap resonance. The wave amplification in the gap, the heave motions of the lee-side box, the reflection, transmission, and energy loss coefficients, and the response time and damping time are concerned. Results show that the gap resonance frequency is insensitive to the variation of the breadth ratio. As the breadth ratio increases from 0.4 to 1.4. The maximum wave height amplification in the gap reduces more than 25%. The maximum response time of the fluid resonance occurs when two boxes have equal breadth.

Seismic fragility estimation through real-time hybrid simulation and surrogate-based multi-fidelity Monte Carlo predictor

The intricate dynamic responses of structures, particularly when confronted with uncertainty, poses significant challenges in constructing reliable analytical models, thus resulting in unreliable fragility assessment. This study presents an approach that integrates data-driven techniques with real-time hybrid simulation (RTHS) to achieve reliable fragility analysis. RTHS data and imprecise surrogate models are fused using a multi-fidelity Monte Carlo predictor (MFMC), yielding unbiased estimations for parameters of fragility curve. The MFMC predictor can be considered as an optimized forecasting tool that harnesses the computational efficiency of low-fidelity simulations and the precision of high-fidelity experiments. RTHS is utilized for obtaining high-fidelity experimental data, part of which is leveraged to establish low-fidelity surrogate models, ultimately enabling multi-fidelity analysis through direct data-to-data analysis. A two-story steel moment-resisting frame equipped with self-centering viscous dampers is used as a proof-of-concept to demonstrate the proposed method. Structural and ground motion uncertainties are incorporated to reflect practical engineering environment. The low-fidelity Kriging models were built using portion of high-fidelity RTHS data. Simulation and experimental investigations were conducted to evaluate the effectiveness and efficiency of the proposed approach. The results suggest that the proposed method provides a promising tool for accurately evaluating seismic fragility, particularly when simulation-based methods are not feasible.

A frequency-domain hydroelastic approach for modeling and stochastic dynamic analysis of pontoon-type floating bridges

In recent years floating bridges have been extensively studied owing to their potential application for crossing wide and deep fjords. An efficient approach for analyzing stochastic hydroelastic responses of floating bridges is valuable for fast assessment of various load effects for preliminary design. This study develops a frequency-domain approach for hydroelastic analysis of pontoon-type floating bridges subjected to wave loads. The frequency-dependent motion equation of the multi-pontoons is established based on the potential flow theory and linearized Morison’s equation for viscous damping. The structural dynamics of the elastic bridge girder are represented utilizing a beam model with excitations transferred from each pontoon. The proposed method is applied to an end-anchored curved floating bridge, with highlights on the verification against a validated time-domain simulation tool. Convincing agreements are found in terms of the eigenvalues, wave excitation loads, and response statistics. The method shows high fidelity with an accurate capture of the linear wave-frequency responses as well as the low-frequency resonances. A further discussion demonstrates that the wave viscous damping restrains low-frequency resonant responses, whereas the hydrodynamic interactions of pontoons induce more resonant peaks in the wave frequency.