# A New Method to Calculate Additional Damping Ratio considering the Effect of Excitation Frequency

Authors: Weizhi Xu; Dongsheng Du; Shuguang Wang; Weiwei Li
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The additional damping ratio (ADR) is an important indicator for evaluating the damping effect of structures with energy-dissipation devices. Most existing methods for determining the ADR require an analysis of the structural dynamic response and complex iterative calculations. An innovative simplified calculation method for determining the ADR of a structure supplemented by nonlinear viscous dampers is proposed. This method does not require the dynamic response of the structure to be calculated and only requires the structural characteristics, excitation frequency, and damper parameters. In this study, several typical calculation methods for the ADR were analysed. Then, a calculation formula for the ADR was derived with consideration of harmonic excitation under the condition where the excitation frequency is equal to the structural natural frequency, without calculation of the structural dynamic response or an iterative process. The effect of the excitation frequency on the calculated value of the ADR with different damping exponents was studied. Accordingly, the response spectrum average period (RSAP) was considered as the excitation period of ground motion to evaluate the excitation frequency, and a simplified calculation method for the ADR considering the effect of the excitation frequency characterised by the RSAP of the ground motion was established. Finally, the accuracy and effectiveness of the proposed method were verified by comparison with ADRs calculated using other methods.


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## Body

## 1. Introduction

In the past two decades, many major earthquakes have occurred in the world, which have caused many casualties and considerable property damage [1–3]. Both structural and nonstructural failures induced by seismic excitation are responsible for the losses. Therefore, it is vital to improve the seismic performance of building structures. The introduction of passive energy-dissipation technology to building structures provides an effective solution to increase the structural damping and mitigate the effect of seismic action, and the importance of damping has been widely recognised in the field of structural dynamics. Effective damping has three main components [4]: inherent damping; supplemental damping provided by the dampers; the hysteretic damping, which is related to the nonlinear behaviour of the structure.Microscopic internal friction, local plastic deformation, and plastic flow in a range of stresses within the apparent elastic limit account for a major proportion of the energy dissipation due to the inherent damping [5]. In numerical analysis, it is important to select an appropriate inherent damping model. A few damping models have been proposed and widely used in numerical analysis, e.g., viscous damping (VD), Rayleigh damping (RD), mass-proportional damping (MD), and stiffness-proportional damping (SD). The RD model, which consists of MD and SD, is widely used in commercial finite-element platforms because of its mathematical simplicity. Huang [5] reported that the desired energy-dissipation rate was achieved at only two target frequencies and that the mass-proportional term applies damping to rigid-body modes, which cannot dissipate energy in reality. To solve this problem, a versatile frequency-insensitive damping modelling method was developed, which can achieve approximately uniform energy dissipation in the specified frequency range and make the damping orthogonal to the rigid-body motion. In a few studies [6–8], the damping leakage phenomenon in structural systems was investigated, revealing that the damping leakage can result in a significant underestimation of the first-mode response and/or higher-mode responses. To mitigate the problems associated with the improper damping models in the base-isolated structure, Anajafi et al. [6] proposed a damping modelling approach that involves removing the mass-proportional component of superstructure damping from the global damping matrix.In addition to the inherent damping, the additional damping ratio (ADR), which consists of contributions from various energy-dissipation devices (EDDs), has been utilised in different types of structures to mitigate the structural responses induced by seismic excitation [9–16]. The viscous fluid damper (VFD)—a rate-dependent EDD—is favoured by engineers and increasingly used in engineering structures to reduce excessive vibrations because of the absence of supplemental stiffness and a high energy-dissipation rate in different deformation stages. Wu et al. [17] evaluated the seismic performance of a nine-story steel frame supplemented with five typical dampers under near-fault earthquakes. The results indicated that the VFD functioned better than other four types of dampers even under high pulse velocity amplitude (PVA). To quantitatively evaluate the damping effects for structures supplemented with VFDs, the ADR was proposed. Additionally, in China, the ADR is a key structural parameter for engineers in the design process. Firstly, the target ADR is determined; then, the aseismic design is conducted via the mode analysis response spectrum method, considering the total damping ratio, the sum of the inherent damping ratio, and the ADR. Subsequently, the damper parameters and the layout are designed to achieve the target ADR. The design process determines the importance of estimating the ADR of the structures equipped with EDDs.Recently, scholars have proposed various calculation methods for the ADR. Lee et al. [18] proposed an evaluation method for the equivalent damping ratio for systems configured with dampers. In that study, the Lyapunov function was first defined in the form of the modal energy; then, the Riccati matrix and damping ratio parameters were derived. Finally, a formula for the equivalent damping ratio was obtained by analogising the viscous modal damping ratio. Charney et al. [19] used the modal strain energy method, free vibration logarithmic decay method, and eigenvector method for complex eigenvalues to analyse the equivalent VD ratio of a single-story, single-bay frame. The results indicated that higher flexibility of the nodes corresponded to a larger phase difference between the deformation velocity and the relative horizontal velocity of the interlayer devices. However, this conclusion was only applicable to the single-layer, single-span structure. Occhiuzzi et al. [20] proposed a method for calculating the modal damping ratio using an equation for the dynamic system state. An analysis revealed that the first vibration mode significantly affected the damping ratio, whereas the effects of the higher-order vibration modes were negligible. The aforementioned calculation methods, which involve complex theoretical formulas and large amounts of calculations, are difficult for engineers to apply. Therefore, a simpler, direct method needs to be developed.Silvestri et al. [21–23] proposed a practical method for the aseismic design of structures configured with VFDs, in which the mechanical characteristics of the VFDs are easily determined according to the performance objectives of the structure. However, the calculation method adopted for the ADR still cannot be widely used in practical engineering. Diotallevi et al. [24] proposed a method for evaluating the ADR according to the damping index. This method has the advantage of realising direct evaluation of ADR without iteration but is limited to the evaluation of linear-elastic systems. Weng et al. [25] proposed a method for calculating the equivalent ADR of the viscous damper according to the code response spectrum. Landi et al. [26] proposed a simplified method for calculating the ADR according to the capability and demand spectra. Under the condition of confirmed yield acceleration and target displacement of the known structure, the corresponding ductility factor, equivalent total damping ratio, and significant period could be confirmed directly. Kudu et al. [27] compared the modal damping ratios determined with consideration of the measurement time, frequency scope, and sample rate. When the modal damping ratio changed significantly, the natural frequency of the structure did not. Ishimaru et al. [28] presented an approach for calculating the optimal VD ratio and accumulated ductility of a structure configured with a bilinear hysteretic damper and a dynamic quality damper, which has a very high accuracy for calculating the accumulated ductility coefficient. A method for calculating the ADR based on the concept of effective modal damping energy dissipation was proposed by Weng et al. [29], which has a clear physical significance and fully considers the time-varying characteristics of the effective damping ratio. Its advantages include simple calculations and a high accuracy. Landi et al. [30] performed probabilistic seismic assessment of RC structures with and without a damper using a simplified SAC Federal Emergency Management Agency (SAC-FEMA) program. The results showed that the method was effective and could be used as a simplified alternative to nonlinear dynamic analyses for probabilistic assessment purposes. Love and Tait [31] proposed a simple method for predicting the effective damping of a linear structure with nonlinear EDDs. This method exploits the characteristic that the output mean energy of the structure remains unchanged under a wind load to make predictions. However, the error between the prediction results of this method and the calculation results based on time-history analysis was significant. He et al. [32] proposed an effective finite-element analysis method for calculating the modal damping ratio of complex materials and proved the effectiveness and accuracy of the method through a theoretical analysis and a comparison with experimental results.Most of the aforementioned methods for calculating the ADR require the calculation of the structural dynamic response and iteration, making them cumbersome and inconvenient to use directly in practical engineering. Additionally, the solutions of the ADR for supplemental nonlinear VFDs based on the energy principle rarely consider the effect of the excitation frequency on the ADR. Hence, it is necessary to establish a calculation method for the ADR that reflects the effect of the excitation frequency on the dynamic response of the structure with VFDs.The remainder of this paper is organised as follows. Section2 introduces two typical calculation methods for the ADR: the energy-ratio method (ERM) and the damping index method (DIM). Section 3 presents the proposed method for calculating the ADR under harmonic excitation considering the impact of the excitation frequency, which is theoretically derived. Then, the response spectrum average period (RSAP) is used to distinguish the frequency properties of different ground motions. Subsequently, a simplified calculation method for the ADR under ground motion is presented. In Section 4, the accuracy and effectiveness of the proposed calculation method for the ADR are confirmed through a case study. Conclusions and directions for improvement of the proposed method are presented in Section 5.


## 2. Existing Methods for Calculating ADR

### 2.1. ERM

Chopra [33] proposed a classical calculation equation for the ADR based on the principle of hysteresis energy equivalence:(1)ζsd=wD4πwSoωΩ,where wSo represents the maximum strain energy of the damping structure with a viscous damper under the expected displacement, wD represents the energy consumed by the viscous damper in one cycle with the expected deformation, and ω and Ω represent the structural natural frequency and the excitation frequency, respectively.The damping force is given as follows:(2)FD=cαsgnu˙u˙α,where cα is the damping coefficient corresponding to different velocity exponent (α) values and sgn (x) is a sign function.For a single-degree-of-freedom (SDOF) system subjected to a harmonic displacement excitation with the function ofu(t) = u0sin Ωt, the energy consumption of the nonlinear viscous damper is given as follows:(3)wD=∫FDdu=∫02π/Ωcαu˙1+αdt.Integration of equation (3) yields(4)wD=λcαu01+αΩα,where λ is a constant related to α.Whenα = 1, λ=π, and the following equation can be obtained:(5)wD=πc1u02Ω.Linearisation of the nonlinear viscous damper is performed according to the equal dissipating energy of a linear viscous damper, and the equivalent linear damping coefficientc1 can be determined using the following equation:(6)πc1u02Ω=λcαu01+αΩα.By dividing both sides of equation (6) by 2mω, the ADR of nonlinear viscous dampers can be calculated as follows:(7)ζsd=λπcα2mω1Ωuo1−α,where m represents the mass of the structure.This classical method is easy to understand. However, the structural dynamic response must be calculated to determine the maximum displacement, and the calculation process is complex, which is not convenient for practical engineering.


### 2.2. DIM

To overcome the shortcomings of the ERM, Landi [25] proposed a method for evaluating the ADR directly according to the damping index. For an SDOF system under harmonic excitation, the damping index ε is defined by the following relationship:(8)ε=λπcα2mω1−2αΩ1−αa0α−1,where a0 represents the peak acceleration of harmonic excitation.Under the resonance condition, i.e., Ω =ω, equations (7) and (8) can be rewritten as follows:(9)ζsd=λπcα2mω2−α1uo1−α,(10)ε=λπcα2m1ωαu¨g0α−1,respectively, where u¨g0 represents the peak acceleration of the seismic excitation.The relationship between ADRζsd and the damping index ε can be expressed as follows:(11)ζsd=εRdα−1,where Rd=ω2u0/u¨g0, reflecting the amplification of the dynamic response.The acceleration amplification factorRa can be computed via numerical analysis, and the damping index ε can be determined according to the structural and input characteristics. Then, ADR ζsd is calculated using equation (11). Figure 1 shows the ADR-frequency ratio curves with different damping indices (ε) and α = 0.50 under harmonic excitation, and Figure 2 shows the ADR spectra for different damping indices (ε) and α = 0.50 under the El Centro ground motion recorded during the Imperial Valley Earthquake on May 18, 1940. According to the curves in Figures 1 and 2, ADR ζsd is directly obtained when the damping index ε and frequency ratio (Ω/ω) or natural period (T) of the structure are determined. It appears that the ADR can be determined directly, without calculation of the structural response. Figures 1 and 2 were obtained via numerical analysis of the structure with additional dampers.Figure 1
Relationship between the ADR and the frequency ratio for different damping indices (ε) under harmonic excitation.Figure 2
ADR spectra for different values ofε under the excitation of the north-south component of El Centro ground motion (α = 0.50).


## 2.1. ERM

Chopra [33] proposed a classical calculation equation for the ADR based on the principle of hysteresis energy equivalence:(1)ζsd=wD4πwSoωΩ,where wSo represents the maximum strain energy of the damping structure with a viscous damper under the expected displacement, wD represents the energy consumed by the viscous damper in one cycle with the expected deformation, and ω and Ω represent the structural natural frequency and the excitation frequency, respectively.The damping force is given as follows:(2)FD=cαsgnu˙u˙α,where cα is the damping coefficient corresponding to different velocity exponent (α) values and sgn (x) is a sign function.For a single-degree-of-freedom (SDOF) system subjected to a harmonic displacement excitation with the function ofu(t) = u0sin Ωt, the energy consumption of the nonlinear viscous damper is given as follows:(3)wD=∫FDdu=∫02π/Ωcαu˙1+αdt.Integration of equation (3) yields(4)wD=λcαu01+αΩα,where λ is a constant related to α.Whenα = 1, λ=π, and the following equation can be obtained:(5)wD=πc1u02Ω.Linearisation of the nonlinear viscous damper is performed according to the equal dissipating energy of a linear viscous damper, and the equivalent linear damping coefficientc1 can be determined using the following equation:(6)πc1u02Ω=λcαu01+αΩα.By dividing both sides of equation (6) by 2mω, the ADR of nonlinear viscous dampers can be calculated as follows:(7)ζsd=λπcα2mω1Ωuo1−α,where m represents the mass of the structure.This classical method is easy to understand. However, the structural dynamic response must be calculated to determine the maximum displacement, and the calculation process is complex, which is not convenient for practical engineering.


## 2.2. DIM

To overcome the shortcomings of the ERM, Landi [25] proposed a method for evaluating the ADR directly according to the damping index. For an SDOF system under harmonic excitation, the damping index ε is defined by the following relationship:(8)ε=λπcα2mω1−2αΩ1−αa0α−1,where a0 represents the peak acceleration of harmonic excitation.Under the resonance condition, i.e., Ω =ω, equations (7) and (8) can be rewritten as follows:(9)ζsd=λπcα2mω2−α1uo1−α,(10)ε=λπcα2m1ωαu¨g0α−1,respectively, where u¨g0 represents the peak acceleration of the seismic excitation.The relationship between ADRζsd and the damping index ε can be expressed as follows:(11)ζsd=εRdα−1,where Rd=ω2u0/u¨g0, reflecting the amplification of the dynamic response.The acceleration amplification factorRa can be computed via numerical analysis, and the damping index ε can be determined according to the structural and input characteristics. Then, ADR ζsd is calculated using equation (11). Figure 1 shows the ADR-frequency ratio curves with different damping indices (ε) and α = 0.50 under harmonic excitation, and Figure 2 shows the ADR spectra for different damping indices (ε) and α = 0.50 under the El Centro ground motion recorded during the Imperial Valley Earthquake on May 18, 1940. According to the curves in Figures 1 and 2, ADR ζsd is directly obtained when the damping index ε and frequency ratio (Ω/ω) or natural period (T) of the structure are determined. It appears that the ADR can be determined directly, without calculation of the structural response. Figures 1 and 2 were obtained via numerical analysis of the structure with additional dampers.Figure 1
Relationship between the ADR and the frequency ratio for different damping indices (ε) under harmonic excitation.Figure 2
ADR spectra for different values ofε under the excitation of the north-south component of El Centro ground motion (α = 0.50).


## 3. New Simplified Calculation Method for ADR

To calculate the ADR of a structure equipped with nonlinear viscous dampers more conveniently, a simplified method is proposed on the basis of the foregoing methods. In the proposed method, the structural response need not be calculated; rather, only the spectral characteristics of ground motions must be considered.


### 3.1. ADR of Damping System under Harmonic Excitation

The dynamic equation of the SDOF system with a linear damper under harmonic excitation is(12)mu¨+c1u˙+ku=a0msinΩt,where c1 is the damping coefficient of the linear viscous damper when the structural damping is 0 and a0 represents the peak acceleration of harmonic excitation.The complete solution of equation (12) [33] is(13)ut=e−ξ1ωtAcosωDt+BsinωDt+CsinΩt+DcosΩt,where ωD=ω1−ζ12 represents the frequency of the structure considering damping, ω represents the natural frequency of the structure, and ζ1 represents the damping ratio of the linear viscous damper.Considering the initial conditionsu (0) = 0 and u˙(0) = 0, the following can be obtained:(14)Cζ1,Ωω=a0ω21−Ω/ω21−Ω/ω22+2ζ1Ω/ω22,Dζ1,Ωω=a0ω2−2ζΩ/ω1−Ω/ω22+2ζ1Ω/ω2.According to equation (13), the steady-state dynamic response us is calculated as follows:(15)us=us0sinΩt+φ,where us0=C2+D2 and tanφ=D/C.The steady-state response is taken as the approximate response of the structure, i.e.,(16)u0=us0.Considering the equal energy consumption of linear and nonlinear dampers and the identical peak dynamic responses of the SDOF for linear and nonlinear dampers, the ADRs of the two damping systems should be equal:(17)ζ1=ζsd.According to equations (7), (16), and (17), the ADR can be determined as follows:(18)ζsd=λπcα2mω1ΩC2+D21−α.The ADR can be easily determined using equation (18) without calculating the response of the damping structure but with iterative calculations. Comparisons of the ADRs calculated using equation (18) and the ERM are presented in Figures 3 and 4. As shown, when the frequency ratio Ω/ω was ≤1, the calculation results of equation (18) agreed well with the results obtained using the ERM; when the frequency ratio was >1, the calculation results of equation (18) were larger than those of the ERM owing to the impact of the transient-state solution on the structural response, indicating that the frequency ratio affected the difference between the ADRs calculated using equation (18) and the ERM.Figure 3
Comparison of the ADRs calculated using the two methods with different frequency ratios (α = 0.50).Figure 4
Comparison of the ADRs calculated using the two methods with different excitation amplitudes (Ω/ω = 1.50).Using equation (18), the ADR can be determined without calculating the dynamic response, but iterative calculations are required. Considering that the results obtained via the two calculation methods are similar, particularly when Ω = ω, further analysis is conducted for the resonance condition.When Ω =ω, equation (15) is simplified as follows:(19)us0=a02ω2ζ1.Substituting equation (19) into equation (6) and setting ζ1 = c1/2ωm yields(20)c1=λπ1a0m1−αcα1/α.Then, by dividing both sides of equation (6) by 2mω, the ADR under the resonance condition is obtained:(21)ζsd,res=12mωλπ1a0m1−αcα1/α.Figure5 presents the relationship curves of ζsd,res and a0 for different values of α and cα. As shown, the calculation results of the two methods agreed well, and the difference between them decreased with the increasing damping exponent. Additionally, the ADR increased with the damping coefficient, owing to the considerable energy dissipation.Figure 5
ADRs calculated using the two methods forcα = 1000, 2000, 3000, 4000, 5000, and 6000 kN s/m: (a) α = 0.30; (b) α = 0.40; (c) α = 0.50; (d) α = 0.60; (e) α = 0.70; (f) α = 0.90.
(a)(b)(c)(d)(e)(f)


### 3.2. Simplified Calculation of ADR considering Excitation Spectral Characteristics

Because of the difference between the ADRs calculated using equation (18) and the ERM depended on the frequency ratio, it was important to study the variation of the ADR with different frequency ratios. Figure 6 shows the curves for the relationship between ζsd and Ω/ω obtained via the ERM.Figure 6
Relationship between ADRζsd and the frequency ratio Ω/ω: (a) α = 0.30; (b) α = 0.50; (c) α = 0.70; (d) α = 0.90.
(a)(b)(c)(d)In Figure6, the ADRs on the left and right sides of the resonance point exhibit different trends, and the curve is approximately parabolic when Ω/ω < 1 and approximately linear when Ω/ω > 1. Therefore, the ζsd vs. Ω/ω curve can be defined as a piecewise function containing the structural resonance point (1, ζsd,res) as follows:(22)ζsd=a1Ωω−12+ζsd,res,Ωω<1,ζsd,res,Ωω=1,a2Ωω−1+ζsd,res,Ωω>1,where a1 and a2 are the coefficients that must be determined via fitting.Figures7 and 8 present the curves of a1 and a2 under different damping exponents (α). Numerical fitting revealed that(23)a1=β1eγ1α,where β1 = 109.6 and γ1 = –5.489, and(24)a2=β2eγ2α,where β2 = 10.02 and γ2 = –4.649.Figure 7
Fitting curve ofa1.Figure 8
Fitting curve ofa2.Equation (22) was used to determine the ADR after considering the excitation spectral characteristic without calculating the structural dynamic response. Figure 9 presents the relationship curve between ζsd and Ω/ω under different damping exponents (α). The calculation results of the ERM, equation (18), and equation (22) are compared. As shown, the calculation results of equation (18) were the largest among the three calculation methods when Ω/ω > 1; however, the calculation results of equation (22) and the ERM agreed well, with sufficient precision for practical engineering.Figure 9
ADR (ζsd) vs. frequency ratio (Ω/ω): (a) α = 0.30; (b) α = 0.50; (c) α = 0.70; (d) α = 0.90.
(a)(b)(c)(d)


### 3.3. Calculation of ADR of Damping Structure under Ground Motion Excitation

The spectral characteristics of ground motions affect the structural dynamic response, as well as the ADR. However, the foregoing simplified methods do not consider the impact of the ground vibration frequency on the ADR. Thus, a calculation method for the ADR that does not require calculation of the structural dynamic response but considers the spectral characteristics of the ground motions is derived.Considering the condition of structural resonance under harmonic excitation,u0 is similar to a0/2ω2ζ. If(25)uw=u¨g02ω2ζ1,where uw represents the assumed value of the peak displacement response, which is related to the peak ground acceleration (PGA). There is a relationship between u0 and uw. The ratio between them is defined as f1 = u0/uw and reflects the displacement amplification between the nonresonant and resonant responses of the structure.Then,u0 is determined as follows:(26)u0=f1⋅u¨g02ω2ζ1,where u¨g0 represents the seismic-excitation peak acceleration and ζ1 represents the ADR of the linear viscous damper.For a linear structural system subjected to a seismic excitation, the coefficientf1 is unrelated to the PGA but is related to ADR (ζ1). Therefore, f1 is designated as a unary polynomial of ζ1:(27)f1ζ1=b1ζ13+b2ζ12+b3ζ1+b4,where b1, b2, b3, and b4 are the coefficients of the polynomial, which can be determined via fitting. The ADR of the structure is typically <0.5, thus considering a cubic polynomial can ensure an appropriate accuracy.Considering the different natural cycle frequencies (ω) of the structure, a series of coordinate points (ζ1, f1) can be obtained through time-history analyses of the SDOF system with different ADRs (ζ1). Taking the El Centro seismic record of the Imperial Valley earthquake as an example, considering the different natural cycle frequencies of the structure, ζ1 and corresponding f1 values can be obtained, as shown in Figure 10. Table 1 presents the fitting coefficients for the unary polynomial of ζ1. The fitting coefficients with ω = 5.39 are used in the case study in Section 4.Figure 10
Relationship betweenf1 and ζ1 under El Centro seismic excitation.Table 1
Fitting coefficients of the polynomial for differentω values.
ω (rad/s)Fitting coefficientsb1b2b3b420.23–0.420.300.0140.34–0.860.800.015.390.40–0.890.840.0481.11–2.391.810.04101.76–3.702.660.07121.84–3.993.040.08141.19–3.133.020.02160.50–2.002.730.00180.14–1.282.440.0120–0.05–0.862.270.03By substituting equation (26) into equation (9), the ADR with Ω = ω is obtained:(28)ζsd,res=λπcα2mω2−α1f1u¨g0/2ω2ζ11−α.According to equation (17), equation (28) can be transformed into(29)ζsd,res=λπcα2mω2−α1f1u¨g0/2ω2ζsd,res1−α.Equation (22) is used to determine the ADR when Ω ≠ ω:(30)ζsd=a1Ωmω−12+ζsd,res,Ωmω<1,ζsd,res,Ωmω=1,a2Ωmω−1+ζsd,res,Ωmω>1,where Ωm represents the average frequency of the seismic excitation.The average periodTm of a seismic wave can be calculated as follows [29]:(31)Tm=∑iCi21/Fi∑iCi2,where Ci is the Fourier peak coefficient and Fi represents the discrete frequency of the fast Fourier transform.Then average frequency Ωm of the seismic input ground motion acceleration is calculated as follows:(32)Ωm=2πTm.Figures11 and 12 present the curves of ADR (ζsd) vs. the natural vibration period of the structure (T) and the PGA (u¨g0) with different damping coefficients (ca, in kN·s/m) for the El Centro ground motion obtained using the ERM and equations (29) and (30). Here, the damping exponent was α = 0.50.Figure 11
ADR spectra with different values ofca (kN·s/m) for the El Centro ground motion (α = 0.50).Figure 12
Relationship between the ADR and the seismic-excitation peak acceleration with different values ofca (kN·s/m) for the El Centro ground motion (α = 0.50).As shown in the figures, for Ω =ω, the calculation results of equation (29) and the ERM agreed well, and for Ω ≠  ω, the calculation results of equation (30) were larger than the calculation results of the ERM. The calculation results of equation (30) were more accurate than those of the equation (29) because this equation considered the impact of the ground motion frequency, and the change law before and after the resonance point was consistent with that under harmonic excitation.


## 3.1. ADR of Damping System under Harmonic Excitation

The dynamic equation of the SDOF system with a linear damper under harmonic excitation is(12)mu¨+c1u˙+ku=a0msinΩt,where c1 is the damping coefficient of the linear viscous damper when the structural damping is 0 and a0 represents the peak acceleration of harmonic excitation.The complete solution of equation (12) [33] is(13)ut=e−ξ1ωtAcosωDt+BsinωDt+CsinΩt+DcosΩt,where ωD=ω1−ζ12 represents the frequency of the structure considering damping, ω represents the natural frequency of the structure, and ζ1 represents the damping ratio of the linear viscous damper.Considering the initial conditionsu (0) = 0 and u˙(0) = 0, the following can be obtained:(14)Cζ1,Ωω=a0ω21−Ω/ω21−Ω/ω22+2ζ1Ω/ω22,Dζ1,Ωω=a0ω2−2ζΩ/ω1−Ω/ω22+2ζ1Ω/ω2.According to equation (13), the steady-state dynamic response us is calculated as follows:(15)us=us0sinΩt+φ,where us0=C2+D2 and tanφ=D/C.The steady-state response is taken as the approximate response of the structure, i.e.,(16)u0=us0.Considering the equal energy consumption of linear and nonlinear dampers and the identical peak dynamic responses of the SDOF for linear and nonlinear dampers, the ADRs of the two damping systems should be equal:(17)ζ1=ζsd.According to equations (7), (16), and (17), the ADR can be determined as follows:(18)ζsd=λπcα2mω1ΩC2+D21−α.The ADR can be easily determined using equation (18) without calculating the response of the damping structure but with iterative calculations. Comparisons of the ADRs calculated using equation (18) and the ERM are presented in Figures 3 and 4. As shown, when the frequency ratio Ω/ω was ≤1, the calculation results of equation (18) agreed well with the results obtained using the ERM; when the frequency ratio was >1, the calculation results of equation (18) were larger than those of the ERM owing to the impact of the transient-state solution on the structural response, indicating that the frequency ratio affected the difference between the ADRs calculated using equation (18) and the ERM.Figure 3
Comparison of the ADRs calculated using the two methods with different frequency ratios (α = 0.50).Figure 4
Comparison of the ADRs calculated using the two methods with different excitation amplitudes (Ω/ω = 1.50).Using equation (18), the ADR can be determined without calculating the dynamic response, but iterative calculations are required. Considering that the results obtained via the two calculation methods are similar, particularly when Ω = ω, further analysis is conducted for the resonance condition.When Ω =ω, equation (15) is simplified as follows:(19)us0=a02ω2ζ1.Substituting equation (19) into equation (6) and setting ζ1 = c1/2ωm yields(20)c1=λπ1a0m1−αcα1/α.Then, by dividing both sides of equation (6) by 2mω, the ADR under the resonance condition is obtained:(21)ζsd,res=12mωλπ1a0m1−αcα1/α.Figure5 presents the relationship curves of ζsd,res and a0 for different values of α and cα. As shown, the calculation results of the two methods agreed well, and the difference between them decreased with the increasing damping exponent. Additionally, the ADR increased with the damping coefficient, owing to the considerable energy dissipation.Figure 5
ADRs calculated using the two methods forcα = 1000, 2000, 3000, 4000, 5000, and 6000 kN s/m: (a) α = 0.30; (b) α = 0.40; (c) α = 0.50; (d) α = 0.60; (e) α = 0.70; (f) α = 0.90.
(a)(b)(c)(d)(e)(f)


## 3.2. Simplified Calculation of ADR considering Excitation Spectral Characteristics

Because of the difference between the ADRs calculated using equation (18) and the ERM depended on the frequency ratio, it was important to study the variation of the ADR with different frequency ratios. Figure 6 shows the curves for the relationship between ζsd and Ω/ω obtained via the ERM.Figure 6
Relationship between ADRζsd and the frequency ratio Ω/ω: (a) α = 0.30; (b) α = 0.50; (c) α = 0.70; (d) α = 0.90.
(a)(b)(c)(d)In Figure6, the ADRs on the left and right sides of the resonance point exhibit different trends, and the curve is approximately parabolic when Ω/ω < 1 and approximately linear when Ω/ω > 1. Therefore, the ζsd vs. Ω/ω curve can be defined as a piecewise function containing the structural resonance point (1, ζsd,res) as follows:(22)ζsd=a1Ωω−12+ζsd,res,Ωω<1,ζsd,res,Ωω=1,a2Ωω−1+ζsd,res,Ωω>1,where a1 and a2 are the coefficients that must be determined via fitting.Figures7 and 8 present the curves of a1 and a2 under different damping exponents (α). Numerical fitting revealed that(23)a1=β1eγ1α,where β1 = 109.6 and γ1 = –5.489, and(24)a2=β2eγ2α,where β2 = 10.02 and γ2 = –4.649.Figure 7
Fitting curve ofa1.Figure 8
Fitting curve ofa2.Equation (22) was used to determine the ADR after considering the excitation spectral characteristic without calculating the structural dynamic response. Figure 9 presents the relationship curve between ζsd and Ω/ω under different damping exponents (α). The calculation results of the ERM, equation (18), and equation (22) are compared. As shown, the calculation results of equation (18) were the largest among the three calculation methods when Ω/ω > 1; however, the calculation results of equation (22) and the ERM agreed well, with sufficient precision for practical engineering.Figure 9
ADR (ζsd) vs. frequency ratio (Ω/ω): (a) α = 0.30; (b) α = 0.50; (c) α = 0.70; (d) α = 0.90.
(a)(b)(c)(d)


## 3.3. Calculation of ADR of Damping Structure under Ground Motion Excitation

The spectral characteristics of ground motions affect the structural dynamic response, as well as the ADR. However, the foregoing simplified methods do not consider the impact of the ground vibration frequency on the ADR. Thus, a calculation method for the ADR that does not require calculation of the structural dynamic response but considers the spectral characteristics of the ground motions is derived.Considering the condition of structural resonance under harmonic excitation,u0 is similar to a0/2ω2ζ. If(25)uw=u¨g02ω2ζ1,where uw represents the assumed value of the peak displacement response, which is related to the peak ground acceleration (PGA). There is a relationship between u0 and uw. The ratio between them is defined as f1 = u0/uw and reflects the displacement amplification between the nonresonant and resonant responses of the structure.Then,u0 is determined as follows:(26)u0=f1⋅u¨g02ω2ζ1,where u¨g0 represents the seismic-excitation peak acceleration and ζ1 represents the ADR of the linear viscous damper.For a linear structural system subjected to a seismic excitation, the coefficientf1 is unrelated to the PGA but is related to ADR (ζ1). Therefore, f1 is designated as a unary polynomial of ζ1:(27)f1ζ1=b1ζ13+b2ζ12+b3ζ1+b4,where b1, b2, b3, and b4 are the coefficients of the polynomial, which can be determined via fitting. The ADR of the structure is typically <0.5, thus considering a cubic polynomial can ensure an appropriate accuracy.Considering the different natural cycle frequencies (ω) of the structure, a series of coordinate points (ζ1, f1) can be obtained through time-history analyses of the SDOF system with different ADRs (ζ1). Taking the El Centro seismic record of the Imperial Valley earthquake as an example, considering the different natural cycle frequencies of the structure, ζ1 and corresponding f1 values can be obtained, as shown in Figure 10. Table 1 presents the fitting coefficients for the unary polynomial of ζ1. The fitting coefficients with ω = 5.39 are used in the case study in Section 4.Figure 10
Relationship betweenf1 and ζ1 under El Centro seismic excitation.Table 1
Fitting coefficients of the polynomial for differentω values.
ω (rad/s)Fitting coefficientsb1b2b3b420.23–0.420.300.0140.34–0.860.800.015.390.40–0.890.840.0481.11–2.391.810.04101.76–3.702.660.07121.84–3.993.040.08141.19–3.133.020.02160.50–2.002.730.00180.14–1.282.440.0120–0.05–0.862.270.03By substituting equation (26) into equation (9), the ADR with Ω = ω is obtained:(28)ζsd,res=λπcα2mω2−α1f1u¨g0/2ω2ζ11−α.According to equation (17), equation (28) can be transformed into(29)ζsd,res=λπcα2mω2−α1f1u¨g0/2ω2ζsd,res1−α.Equation (22) is used to determine the ADR when Ω ≠ ω:(30)ζsd=a1Ωmω−12+ζsd,res,Ωmω<1,ζsd,res,Ωmω=1,a2Ωmω−1+ζsd,res,Ωmω>1,where Ωm represents the average frequency of the seismic excitation.The average periodTm of a seismic wave can be calculated as follows [29]:(31)Tm=∑iCi21/Fi∑iCi2,where Ci is the Fourier peak coefficient and Fi represents the discrete frequency of the fast Fourier transform.Then average frequency Ωm of the seismic input ground motion acceleration is calculated as follows:(32)Ωm=2πTm.Figures11 and 12 present the curves of ADR (ζsd) vs. the natural vibration period of the structure (T) and the PGA (u¨g0) with different damping coefficients (ca, in kN·s/m) for the El Centro ground motion obtained using the ERM and equations (29) and (30). Here, the damping exponent was α = 0.50.Figure 11
ADR spectra with different values ofca (kN·s/m) for the El Centro ground motion (α = 0.50).Figure 12
Relationship between the ADR and the seismic-excitation peak acceleration with different values ofca (kN·s/m) for the El Centro ground motion (α = 0.50).As shown in the figures, for Ω =ω, the calculation results of equation (29) and the ERM agreed well, and for Ω ≠  ω, the calculation results of equation (30) were larger than the calculation results of the ERM. The calculation results of equation (30) were more accurate than those of the equation (29) because this equation considered the impact of the ground motion frequency, and the change law before and after the resonance point was consistent with that under harmonic excitation.


## 4. Case Study

In 1997, Takewaki [34] presented an optimal vibration-reduction design for a six-story shear frame (Figure 13). The mass was identical for all the floors of the frame, i.e., m1 = m2 = … = m6 = 0.80 × 105 kg, and the lateral stiffness was evenly distributed, i.e., k1 = k2 = … = k6 = 4.00 × 107 N/m. The initial damping coefficient of each VFD was c1 = 1.50 × 106 (N s/m), and the inherent structural damping ratio was taken as ζ0 = 0.05.Figure 13
Six-story shearing frame model with VFDs.According to the foregoing conditions, it is easy to determine the natural periodT1 = 1.1656 s and the first vibration mode of the structure {ϕ}T = {0.2411, 0.4681, 0.6680, 0.8290, 0.9419, 1.0000}. According to the equivalent SDOF method [33], the equivalent mass Mr, equivalent stiffness Kr, and equivalent damping Cr can be obtained as follows:(33)Mr=ϕTMϕϕTM1ϕTMϕ,Cr=ϕTCϕϕTM1ϕTMϕ,Kr=ϕTKϕϕTM1ϕTMϕ,where M represents the mass matrix, K represents the lateral stiffness matrix, C represents the damping matrix corresponding to the linear VFDs, and {ϕ} represents the first vibration mode of the structure.According to equation (33), the equivalent mass is Mr = 3.32 × 105 kg, the equivalent rigidity is Kr = 9.64 × 106 N/m, and the equivalent damping is Cr = 6.08 × 105 (N s/m). Additionally, the equivalent period of the structure is T = 1.1656 s, which is consistent with that of the original frame.The information of 10 ground motions selected from the Pacific Earthquake Engineering Research Center (PEER), as well as the corresponding average frequency Ωm, is presented in Table 2. The PGA of each ground motion was taken as 500 Gal, which is similar to the intensity of a rare 8-degree earthquake, for the time-history analyses. Figure 14 shows the interstory displacement of the shear frame equipped with linear VFDs subjected to the seismic excitation. Clearly, the displacement response of the frame structure tended to be larger under seismic motions with a lower average frequency Ωm.Table 2
Information on the considered earthquake records.
NameLocationYearMsStationPGA (g)Ωmζsd (α = 1)KEATKern County19527.4Athenaeum0.0486.8114.06SATINorthridge19666.7Saticoy0.3417.9714.15KEHOKern County19527.4Hollywood0.0428.3415.52NEWHNorthridge19946.4Newhall0. 5909.0015.93ELCEImperial Valley19407.0El Centro0.34811.2016.97TAFTKern County19527.4Taft0.15911.2218.26TABATabas19787.4Tabas0.85412.1617.95IRARIrpinia19806.9Arienzo0.02712.9918.08IRAUIrpinia19806.9Auletta0.05514.7919.01MACOMammoth19805.7Convict Creek0.17814.9018.98Figure 14
Interstory displacement of the six-story shear frame equipped with linear VFDs.Whenα = 1, the ADR of the linear viscous dampers is calculated as follows using equation (21): ζ1 = Cr/(2Mrω) = CrT/4πMr = 17.00%. The calculation results of the time-history analyses are presented in Table 2, which are close to 17.00%, indicating that the calculation result of equation (28) can be used to quickly evaluate the ADR with consideration of the resonance condition, despite the error.Whenα ≠ 1, Ω = ω is set; then, equation (6) is transformed into(34)c1=λπcαωuo1−α.The nonlinear damping exponent is set asα = 0.5, and the peak deformation is set as u0 = 0.05 m, which is approximately 1/60 of the floor height (3 m). This is close to the interstory displacement limit for a structure under rare earthquakes based on the Chinese seismic code [35]. The other parameters are identical to those of the original structure. According to equation (34), the equivalent damping coefficient is ca = 2.84 × 105 (N s/m).Taking the El Centro seismic record of the Imperial Valley earthquake as an example, according to equations (31) and (32), the average period is Tm = 0.56 s and Ωm = 11.20 rad/s. According to the fitting results in Table 1, f1 in equation (29) can be expressed as follows:(35)f1=0.4ζsd,res3−0.9ζsd,res2+0.84ζsd,res+0.04.By solving equation (29), ADR ζsd,res of the shear frame under the structural resonance condition is calculated as 9.74%. By substituting this value of ζsd,res into equation (30), the ADR under the El Centro seismic excitation considering the spectral characteristics is determined to be 10%. Additionally, the ADR of the nonlinear viscous damper can be determined using Figures 11 and 12 according to the damping coefficient ca, natural vibration period of the structure T, or PGA u¨g0 of the El Centro motion.Taking the 10 input ground motions listed in Table2 as seismic excitations, the ADRs were calculated using five methods: the ERM, two nonlinear time-history analysis methods (NMA [28] and Code [35] methods), and equations (29) and (30). The results are presented in Table 3, and a comparison of the ADRs calculated using three of the methods with different frequency ratios is presented in Figure 15. As indicated in Table 3, the difference between the results obtained using the ERM and equation (29) under the resonance condition was very small. The results obtained using the NMA method were larger than those obtained using the Code method, indicating that the Code method is relatively conservative with regard to structural safety. The calculation results of equation (30) were larger than those of equation (29), which is consistent with the change law of the ADR under harmonic excitation with a frequency ratio of >1 (see Figure 15). Because the spectral characteristics of the seismic excitation significantly affected the ADR of the damping structure, the calculation results of equation (30), which considered the excitation frequency, were more accurate than those of equation (29).Table 3
ADRs calculated using different methods.
ADR (%)Seismic waveΩm = ωΩm ≠ ωERMEquation (29)Nonlinear time-history analysis methodEquation (30)NMA [29]Code method [35]KEAT12.1912.2316.6113.9213.43SATI8.008.139.578.058.79KEHO12.2712.9215.5914.1613.97NEWH9.329.7813.5211.5910.32ELCE9.689.7413.5312.5910.00.TAFT10.3910.6815.2513.7711.74TABA9.6610.0513.9512.3010.52IRAR11.1311.1613.9211.8212.44IRAU13.6813.7418.9815.3515.45MACO13.6414.0716.9514.7615.80Note. ADR of NMA = (energy dissipated by VFDs/energy dissipated by modal damping) × inherent structural damping ratio.Figure 15
Comparison of the calculation results of equations (29) and (30) and the ERM.Figure16 presents a comparison of the roof-displacement time histories of the shear frame with ADRs calculated using different methods. As shown, the displacement responses calculated using the ERM and equation (30) were similar, and the displacement responses obtained using the Code method were smaller than those for the other two methods, indicating that the proposed calculation method is relatively conservative.Figure 16
Roof-displacement time histories of the shear frame with ADRs calculated using different methods.


## 5. Conclusions

A new simplified calculation method for the ADR of a damping structure equipped with a nonlinear viscous damper was developed on the basis of previous studies. The accuracy of this method was verified by analysing a six-story RC frame structure. According to the calculation results, the following conclusions are drawn:(1)
The average frequency Ωm was introduced to characterise the seismic excitation, and the ADR calculated using the proposed method considering the effect of the average frequency was larger than that calculated using the ERM. Moreover, as the earthquake intensity increased, the ADR decreased.(2)
Studying the effect of the excitation frequency on the calculated value of the ADR with different velocity exponents revealed that the ADR of the damping structure was the smallest for Ωm/ω = 1, and it decreased with the increasing frequency ratio for Ωm/ω < 1. Additionally, the ADR of the damping structure decreased with an increase in the seismic-excitation peak acceleration.(3)
According to the case study of a shear frame, the displacement response of the frame structure tends to be larger under seismic motions with a lower average frequency Ωm, indicating the significant impact of the frequency characteristics of the seismic excitation on the structural response. Compared with other methods tested, the calculation results of the proposed simplified method were closer to those of the Code method, confirming the effectiveness of the proposed method.When the ground motion characteristics, structural characteristics, and damper parameters are known, the proposed method can be used to calculate the ADR of the damping structure conveniently. However, the method for calculating the ground motion frequency using the RSAP requires further improvement.



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