Research Article  Open Access
Weizhi Xu, Dongsheng Du, Shuguang Wang, Weiwei Li, "A New Method to Calculate Additional Damping Ratio considering the Effect of Excitation Frequency", Advances in Civil Engineering, vol. 2020, Article ID 3172982, 15 pages, 2020. https://doi.org/10.1155/2020/3172982
A New Method to Calculate Additional Damping Ratio considering the Effect of Excitation Frequency
Abstract
The additional damping ratio (ADR) is an important indicator for evaluating the damping effect of structures with energydissipation 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.
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 energydissipation 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), massproportional damping (MD), and stiffnessproportional damping (SD). The RD model, which consists of MD and SD, is widely used in commercial finiteelement platforms because of its mathematical simplicity. Huang [5] reported that the desired energydissipation rate was achieved at only two target frequencies and that the massproportional term applies damping to rigidbody modes, which cannot dissipate energy in reality. To solve this problem, a versatile frequencyinsensitive damping modelling method was developed, which can achieve approximately uniform energy dissipation in the specified frequency range and make the damping orthogonal to the rigidbody 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 firstmode response and/or highermode responses. To mitigate the problems associated with the improper damping models in the baseisolated structure, Anajafi et al. [6] proposed a damping modelling approach that involves removing the massproportional 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 energydissipation 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 ratedependent 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 energydissipation rate in different deformation stages. Wu et al. [17] evaluated the seismic performance of a ninestory steel frame supplemented with five typical dampers under nearfault 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 singlestory, singlebay 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 singlelayer, singlespan 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 higherorder 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 linearelastic 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 timevarying 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 (SACFEMA) 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 timehistory analysis was significant. He et al. [32] proposed an effective finiteelement 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. Section 2 introduces two typical calculation methods for the ADR: the energyratio 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:where represents the maximum strain energy of the damping structure with a viscous damper under the expected displacement, 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:where c_{α} is the damping coefficient corresponding to different velocity exponent (α) values and sgn (x) is a sign function.
For a singledegreeoffreedom (SDOF) system subjected to a harmonic displacement excitation with the function of u(t) = u_{0}sin Ωt, the energy consumption of the nonlinear viscous damper is given as follows:
Integration of equation (3) yieldswhere λ is a constant related to α.
When = 1, , and the following equation can be obtained:
Linearisation of the nonlinear viscous damper is performed according to the equal dissipating energy of a linear viscous damper, and the equivalent linear damping coefficient c_{1} can be determined using the following equation:
By dividing both sides of equation (6) by 2mω, the ADR of nonlinear viscous dampers can be calculated as follows: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:where a_{0} represents the peak acceleration of harmonic excitation.
Under the resonance condition, i.e., Ω = ω, equations (7) and (8) can be rewritten as follows:respectively, where represents the peak acceleration of the seismic excitation.
The relationship between ADR ζ_{sd} and the damping index ε can be expressed as follows:where , reflecting the amplification of the dynamic response.
The acceleration amplification factor R_{a} can be computed via numerical analysis, and the damping index ε can be determined according to the structural and input characteristics. Then, ADR ζ_{sd} is calculated using equation (11). Figure 1 shows the ADRfrequency 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 ζ_{sd} 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.
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 iswhere c_{1} is the damping coefficient of the linear viscous damper when the structural damping is 0 and a_{0} represents the peak acceleration of harmonic excitation.
The complete solution of equation (12) [33] iswhere represents the frequency of the structure considering damping, ω represents the natural frequency of the structure, and ζ_{1} represents the damping ratio of the linear viscous damper.
Considering the initial conditions u (0) = 0 and (0) = 0, the following can be obtained:
According to equation (13), the steadystate dynamic response u_{s} is calculated as follows:where and .
The steadystate response is taken as the approximate response of the structure, i.e.,
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:
According to equations (7), (16), and (17), the ADR can be determined as follows:
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 transientstate solution on the structural response, indicating that the frequency ratio affected the difference between the ADRs calculated using equation (18) and the ERM.
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:
Substituting equation (19) into equation (6) and setting ζ_{1} = c_{1}/2ωm yields
Then, by dividing both sides of equation (6) by 2mω, the ADR under the resonance condition is obtained:
Figure 5 presents the relationship curves of ζ_{sd,res} and a_{0} for different values of α and c_{α}. 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.
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(b)
(c)
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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 ζ_{sd} and Ω/ω obtained via the ERM.
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In Figure 6, 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 ζ_{sd} vs. Ω/ω curve can be defined as a piecewise function containing the structural resonance point (1, ζ_{sd,res}) as follows:where a_{1} and a_{2} are the coefficients that must be determined via fitting.
Figures 7 and 8 present the curves of a_{1} and a_{2} under different damping exponents (α). Numerical fitting revealed thatwhere β1 = 109.6 and γ1 = –5.489, andwhere β2 = 10.02 and γ2 = –4.649.
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 ζ_{sd} 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.
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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, u_{0} is similar to a_{0}/2ω2ζ. Ifwhere represents the assumed value of the peak displacement response, which is related to the peak ground acceleration (PGA). There is a relationship between u_{0} and . The ratio between them is defined as f_{1} = u_{0}/ and reflects the displacement amplification between the nonresonant and resonant responses of the structure.
Then, u_{0} is determined as follows:where represents the seismicexcitation peak acceleration and ζ_{1} represents the ADR of the linear viscous damper.
For a linear structural system subjected to a seismic excitation, the coefficient f_{1} is unrelated to the PGA but is related to ADR (ζ_{1}). Therefore, f_{1} is designated as a unary polynomial of ζ_{1}:where b_{1}, b_{2}, b_{3}, and b_{4} 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 (ζ_{1}, f_{1}) can be obtained through timehistory analyses of the SDOF system with different ADRs (ζ_{1}). Taking the El Centro seismic record of the Imperial Valley earthquake as an example, considering the different natural cycle frequencies of the structure, ζ_{1} and corresponding f_{1} values can be obtained, as shown in Figure 10. Table 1 presents the fitting coefficients for the unary polynomial of ζ_{1}. The fitting coefficients with ω = 5.39 are used in the case study in Section 4.

By substituting equation (26) into equation (9), the ADR with Ω = ω is obtained:
According to equation (17), equation (28) can be transformed into
Equation (22) is used to determine the ADR when Ω ≠ ω:where Ω_{m} represents the average frequency of the seismic excitation.
The average period T_{m} of a seismic wave can be calculated as follows [29]:where C_{i} is the Fourier peak coefficient and F_{i} represents the discrete frequency of the fast Fourier transform.
Then average frequency Ω_{m} of the seismic input ground motion acceleration is calculated as follows:
Figures 11 and 12 present the curves of ADR (ζ_{sd}) vs. the natural vibration period of the structure (T) and the PGA () with different damping coefficients (c_{a}, 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.
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 vibrationreduction design for a sixstory 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 c_{1} = 1.50 × 106 (N s/m), and the inherent structural damping ratio was taken as ζ_{0} = 0.05.
According to the foregoing conditions, it is easy to determine the natural period T_{1} = 1.1656 s and the first vibration mode of the structure {}T = {0.2411, 0.4681, 0.6680, 0.8290, 0.9419, 1.0000}. According to the equivalent SDOF method [33], the equivalent mass M^{r}, equivalent stiffness K^{r}, and equivalent damping C^{r} can be obtained as follows:where M represents the mass matrix, K represents the lateral stiffness matrix, C represents the damping matrix corresponding to the linear VFDs, and {} represents the first vibration mode of the structure.
According to equation (33), the equivalent mass is M^{r} = 3.32 × 105 kg, the equivalent rigidity is K^{r} = 9.64 × 106 N/m, and the equivalent damping is C^{r} = 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 Ω_{m}, 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 8degree earthquake, for the timehistory 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 Ω_{m}.

When α = 1, the ADR of the linear viscous dampers is calculated as follows using equation (21): ζ_{1} = C^{r}/(2M^{r}ω) = C^{r}T/4πM^{r} = 17.00%. The calculation results of the timehistory 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.
When α ≠ 1, Ω = ω is set; then, equation (6) is transformed into
The nonlinear damping exponent is set as α = 0.5, and the peak deformation is set as u_{0} = 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 c_{a} = 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 T_{m} = 0.56 s and Ω_{m} = 11.20 rad/s. According to the fitting results in Table 1, f_{1} in equation (29) can be expressed as follows:
By solving equation (29), ADR ζ_{sd,res} of the shear frame under the structural resonance condition is calculated as 9.74%. By substituting this value of ζ_{sd,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 c_{a}, natural vibration period of the structure T, or PGA of the El Centro motion.
Taking the 10 input ground motions listed in Table 2 as seismic excitations, the ADRs were calculated using five methods: the ERM, two nonlinear timehistory 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).
 
Note. ADR of NMA = (energy dissipated by VFDs/energy dissipated by modal damping) × inherent structural damping ratio. 
Figure 16 presents a comparison of the roofdisplacement 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.
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 sixstory RC frame structure. According to the calculation results, the following conclusions are drawn:(1)The average frequency Ω_{m} 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 Ω_{m}/ω = 1, and it decreased with the increasing frequency ratio for Ω_{m}/ω < 1. Additionally, the ADR of the damping structure decreased with an increase in the seismicexcitation 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 Ω_{m}, 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.
Data Availability
Some or all data and models used during the study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (grant nos. 51678301 and 51678302) and the National Key R&D Program of China (grant no. 2017YFC0703600).
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Copyright © 2020 Weizhi Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.