IntroductionLarge-span structures and high-rise buildings have low vibration frequencies and often low inherent damping, e.g., bridges, cables (Chen et al. 2020b), and skyscrapers (Shoraka et al. 2017). Under wind or seismic loading, such structures can experience large-amplitude vibrations and the vibrations typically involve several modes of the structure. The vibrations threaten the long-term structural safety and users’ comfort (Petrini et al. 2020). Therefore, supplemental damping for vibration mitigation is a topical area recently.For tall buildings and large-span bridges, damping devices are typically installed between two close structural points/components. The deformation of the damping devices is thus relatively small in structural vibration. For example, dampers are installed between the bridge deck and stay cables for cable vibration suppression in cable-stayed bridges (Main and Jones 2002; Chen et al. 2020a); viscous dampers are installed between the outrigger and the perimeter column for response reduction of tall buildings (Shoraka et al. 2017; Smith and Willford 2007; Chen et al. 2010; Tan et al. 2015; Fang et al. 2019; Zhou and Xing 2021; Xing et al. 2020; Zhou et al. 2021). The outrigger-damper-perimeter column system is often referred to as a conventional damped outrigger (CDO). Different types and configurations of outrigger systems have been considered, and the optimization design was studied (Zhou et al. 2021; Lin et al. 2018; Das and Tesfamariam 2020).Traditional damping devices experience relatively small deformation in structural vibration, particularly when the structure vibrates in lower modes. Hence, the supplemented structural damping is very limited (Main and Jones 2002; Tan et al. 2014). In this regard, negative stiffness has been introduced to increase the deformation amplitude of the damping device installed on flexible structures such that the supplemented damping can be improved. Typical passive negative stiffness devices (NSDs) are designed based on precompressed springs together with amplification mechanisms (Nagarajaiah and Reinhorn 1994; Sarlis et al. 2013; Pasala et al. 2013, 2015; Sarlis et al. 2016). Such NSDs were shown effective in based isolation (Sun et al. 2017b) and seismic protection (Attary et al. 2015; Nagarajaiah and Sen 2020). In addition, adaptive NSDs have been proposed and tested (Nagarajaiah 2009; Wang et al. 2021a), leading to more robust vibration control using tuned mass dampers (Nagarajaiah and Varadarajan 2005; Nagarajaiah and Sonmez 2007; Wang et al. 2021b). Other means to realize passive negative stiffness is also available, e.g., by a group of magnets (Shi and Zhu 2015). NSDs have been introduced to enhance the damping of stay cables (Chen et al. 2015) and high-rise buildings with outrigger systems (Wang et al. 2020, 2019a, b). Multiobjective design of such systems has been investigated considering damper cost (Sun et al. 2021a) and both earthquake and wind loads (Wang et al. 2021c).Another means to equivalently realize negative stiffness by inertial effect has also been explored in the field of structural vibration control. Particularly, the inerters (Smith 2002), which can amplify inertial effects by rotating mass and gear mechanism, are receiving a lot of attention (Lazar et al. 2014). The effects of inerter-based devices on single- or multi-degree-of-freedom systems have been investigated, e.g., Wen et al. (2021). Inerters have been introduced into the damping system of stay cables to enhance the damping provided by dampers installed near cable anchorage (Lazar et al. 2016; Lu et al. 2017) and to improve the dissipation capacity of cable cross-ties (Sun et al. 2017a); rotational mass has been proposed to enhance the damping effect by CDOs to high-rise buildings (Liu et al. 2018). Likewise, Asai and Watanabe (2017) proposed tuned inertial mass electromagnetic transducers for response reduction and energy harvesting in the outrigger system for tall buildings.The NSD directly modifies structural stiffness at the place where the damping devices are installed. Such modification does not depend on vibration frequency. The inerters, however, change the apparent mass of a structure at the installation place, and thus the resultant inertial force sensitively depends on vibration frequency. As such, the inerters need to be tuned to a specific structural mode or frequency. For single-mode or narrow-band vibrations, inerters can achieve the same level of negative stiffness as an NSD and hence comparable damping enhancement when combined with dampers. The frequency-dependent mechanical properties of inerters make them behave differently when structural vibrations involve different modes (Chen et al. 2021; Sun et al. 2021b).In this study, the mechanical properties of NSDs/inerters and combinations of those devices with dampers and springs are discussed. The focus is placed on the frequency-independence (NSDs)/dependence (inerters) and their role in multimode vibration mitigation. Subsequently, tall buildings/bridges with outrigger systems incorporating NSDs and inerters are taken as examples to demonstrate the performance of NSDs and inerters in multimode vibration suppression. The novel contribution of this study is two-fold: (1) the tuning principle based on equal modal damping (Krenk 2005; Krenk and Høgsberg 2013, 2014, 2016) is extended to maximize the damping effect of the damped outrigger system with inerters (IDOs) for tall buildings; and (2) multimode damping effects of IDOs are understood with comparison to that of the outrigger system incorporating NSDs (NSDOs).The remainder of this paper is organized as follows. The frequency-independent(NSDs)/dependent(inerters) negative stiffness and mechanical properties of those elements together with springs and dampers are first discussed. Mathematical models of a tall building with NSDOs/IDOs are then presented, and based on which, the tuning methods for determining optimal device parameters and multimode damping are provided. Typical systems with optimized parameters are chosen for numerical analyses of structural responses under seismic or wind loading for verification of the analytical results. The study is closed with a brief conclusion.Frequency-Independent/Dependent Negative Stiffness and DampingFrequency-Independent/Dependent Negative StiffnessFig. 1(a) shows a typical realization of passive negative stiffness, i.e., by a compressed spring. When the upper end of the compressed spring moves in the direction perpendicular to the line of axial action of the spring, the precompressing force gives a component in that direction to assist the movement, leading to the so-called negative stiffness effect. The device can be designed to have a constant negative stiffness coefficient kNS≤0 when the deformation amplitude is in a certain range (Sarlis et al. 2013). The test results are shown in Fig. 2 from a single spring NSD [similar to Fig. 1(a)] tested at Rice University. The normalized force-displacement behavior from experiments when subjected to harmonic excitation of constant amplitude is shown in Fig. 2 and compared with analytical prediction using the NSD model incorporating the Columb friction behavior.The force provided by the device is represented by (the damping effect is ignored here) (1) f(t)=kNSu(t)orf˜=kNSu˜ in the frequency domainIn the preceding expression, f denotes the resistant force when the device is under deformation u(t) and t = time variable. The expression in the frequency domain is obtained by letting f=f˜eiωt and u=u˜eiωt with ω denote the vibration frequency and i=−1. It is seen from the preceding expression and Fig. 1(a) that the negative stiffness effect is independent of vibration frequency.Fig. 1(b) shows an inerter with rotational mass and the force between the two terminals of the inerter can be represented by (2) f(t)=bu¨(t)orf˜=−bω2u˜ in the frequency domainIn the expression, ()˙ is the time derivative operator and b≥0 is inertance. By comparing the force-deformation relations respectively for the NSD and the inerter in the frequency domain, it can be concluded that the inerter can achieve negative stiffness equivalently, and the equivalent stiffness coefficient is −bω2, which is sensitively dependent on the vibration frequency, as shown in Fig. 1(b).The negative stiffness effect can increase the relative displacement of the structure at the position where an NSD or an inerter is installed. Therefore, a damper, frequently of the viscous type, is placed in parallel with the negative stiffness device, and its energy dissipation capacity is enhanced. Normally, the assembly of the device is used to connect two points of a structure subjected to relative deformation (Ji et al. 2021) or connect the structure to a fixed point. The stiffness of the connection is also of critical importance and the stiffness coefficient is denoted by kc. Typical installations are shown in Figs. 1(c and d). The NSD, the viscous damper, and their flexible support can be modeled as a Maxwell element, as shown in Fig. 1(c), and the force-deformation is governed by (3) f=kcuc=kNS(u−uc)+cd(u˙−u˙c)where uc = deformation of the support. The normalized complex stiffness of the assembly can be solved (4) G(ω¯)=f˜kcu˜=kNS+iωcdkc+kNS+iωcd=η+iω¯c¯d1+η+iω¯c¯d=η+η2+(ω¯c¯d)2(1+η)2+(ω¯c¯d)2+iω¯c¯d(1+η)2+(ω¯c¯d)2The normalized variables in the expression are defined as (5) η=kNSkc,ω¯=ωω0,c¯d=cdω0kcLikewise, the force-deformation of the assembly of the inerter, the viscous damper, and their flexible support, as shown in Fig. 1(d), is governed by (6) f=kcuc=b(u¨−u¨c)+cd(u˙−u˙c)Correspondingly, the normalized complex stiffness is given as (7) G(ω¯)=f˜kcu˜=−bω2+iωcdkc−bω2+iωcd=−b¯ω¯2+iω¯c¯d1−b¯ω¯2+iω¯c¯d=−b¯ω¯2+b¯2ω¯4+(ω¯c¯d)2(1−b¯ω¯2)2+(ω¯c¯d)2+iω¯c¯d(1−b¯ω¯2)2+(ω¯c¯d)2where the inertance is normalized as b¯=bω02/kc.Note that the real part of G(ω¯) denotes the overall stiffness of the assembly, while the imaginary part denotes the energy dissipation capacity. Previous studies show that a negative stiffness can increase the maximal damping effect (Chen et al. 2015), while a positive stiffness would decrease the maximal damping effect (Fang et al. 2019). Figs. 1(e and f) show variations of the real part of G(ω¯), respectively, corresponding to the NSD- and inerter-enhanced damping systems. Normally, the support stiffness is relatively large and hence c¯d is small, so the plots show the variation in the range of c¯d∈[0,0.2]. Besides, the absolute value of kNS is much smaller than the support stiffness and thus η+1>0. For the NSD-enhanced damper, letting the real part of Re(G(ω¯))=0 one obtains the relation ω¯=−η−η2/c¯d. The curve of ω¯=−η−η2/c¯d is also plotted in Fig. 1(e), and in the upper side of this curve, Re(G(ω¯)) is positive; and below this curve, Re(G(ω¯)) is negative. From Eq. (4), one finds that Re(G(ω¯))=η/(1+η) when ω¯=0 or c¯d=0. For a given η>−1 and a given ω¯, Re(G(ω¯)) is negative when c¯d∈[0,−η−η2)/ω¯). For the inerter-enhanced damper, equivalent negative stiffness is realized only in the region below the curve ω¯=b¯−c¯d2/b¯. Besides, relatively large negative stiffness is only achieved near the point (b¯−0.5,0,0). Therefore, for a mode with a frequency much smaller than b¯−0.5, the inerter has almost no effect on the damping enhancement. More importantly, for a structural mode with a frequency larger than b¯−0.5, the damping of this mode is degraded by the equivalent positive stiffness of the inerter-based damper.Note that only linear behaviors of an NSD are focused on considering relatively small deformation of the device, while adaptive stiffness behaviors can be achieved by combining precompressed springs and nonlinear springs. Fig. 1(g) shows an example of a precompressed spring and a nonlinear spring installed in parallel, as studied by Pasala et al. (2013) and Sarlis et al. (2013).Rate-Independent DampingIt is also noted from Fig. 2 that the linear hysteretic damping [butter fly damping] mechanism is evident in the NSD force-displacement loop in addition to the negative stiffness behavior. The hysteric behavior is more close to linear hysteric damping than Columb damping (Muravskii 2004). Structural testing supports damping that is frequency-independent to closely represent the actual energy dissipation mechanism, which is often approximated by viscous damping. Hence, a hysteretic energy dissipation mechanism that provides forces that are proportional to displacement, u(t), and are in phase with the velocity, u˙(t), is more appropriate (Caughey and Vijayaraghavan 1970; Inaudi and Makris 1996). That is (8) f(u,u˙)=ζk|u(t)|u˙(t)|u˙(t)|where ζ,k are two factors. A damping force that is proportional to both displacement and velocity, and is in phase with velocity is also appropriate (Reid 1956; Makris 1997) (9) Fig. 3 shows the force-displacement behavior of damping models in Eqs. (8) and (9), wherein it is evident that the damping force is proportional to displacement, in phase with velocity, and independent of frequency (10) −Mω2+k(1+iζ)Reiωt=Mω2Zeiωtwhere M = mass; and R and Z = amplitude variables. The term k(1+iζ) illustrates why this form of damping is called complex-stiffness damping. The transmissibility ratio for complex-stiffness damping is (Gavin 2014) (11) Trcs(ω)=k(iζ+1)−mω2+k(iζ+1)=iζ+1−(ω/ωn)2+iζ+1where ωn2=k/M.A negative stiffness device with rate-independent (frequency-independent) damping (NSD-RID) can also be developed based on the classical Maxwell–Weichert Model (Biot 1958; Makris 1997) as shown in Fig. 4(a). The practical/implementable NSD-RID consists of three Maxwell elements (spring and fluid damper in series)—tuned to three separate frequencies—to achieve adequate frequency bandwidth where the damping is nearly constant, in parallel with the NSD as shown in Fig. 4(b). By introducing NSD-RID, both frequency-independent stiffness and a damping system that can help mitigate vibrations in several modes are envisioned. NSD-RID is the subject of a future study. In this study, we study NSD with a fluid damper and ID with a fluid damper.Mathematical Modeling of a Typical Flexible Structure under ControlTo investigate the damping effect of the NSD- and inerter-enhanced dampers for multimode response mitigation of a flexible structure, a tall building with a damped outrigger system is considered. The first several modes of the building are concerned as they can be excited under seismic or wind loading. The mathematical models for dynamic analysis and parametric study are introduced in this section.System DescriptionFig. 5 shows typical flexible structures with outrigger systems incorporating NSDs and inerters. The systems in Figs. 5(c and d) are similar to the strengthen system studied in (Neff Patten et al. 1999) where a semiactive control strategy was applied to the I-35 Walnut Creek Bridge, see Fig. 6(a). Outriggers for highway bridges have also been studied by Agrawal et al. (2009), see Fig. 6(b), and have been used for bridge retrofit (LUSAS 2021). For long-span bridges, the application of damped outriggers will be further explored by (Chen et al. 2022). The systems in Fig. 5 can be analyzed using the same methods. In this paper, the tall buildings in Figs. 5(a and b) are taken as examples, where EI denotes the bending stiffness of the tall building when modeled as an Euler beam, m is the uniformly distributed mass in kg/m, kc denotes the axial stiffness of the perimeter column, and H is the height of the building. In Fig. 5(a), the outrigger is connected to the perimeter columns by an NSD placed in parallel with a viscous damper, and the negative stiffness coefficient is denoted by kNS and cd is the viscous coefficient. This outrigger system is referred to as NSDO hereafter, and it reduces to a CDO when kNS=0. In Fig. 5(b), the viscous damper (with coefficient cd) is placed in parallel with an inerter, and the outrigger is connected by the assembly to the columns. The inertance is denoted by b. The column is modeled as a spring with stiffness coefficient kc. Hereafter, an outrigger system incorporating an inerter is referred to as IDO for convenience.Complex Modal AnalysisFollowing previous studies on tall buildings with damped outrigger systems (Tan et al. 2015; Wang et al. 2020), the building is modeled as a cantilever beam and the outrigger is modeled as rotational inerto-viscoelastic springs with the complex stiffness coefficient denoted by kθ. With such assumptions, the frequency of the system can be solved by complex modal analysis (CMA). Correspondingly, the frequency equations have been derived by Liu et al. (2018) and Wang et al. (2020), respectively. Therefore, the CMA method is introduced here concisely.When the cantilever beam is attached to the NSDOs, using Eq. (4), the stiffness coefficient kθ is expressed (Wang et al. 2020) as (12) kθ=2r2kc(kNS+iωcd)kc+kNS+iωcd=EIαHβ2λ2cαβi+η1+2λ2cαβi+ηwhere r = length of the outrigger; and αH = position of the outriggers. Other parameters in the preceding equation are defined here as (13) λ4=mH4ω2EI,c=cdr2HmEI,β=EI2EcAcr2,kc=EcAcαHwhere λ = wave number. Likewise, for the system with an inerter, using Eq. (7), one obtains (14) kθ=2r2kc(−ω2b+iωcd)kc−ω2b+iωcd=EIαHβ2λ2cαβi−2μαβλ41+2λ2cαβi−2μαβλ4Note that the nondimensional inertance is defined as μ=br2/(mH3). Following the derivation in Liu et al. (2018) and Wang et al. (2020), the frequency equations are generally expressed in terms of ks=kθH/(λEI), as (15) ks[cos(λ−2λα)sinh(λ)+cosh(λ−2λα)sin(λ)+2cos(λα)sinh(λα)+2cosh(λα)sin(λα)+cos(λ)sinh(λ)+cosh(λ)sin(λ)−2cos(λ−λα)sinh(λ−λα)−2cosh(λ−λα)sin(λ−λα)]+4[cos(λ)cosh(λ)+1]=0Define a characteristic frequency associated with the IDO and its normalized form as (16) ωIDO=kcb=12αβμEImH4,ω¯IDO=ωIDOω0=12αβμwhere ω0=EI/(mH4) = fundamental circular frequency of a pinned-pinned beam with length H, bending stiffness EI, and mass per unit length m. Similarly, the system frequency is normalized as ω¯=λ2.After solving the complex wave number from Eq. (15), modal damping can be obtained by (17) where Im(ω¯) = imaginary part of the frequency. The damped frequency of the system is given as Re(ω¯). For given system parameters, Eq. (15) is a nonlinear equation of ω¯=λ2 with an infinite number of solutions. For numerical solutions, it is first rewritten by isolating the viscous coefficient, following the methods in Main and Jones (2002) and Chen et al. (2021), as (18) ic=12λ2αβ(λαβ−C/D−λαβ−η)andic=12λ2αβ(λαβ−C/D−λαβ+2μαβλ4)respectively, for the system with the NSDOs and the IDOs. Providing parameters (β,α,η) or (β,α,μ), all possible frequencies of the system can be solved from the real part of the preceding equations, and afterward, the corresponding viscous coefficients are obtained from the imaginary part of the preceding equations, i.e. (19) where Ξ = right-hand side of Eq. (18). Note that when kNS=0 (or b=0) and cd=0, the system reduces to a cantilever beam whose modal frequencies are well-known. These special frequencies are denoted by ω¯0n with n = mode index. The values of ω¯0n for the first five modes are listed in Table 1.Table 1. Normalized frequencies of a uniform cantilever beamTable 1. Normalized frequencies of a uniform cantilever beamMode No.SymbolNormalized frequency1ω¯013.51562ω¯0222.03363ω¯0361.70104ω¯04120.91205ω¯05199.8548Finite Element ModelingFor verifying the CMA method and also for analyzing dynamic responses of the system under seismic loading, a finite element model (FEM) of the system is introduced. The cantilever beam is discretized by using a number of beam elements and the outriggers are located at one node. The stiffness matrix of the beam elements and the modeling details can be found in Wang et al. (2020). The model is implemented in an in-house code for eigenanalysis and dynamic analysis of the building under wind and seismic loads.Parametric Study on Multimode Damping EffectsTuning IDO Based on the Principle of Equal Modal DampingAs shown in Fig. 1(f), the negative stiffness behavior of an IDO is sensitive to frequency. Hence, tuning the IDO for maximal damping is first addressed. In the following, the relative height and stiffness of the perimeter columns are set as β=5.0 and α=0.55 (Wang et al. 2020). Using the CMA method, for a specific value of μ, the frequency curves can be solved. First, the first mode is targeted for adjusting the parameter μ for maximal modal damping. Using Eq. (16) and substituting ω¯IDO by ω¯01, the value of μ for tuning the IDO to the first building mode can be estimated, i.e., 1/(2αβω¯012)≈0.015. Then, by varying μ around this value, the optimal μ for maximizing the first modal damping can be approached. Three typical frequency curves corresponding to the first mode and the mode associated with the IDO are demonstrated in Fig. 7(a). Note that the frequency curve associated with the IDO mode is always connected to the point (0, 1) on the (ω¯,ξ) plane as the mode tends to be fully damped as the damping coefficient increases. It is noted when μ≈0.010472 the frequency curve corresponding to the IDO mode intersects that of the first mode of the building when the maximal damping of the first mode is achieved, i.e., ζ1≈0.2057. Note that the values are numerically computed and hence the two curves might not exactly intersect at μ=0.010472. When the maximal damping is attained the viscous coefficient is about c=0032833. Likewise, it can be found that when μ=0.00036454 (found by varying μ around 1/(2αβω¯022)≈0.000375), the maximal damping of the second building mode can be achieved, and correspondingly c=0.001090. Typical frequency curves of the second building mode and the IDO mode for varying μ are plotted in Fig. 7(b). In the same way, the optimal μ can be determined to maximize the damping of the third mode. The optimal values of (μ,c) and corresponding damping ratios are listed in Table 2 for the first three modes of the tall building.Table 2. Inertances and viscous coefficients determined by tuning the IDO to the first three modes based on the principle of equal modal dampingTable 2. Inertances and viscous coefficients determined by tuning the IDO to the first three modes based on the principle of equal modal dampingTarget mode no.Maximal dampingInertance, μOptimal, cηOptimal, c10.20570.010472000.032833−0.69040.01143520.03360.000364540.001090−0.95620.00030030.04910.000046350.000566−0.93390.000157For comparison, a building of 160 m high is considered (Gamaliel 2008) and analyzed by using the FEM is modeled with the previously described cantilever beam analogy and analyzed using FEM. The bending stiffness is EI=4×1012N·m2, and the distributed mass is m=85,000  kg/m. The outrigger system is installed at αH=0.55×160=88  m, and the axial stiffness of the perimeter columns is kc=EI/(2r2βαH)=2.02×107N/m with r=15  m. The building is divided into 40 uniform elements. For specified IDO parameters, i.e., b and cd, the complex frequencies are obtained by eigenanalysis. The computed damping and frequencies for the first mode for different values of inertance and damper coefficients are plotted in Fig. 8. It is seen that the results obtained by the CMA method and the FEM are pretty consistent for both the damping and the damped frequency. Only when the frequency changes dramatically as the damping coefficient increase, the differences in the frequency results computed by the two methods become visible. The difference may be attributed to the discretization error in the FEM. In the following, the CMA method is used for parametric study, and the FEM is used for computing structural responses under seismic loading.Comparison of Multimode Damping EffectsFor comparing the multimode damping of the building with NSDOs and that with IDOs, the stiffness coefficient of the NSDO is determined such that the same amount of damping is achieved to the same target mode. For example, when the first mode is targeted, the negative stiffness coefficient is numerically determined such that the first-order modal damping is 0.2057 (see Table 2). Using the definitions in Eq. (13), the ratio bω2/kc is expressed using dimensionless variables as (20) By substituting ω¯0n in Table 1 into the preceding relation, an equivalent normalized stiffness (a minus sign is added) can be first estimated. Then, by varying η around this value to solve Eq. (18), the exact equivalent NSDO can be found. For the IDO listed in Table 2, the parameters of the equivalent NSDO are obtained and also listed in the table.When the IDO is tuned to the first mode, damping curves and frequency variations for the first four modes are plotted in Fig. 9. The damping and frequency variations corresponding to the CDO and also the equivalent NSDO are also plotted in the figure. In Fig. 9(a), it is seen that by using the IDO and the NSDO, the maximal damping ratio of the first mode is of the same, and the damping is much larger than that can be achieved by a CDO. Interestingly, the optimal coefficient of the damper is much smaller when NSDOs are used as compared to that corresponding to the system with IDOs. Besides, the damping is much more sensitive to the viscous damping coefficient when it is close to the optimal one when the IDO is used as compared to the cases of the system with CDOs or NSDOs. More importantly, it is seen from Figs. 9(c–h), in this case, the damping provided by the IDO to the modes other than the first mode is almost zero because the inertance and hence the inertial force are too large when the building vibrates in higher modes. In other words, the relative motion between the outrigger and the perimeter columns is locked and hence no damping can be supplemented regardless of the damper coefficient. The locking effects can also be seen from the frequency variations in Figs. 9(d, f, and h) as the frequency tends to be a constant with respect to increasing damper coefficient. However, a comparison of the multimode damping achieved by the NSDO and the CDO shows that the damping ratios of all modes are improved to a similar extent.When the IDO is tuned to the second mode, damping curves and frequency variations for the first three modes are plotted in Fig. 10. In this case, as the inertance is much smaller, μ=0.00036454, the IDO has a very limited effect on the damping of the first building mode when comparing the damping curve to that of the system with CDOs, as seen in Figs. 10(a and b). However, the inertance again is too large for the third mode and hence the damping of the third mode is almost zero, as shown in Figs. 10(e and f). When an equivalent NSDO is used, the damping for each mode is increased considerably. Particularly, as the absolute value of the negative stiffness is larger than that in the previous case (in Fig. 9), the maximal damping for the first mode is larger than that in Fig. 9(a) and approaches 1 for a small damper coefficient, meaning that the first mode is critically damped. In this case, a larger negative stiffness (absolute value) is required because the damping is improved by a larger extent, from 0.0011386 (maximally achievable by CDOs) to 0.0336. Generally, the damping is relatively small because the outrigger location is close to the antinode (zero rotation points) of the second mode of the building (without any damping system).Similar characteristics can be found in the case when the IDO is tuned to the third mode. The damping curves and frequency variations are plotted in Fig. 11 for the second to fourth modes. The small inertance has almost no effects on the damping of the first two modes and again it has an adverse effect on the damping of the fourth mode as compared to the effect of CDOs. The effects of the NSDOs are the same as in the previous two cases.The damping curves for different modes are plotted together in Fig. 12. Because the damping and the optimal coefficients can be quite different in different modes they are plotted in log scale in the figure. By comparing Figs. 12(a and b), it is seen that the NSDO decreases the optimal coefficient for each mode to a considerable extent for a given η. However, the IDO has varied effects on the optimal damper coefficient for different modes, as shown in Figs. 12(c and d), which is disadvantageous for multimode response mitigation because for a passive damper, its coefficient normally does not change too much with respect to frequency (Chen et al. 2020b).From the previous comparison, it can be concluded that when the IDO is designed targeting the first mode of a tall building no damping can be provided to the higher modes. For higher modes, the damping attained is smaller than that can be provided by CDOs installed at the same height. When the IDO is tuned to a higher mode, it has almost no effects on a mode lower than the target mode because the inertial force is too small when the building vibrates at a low frequency. For a mode higher than the target mode, the IDO again has an adverse effect. The NSDOs can improve the damping of each mode to the same extent. It needs to mention that in some cases when the outrigger system is close to the node of a higher mode of the building, it could be better to use IDOs with small inertance to improve the damping instead of using the NSDOs as a larger negative stiffness (absolute value) is required.Effect of the Location of the Outrigger SystemThe effect of the outrigger system location is examined in this subsection. The parameter setting β=5 is the same as in the previous cases. For each location α∈(0,1] and a specific mode, the inertance μ is varied to find the optimal value that achieves the maximal damping (by equal modal damping principle). The optimal inertance μoptω¯0n2 is plotted in Fig. 13(a) for the first three modes, and the corresponding maximal damping ratios are plotted in Fig. 13(b). It is seen that the optimal inertance decreases as the IDO location moves from the bottom to the top of the building. From the perspective of practical implementation, it is better to install the IDO at a higher position as a relatively smaller inertance is able to achieve a comparable damping effect. When the IDO approaches the antinode of the mode shape of the cantilever, i.e., zero rotation points, the supplemental damping is zero regardless of the parameters of the IDO and hence the optimal inertance is undefined for IDOs at those positions. It is interesting to observe that the curves of μopt versus α corresponding to the first three modes are quite close, which can be used to guide IDO tuning. As the frequencies of different modes of the cantilever are widely separated, as listed in Table 1, the optimal μopt for different modes are hence much different, indicating that it is nearly impossible to use one IDO to improve multimode damping effects.For the purpose of comparison, when β=5, the damping supplemented respectively by the CDOs and the NSDOs (η=−0.75) versus the location α are plotted in Figs. 14(a and b). For all the devices, supplemental damping changes in a similar manner versus device location. In other words, when the device approaches an antinode of the mode shape (when no attachment is on the cantilever), the damping for the corresponding mode decreases to zero; between two adjacent antinodes of the mode shape, the supplement damping increases when the device moves always from the antinodes and attains a maximum when the device is close to the node of the mode shape. Not surprisingly, the damping provided by the CDOs is small, less than 0.05 for the first three modes and for all the locations. The NSDOs improve the damping for each mode comparably. For example, the damping of the first three modes is improved by about six times when η=0.75. Interestingly, when η=−0.75, NSDOs outperform the IDOs in terms of supplement damping for the first mode; however, for the second and third modes, the IDOs’ performances are superior. In other words, for higher modes, a larger negative stiffness (absolute value) is required, while a much smaller inertance is sufficient to achieve damping enhancement of the tall building with an outrigger system. Note that these discussions apply to the case where a single mode is targeted.Numerical AnalysisIn this section, numerical analyses of a tall building with damped outriggers are performed to assess the multimode control effects of NSDOs and IDOs. The parameter optimization method discussed in the preceding two sections is used.Building Information and Parameter SettingA simplified model of the 60-story St. Shangri-La Palace, Philippines, see Fig. 15, is used here (Infanti et al. 2008; Fang et al. 2019). The structure is H=210  m high (Wang et al. 2020). The bending stiffness of the main structure is EI=1.828×1013N·m2, and the mass per unit length is m=9.0308×104  kg/m; the length of the outrigger is r=4  m. The Rayleigh type of inherent damping is considered, and for the first two modes, a damping ratio of 0.01 is assumed. The tall building model is considered in the linear elastic range.As multimode damping effect is concerned, the outrigger system is placed at α=1. Note that the maximally achievable damping is small for a mode when the outrigger is near antinodes of the corresponding mode shape (see Figs. 13 and 14). The stiffness ratio is assumed to be β=5.0. In practice, the first mode is usually of primary importance, and hence, the optimal inertance and viscous damper coefficient of the IDO is determined by targeting the first mode. Correspondingly, the equivalent NSDO is determined such that the same level of modal damping is provided by the NSDOs as that achieved by the IDOs. The values of the IDO and NSDO parameters are listed in Table 3. The damping curves of the building with the IDOs and the NSDOs for the first modes are plotted in Fig. 16 with respect to the viscous coefficient. Apparently, for the first mode, the optimal damping is the same. For the second and third modes, the IDOs provide almost zero damping. When the damper coefficient of the NSDO is optimized with respect to the first mode, the damping ratios for the second and third modes are 0.015 and 0.002, respectively. Notably, the damping for the second and third modes can be improved when the damper coefficient is decreased. A change of the damper size in the IDOs in this case would not nearly affect the damping of the second and the third modes.Table 3. Parameter setting in numerical analysesTable 3. Parameter setting in numerical analysesTarget mode no.Maximal dampingInertance, μOptimal, cηOptimal, c10.168450.0060455500.015136−0.712390.0050226Structural Responses under Seismic LoadingStructural responses of the building respectively with the IDOs and NSDOs in Table 3 under seismic loading are then computed using the FEM. Two typical earthquake records are considered as the input excitation as follows: •El-Centro earthquake: north-south component of the 1940 Imperial Valley, California, earthquake that was recorded at the Imperial Valley Irrigation District substation in El-Centro, California (far-field ground motion), which is prolonged by an extra 30 s to observe free vibrations.•Northridge earthquake: north-south component of the 1994 Northridge earthquake that was recorded at the Sylmar Olive View far-field (FF) in Sylmar, California (near-fault ground motion), which is prolonged by an extra 20 s to observe free vibrations.The pseudo-acceleration spectra of the selection ground motions and the first three periods of structures are depicted in Fig. 17. It is clearly observed that the higher vibration modes of tall buildings can be excited by the ground motions.Response EnvelopesSeismic response envelopes of the building with different outrigger systems are shown in Figs. 18 and 19. It is clear that the IDOs and the NSDOs are effective in reducing seismic responses including both accelerations and interstory drifts. Specifically, the NSDOs outperform the IDOs in controlling seismic responses even though these two are designed to achieve the same level of damping ratio for the first mode. For example, the NSDOs further decrease the structural drift or acceleration by about 5%–10% (see Table 4) under the seismic loading. The reason lies behinds these results is the fact that higher modes play an important role in the seismic vibration of the building. As shown in Fig. 16, the designed NSDO aiming at the first mode is still able to supplement damping to higher modes, while the designed IDO decreases the high-order modal damping to zero, leading to the observed difference in seismic responses of the building.Table 4. Maximum seismic responses of tall buildings with different outrigger systemsTable 4. Maximum seismic responses of tall buildings with different outrigger systemsEarthquake recordBare systemIDONSDOBare systemIDONSDOEl-Centro1.1211.0870.9884.8994.1994.048Northridge1.8271.8001.6426.1095.7075.284Time-History ResponsesFigs. 20 and 21 show typical time histories of the structural responses of the building respectively equipped with the IDOs and the NSDOs subjected to the selected earthquake records. Note that the input earthquakes are prolonged by 30 s for the El-Centro earthquake and 20 s for the Northridge earthquake, respectively. Therefore, the time-history results include both earthquake-excited vibrations and free vibrations. For example, in Fig. 20, the first 30 s time histories represent earthquake-excited responses, while the last 30 s time histories correspond to structural free vibrations.From these figures, it is observed that the earthquake-excited responses are mainly dominated by higher mode vibrations. The response period is short, with a value of 0.8 s to 0.9 s, which is close to the structural period of the 2nd mode. During free vibrations of the building with the NSDOs under the El-Centro earthquake, a longer vibration period is seen from the top drift response, and the response period is about 9 s, which is close to the fundamental period of the building. However, during the free vibrations of the building equipped with the IDOs (the last 30 s time histories in Fig. 20), the 2nd mode vibration is still observed, indicating the designed IDO is not effective in suppressing the 2nd mode vibration. This confirms the superiority of the NSDOs as compared to the IDOs in the control of high-order vibrations.Damper PerformancesIn fact, neither NSDs nor inerters dissipate seismic energy. As the only energy dissipation device, the performance of a viscous damper installed at the end of the outrigger is vital for response reduction of the tall building. Fig. 22 compares the force-deformation curves of the dampers. Although the viscous coefficient in the NSDO is much smaller than that of the IDO (about 1/3 times smaller, see the optimal c in Table 3), the damper in the NSDO undergoes a larger deformation and thus a much larger damping force as compared to that in the IDOs, leading to a much greater energy dissipation. In other words, the NSDO is more efficient than the IDO in energy dissipation in these cases.Response SpectraTo further compare the performances of the NSDO and the IDO in suppressing multimode structural responses of the building under seismic loading, Fig. 23 plots the power spectra density (PSD) of top floor acceleration responses of the building respectively equipped with the IDO and the NSDO under the two earthquakes. It is shown that for the 1st mode vibration whose frequency lies in 0.1 Hz to 0.2 Hz, the NSDO and the IDO perform similarly in structural response reduction, as expected, because these two outrigger systems are both designed to target the 1st mode with the same achieved damping enhancement (see Table 3). Generally, the amplitude of the structural responses corresponding to the first mode is small. In contrast, it is observed that the response components with a frequency close to the 2nd or 3rd modal frequency (with a frequency of about 1.15 Hz) are the most prominent peaks in the PSD curves, indicating the earthquake-excited responses of the building are dominated by the 2nd mode rather than the 1st mode. In terms of the 2nd mode, the NSDOs substantially reduce the response amplitude as compared to that achieved by the IDO. For example, the 2nd mode amplitude of the building with the NSDOs is decreased by about 82% [from 0.1535 to 0.02707, see Fig. 23(a)] as compared to that of the building when equipped with the IDO under the El-Centro earthquake. In the case of the Northridge earthquake, a similar performance is seen in Fig. 23(b).The presented discussion assumes linear behaviors of the NSD in the NSDO. In fact, it is known that the typical NSD realized by precompressed springs exhibits elastic nonlinear behaviors (stiffening effect in particular) at large deformations. This favorable characteristic has been well utilized to reduce the inelastic response of the structure, e.g., by Nagarajaiah and Reinhorn (1994), Sarlis et al. (2013), Pasala et al. (2015), and further by Wang et al. (2021a) to address the robustness of braced-damper in yielding structures.The influences of ground motion parameters were investigated in previous studies, where three sets of earthquakes, 44 FF ground motions, 28 near-fault nonpulselike (NFNP) ground motions, and 28 near-fault pulselike (NFP) ground motions are selected to match the ASCE7-10 code spectrum with characteristic values of Sds=1.5  g and Sd1=0.75  g (Fig. 24). The performance of the negative stiffness amplifying damping concept has been validated for both elastic and plastic structures. Fig. 25 presents the Sd−Sa spectra of the elastic structures. It is clear that the proposed concept is validated for a wide range of ground motion types. Readers are suggested to refer to (Wang et al. 2019b, 2020) for more details on the negative stiffness amplifying damper (NSAD) and its applications.Structural Responses under Wind ExcitationDuring long-term operation, wind is another common hazard for tall buildings. This section focuses on the performance of different outrigger systems in reducing wind-induced structural vibrations. It is noteworthy that the section focuses on dynamic components of the wind-induced responses. The static deformation of the building induced by mean wind can be addressed by using some well-developed strategies like adding conventional outrigger systems (Wang et al. 2020) or using gap spring assembly (Nagarajaiah and Reinhorn 1994; Sarlis et al. 2013, 2016; Pasala et al. 2013, 2015) in the NSDO, as shown in Fig. 1(g).Wind LoadConsider the structure is subjected to a given wind load with the PSD matrix Sf(Ω), with Ω denoting the frequency in rad/s. The PSD matrix of wind-induced structural responses can be computed as follows: (21) Su(Ω)=H*(Ω)Sf(Ω)H(Ω),Su¨(Ω)=Ω4Su(Ω)where H(Ω) = transfer function matrix; and superscript * = conjugate transposition; and Su(Ω) and Su¨(Ω) = the PSD matrices of structural displacement and acceleration, respectively.Then, the variance of displacement and acceleration could be obtained, respectively, as (22) σuk=[∫0∞Suk(Ω)dΩ]1/2,σu¨k=[∫0∞Su¨k(Ω)dΩ]1/2where Suk(Ω) and Su¨k(Ω)=kth diagonal element of Su(Ω) and Su¨(Ω), respectively. The widely used power-law function is used to describe the mean wind profile in the vertical direction. The wind gust is considered by using the Davenport spectrum (Davenport 1961). Moreover, the Vickery model (Vickery 1970) is selected for the correlation along wind direction. As for crosswind, the model proposed by Liang et al. (2002) that also takes wind turbulence, vortex shedding, and spatial correlation into account is adopted.Along Wind-Induced Structural ResponsesFig. 26 shows the wind-induced response envelopes of tall buildings with different outrigger systems along the wind direction. It is clear that both NSDO and IDO substantially mitigate the vibration of tall buildings in along wind direction due to the extra energy dissipation provided by them. A comparison between the NSDO and the IDO indicates that the NSDO is more efficient in reducing along wind acceleration but may slightly amplify the drift demand. Fig. 27 presents the results of spectra analysis. The vibration of the tall building along the wind is dominated by its first mode, and both the NSDO and the IDO perform well in reducing the 1st mode spectra peak. However, as mentioned before, the NSDO is able to provide larger damping for higher modes as compared to the IDO. Therefore, a more significant reduction is observed for the NSDO case at the 2nd mode peak.Crosswind Induced Structural ResponsesGenerally, the responses of tall buildings with large aspect ratios in the crosswind direction are usually much larger than those along the wind direction (Liang et al. 2002). Thus, the crosswind vibration is more critical for the wind-induced performance of tall buildings. The results shown in Fig. 28 also demonstrate that the crosswind response is much larger. Moreover, although both the NSDO and the IDO considerably reduce the crosswind responses, it is clear that the NSDO performs better in the crosswind direction for both acceleration and drift responses. The top acceleration spectra shown in Fig. 29 again stress the fact that the NSDO achieves a larger damping ratio and lower spectral amplitude for higher modes (particularly the 2nd mode).SummaryThe numerical results show that structural responses of tall buildings under seismic and wind loads involve multimode components, suggesting the importance of multimode damping enhancement. The response reduction performances of different outrigger systems confirm the analytical findings in the preceding sections. The frequency-independent negative stiffness achieved by the NSD makes the NSDO performance less sensitive to frequency variation as compared to the IDO, which is frequency sensitive. Thus, the NSDO designed for the 1st mode is still able to provide considerable damping for higher modes, which is substantially different from the IDO and other inerter-based damping systems.References Agrawal, A., P. Tan, S. Nagarajaiah, and J. 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