The general equations of motion of the stand-alone, two-level, 3D shear-type system can be formally written as in Equation 1: M x ̈ + C x ̇ + K x = − M r u ̈ g t , (1) where M , C , and K are the mass, damping, and stiffness matrices of dimension 6 × 6 , respectively, the vector x = u s 1 , v s 1 , ϕ s 1 , u s 2 , v s 2 , ϕ s 2 T accounts the six Lagrangian parameters, and u g is the seismic input. As observed, the seismic excitation is applied along a direction inclined of angle ζ respective to the horizontal line (see Figure 1). In the paper, ζ is always set to ζ = 4 5 o . Since the mass matrix M is diagonal, the projection vector r = cos ( ζ ) , sin ( ζ ) , 0 , cos ( ζ ) , sin ( ζ ) , 0 T . Although assuming a fixed angle may appear restrictive, preliminary analyses indicated that variations in this parameter have only a minor influence on the resulting maps, which remain qualitatively consistent. Therefore, only the results obtained with ζ = 4 5 o are presented herein.

The damping matrix C is obtained from the stand-alone, two-level, 3D shear-type system, by a Rayleigh approach. Specifically, C is a linear combination of mass and stiffness matrices as: C = α M + β K , (2) where coefficients α and β are: α = ξ 2 ω 1 ω 2 ω 1 + ω 2 , β = ξ 2 ω 1 + ω 2 . (3)

In Equation 3, ω 1 and ω 2 are the circular frequencies of the first two oscillation modes of the structure, whereas ξ is the modal damping of such first two modes. In the analyses performed, it is always set ξ = 0.05 . The damping matrix M obtained from Equation 2 is used also to describe the damping properties of the 3D frame structure in System 1 and System 2.

The equations of motion are obtained by a Lagrangian approach. To write the kinetic T and potential V energies, the displacements of the points labeled from A to D (vertexes of the first level) and from P to S (vertexes of the second level) needed to be written. In fact, the elastic springs and the virtual masses of the inerters are all applied in these points (see Figure 1a). With respect to coordinate systems that have origin in the mass centers of the two levels (intersection of the diagonals), such displacements read (Equation 4): x A = x g + u s 1 + ϕ s 1 l y , y A = y g + v s 1 − ϕ s 1 l x , x B = x g + u s 1 + ϕ s 1 l y , y B = y g + v s 1 + ϕ s 1 l x , x C = x g + u s 1 − ϕ s 1 l y , y C = y g + v s 1 − ϕ s 1 l x , x D = x g + u s 1 − ϕ s 1 l y , y D = y g + v s 1 + ϕ s 1 l x , ⋯ x P = x g + u s 2 + ϕ s 2 l y , y P = y g + v s 2 − ϕ s 2 l x , x Q = x g + u s 2 + ϕ s 2 l y , y Q = y g + v s 2 + ϕ s 2 l x , x R = x g + u s 2 − ϕ s 2 l y , y R = y g + v s 2 − ϕ s 2 l x , x S = x g + u s 2 − ϕ s 2 l y , y S = y g + v s 2 + ϕ s 2 l x , (4) where x g = u g cos ( α ) and y g = u g ⁡ sin ( α ) are the components of the seismic input along the horizontal and vertical directions, respectively. The kinetic energy then reads (Equation 5): T = 1 2 m s 1 x ̇ g + u ̇ s 1 2 + y ̇ g + v ̇ s 1 2 + I s 1 ϕ ̇ s 1 2 + m s 2 x ̇ g + u ̇ s 2 2 + y ̇ g + v ̇ s 2 2 + I s 2 ϕ ̇ s 2 2 + M 1 x ̇ B − x ̇ g 2 + x ̇ D − x ̇ g 2 + M 2 y ̇ C − y ̇ g 2 + y ̇ D − y ̇ g 2 , (5) where I s 1 and I s 2 are the polar moment of inertia of the two levels of the 3D frame structure. The potential energy reads (Equation 6): V = 1 2 k x A x A − x g 2 + k x C x C − x g 2 + k y A y A − y g 2 + k y B y B − y g 2 + k xAP x P − x A 2 + k xCR x R − x C 2 + k yAP y P − y A 2 + k yBQ y Q − y B 2 , (6) where k x A and k x C represent the sum of the stiffness of all the columns below the first level along horizontal lines passing through points A and C , respectively. Similarly, k y A and k y B represent the sum of the stiffness of all the columns below the first level along vertical lines passing through points A and B , respectively. The same meaning is associated to the other stiffnesses appearing in the potential energy. Specifically, k xAP and k xCR represent the sum of the stiffness of all the columns below the second level along horizontal lines passing through points P and R , respectively. Similarly, k yAP and k yBQ represent the sum of the stiffness of all the columns below the second level along vertical lines passing through points P and Q , respectively.

To account for the non-conservative forces due to the damping of the structure, their virtual work δ W needs to be computed (Equation 7). It reads: δ W = C x ̇ T δ x , (7) where C is the damping matrix and x is the vector containing the six Lagrangian parameters, already described in Section 2.1. Finally, δ x represents the virtual variation of the Lagrangian parameters.

By introducing the Lagrangian function L = T − V , the six equations of motion are then obtained through Equation 8: d d t ∂ L ∂ x ̇ − ∂ L ∂ x = C x ̇ . (8) The equations of motion are omitted for brevity, but all the equations presented in this section permit their derivation.

To derive the equations of motion of the System 2 (see Figure 1b), the displacements of all the vertexes of the 3D frame structure need to be evaluated. Their displacements are exactly equal to those written for System 1 and reported in Equation 4. Moreover, also the displacements of all the vertexes of the external auxiliary structures have to be written. With respect to coordinate systems that have origin in the mass centers of the two external structures (intersection of the diagonals, see Figure 1b), such displacements read as in Equation 9: x L = x g + u 1 + ϕ 1 l y , x M = x g + u 1 − ϕ 1 l y , x N = x g + u 1 + ϕ 1 l y , x O = x g + u 1 − ϕ 1 l y , ⋯ y E = y g + v 2 − ϕ 2 l x , y F = y g + v 2 + ϕ 2 l x , y H = y g + v 2 − ϕ 2 l x , y I = y g + v 2 + ϕ 2 l x , (9) Then kinetic energy T reads (Equation 10): T = 1 2 m s 1 x ̇ g + u ̇ s 1 2 + y ̇ g + v ̇ s 1 2 + I s 1 ϕ ̇ s 1 2 + m s 2 x ̇ g + u ̇ s 2 2 + y ̇ g + v ̇ s 2 2 + I s 2 ϕ ̇ s 2 2 + m 1 x ̇ g + u ̇ 1 2 + I 1 ϕ ̇ 1 2 + m 2 y ̇ g + v ̇ 2 2 + I 2 ϕ ̇ 2 2 + M 1 x ̇ L − x ̇ g 2 + x ̇ M − x ̇ g 2 + M 2 y ̇ E − y ̇ g 2 + y ̇ F − y ̇ g 2 , (10) where I 1 and I 2 are the polar moment of inertia of the two auxiliary structures. The potential energy reads as in Equation 11: V = 1 2 k x A x A − x g 2 + k x C x C − x g 2 + k y A y A − y g 2 + k y B y B − y g 2 + k xAP x P − x A 2 + k xCR x R − x C 2 + k yAP y P − y A 2 + k yBQ y Q − y B 2 + k x N x N − x g 2 + k x O x O − x g 2 + k y H y H − y g 2 + k y I y I − y g 2 + k xBL x L − x B 2 + k xDM x M − x D 2 + k yCE y E − y C 2 + k yDF y F − y D 2 , (11) where k xBL , k xDM , k uCE , and k yDF are the stiffness of the coupling visco-elastic devices that connects the 3D frame structure to the auxiliary structures (see Figure 1b).

To account for the non-conservative forces due to the damping of the structure, their virtual work δ W needs to be computed as in Equations 12, 13: δ W = C s + C ext x ̂ ̇ T δ x ̂ , (12) where C s = C 0 0 0 , x ̂ = x x ext . (13) Quantities C and x are the damping matrix and vector of the Lagrangian parameters of the 3D frame structure already introduced in Equation 7, and C s is a 10 × 10 matrix. Moreover, C ext is still a 10 × 10 matrix and accounts for the damping of the auxiliary structures and coupling visco-elastic devices. Vector x ext = u 1 , ϕ 1 , v 2 , ϕ 2 T contains the Lagrangian parameters associated to the auxiliary structures. Due to the large dimensions of Matrix C ext and by exploiting the symmetry properties, only its symmetric components c i j are reported in Equation 14: c 11 = − c xBL − c xDM c 21 = 0 ,   c 22 = − c yCE − c yDF c 31 = c xDM l y − c xBL l y ,   c 32 = c yCE l x − c yDF l x ,   c 33 = − c xBL l y 2 − c xDM l y 2 − c yCE l x 2 − c yDF l x 2 c 41 = 0 ,   c 42 = 0 ,   c 43 = 0 ,   c 44 = 0 c 51 = 0 ,   c 52 = 0 ,   c 53 = 0 ,   c 54 = 0 ,   c 55 = 0 c 61 = 0 ,   c 62 = 0 ,   c 63 = 0 ,   c 64 = 0 ,   c 65 = 0 ,   c 66 = 0 c 71 = c xBL + c xDM ,   c 72 = 0 ,   c 73 = c xBL l y − c xDM l y ,   c 74 = 0 ,   c 75 = 0 ,   c 76 = 0 ,   c 77 = − c xBL − c xDM − c x N − c x O c 81 = c xBL l y − c xDM l y ,   c 82 = 0 ,   c 83 = c yDF l x − c yCE l x ,   c 84 = 0 ,   c 85 = 0 ,   c 86 = 0 ,   c 87 = − c xBL l y + c xDM l y − c x N l y + c x O l y ,   c 88 = − c xBL l y 2 − c xDM l y 2 − c x N l y 2 − c x O l y 2 c 91 = 0 ,   c 92 = c yCE + c yDF ,   c 93 = c yDF l x − c yCE l x ,   c 94 = 0 ,   c 95 = 0 ,   c 96 = 0 ,   c 97 = 0 ,   c 98 = 0 ,   c 99 = − c yCE − c yDF − c y H − c y I c 101 = 0 ,   c 102 = c yDF l x − c yCE l x ,   c 103 = c yCE l x 2 + c yDF l x 2 ,   c 104 = 0 ,   c 105 = 0 ,   c 106 = 0 ,   c 107 = 0 ,   c 108 = 0 ,   c 109 = c yCE l x − c yDF l x + c y H l x − c y I l x ,   c 1010= − c yCE l x 2 − c yDF l x 2 − c y H l x 2 − c y I l x 2 (14)

Damping coefficients c x N , c x O , c y H , and c y I are associated to the auxiliary structures and represent the viscous characteristic of the devices applied to the vertexes that correspond to the last subscript. As an instance, c x N is the damping coefficient that complete the description of the visco-elastic device applied at point N and having stiffness k x N . Note that in Figure 1 the visco-elastic properties associated to the elements of the structures are drawn only with springs for simplicity. Such dampings are conventionally defined as in Equation 15: c x N = c x O = 2 ω e 1 m 1 ξ e , c y H = c y I = 2 ω e 2 m 2 ξ e , (15) where ω e 1 = ( k x O + k x N ) / m 1 and ω e 2 = ( k y H + k y I ) / m 2 are the frequencies of the two auxiliary structures assumed isolated. It is always considered ξ e = 0.05 .

Damping coefficients c xBL , c xDM , c yCE , and c yDF are instead associated to the coupling devices that connect the couple of vertexes corresponding to the last two subscripts. As an instance, c xBL is the damping coefficient that complete the description of the visco-elastic device connecting points B and L and of stiffness k xBL . Such dampings are conventionally assumed as in Equation 16: c xBL = c xDM = 2 ω e 1 m 1 ξ c , c yCE = c yDF = 2 ω e 2 m 2 ξ c , (16) where it is always considered ξ c = 0.10 .

By introducing the Lagrangian function L = T − V , the ten equations of motion are then obtained through Equation 17: d d t ∂ L ∂ x ̂ ̇ − ∂ L ∂ x ̂ = C s + C ext x ̂ ̇ , (17) and can be synthetically written as in Equation 18: M ̂ x ̂ ̈ + C ̂ x ̂ ̇ + K ̂ x ̂ = − M ̂ r ̂ u ̈ g t . (18)

The explicit expressions of such equations are presented in the Supplementary Appendix.

An inerter is a mechanical device designed to exert a force, F I D , that is directly proportional to the relative acceleration between two connection points, or terminals. A common design approach involves using a rack-pinion-flywheel assembly, as depicted in Figure 3. In this configuration, the device’s primary components include two or more flywheels with radius R i and mass m ω i , which are linked to a linear rack through a pinion-gear mechanism. This arrangement allows the inerter to effectively translate relative accelerations into force outputs. The resisting force F I D of the inerter device is defined as in Equation 19 (Makris and Kampas, 2016; Thiers-Moggia and Málaga-Chuquitaype, 2018): F I D = m I u ̈ d x − u ̈ s x . (19) Here, m I represents the virtual mass (or inertance) of the inerter. This virtual mass can be increased by incorporating additional flywheels into the device. For an inerter device comprising N flywheels, m I is given by Equation 20: m I = 1 2 m ω 1 R 1 2 ρ 1 2 + 1 2 m ω 2 R 1 2 R 2 2 ρ 1 2 ρ 2 2 + ⋯ + 1 2 m ω i R 1 2 R 2 2 ⋯ R i 2 ρ 1 2 ρ 2 2 ⋯ ρ i 2 + ⋯ + 1 2 m ω N R 1 2 R 2 2 ⋯ R i 2 ⋯ R N 2 ρ 1 2 ρ 2 2 ⋯ ρ i 2 ⋯ ρ N 2 . (20)

The first two terms of Equation 20 define the virtual mass m I of the device shown in Figure 4. Importantly, in this type of inerter device, kinetic energy can transfer freely between the inerter and the structure due to the absence of clutches or other mechanisms that could restrict such energy exchange, as seen in (Makris and Kampas, 2016).

After selecting the frame characteristics for the parametric analysis, the fundamental parameter to consider as a variable is the value of the virtual mass provided by the inerters. The total virtual mass M acting in both x and y directions thanks to the couples of equal inerters of virtual mass M 1 (acting along the x direction) and M 2 (acting along the y direction) (see Figure 1a) is parametrized with respect to the mass of the first level m s 1 as in Equation 21: M = η m s 1 . (21)

Therefore, for equal inerters along the x and y directions, follows that M 1 = M 2 = M / 2 . It is worth observing that, when the distribution of the virtual masses is uniform along the sides of the structures, as considered in this paper, the mass centers of both System one and System two coincide with the center of mass of the stand-alone frame structure.

For System 2, a higher number of variable parameters than System 1 are investigated to analyse the performance of the protection method. The variable parameters considered in the analysis are.

• The total virtual mass M of the inerters in both the x and y directions (parametrized as before, Equation 22)

For equal inerters along the x and y directions, follows that M 1 = M 2 = M / 2 .

• The total stiffness K e of the external auxiliary structures along their shorter directions (assumed to be equal in both, Equation 23)

In all the analyses the mass of the auxiliary structures are considered fixed quantities, m 1 = m 2 = 0.1 m s 1 .

The parametric analyses are performed on two different frame structures. Table 1 shows the original geometrical characteristics of the studied M-DOF frame structures, that are then modeled by two-level, 3D shear-type systems. Specifically, the columns Storeys, Area, 2 l x , and 2 l y refer to actual the M-DOF frame structures. The structure labeled as Frame 1 (first row of the Table) has four identical rectangular levels, each with an area of 300 m 2 and dimensions 20.0 × 15.0 m 2 , while the other, labeled as Frame 2 (second row of the Table), has five identical rectangular levels, each measuring 350 m 2 and with dimensions 25.0 × 14.0 m 2 . Both structures are assumed to have levels with uniform stiffness in both directions. Additionally, the mass centers at each level coincide, while the stiffness centers, although aligned with each other at the two levels, are offset from the mass centers.

The remaining columns of Table 1 refer to the mechanical characteristics of the two-level, 3D shear-type systems, which are dynamically equivalent to the actual structures. The previously mentioned dynamic equivalence criterion ((Fabrizio et al., 2017a; Fabrizio et al., 2019; Fabrizio et al., 2017b)), used to assess the characteristics of the 3D shear-type systems, determines the values of masses and stiffnesses such that the first level corresponds to the first level of the actual structure, while the second level corresponds to the top level of the actual structure. Consequently, the displacement differences between the two levels serves as a measure of the drift in the actual structure. Columns m s 1 and m s 2 provide the masses associated to each level. It is worth noticing that in Frame 1, m s 2 is three times m s 1 since the second level of the 3D shear-type system has to represent all the levels above the first. The same thing occur for the Frame 2, where m s 2 is four times m s 1 . Columns K 1 x = K 1 y and K 2 x = K 2 y are the total stiffness of the two levels of the 3D shear-type system along directions x and y . Finally, the last column, Period, provides the fundamental oscillation period of the structures. The applied equivalence criterion ensures that both the actual structures and the 3D shear-type systems have the same oscillation period.

The position of the stiffness centers k s of the 3D shear-type systems, assumed coincident at each level, is reported in Table 2. The two columns provide the coordinates of k s with respect the structural vertex A ≡ P (see Figure 1).

The eccentricity of the stiffness center with respect to the mass center is caused by the non-uniform distribution of stiffness among the columns along the two alignments parallel to the x − and y − axes. Specifically, the position of the stiffness center in Table 2 is determined by the following stiffness distribution (Equation 26): k x A = 0.60 K 1 x , k x C = 0.40 K 1 x k y A = 0.60 K 1 y , k y B = 0.40 K 1 y … k xAP = 0.60 K 2 x , k xCR = 0.40 K 2 x k yAP = 0.60 K 2 y , k yBQ = 0.40 K 2 y (26)

To evaluate the performance of the proposed methods, two performance indexes are introduced (Equation 27). They read: α 1 = max s D t max s ̃ D t , α 2 = max Δ s D S t max Δ s ̃ D S t (27) where s D = u D 2 + v D 2 is the displacement of point D of System 1 or System 2, whereas s ̃ D is the displacement of the same point D of the stand-alone frame structure. Similarly, Δ s D S = ( u S − u D ) 2 + ( v S − v D ) 2 is the relative displacement (total drift) between points S and D of System 1 or System 2 (drift), whereas Δ s ̃ D S is the same drift evaluated on the stand-alone frame structure. Therefore, α 1 refers to the displacement of the first level of the structures, whereas α 2 refers to the drift between the top and the first levels.

Two additional performance indexes related to the torsion of the structure are used to assess the performance of the protection methods (Equation 28). They are defined as: β 1 = max ϕ s 1 t max ϕ ̃ s 1 t , β 2 = max ϕ s 2 t − ϕ s 1 t max ϕ ̃ s 2 t − ϕ ̃ s 1 t , (28)

where ϕ s 1 and ϕ s 2 are the rotations of the two levels of the System 1 or Sytem 2, whereas ϕ ̃ s 1 and ϕ ̃ s 2 are the corresponding rotations of the stand-alone frame structure. It is worth observing that β 1 refers to the torsion of the first level of the structures, whereas β 2 refers to torsional drift between the top and first levels.

With this formulation, the effectiveness of the external apparatus improves as the values of α 1 , α 2 and β 1 , β 2 decrease relative to unity. Although no absolute engineering threshold is imposed to quantify the effectiveness of the protection systems, a response ratio significantly lower than unity (typically below 0.8) can be considered as indicative of substantial improvement, with values approaching 0.6 or lower denoting particularly effective performance.

Both the stand-alone 3D frame structure and System 1 are 6-DOF mechanical systems, therefore they admit six frequencies and oscillation modes. The results of the spectral analysis is reported in Figure 4, which shows the periods and modal shapes of the stand-alone frame structure (left columns), and those of the System 1 with two different total virtual mass M . Specifically, the middle column refers to the case with M = 10 m s 1 and the right column shows the case with M = 50 m s 1 . In the modal shapes, the dashed line refers to the undeformed configuration, the thick solid line shows the modal displacement of the first level, whereas the thin solid line refers to the second level.The first observation concerns the fact that the mode periods increase as the virtual mass increases, producing effects similar to those achieved through base isolation (Mazza and Labernarda, 2022; Mazza et al., 2024). The other notable effect is related to the role played by the virtual mass M on the modal shapes. As M increases, a separation of effects on the oscillation modes can be observed. Specifically, for high values of M , only the first level exhibits significant modal displacement in the first three modes, while the second level is nearly superimposed on the first. In the second set of three modes, only the second level shows significant modal displacement, with the first level nearly superimposed on the undeformed configuration. This separation of effects between the two levels of the system clarifies the structure’s dynamic response, enabling a more straightforward design approach.

The spectral analysis on System 2 is performed under the hypothesis that γ e = γ c = 1 ( i . e ., K e = K c = 1.0 K xs1 , see Equations 23 and 24). Moreover, the stiffness of the coupling devices ( k xBL , k xDM and k yCE , k yDF ) are distributed to align the stiffness and mass centers of the first level of the coupled structure. This can be obtained with specific values of α and β in Equation 25: α = 0.4 , β = 0.6 .

Figure 5 shows the frequencies and modal shapes of the stand-alone frame structure and of System 2 for two different values of M . The same graphical conventions as in Figure 4 are used: dashed lines represent the undeformed configuration, while thick and thin solid lines represent the first and second levels, respectively. Since the stand-alone and System 2 admit different number of modes (six and ten, respectively), in Figure 5 the modes of the stand-alone system and System 2 with similar shapes are arranged side by side.

For the smallest value of M , M = 20 m s 1 , the first three modes of System 2 (middle column) show a slight increase in periods and exhibit shapes similar to those of the first three modes of the stand-alone structure. However, the modal amplitudes of the frame structure of System 2 are smaller than those of the stand-alone structure, with the predominant modal displacements occurring primarily in the external structures. The last three modes exhibit periods and modal shapes close to the last three of the stand-alone structure, with a negligible participation of the external auxiliary structures.

An increase of the virtual mass up to the value of M = 100 m s 1 results in a significant increase in the periods of the first three modes. Additionally, these modes primarily affect the first level, as the modal displacements at the second level are negligible, with predominant effects mainly occurring in the external structures. Modes 5, 6, and seven exhibit periods similar to those of the first three modes of the stand-alone structure, with the predominant modal displacements concentrated at the second level. This indicates a distinct separation of modal effects between the first and second levels, as previously discussed for System 1. Finally, the last three modes of System two have periods that closely match those of the last three modes of the stand-alone structure, albeit with slightly higher modal displacements at both levels. It is noteworthy that in modes 5 to 10, the external auxiliary structures are minimally involved due to their small modal displacements. As a final remark, mode four of System two does not have a direct correspondence with the modes of the stand-alone structure. It involves the frame structure through translational displacement and the external auxiliary structures through pure torsional displacement. Also in this case, distinguishing the modal effects at each system level enhances clarity in analyzing the structure’s dynamic response, thereby facilitating a more streamlined design process.

The following earthquake records are considered in this study (Figure 6).

(a) Kobe, 1995 Japan earthquake, Takarazuka station, 0 deg, ground level, position of the station: 34.8090N, 135.3440W;

For ease of reference, each record is labeled using the underlined name provided in the list.

The use of three natural earthquake records (Kobe, Newhall, and Parkfield) is driven by the need for an initial evaluation of the protection system’s performance under seismic inputs characterized by distinct spectral contents. These records are selected based on their effective use in prior studies by the authors, where they demonstrated marked differences in system dynamics due to variations in frequency content. This distinction is further supported by their pseudo-acceleration spectra, which highlight how different excitations can engage different modal responses of the structural system.

In this preliminary phase of the study, the earthquake records are used without scaling to a specific design spectrum or code-based intensity level. The aim is to qualitatively assess the sensitivity of the protection system to differences in seismic frequency content using natural, unmodified ground motions, thus preserving the intrinsic variability present in real earthquake data. This choice avoids potential distortions introduced by artificial scaling and is justified in the context of the linear structural model adopted in the study. Since the performance of the proposed systems is expressed through normalized response ratios (e.g., α 1 and α 2 ), applying uniform scaling to the input ground motions would not affect the results since both the numerator and denominator of the ratios would scale proportionally.Nonetheless, for a comprehensive assessment of practical applicability, especially for a specific structure or site, ground motions should be selected and scaled in accordance with seismic code provisions, hazard levels, and target spectra. Future developments of this research will include such extensions to enable a more rigorous, code-compliant performance evaluation of the proposed protection strategies.

The parametric seismic analysis involves evaluating the performance indices, α 1 and α 2 , by varying the total virtual mass M and using a single earthquake record. As a result, performance curves plotting the performance indexes against the total virtual mass M are generated. Each graph also includes a line at unity to highlight the range of M values where the proposed method effectively reduces the seismic effects on the frame structure. Specifically, in regions where the performance curves fall below this line, System one exhibits better seismic behavior than the stand-alone frame structure.

Figure 7 shows the results obtained by exciting Frame 1 (see Table 1) with the three selected earthquake records. Each graph displays two performance curves: the solid curve corresponds to the α 1 index, while the dashed curve represents the α 2 index. As can be observed, there are large ranges of η ( i . e ., of M ) where both the indexes are less than unity, assuring good performances of System 1 respective to the stand-alone frame structure. Notably, the best performances of the two levels (identified by the minimum values of α 1 and α 2 ) occur for different virtual mass M . Specifically, to ensure optimal seismic response at the first level, the value of M must be higher than that required to achieve the best response at the second level. This can be explained by noting that, at the highest values of M , the system exhibits a separation in the dynamics between the first and second levels, as previously discussed. Furthermore, the modes primarily involving the first level acquire longer periods, thereby shifting the dynamics into the range of periods where the earthquakes exhibit lower spectral power. In contrast, the seismic response of the second level is associated with modes that have periods falling within the range where the earthquakes exhibit the highest spectral power. Conversely, the optimal seismic response at the second level, represented by α 2 , necessitates small values of M . However, these values are found in regions where the performance index α 1 remains significantly greater than unity. Therefore, selecting an optimal value for the virtual mass M must consider both performance indexes simultaneously. It is important to note that developing an optimal procedure for choosing the value of M is not the focus of this paper. Finally, based on the results, it can be concluded that the performance trends of System one are qualitatively consistent across the three selected earthquake records, as evidenced by the similar shapes of the performance curves and the comparable ranges of the parameter η in which optimal responses are observed in Figure 7. Nonetheless, some variation in the amplitude of the performance indexes is apparent, most notably for the Parkfield earthquake, where the range of η values for which α 1 exceeds unity is significantly narrower than in the other two cases.

The time-histories of the total displacement s D and drift Δ s D S (see Section 3.4), for both System one and the stand-alone structure are obtained and compared in Figure 8. This figure shows the time-histories of the absolute value of s D and Δ s D S that refer to points labelled with letters P 1 and P 2 in Figure 7. Specifically, at point P 1 , the performance index α 2 reaches its minimum value, while at point P 2 , the index α 1 attains its minimum value. Each graph in Figure 8 contains two time-histories: one representing the stand-alone structure (thick grey line) and the other representing System 1 (thin black line). The system’s behavior at point P 1 (graphs in the first row of Figure 7) demonstrates that the maximum drift Δ s D S of System 1 is smaller than that of the stand-alone structure. Conversely, the maximum displacement s D is larger for System 1, aligning with the performance curves in Figure 7. The time-histories referring to point P 2 (second row of Figure 8) show that the maximum values of both s D and Δ s D S for System 1 are smaller than those for the stand-alone structure, consistent with the curves in Figure 7.

The performance curves obtained from Frame 2 (see Table 1) and shown in Figure 9 are qualitatively similar to those referring to Frame one that are shown in Figure 7. Independently from the earthquake record, the performance index α 1 acquires the minimum value for high values of the parameter η . Contrarily, the performance index α 2 reaches its minimum value for small values of η . As a general observation, the efficiency of using inerter devices to enhance seismic behavior decreases as the dimensions of the frame structure, and in particular its height, increase. The observed reduction in performance is primarily associated with the relatively low installation level selected for the connection of the protection system. Inerters or auxiliary structural elements exert a direct influence on the displacement of the first storey, contributing to a modal regularization that effectively limits top-storey displacements. However, as the height of the structure increases (i.e., with a greater number of storeys), the influence of this indirect effect tends to diminish due to the increasing structural flexibility and mode complexity.

The parametric seismic analysis involves mapping the values of the performance indexes, α 1 and α 2 (or β 1 and β 2 ), onto the parameter plane defined by γ e and γ c (see Equations 23 and 24, respectively) using a single earthquake record and for a fixed value of the virtual mass M . This process generates performance maps, which are contour plots illustrating the distribution of α 1 and α 2 across the γ e − γ c parameter’s plane. To enhance readability, the contour levels are delineated using a specific greyscale scheme, where lighter shades correspond to smaller values of the performance indices. A better seismic performance of System two than the stand-alone frame structure is observed within regions of the maps characterized by clearer greyscale tones. These areas indicate a superior system performance, as denoted by lower values of the performance indexes α 1 and α 2 .

The first performance maps presented are those referring to Frame 1 (see Table 1), excited by Parkfield earthquake, are obtained. The stiffness of the coupling devices ( k xBL , k xDM and k yCE , k yDF ) are distributed to align the stiffness and mass centers of the first level of the coupled structure. Since, the sum K c of these stiffnesses in each direction is related to the first-level stiffness, K x 1 (or K y 1 ) through the parameter γ c (see Equation 24), the distribution of the coupling stiffnesses directly depends on this parameter. Specifically, the parameters α and β appearing in Equation 25 read (Equation 29): α = 0.5 − 0.1 γ c , β = 0.5 + 0.1 γ c . (29)

Figure 10 shows the performance maps, arranged in matrix form. Specifically, along the rows the maps refer to different values of M , whereas along the columns the maps refer to the two performance indexes α 1 and α 2 . In the regions colored with uniform grey the performance indexes are higher than unity, therefore no advantage in using the proposed method occurs. It should be noted that the grayscale levels used in the performance maps are independently scaled for each figure, as they are adapted to the specific range of values of the performance index displayed. Consequently, no uniform colormap bar is shown. These maps are intended to qualitatively highlight regions of improved performance, with lighter shades indicating lower index values relative to the stand-alone case, rather than to serve as absolute design charts. The analysis reveals that the size of the advantage regions, where the performance indexes are less than unity, as well as the values of the indexes themselves, depend on the virtual mass M . An increase in this parameter has opposing effects on the two indexes. Specifically, as M increases, the advantage regions in the α 1 map expand, while those in the α 2 map contract. Additionally, an increase in M generally leads to a decrease in the values of α 1 and an increase in those of α 2 . The spectral analysis in Section 4 provides insights into the dependence of the maps on the value of M . For high values of M , there is a form of modal decoupling between the two levels of the frame system. The modes that primarily involve the first level acquire a longer period, shifting their dynamics outside the range where the earthquake has higher spectral power. Consequently, the seismic effects on the first level are less severe. Conversely, the modes involving the second level acquire periods that fall within ranges where the earthquake has higher spectral power, leading to more intense excitation. Therefore, in general, it is not possible to find a pair of values for the parameters γ c and γ e that minimize both performance indices simultaneously. The optimal results should be sought for moderate values of M , where the best trade-off between the performance indices can be achieved.

It is of great interest to evaluate the effects occurring from the coincidence between the stiffness and mass centers at the first level. Therefore, while the results shown in the previous Figure 10 are obtained considering this coincidence, in Figure 11 the performance maps with and without such coincidence are compared. The results refer to Frame 1, excited by Parkfield earthquake, and fixed M = 30 m s 1 . Also in this figure a matrix arrangement of the maps is used. The first two rows (Figure 11a) refer to maps obtained with and without coincidence between the stiffness and mass centers. As observed, this coincidence primarily affects the drift of the frame structure, as the advantage region in the α 2 map expands and the values of α 2 decrease.

The coinciden\ce between the stiffness and mass centers at the first level has a limited effect on the α 1 index, as the corresponding maps remain qualitatively similar in their overall trends and advantage regions. However, notably, the case without alignment shows a local minimum in the region of higher γ e and γ c values, suggesting the emergence of particularly favorable configurations in that portion of the parameter space. The performance indexes β 1 and β 2 provide interesting information about the torsion of the first level and the torsional drift of the second level of the frame structure. Figure 11b compares the performance β 1 and β 2 maps obtained with and without the coincidence between stiffness and mass centers. The results show that this coincidence significantly reduces the torsion at the first level, as the advantage region increases and the values of β 1 decrease. Conversely, the torsional drift, measured by β 2 , remains largely unchanged. As expected, since the coincidence between the stiffness and mass centers occurs only at the first level, the improvement in torsional effects is primarily noticeable at the first level. Nevertheless, there is also a simultaneous, notable improvement in the overall drift of the frame structure.

To further assess the effectiveness of System two in mitigating seismic effects, Frame 1 with a total virtual mass of M = 30   m s 1 is also subjected to excitation from the Kobe and Newhall earthquake records. The corresponding performance maps, reported in Figure 12, complement the results previously obtained under the Parkfield earthquake and shown in Figure 10 (second row) and Figure 11a (second row). These additional maps confirm that System two maintains its protective capabilities across earthquakes characterized by distinct spectral contents. TextcolormagentaConsistently across the three seismic inputs, the maps reveal a broad region within the γ c – γ e parameter space where both performance indices α 1 and α 2 are significantly below unity, denoting a simultaneous reduction of first-level displacements and inter-storey drift. Although slight variations in the extent and location of these favorable regions are observed, likely due to the different dominant frequency bands of the earthquakes, the overall trend supports the robustness of the adopted protection strategy. TextcolormagentaMore specifically, the Kobe earthquake induces slightly larger values of the performance indices compared to Parkfield, with optimal regions shifting slightly toward higher values of γ c . In contrast, the Newhall excitation produces performance contours similar in structure to those from the Parkfield case, although the drift index α 2 exhibits more sensitivity to variations in the external structure stiffness γ e . These findings emphasize the adaptability of System two under real earthquake conditions and reinforce the idea that optimal tuning of the auxiliary system can be maintained within practical ranges of stiffness ratios.

The performance maps for Frame 2, subjected to seismic excitation from the three selected earthquakes, are presented in Figure 13. All results are obtained for a fixed value of the total virtual mass M = 50   m s 1 . These maps provide a comprehensive overview of the performance of System two when applied to a taller frame structure. Across all earthquake scenarios, System two consistently demonstrates its ability to reduce both the displacement at the base level and the inter-storey drift, confirming its applicability beyond the smaller prototype frame.

Nonetheless, the maps also reveal that the regions of optimal performance vary depending on the specific performance index considered. In particular, the minima of the displacement performance index α 1 and the drift performance index α 2 do not coincide in the γ c - γ e parameter space. This divergence indicates that no single combination of coupling and external stiffness values simultaneously optimizes both metrics. As a result, the design strategy for System two must account for this inherent trade-off, selecting parameter configurations that balance the reduction in first-level displacements with the control of structural drift. Such balance is especially critical in taller buildings, where different vibration modes may dominate the response depending on the excitation frequency content and structural configuration.

Finally, to understand how the coupling with external auxiliary structures works in reducing the seismic effects on frame structures, the time-histories of systems represented by labels P 3 and P 4 in Figure 13 are obtained and discussed. Figure 14 show such time-histories, arranged in matrix form. Specifically, rows refer to the different systems labeled with P 3 and P 4 , while columns refer to displacements along the x − or y − axis. Each graph contains three curves that represent the following: the displacement of point D at the first level of the frame structure ( u D in the x -direction and v D in the y -direction), the displacement of point M on one of the external structures ( u M in the x -direction and v M in the y -direction), and the drift between point S at the second level of the frame structure and point D ( Δ u D S in the x -direction and Δ v D S in the y -direction). These points are all clearly indicated in Figure 1. As can be observed, at point P 3 on the map, the minimum value of the performance index α 1 occurs. The time-histories show that the two levels of the frame structure move in-phase with each other, while the external auxiliary structure moves almost in counter-phase to the entire frame structure. In this case, the external structure works as a tuned mass damper for the structure being protected. Conversely, P 4 is located at a point of minimum value for the index α 2 . Here, the time-histories reveal that the first level and the external structure are in-phase with one another, whereas the second level moves primarily in counter-phase relative to all other components of System 2. Therefore, it can be concluded that both the first level and the external structure act as tuned mass dampers for the second level.