Consider N sets of SDOF building models for covering a broad frequency range. Let ω 1 , … , ω N , h 1 , … , h N denote the natural circular frequencies and the damping ratios, respectively. The ground acceleration u ¨ g d t , … , u ¨ g N d t d t , , u ¨ g N d t + 1 d t , … , u ¨ g N d t + Δ N d t d t is defined in a discrete time system with time step d t . Note that N d t d t is the effective duration of the generated ground motion and u ¨ g N d t + 1 d t , … , u ¨ g N d t + Δ N d t d t are set to 0. The displacement and velocity responses d ω i , h i , N d t + Δ N d t d t , d ˙ ω i , h i , N d t + Δ N d t d t at t = N d t + Δ N d t d t are expressed as d N d t + Δ N d t d t d ˙ N d t + Δ N d t d t = G u ¨ g d t ⋮ u ¨ g N d t d t (1) where d ⁡ N d t + Δ N d t ⁢ d t = d ω 1 , h 1 , N d t + Δ N d t d t , … , d ω N , h N , N d t + Δ N d t d t T (2) d ˙ ⁢ N d t + Δ N d t ⁢ d t = d ˙ ω 1 , h 1 , N d t + Δ N d t d t , … , d ˙ ω N , h N , N d t + Δ N d t d t T (3) G = g d ω 1 , h 1 , N d t + Δ N d t − 1 d t ⋯ g d ω 1 , h 1 , Δ N d t d t ⋮ ⋮ g d ω N , h N , N d t + Δ N d t − 1 d t ⋯ g d ω N , h N , Δ N d t d t g d ˙ ω 1 , h 1 , N d t + Δ N d t − 1 d t ⋯ g d ˙ ω 1 , h 1 , Δ N d t d t ⋮ ⋮ g d ˙ ω N , h N , N d t + Δ N d t − 1 d t ⋯ g d ˙ ω N , h N , Δ N d t d t (4) g d ω , h , Δ t , g d ˙ ω , h , Δ t are the displacement and velocity responses after Δ t seconds for the SDOF model with natural circular frequency ω and damping ratio h under the triangular wave, which has a unit acceleration only at t = 0 and zero acceleration at all other time (Akehashi and Takewaki, 2022b). When 1 / d t is multiplied by the acceleration amplitude and the operation d t → 0 is taken, the triangular wave becomes equivalent to the impulsive ground motion u ¨ g = δ t . g d ω , h , Δ t , g d ˙ ω , h , Δ t can be also regarded as the unit impulse response in the discrete time system. Note that G is 2 N × N d t dimensional matrix. The singular value decomposition of G is expressed as G = U σ V T = u 1 , … , u n σ 1 O ⋱ O σ n v 1 , … , v n T (5) where U , V are matrices composed of the left-singular vectors and right-singular vectors, and σ is a n × n diagonal matrix composed of the singular values, and n = min 2 N , N d t . u 1 , … , u n are orthogonal to each other, and v 1 , … , v n are also orthogonal to each other. The substitution of Equation 5 into Equation 1 and the pre-multiplication of V σ − 1 U T leads to V σ − 1 U T d N d t + Δ N d t d t d ˙ N d t + Δ N d t d t = V V T u ¨ g d t ⋮ u ¨ g N d t d t (6)

It should be pointed out that V V T = v 1 v 1 T + … + v n v n T is the projection matrix into the subspace composed by v 1 , … , v n . Then, u ¨ g d t , … , u ¨ g N d t d t T can be obtained as u ¨ g d t ⋮ u ¨ g N d t d t = ∑ i = 1 n 1 / σ i u i T d N d t + Δ N d t d t d ˙ N d t + Δ N d t d t v i + u ¨ g , r (7) where u ¨ g , r is a residual vector and it is composed of the orthogonal vectors to v 1 , … , v n . u ¨ g , r is usually set to 0 . When 2 N ≤ N d t , the responses under u ¨ g d t , … , u ¨ g N d t d t T correspond to d , d ˙ . When 2 N > N d t , u ¨ g d t , … , u ¨ g N d t d t T provides an approximation.

Note that the acceleration responses d ¨ do not have to be taken into the formulation because d ¨ are derived by solving the equation of motion.

It is also noted that the proposed method works as a response-based filter, and the response and the obtained signals satisfy the equation of motion for every time step. In relation to this, the application to down-sampling of input signals is explained in Supplementary Appendix SC.

Let us consider a simple example for obtaining u ¨ g d t , u ¨ g 2 d t from the target responses d 3 d t , d ˙ 3 d t of an undamped SDOF model. In this example, N = 1 , N d t = 2 , Δ N d t = 0 . Figure 2 illustrates the target responses d 3 d t , d ˙ 3 d t and g d d t , g d ˙ d t , g d 2 d t , g d ˙ 2 d t in complex plane. Another part of the complex conjugate solutions is not illustrated here. The red lines correspond to d 3 d t , d ˙ 3 d t , g d d t , g d ˙ d t , g d 2 d t , g d ˙ 2 d t , and the lines rotate counterclockwise with the model’s natural circular frequency. g d d t , g d ˙ d t , g d 2 d t , g d ˙ 2 d t and u ¨ g d t , u ¨ g 2 d t can be interpreted as the component responses and the weighting coefficients because the target responses satisfy d 3 d t = u ¨ g d t g d 2 d t + u ¨ g 2 d t g d d t and d ˙ 3 d t = u ¨ g d t g d ˙ 2 d t + u ¨ g 2 d t g d ˙ d t . In other words, when the target responses d 3 d t , d ˙ 3 d t are geometrically expressed in the two-dimensional complex plane, g d d t , g d ˙ d t , g d 2 d t , g d ˙ 2 d t correspond to the component vectors.

Next, the relation between the phase angle of the target response, the duration and amplitude of the generated input ground motion is investigated.

Consider an SDOF building model with mass m , natural circular frequency ω = 1 and damping ratio h = 0 . Let ϕ / ω , d ϕ / ω = − sin ⁡ θ , d ˙ ϕ / ω = − cos ⁡ θ denote the duration of the generated ground motions, the target displacement and velocity responses, where θ is the phase angle of d ϕ / ω (Figure 3). When ϕ = 2 π , the duration is equal to the natural period of the SDOF model. It should be noted that the difference between the phase angles of d ϕ / ω and d ˙ ϕ / ω is π / 2 since h = 0 . It should also be pointed out that the mechanical energy at t = ϕ / ω is 1 / 2 m , regardless of θ .

Figures 4, 5 show the relationship between ϕ , θ and the peak ground acceleration (PGA). It can be observed that when θ is constant, PGA decreases almost monotonically as ϕ increases, namely, as the duration becomes longer. On the other hand, when ϕ is constant, PGA varies almost periodically with respect to θ . However, in the range where ϕ ≥ π , θ hardly influences PGA. From these results, to reduce PGA of the generated ground motion, it is recommended to set the duration longer than half the natural period of the building model.

Note that the setting ω = 1 , h = 0 is just for explanation. This discussion is extended to any combination of ω , h to arrange d ˙ ϕ / ω = − ω ⁡ cos θ + θ ′ , where θ ′ = arctan h / 1 − h 2 and the difference between the phase angles of d ϕ / ω and d ˙ ϕ / ω is π / 2 + θ ′ .

The formulations presented in Section 3.1 are extended to multiple-point input ground motions. Specifically, the input ground motion is expressed in terms of the ground velocity and displacement rather than the ground acceleration.

The equation of motion for an SDOF model with natural circular frequency ω and damping ratio h excited by a ground motion is expressed by d ¨ a + 2 h ω d ˙ a + ω 2 d a = 2 h ω u ˙ g + ω 2 u g (8) d a = d + u g (9) where d a , d , u ˙ g , u g denote the absolute and relative displacement responses, and the ground velocity and displacement. Following the manner of Equation 9, Equation 1 can be rewritten as d a N d t + Δ N d t d t d ˙ a N d t + Δ N d t d t = G a u ˙ g d t ⋮ u ˙ g N d t d t u g d t ⋮ u g N d t d t = G a P PP u ¨ g d t ⋮ u ¨ g N d t d t (10) where d a ⁢ N d t + Δ N d t ⁢ d t = d a ω 1 , h 1 , N d t + Δ N d t d t , … , d a ω N , h N , N d t + Δ N d t d t T (11) d ˙ a ⁢ N d t + Δ N d t ⁢ d t = d ˙ a ω 1 , h 1 , N d t + Δ N d t d t , … , d ˙ a ω N , h N , N d t + Δ N d t d t T (12) G a = 2 h 1 ω 1 O ⋱ O 2 h N ω N G d a ω 1 2 O ⋱ O ω N 2 G d a 2 h 1 ω 1 O ⋱ O 2 h N ω N G d ˙ a ω 1 2 O ⋱ O ω N 2 G d ˙ a (13) G d a = g d a ω 1 , h 1 , N d t + Δ N d t − 1 d t ⋯ g d a ω 1 , h 1 , Δ N d t d t ⋮ ⋮ g d a ω N , h N , N d t + Δ N d t − 1 d t ⋯ g d a ω N , h N , Δ N d t d t (14) G d ˙ a = g d ˙ a ω 1 , h 1 , N d t + Δ N d t − 1 d t ⋯ g d ˙ a ω 1 , h 1 , Δ N d t d t ⋮ ⋮ g d ˙ a ω N , h N , N d t + Δ N d t − 1 d t ⋯ g d ˙ a ω N , h N , Δ N d t d t (15) g d a ω , h , Δ t , g d ˙ a ω , h , Δ t are the absolute displacement and velocity responses after Δ t seconds for the SDOF model with natural circular frequency ω and damping ratio h excited by the triangular lateral load, which has a unit amplitude only at t = 0 and zero amplitude at all other time. Note that G a and P , PP T are 2 N × 2 N d t and 2 N d t × N d t dimensional matrices. P is a N d t × N d t dimensional matrix which operates time integration. When the trapezoidal rule is adopted, P is expressed as P = d t 1 / 2 O 1 ⋱ ⋮ ⋱ ⋱ 1 ⋯ 1 1 / 2 (16) u ¨ g d t , … , u ¨ g N d t d t is obtained through the singular value decomposition of G a P , PP T , and then u ˙ g d t , … , u ˙ g N d t d t , u g d t , … , u g N d t d t are obtained through multiplying P , PP T to u ¨ g d t , … , u ¨ g N d t d t .

When dealing with multiple-point input ground motions, the above procedure is repeated for each input point. Moreover, the proposed method can be extended to lateral wind loads in a similar manner.

An artificial ground motion OS1 is treated. OS1 is often used in Japan as a representative of long-duration, long-period ground motions. The duration is about 650 s, and the trimming range is set to 0 ≤ t ≤ 60 s. Four SDOF models with natural periods T 1 , … , T 4 = 1 , 2 , 4 , 8 s, and damping ratios h 1 , … , h 4 = 0.05 are adopted for generating an input ground motion with a duration of 8s. Consequently, the total reduced time is 52s and it corresponds to about 10% of the duration of the original motion. The ground acceleration of the original and generated motions is shown in Figure 1.

Figures 6a,b show the displacement responses of the four SDOF models under the original and generated motions. Figure 6c shows the generated ground motion. It can be observed that the displacement responses under the generated motions closely match those under the original motion in the range of t ≥ 60 s.

An 8-story, 20-bay planar steel moment frame is treated (Figure 7). The common story height is 4 m, and the common span length is 8 m. The total mass of each floor is 1280 t. A mass of 32 t is allocated to the top nodes of the corner columns, and a mass of 64 t is allocated to the top nodes of the interior columns. The cross-sections dimensions, moment of inertia and cross-sectional area of the frame are listed in Tables 1, 2. H-shape sections are applied to beam sections except for the footing beams, and square box sections are applied to column sections and the footing beams. The cross-sections of columns and beams are common within each story. The moment of inertia and cross-sectional area of the columns and beams are also shown in Figure 7. Young’s modulus is set to 2.05 × 10 5 N/mm². The damping ratio of the fundamental eigenmode is 0.02 (stiffness-proportional type). The dynamic parameters of the frame are listed in Table 3.

The input ground motion is explained. The ground motion is specified at the engineering bedrock with S-wave velocity V s = 400   m / s based on the design response spectrum (Ministry of Construction, 2000). The signal phase of Hachinohe EW component during the 1968 Tokachi-oki Earthquake is employed. Note that Hachinohe EW component contains low-frequency components. The ground consists of a single sandy soil layer with S-wave velocity V s = 200   m / s , density of 1.73 g/cm³, and thickness of 30 m. The nonlinear characteristics of the soil layer is considered by the equivalent linearization technique (Schnabel et al., 1972; Koyamada et al., 2005). The time phase difference of ground motion inputs at both ends of the frame is set to 0.5s.

The duration of the ground motion at each input point is 120 s, and the time-history response analysis is performed in the range of 0 ≤ t ≤ 120.5   s . The trimming range is set to 0 ≤ t ≤ 17   s and the duration of the generated motion is 2 s. Consequently, the total reduced time is 15s, corresponding to about 12.5% of the duration of the original motion. Note that the time range of trimming is common for all the input points. Figure 8 shows the ground displacement and the displacement, velocity, and acceleration response spectrums S D , S V , S A .

When the horizontal displacements of nodes in the same floor are assumed to be identical, the 1-6 th natural circular frequencies and damping ratios are ω 1 , … , ω 6 = 6.01 , 16.0 , 28.6 , 44.6 , 64.5 , 87.1 and h 1 , … , h 6 = 0.02 / ω 1 × ω 1 , … , ω 6 . These parameters are adopted to create SDOF models, which are then used to generate ground motions. It should be pointed out that this assumption is not considered in the time-history response analysis. Note that the sum of 1-6th effective modal mass is larger than 99% of the total mass of the frame.

Figure 9 shows the generated ground displacement, the 5th floor and top acceleration responses of the moment frame under the original and generated motions. The reference nodes are also illustrated. It can be observed that the displacement responses under the generated motions closely match those under the original motion in the range of t ≥ 17 s.

A full-scale 35-story elastic-plastic building model with a strong back core frame is treated to compare the proposed method with the previous paper (Akehashi and Fujita, 2025). The building model and the ground motion OS1 treated in the previous paper are also used in this paper. The main frame is a reinforced-concrete moment-resisting frame and the subframe is a reinforced-concrete shear-wall structure (Figure 10a). The details of the model are explained in the previous papers (Kawai et al., 2021; Akehashi and Fujita, 2025). PGA of OS1 is set to 2 m/s². The ranges where the ground acceleration power is below 1% or exceeds 96% of the total are trimmed. Although the previous paper adds single impulse input as a correction process after trimming of the leading part of the ground motion, a generated ground motion whose duration is equal to the fundamental natural period is connected in this paper. The natural eigenmodes are selected in descending order of the effective modal mass until the cumulative mass is larger than 90% of the total mass of the frame. Table 4 shows the natural circular frequencies, the damping ratios and the effective modal mass ratios for x-directional ground motion.

Figure 10b presents the maximum interstory drift angle and the maximum floor accelerations. It can be observed that the maximum interstory drift angle and the maximum floor accelerations evaluated by the proposed method show better correspondence with the results for the original motion compared to those evaluated by the previous method. It should be pointed out that the generated ground motion accurately evaluates the higher-mode responses, although the impulse input at the initial time step strongly excites the higher-mode response. Note that it took 75484 s for the analysis under the original motion, and it took 23623, 24084 s for the evaluation by the previous and proposed method.