Analysis of seismically‐isolated two‐block systems using a multi–rocking‐body dynamic model

A novel multibody rocking model is developed to investigate the dynamic response of two stacked rigid blocks placed on a linear base isolation device. The model is used to investigate the dynamic response of a realistic statue‐pedestal system subject to pulse‐like ground motions. The analysis shows that, in general, base isolation increases the safety level of the rocking system. However, for large period pulses or small size blocks, the isolator can amplify the ground motion, resulting in a lower minimum overturning acceleration than for the nonisolated system. Further, the amplification or shock spectrum of a linear mass‐dashpot‐spring oscillator, was found to be the reciprocal of the minimum nondimensional overturning acceleration of the investigated rocking system. Novel rocking spectra are obtained by normalizing the frequency of the pulse by the frequency of the isolator. The analysis also demonstrates how the dynamic response of the two stacked blocks is equivalent to that of a single‐block configuration coincident with the whole system assumed monolithic or the upper block alone, whichever is more slender.


INTRODUCTION
Statues, obelisks, electrical equipment, computer servers, medical equipment, storage systems, and transformers are just a few examples of objects whose behavior can be modeled as rigid rocking objects when subjected to seismic actions (Housner, 1963;Yim et al., 1980;Zhang & Makris, 2001). This study is motivated by the need to protect these nonstructural elements from earthquakes; base isolation systems may be a suitable strategy for this purpose. The problem can be framed into two subjects: (1) rocking response of rigid blocks and (2) base isolation technology to mitigate the response to seismic actions. Regarding the first subject, the dynamics of rigid blocks is a topic extensively investigated in the last decades, of the rocking spectra for sine pulses. Moreover, rigid block modeling has recently gained attention for the assessment of the seismic vulnerability of nonstructural elements (Dar et al., 2016;Vlachakis et al., 2021).
There are also cases where one block is placed on top of another, and the single rigid body model is unable to adequately define the dynamic response. Stacked servers, medical shelves, a piece of machinery on a base and ancient columns are real-life examples. Particularly rele-F I G U R E 1 Example of a real-life two stacked rigid block system: A Roman bust placed on a pedestal. vant is the case of two blocks one on top of the other, such as an art object placed on a pedestal (Figure 1). The response of this type of system was explored by Psycharis (1990), Spanos et al. (2001), Chatzis et al. (2018), and Diamantopoulos et al. (2022). The authors draw the conclusion that the formulation of this extremely nonlinear analytical model is challenging, due in part to highly nonlinear equations of motion, impacts, and continually changing contacts (Bao & Konstantinidis, 2020). Further studies on the two-block assemblies were conducted in the field of out-of-plane mechanisms for masonry structures (Lourenço et al., 2011), with the goal of investigating a two-block vertically spanning wall mechanism (DeJong & Dimitrakopoulos, 2014;Prajapati et al., 2022;Sorrentino et al., 2008).
Further, a variety of experimental tests have been conducted to investigate the response of rocking objects, such as medical shelves (Kuo et al., 2011), art objects (Huang et al., 2022), display cases (Prota et al., 2022), and hospital cabinets (Di Sarno et al., 2019).  carried out a large experimental campaign on several bustpedestal systems placed on a base isolation device. Despite the above-mentioned test, an analytical model capable of capturing the response of a seismically isolated two stacked rigid blocks system ( Figure 2) has not been developed to date.
In this paper, a new computational model is developed to analyze the response of two stacked rocking blocks resting on flexible elements (such as a base isolation device). There are similarities between the proposed model F I G U R E 2 Motion patterns of the seismically isolated two stacked rigid blocks system (patterns 1b, 2b, 3b, 4b are reversed with respect to 1a, 2a, 3a, 4a, respectively). and multibody dynamics (MBD), which is the topic of computer modeling and analysis of constrained bodies subject to large displacements and rotations (Shabana, 1997). This paper illustrates in detail a multi-rocking-body dynamic (MRBD) model capable of describing the dynamics of a three degree-of-freedom (DOF) system; the model is then used to examine the response of the two stacked blocks placed on an isolator when subject to sine-pulse ground motion.
The real-life objects at the base of the model in Figure 2 are 3D in nature. However, the investigation of the 3D dynamic response, including base isolation, would typically be investigated numerically, for example by means of the discrete element method (Malomo et al., 2021) or the finite-discrete element method (AlShawa et al., 2017), because deriving the analytical equations of motion for a range of 3D geometries is impractical. Nonetheless, analytical equations provide a powerful tool to identify general trends of the response for a large range of the values of the system parameters, and therefore can guide the design of base isolation intervention. Therefore, analytical equations are derived in the following for a representative 2D system.
The first part of the study focuses on the description of the kinematic quantities, energies, and nonconservative forces, followed by the derivation of the equations of motion (via a Lagrangian approach) for each possible rocking mode. Following the description of all possible transitions between rocking modes (or patterns), the solution algorithm is presented as well using pseudo-code.
In the second part of the paper, the effect of base isolation on a realistic statue-pedestal system is investigated, and significant results for both the two stacked blocks system and the single rocking block have emerged. The response of this type of system to pulse-like ground motion can be expressed using rocking spectra, which are typically built by assuming the ratio between the pulse period and period of the isolator is constant (Vassiliou & Makris, 2012). In this study, rocking spectra derived by assuming the period of the base isolation as constant are compared to the previous typology, and the benefits and drawbacks of the two types of spectra are discussed. Interesting results are also obtained using shock spectra for linear mass-dashpotspring (LMDS) oscillators subjected to pulse-like ground motion; a direct relation between minimum overturning nondimensional acceleration of rigid blocks and maximum amplification of the LMDS oscillator is discovered. Moreover, minimum acceleration overturning spectra are derived using a different normalization criterion for the pulse frequency, which has previously been normalized with respect to the frequency parameter of the rigid block (Pellecchia et al., 2022;Vassiliou & Makris, 2012;Zhang & Makris, 2001). The use of this novel criterion allows the response to be expressed in terms of a rocking spectrum independent of the isolator period, allowing other parameters to be investigated. The previous observations are generally applicable to both single-block and two stacked block models, hence additional analysis on the two-block assembly was performed. Previous research on this topic has focused on the dynamics and energy dissipation of the system (Chatzis et al., 2018;Psycharis, 1990;Spanos et al., 2001); here, dynamic equivalence with a single rocking block is found as long as a specific configuration is used. Finally, the paper concludes with practical observations for designing base isolation devices for rocking objects and enhancing the safety level of the system.

EQUATIONS OF MOTION
The MRBD system of this study is an assembly of two stacked rigid blocks, both free to rock, placed on a seismic isolation device. Friction is assumed to be sufficiently large to prevent sliding between the two blocks and between the lower block and the isolator. Sliding is usually beneficial, provided that the block does not impact other objects or walls, and provided that the upper block does not fall from the lower one (Kounadis, 2018). Hence, an isolator design based on the no-sliding assumption is generally conservative. Here, the behavior of the isolator is assumed to be linear, hence the base isolation system adds one DOF and it can be described by the mass , the stiffness , and the viscous damping (all symbols presented herein are listed in Table D1 of Appendix D). The lower and upper blocks have thicknesses 2 1 and 2 2 , respectively, while the height of the lower block is 2ℎ 1 and that of the upper block is 2ℎ 2 . Further, the lower block has mass 1 , while the upper has mass 2 . If the joint between the blocks is closed and the two blocks move together, the following parameters can be defined: (1) the total mass of the blocks, = 1 + 2 ; (2) the height, ℎ, of the combined center of mass. The system is characterized by five modes or patterns (Figure 2), and the equation of motion must be formulated for each one of them. Due to the several variables and to the different motion patterns, variational methods are adequate for deriving the equations of motion of the MRBD model, avoiding the calculation of internal forces (Fabien, 2008;Humar, 1990;Meirovitch, 1970). The system studied here has three DOFs, the rotations 1 and 2 of the lower and of the upper block, respectively, and the horizontal displacement of the isolator .
The analytical equation of motion for each pattern is obtained using a Lagrangian approach, deriving with respect to the time the kinetic energy, the potential energy, and the dissipation function, as well as calculating the nonconservative forces.
As shown in Figure 2, each pattern is characterized by a specific number of Lagrangian parameters, and the same number of equations of motion, one for each generalized coordinate.

Kinematic analysis
Once the generalized coordinates are defined, it is convenient to define displacement, position, and velocity vectors to describe the motion of the center of mass of In this case, the displacement of the generic point P, belonging to the upper body rotating with angle θ i about Q, can be described, with reference to the global axes x-y, by means of the displacement vector : (1) where = + ⋅ , is the known displacement vector of point Q, is the known displacement vector of point R belonging to the LMDS oscillator, is the displacement vector transformation matrix of the i-th block, d P,Q ( , ) is the vector that contains the coordinates, and , of the generic point P (Q) with respect to the center of rotation Q (R) in the global reference system x-y.
Similarly, it is possible to define a position vector , , used to evaluate the velocity of any point and the angular momentum of a rigid body: where , is the known position vector of the point Q, is the transformation matrix for the position vector, which allows the expression of the position vector of the generic point as a function of the main reference system. Furthermore, to compute the kinetic energy, the velocity of the centers of mass must be calculated for all bodies. In the case of the generic point P, which belongs to a rigid body that rotates with angular velocitẏabout the point Q, the generic velocity vector is: where is the known velocity vector of the point R, =̇× , is the relative velocity vector of point Q,i s the angular velocity vector, and × is the vector product operator.
These vectors can be obtained knowing only the Lagrangian parameters and the coordinates of the points, allowing a certain convenience in computing the equation of motion for MRBD systems. Once these vectors are known, the kinetic and potential energy as well as the nonconservative forces, used to assemble the Lagrangian equations of motion, can be easily derived for any kind of MRBD system.

Lagrange's equation
The kinetic energy T can be derived as: where | | is the magnitude of the velocity vector of the i-th center of mass, is the mass moment of inertia of the i-th block with respect to its centroid G, is the mass of the i-th block, the b subscript is related to the base isolator, and the dot superscript indicates derivative with respect to time.
The potential energy, V, and the dissipation function, , can be obtained as: where , is the vertical component of the position vector of the center of mass of the i-th body.
The last step to build the equations of motion is the applied effort analysis, which provides the nonconservative forces (one for each generalized coordinate) using the virtual work : where , and , are, respectively, the horizontal and vertical components of the virtual displacement vector of the i-th center of mass,̈and̈are, respectively, the horizontal and vertical components of the seismic ground acceleration. Writing , , , and , as functions of the virtual generalized coordinates provides the generalized forces.
The approach previously explained is used for the patterns characterized by three DOFs (patterns 1 and 2). In the case of pattern 3, it is necessary to assume the rotations of the upper and lower blocks equal. Regarding pattern 4, it is sufficient to assume the rotation of the lower block equal to zero. For pattern 0, both the rotations 1 and 2 must be assumed equal to zero. A practical example can be found in Appendix A.

PATTERN CHANGE
The two-block system can change pattern during motion due to impacts or acceleration thresholds exceeded (Prajapati et al., 2022;Psycharis, 1990;Spanos et al., 2001). In this case, where the blocks are placed on a seismic isolation device, three major differences in the formulation of pattern changes are found in comparison to the nonisolated system. The first concerns the beginning of motion: differently from the nonisolated system, the motion is not triggered by a threshold acceleration, and the base isolation device is activated for any amplitude of the ground motion. In the isolated case, the system begins its motion in pattern 0, then for each step of the integration, a check of rocking motion initiation is performed.
The second difference regards the pattern change due to acceleration thresholds exceeded. Although the procedure is the same as the one proposed in Spanos et al. (2001) and Prajapati et al. (2022), where internal and external moments are set equal, here the external accelerations of the isolation device must be included in the computation of the external moments.
The third difference is in the calculation of the coefficient of restitution after collision occurs: In the isolated system, the lower block impacts with a device of finite mass (the isolator) instead of a rigid foundation.

Pattern change due to acceleration thresholds exceeded
If the ground motion is large enough, rocking will initiate and the system will change its pattern. The problem is generally solved by setting the external moment equal to the internal moment in order to find a threshold acceleration to detect the change (Spanos et al., 2001). In the case at hand, the procedure for evaluating this threshold acceleration is similar to that in Prajapati et al. (2022), but has been appropriately modified to account for the effect of the base isolation system. External moments are generated by inertia forces originating from the ground (and isolator) motion, which are defined in the external acceleration vector : Then, the external moment vector can be calculated as follows: where + is the position vector of the i-th center of mass with respect to the center of rotation associated with the pattern to which the system could change if the threshold acceleration is reached. The patterns before and after the change are denoted by − and + , respectively. To compute the internal moments, the normal and tangential accelerations of the masses must be calculated in each instant prior the possible pattern change; these accelerations are computed with respect to the pattern before the possible change as follows: where , and , are, respectively, the normal and the tangential acceleration vector of the i-th center of mass,ï s the angular acceleration vector, and − is the position vector of the i-th center of mass with respect to the pattern before the possible change.
The internal moment vector can then be calculated with respect to the pattern in which the system could change its motion as follows: Finally, by setting internal and external moments to be equal, the minimum acceleration̈, for which the system can transition from pattern to pattern can be determined (Equation 11).
where , and , are, respectively, the rotational component of the internal and external moment vectors.
The principal difference between the isolated and nonisolated cases is the beginning of rocking motion. The uplift acceleration, which causes rocking motion, can be viewed as a further pattern change caused by an external acceleration and calculated using the previous equations. In Appendix B, more details on pattern changes without impacts can be found.

Pattern change due to impacts
An impact can occur between the upper and lower blocks (middle impact) or between the lower block and the isolator (lower impact); if this happens, the system starts to move in another pattern with consequent energy loss.
Here, impacts are assumed to be instantaneous, moreover only one impact can occur at a given instant. Further, if the impact is assumed to be inelastic (no bouncing), the rotation continues smoothly from one corner to the other, and the angular momentum about is conserved as customary in rocking dynamics (Housner, 1963) and reasonably confirmed by experiments (Sorrentino et al., 2011). As previously described in Spanos et al. (2001) and Prajapati et al. (2022), the analytical coefficient of restitution must be estimated for each type of pattern change due to impacts. The distinction between the nonisolated case and that where the system is placed on an isolator is in the calculation of the postimpact velocities. When an impact occurs, the base isolation also undergoes a change in velocity (Vassiliou & Makris, 2012). For the model of two stacked rigid blocks placed on a finite mass isolator, the new velocities after impact must be evaluated for all three Lagrangian parameters. The postimpact velocities are computed by solving a system of three equations (Equation 12), obtained using conservation of momentum principles, and more specifically: (1) conservation of angular momentum of the entire system , (2) conservation of angular momentum of the top block only , and (3) conservation of linear momentum in the horizontal direction , . The momentums must be calculated with respect to the pattern before (−) and after (+) the impact.
The angular momentum of the entire system is obtained as follows: The angular momentum of the upper body alone can be derived using Equation (14).
The linear momentum in the horizontal direction can be obtained as follows: Further, after each type of impact, whether lower or middle, the system can transit to an a priori assumed pattern, after which angular velocities are controlled to assess kinematic admissibility. If it is not satisfied, the system will switch to an alternative pattern (Prajapati et al., 2022). Further information about transitions due to impacts can be found in Appendix C, while the postimpact velocities for every possible pattern change are presented in Destro Bisol et al. (2023).

SOFTWARE DEFINITION
Once the equations of motion are obtained and the pattern changes are defined for each possible transition, the response of the system to ground motion was simulated. The solution algorithm, implemented in MATLAB (2018), is briefly summarized in Figure 4 using a pseudocode scheme The analysis can start once the input data, that is, the geometric and mechanical properties and ground motion excitation, are defined. The first stage is the integration of the equation of motion for pattern 0; the integration continues until an event occurs. From pattern 0, two possible events can occur: (a) rocking motion starts without impact; (b) the system stops, that is, comes to rest. If the threshold acceleration of Equation (11) is overcome, rocking motion starts (event a) and the pattern in which the system is moving must be updated, a new case must be assumed for the switch function (Figure 4, line 6) and the integration of the specific equation of motion is restarted.
Once the system is moving in a rocking pattern (either 1, 2, 3, or 4), the following events can occur: continues with the new pattern, all these possible transitions are defined in Table B1. If event (b) takes place, impact occurs, and the postimpact velocities must be calculated (Figure 4, line 13) in relation to the primary pattern (Table C1) in which the system can move; then the kinematic check must be performed. If the kinematic check is satisfied, the integration can proceed in the new pattern with the newly calculated postimpact velocities. If it is not satisfied, the postimpact velocities must be calculated (Figure 4, line 17) with respect to the secondary pattern (Table C1) in which the system can switch, and the integration continues with the new pattern. If event (c) occurs, the system (or part of it) has overturned: In this case, the analysis stops, and the while cycle is interrupted.

RESPONSE TO PULSE-TYPE GROUND MOTION
The developed computational software can be employed for any type of ground excitation, with earthquake ground motion being the most important in terms of potential damage to the blocks investigated here. However, record selection can be performed only with a specific system at a specific location in mind and usually would involve modeling the building that houses the system (D'Angela et al., 2021Giouvanidis & Dimitrakopoulos, 2018). Therefore, in order to get a general understanding of the dynamic response of the MRBD model, as well as to derive general trends in its behavior, a pulse-like ground motion is considered. Conveniently, the response to this input can be expressed in dimensionless plots using rocking spectra (Zhang & Makris, 2001). These spectra are nondimensional, allowing exploration of a range of pulses with different amplitude and period, or a range of systems with different slenderness and size. For this purpose, the slenderness ( 2 ), the size parameter ( 2 ), and the frequency parameter ( 2 ) are introduced for two cases: (a) two-block assembly considered as monolithic, that is, interface between upper and lower block is closed (Equation 16); (b) upper block alone (Equation 17).
The previous equations are derived under the assumptions that both blocks are homogeneous and share the same density. The base isolation device can be described using the period , the mass ratio , and the damping ratio as described in Equation (18). Additionally, the pulse amplitude is denoted as , the pulse circular frequency is named , and the pulse period is denoted as .
where is circular frequency of the isolator, and = + is the total mass. The behavior of the system and its sensitivity to the different parameters are described starting from an application where the real-life system in Figure 1 is approximated using the model in Figure 5a, with appropriate parameters. The lower block of the investigated system has a height of 2ℎ 1 = 1.0 m, and a thickness of 2 1 = 0.4 m, whereas the upper block has a height of 2ℎ 1 = 0.75 m and a thickness of 2 2 = 0.22 m; it is assumed that the two blocks have the same density. The mechanical properties of the base isolation for the investigated system can be summarized as follows: (a) period = 2.0 s; (b) mass ratio = 0.9; (c) damping ratio = 0.05. This scenario is referred to as the "reference system" in this paper; variations to this reference system will be considered later in the paper.
The response of the reference system to a sine-pulse ground motion (Figure 5b) is shown in terms of displacement of the base isolator (Figure 5c), of rotation of each of the two blocks (Figure 5d), and in terms of pattern (Figure 5e). The system begins to move in pattern 0 at the onset of the ground motion, then the system starts to rock in pattern 3 due to a large acceleration. As the acceleration increases, another pattern change without impact occurs, and the two blocks start to move with different angular velocities. Following that, numerous middle and base impacts occur, and the system changes continuosly its pattern until the system stops rocking. Finally, the system continues its motion in pattern 0, until it comes to rest.

5.1
Overturning spectra: Constant period of the isolator When the sine-pulse analysis is used to obtain overturning spectra, it can be an effective tool to understand the dynamic response of rocking systems. This type of spectra is created by subjecting the system to a wide range of sine pulses that differ in frequency and amplitude, and then determining whether or not overturning occurs. When a single rocking rigid block undergoes a sine-pulse ground motion, it can overturn in two manners: (a) with no impact and (b) following an impact between lower block and foundation (Dimitrakopoulos & DeJong, 2012). In this case, where two blocks are stacked one on top of the other, overturning can occur in any of the above-mentioned patterns and can involve a variety of impacts. The overturning mechanisms can be divided in four different types: (a) no impact; (b) middle impact, where an impact between the two blocks occurs; (c) base impact, in which the lower block impacts the base isolation device one or more times prior to the collapse; and (d) base and middle impact, where the two blocks impact one another and the lower block impacts with the isolator prior to the overturning.
Here, the investigated system ( Figure 5a) is used to evaluate the performance of the isolator on the overall dynamic response. Three rocking spectra, for different periods , are derived to study the effect of the isolation system ( Figure 6). Overturning spectra are generally built by normalizing the frequency of the pulse with respect to the inverted pendulum frequency parameter, and the amplitude of the pulse by the minimum acceleration that triggers the rocking motion (Housner, 1963), also called the uplift acceleration. Because the upper block can be seen as the valuable object to be protected using the base isolation technique, its response can be used as a benchmark to evaluate the efficiency of the isolator and the influence of the other system parameters. Hence, the frequency parameter 2 and the acceleration tan( 2 ) (Equation 17), related to the upper block alone, are used. The analytical rocking spectra for the upper block alone are computed using the equations in Dimitrakopoulos and DeJong (2012) and are shown with a black line in Figure 6. The overturning spectra in Figure 6 show how strongly the period influences the global response of the system: As increases, the overturning area shrinks, implying that the efficiency of the base isolation device increases as well. If the overturning spectrum for short periods ( Figure 6a) is compared with the response of the upper block alone, the overturning area is larger than the analytical spectra, meaning that the addition of the isolator decreases the safety level of the art object. When is increased (Figure 6b,c), the overturning area shrinks and is generally contained within the analytical spectrum, implying that the use of the seismic isolation device increases the safety level of the system. This effect is particularly evident for large values of or small values of 2 . The beneficial effect provided from the isolator vanishes when is small (or 2 is large), and in this region, an overturning area below the upper block spectrum is noticed ( Figure 6). The base isolation device is ineffective in this region because the upper block alone overturns for larger sine-pulse amplitudes than the isolated system. This peculiar behavior indicates that the isolator amplifies the ground motion: This amplification depends on , the ratio between the pulse period, , and the period of the isolator, . Similar conclusions can be drawn when the analytical rocking spectra for the nonisolated system, considered as monolithic, is compared with the response of the isolated system ( Figure 6). In this case, the overturning area is contained within the analytical spectrum only when long period isolators are used (Figure 6c), implying that to increase the safety level of the system, it is necessary to use base isolation devices with longer period if compared with the case of the upper block alone. Indeed, if the two analytical spectra are compared (Figure 6), it is possible to observe that the addition of the pedestal improves the performance of the system for large blocks or small period pulses. This behavior is slightly reversed for small blocks or for large period pulses; indeed in this region, the analytical spectrum of the nonisolated monolithic system is just below the spectrum of the nonisolated upper block alone.
Shock spectra (Irvine, 2002) can be a powerful tool to enhance the understanding of the dynamic response of LMDS oscillators (such as the base isolation in this study). This type of spectra is created by subjecting the LMDS to a wide range of sine pulses that differ in frequency and amplitude, and then determining the ratio of the maximum (pos) or minimum (neg) LMDS acceleration and the amplitude of the pulse. A sample shock spectrum ( Figure 7a) is built using a full sine-pulse ground motion and an LMDS oscillator with damping ratio = 0.05, the same as in the linear isolator of the investigated system. The spectrum highlights how the LMDS system can amplify the ground motion, because the oscillator tends to become in phase with the second half of the full sine pulse. This phenomenon has a significant influence on the response of isolated rocking-block systems; the overturning spectrum of Figure 6b, after zooming in on the region below the analytical spectrum of the upper block (see Figure 7b), mimics a mirrored shock spectrum (Figure 7a). Further observations on the relationship between normalized minimum overturning acceleration and amplification of the LMDS system can be found in the following sections ( Figure 11).
Shock spectra were also constructed using a half sinepulse ground motion and an LMDS oscillator with varying damping factors (Figure 7c). The spectrum for ξ b = 0.05 shows smaller amplitudes compared with Figure 7a, because there is no synchronicity anymore between the response of the oscillator and the second half of the full sine pulse. Given the simpler nature of this spectrum, for which no minimum (negative) amplification is present any longer, it is possible to expand on the investigation of the damping effect. The shock spectra show how, as damping is increased, the amplification of the ground motion decreases in general. This behavior is weakly inverted for large damping factors and for long period pulses (or short LMDS oscillator periods). In this case, the beneficial effect of the damping vanishes, and the acceleration amplification can be slightly reduced for smaller values of the damping ratio (Figure 7c).
Finally, minimum acceleration rocking spectra (Figure 7d) for various damping factors and isolator periods are plotted. These spectra are built using only the first amplitude of the sine-pulse, which causes overturning, neglecting the information above this curve. The response of the system in Figure 7d for different damping ratios confirms the results obtained with the shock spectra ( Figure 7c), indicating that as the damping ratio increases, so does the minimum acceleration necessary to overturn the system. Similar to the shock spectra, for large values of damping ratio and for large pulse circular frequency (or small frequency parameter 2 ), the normalized minimum overturning acceleration decreases compared to the cases with smaller damping ratio. Observing, for example, the case with damping = 0.4 in Figure 7d, it is possible to see that the curve, at approximately / 2 = 1.25, becomes almost straight and intersects the smaller damping curves. This behavior is due to the following: (a) The amplification of the LMDS oscillator becomes linear in this range of periods; (b) collapse occurs following only a middle impact (while in the previous stage, collapse occurs for both middle and base impacts). Thus, it is possible to conclude that adding damping to the isolator is generally beneficial for the system, but this may not be true for short period isolators with large damping. Moreover, when the period is increased while keeping the damping constant, the area encompassed by the spectrum tends to shrink, but the minimum normalized accelerations of the spectra remain constant (Figure 7d).

Overturning spectra: Constant period ratio
The previous rocking spectra in Figure 6 do not have the typical shape of a single rigid block rocking spectrum (Zhang & Makris, 2001), in which two areas can be distinguished: (a) a region where the overturning occurs without impact and (b) a bubble-shaped region where overturning F I G U R E 7 LMDS oscillator shock spectra for: (a) full sine-pulse ground motion; (b) detail of the overturning spectra in Figure 6b; (c) half sine-pulse ground motion for different damping ratios; (d) minimum acceleration overturning spectra for different base isolation damping ratios and periods.
occurs following an impact with the foundation. The reason for this difference can be found in the altering effect of the base isolation on the sine-pulse ground motion. As previously stated, the ratio plays an important role in the response of the system; if the period of the base isolation is kept constant while building the rocking spectra, then is allowed to vary. Thus, the amplification of the linear isolator is different for each step of the analysis, implying that the response is significantly altered when compared to the simple rigid block placed on a rigid foundation.
Alternatively, the rocking spectra can be built assuming that the period ratio remains constant (Figure 8) for all analyses, which means that the amplification of the ground motion caused by the linear isolator remains constant. For these reasons, two distinct rocking spectra are built for two different period ratios: (1) = 3 ( Figure 8a) and (2) = 1/3 (Figure 8b). The results in this case show that by keeping the ratio constant during spectrum construction, the typical bubble-shaped rocking spectra (Vassiliou & Makris, 2012) and the distribution of the overturning mechanisms (Chatzis et al., 2018) are restored. When the two rocking spectra are compared, it is noticeable that the minimum overturning acceleration increases as the period ratio decreases. The reason for this behavior is again illuminated by the full sine-pulse shock spectrum ( Figure 8c); indeed, the use of a constant period ratio implies that the amplification of the linear isolator is constant. If the period ratio is assumed to be 3, is amplified by approximately 1.30 times, whereas if is 1/3, the ground motion is amplified by approximately 0.55 times (i.e., deamplified). Finally, the rocking spectra are compared to the response of the upper block on a rigid foundation (expressed in terms of analytical spectra). The base isolation system is generally ineffective when = 3, because the isolated system overturns for a smaller amplitude of the ground motion than the upper block alone. When is assumed to be 1/3, the isolator appears to be effective for small values of (or for large values of 2 ), but ineffective for small frequency pulses or blocks of large size (small values of 2 ). The last observations lead to the conclusion that isolation technology may be ineffective in some circumstances (e.g., for short period pulses) in preventing overturning of rocking systems. However, it is important to note that the period of the base isolation is different for each simulation when building this type of spectrum (Figure 8a,b). The use of isolators having a small period, for example, T b < 2.0 s, is not frequent in practice and it is well known that large values are usually effective (Figure 7), although they require that larger displacements are accommodated.

Minimum acceleration overturning spectra: Normalization studies
Previously the effect of the period of the isolator on the global behavior of the system was investigated. The sinepulse analysis and the minimum acceleration rocking spectra are used in the following sections to understand the influence of various system parameters. For this purpose, the use of minimum acceleration overturning spectra (built using only the minimum acceleration that causes the overturning for each frequency) allows the response to be expressed with only one curve, making comparison easier. Furthermore, in this section, a novel normalization criterion is used to express the response of the system independent of the period of the isolator, and to investigate different parameters.
Here, 11 spectra are derived for 11 different values (Figure 9), and the results confirm the previous observation: As the period increases, so does the safety level. When the spectra are built by normalizing the pulse frequency, , with respect to the frequency of the base isolation, , (rather than 2 ), all the curves collapse into one. Consequently, it can be stated that (for a given value of 2 ) by using the aforementioned normalization, the response of the system can be expressed independently of the period of the base isolation, facilitating the investigation of other parameters.
So far, normalization has always been referred to the upper block, with the goal of establishing the effective-ness of the base isolation system in protecting the upper rocking object; it is now necessary to evaluate the response of the system accounting (when necessary) for the presence of a lower block. The response is now evaluated in relation to two distinct slenderness values that characterize the system: (a) slenderness of the two blocks considered as monolithic and (b) slenderness 2 of the upper block alone. Further, the slenderness ratio is introduced: Therefore, different minimum acceleration overturning spectra are derived for various slenderness ratio values. In this case, the curves are built again for 11 periods to maintain independence from the period , and the median values are assumed to define the spectrum of each different slenderness ratio. The amplitude of the pulse, , is now normalized in two ways. First, it is normalized with respect to the slenderness of the monolithic system, (Figure 10a), and then to the slenderness of the upper block alone, 2 (Figure 10b). If the pulse amplitude is normalized with respect to (Figure 10a), then the response of the system (when the slenderness ratio is greater than one) collapses into a single curve, whereas when < 1, the minimum acceleration for which overturning occurs decreases. On the other hand, if is normalized with respect to 2 (Figure 10b), the response of the system when < 1 collapses into one curve, while decreasing F I G U R E 9 Minimum acceleration overturning spectra for 11 different isolator periods: a) pulse frequency normalized with respect to frequency parameter of the upper block; (b) pulse frequency normalized with respect to the frequency of the isolator.
F I G U R E 1 0 Minimum acceleration overturning spectra for different slenderness ratio: (a) pulse amplitude normalized with respect to the minimum uplift acceleration of the blocks considered monolithic; (b) pulse amplitude normalized with respect to the minimum uplift acceleration of the upper block alone; (c) pulse amplitude normalized with respect to the minimum uplift acceleration of the slenderest configuration.
the slenderness ratio ( > 1), the minimum overturning acceleration decreases as well.
The results in Figure 10 highlight a useful phenomenon. The response of the system is governed by the slenderest configuration between (a) the whole system considered as monolithic and (b) the upper block alone. If the smaller slenderness is assumed to normalize the pulse amplitude, all the curves collapse into one (Figure 10c), allowing the response of the system to be described by considering only one slenderness parameter and ignoring the interaction between the two blocks. Thus, the slenderness can be defined as in Equation (20). The same criterion can be applied to the frequency parameter of the system, and a parameter can be defined in the same way as for .
To validate the previous findings and to better understand the linear isolator amplification, the response in terms of shock spectra for an LMDS oscillator subjected to a full sine pulse is compared to the minimum acceleration overturning spectrum (Figure 11). Based on the results presented above, the spectra (Figure 11b) built for F I G U R E 1 1 Comparison between the ground motion amplification of the LMDS oscillator and the normalized minimum overturning acceleration of the system: (a) LMDS oscillator shock spectra for the full sine-pulse ground motion in terms of frequencies; (b) minimum acceleration overturning spectra normalized with respect to the slenderest configuration.
the investigated system are normalized with respect to the slenderest configuration (the two blocks considered as monolithic in this case). Further, to facilitate comparisons, the shock spectrum ( Figure 11a) is expressed in terms of the ratio between the frequency of the pulse and the frequency of the LMDS oscillator. When the two spectra are compared, the correlation between the dynamic amplification of the LMDS oscillator and the minimum nondimensional overturning acceleration is evident, especially now that the response of the system has been normalized with respect to the proper configuration. Thus, some specific peaks of the response are investigated for both spectra (points A, B, and C in Figure 11a,b) to prove the exact correspondence between the LMDS oscillator amplification and overturning acceleration. When the peak amplification ( Figure 11a) is compared to the minimum overturning acceleration (Figure 11b) in points A, B, and C, it is possible to observe that these two values are approximately the reciprocal of each other.
These final observations lead to three considerations: (a) The minimum overturning acceleration is an exact function of the LMDS oscillator dynamic amplification, and it is governed by the ratio between the pulse frequency (period) and the frequency (period) of the linear isolator; (b) the final behavior depends on the envelope of the maximum response between the positive and negative amplification of the LMDS oscillator; and (c) the correspondence between and further justifies the normalization with respect to the more slender configuration. Indeed, when the response of the system is normalized using the characteristics of the upper block (Figure 7b), it is possible to observe a slight difference between the minimum overturning acceleration and the amplification of the LMDS oscillator (Figure 11a).
The influence of the mass ratio on the response of the system is also investigated here; for this purpose, the F I G U R E 1 2 Minimum acceleration overturning spectra for different mass ratios: (a) pulse frequency normalized with respect to the frequency parameter of the slenderest configuration; (b) pulse frequency normalized with respect to the frequency of the isolator. response of the system varying the mass ratios is analyzed ( Figure 12).
Minimum acceleration overturning spectra (Figure 12a) are obtained, assuming a constant value of = 2 s, for various mass ratios ranging from = 0.9 (light isolator mass) to = 0.001 (very heavy isolator mass). The results show that this parameter has little effect on the response of the system, which remains approximately constant for different mass ratios.
The response of the system to different is then investigated for different periods of the base isolator ( Figure 11b). As previously observed (Figure 8), if the pulse frequency is normalized with respect to the frequency of the base isolator, the normalized minimum acceleration overturning spectra collapse into one; the mass ratio has no effect also in this case, and the response is approximately the same for different and different (Figure 12b). In F I G U R E 1 3 (a) Three-dimensional normalized minimum acceleration overturning spectra for the isolated and for the nonisolated system; (b) minimum acceleration overturning spectra for the isolated (for = 3.0 s), numerical, and closed form, as well as for the nonisolated system; (c) values in terms of normalized pulse frequency, period of the isolator, and normalized minimum overturning acceleration for the points A, B, and C. conclusion, the results indicate that the ratio between the mass of the blocks and the total mass (including the isolator) does not significantly influence the response.
Once the influence of parameters such as the damping ratio ( Figure 7) and the mass ratio ( Figure 12) is understood, and the response of the system is reduced to only one slenderness parameter , a three-dimensional overturning spectrum is derived. The spectrum is built subjecting the system to a wide range of pulses that differ in amplitude and frequency, while also varying the period of the isolator. The spectrum (Figure 13a) confirms the previous observations, that is, that increasing the period of the isolator increases the safety level of the system. Figure 13 also shows the analytical overturning spectrum when the system is considered as monolithic and placed on a rigid foundation. The surface describing the response of the nonisolated system obviously does not depend on the period of the isolator, so it remains constant for any T b . When the two surfaces, that is, the isolated and nonisolated responses, are compared, a region can be identified in which the response of the system standing on a rigid foundation is safer than when it is isolated. This region is larger for small periods T b , where the isolation is almost ineffective, and it shrinks as the period of the isolator increases (as does the isolator efficiency). Further, the efficiency of the base isolation is determined not only by its period, but also by the frequency of the pulse and the frequency parameter of the rocking object.
Three cases are investigated here (points A, B, and C in Figure 13a,b), using the three-dimensional overturning spectrum (Figure 13a) and the two-dimensional overturning spectra (Figure 13b) for a constant period = 3.0 s. In the case of a long period pulse or a small block (point A), the response of the system is safer in the nonisolated case; indeed, the amplification of the linear isolator is particularly evident in this region. Decreasing the pulse period or increasing the size of the block (point B) increases isolator effectiveness, and the isolated system becomes safer than the nonisolated. In the case of a large block or a short period pulse (point C), the beneficial effect of the base isolation device further increases, which means that the minimum overturning acceleration increases as well. The outcomes demonstrate that the dynamic response of the system is systematic. For this reason, the closed-form fit of the overturning spectrum in Figure 13b can have practical applications, and it is presented as follows: To maximize efficiency of the base isolation technology, the response of the system should be moved as far away as possible from the region where the ground motion is amplified by the linear isolator. This outcome could be obtained by increasing the period of the isolator. For example, assuming = 0.8 s (Makris, 2000) and = 3.1 rad/s (as for the system in Figure 5a), the isolator is effective for > 2.0 s. Nevertheless, for long period pulses or small blocks the minimum overturning acceleration of the isolated system can be lower than the nonisolated system. Assuming = 0.8 s and using a very small block (i.e., a household brick for which ≈ 8.0 rad/s), the isolator becomes efficient only for > 5.0 s. This potentially dangerous phenomenon disappears for small periods of the pulse or for large blocks. Indeed, assuming a large block (i.e., a computer server ≈ 2.7 rad/s) and considering the same period pulse of the previous example, the isolator becomes efficient for > 1.6 s. Therefore, the safety level of the system may be enhanced by increasing the size of the lower block (pedestal) while keeping constant. This solution might become impractical if the upper block needs to remain at a reasonable height, such as for an art object in an exhibition. In that case, a possible alternative is restraining the upper block to a squat lower block, but this solution might involve large internal stresses that may be incompatible with brittle historical artifacts (Sorace & Terenzi, 2015).

CONCLUSION
A dynamic two rigid body rocking model isolated at the base was developed. The friction was assumed large enough to prevent any sliding, the two blocks can rock only about the corners and the impacts are assumed to be instantaneous. Then, the computational method used to solve the governing equations was presented in terms of pseudocode, emphasizing the events marking the transition between the five possible patterns. The model was used to derive rocking spectra for twoblock systems subjected to sine-pulse ground motion, described by its amplitude and frequency. The main findings, can be summarized as follows: 1. The base isolation device is effective for long period devices.
2. As an alternative or concurrent approach to the design of a long period device, the size of the system could be increased to enhance the effectiveness of the base isolation. 3. Depending on the size of the system, if long period pulses are expected, it could be impossible to deliver an effective base isolation with current technologies. 4. The amplification caused by the isolator depends on the ratio between the pulse period and the period of the isolator, and the amplification of the LMDS oscillator, which can be obtained using shock spectra for specific pulses, is the reciprocal of the normalized minimum overturning acceleration of the system. 5. Overturning spectra can be obtained assuming the isolator period to be constant or assuming a constant ratio of the periods (between the pulse and that of the isolator). The latter type of spectra must be carefully interpreted because each frequency of the pulse corresponds to a different isolator period, meaning that is constantly varying along the x-axis. 6. Minimum overturning acceleration spectra can be used to describe the response of the system; when the frequency of the impulse is normalized with respect to the frequency of the isolator, the response for different isolator periods collapses to a single curve. 7. The response of two stacked rocking blocks can be understood using only one configuration, the slenderest between the two blocks considered as monolithic and the upper block alone. This means that, using the proper configuration, the response of the two-block system to sine-pulse ground motion can be approximated well by neglecting block interaction.
Potential future developments involve the consideration of alternative excitations, such as symmetric and antisymmetric Ricker wavelets and recorded accelerograms, in order to consider either more dangerous or more realistic excitations. Additionally, the no-sliding hypothesis and homogeneous density assumption could be removed, while configurations involving three or more bodies could be investigated.

A C K N O W L E D G M E N T S
This work was partially carried out under the MONALISA research project funded by Lazio Regional Government and Italian Ministry of University and Research. The opinions expressed in this publication are those of the authors and are not necessarily endorsed by the funding bodies.

APPENDIX B
Here, a pattern change due to acceleration thresholds exceeded is investigated. For conciseness, only the pattern change from 3a to 1a is reported. In this case, the threshold acceleration that causes the pattern change is defined as follows: TA B L E C 1 Pattern change due to impacts.