The dynamic behaviour of flexible oscillators rocking and sliding on concentrated springs

This study presents the Flexible Rocking Model on Concentrated Springs (FRMCS), developed to investigate 2D laterally flexible oscillators rocking and sliding on deformable support media during ground excitations. In this model, concentrated vertical springs and viscous dampers simulate the contact forces from support medium at the corners of the body; the tensionless vertical contact element is linear in compression. Horizontal concentrated springs and linear viscous dampers simulate the frictional behaviour at the corners; the constitutive law for the springs models elastic deformations and sliding (according to Coulomb’s friction law). With these elements, FRMCS can model the response of a rocking body which can experience sliding and free-flight phases of motion. The consideration of the flexibility of the support medium enables the evaluation of the forces exerted by the support medium on the structure during an impact. In this study, the FRMCS response is first compared to a previous model where the support medium deformability and the effects of sliding and free-flight are ignored. Then, the responses of four configurations, which feature either stiff or soft lateral springs and stiff or soft high-grip support media, are examined under the influence of pulse excitations. Finally, to understand the potential influence of sliding, a configuration with a low-grip support medium is explored. The comparative influence of lateral flexibility and support medium deformability and sliding is quantified with stability diagrams and various response spectra, describing structural force and moment demands.

Observing the foundations of certain structures uplift during earthquakes has led researchers to study the rocking phenomenon.To describe rigid bodies rocking on rigid support media, Housner 1 proposed the inverted pendulum model (IPM) where the motion of the rigid body is idealised with pure rocking motion and a single rotational degree of freedom.To achieve the idealisation, the flexibility of the body, the deformability of the ground and the effects of free-flight and sliding had to be considered as negligible.Following this pioneering work, researchers [2][3][4][5][6][7] analytically and experimentally studied the effects of sliding and free flight motion.The presence of these phases of motion influences the energy loss at impact and consequently the stability of the rigid body.Considering that large structures are often laterally flexible, researchers [8][9][10][11][12][13] proposed models for idealised flexible oscillators rocking on rigid support media.These studies highlighted the coupling between the vibrations of the flexible body and the rocking motion, and investigated the feasibility of rocking as a structural seismic response control strategy.However, rocking simulations are significantly influenced by the treatment of impact.The aforementioned studies on flexible oscillators have all neglected the sliding and free flight phases of motion, and have adopted divergent assumptions at impact, as recently reviewed by Zhu et al. 13 Other researchers have investigated rocking on deformable support media.Some of these studies investigated rigid blocks while others explored flexible oscillators.Psycharis and Jennings 8 investigated foundation deformability with either concentrated springs and dampers (CSM) at the corners or a distributed Winkler spring medium underneath the structure.The midpoint of the base was assumed not to move horizontally, and consequently sliding behaviour was not considered.The equations of motions were applicable for small rotations.Despite using a spring-damper foundation, Psycharis and Jennings 14 introduced a Housner-like energy coefficient to dissipate energy instantaneously upon contact, which was later adopted by Palmeri and Makris 15 for their treatment of the same problem for large rotations.This approach features an inconsistency, since springs and dampers cannot produce impulses over zero-time duration.As noted later by Psycharis, 8 Yim and Chopra 10,16 energy dissipation can instead be modelled using viscous dampers alone.
Chatzis and Smyth 17 proposed a model that takes into consideration sliding, free flight, geometric nonlinearity and support flexibility with CSM or Winkler models.Different from earlier studies, sliding was modelled by using horizontal non-linear springs and dampers that exhibit a Coulomb frictional behaviour.This paper explored the effects of ground deformability, sliding and free-flight on rocking bodies.However, the rocking bodies were assumed to be rigid.
In this study, a new analytical model called the Flexible Rocking Model on Concentrated Springs (FRMCS) is developed and presented in Section 2 (see Section 2).The model aims to explore the influence of the support medium deformability on the dynamic behaviour of flexible rocking oscillators, while accounting for possible sliding and free flight motions.Importantly, the consideration of support medium deformability enables an investigation of the forces exerted by the support medium on the structure during impact, and removes the requirement for determining arbitrary parameters during impact as in the case of hard rocking models.After conducting model parameter normalisation in Section 3, FRMCS is compared with the Flexible Rocking Model (FRM) 13 in Section 4 considering the response of a slender body on a high-grip stiff support medium where the effects of sliding and fee-flight are minimal.This is to see if FRMCS can yield consistent results with FRM because the two models feature different energy dissipation mechanisms.Then, two different rocking configurations covering soft oscillators with either stiff or soft support media along with one configuration that focuses on the effect of sliding are subjected to sinusoidal pulse excitations to preliminarily explore the effects of support deformability, lateral flexibility and sliding on the rocking response (see Section 5.1).To generalise the results, Sections 5.2-5.7 consider two more configurations covering stiff oscillators with either stiff or soft support media, together with the previous three configurations.The five configurations, in total, are subjected to a wide range of sinusoidal pulse excitations and their force and displacement response spectra are examined.The findings are summarised in Section 6.

MODEL INTRODUCTION
A schematic drawing for FRMCS is shown in Figure 1A.The model consists of a top point mass   and a rigid bottom body of mass  with a centre of mass located at point B and a mass moment of inertia about B,   .The bottom mass can take any shape that is bilateral symmetric and to simplify the figure, it is represented as a box.The top mass   is connected to the bottom mass through springs and dampers, and is constrained to move laterally, in a direction parallel to the line defined by the two bottom corners ZY.When the spring is undeformed, the top mass is located at point P. The width of the bottom body is denoted as b, height as h.Its centre of mass is located at height ℎ  .The centre of mass of the whole body is located at point C when the body is at rest at height ℎ  .The slenderness of the top mass, the bottom mass and the whole system are described with   (tan   = ∕(2ℎ)),  (tan  = ∕(2ℎ  )) and   (tan   = ∕(2ℎ  )), respectively.  ,   and  are the distance from points P, C, B to either of the bottom corners.FRMCS can be used to represent rocking structures with lateral flexibility.For example, a tall rocking bridge pier can be idealised with FRMCS where the top mass would be the supported mass in the deck and the bottom mass the bridge foundation mass.The bridge pier mass can be distributed to the top and bottom masses.Since the bottom mass represents structural components above the rocking interface, it may be located at a non-zero height.It should be noted that FRMCS is a simplified representation, because structures normally have more than one lateral mode but FRMCS possesses only one lateral degree of freedom.
Figure 1B shows the deformed position of the system.In this paper, this will largely be due to the influence of horizontal ẍ and vertical ÿ ground excitations.The FRMCS has five degrees of freedom: , , ,  and   . and  are the  coordinates of the centre of mass of the bottom body with respect to an absolute frame with origin, O, located on the undeformed ground surface. is elongation of the lateral spring and describes the ⃗  coordinate of the body in the rotating  coordinate system with the origin at B. The rotation of the bottom body is described by .  is the frictional spring force that has to be tracked to describe current states of the model; it is introduced in depth in Section 2.3.The positive sign convention is demonstrated with the unit vector pairs shown in Figure 1B.
Contact elements at the corners consists of a spring (  ) and a damper (  ) in the vertical direction.In the horizontal direction, a spring (  ) and a damper (  ) are used to model horizontal ground deformability and sliding behaviour according to Coulomb's law of friction.The springs and dampers are illustrated in Figure 1.

Equations of motion
The free-body diagrams of the top mass and the bottom body are shown in Figure 2.   and   represent the total forces due to the springs and dampers acting on the top mass, respectively.  represents the normal reaction force between the top and the bottom mass that constrains the top mass to move in the  direction. 1 and  1 ("1" refers to the horizontal direction in the inertial reference frame) are the horizontal contact forces at the two corners due to the frictional springs and dampers, and  2 and  2 ("2" refers to the vertical direction in the inertial reference frame) are the vertical contact forces due to the ground springs and dampers at the two corners.The vertical and horizontal contact forces become zero when the support spring goes into tension.The equations of motion of the top mass are given by   (( ẍ + ẍ) cos  + ( ÿ + ÿ) sin  + ü −  θ2 − (ℎ − ℎ  ) θ) =   +   −    sin (1) Equations ( 1) and ( 2) are the equations of motion of the top mass along the directions of the rotating vectors ⃗  and ⃗ , respectively.Equations (3)-( 5) are those of the bottom body along ⃗ , ⃗  and , respectively.

Constitutive relation and numerical implementation
The vertical contact forces  2 and  2 are exerted by the vertical support springs and dampers, as follows: where   and   are the vertical coordinates of the corners Z and Y, respectively, in the inertial reference frame.If either of the coordinates are larger than zero, then the corresponding corner is above the ground surface (refer to Figure 1B) and there is no contact force.The horizontal contact forces  1 and  1 are the sum of the horizontal spring and damper forces at each corner.They are defined using the Coulomb's friction model, illustrated in Figure 3A, where   is the normal contact force (i.e. 2 for the left corner and  2 for the right corner) and  is the coefficient of friction.The horizontal force in a contact spring is generically described as   in the figure; it refers to  1 if the computation is done for the left corner and  1 for the right corner.Figure 3B illustrates the elastic-predictor and plastic-corrector scheme employed in the numerical implementation of the model.To calculate the corner friction force (i.e. 1 and  1 ) governed by the Coulomb's friction model at each time step, the frictional spring force (  ) is treated as an additional state; it is tracked and corrected using the elastic-predictor and plastic-corrector scheme: where * symbol indicates a trial quantity which will be corrected if it violates Coulomb's friction limit.  is the tracked frictional spring force and ẋ is the horizontal velocity of each corner (e.g.ẋ and ẋ ).Equation ( 9) represents the elastic-predictor step and Equation ( 13) the plastic-corrector step.They are graphically illustrated in Figure 3B for better understanding.Numerical simulations were carried out in MATLAB for this study.ode45 was chosen as the numerical solver and had been modified to include the aforementioned elastic-predictor and plastic-corrector scheme for the implementation of penalty Coulomb's friction model.Relative tolerance and absolute tolerance are modified so that the simulation results are not sensitive to them.However, they cannot be too small otherwise they would reach the computational hardware limit and errors would happen.For soft support springs (defined in Section 5), error tolerance in the order of e-06 returned converged results; while for stiff support springs, it was in the order of e-10.The stricter requirement of error tolerance for stiff support springs is a consequence of solving a relatively stiffer differential equations compared to the soft support springs ones.In addition, event function was used to switch between different governing equations, which accounted for the different states of the   ; for example,   will be zero if the body is in air and hence disappear from the state variables.

Definition of base shear, axial force and moment quantities
The positive directions of the base shear force (), axial force () and moment () are illustrated in Figure 4.These are computed using the contact forces:

MODEL PARAMETER NORMALISATION
In this paper, horizontal pulses are used to examine the response of FRMCS to earthquakes.9][20][21] To present the model input parameters in a more compact way, dimensional analysis is utilised to obtain dimensionless groups.The model governing equation can be re-written to the following mathematical form: (, , , ,   , ,   ,   , , ,   ,   ,   ,   ,   ,   , , , , , ,   ) =  (17)   where  and  are the magnitude and the frequency of the horizontal acceleration pulse, respectively.
is the damping ratio associated with the top lateral spring if the bottom body was fixed, while   =   ∕(2 and   =   ∕(2 √   (  + )) are the ground damping ratio of the system of two masses rigidly moving in the vertical and horizontal direction, respectively. is the coefficient of friction.  ,   and   are frequency parameters for the SDOF systems that would correspond to the isolated movements of the top mass in lateral direction and the two masses rigidly moving together in the vertical and horizontal direction, respectively.A larger value of those frequency parameters implies a stiffer spring.They are defined by the following equations: TA B L E 1 Input Π groups. groups Aside from the output parameters , , ,  and   , there are 17 independent input parameters and according to the Buckingham's  theorem, they can be expressed in terms of 14 dimensionless groups using three repeating variables.The choice of repeating variables is not unique as long as themselves do not form a dimensionless group.In this paper,   ,  and  are chosen as the repeating variables, where  is the rocking frequency parameter ( = √ ( +   )  ∕(  +    2  +  2 )).The 17 input parameters are expressed in terms of the 14 dimensionless groups.The resulting dimensionless groups are adjusted by conducting algebraic and trigonometric operations to achieve a new set of more intuitive dimensionless parameters.The final set of dimensionless parameters is listed in Table 1.
A clarification is necessary for the dimensionless groups Π 2 , Π 4 , Π 6 and Π 8 .The unit of  terms in these dimensionless groups is rad/s while that of  is 1/s.

MODEL COMPARISON FOR EFFECTIVELY RIGID SUPPORT MEDIA
In this section, FRMCS simulations are compared to those from the FRM.A brief review of the FRM is presented in Section 4.1.Free vibration tests and stability analysis under pulse excitations are conducted in Sections 4.2 and 4.3, respectively.

Review of FRM
The FRM was proposed by Zhu et al. 13 to describe the rocking of laterally flexible oscillators on a rigid support medium.It consists of a top mass and bottom rigid body, similar to the FRMCS.However, sliding and free flight are not accounted for in the FRM.To enable comparisons, the geometric parameters of FRM have been adopted to be compatible with FRMCS.F I G U R E 6 (A) Impulses during impact for the top mass and (B) the bottom mass from Zhu et al. 13 .
Due to the rigid treatment of the support medium in FRM, the impact between the bottom mass and the support medium has an infinitesimal duration.This gives rise to a vertical impulsive force  1 between the top mass and the bottom body, and vertical   and horizontal   impulsive forces between the bottom body and the support medium during impact.
Zhu et al. 13 argued that there cannot be a horizontal impulse between the top mass and the bottom body during an infinitesimal duration impact.The lateral force that can be exerted by springs and dampers during an infinitesimal duration impact is either zero or a finite value; therefore, the corresponding horizontal impulse  1 (shown in Figure 6A) is zero.The horizontal position of the vertical support medium impulse   can be anywhere between the future rocking corner and the midpoint of the base.The horizontal location of   is described by b and illustrated in Figure 6B.A value of  must be assumed in FRM.Zhu et al. 13 investigated a bottom body with feet at the corners, where  can be reasonably assumed as zero.

Free vibration comparison
In this section, the free vibration responses from FRMCS and FRM are compared for a case where the support medium is practically undeformable and, the effects of sliding and free-flight are minimal.The common parameters that describe the geometry, lateral flexibility and damping ratio of models are presented in Table 2. To enable a like for like comparison, the vertical spring stiffness (  ), the viscous damping factor (  ), the frictional spring stiffness (  ), the horizontal viscous damping factor (  ) and the coefficient of friction () of the support media should be taken such that (i) the contact is effectively rigid, (ii) the effects of sliding and free flight are minimal.The parameters were obtained by gradually increasing the magnitude from initial guesses until responses stopped changing in a noticeable manner (i.e. when   = 2.414 ⋅ 10 value of coefficient of friction minimizes the effects of sliding, but small amounts of sliding always occur in FRMCS as the normal contact force tends to zero.Three bodies, each with a different lateral spring stiffness (Case A:   = 9.8225, Case B:   = 4.9599, Case C:   = 2.4313), were examined as in Zhu et al. 13 These values of   represent reasonable lateral vibration frequency values for stiff, intermediately stiff and flexible structures.For all three bodies, the free vibration motion is initiated with  0 = 0.8  .For the FRM, the initial values u0 , θ0 and  0 are set to zero.The FRMCS adopts the same initial parameters but also requires the application of external forces to ensure static equilibrium.The force  1 is applied at the corner of the body above the support medium, and an external horizontal force  2 is applied to the corner of the body embedded in the ground so that the bottom body is in static equilibrium.The free body diagram is shown in Figure 7.The initial rotation  0 is taken as positive and hence,  1 and  2 are applied to the left and right corner, respectively.The release of  1 and  2 initiates the movement of the body. 2 0 is the initial support vertical spring force.  0 is the initial normal force exerted on the bottom mass by the top mass, which is the component of the top mass weight that is normal to the top surface.Because  0 and u0 are set to zero, there is no lateral force on the bottom mass due to the top springs and dampers. 2 0 ,  1 and  2 are found by setting the body at static equilibrium with  0 and assuming that the frictional springs are undeformed.Mathematically, this leads to The initial vertical displacement of the right corner,  0 , then can be solved using the obtained value of  2 0 .Finally,  0 and  0 are obtained with  0 and  0 using the following kinematics relationships: The free vibration time history plots are presented in Figure 8.In each column, the normalised rotation (∕  ), normalised top mass relative elastic displacement (/  ) and normalised energy (/  ) are presented for a specific lateral spring configuration; in each sub-figure, the response obtained from both FRMCS and FRM are plotted. is the total energy and   is the potential energy difference between the unstable and stable equilibrium location of a rigid rocking  ℎ)).It can be seen from Figure 8 that the responses from the FRMCS and FRM are in excellent agreement.In FRM, pre-impact state variables and the impulse-momentum equations are used to define the post-impact parameters.For the free vibration simulations shown in Figure 8, Zhu et al. 13 assumed that the vertical impulse occurs at the future-rocking corner, similar to Housner's IPM.The proposed FRMCS does not require appropriate definition of impulsive force location to define energy loss at impact.Instead, energy is dissipated through viscous damping at contact elements.Therefore, the agreement between these two fundamentally different models is instructive.It demonstrates that the Zhu et al. 13 assumptions concerning impulse transmission through the springs and support medium force concentration at the edges are consistent with the FRMCS predictions for effectively rigid support media.The excellent agreement between the time histories indicates that the assumptions hold for small and large rocking motion and for different lateral flexibility values.Considering that the FRM yields equivalent results to IPM for effectively rigid bodies, it can also be concluded that FRMCS also can produce equivalent results to IPM.This aspect was examined numerically by the authors but is not presented for brevity.

Stability analysis comparison
In stability analysis, a rocking body is subjected to horizontal ground accelerations in the form of single cycle pulses with different magnitudes and frequencies.Magnitude and frequency pairs, which cause overturning, are recorded.In this study, the overturning event is recognized when  is equal to ninety degrees.In general, three areas can be observed in a stability plot for symmetric rocking bodies subjected to single cycle horizontal acceleration pulses when sliding and free flight are neglected.Taking Figure 9A for example, typically, the uppermost line in the diagram corresponds to the minimum value of pulse magnitude for a specific frequency  for which the body will overturn without experiencing any impacts.Above this line, overturning occurs without impact; this is referred to as Mode II failure area.Below the Mode Cases A (stiff) and C (flexible) from previous free vibration comparison are used for the stability analysis.The results are presented in Figure 9; the black line represents the FRMCS response and the red line the FRM.In Figure 9A, it can be seen that the FRMCS is in excellent agreement with the FRM for Case A. There is a small region of disagreement around ∕ equal to 9.5.Similar observations can be made for Figure 9B, which examines a more flexible oscillator.The small regions of disagreement are due to the finite values adopted for the support medium flexibility and numerical simulation errors.
Comparing Figure 9A and Figure 9B, it can be seen that the Mode I overturning area shrinks and sinks with increasing lateral flexibility, reproducing earlier observations from Acikgoz and DeJong. 11The combined influence of the lateral spring and the support medium deformability is investigated next.

Time history evaluation
To understand the influence of support medium deformability and the coefficient of friction on the structural response, five configurations in total are studied.The values of the common dimensionless groups of the five rocking configurations are presented in Table 3 while the case-specific groups are reported in Table 4. First, high grip surfaces with either soft or  stiff lateral spring and vertical contact elements are considered (Cases 1-4).The influence of a lower surface grip is then examined by reducing the horizontal support spring coefficient of friction (Case 5).
The pulse excitation time histories of some of the configurations in Table 4 are comparatively examined in this section to obtain an intuitive understanding of response.These are soft(lateral spring)-stiff(support spring) (Case 2), soft-soft (Case 4) and Case 5. Case 5 is the same as Case 4 except that it uses a low-grip surface with  = 0.14.This is defined as a low-grip surface since  <   , where sliding is expected to be dominant.Π 2 is 2.431 for the soft lateral spring; Π 4 is 563.865 and 56.387 for the stiff and soft support spring configurations.The three configurations share the same geometric properties presented in Table 2.It should be noticed that the Π 4 value adopted for the stiff support spring configuration is five times smaller than the values used in Section 4, which were meant to simulate rigid support media.
The three configurations are subjected to a single cycle sinusoidal acceleration pulse with Π 8 and Π 9 equal to six and three, respectively.Figure 10  In Figure 10A,B, the rocking and lateral elastic displacement responses of Cases 2 and 4 appear to be relatively similar.However, the forces exerted by the support media on the structures are very different.Figures 10C and 10D present the contact forces at the impacting corner following the first major rocking cycle, around t = 1 s.This brief time window describes when maximum vertical contact forces are experienced by both bodies.However, the maximum vertical force for Case 2 is almost 10 times that of Case 4. This observation indicates how the support stiffness may have a significant influence on the contact forces while having no apparent effects on the maximum displacement response.It is noteworthy that an increase in contact force is likely to indicate an increase in base force and moment demands in the structure (see Section 2.4); this aspect will be examined in detail in Section 4.2.
In Figure 10, impact can be assumed to initiate when the future-rocking corner hits the support (i.e. when  2 starts to compress).It ends when the opposite corner is in the air (i.e. when  2 is zero) and the vertical velocity of the futurerocking corner becomes zero.This corresponds to the interval 1.01858-1.03924s in Figure 10C.The impulses due to normal contact forces at points Y and Z can be calculated as the integral of force over the impact duration.The ratio of the magnitude of left corner (i.e.future rocking corner) impulse to the right corner impulse is equal to 23.Ratios of similar orders of magnitude are observed for other impacts.Similarly to Zhu et al. 13 and Chatzis and Smyth, 17 this indicates that for bodies which can be modelled using the CSM, for example, those that have feet at their bottom corners, the vertical impulse is practically concentrated at the future rocking corner for stiff support media when the effects of sliding and free flight are small.However, a similar impulse concentration is not observed for Case 4, where the support medium is soft.Using the same definition of impact and assuming that it lasts from 1.02874 to 1.49571 s, a future to past rocking corner impulse ratio of 3.856 is obtained.Ratios of similar order of magnitude are observed for other impacts.Such smaller ratios indicate that for soft support media, impulses cannot be assumed to be concentrated at the future rocking corner.
The following Figure 11 presents a similar comparison between Case 4 (soft-soft) and Case 5.This figure additionally examines the base axial force  and base shear force  responses.
Figure 11 compares the time history responses of Cases 4 (soft-soft with high-grip, examined already in Figure 10) and 5 (soft-soft with low-grip).Figure 11A shows the horizontal position  of the centroid of the bottom body.At the end of the time history, a residual displacement can be observed for Case 5 due to sliding of the body.Despite the sliding, Figure 11B,C shows that the rotational response  and the lateral displacement response  are similar between the two cases.
The impact of sliding on the contact forces is examined in Figure 11E,F.The base axial force  (see Equation 25) is normalised with respect to the weight of the structure.The base shear force  is normalised with respect to the static horizontal force needed at the centre of mass to uplift an equivalent rigid rocking body.

𝑁 𝑛𝑜𝑟𝑚 = 𝑁 (𝑚 + 𝑚 𝑝 )𝑔
(26) In Figure 11E, the axial force response patterns and maximum peak values are similar for the two cases.However, sliding has a significant influence on the base shear force response.Figure 11F shows that the maximum shear force response decreased by almost six times in Case 5 compared to Case 4.
It is useful to summarise the observations from the examined time histories: (a) the support spring stiffness has a significant effect on the maximum vertical support force response (and consequently base force and moment demands in the structure); (b) for bodies with feet at the bottom corners, the vertical impulses from the support medium concentrate at the future rocking corner when the support medium is stiff, while this may not true for soft support media and (c) sliding between the structure and support medium can significantly reduce the maximum base shear force demand in the structure.

Response spectra evaluation
Previous time history studies are based on three rocking configurations subjected to a pulse excitation of specific amplitude and frequency.A more exhaustive investigation is conducted in this section to see if the observed trends are applicable to other configurations and single cycle excitations with different amplitudes and frequencies.To achieve this, different values of dimensionless groups are used in the following sections to investigate a wide range of support and lateral spring stiffness.Therefore, in addition to Cases 2, 4 and 5 from the previous section, another two cases are examined below: stiff(lateral spring)-stiff(support spring) (Case 1) and stif-soft (Case 3) configurations (see Tables 3 and 4).

Case one (stif-stiff)
In this section, response spectra referring to various quantities are presented in Figure 12A-E.These spectra report the maximum magnitudes of the examined quantities for single cycle sinusoidal acceleration pulses with the corresponding of a given frequency and amplitude.The spectra are plotted for varying frequency in the horizontal axis and amplitude in the vertical while indicating the maximum values of the quantity of interest using colours as indicated by colour bars.The stability diagram is also plotted in red in Figure 12A for ease of reference.The response spectra for pairs of frequency and amplitudes that would lie within the Mode II failure is not of interest and so the corresponding regions are left uncoloured in the figures.
The non-dimensional pre-impact rotational velocity parameter θ∕ is obtained using the maximum rotational velocity in the time history, just before the rocking body hits the support medium.The discretised contours in Figure 12A refer to this parameter.Figure 12B  the base moment needed to uplift a rigid assembly of the two masses statically: Normalised base axial force in Figure 12C was introduced earlier in Equation ( 26).The lateral displacements in Figure 12D are normalised with respect to   similar to earlier sections to obtain   .
Figure 12E represents the lateral moment,   .  is calculated as the product of the top shear force   (sum of   and   ) and the bottom body height ℎ.This represents the part of the base moment demand caused by the lateral springs and dampers.In Figure 12F, the maximum normalised pre-impact rotational velocity is plotted against the maximum   or   from different time analyses.To relate various force demands in the system, Figure 13A presents the ratio between the maximum internal forces (i.e. the base axial and shear force) to the maximum ground forces (i.e. the vertical and the horizontal support force) from the same excitation.During this specific numerical experiment, it was noted that the maximum internal force and the maximum external force occurred at the same time instance, therefore this ratio qualitatively shows whether the maximum internal force occurs during impact.For example, if it occurs during impact then  is close to zero and hence the ratio should be close to 1; if it occurs outside the impact time window, the ratio will not be 1 according to equations in Section 2.4. Figure 13B presents the graphical relationship between the maximum preimpact rotational velocity and the maximum shear forces applied between the two masses and at the base of the bottom mass.The top shear force   is calculated as the negative sum of the lateral spring force (  ) and the lateral damper force (  ).The lateral moment   and the top shear force   are normalised in the same way as  and  to obtain    and    .
Figure 12A has two white areas, one above the coloured contour map and one below.The white area above the coloured contour map is the area of overturning without impact and hence the pre-impact θ is undefined; the white area below the map represents the area where uplifting from ground did not occur and hence the pre-impact θ∕ is undefined either.Figure 12A shows that the maximum pre-impact θ increased with the single cycle pulse amplitude given a fixed pulse frequency outside the overturning zone.However, inside the overturning zone (Mode I failure zone, coloured area enclosed by the red line), the maximum pre-impact θ first increased and then decreased with pulse amplitude for pulse frequency ranging from 0.2p to 5p.The base moment and the base axial force spectra (Figure 12B,C) demonstrate similar response trends to those of the pre-impact rotational velocity (Figure 12A).In areas where overturning is not experienced, forces and moments appear to increase with increasing pulse amplitude and decreasing pulse frequency.However, highest force and moment demands coincide with regions of Mode I failure as can be seen from the yellow area.Figure 12F reveals that there is an almost linear relationship between the maximum pre-impact rotational velocity and the maximum base forces.Figure 13A shows that the maximum internal forces (i.e. the base shear and the base axial force) are the same as the maximum external forces (i.e. the vertical and the horizontal support force) for all of the excitations.This demonstrates that the maximum base forces occur around impact where the base is almost horizontal.Earlier, Section 4.3 showed that for Case A (stif-stiff case), the vertical impulse at impact concentrates at the future-rocking corner.Therefore, the maximum base moment should be approximately equal to the product of maximum base axial force and half the base width.This explains the similarity between Figure 12B and Figure 12C.
Figure 12D shows that the lateral displacement of the top mass does not vary significantly across a wide range of pulse excitations.Correspondingly, the top shear force in Figure 13B does not linearly increase with the maximum pre-impact rotational velocity and features a plateau.Similar observations were made by Yim and Chopra, 10 Psycharis 22 and Acikgoz and DeJong. 23However, Figure 13B highlights that the maximum top shear force is in general much smaller than the maximum base shear force.Similarly, it can be seen that the lateral moment   is also much smaller than the base moment  by comparing Figures 12B and 12E.This shows that the maximum lateral displacement alone is not an indicator of reducing the applied moments on the base of the body.These observations emphasise the need for caution when interpreting the reduction of base shear force and moment demands that can be achieved with a rocking system by judging top shear forces only.For the investigated body, the maximum base axial force can reach 30 times the system self weight; the maximum base shear force can reach 40 times the static uplifting force and the maximum base moment can reach 30 times the moment needed to uplift the structure statically.

5.4
Case two (soft-stiff) The following Figure 14 presents the same type of response spectra for the soft(lateral spring)-stiff(support spring) configuration to those presented in Figure 12.
Figure 14 shows that the spectra of the pre-impact normalised θ,   ,   , and   parameters are still correlated.This is evident in Figure 14F where a linear relationship is observed between maximum normalised pre-impact θ and maximum normalised base forces.However, due to the interaction between the lateral vibrations of the top mass and the rocking motion, the response spectra appear to be more irregular in Figure 14 than Figure 12.For example, in Figure 14A, the rocking motion amplitudes (and consequently the maximum pre-impact rotational velocities) do not tend to increase with increasing pulse amplitude and decreasing pulse frequency as in Figure 12A.As a result, in areas where overturning is not experienced, the maximum forces and moments do not necessarily increase with increasing pulse amplitude and decreasing pulse frequency.Furthermore, the highest force and moment demands do not coincide with regions of Mode I failure as can be seen that the green area also appeared outside the Mode I failure zone as well.In the   spectrum in Figure 14D, the contour lines appear to be more evenly scattered, highlighting a more uniform dependence on the excitation amplitude and frequency.The normalised lateral moment    shown in Figure 14E is much smaller compared to the normalised base moment   in Figure 14B, highlighting again the need for caution when interpreting the force reduction effect that can be achieved with rocking systems.Additionally, it is noteworthy that the normalised maximum force and the normalised moment response magnitudes are similar in Figures 12 and 14, despite an increase in structural flexibility in the latter case.

5.5
Case 3 (stiff-soft) The following Figure 15 plots results for the stiff(lateral spring)-soft(support spring) configuration.
From Figure 15F, it can be seen that the strongly linear relationship between the maximum normalised pre-impact θ and the maximum normalised base forces also holds for the soft support medium.Second, Figure 15A shows that while the response range for the maximum normalised pre-impact θ is the similar to Cases 1 and 2 shown in Figures 12A and 14A, the maximum normalised force and moment magnitudes in Figures 15B,C,F are significantly reduced.More specifically, the maximum   and   response are reduced by 10 times and the   by three times in comparison to corresponding values in Figure 12B,F, demonstrating that the support spring stiffness can significantly reduce the maximum base force and moment demands when the lateral spring is stiff.In addition, because the significant reduction in   the difference between    and   becomes smaller compared to the previous two cases.Third, by comparing Figure 15D and Figure 12D, it can be seen that when the lateral spring is stiff, the reduction of the stiffness of the support results in slightly larger maximum top mass displacement.Lastly, the similarity of response patterns between the   and the   , for example, previously observed in Figure 12, is not present in Figure 15.The differences between axial force and moment demand patterns are to be expected as the net support reaction forces are not concentrated near the future-rocking corner for soft support media.
To verify this argument, the   is plotted against the   in Figure 16A for Case 3, and Figure 16B for Case 1.
A reference line with slope of one and passing through point (1,1) is also plotted.During impact, if the net reaction force is concentrated at the future-rocking corner, then the base moment should be equal to the product of the axial force (i.e. the net vertical support force) and the half the base width (1 m).This corresponds to the slope of one for the reference line.It can be seen from Figure 16B that the blue line follows closely to the reference line for Case 1.It falls slightly below the reference line when the maximum axial force is small then goes slightly above as the maximum axial force increases.In contrast, it can be clearly seen in Figure 16A that in the soft support rocking, the blue markers are located significantly below the reference line.This supports the argument that even for a body with feet placed at its bottom corners that during impact, the net vertical support force is not concentrated near the future rocking corner when the support medium is softer.

Case 4 (soft-soft)
The following Figure 17 presents the results for the soft(lateral spring)-soft(support spring) configuration.
As can be seen in Figure 17F, a strong linear trend between the maximum normalised pre-impact θ and the maximum normalised base forces is still observed.It is likely that this linear trend applies to other (e.  and   spectra (Figure 17B,C) do not mirror one another, as the support forces at impact are not concentrated in one corner.
As can be seen in Figure 17B,C, the range of the maximum normalised response indicates that the soft lateral spring slightly reduces the maximum response of   and   compared to Case 3 (stif-soft).However, this force reduction effect is smaller than the reduction achieved by softening the support medium between Case 2 (soft-stiff) and Case 1.A similar observation was qualitatively made with time history investigations in Section 5.
In terms of the normalised top mass displacement   , it can be seen in Figure 17D that the soft support spring does not result in any noticeable change (an increase of 3.0%) under different pulse excitations by comparing Figure 17D and Figure 14D.A similar observation was made in Section 5.However, earlier, a noticeable increase of 23.4% in the maximum   response was observed when comparing Cases 1 and 3.This shows that the effect of the compliance of the support medium on increasing the maximum   may becoming more or less pronounced depending on the stiffness of the lateral spring.In addition, the same trends highlighted in Figures 13A and 13B are also observed across all the four cases, although the corresponding figures were not shown for Cases 2-4 for brevity.Therefore, it is likely that for all different rocking configurations, the maximum base forces happen during impact and that the maximum top shear force is smaller than the maximum base shear force.

Case 5 (Case 4 with sliding)
Case 5 investigates the influence of sliding on a soft lateral spring and soft support medium by adopting Case 4 geometry and material properties but using two lower values of coefficient of friction .The sliding   is defined by combining sliding displacements at each support spring using the following equations: =   (−  ) +   (  ) where   and   are the sliding, that is, the inelastic parts of the displacement of corners Z and Y, illustrated as the "slide" part in Figure 3A;   is the elastic deformation of the frictional spring at corner Z (i.e.  =   ∕  );   is the initial horizontal coordinate of corner Z and  is the Heaviside function (notice   will never exactly equal to zero in numerical simulations so the Heaviside function is well defined).  uses the information from the spring in contact with the ground, which experiences negligible transnational motion due to rocking motion of the body.Figure 18 shows response spectra under different values of .In the spectra,   is normalised by the base width (2 m), and the maximum absolute values attained by this parameter during the time history results are reported.Spectra exclude the unsafe area (i.e.Mode I and II areas) where large sliding is often experienced as the rocking angle tends to ∕2.This sliding near collapse becomes less of interest as the body would tend to fail anyway by overturning and hence the associated sliding that occurs before failure will not be examined in the paper.In Figure 18, decreasing  from 0.8 to 0.3 leads to some, albeit small magnitude, sliding.This is accompanied by a 50% reduction in the maximum   and negligible change in the maximum   and the maximum   .It is also noteworthy that the white area, where overturning happens, shrinks in the high frequency region (i.e.around Π 8 larger than 7).These observations suggest that, in addition to increasing support flexibility, allowing a small amount of sliding can notably reduce the maximum base shear force demand.A similar observation was qualitatively made with time history investigations in Section 5.
The response spectra for  = 0.14, which are shown in Figure 18C,F,I,L, feature a distinct overturning region compared to the other examples.The presence of an overturning failure area although  is smaller than (  ) = 0.15 is instructive -for a similar rigid body rocking on a rigid surface, no overturning would be expected, as the body would experience pure sliding when subjected to pulse excitation.As Greenbaum 24 showed for rigid bodies on deformable supports and Zhu et al. 13 showed for flexible oscillators on rigid support media, not only uplift but even overturning can be observed for situations where a rigid body on a rigid ground would only be predicted to experience pure sliding, when deformability of the support medium or the structure are considered.Figure 18I indicates a drastic increase in   ; sliding displacements in the order of metres can be observed.The accompanying maximum base shear force also decreases.Due to the significant interaction with sliding, the maximum top mass displacement demand also reduces noticeably.The results appear to suggest that, for this type of excitation, adopting very low grip support media may help to reduce the force and the displacement responses at the expense of increasing sliding.
Finally, the stability diagrams of the five cases are plotted in the following Figure 19.As can be seen in Figure 19A, the Mode II collapse boundary is not significantly affected by the support medium and the lateral spring stiffness for large pulse frequencies; however, for small pulse frequencies ranging from 0.2p to 2p, softer lateral spring decreased the pulse amplitude needed to overturn the structure as can be seen from the observation that the purple line is below the blue line and, the red line is below the green line.Mode I is more strongly affected.Comparison of Case 1 and Case 2 lines shows that when the support medium spring is stiff, reducing the lateral spring stiffness can reduce the Mode I overturning area while also causing it to extend to the low frequency and low amplitude excitation region.The same effect was observed in Ref. 11 and was described as the "shrinking and sinking" of the Mode I area.A similar "shrinking and sinking" effect can also arise due to increasing support flexibility, compare stability diagrams for the Cases 3 and 4.
The stability plots of Cases 4 and 5 (0.3 ) in Figure 19B indicate that small amount of sliding slightly increases the Mode I overturning area but shrinks the the Mode II overturning area in the high frequency excitation range (i.e.around Π 8 larger than 7).With more sliding (see Case 5 (0.14 )), the Mode II area decreases starting from Π 8 equal to 3 but slightly increases in the low frequency excitation range (Π 8 = 0.4-1).At the same time, the Mode I area enlarges from around Π 8 equal to 3.5.In summary, sliding has both positive and negative effects on the rocking stability depending on the pulse frequency.

CONCLUSION
A new model for the laterally flexible oscillator on a deformable support medium is developed in this paper.The support medium is represented by concentrated support springs and the model is called FRMCS.After presenting the equations of motion and the constitutive model for the support springs, the free vibration and pulse excitation of various configurations are examined.First, FRMCS is used to model the behaviour of a flexible oscillator, rocking on a very stiff (effectively rigid) support medium with the dimensions and coefficient of friction chosen so that the effects of sliding and free-flight are minimal.The model results are compared to those from the FRM (Zhu et al. 13 ).The two models are shown to be in excellent agreement.This agreement highlights that when the body can be modelled with FRMCS, for example, when it has feet at its bottom corners, the assumption that the impulsive forces are concentrated at the future-rocking corner is reasonable when the support is very stiff.
After the comparison of the models for that limit case, model configurations covering laterally stiff and flexible oscillators with either stiff or soft support media are considered; the influence of surface grip (i.e.different coefficients of friction) is also considered.In total, five rocking configurations are examined, by subjecting the model to a wide range of horizontal single cycle pulses and comparatively exploring their response spectra (describing the displacement and force demands in the systems) and stability diagrams.
The investigations on the response spectra revealed important trends, which are summarised here.First, during impact, the net vertical support force does not necessarily concentrate near the future-rocking corner for soft support media.Second, it generally appears that the maximum base force demands grow with increasing maximum pre-impact rotational velocity during single cycle pulse excitation.Third, reducing the support spring stiffness can limit the maximum base forces efficiently.The comparative influence of the lateral spring flexibility on the maximum base forces appears to be smaller than the support spring flexibility for the cases studied.
Stability diagram comparisons show that a soft lateral spring has a significant "shrinking" and slight "sinking" effect on the Mode I failure area.Increasing support medium flexibility leads to a similar outcome in terms of stability.In addition, when subjected to single cycle pulse, for flexible oscillators on soft support media, the maximum base force demands reduce if sliding is allowed; even a small amount of sliding can result in a large reduction in force demands.However, sliding affects the rocking stability both positively and negatively depending on the pulse frequency.

A C K N O W L E D G E M E N T S
The authors have nothing to report.

D ATA AVA I L A B I L I T Y S TAT E M E N T
Data are available upon reasonable requests.

F
I G U R E 1 (A) Schematic drawing and (B) displaced configuration of FRMCS.FRMCS, Flexible Rocking Model on Concentrated Springs.

F I G U R E 4
Free body diagram of the base section.

Figure 5 F I G U R E 5
shows the displaced configuration of FRM, with the ground excitation ẍ and ÿ .The model has two degrees of freedom:  and . is the relative displacement of the top mass and  is the rotation of the bottom mass.The positive sign convention is shown on the top right of Figure 5. Displaced configuration of FRM.FRM, Flexible Rocking Model.

F I G U R E 8
Free vibration response comparisons for (A) Case A (  = 9.8225), (B) Case B (  = 4.9599) and (C) Case C (  = 2.4313) from FRMCS and FRM simulations.FRM, Flexible Rocking Model; FRMCS, Flexible Rocking Model on Concentrated Springs.block with similar geometric characteristics, as defined in Acikgoz and DeJong. 11  is defined as the top mass displacement for which uplift would be expected to occur for a structure with zero bottom mass height (ℎ  = 0) and no damping (  = [   +  ]∕(2 2

F I G U R E 9
Stability plots for (A) stiff body rocking on rigid support and (B) soft body rocking on rigid support.II impact boundary, Mode I failures may be encountered, where the body overturns after experiencing impacts.Areas outside Mode I or II failures are safe.
compares the response of Case 2 (soft-stiff) and Case 4 (soft-soft) configurations.Time history responses of ,  and vertical support forces at two corners are shown.The black, the blue and the red lines represent Cases 2, 4 and 5, respectively.
presents the spectra for the normalised base moment   .The parameter is normalised by F I G U R E 1 2 Response spectrum for Case 1 (  = 9.822,   = 563.865).F I G U R E 1 3 Maximum normalised pre-impact θ versus (A) maximum normalised internal and external forces and (B) maximum normalised shear forces.

F I G U R E 1 6
Maximum   versus maximum   for (A) Case 3 and (B) Case 1.
g. intermediate lateral spring and support stiffness) configurations that have not been examined in this paper.Similar to Case 3, trends observed in F I G U R E 1 7 Response spectrum for Case 4 (  = 2.431,   = 56.387).

F I G U R E 1 8
Response spectrum comparison for different values of  (  = 2.431,   = 56.387).F I G U R E 1 9 Comparison of stability plots for (A) Cases 1-4 and (B) Cases 4 and 5.
TA B L E 2 F I G U R E 7Free body diagram for the initial condition.
TA B L E 3