# Experimental Validation of Numerical Model for Bi-Tilt-Isolator

Authors: Ming-Hsiang Shih; Wen-Pei Sung; Chia-Yu Ho
Journal: Shock and Vibration
 (2018)

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## Abstract

Bi-Tilt Isolator (BTI) is composed of bi-tilt beveled substrate and slider. The advantages of BTI are that the maximum upload seismic force of structure can be easily controlled and displacement of isolation layer will be reduced. Sliding force, friction force, and impulse force are caused in the slanting process of BTI, nonlinear behavior. A nonlinear mathematical model is derived based on the sliding upwards, sliding downwards, and transition stages. Then, BTI element of nonlinear analysis program, GENDYN, is developed by the fourth-order Runge-Kutta method, the discretized ordinary differential equation for three movement stages of BTI. Then, test set-up of superstructure installed with BTI is tested and recorded the real displacement and acceleration responses under conditions of full lubrication, mild lubrication, and without lubrication between interface of bi-tilt beveled substrate and slider with three various initial displacements. The comparison of simulation results and test results shows the following:(1) root mean square error is below 1.35% for WD40 sprayed, 0.47% for WD40 whipped, and 0.54% for without lubrication, respectively; (2) the maximum root mean square error for simulating with cubic polynomial function of friction is much less than those of constant friction except conditions of full lubrication, which are not affected by kinetic friction force; (3) application of cubic polynomial function for simulating friction of BTI with three different lubricated conditions can perform very fine simulation results, compared with the test results. This proposed mathematical model and BTI element of GENDYN program, using cubic polynomial function of friction, perform fine simulation capability to assess nonlinear isolation effect of structure installed with BTI.


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## Body

## 1. Introduction

Earthquakes and strong winds are unavoidable natural disasters on the planet. Recently, the intensity of natural disasters has been enhanced according to many factors; strong earthquakes, such as 9.0 magnitude (Richter scale) earthquakes, happened on 2004 in India and 2011 in Japan and triggered severe tsunami, resulting in heavy losses of life and property. In addition, some strong earthquakes occurred in the world. For example, 7.7 magnitude (Richter scale) earthquake in Pakistan on 2013 and 7.8 and 7.3 magnitude (Richter scale) earthquakes in Nepal on 2015 caused a lot of casualties. Taiwan is located in the Circum-Pacific Seismic Zone and also at the junction of the Eurasian plate and the Philippine Sea plate, witnessing sensible earthquakes every year. In particular, Chi-Chi earthquake (7.3 magnitude on the Richter scale) happened on 1999 to result in great damage to buildings and bridges. Moreover, a 6.4 magnitude (Richter scale) earthquake happened in southern Taiwan and led to the 16-story building collapsing on the ground, causing heavy casualties in 2016, before Chinese New Year. The main reason for building collapse is the lack of earthquake resistance capability.To maintain the safety of buildings to resist seismic force and external force, structural control theorems and equipment are widely applied in Architecture and Civil Engineering. Structural control techniques [1–3] have been divided into passive control (isolation, shock absorption, and energy dissipation) [4–11], active control [12–23], and semiactive control [24–27]. In this study, a newly developed Bi-Tilt Isolator is proposed as an isolation system for building. Traditionally, soft isolation layer is used as base isolation, for example, lead rubber bearing, LRB [28–33], and rubber bearing, RB [34, 35]. The purposes of LRB and RB are applied to extent structural period and isolate seismic waves into the structures to reduce the horizontal seismic force. The defect of these kinds of isolation systems is a large displacement of isolation layer, affecting the practicality.To improve shortcomings of base isolation system, Bi-Tilt Isolator, BTI, composed of bi-tilt beveled substrate and slider, as shown in Figure1, is proposed in this study. Although the proposed BTI and Friction Pendulum System (FPS) look quite similar, isolation efficiency and practical performance between these two devices have significant differences.Figure 1
The proposed Bi-Tilt Isolator.FPS is composed of a concave plate with spherical surface and an articulated slider with the same radius of curvature as the substrate [36–38]. These two devices provide functions of autohoming and frictional energy dissipation mechanism to meet most of the requirements for structural isolators. They have an advantage over flexible vibration isolator, such as performance of High Damping Rubber Bearing (HDRB), but they do not cause an eccentric effect. But, the maximum force of BTI is determined by its bevel angle and friction coefficient, independent of the relative displacement of the isolation layer. This isolator has no isolation effect under small earthquakes, and only large earthquakes can limit the maximum upload earthquake force. Action force of FPS is related to the displacement of the isolation layer. When surface seismicity exceeds design seismicity, the seismic force uploaded to the superstructure may exceed the design value, and the deformation of the structure is hard to estimate. BTI does not have the characteristic frequency of linear spring, and there is no resonance phenomenon in isolation layer. The energy dissipation mechanism of BTI has a friction force and an impact force on a bevel transition, which enhances the energy dissipation effect. Nevertheless, this impact force on bevel transitions causes high frequency and transient acceleration response but does not apply to noise sensitive situations, such as high-tech plant.Actually, friction force and slop restoring force of BTI happened between bi-tilt symmetrical beveled substrate and slider. When external force to building is less than combination of these two forces, there is no sliding phenomenon of BTI. Thus, isolation effect of BTI does not occur in small earthquakes or intermediate strong winds. Particularly, to overcome strong winds, BTI cannot begin to slide under the maximum consideration of wind force. Otherwise, uploaded load to superstructure, induced by seismic force, is limited by characteristic of BTI. Therefore, upload seismic force to superstructure can be easily controlled to maintain the safety of building. On the other hand, displacement of isolation layer has been restricted by the sliding range at the interface of bi-tilt beveled substrate and slider, which makes it impossible to produce a big displacement. However, friction force and impulse force of BTI happened in sliding process. In order to establish the numerical analysis method for structural engineers to apply this BTI to buildings, mathematical model is derived in this study based on the moving process of BTI. Then, analysis model of BTI element is developed for nonlinear shear building analysis program, GENDYN, based on the second-order dynamic equation, reduced to the first-order differential equation at state space. The fourth-order Runge-Kutta method [39–41] is applied to solve the nonlinear dynamic equation. Finally, test results and simulation results of displacement and acceleration responses of BTI under conditions of full lubrication, mild lubrication, and without lubrication are compared to test and verify the analysis accuracy of this proposed mathematical model, analysis method, and nonlinear analysis program, GENDYN, in this study.


## 2. Mathematical Model for BTI Elements

Bi-Tilt Isolator is composed of two parts: bi-tilt beveled substrate and slider components, where double beveled substrate is connected with structural foundation, while the sliders are connected with the upper structure. In order to adjust the size of friction between the double beveled substrate and sliders, a specific material, such as Teflon, can be embedded in sliders to reduce friction. Sliders are subjected to the weight of superstructure. Therefore, there are considerable normal force and friction between double beveled substrate and sliders. When superstructure is subjected to horizontal forces, such as seismic force and wind force, impact force, or base distribution caused by earthquake, the horizontal force between double beveled substrate and sliders would be increased. When this force is less than the sum of friction and inclined forces between double beveled substrate and sliders, it does not generate the relative displacement. But, when this force is greater than the sum of friction and inclined forces, the relative displacement would occur. Therefore, uploading seismic force to superstructure, caused by seismic disturbance, can be restricted to improve the seismic proof capability of structure when the structure is provided with Bi-Tilt Isolator.Basic assumptions of this Bi-Tilt Isolator: the foundation of structure can be assumed as a fixed end without displacement, relative to the ground surface displacement coordinate system between double beveled substrate and structural foundation. Mathematical model is derived based on the above-mentioned assumption or the mass of degrees of freedom relative to the slider mass of degrees of freedom is infinite; Figure2 shows configuration of Bi-Tilt Isolator.Figure 2
Configuration of a Bi-Tilt Isolator.Figure3 shows the relative motion between double beveled substrate and sliders; the action force between these two parts can be divided into three states.Figure 3
Action force between double beveled substrate and sliders in the process of movement.
(a)
Sliding upwards stage (b)
Sliding downwards stage


### 2.1. Sliding Upwards Stage

Referring to free body diagram of Figure3(a) on the left of the slider, deducting force component on normal direction of slope, tilt force of sliders, and directing to neutral position of slope can be expressed as follows:(1)R=Wsin⁡θ+μWcos⁡θ,where R is the tilt force, directing to neutral position of slope on slider; W is summation of self-weight of structure and slider, acted on the slider; θ is slope angle; μ is the coefficient of friction.Resultant force of the tilt acceleration of slider on the tendency of oblique direction of (1) can be expressed as follows:(2)as=Wsin⁡θ+μWcos⁡θmb,where as is the tilt acceleration of slider and mb is slider mass of degrees of freedom (with isolation layer mass).In analysis mode of shear building, the horizontal degree of freedom is only considered for each floor. Thus, the horizontal component of tendency of oblique acceleration is taken from (2) as follows:(3)ah=Wsin⁡θ+μWcos⁡θmbcos⁡θ.Therefore, action force of bi-tilt substrate effect on the slider can be expressed as (4). The product of acceleration and the mass of slider can be obtained as follows:(4)Fe=Wsin⁡2θ2+μWcos2⁡θ.The slider is shown in the left hand side of neutral position of bi-tilt substrate in Figure 3(a), but the direction of action force of bi-tilt substrate is opposite to that in (4).


### 2.2. Sliding Downwards Stage

Referring to free body diagrams on the right side of the slider in Figure3(b), deducting force component on normal direction of slope, tilt force of slider, and point to the slant neutral position can be expressed as follows:(5)R=Wsin⁡θ-μWcos⁡θ.According to the same way of the above-stage procedure, action force of bi-tilt bevel effect on the slider can be expressed as follows:(6)Fe=Wsin⁡2θ2-μWcos2⁡θ.If the slider is located on the left hand side of neutral position of bi-tilt bevel in Figure 3 and slide to the top left hand, the direction of action force of bi-tilt substrate is opposite to that in (6).


### 2.3. Downwards/Upwards Transition Stage

Slider moves from the left hand side to the right hand side of neutral position, and movement direction of slider changes discontinuously. Considering rigid body collision, slider bears the pulse effect in the collision, and the action force approaches infinity result in a discontinuity on the analysis. Figure4 shows instant speed vector of slider before and after the collision through the neutral position. Before the collision, the slider passes through the neutral position of BTI, shown in Figure 4(a). Slider moves down to the right; the speed vector (v→0) can be divided into parallel (v0t) and perpendicular (v0n) on the right side bevel component. When the slider contacts the right side bevel, the perpendicular component changes but the parallel component does not change. From experimental observation, separation (or jump) phenomenon between the slider and bi-tilt bevel does not happen when slider transits from the neutral position to the other side. Therefore, the assumption of this study is that the perpendicular component of momentum in collisions is “inelastic collision.” Otherwise, assuming that the bi-tilt bevel substrates are fixed points, thus, the speed components of sliders (v0n) before the collision, perpendicular to the right side of slope, disappeared to be zero after the collision. Speed vectors of sliders (v→1) are the only the parallel component, as shown in Figure 4(b). Therefore, the speed vector of slider before the collision is shown as follows:(7)v→0=vott→-v0nn→,where t→ and n→ represent the unit tangent vector and unit normal vector, respectively.Figure 4
Instant speed vector of slider before and after the collision through the neutral position.
(a)
Prior to collision (b)
Posterior to collisionThen, speed vector after the collision is expressed as follows:(8)v→1=vott→.Thus, the horizontal component ratio of movement speed of slider before and after the collision can be expressed as follows:(9)rv=v1cos⁡θv0cos⁡θ=v0tv0=cos⁡2θ,where θ is the angle between the slope and the horizontal plane; 2θ is the angle between the right and left bevel.Assuming the horizontal momentum loss rate isη of slider in transition stage when shear building is applied to analyze. The horizontal momentum loss rate is expressed as follows:(10)η=1-cos⁡2θ.Thus, sliders bear impulse force I in transition stage, shown as follows:(11)I=-ηmbv0h,where v0h is the horizontal speed component of slider before the collision.Therefore, assuming sliders bear the action forces of bi-tilt substrate in transition stage is shown as follows:(12)Fi=ηmbv0hΔt,where Δt is collision time. It is a very short time, assuming Δt=0.001sec.Assuming that sliders move distanced between the right and left hand side of the neutral position, typical action force (which to the left is positive) of bi-tilt substrate will have an effect on slider. Thus, force-relative displacement relation of the BTI element can be derived, as shown in Figure 5.Figure 5
Force-relative displacement relation of the BTI element.Kinetic friction force in (4) and (6) is defaulted as constant. But, some studies [42] reveal that friction force is the function of contacted normal force and sliding velocity. Experimental results also display that kinetic friction force of BTI slider, use of Teflon slider, is highly related to sliding speed. Therefore, to appropriately describe friction with sliding velocity change, all analysis equations must be based on cubic polynomial. Therefore, the friction responses in (4) and (6) of part of friction force should be simulated based on the actual calibration function. Constant friction force and cubic nonlinear friction function are used to simulate the friction responses of numerical simulation in this study and to compare the correlation time history of displacement responses between experiment results and simulation results to verify the reasonableness and accuracy of this proposed mathematical model. At this point, constant friction refers to the minimum root mean square error of constant friction to obtain the best simulation results.


## 2.1. Sliding Upwards Stage

Referring to free body diagram of Figure3(a) on the left of the slider, deducting force component on normal direction of slope, tilt force of sliders, and directing to neutral position of slope can be expressed as follows:(1)R=Wsin⁡θ+μWcos⁡θ,where R is the tilt force, directing to neutral position of slope on slider; W is summation of self-weight of structure and slider, acted on the slider; θ is slope angle; μ is the coefficient of friction.Resultant force of the tilt acceleration of slider on the tendency of oblique direction of (1) can be expressed as follows:(2)as=Wsin⁡θ+μWcos⁡θmb,where as is the tilt acceleration of slider and mb is slider mass of degrees of freedom (with isolation layer mass).In analysis mode of shear building, the horizontal degree of freedom is only considered for each floor. Thus, the horizontal component of tendency of oblique acceleration is taken from (2) as follows:(3)ah=Wsin⁡θ+μWcos⁡θmbcos⁡θ.Therefore, action force of bi-tilt substrate effect on the slider can be expressed as (4). The product of acceleration and the mass of slider can be obtained as follows:(4)Fe=Wsin⁡2θ2+μWcos2⁡θ.The slider is shown in the left hand side of neutral position of bi-tilt substrate in Figure 3(a), but the direction of action force of bi-tilt substrate is opposite to that in (4).


## 2.2. Sliding Downwards Stage

Referring to free body diagrams on the right side of the slider in Figure3(b), deducting force component on normal direction of slope, tilt force of slider, and point to the slant neutral position can be expressed as follows:(5)R=Wsin⁡θ-μWcos⁡θ.According to the same way of the above-stage procedure, action force of bi-tilt bevel effect on the slider can be expressed as follows:(6)Fe=Wsin⁡2θ2-μWcos2⁡θ.If the slider is located on the left hand side of neutral position of bi-tilt bevel in Figure 3 and slide to the top left hand, the direction of action force of bi-tilt substrate is opposite to that in (6).


## 2.3. Downwards/Upwards Transition Stage

Slider moves from the left hand side to the right hand side of neutral position, and movement direction of slider changes discontinuously. Considering rigid body collision, slider bears the pulse effect in the collision, and the action force approaches infinity result in a discontinuity on the analysis. Figure4 shows instant speed vector of slider before and after the collision through the neutral position. Before the collision, the slider passes through the neutral position of BTI, shown in Figure 4(a). Slider moves down to the right; the speed vector (v→0) can be divided into parallel (v0t) and perpendicular (v0n) on the right side bevel component. When the slider contacts the right side bevel, the perpendicular component changes but the parallel component does not change. From experimental observation, separation (or jump) phenomenon between the slider and bi-tilt bevel does not happen when slider transits from the neutral position to the other side. Therefore, the assumption of this study is that the perpendicular component of momentum in collisions is “inelastic collision.” Otherwise, assuming that the bi-tilt bevel substrates are fixed points, thus, the speed components of sliders (v0n) before the collision, perpendicular to the right side of slope, disappeared to be zero after the collision. Speed vectors of sliders (v→1) are the only the parallel component, as shown in Figure 4(b). Therefore, the speed vector of slider before the collision is shown as follows:(7)v→0=vott→-v0nn→,where t→ and n→ represent the unit tangent vector and unit normal vector, respectively.Figure 4
Instant speed vector of slider before and after the collision through the neutral position.
(a)
Prior to collision (b)
Posterior to collisionThen, speed vector after the collision is expressed as follows:(8)v→1=vott→.Thus, the horizontal component ratio of movement speed of slider before and after the collision can be expressed as follows:(9)rv=v1cos⁡θv0cos⁡θ=v0tv0=cos⁡2θ,where θ is the angle between the slope and the horizontal plane; 2θ is the angle between the right and left bevel.Assuming the horizontal momentum loss rate isη of slider in transition stage when shear building is applied to analyze. The horizontal momentum loss rate is expressed as follows:(10)η=1-cos⁡2θ.Thus, sliders bear impulse force I in transition stage, shown as follows:(11)I=-ηmbv0h,where v0h is the horizontal speed component of slider before the collision.Therefore, assuming sliders bear the action forces of bi-tilt substrate in transition stage is shown as follows:(12)Fi=ηmbv0hΔt,where Δt is collision time. It is a very short time, assuming Δt=0.001sec.Assuming that sliders move distanced between the right and left hand side of the neutral position, typical action force (which to the left is positive) of bi-tilt substrate will have an effect on slider. Thus, force-relative displacement relation of the BTI element can be derived, as shown in Figure 5.Figure 5
Force-relative displacement relation of the BTI element.Kinetic friction force in (4) and (6) is defaulted as constant. But, some studies [42] reveal that friction force is the function of contacted normal force and sliding velocity. Experimental results also display that kinetic friction force of BTI slider, use of Teflon slider, is highly related to sliding speed. Therefore, to appropriately describe friction with sliding velocity change, all analysis equations must be based on cubic polynomial. Therefore, the friction responses in (4) and (6) of part of friction force should be simulated based on the actual calibration function. Constant friction force and cubic nonlinear friction function are used to simulate the friction responses of numerical simulation in this study and to compare the correlation time history of displacement responses between experiment results and simulation results to verify the reasonableness and accuracy of this proposed mathematical model. At this point, constant friction refers to the minimum root mean square error of constant friction to obtain the best simulation results.


## 3. Analysis Program of the Proposed Model, GENDYN Analysis Procedure

The developed nonlinear shear building analysis program, GENDYN, is applied to process numerical simulation to verify the accurate degree of the proposed model for BTI. The second-order dynamic equation in GENDYN is reduced to the first-order differential equation at state space. Then, the fourth-order Runge-Kutta method is applied to solve the nonlinear dynamic equation.


### 3.1. GENDYN Analysis Procedure

The first-order ordinary differential equation of GENDYN program is discretized as follows.(13)x˙=Ax+Bu,where x is 2n dimensional state vector; A is 2n×2n system matrix of state space; B is 2n×n force distribution matrix of state space; n is degree of freedom.State vector is composed of displacement vector and velocity of degree of freedom as follows:(14)x=yy˙,where y is n dimensional displacement vector; y˙ is n dimensional velocity vector.System matrixA represents the role of linear components, such as mass, linear spring, and damping as follows:(15)A=0I-M-1K-MC,where M is n×n the mass matrix of the second-order ordinary differential equation of motion; K is n×n the stiffness matrix of the second-order ordinary differential equation of motion; C is n×n the damping coefficients matrix of the second-order ordinary differential equation of motion; I is n×n unit vector.B matrix is used for the conversion of the external force and each acceleration of degree of freedom; for a lumped mass system, B matrix can be defined as(16)B=0M-1.External force vector u is the summation of external force and nonlinear component force. Equation (13) is solved by numerical analysis method. Two items at the right hand side of equal sign can be combined into one to get (17) as follows:(17)x˙=I00M-1y˙f,where f is the summation of linear component force, nonlinear component force, and external force.That is,(18)f=u-Kx-Cx˙.Thus, (18) can be explained as the summation of internal force and external force, acting on each mass of degree of freedom. It is analyzed by stepwise integration method, and the basic procedure at each time step is as follows:Time step_i:
Given:xi-1,fi-1(xi-1),fi(xi-1)Step integral by appropriate methodUpdate parameters of nonlinear componentsThe fourth-order Runge-Kutta is analyzed by stepwise integration method to process four times calculation of internal force of component at each step. Nonlinear component parameters for calculation of internal forces at these steps can be assumed as unchanged. Only in the end of each calculation step before it moves on to the next step can the nonlinear component parameters be renewed based on the latest status.


### 3.2. The Fourth-Order Runge-Kutta Method

In order to find solution of ordinary differential equations by stepwise time step, the fourth-order Runge-Kutta method is applied in this study.The Runge-Kutta methods are a family of implicit and explicit iterative methods, which include the well-known routine called the Euler Method, used in temporal discretization for the approximate solutions of ordinary differential equations [39]. Key concept of fourth-order Runge-Kutta algorithm is described as follows [40, 41].The first-order ordinary differential equation is defined by(19)dytdt=fyt,t.To progress from a point at t=t0,y∗(t0), by one time step, h, the steps are as follows:(1)
Approximate derivative att=t0, k1=f(y∗(t0),t0).(2)
Intermediate estimate of function att=t0+h/2, using k1,(20)y1t0+h2=y∗t0+k1h2.(3)
Estimate of slope att=t0+h/2(21)k2=fy1t0+h2,t0+h2.(4)
Another estimate of slope att=t0+h/2, using k2,(22)y2t0+h2=y∗t0+k2h2.(5)
Another estimate of slope att=t0+h/2(23)k3=fy2t0+h2,t0+h2.(6)
An estimate of function att=t0+h/2, using k3,(24)y3t0+h=y∗t0+k3h.(7)
Estimate of slope att=t0+h(25)k4=fy3t0+h,t0+h.(8)
Estimate ofy(t0+h)(26)y∗t0+h=y∗t0+k1+2k2+2k3+k46h.


### 3.3. State Updating Procedure

Bevel force, impact force, and friction force are three nonlinear force components of BTI. These forces should be renewed by analyzing the state vector ofxi. Analysis method is described as follows:(1)
The summation of bevel force and friction forceFe: value of bevel force is related to displacement and velocity based on (4) and (6) as follows:(27)Fe=-Wsin⁡2θ2sign⁡yrel,i-μWcos2⁡θsign⁡y˙rel,i,whereyrel,i is the relative displacement of the double elements; y˙rel,i is the relative velocity of the double elements.(2)
Impact forceFi: collision time between bevel tilt substrate and slider is very short, happening at the moment when slider moves from the left hand side of neutral point of BTI to the right hand side and occurred in the opposite process. Therefore, assume collision time is Δt. Effect of impact force is used to simulate impulse of slider. In order to avoid lack of impulse, Δt must be set as an integer multiple of setting the time stride for analyzing settings of GENDYN. For example,(28)Nti=Δth,whereh is setting of the time stride to analyze settings of GENDYN.Therefore, we detect the following conditions:(29)yrel,i·yrel,i-1<0.Then, impact force can be assumed as follows:(30)Fi=-ηmby˙rel,iΔtsign⁡y˙rel.Time interval of impact force must remain total length of time Δt. Therefore, time of impact points must be stored for checking the analysis time point to calculate the force point and start time of impact force. When the time difference between these two points is less than Δt, impact force is constant as in (30). When the time difference between these two points is greater than Δt, impact force then becomes zero.


## 3.1. GENDYN Analysis Procedure

The first-order ordinary differential equation of GENDYN program is discretized as follows.(13)x˙=Ax+Bu,where x is 2n dimensional state vector; A is 2n×2n system matrix of state space; B is 2n×n force distribution matrix of state space; n is degree of freedom.State vector is composed of displacement vector and velocity of degree of freedom as follows:(14)x=yy˙,where y is n dimensional displacement vector; y˙ is n dimensional velocity vector.System matrixA represents the role of linear components, such as mass, linear spring, and damping as follows:(15)A=0I-M-1K-MC,where M is n×n the mass matrix of the second-order ordinary differential equation of motion; K is n×n the stiffness matrix of the second-order ordinary differential equation of motion; C is n×n the damping coefficients matrix of the second-order ordinary differential equation of motion; I is n×n unit vector.B matrix is used for the conversion of the external force and each acceleration of degree of freedom; for a lumped mass system, B matrix can be defined as(16)B=0M-1.External force vector u is the summation of external force and nonlinear component force. Equation (13) is solved by numerical analysis method. Two items at the right hand side of equal sign can be combined into one to get (17) as follows:(17)x˙=I00M-1y˙f,where f is the summation of linear component force, nonlinear component force, and external force.That is,(18)f=u-Kx-Cx˙.Thus, (18) can be explained as the summation of internal force and external force, acting on each mass of degree of freedom. It is analyzed by stepwise integration method, and the basic procedure at each time step is as follows:Time step_i:
Given:xi-1,fi-1(xi-1),fi(xi-1)Step integral by appropriate methodUpdate parameters of nonlinear componentsThe fourth-order Runge-Kutta is analyzed by stepwise integration method to process four times calculation of internal force of component at each step. Nonlinear component parameters for calculation of internal forces at these steps can be assumed as unchanged. Only in the end of each calculation step before it moves on to the next step can the nonlinear component parameters be renewed based on the latest status.


## 3.2. The Fourth-Order Runge-Kutta Method

In order to find solution of ordinary differential equations by stepwise time step, the fourth-order Runge-Kutta method is applied in this study.The Runge-Kutta methods are a family of implicit and explicit iterative methods, which include the well-known routine called the Euler Method, used in temporal discretization for the approximate solutions of ordinary differential equations [39]. Key concept of fourth-order Runge-Kutta algorithm is described as follows [40, 41].The first-order ordinary differential equation is defined by(19)dytdt=fyt,t.To progress from a point at t=t0,y∗(t0), by one time step, h, the steps are as follows:(1)
Approximate derivative att=t0, k1=f(y∗(t0),t0).(2)
Intermediate estimate of function att=t0+h/2, using k1,(20)y1t0+h2=y∗t0+k1h2.(3)
Estimate of slope att=t0+h/2(21)k2=fy1t0+h2,t0+h2.(4)
Another estimate of slope att=t0+h/2, using k2,(22)y2t0+h2=y∗t0+k2h2.(5)
Another estimate of slope att=t0+h/2(23)k3=fy2t0+h2,t0+h2.(6)
An estimate of function att=t0+h/2, using k3,(24)y3t0+h=y∗t0+k3h.(7)
Estimate of slope att=t0+h(25)k4=fy3t0+h,t0+h.(8)
Estimate ofy(t0+h)(26)y∗t0+h=y∗t0+k1+2k2+2k3+k46h.


## 3.3. State Updating Procedure

Bevel force, impact force, and friction force are three nonlinear force components of BTI. These forces should be renewed by analyzing the state vector ofxi. Analysis method is described as follows:(1)
The summation of bevel force and friction forceFe: value of bevel force is related to displacement and velocity based on (4) and (6) as follows:(27)Fe=-Wsin⁡2θ2sign⁡yrel,i-μWcos2⁡θsign⁡y˙rel,i,whereyrel,i is the relative displacement of the double elements; y˙rel,i is the relative velocity of the double elements.(2)
Impact forceFi: collision time between bevel tilt substrate and slider is very short, happening at the moment when slider moves from the left hand side of neutral point of BTI to the right hand side and occurred in the opposite process. Therefore, assume collision time is Δt. Effect of impact force is used to simulate impulse of slider. In order to avoid lack of impulse, Δt must be set as an integer multiple of setting the time stride for analyzing settings of GENDYN. For example,(28)Nti=Δth,whereh is setting of the time stride to analyze settings of GENDYN.Therefore, we detect the following conditions:(29)yrel,i·yrel,i-1<0.Then, impact force can be assumed as follows:(30)Fi=-ηmby˙rel,iΔtsign⁡y˙rel.Time interval of impact force must remain total length of time Δt. Therefore, time of impact points must be stored for checking the analysis time point to calculate the force point and start time of impact force. When the time difference between these two points is less than Δt, impact force is constant as in (30). When the time difference between these two points is greater than Δt, impact force then becomes zero.


## 4. Experimental Design and Data Acquisition

Bi-tilt beveled substrate and sliders form Bi-Tilt Isolator, BTI. Sliding force will be caused by the shift of slider. Actually, bi-tilt beveled substrate has a symmetrical slope surface; therefore, the slope resorting force is constant. When the superstructure on BTI is subjected to a certain level and above seismic force to overcome the slope resorting force, the sliding force will occur between bi-tilt beveled substrate and sliders. This force is equal to the product of the horizontal component of the sliding down weight of isolator and kinetic friction force. Therefore, the maximum upload force to structure, caused by external force, can be easily predicted and controlled. Otherwise, the displacement at the isolator layer should be reduced to avoid the defects of base isolation. That is, the external force at slider, installed on the base of superstructure, is greater than the sum of friction force and inclined force. Slip phenomenon will be stimulated on BTI to cause displacement and upload external force to structure. There are three stages of the relative motions between bi-tilt beveled substrate and slider: sliding upwards stage, sliding downwards stage, and downwards/upwards transition stage. Action forces between the bi-tilt beveled substrate and slider are self-weight of superstructure and slider, friction force, and impact force at the collision. In order to capture the real reactions of BTI in the process of movement, experimental design is planned as follows:(1) experimental model: superstructure, self-weight is 20.78 kg, made of aluminum extrusion; isolation layer, made of aluminum extrusion; sliding interface, steel (polished) with Gasket, made of Teflon (Polytetrafluoroethylene); (2) displacement detecting: the developed noncontact measurement technology, Digital Image Correlation (DIC) method is proposed to measure the displacement variation; and (3) acceleration detecting: a smart recorder. Experimental set-up is shown in Figure 6. High speed digital camera is applied to measure all dynamic displacement responses with marks on this test set-up. A smart recorder consists of (1) Arduino Nano, (2) MPU 9250, and (3) SD module. MPU 9250 breakout board is equipped with nine degrees of freedom in inertial measurement unit, IMU and 16-bit ADC (Analog-to-Digital Converter) to detect the acceleration responses in the process of the dynamic test.Figure 6
Set-up for experimental design and data acquisition.Experimental parameters are three tests for WD40 sprayed, WD40 whipped, and without lubrication with initial displacement of −50 mm, 100 mm, and 150 mm, respectively. Each parameter combination processes two tests to record displacement and acceleration responses by dynamic DIC and MPU9250 accelerometers.


## 5. Test and Analysis Results and Discussions

All experimental records are analyzed by dynamic analysis to acquire the corresponding optimal parameters based on the optimization procedure. Normal vector equation method is adapted to process parameter optimization. Four friction parameters are used as independent variable, assuming friction is cubic polynomial functions of speed. The minimum root mean square error of time history of displacement for three seconds (901 records) is targeted to process iterative analysis. Parameter analysis of the optimization friction function with the maximum initial displacement (150 mm) for three different lubrication conditions is used to analyze the other displacement responses with various initial conditions and compare with the experimental data. If the optimization parameters by the initial displacement 150 mm are applied in the other initial displacements, they can also obtain high-precision displacement reactions. It demonstrates practical application value of this proposed model and program.Then, assuming that friction is a constant to obtain the minimum root mean square error for simulation and experiment of displacement responses within three seconds, the optimization friction forces for the state of the maximum initial displacement are iterated with three various lubrication conditions. Next, this friction force substitutes into state of the other initial displacements within three seconds to calculate the time history of displacement reactions and compare with the time history of displacement responses of experimental and simulation results, acquired by the cubic polynomial function simulation of friction.


### 5.1. Test and Analysis Results

Table1 lists all analysis results of the root mean square error of different initial displacement and lubrication conditions with optimal parameters of friction force. Figures 7~9 are analysis results of function of individual best fit and friction force and function of overall best fit and friction and friction force for three different conditions of lubrication with various initial situations. Figure 10 is the comparison of function of overall best fit and friction force with various lubricated conditions. Figures 11~13 reveal simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function under conditions of WD40 sprayed with initial displacement 58.68 mm, WD40 whipped with initial displacement 47.98 mm, and without lubrication with initial displacement of 54.05 mm, respectively. Figures 14~16 show simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function under conditions of WD40 sprayed with initial displacement of 106.76 mm, WD40 whipped with initial displacement 96.87 mm, and without lubrication with initial displacement of 106.98 mm, respectively. Figures 17~19 display simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function under conditions of WD40 sprayed with initial displacement of 152.56 mm, WD40 whipped with initial displacement of 151.83 mm, and without lubrication with initial displacement of 154.76 mm, respectively. Figure 20 reveals the comparison of RMS displacement errors for simulation results with cubic polynomial friction function and constant friction force.Table 1
Root mean square error of different initial displacement and lubrication conditions with optimal parameters of friction force.
Initial displacement WD40 sprayed, m WD40 whipped, m W/O lubrication, m 0.05 m∗ (0.048~0.059 m) 0.00028 0.00021 0.00029 0.00041 0.00033 0.00079 0.00156 0.00225 0.00453 0.47% 0.44% 0.54%  0.10 mm∗ (0.097~0.107 m) 0.00055 0.00046 0.00034 0.00056 0.00082 0.00094 0.00182 0.00438 0.00649 0.51% 0.47% 0.32%  0.15 mm∗ (0.152~0.155 m) 0.00206 0.00057 0.00061 0.00206 0.00057 0.00061 0.00229 0.00829 0.00874 1.35% 0.37% 0.40% ∗There is slight difference for actual initial displacement in the process of test; the optimization solution with cubic polynomial of friction; the error of the optimum parameters for using the maximum initial displacement simulation; simulation errors with constant friction; the ratio of error of the optimum solutions and initial displacement.Figure 7
Function of individual best fit and friction force and function of overall best fit and friction and friction force (WD40 sprayed) with various initial displacements.Figure 8
Function of individual best fit and friction force and function of overall best fit and friction and friction force (WD40 whipped) with various initial displacements.Figure 9
Function of individual best fit and friction force and function of overall best fit and friction and friction force (without lubrication) with various initial displacements.Figure 10
The comparison of function of overall best fit and friction force with various lubricated conditions.Figure 11
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 sprayed with initial displacement of 58.68 mm).Figure 12
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 whipped with initial displacement of 47.98 mm).Figure 13
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (without lubrication with initial displacement of 54.05 mm).Figure 14
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 sprayed with initial displacement of 106.76 mm).Figure 15
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 whipped with initial displacement of 96.87 mm).Figure 16
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (without lubrication with initial displacement of 106.98 mm).Figure 17
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 sprayed with initial displacement of 152.56 mm).Figure 18
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 whipped with initial displacement of 151.83 mm).Figure 19
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (without lubrication with initial displacement of 154.76 mm).Figure 20
The comparison of RMS displacement errors for simulation results with cubic polynomial friction function and constant friction force.


### 5.2. Discussions

The purpose of this study is to explore the experimental curves of friction within the speed range on optimization of parameters calculation. Even through friction parameters are significantly different, friction force does not have obvious differences. Analysis results for the function of friction force and velocity display that there is little difference between the function of individual best fit and friction and function of overall best fit and friction under condition of full lubrication and slightly large difference happening at high velocity under condition of mild lubrication with large initial displacement. Results for condition of without lubrication show divergence between the analysis results of the function of individual best fit and friction and function of overall best fit and friction with large initial displacement and those of the other two. Then, the comparison results of different lubrication conditions reveal that the function curve of friction force and velocity is very gentle for condition of full lubrication, slightly steep for condition of mild lubrication, and steep for condition of without lubrication. These results show that movement behavior of BTI for function of friction force and velocity can be simulated by this proposed model.Root mean square error between the experimental data and individual optimization parameters simulation results of displacement reactions are below 1.35% (WD40 sprayed), 0.47% (WD40 whipped), and 0.54% (without lubrication) of initial displacement, respectively. However, the results of optimization parameters for the two tests still have significant differences in exactly the same combination of experimental parameters. Therefore, it can identify that speed responses of the isolated layers are not high. Although four friction parameters are too many, results of this study confirm that BTI element of GENDYN program does have the capability to simulate the nonlinear responses of structure with combined effect of frictional sliding and collision.The maximum root mean square error between simulation results and experimental data are 0.00229 m (WD40 sprayed), 0.00829 m (WD40 whipped), and 0.00874 m (without lubrication), using constant friction for Bi-Tilt Isolator with three various conditions. The maximum root mean square errors are 0.00206 m (WD40 sprayed), 0.00082 m (WD40 whipped), and 0.00094 m (without lubrication), respectively, using cubic polynomial function of friction for Bi-Tilt Isolator with three various conditions.The optimization parameters, set out by the time history of displacement responses with the maximum initial displacement for each condition, are substituted for the rest of the initial displacement conditions to simulate displacement reactions. The comparison of simulation displacement responses and experimental data indicates a high correlation degree. BTI element of GENDYN program provides very good numerical simulation capability for Bi-Tilt Isolator. It can be used to assess structural seismic responses and effect of vibration isolation of structure, installed with BTI.Simulation results of time history of displacement, by BTI element of GENDYN program, using cubic polynomial function of friction for Bi-Tilt Isolator with three various conditions, are very close to the displacement responses of test. But, when friction parameter sets as constant, analysis precision is relative to the variation of friction and velocity, affected by kinetic friction force.


## 5.1. Test and Analysis Results

Table1 lists all analysis results of the root mean square error of different initial displacement and lubrication conditions with optimal parameters of friction force. Figures 7~9 are analysis results of function of individual best fit and friction force and function of overall best fit and friction and friction force for three different conditions of lubrication with various initial situations. Figure 10 is the comparison of function of overall best fit and friction force with various lubricated conditions. Figures 11~13 reveal simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function under conditions of WD40 sprayed with initial displacement 58.68 mm, WD40 whipped with initial displacement 47.98 mm, and without lubrication with initial displacement of 54.05 mm, respectively. Figures 14~16 show simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function under conditions of WD40 sprayed with initial displacement of 106.76 mm, WD40 whipped with initial displacement 96.87 mm, and without lubrication with initial displacement of 106.98 mm, respectively. Figures 17~19 display simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function under conditions of WD40 sprayed with initial displacement of 152.56 mm, WD40 whipped with initial displacement of 151.83 mm, and without lubrication with initial displacement of 154.76 mm, respectively. Figure 20 reveals the comparison of RMS displacement errors for simulation results with cubic polynomial friction function and constant friction force.Table 1
Root mean square error of different initial displacement and lubrication conditions with optimal parameters of friction force.
Initial displacement WD40 sprayed, m WD40 whipped, m W/O lubrication, m 0.05 m∗ (0.048~0.059 m) 0.00028 0.00021 0.00029 0.00041 0.00033 0.00079 0.00156 0.00225 0.00453 0.47% 0.44% 0.54%  0.10 mm∗ (0.097~0.107 m) 0.00055 0.00046 0.00034 0.00056 0.00082 0.00094 0.00182 0.00438 0.00649 0.51% 0.47% 0.32%  0.15 mm∗ (0.152~0.155 m) 0.00206 0.00057 0.00061 0.00206 0.00057 0.00061 0.00229 0.00829 0.00874 1.35% 0.37% 0.40% ∗There is slight difference for actual initial displacement in the process of test; the optimization solution with cubic polynomial of friction; the error of the optimum parameters for using the maximum initial displacement simulation; simulation errors with constant friction; the ratio of error of the optimum solutions and initial displacement.Figure 7
Function of individual best fit and friction force and function of overall best fit and friction and friction force (WD40 sprayed) with various initial displacements.Figure 8
Function of individual best fit and friction force and function of overall best fit and friction and friction force (WD40 whipped) with various initial displacements.Figure 9
Function of individual best fit and friction force and function of overall best fit and friction and friction force (without lubrication) with various initial displacements.Figure 10
The comparison of function of overall best fit and friction force with various lubricated conditions.Figure 11
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 sprayed with initial displacement of 58.68 mm).Figure 12
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 whipped with initial displacement of 47.98 mm).Figure 13
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (without lubrication with initial displacement of 54.05 mm).Figure 14
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 sprayed with initial displacement of 106.76 mm).Figure 15
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 whipped with initial displacement of 96.87 mm).Figure 16
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (without lubrication with initial displacement of 106.98 mm).Figure 17
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 sprayed with initial displacement of 152.56 mm).Figure 18
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (WD40 whipped with initial displacement of 151.83 mm).Figure 19
Simulation time history of displacement for the friction function of measuring displacement and individual best fit and overall best fitting function (without lubrication with initial displacement of 154.76 mm).Figure 20
The comparison of RMS displacement errors for simulation results with cubic polynomial friction function and constant friction force.


## 5.2. Discussions

The purpose of this study is to explore the experimental curves of friction within the speed range on optimization of parameters calculation. Even through friction parameters are significantly different, friction force does not have obvious differences. Analysis results for the function of friction force and velocity display that there is little difference between the function of individual best fit and friction and function of overall best fit and friction under condition of full lubrication and slightly large difference happening at high velocity under condition of mild lubrication with large initial displacement. Results for condition of without lubrication show divergence between the analysis results of the function of individual best fit and friction and function of overall best fit and friction with large initial displacement and those of the other two. Then, the comparison results of different lubrication conditions reveal that the function curve of friction force and velocity is very gentle for condition of full lubrication, slightly steep for condition of mild lubrication, and steep for condition of without lubrication. These results show that movement behavior of BTI for function of friction force and velocity can be simulated by this proposed model.Root mean square error between the experimental data and individual optimization parameters simulation results of displacement reactions are below 1.35% (WD40 sprayed), 0.47% (WD40 whipped), and 0.54% (without lubrication) of initial displacement, respectively. However, the results of optimization parameters for the two tests still have significant differences in exactly the same combination of experimental parameters. Therefore, it can identify that speed responses of the isolated layers are not high. Although four friction parameters are too many, results of this study confirm that BTI element of GENDYN program does have the capability to simulate the nonlinear responses of structure with combined effect of frictional sliding and collision.The maximum root mean square error between simulation results and experimental data are 0.00229 m (WD40 sprayed), 0.00829 m (WD40 whipped), and 0.00874 m (without lubrication), using constant friction for Bi-Tilt Isolator with three various conditions. The maximum root mean square errors are 0.00206 m (WD40 sprayed), 0.00082 m (WD40 whipped), and 0.00094 m (without lubrication), respectively, using cubic polynomial function of friction for Bi-Tilt Isolator with three various conditions.The optimization parameters, set out by the time history of displacement responses with the maximum initial displacement for each condition, are substituted for the rest of the initial displacement conditions to simulate displacement reactions. The comparison of simulation displacement responses and experimental data indicates a high correlation degree. BTI element of GENDYN program provides very good numerical simulation capability for Bi-Tilt Isolator. It can be used to assess structural seismic responses and effect of vibration isolation of structure, installed with BTI.Simulation results of time history of displacement, by BTI element of GENDYN program, using cubic polynomial function of friction for Bi-Tilt Isolator with three various conditions, are very close to the displacement responses of test. But, when friction parameter sets as constant, analysis precision is relative to the variation of friction and velocity, affected by kinetic friction force.


## 6. Conclusions

Bi-Tilt Isolator (BTI) is a new base isolator for building. Upload force, induced by external force, for superstructure with BTI, can be controlled easily to maintain the safety of structures. Otherwise, displacement of superstructure will be reduced according to the limit displacement of isolation layer. The friction force and impact force between double beveled substrate and slider will be caused in the process of relative movement and nonlinear behavior. In this study, a mathematical model is derived based on the three real movement stages of BTI, sliding upwards stage, sliding downwards stage, and downwards/upwards transition stage. Due to the complicated numerical calculation, nonlinear analysis program, GENDYN, applying the fourth-order Runge-Kutta method, is developed to solve the nonlinear dynamic responses of BTI. Analysis procedure of GENDYN program, discretized ordinary differential equation, is derived and applied for nonlinear components of bevel force, impact force, and friction force of sliding upwards stage, sliding downwards stage, and downwards/upwards transition stage. Then, all dynamic test results of three different lubricated conditions with the maximum initial displacements are analyzed to acquire optimal parameters and assume friction function as cubic polynomial function of speed. These optimal parameters are used to simulate the displacement responses for test model with various initial conditions and also comparison with the test results. In order to test and verify the analysis accuracy of the derived mathematical model and nonlinear analysis program, GENDYN, the relationship of friction force and velocity for conditions of mild lubrication, full lubrication, and without lubrication, with different initial displacement, is simulated by the function of individual best fit and friction force and function of overall best fit function and friction force. Then, the comparison of time history of measuring displacement, individual best fit, and overall best fit function under conditions of three various lubricated conditions is discussed. The conclusions from a series of test and analysis results are summarized as follows:(1)
The optimal parameters, acquired by the maximum initial displacement in the process of optimization, can be used to the other initial conditions. The comparison of simulation time history of displacements and test data provides high correlation degree.(2)
Analysis results show that root mean square error between the experimental data and simulation results of displacement responses are below 1.35% for WD40 sprayed, 0.47% for WD40 whipped, and 0.54% for without lubrication with three different initial displacements, respectively.(3)
The maximum root mean square error between simulation results and test results for simulating with cubic polynomial function of friction is much less than those of constant friction. Otherwise, there is little difference of the maximum root mean square error between simulation results and test results under condition of full lubrication, not affected by the kinetic friction force.(4)
Analysis accuracy of this proposed model and program is influenced by the variation of friction force and velocity, caused by the kinetic friction force when the friction force set as constant.(5)
Application of cubic polynomial function for simulating friction of BTI with three different lubricated conditions can perform very fine simulation results, compared with the test results.All test and analysis results reveal that this proposed mathematical model and BTI element of GENDYN program, using cubic polynomial function of friction, perform fine simulation capability to assess nonlinear isolation effect of structure installed with BTI. This proposed GENDYN program can be widely applied for structure engineers to design structure, installed with BTI.



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*Source: 7163516-2018-03-14.xml*