Development and Analysis of the Control Effect of a Reid Damper with Self-Centering Characteristics

To improve the recoverability of structures following an earthquake, a Reid friction damper with self-centering characteristics is proposed and its hysteretic behavior is studied by theoretical analysis and experimental research.,emain parameters of the damper are the equivalent stiffness and energy dissipation coefficient. Based on a 10-story steel frame structure, 10 energy dissipation design schemes using the proposed Reid damper are proposed. ,e additional equivalent damping ratios of the 10 schemes are equal, whereas the energy dissipation coefficients of the dampers are different. ,e vibration control effects of the energy dissipation structures are analytically investigated under four earthquake loads. ,e experimental results of the friction damper are in good agreement with the theoretical results, and the hysteretic behavior of the damper follows that of a typical Reid model. ,e seismic response and structural damage can be reduced using any of the 10 design schemes; however, the effects are different. When the energy dissipation coefficient is in the range of 0.1–0.3, the control effect on the interstory drift is better; however, the structural acceleration response and damping force of the dampers increase. When the energy dissipation coefficient is in the range of 0.6–1.0, the energy dissipation effect of the dampers is good; however, the self-centering ability is poor. ,erefore, the optimum range of the energy dissipation coefficient of a Reid damper intended for energy dissipation structures should be 0.3–0.6.


Introduction
Currently, the performance-based seismic design has been widely recognized, and scholars from various countries have conducted numerous studies on the same [1].Scholars in the United States and Japan have proposed the resilient city as a general direction for cooperation in earthquake engineering [2].Under the premise that a structure meets the performance target, a postearthquake structure can be quickly repaired and restored to its normal function.is has become one of the important research directions for the sustainable development of earthquake resistant projects.e difficulty in restoring earthquake-hit structures and the associated economic cost is directly related to the residual deformation of the structures.erefore, restricting their residual deformation has become a key point for structural recovery [3].
e SMA is used to provide resilience for the dampers, owing to its shape memory property.Another important property of SMAs is pseudoelasticity (superelasticity), which implies that the large strain produced during loading will be recovered and the energy will be dissipated simultaneously.Based on the excellent properties of SMA, many SMA dampers with selfcentering characteristics have been designed [10,[15][16][17].
For self-centering dampers, the idealized bilinear elastic model is generally used to simulate the restoring force, whereas the bilinear elastoplastic model is used to simulate the dissipated energy.Hence, the hysteretic models of selfcentering dampers are ag-shaped [18] or improved agshaped [6,[19][20][21], which have attracted signi cant attention.Another model that can describe the self-centering dampers is the Reid model [22][23][24].is model is used to describe the damping force, which is proportional to the displacement amplitude and in phase with the velocity [25].One of its important characteristics is that it can consider the varying amplitude damping force with the increase in the structural deformation and can better meet the demand of shock absorption under earthquakes of varying intensity.Currently, the damper that can realize the Reid model mainly comprises a variable friction device and a restoring force device.e restoring force device uses the same parts as those in other self-centering dampers.e main methods of achieving variable friction include piezoelectricity [26], variable contact surface friction coe cient [27], and variable friction contact surface [28].Nims et al. developed an energy dissipating restraint (EDR), whose damping force is a combination of spring elasticity and friction, and the friction is proportional to the displacement [29,30].e force-displacement hysteretic curve of the EDR is consistent with the Reid model.
Nims et al. studied the vibration control e ect of Reid dampers in a single degree of freedom system by numerical simulation [29] and carried out shaking table tests of a simple steel structure with Reid dampers [30].e results show that Reid dampers are e ective to control the vibration response of the structure, but the control e ect for the multidegree of the freedom structure and the structure in the nonlinear stage is not involved.In this paper, the hysteresis curves and energy dissipation characteristics of a Reid model are rst analyzed.A Reid friction damper is then proposed and studied by theoretical analysis and experimental research.Based on a 10-story steel frame structure, 10 energy dissipation design schemes using the Reid dampers are determined.e additional equivalent damping ratios of the 10 schemes are equal, whereas the energy dissipation coe cients of the dampers are di erent.Compared to an uncontrolled structure, the vibration control e ects on the displacement, acceleration, energy dissipation, structural damage, and residual deformation of the energy dissipation structures are analytically investigated under four earthquake loads.Finally, based on the research results, the design of the Reid damper for energy dissipation structures is proposed.

Reid Hysteresis Model
Figure 1(a) shows the force-displacement relationship of the Reid hysteretic model.e two diagonal triangles, which are symmetric about the origin, in the rst and third quadrants constitute the hysteresis loop.It is assumed that the transition between the loading and unloading states is instantaneous.For the Reid hysteresis model, the relationship between the damping force f d and the displacement u d can be expressed as follows: where k r1 and k r3 denote the loading sti ness and unloading sti ness, respectively, and _ u d denotes the velocity.As shown in Figures 1(b) and 1(c), the Reid model can be decomposed into a linear hysteretic damping model, whose force is proportional to the displacement amplitude and in phase with the velocity, and a linear spring model.
Accordingly, the relationship between the damping force and displacement can be rewritten as follows: where k re and η denote the equivalent elastic sti ness and energy dissipation coe cient, respectively, and η is less than 1.Furthermore, from Equations ( 1) and ( 2), it follows that e Reid hysteretic loop shows that the damper will automatically return to its initial position after unloading and will not generate residual deformation.With regard to the energy dissipation capacity, the energy dissipated when repeatedly loading a cycle for a displacement u d can be obtained as follows: e frictional force of the constant friction damper is independent of the displacement but is in phase with the velocity.To realize a linear hysteretic damping model, the friction magnitude should vary linearly with respect to the displacement.In current piezoelectric friction dampers, the positive force between the frictional contact surfaces can be changed by applying a voltage.However, they are semiactive devices, and the construction is more complex.
Figure 2 shows the schematic of the passive variable friction damper, which can realize the Reid model.In Figure 2, μ 0 denotes the friction coe cient between the sliding block and the friction plate, μ 1 denotes the friction coe cient between the sliding block and the extrusion block, and θ denotes the angle between the two frictional contact surfaces.During the loading, the interaction force between the extrusion block and the sliding block will increase, consequently, increasing the friction between the sliding block and the friction plate.
At the same time, the compression force acting on the spring increases.During the unloading, the interaction force between the extrusion block and the sliding block and the friction between the sliding block and the friction plate decrease.e transition between the loading and unloading stages of the damper is assumed to be instantaneous.As shown in Figure 3(a), the sliding and extrusion blocks are isolated.During the loading, the damping force is equal to the sum of the spring compression reaction force and the frictional force.During the unloading, the damping force is equal to the di erence between the spring compression reaction and the frictional force.
e damping force is obtained as follows: where F d , F s , and F f denote the damping force, spring elastic force, and frictional force, respectively.For the loading stage, as shown in Figure 3(b), the force balance relationship can be obtained by considering the extruded block as an isolated body: where N 1 and f 1 denote the positive pressure and frictional force acting on the contact surface between the extrusion block and the sliding block on the left side, respectively, and similarly, N 2 and f 2 denote the same for the right side.
With the sliding block considered as an isolated body, the frictional force and positive pressure can be obtained by balancing the forces as follows: where N denotes the positive pressure acting on the contact surface between the friction plate and the sliding block.e relationships between the positive pressure and the frictional force are as follows: F f 2μ 0 N, f 1 μ 1 N 1 , and f 2 μ 1 N 2 .Combined with Equations ( 6)-( 8), the damping force can be obtained as follows: Equation ( 9) can be rewritten as follows: e damping force for the unloading stage can be obtained using the same method: As the slip deformation between the extrusion block and the sliding block is relatively small, its contribution to the axial deformation of the damper can be ignored.e following equations are obtained by substituting F s k s u d into Equations (10) and (11): To simplify the design, it is assumed that the friction coe cients between the sliding block and the extrusion block and that between the sliding block and the friction plate are equal, i.e., μ 0 μ 1 μ.erefore, we have the following equation: Shock and Vibration 3 Figure 4(a) shows the hysteresis curves of the dampers for a spring sti ness (k s ) of 100 kN/m, a friction coe cient (μ) of 0.15, and angles (θ) of π/9, π/6, and π/3. Figure 4(b) shows the variation curve of the energy dissipation coe cient η with respect to θ for di erent values of the friction coe cient.e following are the analysis results: (1) From the theoretical analysis of the simpli ed mechanical model of the variable friction damper, we prove that the damper can realize the Reid damping hysteretic model (2) Under the same friction coe cient μ, the energy dissipation coe cient η decreases with the increase in θ; and under the same θ, the energy dissipation coe cient η increases with the increase in the friction coe cient μ (3) e equivalent sti ness k re of the damper is slightly lower than the spring sti ness k s

Construction Scheme of Reid Damper.
Figure 5 shows the detailed construction scheme of the Reid variable friction damper, established on the basis of the above theoretical analysis on friction dampers.e main components include a friction plate, sliding blocks, extrusion blocks, a spring, a transmission rod, and end plates (which are used to x the entire damper).It needs to be added that the spring does not require prestress, which is one of the characteristics of the damper.In addition, if the spring is prepressed, the damper hysteresis curve can realize the double ag type.e damper is a symmetric structure.When the intermediate transmission rod moves from the initial position to the right, only the right-side extrusion block, sliding block, and spring start to bear the force.When the middle transmission rod moves from the initial position to the left, only the left-side extrusion block, sliding block, and spring start to bear the force.

Performance Experiment on Damper.
Based on the above damper design, a small-scale Reid variable friction damper is established using Q235 steel.e height, width, and total length of the damper test specimen are 70, 79, and 277 mm, respectively.e angle θ between the two frictional contact surfaces is π/6.
e friction coe cients μ 1 between the sliding block and the friction plate and that between the sliding block and the extrusion block (μ 2 ) are equal, i.e., 0.12.
e compression springs used in the damper are red mold springs made according to Japanese Industrial Standards.
ey are made of high-performance chrome alloy steel.e outer diameter of the spring is 30 mm, the inner diameter is 15 mm, the length is 50 mm, the compression capacity is 16 mm in 300,000 cycles, and the bearing capacity is 1800 N.
As shown in Figure 6(a), the spring performance test results show that the spring force is linear with the displacement, the elastic sti ness is 100 kN/m, and the ultimate displacement can reach 20 mm.After the unloading is completed, the spring returns to the initial position and the residual deformation is substantially zero.In summary, the mold spring performance is stable and can provide better self-recovery ability.
e mold spring used in this test is small in size, and the damper applied in the engineering structure often requires a spring with a large bearing capacity.erefore, it is necessary to process and manufacture larger-sized mold springs and perform corresponding performance tests to meet engineering needs.Figure 6(b) shows the test specimen, test loading device, and sensor layout.
e low-frequency reciprocating test is loaded for a displacement amplitude of 15 mm. Figure 7 shows 30 hysteresis loop curves of the dampers under cyclic loading.e theoretical result obtained using Equation ( 12) is represented using dotted lines.e experimental results are in good agreement with the theoretical results, proving that the design scheme of the Reid damper is feasible.

Structural Analysis Model and Energy
Dissipation Design Using Reid Dampers

Design Parameters of Structural Analysis Model.
A 10story steel frame structure is selected as the analysis model.e structure is designed according to the Chinese code [29].e precautionary intensity of the structure is 8 degrees, and   Shock and Vibration the design peak of the ground motion is 0.2 g. e site condition is category II, and the classi cation of the design earthquake is the rst group.e characteristic period of the site is 0.35 s. e frame is simpli ed to a two-dimensional plane model, as shown in Figure 8.
e steel frame structure has ve 8 m spans.e height of each layer is 4 m, and the total height is 40 m. e same square steel tubes are used for the frame columns of each layer of the structure, and the same type-I steel is used for the frame beams.Figure 8 shows the speci c parameters of the section of the frame.e entire structure is made of Q345 steel, and the additional mass of each oor is 320 t. e structure satis es the assumption of the rigid diaphragm.In other words, the horizontal degrees of freedom of the nodes on the same oor are coupled.

Energy Dissipation Design Using Reid Dampers.
As shown in Figure 9, the dampers are diagonally arranged on the outer frames.e relationship between the interstory displacement Δu i of story i and the axial displacement Δu di of the dampers on story i is given as follows: where θ d denotes the angle between the dampers and the frame beam.e equivalent damping ratio of the dampers attached to the structure can be obtained as follows: where ξ eq denotes the equivalent damping ratio, W d denotes the energy dissipated by the dampers under the speci ed displacement, and E s denotes the structural elastic potential energy.e additional equivalent damping ratio of the entire structure is determined based on the seismic performance requirements.e energy dissipation requirements of the dampers are obtained to optimize the design parameters of the damper.Suppose the additional equivalent damping ratio of each story is ξ add , according to Equation (15), it can be obtained as follows: where W di and E si denote the energy dissipated by the dampers and the elastic potential energy of story i, respectively.E si is calculated as follows: where k i denotes the shear sti ness of story i. 6

Shock and Vibration
When the Reid dampers are used, the energy dissipated by the dampers on story i is obtained as follows: where n denotes the number of dampers on story i and k rei and η i denote the equivalent elastic stiffness and energy dissipation coefficient on story i, respectively.Combined with Equations ( 14)-( 18), the Reid damper parameters for story i can be calculated as follows: Taking the stiffness ratio as λ i � k rei /k i , we have the following equation: According to Equation (20), with the increase in the damping ratio, the energy dissipation coefficient is inversely proportional to the stiffness ratio.For the selected steel frame structure, 10 different schemes are established with an additional damping ratio of 0.05.Table 1 lists the mechanical parameters of each scheme of the Reid dampers.Table 2 lists the detailed damper parameters k re for each story of all schemes.

Finite Element Model of the Structure.
e finite element software ABAQUS is used to establish the plane analysis model of the steel structure, and the nonlinear time history analysis of the structure model under earthquake is carried out.e frame beams and columns are simulated using twodimensional fiber beam elements (B21).
e horizontal degrees of freedom of the nodes on the same story are coupled using the "rigid" command to simulate the rigid baffle assumption.
e load acting on the structure is converted to the equivalent density of the frame beam.e constitutive relationship of the steel is simulated using the bilinear hardening model.e initial elastic modulus of steel is 2.06E11 Pa, the elastic modulus after yielding is 1.03E9 Pa, and the yield strength is 345 Mpa.
e first three order frequencies of the structure are 0.33, 1.02, and 1.90 Hz, respectively.Figure 10 shows the corresponding vibration modes.e Rayleigh damping model is employed for the structural damping.
e mass damping coefficient α and stiffness damping coefficient β are calculated using Equation (21).e damping ratio ξ of the structural model is 0.04.ω 1 and ω 2 denote the first and second order circular frequencies of the structure, respectively: e Reid damper model is implemented using the user material subroutine interface (UMAT) provided by ABA-QUS. Figure 11 shows the test results of the Reid damper model (η � 0.6, k re � 1E4(kN/m)).

Earthquake Records and Analysis Conditions.
To evaluate the proposed control strategies, two far-field and two near-field earthquake historical records are selected: El Centro, Hachinohe, Northridge, and Kobe [32].Table 3 lists the detailed information of the earthquake records.Figure 12 shows the acceleration-time histories of the earthquakes.Additionally, this vibration control study considers various levels of earthquake records, including 0.3 g and 0.51 g, which correspond to fortification earthquake and rare earthquake, respectively.

Analysis of Vibration Control Effect
e main evaluation criteria include structural deformation, level acceleration, energy consumption ratio, structural damage, and state of the Reid dampers.Additionally, the control effect of the Reid dampers on the structural residual deformation is analyzed.

Control Effect on Structural Deformation.
Figure 13 shows the interstory drift ratio results of the different controlled schemes and that of the uncontrolled structure under the condition that the peak value of the earthquake is 0.3 g. e interstory drifts of the uncontrolled structure under the excitation of different earthquakes are clearly different.ere is an obvious deformation mutation between the upper and lower stories of the structure, particularly under the action of the Kobe earthquake.e maximum peak interstory drift ratios of the uncontrolled structure, mainly in the third and eighth story, are 1/75, 1/50, 1/111, and 1/73 under the four groups of earthquakes, respectively.e different schemes have an effective control effect on the structural deformation response and show some discreteness.Overall, the deformation distributions of the different controlled structures are similar, thus effectively solving the deformation mutation problem of the structure.is is most obvious in the Kobe earthquake condition.
Figure 14 shows the interstory drift ratio results of the different controlled schemes and that of the uncontrolled structure under the condition that the peak value of the earthquake is 0.51 g.Compared to the 0.3 g condition, the interstory drift response of the uncontrolled structure is aggravated, and the overall deformation shape of the structure is consistent.e maximum peak interstory drift ratios of the uncontrolled structure increase to 1/49, 1/38, 1/68, and 1/50.
Figure 15 shows the reduction rates of the maximum interstory drift ratios of the different schemes, wherein Figure 15(a) represents the 0.3 g condition, and Figure 15(b) represents the 0.51 g condition.Overall, the control effect of the maximum interstory drift ratio is good, and the average reduction rates of the four earthquakes for each scheme are in the range of 21-33% under the 0.3 g conditions.Compared to the 0.3 g conditions, the control effect of the maximum interstory drift ratio is similar under the 0.5 g conditions.However, the control effect is poor, and the average reduction rates of the four earthquakes for each         10 Shock and Vibration scheme are in the range of 15-24%.Nevertheless, the control e ect of the rst scheme is the best.However, the equivalent sti ness of the rst scheme added to the structure is clearly higher, thus signi cantly increasing the acceleration response.e reasons for the same are explained in the next section.

Shock and Vibration
Figure 16 shows the time histories of the top-story displacements of the sixth scheme controlled structure and uncontrolled structure under the action of the El Centro earthquake.
e displacement amplitudes of the uncontrolled structures are 0.354 and 0.546 m, respectively, under the 0.3 g and 0.51 g conditions, whereas those of the controlled structure are 0.318 and 0.478 m, respectively, and the corresponding reduction rates are 10.2% and 12.5%.Further, by comparing the displacement time responses of the controlled and uncontrolled structures, we found that the displacement amplitude of the controlled structure can be e ectively suppressed in the early stages of the earthquake, and the displacement response of the structure can be quickly attenuated to reach the static state in the later stages of the earthquake.

Control E ect on Acceleration Response.
e control e ects on the acceleration responses of each story in the controlled and uncontrolled structures are similar under the 0.3 g and 0.51 g conditions.erefore, only the contrast results under the 0.51 g conditions are given, as shown in Figures 17, and 18 shows the reduction rates of the maximum story accelerations for the di erent schemes.For the rst scheme, the equivalent sti ness of the Reid damper attached to the structure is clearly higher, resulting in a signi cant increase in the acceleration response of the structure.With respect to the average value of the acceleration reduction rates of the four earthquakes, the acceleration responses of the nine other schemes do not signi cantly increase.

Analysis of Energy Consumption Ratio.
For the uncontrolled structure, the input energy of the earthquake is largely re ected in the structural damping and plastic deformation of the structure.With the increase in the earthquake intensity, the input energy increases, and the energy dissipation associated with the structural plastic deformation increases.
e structural damage is directly related to the energy dissipation associated with the structural plastic deformation.e greater the energy dissipation associated with the plastic deformation, the greater the structural damage.Figure 19 shows the energy-time history curves of the controlled structure and sixth scheme controlled structure under the action of the 0.51 g El Centro wave.With the installation of the Reid dampers, the plastic energy consumption of the structure decreases from 796.4 kJ to 339.5 kJ, reducing by 57.4%. is shows that the plastic damage of the structures can be e reduced using the Reid dampers.
e dynamic characteristics of the structure can change depending on the arrangement of the Reid dampers in the structure.Even under the action of the same earthquake, the input energy of the earthquake is di erent.erefore, to compare the e ects of the di erent schemes on the energy consumption of the structure, the ratios of the damping energy, plastic deformation energy, and energy consumption of the damper to the total input energy are qualitatively evaluated.Figure 20 shows the energy consumption ratios of the controlled and uncontrolled structures under the 0.51 g El Centro wave and 0.51 g Northridge wave.On the one hand, the plastic damage energy consumption ratio of the uncontrolled structure is greater than that of the controlled structure; on the other hand, the greater the energy dissipation factor of the Reid dampers, the greater the energy consumption of the dampers.e total energy dissipated by the Reid dampers, designed for an additional damping ratio of 0.05, is slightly lower than that of Rayleigh damping (0.04). is is largely because the damping ratio of the higher order modes in the Rayleigh damping model is greater than 0.04.

Structural Damage and State of Reid Dampers.
e damage degree of the entire structural component is evaluated in terms of the strain state of the ber element, using which the beam and column are simulated.When the maximum strain of the component exceeds the yield strain of the material, the yield of the component is determined, and a plastic hinge is formed.
e damage degree is determined from the ratio of the maximum strain to the yield strain.When ε y < ε < 2ε y , the damage degree is level 1, indicated using blue-colored plastic hinges; when 2ε y < ε, the damage degree is level 2, indicated using green-colored plastic hinges.
Figure 21(a) shows the damage state of the uncontrolled structure under the 0.51 g El Centro wave.Plastic hinges are observed in 84% of the frame beams.
e main damage degree is level 1, and the damage degree of the beams on story 7 is level 2. Figure 21(b) shows the damage state of the sixth scheme controlled structure under the 0.51 g El Centro wave.e proportion of frame beams that exhibit a plastic hinge is 66%, and the main damage degree is level 1. e Reid dampers in their particular layout could e ectively reduce the degree of structural damage.Under the premise of ensuring the damping e ect, the lower the maximum damping force, the fewer the design requirements for the joints.For this purpose, the maximum damping forces of the Reid dampers under di erent schemes are investigated.Figure 22 shows the maximum damping forces of the Reid dampers on the rst, fth, eighth, and tenth stories under the action of the 0.51 g El Centro wave.
e maximum output force of the dampers decreases with the increase in the energy dissipation coe cient.When the energy dissipation coe cient is too low, such as 0.1, the excessive damping force is not conducive to the structural design.
Figure 23 shows the hysteretic curve of the Reid dampers on the rst, fth, and tenth stories under the action of the 0.51 g El Centro wave.From the comparison of the results shown in Figure 23(a) and 23(b), for the damper on the same story, if the energy dissipation coe cient of the Reid damper is larger, the hysteresis curve of the damper will be fuller.In addition, it can be seen that the dampers of each stories will return to the original position after being completely

Shock and Vibration 13
unloaded, thereby controlling the residual deformation of the structure.

Residual Interstory Drift.
is section mainly analyzes the control e ect of the Reid dampers on the residual interstory drift of the structure.e limit of residual interstory drift is set to 0.5%, and the structure cannot be repaired if the value exceeds this limit [33].e analysis results show that the maximum residual interstory drift of the uncontrolled structure is 1.47% under the action of the 0.51 g Hachinohe wave, and the maximum residual interstory drifts under the other three seismic wave conditions are lower than 0.5%.Figure 24 shows the maximum residual interstory drifts of the rst, third, fth, seventh, and tenth stories of the controlled structure and those of the uncontrolled structure under the 0.51 g Hachinohe wave condition.
e residual deformation of each story of the 14 Shock and Vibration uncontrolled structure is greater than the limit value.All the shock absorption schemes using the Reid dampers have better control e ect on the residual deformation of the structure; however, the control e ect of each scheme is di erent.e trend is that the control e ect decreases with the increase in the energy dissipation coe cient. is is largely because of the decrease in the equivalent sti ness coe cient with the increase in the energy dissipation coe cient of the damper.Moreover, the elastic resilience provided by the damper decreases, thus reducing the self-centering capacity of the structure.In summary, when a Reid damper is used to control the residual interstory drift of a structure, the equivalent sti ness of the damper must be guaranteed.

Shock Absorption Design when Using Reid Dampers.
e control e ects of the 10 types of Reid damper shock absorption schemes with the same additional damping ratio are compared.e lower the energy dissipation factor of the Reid damper, the greater the equivalent sti ness coe cient.
First, the deformation control e ect of the structure decreases with the increase in the energy dissipation   Shock and Vibration coe cient, and the control e ect is similar when the energy dissipation coe cient is in the range of 0.3-1.0.In this range, no obvious ampli cation in the acceleration response of the structure is observed, and the energy dissipation capacity of the dampers is good, thus e ectively controlling the plastic damage energy of the structures.
Second, when the energy dissipation coe cient is in the range of 0.1-0.6, the damper has a relatively high equivalent sti ness and good self-centering ability, thus e ectively controlling the residual deformation of the structure.However, a greater damping force will act on the damper because of the excessive elastic sti ness, making it di cult to design connections and joints.
In summary, when a Reid damper is designed for energy dissipation, the energy dissipation coe cient of the damper should be in the range of 0.3-0.6.

Conclusion
By conducting a theoretical analysis and a performance experiment, we developed a passive self-centering friction damper, which can realize the Reid model, and analyzed the control e ect of a steel frame structure with Reid dampers.e following are the conclusions drawn from this study:  e Reid damper has the potential to replace viscous dampers that are employed in tuned mass damper structures and isolation structures. is research will be carried out in the future.

Figure 2 :
Figure 2: Schematic of the passive variable friction damper.

Figure 3 :
Figure 3: Force balance of the passive variable friction damper: (a) overall force balance of the slider and extrusion block; (b) force balance of the isolated sliding and extrusion blocks.

Figure 4 :Figure 5 :Figure 6 :
Figure 4: Hysteretic loop and parameter relationship of passive variable friction dampers: (a) damper theory hysteretic loop; (b) variation in the energy dissipation coe cient η with θ.

Figure 10 :
Figure 10: Vibration modes of the steel frame structure.

Figure 11 :
Figure 11: Test result of Reid element subroutine.

Figure 13 :
Figure 13: Control e ect of structural interstory drift under 0.3 g peak value conditions (dashed lines indicate di erent schemes, blue solid lines indicate the mean value of the di erent schemes, and green solid lines indicate the uncontrolled structures): (a) El Centro; (b) Hachinohe; (c) Kobe; (d) Northridge.

Figure 16 :
Figure 16: Top-story displacement-time histories of the structures under the action of the El Centro wave: (a) 0.3 g conditions; (b) 0.51 g conditions.

Figure 17 :
Figure 17: Control e ect of structural maximum acceleration under 0.51 g conditions (dashed lines indicate di erent schemes, blue solid lines indicate the mean value of di erent schemes, and green solid lines indicate uncontrolled structures): (a) El Centro; (b) Hachinohe; (c) Kobe; (d) Northridge.

Figure 20 :
Figure 20: Proportional relationship between structural damping energy dissipation, plastic deformation energy dissipation, and damper energy dissipation (E-damper, E-plastic, and E-Rayleigh denote the energy dissipation ratio of the Reid dampers, structural plastic deformation, and structural Rayleigh damping, respectively, η 0.1 ∼ 1.0 indicate the code names of the 10 schemes, respectively, and UC indicates the uncontrolled structure): (a) Hachinohe; (b) Northridge.

Figure 23 :Figure 22 :
Figure 23: Hysteretic curve of Reid dampers in the structure under the action of the 0.51 g El Centro wave: (a) η 0.3; (b) η 0.6.
e performance experiment results of the passive self-centering friction damper proposed in this paper are in good agreement with the theoretical analysis results. is shows that the damper can realize the Reid model.A theoretical basis for the design of the damper is provided.(2)Considering the control e ect on the structural displacement response, acceleration response, damper energy dissipation e ect, failure mode, and residual deformation, it is suggested that the energy dissipation coe cient should be in the range of 0.3-0.6 when Reid dampers are used for energy dissipation.
According to the classical friction theory, the sliding friction magnitude is related to only the magnitude of the pressure F N and friction coe cient μ of the contact surface.e greater the pressure and friction coe cient, the greater the sliding friction.e formula for calculating the frictional force of the constant friction damper is as follows: 3.1.eoretical Analysis of Reid Damper.

Table 1 :
Mechanical parameters of the Reid dampers used in the 10 schemes.

Table 2 :
Detailed damper parameters k re for each story of all schemes (unit: kN/m).

Table 3 :
Information about earthquake records.