Application of computational fluid dynamics in design of viscous dampers - CFD modeling and full-scale dynamic testing

Hassan Lak, Seyed Mehdi Zahrai, Seyed Mohammad Mirhosseini and Ehsanollah Zeighami

1. Department of Civil Engineering, Collage of Engineering, Arak Branch, Islamic Azad University, Arak 3836119131, Iran

2. School of Civil Engineering, College of Engineering, University of Tehran, Tehran 1417935840, Iran

Abstract: Computational fluid dynamics (CFD) provides a powerful tool for investigating complicated fluid flows. This paper aims to study the applicability of CFD in the preliminary design of linear and nonlinear fluid viscous dampers. Two fluid viscous dampers were designed based on CFD models. The first device was a linear viscous damper with straight orifices. The second was a nonlinear viscous damper containing a one-way pressure-responsive valve inside its orifices. Both dampers were detailed based on CFD simulations, and their internal fluid flows were investigated. Full-scale specimens of both dampers were manufactured and tested under dynamic loads. According to the tests results, both dampers demonstrate stable cyclic behaviors, and as expected, the nonlinear damper generally tends to dissipate more energy compared to its linear counterpart. Good compatibility was achieved between the experimentally measured damper force-velocity curves and those estimated from CFD analyses. Using a thermography camera, a rise in temperature of the dampers was measured during the tests. It was found that output force of the manufactured devices was virtually independent of temperature even during long duration loadings. Accordingly, temperature dependence can be ignored in CFD models, because a reliable temperature compensator mechanism was used (or intended to be used) by the damper manufacturer.

Keywords: fluid viscous damper; passive control; dynamic testing; energy dissipation device; computational fluid dynamic; thermography

1 Introduction

Fluid viscous dampers as well as oil dampers are among the most well recognized and widely used energy dissipation devices that apply velocity-dependent controlling forces to an equipped structure. These highly engineered velocity-dependent devices commonly have little or no effect on lateral stiffness and inject pure supplemental viscous damping to a structure regardless of vibration amplitude. In other words, viscous and oil dampers can suppress both small amplitude windinduced vibrations (Dinget al., 2016) and high amplitude seismic-induced vibrations (Seleemah and Constantinou, 1997; Hwanget al., 2005, 2006; Kasai and Matsuda, 2014). In addition, earlier case histories(Kasaiet al., 2012), shaking table tests (Kasai and Matsuda, 2014) and comparative numerical assessments(Wang and Mahin, 2018) have indicated that velocitydependent dampers generally outperform displacementdependent dampers. By far, viscous and oil dampers have more complicated details, and consequently they are more expensive compared to other passive energy dissipation devices with the same capacity. Nonetheless,because of their superior performance, the application of viscous and oil dampers has increased in recent years. For example, according to a survey carried out by Japan Society of Seismic Isolation (JSSI), most of the recent vibration-controlled buildings in Japan have been equipped with either viscous or oil dampers (Nakamura and Okada, 2019).

The effect of viscous dampers on the seismic behavior of structures has been comprehensively investigated in prior shaking table tests (Seleemah and Constantinou, 1997; Hwanget al., 2005, 2006; Kasai and Matsuda, 2014; Constantinou and Symans, 1992;Reinhornet al., 1995; Antonucciet al., 2006; Yiet al., 2018) and numerical studies (Kruep, 2007; Smith and Willford, 2007; Chen and Bao, 2012; Mousavi and Ghorbani-Tanha, 2012; Miyamoto and Gilani,2013; Laiet al., 2015; Yanget al., 2016; Mousaviet al., 2018; Shenet al., 2020; Esfandiyariet al., 2020).In addition, the beneficial contribution oil and viscous dampers make has been reported in earlier case histories such as Kasaiet al.(2012) and Taylor (2003). These documents all indicate that viscous and oil dampers can reduce story shears, story drifts, plastic demands on beams/columns/connections, and story accelerations.As suggested by Kasaiet al.(2012), story accelerations may not always be achieved by displacement-dependent dampers. Therefore, viscous and oil dampers not only improve the seismic behavior of structural elements but also enhance the seismic behavior of accelerationsensitive non-structural components (Esfandiyariet al.,2020). Parametric studies also have been carried out by Dall′Astaet al.(2016) and Scozzeseet al.(2021) to investigate the effect of nonlinearity and the ultimate capacity of viscous dampers on the seismic behavior of different structures. Nonlinear viscous dampers refer to viscous dampers with nonlinear force-velocity curves,which can result in some advantages compared to linear viscous dampers (Lin and Chopra, 2002). This finding also is experimentally verified in this study.

In most of the aforementioned studies, the main objective has been to investigate the performance of the damped structure, with less attention to the damper mechanism itself. Although the dynamic behavior of individual viscous and oil dampers has been experimentally investigated (Mousaviet al.,2018; Esfandiyariet al., 2020; Aiken and Kelly,1996; Infantiet al., 2003; Black and Makris, 2006;Yamamotoet al., 2016), but there is extremely limited information regarding the internal details of viscous and oil dampers, such as orifice shapes and dimensions,piston dimensions, viscosity of the fluid that is used, etc.Figure 1 shows the schematic internal details of some of the previously tested viscous and oil dampers. Such schematic drawings are not enough to fully understand or investigate the behavior of the dampers. To the authors′ knowledge, orifice details and the dimensions of viscous and oil dampers have never been disclosed in open literature. Such details are commonly not known even to associated researchers. For example,during a comprehensive experimental program, Black and Makris (2006) noticed non-homogeneous viscous heating distributions in their considered dampers. As expressed in their report, the internal details of the tested dampers were not known to them, but they believed that such non-homogeneous temperature distributions were probably due to orifice locations in the piston head, a detail of which they were not aware.

Computational fluid dynamics (CFD) is regarded as a powerful tool for simulating complicated fluid flows,particularly for those lacking a closed-form solution.There are a limited number of studies on a CFDbased investigation of fluid viscous dampers. Using two-dimensional CFD models, Syrakoset al.(2018),theoretically investigated the behavior of fluid dampers that had an annular gap orifice filled with silicone oil containing extremely high viscosity—the details of which are not common in current fluid viscous dampers used in the construction industry. In another study, as shown in Fig. 1(d), Mousaviet al.(2018) experimentally investigated the dynamic behavior of bypass viscous dampers and briefly presented results from a CFD model of one of the tested dampers, with no discussion of the CFD modeling. Lak and Zahrai (2023) used a thermal camera to monitor temperature rise during different loads applied to viscous dampers. They found that the temperature rise of the dampers can be neglected in the case of short-duration seismic loads, but the self-heating can be significant during long-duration loading.

Fig. 1 Details of different viscous and oil dampers. (a) Viscous damper with fluidic control orifice (Constantinou and Symans,1992), (b) uniflow type oil damper (AIJ, 2016), (c) multiple unit oil damper (Yamamoto et al., 2016), (d) bypass viscous damper (Mousavi et al., 2018)

In the present paper, the first conceptual design of two linear and nonlinear viscous dampers are presented.Next, a CFD-based design is adopted in which details and dimensions of the dampers are finalized based on results obtained from CFD simulations. The modeling procedure and details of the damper orifices are presented. Full-scale specimens of the designed dampers were manufactured and tested under different dynamic loads. The cyclic behavior of the dampers is presented for harmonic and seismic loading protocols and obtained results from the linear damper are compared with those taken from the nonlinear damper. Using a thermography camera, the rise in temperature of the dampers also is investigated. Finally, obtained experimental results are compared with numerical results from CFD simulations and simplified Maxwell models, which are widely used by professional engineers in design offices. To the authors′ knowledge, this is the first study in which CFD results are presented for full-scale tested viscous dampers with realistic internal details manufactured by a commercial manufacturer.

2 Conceptual design

2.1 Linear fluid viscous damper

Internal details of a fluid viscous damper, typically used in the construction industry, are shown in Fig. 2.During a dynamic excitation, the piston and the piston rod reciprocate inside the damper cylinder, which in turn pressurizes the fluid on one side of the piston, while the pressure on the other side is virtually zero (neglecting the effect of fluid thermal expansion). Because of the generated pressure difference, fluid flows through the orifice(s) of the piston, which creates a streamline from one side of the piston to the other, as shown in Fig. 2.

Fig. 2 Details of a typical fluid viscous damper and fluid flow through its orifice (Courtesy of IRANDAMP Co.)

Bernoulli′s equation can be used to estimate the generated pressure difference in the damper. Assuming an incompressible and steady fluid flow, Bernoulli′s equation for the defined streamline from Point 1 to Point 2 can be written as White (2011):

wherePis the fluid pressure,Vis the fluid velocity,andZis elevation of the fluid from a reference plane.Fluid density is represented byγ, and g is the gravity acceleration. In Eq. (1)hLrepresents total hydraulic losses (also called hydraulic head loss) and subscripts 1 and 2 represent the fluid parameters at points 1 and 2, respectively. Assuming the same velocity (due to principle of mass conservation) and elevation at both points, Eq. (1) can be simplified to:

As a result, the generated pressure difference, as well as the developed damper force, is directly related to the hydraulic loss along the streamline. Damper force (F) is simply the pressure difference multiplied by the effective piston area. Therefore, Eq. (2) can be rewritten as:

where,DcandDsare piston diameter (≈ cylinder diameter)and diameter of the piston rod (shaft), respectively.Considering both local and global hydraulic losses,hLcan be written as (White, 2011):

where,d,LandVare orifice diameter, orifice length and fluid velocity inside the orifice.KLis the local head loss coefficient, which occurs during contraction/expansion of the flow when it enters/exits the orifice. Contraction and expansion loss coefficients are highly dependent upon the geometry of the orifice entrance. Considering the typical cylinder and orifice diameters of viscous dampers, local loss coefficients of sudden expansion and contraction are 1 and 0.5, respectively (White, 2011). As a result,KL=1.5 is rational for an orifice with 90-degree sharp edge entrances. On the other hand, orifices with a chamfer/fillet edge entrance show a negligible contraction coefficient, so in such cases,KL=1 is more suitable. In Eq. (4),fis the Darcy′s friction factor, which for a laminar flow can be written as:

whereReis the orifice Reynolds number andμandρ,respectively, are the dynamic viscosity and density of the fluid. Note that the Reynolds number of the considered dampers is expected to be less than 1000 during all the considered loadings. Accordingly, the fluid flow would remain laminar. Combining Eqs. (3) to (5), the damper force can be estimated as:

Equation (6) expresses damper force as a function of fluid velocity inside an orifice, which is significantly higher than the fluid velocity in the damper cylinder (the velocity of the piston movement). Using the principle of mass conservation, piston velocity easily can be related to orifice fluid velocity. Accordingly, Eq. (6) can be rewritten in terms of piston velocity (u˙) as:

wherenis the number of orifices. Equation (7) indicates that damper force has a linear term and another nonlinear term. The first linear component pertains to frictional(major) hydraulic losses, while the second nonlinear component reflects the effects of local (minor) hydraulic losses. The behavior of fluid viscous and oil dampers are typically expressed by:

whereCis the damping coefficient andαis the velocity exponent of the damper. In the case of a linear damper,α=1, the damping coefficient of a viscous damper withnstraight orifices can be expressed as:

Obviously, such a viscous damper is not absolutely linear, as evidenced by the second term in Eq. (9).However, the second term is significantly smaller compared to the first linear term, especially in the case of lower damper velocities, dampers with a higher number of orifices (highern), and fillet/chamfer edge orifice entrances (smallerKL). Note that this closedfrom relation is obtained for a viscous damper withnstraight orifices inside a piston that has a steady laminar flow with incompressible Newtonian fluid. According to this simplified conceptual design, the linear viscous damper that was considered for this study has a cylinder diameter (≈ piston diameter) ofDc= 125 mm and a piston rod diameter ofDs= 50 mm. Two straight orifices (n=2)with a diameter ofd=2.7 mm and a length ofL= 100 mm are placed inside the piston. Chamfer edge entrances are considered for the orifices so thatKL≈1. The damper is filled with a silicone-based oil with a dynamic viscosity of 0.35 Pa.s and a density of 1000 kg/m3. For such details, expected damping coefficient is calculated as(units: kN.s/mm):

2.2 Nonlinear fluid viscous damper

Fig. 3 Mechanism for the one-way pressure-responsive valves of a nonlinear viscous damper (oil damper)

A previous subsection indicated that nonlinear viscous dampers cannot be achieved using fixedgeometry straight orifices inside the piston. Therefore,more complicated orifices are required for a nonlinear viscous damper with a velocity exponent ofα<1. Note that in this study, the term nonlinear viscous fluid damper refers to any fluid damper with a velocity exponent smaller than 1, which is the most common type of nonlinear damper used in the construction industry.To reach a nonlinear damping, a one-way pressureresponsive valve is placed within each orifice, as shown in Fig. 3. This type of viscous damper is sometimes called an “oil damper”, as the flow resistance will mainly be provided by the pressure-responsive valve rather than the viscosity of the fluid. When the piston moves inside the damper, one of the valves is closed and the other opens. The greater the velocity of the piston, the more pressure difference is generated, making the valve open more while providing less resistance to the flow.Note that the valve opening shown in Fig. 3 is extremely exaggerated. And observe that according to the details illustrated in Fig. 3, by pre-compressing springs of the valves there is an option to reach a fluid damper with a non-zero activation force. The activation force is defined as the force which is required to open the valve and have damper act like a stiff link for forces less than this activation force. The fluid pressure required to open the valve and, consequently, the activation force of the damper, can be adjusted by the valve spring stiffness and its initial pre-compression. The activation force can be zero or non-zero and it can be adjusted according to the estimated service-level excitations, for example,with frequent wind loads. Obviously, no closedform formulation is available for such a complicated mechanism due to the fluid-valve interaction. So, the design of the presented nonlinear viscous damper is solely based on CFD simulations, which are presented in the next section. Details of the nonlinear damper also are presented in the following section.

3 CFD-based design

3.1 CFD modeling of the linear damper

Abaqus/CFD (Dassault Systems Simulia, 2015) is used for CFD analyses. The CFD model of the linear damper is shown in Fig. 4. Since there is no fluid-solid interaction in this damper, only the fluid parts of the damper are modeled, with 8-node linear fluid brick elements. The effects of solid components (piston,piston rod and cylinder) are implicitly accounted for by defining the associated boundary conditions, as shown in Fig. 4(c), in which the first boundary condition, i.e.,BC1, represents the input piston velocity. Adopted mesh sizes are shown in Fig. 4(b), which are selected after mesh size sensitivity investigations take place. A temperature compensator mechanism is intended to be used in the damper specimens. Therefore, the temperature dependency of the fluid is neglected. The validity of this assumption will be investigated in subsequent sections.

Different constant input velocities are monotonically applied to the model and the damper force is calculated from the generated fluid pressure presented in each case.As a result, the force-velocity curve of the damper can be obtained directly from CFD results. Figure 5 shows some of the main results from the CFD simulations of the linear damper. Figure 5(a) indicates that orifice fluid velocity is substantially higher than damper (piston)velocity and it can easily reach values of about 170 m/s(610 km/h) or even more. This is an expected result of the mass conservation principle. Note that after the orifice,fluid velocity is still high because the illustrated fluid velocity profile belongs to a specific streamline, and it does not represent the average velocity along the length of the damper. The associated local fluid turbulence,which occurs at the fluid expansion at the end of the orifice, also is clear in Fig. 5(a). Figure 5(b) illustrates variation in the fluid pressure in which the maximum fluid pressure drops to zero (more precisely, it drops to the ambient pressure) as the fluid flows through the orifice. This indicates that in the linear viscous damper,virtually all energy dissipation would occur along the orifice. Also note that the greater the input velocity, the higher the generated fluid pressure, which is an obvious result of a velocity-dependent damper. The estimated force-velocity curve of the linear damper is shown in Fig. 5(c). CFD results are in good agreement with the results estimated by using Eq. (10) from Bernoulli′s equation. The estimated force-velocity curve will be compared with the experimental results in subsequent sections.

3.2 CFD modeling of the nonlinear damper

As discussed earlier, no closed-form solution is available for the described nonlinear damper. Therefore,selected dimensions and details for the nonlinear damper are solely based on CFD simulations. The final selected dimensions and the associated CFD model are shown in Fig. 6. Since the input piston velocity is monotonic, only one of the one-way valves is required in the model, as the other valve would be closed and therefore prevent fluid flow. Similar to the linear damper, the nonlinear damper cylinder diameter (≈ piston diameter) and piston rod diameter are 125 mm and 50 mm, respectively.The nonlinear damper is filled with silicone-based oil containing a dynamic viscosity of 2 Pa.s and a density of 1000 kg/m3. Details of the orifice and its one-way pressure-responsive valve are shown in Fig. 6.

Fig. 5 Results of CFD simulations for the linear viscous damper. (a) Variation of fluid velocity, (b) variation of generated fluid pressure, (c) obtained damper force-velocity curve: the dotted line is curve fitted based on CFD results

Fig. 6 CFD model of the nonlinear viscous damper (only one of the orifices is shown)

The procedure described in the previous subsection was the simplest case in the CFD modeling of fluid viscous dampers. To simulate behavior of the proposed nonlinear damper, a solid-fluid Co-simulation should be used to account for the flow-valve interaction. As shown in Fig. 7, this requires two models in Abaqus. The liquid components of the damper are modeled in Abaqus/CFD and the solid valve with its pre-compressed spring is modeled in Abaqus/Explicit. Additional details of the model are shown in Fig. 7.

Fluid velocity and fluid pressure associated with different piston velocities are shown in Fig. 8. Orifice fluid velocity is significantly higher than that of the piston velocity, as shown in Fig. 8(a). However, its maximum value is smaller than that of the linear damper. This is because of the larger diameter used for the orifice of the nonlinear damper. Moreover, fluid velocity suddenly drops after passing through the valve. This is due to the fact that diameter of the valve reservoir is higher than that of the orifice. Local turbulences of the fluid also are visible at the one-way valve and at the orifice ends where the fluid contracts and expands. Figure 8(b) indicates that most of the pressure drop occurs within the one-way valve rather than in the orifice. In other words, most of the hydraulic loss is due to valve resistance and not the viscosity of the fluid. This indicates that the designed nonlinear viscous damper is not so sensitive to fluid viscosity. Although not presented in this paper, CFD cosimulations have also supported this claim. The forcevelocity curve of the damper is presented in Fig. 8(c),which will be compared with the experimental results in subsequent sections. Finally, the opening of the pressureresponsive valve under different piston velocities is illustrated in Fig. 9, indicating that, as expected, the greater the input velocity, the larger the valve opening.The early transient fluctuations in the valve opening are due to the assumption of instantaneous input velocity in CFD simulations. Such fluctuations decay rapidly and generally are not expected to occur during seismic excitations. It should be noted that variations of fluid velocity and fluid pressure largely depend on the internal details of the damper, and these results may not hold true for other fluid dampers, which have significantly different internal details.

Fig. 7 Details of the two fluid/solid models required for the co-simulation of the nonlinear damper

Fig. 8 Results of the CFD co-simulations for a nonlinear viscous damper. (a) Variation of fluid velocity, (b) variation of the generated fluid pressure, (c) obtained damper force-velocity curve

4 Experimental study

Full-scale specimens of the designed linear and nonlinear viscous dampers were manufactured by the IRANDAMP Co. and tested under different dynamic loading protocols. As shown in Fig. 10, a unique rocking set-up is used for the tests. In this set-up the dynamic actuator was connected to the viscous damper by use of a rocking support. The rocking support was pinned to the strong floor and its main role was to directly apply actuator displacements to the damper while restraining out-of-plane displacements. This rocking set-up was adopted due to the requirements of the laboratory,which enforced pinned connection at both ends of the dynamic actuator. Typical measurement devices such as LVDT and load cell were used to measure the applied displacements and developed forces. In addition, a thermal camera and an infrared thermometer were used to measure the temperature on the exterior surface of the damper (no internal temperature was measured during the tests).

Cyclic loads with different frequencies and varying amplitudes were applied to the specimens, as presented in Table 1. Harmonic protocols are in general compliance with those recommended by ASCE 7-16 (2017) for velocity-dependent dampers. In addition to code-based harmonic protocols, three seismic excitations were applied to the dampers. These seismic loads were damper deformation time histories obtained from a numerical model of an eight-story building under three earthquake records, specifically, the Cape Mendocino, Manjil and Erzincan earthquakes.

Fig. 9 Opening of the one-way pressure-responsive valve under different input velocities

Fig. 10 (a) Rocking set-up, (b) measurement apparatus, (c) rocking set-up in its deformed shapes

4.1 Experimental results of the linear damper

Figure 11 shows experimental results in terms of the cyclic behavior of the linear viscous damper under different protocols. Stable behavior is obtained for the damper, and acceptance criteria of ASCE 7 is satisfied.No yielding, leakage, or deterioration of any kind was observed during the tests.

Table 1 Dynamic loading protocols used during the experimental phase of the study

Fig. 12 Displacement, velocity, and force time histories of the linear damper under a harmonic load with a frequency of 0.25 Hz and an amplitude of 90 mm (actuator velocity saturation has occurred)

Theoretically, the cyclic behavior of a pure linear viscous damper should be a horizontal ellipse. However,obtained results indicated that the damper resulted in inclined elliptical behavior during higher frequencies and, in some cases, resulted in non-elliptical (rectangularshape) behaviors. The former result is mainly due to fluid compressibility and the flexibility of the solid components of the damper and its connections, a condition noted in earlier studies such as Reinhornet al.(1995), Infantiet al.(2003), Black and Makris (2006). The latter is a result stemming from a referred to as the velocity saturation of the actuator, which occurs when the actuator reaches its speed capacity, i.e., the maximum velocity that the actuator can attain. When velocity saturation occurs, the actuator cannot no longer apply exact sinusoidal loading and there would be a time zone with a constant velocity in the applied velocity time history, as shown in Fig. 12.In this velocity-saturated time zone, damper force also remains constant, and a rectangular-shaped cyclic behavior would be obtained. Due to actuator velocity saturation, the nonlinearity of a viscous damper cannot be judged solely based on the shape of cyclic behavior.The force-velocity curve of the linear viscous damper is shown in Fig. 13, which indicates that the tested damper is defiantly linear despite its rectangular-shaped behaviors under some of the harmonic loadings. Note that scattering of the data points, shown in Fig. 13, is due to the minor dependency of the damper on other factors such as loading frequency, loading amplitude,and a temperature rise due to self-heating. This is the case for virtually all of the energy dissipating and seismic isolation devices, and it will be used to estimate the lower- and upper-bound characteristics of the device.

Fig. 13 The force-velocity curve of the linear viscous damper and its best fitted curve

The linear damper also is tested under an additional small amplitude, high-duration harmonic wind protocol with 1300 cycles, an amplitude of 8 mm, and a frequency of 0.83 Hz. The rather high frequency selected for this protocol was due to the working time limitation of the actuator. Results of this protocol are shown in Fig. 14,indicating that the damper demonstrates a stable behavior with no deterioration even after 1300 cycles and a temperature rise of more than 50°C. Accordingly,the damper is not temperature sensitive and during CFD simulations, the temperature dependence of the fluid can be neglected in that a reliable temperature compensator was used. Again, note that this is the external temperature of the damper, and its internal temperature was probably much higher (Black and Makris, 2006).

Fig. 14 Thermal and force-displacement behavior of the linear viscous damper under 1300 cycles with a frequency of 0.83 Hz and an amplitude of an 8-mm wind protocol)

4.2 Experimental results of the nonlinear damper

The same loading protocols were used to the investigate the behavior of the nonlinear viscous damper. Obtained cyclic behaviors are presented in Fig. 15. During some protocols there is limited asymmetry between maximum positive (tensile) and negative (compressive) forces, which is due to the minor difference between spring stiffness and the precompression load of the pressure-responsive valves.Unlike the linear damper, the nonlinear damper exhibits rectangular-shape cyclic behaviors even during small amplitude, low velocity protocols, indicating that this shape is not formed because of the velocity saturation of the actuator. The force-velocity curve of the nonlinear damper is shown in Fig. 16. Note that the damping coefficient and the velocity exponent are obtained after fitting a power function to the force-velocity data points.No yielding, leakage, or deterioration of any kind was observed during the tests.

A thermal camera also is used to monitor the temperature rise of the nonlinear damper. Figure 17 illustrates thermal images of the nonlinear damper before and after the seismic protocols. Results indicate that the temperature rise during three seismic excitations is only about 3°C, which is significantly less than that measured during the high duration wind protocol.

4.3 Comparison of the linear and nonlinear dampers

Some of the test results from the linear and the nonlinear dampers are compared with each other in this subsection. Figures 18 and 19, respectively, compare force-displacement loops and associated dissipated energies of dampers under the same harmonic and seismic protocols. According to Fig. 18, under low-speed protocols, higher forces and more energy dissipation are measured for the nonlinear damper. During higher speed protocols, on the other hand, maximum forces of both dampers are virtually the same, and, in some cases, higher forces are developed in the linear damper. Generally,the nonlinear damper tended to dissipate more energy,especially during lower loading speeds. Figure 19 shows that the linear damper, despite its higher maximum forces, dissipated less energy compared to the nonlinear damper. During high-speed seismic pulses, lower force peaks are generated in the nonlinear damper, which is another advantage of employing nonlinear dampers.Force-velocity curves of both dampers are compared in Fig. 20, where the higher forces of the nonlinear damper during lower speeds is obvious.

Fig. 16 Force-velocity curve of the nonlinear viscous damper and its best fitted curve

Fig. 17 Temperature rise in the nonlinear viscous damper during seismic protocols

Fig. 18 Comparison of cyclic behaviors and the associated dissipated energies of linear and nonlinear dampers under the same harmonic protocols

Fig. 19 Comparison of cyclic behaviors and the associated dissipated energies of linear and nonlinear dampers under the same seismic protocols

Fig. 20 Comparison of linear and nonlinear viscous dampers in terms of force-velocity curves

5 Model validation

Accuracy of the completed CFD simulations in estimating the force-velocity curve of the dampers is presented in Fig. 21. Good agreement is achieved between the CFD and experimental results, indicating that CFDbased modeling is well suited for the preliminary design of linear and nonlinear viscous dampers. Table 2 presents the damping coefficient and the velocity exponent of the dampers from CFD and experimental results. The authors would like to clarify that the final characteristics of linear and nonlinear viscous dampers should always be obtained from prototype tests, as indicated by ASCE 7-16 (2017).

Behavior of fluid viscous dampers is commonly simulated by use of the simplified Maxwell model in design offices. Earlier studies have reported that the Maxwell model is a reliable analytical tool for simulating the dynamic behavior of fluid viscous dampers (Reinhornet al., 1995; Yamamotoet al., 2016). This model consistsof an exponential damper in a series with a linear spring.The former term is responsible for generating velocitydependent, out-of-phase damping forces, while the latter is used to account for fluid compressibility and flexibility of solid components and connections of the damper, if not explicitly modeled. Using the SAP 2000 CSI, Structural Analysis Program (2019), the accuracy of this numerical technique is investigated. Figure 22 shows that Maxwell′s model can capture the seismic behavior of the linear and nonlinear viscous dampers,and its accuracy is once again verified in this study.

Table 2 Damping coefficient and velocity exponent of the dampers from CFD and test results

6 Conclusions

Fig. 21 Comparison of force-velocity curves obtained from CFD simulations and dynamic tests

Fig. 22 Comparison of experimental and numerical (Maxwell′s model) seismic behaviors of the (a) linear and (b) nonlinear dampers

Computational Fluid Dynamics (CFD) is used to design two linear and nonlinear viscous dampers. Details of the dampers, including orifice shapes and dimensions,are presented in the paper. Full-scale specimens of the designed dampers are manufactured and tested under different dynamic loads. Both dampers have the same dimensions and force/stroke capacities. The main findings of this study are as follows:

- Using Bernoulli′s equation, a closed-form formulation can be obtained to estimate the damping coefficient of linear viscous dampers with fixed-geometry,straight orifices. The damping coefficient estimated by the derived formulation is in good agreement with that estimated by using CFD simulations. However,in the case of nonlinear viscous dampers with more complicated orifices, no closed-form formulation can be obtained from the use of classical fluid mechanics.

- A non-linear viscous damper with one-way pressure-responsive valves is proposed and designed based on CFD simulations.

- From CFD results, it is found that fluid velocity inside the orifice can be substantially higher than the input piston velocity of the damper. Variation of the fluid pressure indicates that in the case of the linear damper,fluid pressure linearly drops from its maximum value to zero as the fluid pass through the straight orifices.However, in the case of the nonlinear viscous damper,the main pressure drop occurs when the flow reaches the pressure-responsive valve. This indicates that flow resistance is mainly provided via pressure-responsive valves rather than via fluid viscosity. Therefore, the designed nonlinear damper also can be categorized as an oil damper.

- Stable cyclic behaviors were obtained from dynamic tests of the manufactured dampers. As expected, higher forces were developed in the nonlinear damper under low-speed protocols. However, in the case of high-speed seismic impulses, the maximum force of the nonlinear damper is smaller compared to that of the linear damper. During virtually all loading protocols, the nonlinear damper dissipated more energy compared to its linear counterpart.

- Because of actuator velocity saturation,rectangular-shape behaviors were obtained even for the linear viscous damper. Accordingly, nonlinearity of a viscous damper cannot be judged solely based on its cyclic behavior shape.

- Using a thermal camera, the temperature of the viscous dampers was monitored during the tests.It was seen that under high duration wind protocols,the temperature rise of the damper can be substantial(51°C under 1300 cycles with an amplitude of 8 mm, a frequency of 0.83 Hz, and a peak damper force of about 70 kN). However, the temperature rise during seismic protocols is very small (3°C under three earthquakes with a return period of 475 years). Due to the use of the temperature compensator mechanism, no deterioration is observed in the output force of the tested dampers.

- Force-velocity curves of the dampers estimated from CFD simulations show good consistency with the test results. Accordingly, a CFD-based design can be accepted for the preliminary design of fluid viscous dampers. However, finalized properties of the manufactured dampers should always be reported based on prototype tests.