Dynamic behavior of a magnetorheological fluid radial-flow-based energy absorber considering apparent slip boundary condition

Abstract

Accurate theoretical models play a key role in the development of magnetorheological energy absorbers (MREAs). Traditionally, the apparent slip caused by non-uniformity interface (i.e., the separation of carrier fluid and magnetic particles at the wall) of the working medium is often ignored for simplifying theoretical models. However, the apparent slip influence the cross-sectional flow and decreased the theoretical accuracy of the damping force for MREA applied in the impact absorption area. In this study, a dynamic model with apparent slip boundary condition for radial-flow-based MREA was proposed. The effect of apparent slip layer thickness on the dynamic behavior of an MREA was qualitatively and quantitatively compared in terms of the following three aspects: (1) with the Power-Law constitutive model, theoretical models of the dynamic characteristics of the magnetorheological fluid (MRF) squeeze and MREA are presented based on the apparent slip boundary condition with the MRF behavior of the Navier–Stokes equation; (2) the accuracy of the theoretical model to predict MREA peak force and boundary velocity with different slip-layer thicknesses; and (3) global agreement of the dynamic force and range curves between the modeling and experimental results. The results show that the absolute values of the relative errors in the two models with a 2- and 0-µm thick slip layer were less than 3.73 and 7.41%, respectively, and that a 2-µm thickness can help predict the actual dynamic characteristic more efficiently under accidental collision loading conditions.

Keywords

Magnetorheological energy absorber radial flow model apparent slip dynamic behavior

1. Introduction

Magnetorheological energy absorbers (MREAs) have been widely investigated for applications in vehicle suspensions (Bai and Yang, 2019; Nguyen et al., 2018; Sassi et al., 2018), building earthquake mitigation (Chong et al., 2013; Kim et al., 2011), cable-stayed bridges (Duan et al., 2005; Weber and Distl, 2015; Wu and Cai, 2010), engine mounts (Liao et al., 2013; Nguyen et al., 2013), and tunnel construction (Yang et al., 2019), among others, for their high performance and wide modulation range. Most studies, including the above-mentioned examples, have focused on the applications of vibration mitigation under low piston speeds. However, high-speed impact and collision at high piston speeds may cause failures in systems such as automotive bumpers (Ahamed et al., 2014; Ahmadian, 2017; Sassi et al., 2018), aircraft landing gears (Saleh et al., 2018; Yoon et al., 2020), and artillery recoil systems (Li et al., 2019; Qing, 2016). Researchers have proposed MREAs for those accidental collision buffers.

Working models of MR valves are of great significance in the development of MREAs. These valves can be broadly classified into two models depending on the structure of the working channel with the magnetorheological fluid (MRF): annular hybrid flow model and radial squeeze flow model. In the annular hybrid flow model, the magnetic chain is broken by the piston shear force. In the second model, it is broken by the MRF flow squeeze force (Deng et al., 2018; Imaduddin et al., 2014; Shou, 2019; Yu et al., 2016). Under the same external excitation, the radial squeeze flow model produces the largest response for effectively lengthening the magnetic flow channel and demonstrating extensive application potential (Bai et al., 2013; Dogruer et al., 2008; Hu et al., 2019; Liao et al., 2012). In our previous studies (Fu et al., 2017; Li et al., 2018), we proposed a new MREA based on the radial squeeze flow model for its application as a transient shock buffer. It achieved a maximum damping force of up to 151.06 kN at an impact velocity of 4.2 m s⁻¹. This is because the model improved the active length of the damping channel, owing to the perpendicularity of the magnetic field to the radial flow direction (Fu et al., 2017; Liao et al., 2012). Moreover, a high-viscosity linear polysiloxane-based MRF (HVLP MRF) featuring excellent long-term suspension stability was used (Xie et al., 2015, 2016). In addition, the large viscosity (22 Pa s at a shear rate of 400 s⁻¹) of the fluid increased the off-state damping force. However, the long radial flow mode also resulted in a relatively more complex flow characteristics, which leads to a significant difference between the section flow and the real flow. This makes it more difficult to model the characteristics of an MREA, especially in the area of impact absorption.

In addition to the working model of an MR valve, the flow model of the MRF also affects the absorber’s dynamic characteristics as it controls the output MREA force by modulating the input current during impact (Zheng et al., 2014). The conventional model of MREAs considers the MRF as a fully dispersed particle concentrate and omits the effect of the magnetic field on the MRF concentration. Consequently, those models are capable of accurately describing the MRF’s flow behavior in an annular hybrid flow channel, for an MRF with sufficient particle concentration in applications that frequently use it. However, such models are not compatible with MRFs in an infrequently used radial-flow-based valve. To improve the accuracy of the model, studies have considered in the MREA models other influencing factors such as inertial force, minor losses, apparent slip, and wall slip, which effectively increased the accuracy of damping force prediction (Fu et al., 2017; Zhang et al., 2019). In our previous work, to improve the radial flow characteristics of the MRF during accidental collisions, we adopted a Power-Law constitutive model to accurately describe the fluid’s rheological behavior. Additionally, we established the dynamic model of an HVLP-MRF-based MREA with the radial flow mode (Li et al., 2018). However, the model only considered the factors of minor losses or underlining the inertial effect, omitting the physical properties of the apparent slip caused by non-uniformity interface (i.e., the separation of carrier fluid and magnetic particles at the wall) of the MRF at the fluid channel boundary. Moreover, the essential reason for the non-uniform interface is the non-uniform quality of magnetic media, while gravity and magnetic field accumulation are the main reasons for the uneven medium quality and boundary stratification in the working channel. This, in turn, leads to a prediction accuracy degradation, or even failure, of the model in areas where MREAs are infrequently used for impacat absorption. Therefore, it is important to extend the mathematics of the MRF radial squeeze flow mechanism for enhancing the prediction accuracy of the MREA’s damping force.

The variations in continuous shear stress with respect to the shear rate of an MRF caused by apparent slip for a non-uniform interface has not been described in the literature. The relationship between the apparent slip-layer thickness and the MREA damping force should also be extended. Hence, it is necessary to codify the apparent slip phenomenon causing the non-uniformity of the MRF boundary. And the influence of apparent slip on the dynamic characteristics of a MRF radial squeeze flow also requires a more comprehensive theoretical model, especially in the applications of high-speed impact conditions.

In this study, we investigated the effects of the slip-layer thickness on the dynamic behavior of an MRF radial squeeze flow in an MREA through theory and experiment. To this end, we proposed a mathematical model for the radial squeeze flow based on the Power-Law model that incorporates the apparent slip boundary condition. According to the design features of the MREA, the effect of the apparent slip-layer thickness on the absorber’s dynamic behavior was qualitatively and quantitatively evaluated from the following three aspects: (1) with the Power-Law constitutive model and the Navier apparent slip condition, the MRF squeeze dynamic characteristic theoretical model and the MREA dynamic behavior theoretical are presented based on the apparent slip boundary condition where the MRF behavior follows the Navier–Stokes (N–S) equation; (2) the accuracy of the theoretical model with different slip-layer thicknesses to predict the MREA peak force and boundary velocity; and (3) the global agreement of dynamic force and dynamic range curves between the model and the experimental results. Especially, three parameters (i.e., relative error of peak force, absolute error of dynamic force curves, and relative error of dynamic range) were measured to quantitatively evaluate the energy absorption performance of the MREA.

2. MREA configuration

In our previous work (Fu et al., 2017; Li et al., 2018), we designed and demonstrated MREAs based on the MRF radial flow mode. The Power-Law constitutive equations can accurately represent the HVLP MRF with radial flow characteristics. The dynamic characteristic equations were obtained based on the assumption that the MRF flow velocity near the wall is zero. However, the microscale distance between magnetic particles would be reduced by gravity settling accumulation and magnetic-field chain rearrangement. Thus, any small MRF flow rate would induce a pressure gradient in the flow channels, which will cause significant changes in the damping force for an MREA in impact applications. Therefore, to improve the prediction accuracy, an HVLP MRF radial flow model considering the apparent slip boundary condition was proposed in this study.

The active instantaneous stiffness of the MR valve was greatly reduced by installing a corrugated tube between the upper end cup and the MR valve. With this, the MREA damper-force curve was significantly smoothed at the initial stage of the impact. However, the elastic force produced by the corrugated tube made an insignificant contribution to the buffer force of the MREA. Therefore, to comprehensively analyze the HVLP MRF radial Power-Law flow characteristics, we simplified the structure (Figure 1) to a piston cylinder. When an axial impact occurs, the HVLP MRF in the piston cylinder is pushed into the radial channel through the entrance, and then passes through the annular flow channel to the next radial channel for dissipating the impact energy. When a magnetic field is applied, the pressure reduction across the radial channels can be changed to support the HVLP MRF’s rapid transient change from a fluid-like phase to a semi-solid-like phase by varying the field. The pressure drop in the channels for the wall velocity variation also changes. Hence, a full-scale, HVLP MRF radial Power-Law flow-mode-type MREA considering the apparent slip-boundary condition was developed to deal with high-speed impacts.

Based on this analysis, the total MREA force $F_{total}$ is equal to the sum of the force of the hydraulic force $F_{valve}$ and the corrugated tube force $F_{ct}$ :

F_{total} = F_{valve} + F_{ct}

(1)

where (Fu et al., 2020)

F_{ct} = K_{ct} \cdot [\frac{η_{p} G}{E L_{ct}^{σ - 1}} {(x - x_{t_\max})}^{σ} + x_{t_\max}]

(2)

where $K_{ct}$ is the impact stiffness of the corrugated tube in the elastic stage, $x$ is the displacement, $η_{p}$ is a compensation coefficient, $η_{p}$ is the corrugated segment length, $x_{t_max} = 44.5$ mm is the maximum elastic crush displacement, $E$ , $G$ , and $σ$ are the elastic and strain-strengthening moduli, and plastic strain-strengthening exponent, respectively.

We assume the apparent slip effect force caused by the HVLP MRF flowing in the entrance and exit is negligible. Thus, $F_{valve}$ can be expressed as

F_{valve} = F_{MR} + F_{vf} + F_{te}

(3)

where $F_{MR}$ represents the damping force resulting from the yield stress and apparent slip effect of the MRF in the radial channel, $F_{vf}$ is the force due to the viscous flow of the MRF in the annular channel and piston cylinder, and $F_{te}$ is the orifice resistance gap force from the entrance and exit. In addition, the HVLP MRF is assumed to be incompressible with negligible heat loss.

The orifice resistance gap force is defined as follows (Hu et al., 2017):

F_{te} = 8 π K \cdot \frac{L_{ax} R_{ct}^{2}}{R_{ax}^{2}} \cdot V_{i}

(4)

where $K$ is the viscosity of the MRF, $V_{i}$ is the impact velocity, and $R_{ct}$ is the average radius of the corrugated tube.

3. MREA model considering apparent slip

In this section, an MRF flow model considering the apparent slip boundary condition is established in a controlled channel (radial flow channel), uncontrolled channel (annular flow channel), and piston cylinder (Figure 1), respectively. First, the MRF Power-Law flow dynamic models with and without magnetic field were derived under the apparent slip boundary condition. Second, velocity and flow expressions based on the MR behavior were solved simultaneously using the N–S equation. Third, the controlled and uncontrolled damping forces were derived, and a comprehensive expression of the vertical buffer force was obtained.

3.1. MRF apparent slip flow dynamics in controlled channel

According to Liao et al. (2012), the MRF flow layer structure composed of a carrier layer HVLP (apparent slip layer) and an MRF layer is formed under the action of a magnetic field. An axisymmetric MRF radial flow model considering apparent slip, and the rectangular coordinate system are depicted in Figure 2. Here, $r$ , $y$ , and $θ$ are the radial, vertical, and circumferential directions, respectively, $u_{r}$ is the radial flow velocity of the MRF, $2 h$ is the gap between the inner and upper plates, $δ$ is the thickness of the apparent slip layer (HVLP), and $u_{s}$ is the fluid velocity at the slip boundary.

We dismiss the inertial action and focus only on the laminar flow of the HVLP MRF. The N–S equation in the radial channels is described below.

Considering $2 h / R_{ra} << 1$ , it is assumed that $u_{r} = u_{r} (u_{θ}, y)$ , $u_{y} = 0$ , $u_{θ} = 0$ , $Δ P_{r} = Δ P_{r} (r)$ , where $u_{r}$ , $u_{y}$ , and $u_{θ}$ are the velocities in the $r$ , $y$ , and $θ$ directions, respectively; and $Δ P_{r}$ is the pressure drop. The velocity of the fluid in the MRF zone is expressed by the continuity (5) and motion equations (6), under laminar flow and no inertial force.

2 \int_{0}^{h} 2 π rudy = π R_{ct}^{2} \cdot V_{i}

(5)

- \frac{\partial P_{r}}{\partial r} + \frac{\partial τ_{ry}}{\partial y} = 0

(6)

where $τ_{ry}$ is the shear stress on the $roy$ plane, and $\partial P_{r} / \partial r$ is the pressure gradient of the fluid in the MRF zone. In our previous study (Li et al., 2018), the dynamic behavior of an MREA based on the Power-Law constitutive model was experimentally verified. The equations describing this dynamic behavior were used as the MRF constitutive equations in this study.

τ_{ry} = m (H) {\overset{\cdot}{γ}}^{n (H)}

(7)

where $\overset{\cdot}{γ}$ is the shear rate, and $K (H)$ and $n (H)$ are the rheological characteristic parameters varying with the magnetic field intensity $H$ . When $n (H) < 1$ , the viscosity decreases with the increase in $\overset{\cdot}{γ}$ , which describes the phenomenon of shear thinning.

The boundary conditions adopted as follows:

{\begin{matrix} u_{r 1} |_{y = \pm h} = 0 (8 a) \\ u_{r} |_{y = \pm (h - δ)} = u_{s} (8 b) \\ τ_{ry} (y) |_{y = 0} = 0 (8 c) \end{matrix}

(8)

where $u_{r 1}$ is the velocity of the apparent slip layer (HVLP).

With the boundary condition of equation (8c), equation (6) is integrated with respect to $y$ , obtaining,

τ_{ry} = \frac{\partial P_{r}}{\partial r} y

(9)

By substituting equation (7) into equation (9), when $0 \leq y \leq h - δ$ and $- (h - δ) \leq y ≺ 0$ , the radial flow velocity $u_{r}$ can be expressed as

u_{r} = \frac{n}{(n + 1)} \cdot {(- \frac{1}{m} \cdot \frac{\partial P_{r}}{\partial r})}^{\frac{1}{n}} \cdot y^{\frac{n + 1}{n}} + C_{1}

(10)

where $C_{1}$ is the undetermined coefficient.

In Figure 2, the apparent slip layer is separated from the carrier fluid HVLP in the MRF. According to previous results (Xie et al., 2016), with the boundary condition of equation (8a), the boundary apparent slip velocity $u_{s}$ can be expressed as

u_{s} = \frac{1}{4 K} \cdot (- \frac{\partial P_{r δ}}{\partial r}) \cdot (h^{2} - y^{2})

(11)

where $\partial P_{r δ} / \partial r$ is the pressure gradient of the fluid in the apparent slip layer zone.

Substituting the boundary velocity continuity condition of equation (8b) into equation (11), equation (10) can be expressed as

\begin{array}{l} u_{r} = \frac{n}{(n + 1)} \cdot {(- \frac{1}{m} \cdot \frac{\partial P_{r}}{\partial r})}^{\frac{1}{n}} \cdot [{(h - δ)}^{\frac{n + 1}{n}} - y^{\frac{n + 1}{n}}] \\ + \frac{δ (2 h - δ)}{4 K} \cdot (- \frac{\partial P_{r δ}}{\partial r}) \end{array}

(12)

The volume flux through cross-section $Q_{ra}$ is given by the continuity equation

\begin{array}{l} Q_{r a} = π R_{ct}^{2} \cdot V_{i} = 2 \int_{h - δ}^{h} 2 π r u_{s} d y + 2 \int_{0}^{h - δ} 2 π r u_{r} d y \\ = \frac{2 π r}{3 K} \cdot (- \frac{\partial P_{r δ}}{\partial r}) [h^{3} - {(h - δ)}^{3}] + \frac{4 π r n {(h - δ)}^{\frac{2 n + 1}{n}}}{(2 n + 1)} \cdot \\ {(- \frac{1}{m} \cdot \frac{\partial P_{r}}{\partial r})}^{\frac{1}{n}} \end{array}

(13)

and

2 \int_{h - δ}^{h} 2 π r u_{s} dy = π δ (2 h - δ) \cdot V_{i}

(14)

By substituting equation (14) into equation (13), and integrating equation (13) over $r$ , the pressure drop $Δ P_{ra}$ resulting from the flow throttling in the radial flow channels can be expressed as

\begin{matrix} Δ P_{ra} = - \int_{R_{ax}}^{R_{ra}} (\frac{\partial P_{r}}{\partial r} + \frac{\partial P_{r δ}}{\partial r}) dr \\ = \frac{{mV}_{i}^{n} (R_{ra}^{1 - n} - R_{ax}^{1 - n})}{(1 - n) {(h - δ)}^{2 n + 1}} \cdot {(1 + \frac{1}{2 n})}^{n} \cdot \\ {[\frac{R_{ct}^{2}}{2} - \frac{(2 h - δ) (3 h^{2} - 3 h δ + δ^{2})}{3 h - δ}]}^{n} \\ + \frac{3 K (2 h - δ)}{(3 h δ - δ^{2})} \cdot V_{i} \cdot \ln \frac{R_{ra}}{R_{ax}} \end{matrix}

(15)

Therefore, the damping force $F_{MR}$ resulting from the yield stress and apparent slip effect of the MRF in the radial channel can be rewritten as

\begin{array}{l} F_{M R} = \frac{6 π K (2 h - δ) R_{ct}^{2}}{δ (3 h - δ)} \cdot V_{i} \cdot l n \frac{R_{r a}}{R_{ax}} \\ + \frac{2 π m R_{ct}^{2} V_{i}^{n} (R_{r a}^{1 - n} - R_{ax}^{1 - n})}{(1 - n) {(h - δ)}^{2 n + 1}} \cdot {(1 + \frac{1}{2 n})}^{n} \cdot \\ {[\frac{R_{ct}^{2}}{2} - \frac{(2 h - δ) (3 h^{2} - 3 h δ + δ^{2})}{3 h - δ}]}^{n} \end{array}

(16)

3.2. MRF apparent slip flow dynamics in uncontrolled channel

The viscous characteristics of the non-magnetic fluid in the annular flow channel are the main reason for the flow resistance and pressure drops when the HVLP MRFs are extruded. As the gap is far smaller than the radius of the inner plate, the one-dimensional axisymmetric annular channel geometry can be approximated by a rectangular channel. An axial symmetry analysis of the shear mode of this channel is shown in Figure 3(a) and (b).

By describing a HVLP MRF with the Power-Law $τ = η {\overset{\cdot}{γ}}^{α}$ , the axial flow velocity $u_{an}$ of the fluid in the non-magnetic flow channel was obtained with the boundary conditions $u_{an} = u_{s}$ at $r = b / 2$ as

\begin{matrix} u_{an} = \frac{α}{(α + 1)} \cdot {(- \frac{1}{η} \cdot \frac{\partial P_{x}}{\partial x})}^{\frac{1}{α}} \cdot [{(b / 2 - δ)}^{\frac{α + 1}{α}} - r^{\frac{α + 1}{α}}] \\ + \frac{δ (b - δ)}{4 K} \cdot (- \frac{\partial P_{x δ}}{\partial x}) \end{matrix}

(17)

where $η$ and $α$ are the field-off parameters of the HVLP MRF, and $b = R_{an} - R_{ra}$ is the gap width of the annular flow channel.

The volume flux $Q_{an}$ and the pressure drop $Δ P_{an}$ are given by

\begin{matrix} Q_{an} = \frac{2 π}{3 K} \cdot (- \frac{\partial P_{x δ}}{\partial x}) \cdot [{(b / 2)}^{3} - {(b / 2 - δ)}^{3}] \cdot (R_{an} - b / 2) \\ + \frac{4 π α}{(2 α + 1)} \cdot {(b / 2 - δ)}^{\frac{2 α + 1}{α}} \cdot {(- \frac{1}{η} \cdot \frac{\partial P_{x}}{\partial x})}^{\frac{1}{α}} \cdot (R_{an} - b / 2) \end{matrix}

(18)

\begin{matrix} Δ P_{an} = \frac{2 η V_{i}^{α} L_{an}}{{(b - 2 δ)}^{2 α + 1}} \cdot {(2 + \frac{1}{α})}^{α} \cdot {[R_{ct}^{2} - \frac{(2 b - δ) (3 b^{2} - 6 b δ + 4 δ^{2})}{3 b - 2 δ}]}^{α} \\ + \frac{3 K (2 h - δ)}{(3 h δ - δ^{2})} \cdot V_{i} \cdot L_{an} \end{matrix}

(19)

Therefore, the damping force $F_{an}$ resulting from the yield stress and apparent slip effect of the MRF in an annular flow channel can be rewritten as follows:

\begin{matrix} F_{an} = \frac{4 π η R_{ct}^{2} V_{i}^{α} L_{an}}{{(b - 2 δ)}^{2 α + 1}} \cdot {(2 + \frac{1}{α})}^{α} \cdot {[R_{ct}^{2} - \frac{(2 b - δ) (3 b^{2} - 6 b δ + 4 δ^{2})}{3 b - 2 δ}]}^{α} \\ + \frac{6 π {KR}_{ct}^{2} (2 h - δ)}{δ (3 h - δ)} \cdot V_{i} \cdot L_{an} \end{matrix}

(20)

3.3. MRF apparent slip flow dynamics in a piston cylinder

In the MREA, the flow of the HVLP MRF in a corrugated tube was simplified to that in a cylinder to analyze volume flux (Figure 1). The flow throttling in the simplified piston barrel was analyzed with the same method as that for the annular channel. The volume flux through the piston barrel cross-section $Q_{cy}$ and the pressure drop $Δ P_{cy}$ were derived as follows:

\begin{matrix} Q_{cy} = \frac{π δ (R_{ct} - δ) \cdot (1 + δ) \cdot (2 R_{ct} - δ)}{K} \cdot (- \frac{\partial P_{x δ}}{\partial x}) \\ + \frac{π α}{(3 α + 1)} \cdot {(- \frac{1}{η} \cdot \frac{\partial P_{x}}{\partial x})}^{\frac{1}{α}} \cdot {(R_{ct} - δ)}^{\frac{3 α + 1}{α}} \end{matrix}

(21)

\begin{matrix} Δ P_{cy} = \frac{2 η V_{i}^{α} L_{ct}}{{(R_{ct} - δ)}^{3 α + 1}} \cdot {(3 + \frac{1}{α})}^{α} \cdot {[R_{ct}^{2} - 8 (R_{ct} - δ) (1 + δ)]}^{α} \\ + \frac{8 K}{δ (2 R_{ct} - δ)} \cdot V_{i} \cdot L_{ct} \end{matrix}

(22)

Therefore, the damping force $F_{cy}$ resulting from the yield stress and apparent slip effect of the MRF in the cylinder can be rewritten as follows:

\begin{matrix} F_{cy} = \frac{4 π η L_{ct} R_{ct}^{2} V_{i}^{α}}{{(R_{ct} - δ)}^{3 α + 1}} \cdot {(3 + \frac{1}{α})}^{α} \cdot {[R_{ct}^{2} - 8 (R_{ct} - δ) (1 + δ)]}^{α} \\ + \frac{16 π {KR}_{ct}^{2}}{δ (2 R_{ct} - δ)} \cdot V_{i} \cdot L_{ct} \end{matrix}

(23)

The pressure reductions caused by the HVLP MRF flowing in the channels are presented in equations (4), (16), (20), and (23). For the final formulation of damping force, equation (3) was rewritten as follows:

\begin{matrix} F_{d} = \frac{2 π R_{ct}^{2} V_{i}^{n} m (R_{ra}^{1 - n} - R_{ax}^{1 - n})}{(1 - n) {(h - δ)}^{2 n + 1}} \cdot {(1 + \frac{1}{2 n})}^{n} \cdot \\ {[\frac{R_{ct}^{2}}{2} - \frac{(2 h - δ) (3 h^{2} - 3 h δ + δ^{2})}{3 h - δ}]}^{n} + \\ \frac{4 π R_{ct}^{2} V_{i}^{α} η L_{an}}{{(b - 2 δ)}^{2 α + 1}} \cdot {(2 + \frac{1}{α})}^{α} \cdot {[R_{ct}^{2} - \frac{(2 b - δ) (3 b^{2} - 6 b δ + 4 δ^{2})}{3 b - 2 δ}]}^{α} \\ + 2 π R_{ct}^{2} V_{i} K \cdot [\frac{3 (2 h - δ)}{δ (3 h - δ)} \cdot (\ln \frac{R_{ra}}{R_{ax}} + L_{an}) + \frac{8 L_{ct}}{δ (2 R_{ct} - δ)} + \frac{4 L_{ax}}{R_{ax}^{2}}] \\ + \frac{4 π L_{ct} R_{ct}^{2} V_{i}^{α} η}{{(R_{ct} - δ)}^{3 α + 1}} \cdot {(3 + \frac{1}{α})}^{α} \cdot {[R_{ct}^{2} - 8 (R_{ct} - δ) (1 + δ)]}^{α} \end{matrix}

(24)

4. Effect of apparent slip on the MREA

As mentioned above, the magnetic particles rearrange in the magnetic field, which results in the increase of the equivalent particle size and the thickness of the apparent slip layer $δ$ . This trend then affects the flow characteristics of the MRF in the channel. To identify the effect of the thickness $δ$ on the characteristics of the flow field and damping force, we conducted analytical calculations using MATLAB by setting the geometrical design parameters under different impact conditions.

4.1. Effect of apparent slip on the velocity of the flow field in an MREA

To evaluate the effects of the apparent slip layer $δ$ on the flow characteristics in radial channels, we assumed that the gap between the inner and upper plates $2 h = 2$ mm and impact velocity acting on the upper end cup was 2.8 m s⁻¹, for analyzing the radial velocity distribution. As depicted in Figure 4(a) and (b), the apparent slip layer greatly influences the field-dependent radial velocity, whereas the apparent slip layer has a negligible influence (i.e., the boundary velocity variation is less than 1 m s⁻¹, as shown in the inset of Figure 4(a)) on the unexcited current radial velocity. Figure 4(b) illustrates the change in the boundary radial velocity with the apparent slip layer, decreasing by 14.13, 10.56, and 6.49 m s⁻¹ as the slip-layer thickness increases from 0 to 2 µm, 2 to 4 µm, and beyond, respectively. It is noted that a maximum radial velocity change up to 14.13 m s⁻¹ can be obtained compared to other two intervals. Further, the peak radial velocity exhibited the opposite trend, changing by only 12.29 m s⁻¹ in this range. This phenomenon reveals that the model under the apparent slip boundary condition is effective in describing if the slip-layer thickness is set in this range.

Further, to reveal the comprehensive effect of slip layers on the radial flow velocity, two types of radial velocity profiles with slip layers with thicknesses of 0 and 4 µm were proposed for excitation currents of 0, 1, 2, and 3⁻A (Figure 5(a) and (b)). The parameter of velocity ratio $ω_{v}$ is defined as the ratio of the radial flow velocities of various currents to the total change to describe the effect of slip layers on the velocity (Figure 5(a)). Judging from Figures 5(a) and (b), both $ω_{v}$ and boundary velocity follow the same trend. The variations can be divided into two stages (i.e., a distinct cut-off point at the excitation current of 1 A) with an increase in the excitation current. As illustrated in Figure 5(a), the parameter $ω_{v}$ increases first and then decreases with the increase in the current. The specific performance is the ratio of the radial flow velocities without considering the apparent slip, whereas the ratio of flow velocities with different slip-layer thicknesses tends to be consistent with the increase in the current when the latter increases from 1 to 3 A.

Especially, the radial flow velocity ratio only changes by 6.9% without apparent slip, while accounting for 58.5, 76.1, and 83.5% at slip-layer thicknesses of 2, 4, and 6 µm, respectively. The differences between the maximum values of the flow velocity ratio were 32.8, 13.4, and 6.1%. This indicates that, for the considered slip-layer thicknesses, the ratio and thickness variation in Figure 5(a) are consistent with each other. The influence of the slip-layer thickness on the boundary velocity is illustrated in the partially highlighted view of Figure 5(b). In the figure, both $ω_{v}$ and boundary velocity variations change similarly in the same 2–6-µm range of slip-layer thickness. Therefore, the same influencing maximum boundary thickness of 2 µm can be obtained, because describing the radial velocity ratio $ω_{v}$ for the analyzed case is a relevant equation by equation (12). Therefore, it can be predicted that the effect of apparent slip on the flow velocity of magnetorheological cements in a radial channel is coupled with the magnetic field strength, and its influence degree is related to the slip layer thickness and the magnetic field strength and has the maximum value.

4.2. Effect of apparent slip on the dynamic behavior of MREA

It is well known that both peak force and dynamic range are important parameters to characterize the performance of MREA. However, few theories could be found for evaluating the effect of the apparent slip layer $δ$ on the dynamic behavior of the absorber. To solve this problem, two relative parameters, peak force ratio ( $σ_{FS}$ ) and dynamic range ratio ( $σ_{DS}$ ), were proposed to describe the influence of slip-layer thickness on the dynamic behavior. $σ_{FS}$ and $σ_{DS}$ were defined as the ratios of peak force and dynamic range of an MREA considering the apparent slip layer with varying thickness to that without a slip layer, respectively.

Figure 6 illustrates the numerical results for the peak force at an impact velocity of 2.8 m s⁻¹ (@ 3 A). As observed in Figure 6(a), $σ_{FS}$ remains mostly unchanged as the impact velocity increases from 2.8 to 4.6 m s⁻¹, while gradually increasing as the apparent slip-layer thickness continues to increase. This trend gradually plateaus. Figure 6(b) illustrates the same trend, that is, under the condition of an applied field, the influence of slip-layer thickness on $σ_{FS}$ is also gradual, whereas the increasing current has a significant effect. When the thickness was set to 2 µm, the relative change rate of peak force ratio reaches a maximum, which was approximately 79.6%. However, the influence of slip-layer thickness on the peak force was less than 0.3%.

The dynamic range is a key consideration for the performance of an MREA as it can be directly controlled. Therefore, we compared the dynamic range from the MREAs with slip layers of varying thicknesses to that without a slip layer (Figure 7). A distinct separation was observed between the dynamic range lines for different thicknesses, and that for the zero thickness case (without a slip layer). And all the apparent slip dynamic range line are close to coincidence. It is because that the gathered particles glowed apparent slip under the action of magnetic field results lower change of the peak force. The dynamic range will reach the threshold when the range of the slip-layer thickness is limited. Moreover, the slip-layer thickness has a weak effect on the dynamic range; it grows from 1.763 without a slip layer to a maximum of 1.812 at the impact speed 0.8 m s⁻¹, which increases slightly to 0.21% when the impact speed is 4.6 m s⁻¹. When the layer becomes thicker (2–6 µm), the relative change in the dynamic range decreases. Based on previous analysis, we conclude that the optimal thickness of the apparent slip layer is 2 µm, which can lead to the optimal damping force behavior.

5. Experimental study

5.1 MREA and high-speed drop test platform

To experimentally evaluate the effectiveness of the theoretical model, a drop tower test system was constructed. The system consisted of a drop mass, three piezoelectric force sensors (KD3050), a DC power supply (HSPY-200-05), a laser displacement sensor (IL300), an electronic charge amplifier (KD5007G), a data acquisition device (USB-7646B), a computer, an MREA, and a pedestal (Figure 8(a1) and (a2)). The MREA was axially fixed to the support pedestal through a connecting plate and then integrally fastened to the base of the drop tower. The piezoelectric force sensor and electronic charge amplifier were used to measure the MREA force. The power supply was used to control the excitation current transferred to the MREA. The displacement sensor was bolted to the drop tower base to measure the reflection bar displacement, which is equal to the MREA damping stroke. During the test, displacement and damping force data were transmitted to the computer via the data acquisition device.

In the impact test, the constant drop mass was lifted to a height corresponding to the designed impact velocity by a winch rope, and then released to fall freely under the influence of gravity. The impact force and velocity were then applied to the upper end cap of the proposed MREA. Thus, the corrugated pipe was fully compressed, and the HVLP MRF was pushed through the oil storage to the radial flow passage. During the impact, the MREA force was measured by the three piezoelectric force sensors uniformly distributed between the drop tower base and the pedestal. The crush displacement of the corrugated pipe was measured by the laser displacement sensor. The pipe displacement was differentiated once to obtain the velocity and subsequently again to measure the acceleration. The impact velocity was adjusted by varying the lifting height of the mass, and a new corrugated tube was used for each drop test. Thus, a series of tests was conducted to determine the dynamic behavior of the MREA at crush speeds of 2.8, 3.7, and 4.2 m s⁻¹, while maintaining the applied current at a constant level of 0 (off-state) to 3 A (on-state) in increments of 1 A, respectively.

5.2 Analysis of drop test results

Figure 9 summarizes the experimental results. Figure 9(a) to (c) show the time histories of the MREA force at impact velocities of 2.8, 3.7, and 4.2 m s⁻¹ under different excitation field conditions. The MREA force significantly increased both with impact velocity and excitation current. Additionally, the maximum damping force and dynamic range of the MREA were 151.44 kN (at 3.7 m s⁻¹) and 1.34 (at 2.8 m s⁻¹), which proved the controllability of the MREA. As depicted in Figure 9(a), the time taken by the MREA to reach the maximum damping force reduced by 36.8% from 8.7 ms in the on-state to 5.5 ms in the off-state. In contrast to the MREA peak force time, the total impact time history decreased marginally as the applied current was increased (Figure 9(b) and (c)). In addition, the damping forces became consistent in the subsequent stage of the impact process with the increase in the impact velocity. This consistency occupies an increasing proportion of the total impact time history. Furthermore, the MREA force curves in the time histories changed slightly (see the surge point in Figure 9(a)), as shown in Figures 9 (a) to (c). The cause of this phenomenon was the small rebound collision caused by the unbalanced collision energy and filling compactness of magnetorheological cements being related to the impact load of the MREA (corresponding to the MREA damping force in the initial impact stage). The coupling effect of the cement flow and corrugated crushable parts play important roles in the subsequent stage.

5.3 Verification for theoretical model

Figure 10 presents the comparison of force as a function of displacement for MREA between the predicted (taking the slip-layer thickness as 0 (no-consideration apparent slip), 2, 4, and 6 µm, respectively) and experimental results at the impact velocity of 2.8 m s ¹ and applied currents of 0 to 3 A, respectively. As observed in these figures, the experimental curves match well with the theoretical ones. However, we measured a significantly different peak force between the model with ( $δ > 0$ ) and without apparent slip ( $δ = 0$ ). Therefore, the theoretical data of the peak force are greater than the experimental ones when $δ > 0$ , while the peak force is less than that when $δ = 0$ . In other words, the slip layer thickness influences the dynamic behavior of the MREA. This is because the width of the flow channel is narrowed when the apparent slip-layer thickness is considered, while being inversely proportional to the pressure difference generated by the radial flow from equation (10). Further, Figure 10 reveals that the maximum peak force errors of the MREA between experimental and theoretical values (i.e., two models of $δ > 0$ and $δ = 0$ ) are 1.34 and 5.94% at 0 A, 1.95 and 3.95% at 1 A, 1.87 and 3.69% at 2 A, and 1.51 and 5.17% at 3 A, respectively. Furthermore, compared with Figure 10(a) to (d), the force versus displacement curves of the MREA show that the predicted results based on 2-µm slip-layer model concurred with the measured data.

To verify the effectiveness of the model of the proposed MREA with the apparent slip boundary condition, the MREA damping characteristics with 0- and 2-µm apparent layer thicknesses of the two models were compared separately under magnetic field dependency and impact velocity (Figure 11(a) and (b)). As observed in these figures, the experimental curves are better matched with the theoretical curves with 2-µm thickness than the one with 0-µm thickness. Figure 11(a) illustrates the predicted and measured peak forces of the MREA considering thicknesses of 0 and 2 µm under an impact velocity of 2.8 m s⁻¹. Apparently, we observed an insignificant distinction of the peak force when the thickness was 2 µm as the applied current was increased. In contrast to the case without the slip layer (theoretical thickness of 0 µm), the peak force errors of the predicated and measured models decreased and then increased. In addition, we observed that the difference in the MREA peak force between the experimental data and the two thickness models grew significantly as the impact velocity increased (Figure 11(b)). We concluded that the apparent slip performance has little effect on the MREA dynamic characteristics at low impact velocities. The reasoning behind this conclusion is that the increasing velocity of magnetorheological cements in the radial flow channel results in aggravated apparent slip as the impact velocity increases.

To estimate the prediction accuracy of the radial flow model more accurately with the apparent slip boundary condition, the two models with slip-layer thicknesses of 0 and 2 µm must be quantitatively evaluated. Therefore, we employed the relative error of peak force to quantitatively evaluate the accuracy of the models and predict the MREA peak force (Figure 12). We observed that the relative errors for the 2-µm-thick slip layer were only 1.16–3.73% (at 0 A) and 1.05–2.95% (at 3 A), while those for the 0-µm thickness were 5.94–7.41% (at 0 A) and 5.48–6.57% (at 3 A), respectively, when the impact velocity increased from 2.8 to 4.2 m s⁻¹. With or without the apparent slip effect, the errors between the theoretical model and the test results were not greater than 8%. This is consistent with the conclusion drawn in Section 3 that the slip-layer thickness had little effect on the damping force. However, the curves considering the apparent slip model were closer to the experimental results. Therefore, the apparent slip model can better describe the damping behavior of the proposed MREA.

As a key consideration in the MREA performance, the dynamic range directly demonstrates the controllability of an MREA system. Therefore, the dynamic ranges from the two models (0- and 2 µm-thick slip layers) and experimental data were compared (Figure 13). The dynamic range of the 2-µm-thick slip-layer model was more consistent with the experimental data than the 0-µm model. In addition, it becomes gradually closer to the experimental data as the impact velocity is increased as well as the significantly decreasing in this process. This is because the apparent slip layer effect on the uncontrollable damping force (viscous damping force) greatly decreases as the impact velocity increases, while that on the controllable force (Coulomb damping force) determined by the applied current remains unchanged. This means that a more accurate peak force and dynamic damping range can be obtained simultaneously, while requiring different apparent slip layer thicknesses and current input levels associated with those thicknesses under impact field-dependent conditions. The dynamic damping range vs. impact velocity curves of the MREA show that the predicted results of the model with the 2-µm-thick apparent slip layer concurred with the measured data.

6. Conclusions

In our previous work, we studied the radial Power-Law flow dynamics of an HVLP MRF to predict the MREA performance under impact conditions. The effectiveness of this model was verified through high-speed impact tests. However, a crude dynamic model for the suspension boundary layer at the wall increased the risk of failure of the prediction model, and thus degrade the buffer effect. In this study, the radial Power-Law flow model with apparent slip was proposed to improve the theoretical model of the dynamic characteristics of an MREA under impaction. According to the design of the MREA, the Power-Law flow model with damping forces from the apparent slip was explored in three situations with a controlled channel damping force, uncontrolled channel damping force, and piston cylinder force of a simplified corrugated tube. Based on this, the influence of thickness of the apparent slip layer on the radial flow velocity, peak force ratio, and dynamic range of the MREA was simulated on MATLAB.

To evaluate the model, the model and experiment results were qualitatively and quantitatively compared for the global agreement of dynamic force curves, and the accuracy of the theoretical model to predict the MREA peak force and dynamic range. The results show that the peak force relative errors of the two models at $δ > 0$ and $δ = 0$ were in the ranges of 1.34–5.94%, 1.95–3.95%, 1.87–3.69%, and 1.51–5.17%, when the apparent current was increased from 0 to 3 A and the impact velocity was 2.8 m s⁻¹. In addition, absolute values of the relative errors of the peak force in the two models with the 2- and 0-µm-thick slip layers were less than 3.73% (at 2.8 m s⁻¹ and 0 A) and 7.41% (at 2.8 m s⁻¹ and 0 A), respectively. Overall, the 2-µm-thick apparent-slip-layer model predicts the performance of the MREA with higher accuracy than the no-consideration apparent-slip-layer model. Thus, the radial Power-Law flow with the apparent slip boundary condition model can effectively predict the dynamic characteristic and performance of the HVLP-MRF-based MREA under impact conditions.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by the National Natural Science Foundation of China (No. 51905062), by the Science and Technology Research Program of the Chongqing Municipal Education Commission (Grant No. KJQN202000801), by the Scientific Research Project of Chongqing Technology and Business University (No. 1952021), and by the Opening Project of Scientific Research Platform of Chongqing Technology and Business University (No. KFJJ2019077). We also thank the Scientific Research Foundation of Chongqing Technology and Business University for their support (No. 1956021).

ORCID iDs

Zhuqiang Li

Changrong Liao

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