Experimental validation of a magnetorheological energy absorber design analysis

Abstract

A key challenge when designing linear stroke magnetorheological energy absorbers for high-speed impact is that high piston speeds in linear stroke magnetorheological energy absorbers induce high Reynolds number flows in the magnetic valve of the magnetorheological energy absorber, so that achieving high controllable dynamic range can be a design challenge. So far, the research on magnetorheological energy absorbers has typically assumed that the off-state force increases linearly with piston velocity. But at the higher piston velocities occurring in impact events, the off-state damping exhibits nonlinear velocity squared damping effects. This problem was recognized in our prior work, where it was shown that minor losses are important contributing factors to off-state damping. In this study, a nonlinear analytical magnetorheological energy absorber model is developed based on a Bingham-plastic nonlinear flow model combined with velocity squared dependent minor loss factors. This refined model is denoted as the Bingham-plastic nonlinear flow model with minor losses. From this Bingham-plastic nonlinear flow model with minor losses, an effective design strategy is presented for conventional magnetorheological energy absorbers. The Bingham-plastic nonlinear flow model with minor losses is validated via computational fluid dynamics simulation, so that magnetorheological energy absorber performance can be analytically verified before being manufactured. The magnetorheological energy absorber is fabricated and tested up to an effective piston velocity of 5 m/s by using the high-speed drop tower facility at the GM R&D Center. Comparison of our analysis with measured data is conducted, and the effective design of the magnetorheological energy absorber using the Bingham-plastic nonlinear flow model with minor losses is validated.

Keywords

Magnetorheological impact energy absorber Bingham plastic minor losses drop tests

Introduction

Recently, magnetorheological energy absorbers (MREAs) have been proposed as stroking elements because they possess adaptive force capability needed for systems that require control of impact or shock loads for varying payload masses (Hiemenz et al., 2007; Mao et al., 2005, 2007a, 2007b). MREAs can provide adaptive vibration and shock mitigation capabilities to varying payloads, vibration spectra, and shock pulses for varying duration and/or magnitude (Choi and Wereley, 2008; Wereley et al., 2011), as well as other environment factors in a rapid manner (typically 5–10 ms) during the impact event. Typically, the stroking load is a function that increases as a function of payload mass for a fixed available stroke (Wereley et al., 2011). The smallest payload can accommodate the smallest stroking load, so that the off-state stroking force is specified to meet the stroking load threshold of this smallest payload. In contrast, the largest payload can accommodate the largest stroking load, so that the maximum force is tuned to meet the stroking load of this largest payload. MREAs provide a controllable dynamic range, defined as the ratio of MREA stroking force at maximum field to its off-state (zero-field) force, and can meet a specified minimum off-state force level over a relatively high speed range.

To maintain a high dynamic range, a relatively large valve diameter is needed to maintain the Reynolds number, Re, sufficiently low. A large valve diameter requires a large magnetic circuit, so that the MREA may also have a large diameter. However, practical situations impose constraints on the MREA diameter. This trade-off, as well as design variables such as high shaft speed, high dynamic range or tunability, and maximum passive or off-state force levels at high shaft speed, is likely to induce high Reynolds number flows in magnetorheological (MR) valves.

In MREAs subject to impact loads, the Reynolds number is typically much higher than for devices intended for vibration mitigation or isolation applications. The induced high Reynolds number flows in the MR valve can cause significant degradation of dynamic range, as well as undesired high off-state force levels. Ahmadian and Norris (2004) examined the performance of a double-ended MREA subject to impact velocities of up to 6.6 m/s. They achieved a dynamic range of D≈2.75 at 2.2 m/s (86 in/s). However, at a speed of 6.6 m/s (260 in/s), the dynamic range reduced nominally to D≈1. They hypothesized that the transition from controllable (D > 1) to uncontrollable behavior (D≈1) is related to the transition of fluid flow from laminar to turbulent, which is supported by the analysis of Mao et al. (2005). Browne et al. (2009) conducted impact tests of an MREA at stroking velocities ranging from 1.0 to 10 m/s and showed that the MR damper force could be tuned by adjusting the magnetic field. However, a similar trend was found in these two studies—the dynamic ranges of the MR dampers were significantly reduced, and the increase in field-off forces as a function of velocity, which was not linear as is typically assumed, was quadratic, suggesting that nonlinear velocity squared damping effects play a key role. From these two pilot studies, valuable insight on MREA behavior under impact loadings was gained, even though models or qualitative analyses were not addressed to explain this MREA force versus velocity behavior. Later in our own study (Mao et al., 2008), a bifold MREA was designed based on the Bingham-plastic nonlinear flow model (denoted as the BP nonlinear flow model) for nominal piston velocities of up to 6.75 m/s and was drop tested up to nominally 6 m/s. The results showed that a dynamic range of D = 2 could be achieved by restricting the Reynolds number to be below 850 over the tested speed range. However, the measured field-off forces for piston velocities above 2 m/s were much higher than the predicted force levels. It was recognized that minor loss factors or velocity squared effects (White, 1986) played an important role in predicting force levels at higher speeds. This was verified by the substantial agreement of experimental data with modeling results upon adding the minor loss components in the off-state force calculation. Thus, it was concluded that minor losses should be added in the BP nonlinear flow model and associated design strategy for MREAs under high-speed impact conditions.

Even though pressure drop due to minor losses is well known in fluid piping systems (White, 1986), it has been neglected in most MR fluid-based device modeling, analysis, and design in the literature reported so far. To some extent, such an approach to MR damper models and analysis is reasonable because, as shown in Figure 1, most prior work (Dyke et al., 1996; Gavin et al., 1996; Hong et al., 2005; Peel et al., 1996; Wang and Liao, 2009a, 2009b; Wereley, 2008; Wereley and Pang, 1998; Yang et al., 2004) focused on relatively low speed vibration isolation problems, where piston speeds were typically below 1 m/s (resulting in low Re laminar flows), and minor losses can be neglected to simplify design and analysis. A review of many successful modeling techniques that have been used in this context is presented in an excellent review article by Wang and Liao (2011). It is worth noting that, in some studies (Dogruer et al., 2008), minor loss factors were included for calculating the total pressure drop during their design analysis. In addition, it was shown that the modeling results agreed well with the experimental data up to a piston velocity of 0.15 m/s for harmonic excitations. This study showed that the BP linear flow model can accurately handle most MR device modeling issues without consideration of minor losses in low speed (low Re) situations. Few studies to date in the MR literature have correlated minor losses to off-state forces, as well as reported reductions in dynamic range as impact speed increases. A key reason is that application of MREAs to high-speed impact situations has only recently emerged.

Nevertheless, pilot studies on MREAs under impact conditions have so far been conducted on the basis of BP linear flow model (Dyke et al., 1996; Gavin et al., 1996; Peel et al., 1996; Wereley and Pang, 1998; Yang et al., 2004) and BP nonlinear flow model (Mao et al., 2005). In addition, it was shown that such BP linear and nonlinear flow models are not adequate to describe MREA behavior under high-speed impact loadings. Therefore, in this study, a nonlinear analytical MREA model, taking into account the effects of minor loss factors (called the Bingham-plastic nonlinear flow model with minor losses (BPM) model), is developed based on the BP nonlinear flow model. Using this BPM model, an effective design strategy is proposed for MREAs, and candidate designs are developed. Before being manufactured, the MREA off-state performance is further examined using computational fluid dynamics (CFD) simulations using commercial software ANSYS with the FLOTRAN module. An MREA is then fabricated and tested up to an effective piston velocity of 5 m/s by using the high-speed drop tower facility at the GM R&D Center. It will be shown that the BPM model is capable (where the BP model is not) of predicting the MREA passive force versus velocity performance and provides an effective MREA design tool for the entire speed range of impact conditions in this study.

MREA (MR damper) models

BP nonlinear flow model

A schematic diagram of annular duct type MR valve in a typical flow-mode MREA (or interchangeably called MR damper in this study) is presented in Figure 2. According to the BP nonlinear flow model, which considers laminar and turbulent flows (Franzini and Finnemore, 1997; Mao et al., 2005; White, 1986), the damping force, F, of the MR damper is given by

F = Δ P_{η} A_{p} + (Δ P_{MR} A_{p} + F_{f}) sgn (V_{p})

(1)

where

Δ P_{η} = f \frac{ρ {LV}_{d}^{2}}{2 D_{h}}

(2)

and

Δ P_{MR} = \frac{4 L τ_{y}}{D_{h}}

(3)

Here, $Δ P_{η}$ and $Δ P_{MR}$ are the viscous pressure drop and the pressure drop due to MR yield stress, respectively. $F_{f}$ is the friction force due to sealing, $A_{p}$ is the effective piston area, and $V_{p}$ and $V_{d}$ are the piston velocity and the average fluid velocity in the MR gap, respectively. $ρ$ and $τ_{y}$ are the MR fluid density and yield stress, respectively. $L$ is the total active MR valve length (here $L = 3 L_{a}$ , and $L_{a}$ is the active MR valve length associated with one coil). Also, $f$ denotes the Darcy-friction factor and is a piecewise function of the Reynolds number, Re, as follows (Franzini and Finnemore, 1997; White, 1986)

f = \frac{96}{Re} if Re \leq 2000

(4)

f = (1 - α) \frac{96}{2000} + α \frac{1}{{1.8 \log_{10} [{(\frac{ε / D_{h}}{3.7})}^{1.11} + \frac{6.9}{4000}]}^{2}} if 2000 < Re \leq 4000

(5)

\frac{1}{f^{1 / 2}} \approx 1.8 \log_{10} [{(\frac{ε / D_{h}}{3.7})}^{1.11} + \frac{6.9}{Re}] if Re > 4000

(6)

where

α = \frac{Re - 2000}{4000 - 2000}

(7)

and the Reynolds number is defined by

Re = \frac{ρ V_{d} D_{h}}{η}

(8)

Note that here the critical Reynolds number is taken to be Re = 2000 to more favorably ensure a laminar flow range for Re < 2000 (Spurk and Aksel, 2008). Also, $ε$ is the average pipe wall roughness and was chosen to be 0.0016 mm, based on the wall roughness for drawn tubing given by White (1986). Here, $η$ denotes the viscosity of the MR fluid. The average fluid velocity, $V_{d}$ , in the MR gap is given by

V_{d} = \frac{A_{p} V_{p}}{A_{d}} = \bar{A} V_{p}

(9)

where $\bar{A} \equiv A_{p} / A_{d}$ denotes the area ratio between the effective piston area, $A_{p}$ , and the cross-sectional area of the MR gap, $A_{d}$ . $D_{h}$ is the hydraulic diameter given by White (1986)

D_{h} = 2 d

(10)

where $d$ is the effective MR valve gap width. The cross-sectional area of the MR gap, $A_{d}$ , can be computed from

A_{d} = π D_{b} d

(11)

where $D_{b}$ is the effective MR valve diameter (here $D_{b} = D_{in} - d$ , and $D_{in}$ is the inner diameter of the outer tube, as shown in Figure 2). Then the viscous pressure, $Δ P_{η}$ , for a laminar flow in equation (2) is reduced to the most familiar form in the literature as follows

Δ P_{η} = \frac{A_{p} 12 η L V_{p}}{b d^{3}}

(12)

where $b$ is the circumference of the MR valve and here $b = π D_{b}$ .

BPM model

In this section, minor losses will be incorporated into the BP model. For any pipe system, in addition to the Darcy-friction loss, there are additional so-called minor losses due to pipe entrance or exit flows, sudden/gradual contractions or expansions, bends, valves, fittings, and so on. These minor losses represent additional energy dissipation in the flow, usually caused by secondary flows such as flow separation, eddies, and wakes that are generated by changes in flow direction, cross-section, or other pipeline geometry.

Minor losses typically play a minor role in long pipelines and are often neglected. Minor losses play an important role in MREAs (Dixon, 1999), especially in the case of high piston velocity under impact loading, because flow passages in MREAs are relatively short and can be geometrically complicated. Because the flow patterns associated with minor losses are quite complex, the theory is semi-analytical. The minor losses are usually measured experimentally and correlated with the pipe-flow parameters. Factors affecting the value of minor loss include the exact geometry of the component in question, the Reynolds number, and proximity to other fittings. A number of minor loss coefficients for different fittings, components, and flow passages can be found in the literature (Idelchik, 1994; White, 1986).

Adding the pressure drop from minor losses into the BP nonlinear flow model yields the MREA force of the BPM model as follows

F = (Δ P_{η} + Δ P_{ml}) A_{p} + (Δ P_{MR} A_{p} + F_{f}) sgn (V_{p})

(13)

where $Δ P_{ml}$ is the pressure drop resulting from these minor losses given by

Δ P_{ml} = \frac{ρ V_{d}^{2}}{2} \sum_{i} K_{ml_i}

(14)

Here, $K_{ml_i}$ stands for the ith minor loss coefficient associated with the MR valve’s geometries. Note that the minor loss pressure drop is strongly dependent on the exact geometrical profile of the fluid circuit in the MREA.

MREA design using BPM model

MREA configuration and geometric fluid circuit

In this study, MREAs were designed using the BPM model. The schematic of the MREA is depicted in Figure 2. This MREA is double ended (a rod on either side of the piston), and a three-stage electromagnetic coil was placed inside the piston head, which moves together with the piston rod assembly. The two piston rods have the same diameter and protrude through the hydraulic cylinder caps on either side of the MREA. Because no change in volume is induced as the piston rod moves, an accumulator is not required.

The corresponding schematic of the geometric fluid circuit with minor losses regions is also presented in Figure 2. In Figure 2, the numbers from 1 to 9 denote flow regions, and the blue arrows indicate the flow directions. In addition to the Darcy-friction-type viscous force in MR valve segments 2, 4, 6, and 8, there are also following minor losses in this flow system, which are characterized by corresponding minor loss coefficients:

Entrance effect for flow from region 1 to 2;

Sudden expansion for flow from region 2 to 3, 4 to 5, and 6 to 7;

Sudden contraction for flow from region 3 to 4, 5 to 6, and 7 to 8;

Exit effect for flow from region 8 to 9;

Viscous Darcy-friction loss, in passages 3, 5, and 7.

To generalize the design method using the BPM model for an MREA with multiple stage (n-stage) coils, the design analysis is formulated for a single-stage coil and then extended to n-stage coils via the principle of superposition.

Design analysis and equations

As shown in Figure 2, for the fluid flow regions for the segment of one coil, when the fluid flows from region 4 to region 6, the total minor losses pressure drop has three components:

Sudden expansion for flow from region 4 to 5;

Sudden contraction for flow from region 5 to 6;

Viscous Darcy-friction loss in the annulus gap between the inner surface of the outer cylinder and the outer surface of the coil (region 5, referred to as the coil gap).

In Figure 2, $L_{c}$ is the length of the coil, and $d_{c}$ denotes the coil gap thickness. $D_{p}$ and $D_{r}$ are the piston diameter and rod diameter, respectively. Here, $D_{p} = D_{in} - 2 d$ .

The passive pressure drop, $Δ P_{off_1}$ , associated with one coil segment is given by

Δ P_{off_1} = Δ P_{η_1} + Δ P_{ml_1} + Δ P_{c_1}

(15)

where $Δ P_{η_1}$ is the passive viscous pressure drop along the MR gap, and $Δ P_{ml_1}$ and $Δ P_{c_1}$ are the minor loss pressure drops and passive viscous pressure drop induced by the coil gap, respectively. By correlating with the specific geometric dimensions shown in Figure 2, they are derived as

Δ P_{η_1} = f_{η} \frac{ρ L_{a} V_{d}^{2}}{4 d}

(16)

Δ P_{ml_1} = \frac{ρ V_{d}^{2}}{2} (K_{SC} + K_{SE})

(17)

Δ P_{c_1} = f_{c} \frac{ρ L_{c} V_{c}^{2}}{4 d_{c}}

(18)

Here $V_{c}$ is the average fluid velocity in the coil gap and can be computed from the following equation

V_{c} = \frac{A_{p} V_{p}}{A_{c}}

(19)

where $A_{c}$ is the cross-sectional area of the effective coil gap and A_p and A_c are related to the geometric parameters in Figure 2 as given below

A_{p} = \frac{π}{4} (D_{p}^{2} - D_{r}^{2})

(20)

A_{c} = \frac{π}{4} [D_{in}^{2} - {(D_{in} - 2 d_{c})}^{2}]

(21)

$K_{SE}$ and $K_{SC}$ are the sudden expansion and sudden contraction coefficients, respectively. They are determined using empirical formula (White, 1986)

K_{SE} = {(1 - \frac{A_{d}}{A_{c}})}^{2}

(22)

K_{SC} = 0.42 (1 - \frac{A_{d}}{A_{c}})

(23)

Note that alternative semi-empirical formulae for these two minor loss coefficients are available in Dixon (1999). Here, $f_{η}$ and $f_{c}$ are the Darcy-friction factors for the fluid flow in the MR gap and the coil gap, respectively.

For an MREA with n-stage coils, the field-off damper force, $F_{off}$ , in the BPM model can be given by

F_{off} = A_{p} [n Δ P_{off_1} + \frac{ρ V_{d}^{2}}{2} (K_{entry} + K_{exit})]

(24)

Here $K_{entry}$ and $K_{exit}$ are the sharp entrance and exit effects, respectively. In this analysis, $K_{entry}$ was chosen to be the typical constant value of 0.5 and $K_{exit} = 1$ (White, 1986).

Finally, the field-on damper force of the n-stage coils in the BPM model can be given by

F_{on} = F_{off} + F_{MR}

(25)

where $F_{MR}$ is the MR yield force given by

F_{MR} = A_{p} \frac{n 2 L_{a} τ_{y}}{d}

(26)

Proposed design strategy

In the design stage, usually the fluid properties $ρ, η, τ_{y}$ and the MREA performance requirements $F_{off}$ and $F_{MR}$ at the maximum interested piston velocity, $V_{p}$ , are specified. Then the unknown variables in equation (24) are $D_{in}, D_{r}, d, d_{c}, L_{a}, L_{c}$ , all of which are explicit when fully expanding the equation. Mathematically, the design solution is not unique. Thus, we transform the design problem to be an optimization problem that finds the solutions to make the following cost function, Z, with six unknown variables to be close to 0:

Z = \frac{F_{obj} - {\tilde{F}}_{off} (D_{in}, D_{r}, d, d_{c}, L_{a}, L_{c})}{F_{obj}}

(27)

where $F_{obj}$ is the specified design objective force for the off-state, and ${\tilde{F}}_{off}$ denotes the analytical off-state force estimated from equation (24). The feasible design solutions lie in the subspace satisfying Z≈0. However, the feasible design solutions are obtained under the condition that the required magnetic flux density, corresponding to the required yield stress, $τ_{y}$ , is achievable. Therefore, as a final step, the electromagnetic performance of the feasible design solution is checked using electromagnetic finite element analysis (FEA) to ensure that the required magnetic flux density can be supplied by the electromagnetic circuit design.

Since it is impractical to have an explicit closed form of analytical solution for the optimization problem of equation (24), we resort to numerical techniques. Various numerical techniques dedicated to optimization problems can be adopted to solve the problem. Here, we use a rather intuitive and straightforward way to solve the design problem by gradually reducing the number of unknown variables via logical means. To do so, we developed a strategy to determine design variables numerically. Unlike the strategy in our prior work (Mao et al., 2005), this strategy applies to MREA designs with minor loss factors due to velocity squared effects for both laminar and turbulent flows. The procedure that was used is given below

Specify the numerical values of $ρ, η$ , and desired $τ_{y}$ and design specifications of $F_{off}, F_{MR}, V_{p}$ .

Choose the proper sizes of $D_{in}$ and $D_{r}$ based on imposed practical MREA space constraints, material strength requirements, size charts of standard sealing parts involved, and so on. Unknown design variables have been reduced to four.

Choose the number of coil stages, n, with the help of required maximum MR yield force, $F_{MR}$ , and achievable $τ_{y}$ . Usually, the integer number $n$ can be chosen from 1 to 4, but a smaller number may be desirable to achieve a more compact size.

Choose the value of the MR valve gap, $d$ . Typically, the valve gap can be initially chosen in the range of 0.5–2 mm, but a smaller valve gap is desirable for achieving increased magnetic field strength. Larger valve gap is desirable when a lower field-off damper force is required. Thus far, the numerical value of $L_{a}$ can be determined from equation (26). Thus, only two design variables of $d_{c}$ and $L_{c}$ are unresolved.

Define $\bar{d} = d_{c} / d$ (or $\bar{L} = L_{c} / L_{a}$ ), list a reasonable range of $\bar{d}$ (or $\bar{L}$ ) as a vector, and then substitute $d_{c} = \bar{d} \cdot d$ (or $L_{c} = \bar{L} \cdot L_{a}$ ) into equation (24) and solve the equation to obtain corresponding value of $L_{c}$ (or $d_{c}$ ).

Calculate Z with previously obtained values of the six unknowns, the solutions are those satisfying $Z \approx 0$ . Pick one or more sets of the most reasonable solution pairs of $(d_{c}, L_{c})$ with the values of $(d, L_{a})$ to design the magnetic circuit and verify the magnetic flux density for the desired MR yield stress (usually FEA is used). If the solution pair meets the desired magnetic flux density, then the design is done. If not, repeat to choose remaining pair of $(d_{c}, L_{c})$ or repeat steps (4)–(6) or steps (3)–(6) or steps (2)–(6), as necessary.

For clarity, the design procedure is also summarized as a computational flowchart in Figure 3. Note that, although the proposed design strategy and related equations are tailored to the MREA shown in Figure 2, the principle of transforming the design problem using the BPM model into an optimization problem can be applied to the design of any MREA. If the MREA uses a different valve configuration, the designer must examine the geometric profile of the flow system and determine key minor loss components. System equations must be developed for the desired valve geometry, similarly to what was done here in the design analysis using the BPM model. Then, a similar design procedure can be adopted to solve the design parameters with the help of numerical computational software, such as MATLAB.

Implementation of the design strategy

An MREA with an annular valve was designed using the proposed design strategy. The design objectives of the MREA are (1) the field-off force, ${\hat{F}}_{off}$ , should be no more than nominally 15 kN and (2) the maximum field-on force, $F_{MR}$ , should be no less than nominally 21 kN at the peak piston velocity of 4.5 m/s.

Adjusting for the anticipated friction from seals, the design specification for the MREA was modified slightly: ${\hat{F}}_{off} = 14.9 kN$ and $F_{MR} = 6.2 kN$ at $V_{p} = 4.5 m / s$ . The MR fluid properties were $ρ = 3522 kg / m^{3}$ , η = 0.072 Pa s, and $τ_{y} = 60 kPa$ at a magnetic flux density of 0.7 T in the annular gap of the MR valve (refer to Figure 4). These values completed step (1).

For step (2), based on design constraints imposed on the MREA total size and the size chart for standard sealing parts (such as piston ring, U-cap), $D_{in} = 41.28 mm$ (1.625″) and $D_{r} = 15.88 mm$ (0.625″). The number of coil stages, n, was chosen to be 3.

For step (4), given the initial guess of $d = 0.91 mm$ , then $L_{a} = 15.2 mm$ was obtained from equation (26).

For steps (5)–(6), the plot of Z versus $\bar{d}$ and $\bar{L}$ was visualized in Figure 5. The pairs of $(\bar{d}, \bar{L})$ satisfying Z≈0 (within acceptable tolerance) are solutions satisfying the design specifications. From Figure 5, there are many pairs of $(\bar{d}, \bar{L})$ that can satisfy the relationship Z≈0. Two example points of $(\bar{d}, \bar{L})$ are explicitly indicated in Figure 5(b). For the design problem in this study, the pair of (4.03, 1.22) was chosen. This pair is corresponding to $d_{c} = 3.67 mm$ and $L_{c} = 18.5 mm$ , which has a shorter electromagnetic coil and thus a shorter MREA. After magnetic field verification (refer to Figure 6) via analysis and FEA simulation, the final refined parameter values are $D_{in} = 1.625 ″$ , $D_{r} = 0.625 ″$ , d = 0.91 mm, $d_{c} = 3.67 mm$ , $L_{a} = 15.2 mm$ (total MR active length would be 45.6 mm), and $L_{c} = 18.6 mm$ ; the maximum achievable magnetic flux density is 0.7 T corresponding to an MR fluid yield stress of 60 kPa with applied current of 3.25 A for 24 American wire gauge (AWG) magnetic wires.

The predicted MREA performance using the BPM model with these designed geometric values and fluid properties is presented in Figure 7. As seen in this figure, the predicted off-state and maximum field forces of the MREA are $F_{off} = 14.8 kN$ and $F_{on} = 21.0 kN$ at the piston velocity of $V_{p} = 4.5 m / s$ . In the above MREA design using BPM model, the minor loss coefficients are semi-analytical estimates based on formulae from White (1986), and are determined by approximating the gap/duct geometry by equivalent pipe systems. Thus, there is a degree of uncertainty in these values. To further verify the performance of the MREA design, CFD simulation using the commercial software package ANSYS (particularly, FLOTRAN) was carried out to derive the field-off force to compare with the analytical prediction from the BPM model. As shown in Figure 7, the off-state force from CFD simulation substantially agrees with off-state BPM model result. The MREA design was validated using CFD simulation. The MREA (called MREA1) was then manufactured using the geometric dimensions determined using the design procedure.

As further MR damper verification point, we designed a second MREA (MREA2) that has lower desired field-off damper force (i.e. 6 kN at 4.5 m/s) at higher piston velocities than the MREA1. But the maximum MR yield force and the volume size of the MREA2 were kept the same as the MREA1. The designed geometric dimensions of the MREA1 and MREA2 were listed in Table 1. The MREA2 has bigger MR valve gap than the MREA1 so as to decrease the field-off damper force. But, similar to the MREA1, the maximum achievable magnetic flux density of the MREA2 in the MR valve gap is 0.7 T with the applied current of 4.0 A for 24 AWG magnetic wires, which corresponds to the MR yield stress of 60 kPa. The predicted damper force of the MREA2 is presented in Figure 8. As seen in Figure 8, the field-off forces from BPM model and CFD simulation are dramatically reduced at higher piston velocities while having similar MR yield forces of the MREA1. But, different from the design case of the MREA1, the damper force of the MREA2 predicted by the BPM model and CFD simulation shows the discrepancy, especially at higher velocities (i.e. after 1 m/s). One possible explanation is that the values of the minor loss coefficients used for this study were quoted from empirical values of pipes (White, 1986) and thus are slightly different from the values of pipes because of more complicated MR valve configuration. In the next section, it will be identified from experimental test which results between the BPM model and CFD simulation will be in better agreement with the experimental data.

Table 1.

Dimensions of MREA1 and MREA2.

Parameter (mm)	Meaning	MREA1	MREA2
$d$	MR valve gap	0.91	1.21
$d_{c}$	Coil gap thickness	3.67	2.67
$L_{a}$	Active MR valve length of one coil	15.2	16.27
$L$	Total active MR valve length	45.6	48.8
$L_{c}$	Length of one coil	18.6	17.8
$D_{r}$	Piston rod diameter	15.88	15.88
$D_{in}$	Inner diameter of the outer tube	41.28	41.28
$D_{b}$	Effective MR valve diameter	40.37	40.07
$D_{p}$	Piston diameter	39.46	38.86

MREA: magnetorheological energy absorber.

Figure 9 shows the photograph of the designed and fabricated MREA1. Since the MREA2 has the same outer diameter as the MREA1, the appearance of MREA2 looks identical to MREA1 after being assembled. Thus, the MREA2 is not presented in Figure 9.

High-speed drop tower test

High-speed drop tower test setup

The damper forces of the two MREAs (i.e. MREA1 and MREA2) at high speed were evaluated by using the high-speed drop tower facility at the GM R&D Center as shown in Figure 10. The MREA was attached firmly to the top surface of the mounting plate. A small block of aluminum honeycomb was attached using double-sided tape to the small flat plate that had been mounted to the upper end of the MREA rod. The honeycomb block was a pentagonal prism with the bottom two corners having 90° angles, and the longitudinal axes of the hexagonal cells were oriented in the vertical direction of impact. This honeycomb block shaped the deceleration pulse and also served to eliminate the ringing in the load cells due to metal-to-metal impact. The large blocks of aluminum honeycomb positioned to the side of the MREA worked as stoppers to prevent any bottom-out or end-stop impact of the MREA. Four load cells for measuring the damper force of the MREA were positioned beneath the mounting plate, and a linear variable differential transducer (LVDT) was mounted to the MREA to measure shaft motion of the damper.

To conduct a high-speed drop test of the MREA, the drop platform (i.e. drop mass) was raised to the drop height corresponding to a nominal drop speed, $V_{nominal}$ , calculated using the formula

V_{nominal} = \sqrt{2 gh}

(28)

Here $g$ is the gravitational acceleration, and $h$ is the drop height. Note that the nominal drop speed implies the calculated initial drop speed of the drop mass at the moment when the mass touches the MREA. The drop mass was released to freely fall under the action of the gravitation. Data from each of four load cells and the LVDT were recorded at the sampling rate of 10 kHz during tests. A high-speed digital video camera was used to record the motion of the MREA at high speed. More detailed descriptions of the high-speed drop tower facility are given in Browne and Johnson, (2001) and Johnson and Browne (2001).

Test result analysis

The high-speed drop tests were conducted for the nominal drop speed ranges of 1–6 m/s for the MREA1 and for the nominal drop speed ranges of 1–5 m/s for the MREA2 with 1 m/s increments. The applied current varied discretely from 0, 1.0, 2.0 to 3.25 A for the MREA1 and 0, 2 to 4 A for the MREA2, for each nominal drop speed.

Samples of test data for the MREA1 are plotted in Figures 11 and 12. For the nominal drop speeds of 2 m/s, the peak piston velocity at 0 A increases more than the nominal speed and decreases as applied current increases (refer to Figure 11(b)). This peak velocity decrement results from the increased damper force at the applied current input. In addition, it is observed from Figure 12 that the peak piston velocity at a nominal drop speed of 6 m/s only reaches 5 m/s for various current levels. Therefore, in this study, the peak piston velocity rather than the nominal drop speed will be used as a working condition. In addition, the MREA peak force and peak velocity pairs are used as the key metrics to characterize the damper forces of the MREAs.

Figure 13 presents the measured and predicted peak damper forces of the MREA1 with respect to the peak piston velocity. As seen in Figure 13, the BP nonlinear flow model shows significant deviation from the measured damper forces at higher piston velocity. The BPM model and the CFD analysis at 0 A are in good accordance with the measured damper forces. But, the BPM model at 3.25 A shows some difference from the measured damper forces at higher piston velocity. The reason is that the MR yield force model given by equation (26) is derived based on laminar flow model (or lower piston velocity) because the MR yield force behavior at high flow rate has not been explored so far. But, as shown in Figure 13, the measured MR yield force at 3.25 A decreases at high piston velocity. This implies that the MR yield force becomes smaller at high piston velocity different from the MR yield force model given by equation (26).

Figure 14 presents the measured and predicted peak damper forces of the MREA2 with respect to the peak piston velocity. Different from the case of the MREA1 shown in Figure 13, the CFD analysis is continuously in good accordance with the measured damper forces, but the BPM model using the minor losses given by equations (22) and (23) underestimates the measured results as shown in Figure 14(a). Such difference between the BPM model and the measured damper forces is mainly coming from the sudden contraction, $K_{SC}$ , and sudden expansion, $K_{SE}$ . In order to more accurately capture the measured damper forces, the values of the sudden contraction and sudden expansion were adjusted: $K_{SC} = 0.22$ (original from equation (23)) → $K_{SC} = 0.49$ (adjusted) and $K_{SE} = 0.28$ (original from equation (22)) → $K_{SE} = 0.54$ (adjusted). Above adjustment of the sudden contraction and expansion implies that the practical sudden contraction and expansion of the flow inside the MREA2 are much prominent than the initial estimation. As seen in Figure 14(b), the BPM model using the adjusted minor losses shows good agreement with the measured damper forces. But, further theoretical and experimental studies are needed in future work to fully understand this discrepancy. On the other hand, different from the case of the MREA1, the measured MR yield force of the MREA2 is less sensitive to the high piston velocity. The reason is that since the MREA2 has bigger MR valve gap, d, than the MREA1, the flow rate at the gap of the MREA2 is lower than that of the MREA1 at the same piston velocity. In other words, the MREA2 has relatively lower Reynolds number and less turbulent flow in the MR valve gap than the MREA1. It is noted that the CFD result at 4 A was obtained by adding the MR yield force calculated from equation (26) to the CFD result at 0 A.

Conclusion

In this study, a nonlinear analytical model of MREAs taking into account effects of minor loss factors (BPM model) was developed based on a BP nonlinear flow model. An effective design analysis was proposed for a conventional annular valve type MREA, and the MREAs (i.e. MREA1 for higher field-off damper force and MREA2 for lower field-off damper force) were designed and fabricated. Then the damper force performances of both MREAs were tested over the peak higher piston velocities of up to 5 m/s by using the high-speed drop tower facility at the GM R&D Center. The experimental data showed that the BP nonlinear flow model could not predict the measured damper forces of both MREAs. But, the predictions from the BPM model (especially for the off-state) and CFD simulations were in good agreement with the experimental data over the tested high piston velocity range. A key conclusion of this study was that the BPM model could predict the MREA performance at high piston velocity, although the current analysis is more accurate in the off-state. In addition, the BPM model was shown to be useful as an effective MREA design tool.

Footnotes

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research was supported by General Motors Research and Development Center.

Disclaimer

The examples in this study are solely provided for the purpose of scientific discussion of the devices described as they relate to occupant protection efforts in automotive engineering. The work presented here explicitly does not cover all and any engineering design issues around occupant protection efforts; it is not to be construed to being an engineering manual, to provide any specific or ultimate solution nor to represent a certain engineering decision by General Motors LLC, its subsidiaries and affiliates and/or any reasons for such decisions.

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