# Design and Analysis of a New Magnetorheological Damper for Generation of Tunable Shock-Wave Profiles

Authors: Jong-Seok Oh; Tae-Hoon Lee; Seung-Bok Choi
Journal: Shock and Vibration
 (2018)

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## Abstract

A new impact testing system with an integrated magnetorheological (MR) damper is proposed, and its dynamic characteristics are analyzed. The testing system consists of a velocity generator, impact mass, test mass, spring, and MR damper. In order to tune the dual shock-wave profile, a dynamic model was constructed, and the appropriate design parameters of the MR damper were then determined to produce the required damping force. Following this, an impact testing system was constructed to evaluate the design analysis and field-dependent dual shock-wave profiles. The experimental results of impact test showed that the dual shock-wave profile can be altered by changing the intensity of the magnetic field.


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## Body

## 1. Introduction

The components of submarines and other naval vessels are often damaged by shock-waves owing to noncontact underwater explosions (UNDEX) during wartime. Thus, impact tests are conducted to verify shock survivability or the ability of installations to withstand shock loading. To specify the requirements of impact tests, the US formulated military specification MIL-S-901D in 1989 [1]. Based on MIL-S-901D, impact tests are categorized as light- (~550 pounds), medium- (~7,400 pounds), and heavy-weight (7,400 pounds ~). The light- and medium-weight impact tests are performed using a shock machine, and heavy-weight tests are performed using a large floating shock platform and an UNDEX. These methods have several drawbacks (high cost, low efficiency, and environmental damage), and it is often recommended that heavy-weight tests should be conducted using the shock machine method.Several methods have been proposed to solve this problem. Zhaodong et al. [2] proposed a dual-wave shock test machine to simulate an UNDEX and test the shock survivability of shipboard equipment. Wang et al. [3] studied a vertical heavy-weight shock test machine. Mathematical modeling and a mechanistic analysis verified that the proposed test machine can produce shock acceleration wave profiles consistent with the MIL-S-901D criteria (via computer simulation). Kim et al. [4–6] implemented dynamic design analysis for heavy dual shock generation systems. Their results and design analysis showed that the properties of the system components such as the spring and damping coefficients were correctly determined. However, the proposed dual-wave shock testing machines are still being constructed.Many attempts to construct test systems must contend with difficulties such as the controllability of dual shock-wave profiles and dissipation of impact energy over short time periods. In this paper, we propose a heavy-weight impact testing machine equipped with a controllable magnetorheological (MR) damper. By applying a controllable MR damper, the generated acceleration profile can be tuned and the impact energy can be effectively dissipated. It is generally known that MR fluid is a smart material whose rheological properties can be altered by controlling magnetic field. Thus, many methods for energy dissipation have been proposed. Kim et al. [7, 8] evaluated the ride quality of a railway vehicle featuring an MR damper via roller rig and railway field tests, Stone and Cebon [9] evaluated the vibration control performance of an MR damper on heavy vehicles, and Yang et al. [10] proposed a large-scale MR damper to reduce the vibration for civil engineering applications. These studies were implemented on low-velocity and low-frequency semiactive applications; however, in severe conditions that produced high-impact load and velocity, the MR damper did not perform well.Recently, new prediction models for dynamic behavior under high velocity and high frequency conditions have been proposed [11, 12]. To account for the behavior of MR damper at high shear rates, Lee et al. [11] used the Herschel–Bulkley shear model and Mao et al. [12] used a nonlinear Bingham model based on loss factor and hydromechanical analysis. Ahmadian et al. [13–15] studied an MR impact damper in gun recoil dynamics and free-flight drop test facilities. Browne et al. [16] also evaluated the impact damping performance of an MR damper. Although these works focused on the dynamic behavior of MR impact dampers, the maximum damping force range of their proposed MR dampers was relatively low [11–16].In this work, the dynamic characteristics of an MR impact damper are analyzed for high-impact energy dissipation. By using an MR impact damper, the desired dual shock-wave profile can be easily tuned by varying the input current to the MR damper and other physical conditions such as mass, spring constants, and input velocity. Thus, the main contribution of this work is the proposal of a novel impact testing machine using an MR damper for the heavy-weight components of submarines and other naval vessels. First, the impact test system is introduced, and then the dynamic analysis is undertaken. Based on dynamic models, the proper physical coefficients of the test system are determined to generate heavy dual shock-wave profiles. To predict the damping force of the MR damper, a mathematical model is derived based on the Bingham plastic axisymmetric model. Using the geometrical dimensions obtained from the numerical analysis performed with the mechanical model, FE (finite element) magnetic circuit analysis is performed via computer simulation. Then, the damping force of MR damper and input condition such as current is calculated from the FE analysis results and geometrical dimensions. For evaluating the field-dependent dual shock-wave profiles, an impact testing machine system is constructed. By measuring the acceleration on the test mass and damping force of the MR damper, it is confirmed that several dual shock-wave profiles can be obtained by tuning the input conditions of the proposed impact testing system such as input current to MR damper.


## 2. Impact Testing System

In contrast with MIL-S-901, which requires full-scale submarine or naval vessel testing for heavy-weight components, BV-43/85 (developed in Germany) requires that the shock loadings be applied to components that are directly attached to surface shipboard equipment or submarines [1, 17–20]. In BV-43/85, a double half-sine acceleration profile is used to represent the shock loading caused by an UNDEX. As shown in Figure 1, the acceleration profile consists of three parts. The positive half-wave (Part I) shows the effect of the magnitude of impact, and the negative half-wave (Part II) and Part III are related to the energy dissipation system. In Figure 1, the magnitude of acceleration in Part II is relatively lower than that of Part I and the magnitude of residual vibration (Part III) is zero.Figure 1
Shape of dual shock-wave profile in BV-043/85.A hydraulic impact testing system is developed to generate the dual shock-wave profile, as shown in Figure2. The test facility comprises a hydraulic velocity generator, impact mass, polyurethane spring, test mass, air spring, and MR damper. The specimen is attached at the test mass. In the impact testing machine, the hydraulic velocity generator accelerates the impact mass, and the impact mass then strikes the polyurethane spring and test mass with a uniform motion. The high-impact load and velocity are then applied to the air spring and MR damper, which were located between the test mass and a fixed wall. The MR damper is used to reduce the transmitted reaction force (Part II) and residual vibration (Part III). In other words, the generation of a shock-wave profile, particularly for Parts II and III, is the primary function of the MR damper.Figure 2
Schematic of the impact testing machine system.
(a)
Simplified mechanical model
(b)
Experimental apparatusThe method for determining the input velocity of the hydraulic velocity generator and the relation of impact and test mass is represented in Figure3; the equation of motion during contact time is expressed as follows [4]:(1)m1x¨1+k1x1-x2=0,m2x¨2+k1x2-x1=0.Since the 1st impact force is very large and contact time is very short, the motion of the air spring and MR damper is neglected during 1st pulse generation. As the initial conditions are x10=x20=x˙20=0,x˙10=Vg, the acceleration of 1st pulse is expressed as(2)x¨2t=m1ωVgm1+m2sin⁡ωt,ω=m1+m2k1m1m2=πτ1.Supposing that the maximum acceleration, a1, and duration time of 1st pulse wave, t1, are given, the input velocity and the spring constant of polyurethane spring are obtained from (2). To predict the 2nd pulse wave, the dynamic modeling represented in Figure 4 is implemented as follows:(3)m2x¨2+k2x2-x3=0,m3x¨3+cx˙3+k2x3-x2=0,wherex20=x0,x˙20=Vt.m3 is the dummy mass of the air spring. Using the Laplace transform, the responses of the test and dummy masses are expressed as(4)s2X2s=m2Vtsm3s2+cs+k2Ds,s2X3s=m2Vtk2sDswhereDs=m2m3s3+m2cs2+m2+m3k2s+ck2.Suppose that D(s) has one real and two complex roots and can be expressed as (5)Ds=m2m3s3+m2cs2+m2+m3k2s+ck2=m2m3s+αs2+2ςωns+ωn2.Following this, the 2nd pulse wave is obtained as follows [6]:(6)x2t≅-bωn21-ς2e-ςωntsin⁡1-ς2ωnt,whereb=ωn2α-aα,a=2ςωn+α-c/m32.To determine the physical coefficients of the impact testing system, the desired dual shock-wave profile was chosen according to BV-43/84 criteria, as shown in Table 1 [7]. To evaluate the performance of the MR damper, the 1st shock-wave was fixed, and the input and spring velocities of the polyurethane spring remained constant. The impact, test, specimen, and dummy masses were 10,000 kg, 11,000 kg, 5,000 kg, and 8,000 kg, respectively. The spring constants and damping properties of the MR damper were calculated based on (2) and (6). For each 2nd pulse wave profile, there were many solutions of k2 and c. Of these, the proper spring constant and damping constant were determined to meet five shapes of the 2nd pulse and demonstrate the control performance of the MR damper. In other words, using same k2 and c for all test conditions, 2nd pulses were tuned by controlling the damping force of the MR damper; the resulting values are shown in Table 2. From Table 2, the maximum required viscous damping force is 300 kN at 2 m/sec, maximum controllable damping force is 518.3 kN, and total damping force is 818.3 kN.Table 1
Desired profiles of dual shock-waves.
Test number
Peak acceleration,a1 (m/s2)
Duration time,t1 (sec)
Peak acceleration,a2 (m/s2)
Duration time,t2 (sec)
m
1 (kg)
m
2 (kg)
m
3 (kg)
Test #1
750
0.0100
−125
0.0520
10000
11000
8000
Test #2
750
0.0100
−135
0.0540
10000
11000
8000
Test #3
750
0.0100
−140
0.0555
10000
11000
8000
Test #4
750
0.0100
−145
0.0570
10000
11000
8000
Test #5
750
0.0100
−150
0.0585
10000
11000
8000Table 2
Calculated physical coefficients for the impact testing machine.
Test number
V
g (m/s)
k
1 (MN/m)
k
2 (MN/m)
c (kN⋅s/m)
F
M
R (kN)
Input current (A)
Test #1
4.97
516.9
17 MN/m
150 kN⋅s/m
0
0
Test #2
4.97
516.9
17 MN/m
150 kN⋅s/m
171.3
1
Test #3
4.97
516.9
17 MN/m
150 kN⋅s/m
295.1
2
Test #4
4.97
516.9
17 MN/m
150 kN⋅s/m
400.6
3
Test #5
4.97
516.9
17 MN/m
150 kN⋅s/m
518.3
4Figure 3
Simplified model of impact testing system for 1st shock-wave.
(a)
Before impact
(b)
During impact
(c)
After impactFigure 4
Simplified model of impact testing system for 2nd shock-wave.


## 3. Design of MR Damper

In this experiment, a new high-impact MR damper was designed, as shown in Figure5. The MR damper consists of piston, coil, and outer housing. The annular duct is placed between coil and outer housing. During piston movements induced from a high-impact event, the piston moved at a high velocity and the MR fluid flowed through the outer annular duct. Since the pressure drop is caused from fluid flow, viscous damping force is generated. The advantage of MR damper is that damping force can be increased by applying the current input to coil. During MR fluid flow, the yield stress of MR fluid is amplified according to magnetic field. It is generally known that magnetic field is generated on both sides of the coil. By applying multi-magnetic cores, the active length which is calculated as total length of duct minus the total coils length was increased. In other words, the total damping force is bigger than the case of MR damper with single core and same design parameters. With the use of 21 magnetic poles and the consequent increase in the active length, it was identified that the proposed MR damper could provide a more effective control performance.Figure 5
Design configuration of the proposed MR damper.Several shear stress models of MR fluid were used to predict the behavior of MR damper [11–15, 21, 22]. Among them, nonlinear Bingham model showed good performance in shock absorber application [12, 21, 22]. Accordingly, Bingham model was used in this work. Based on the Bingham model, the total shear stress of MR fluid can be expressed as follows: (7)τrx=τyH+ηγ˙=τyH+ηduxrdr,where τyH is the yield stress as a function of the magnetic field; η is the viscous coefficient of the MR fluid; and γ˙ is the shear rate. An axisymmetric annular duct flow model was used to calculate the pressure gradient and damping force, as shown in Figure 6. ux(r) is the fluid velocity in annular duct and x is the longitudinal coordinate. The governing equation of the duct flow model was derived by simplified Navier–Stokes equations with the assumptions that the flow was essentially axial and the effect of fluid inertia can be neglected [10, 23]. Particularly, many researchers obtained reasonable results in shock application by neglecting the effect of fluid inertia in duct [24, 25](8)ddrτrxr+τrxrr=dpdx,τrxr=12dpdxr+C1r,where C1 is the integral constant. The boundary condition of annular duct is uRcore=uRcyl=0. Based on (7), (8), and the boundary condition, the fluid velocity in region 1 was obtained as follows:(9)ux1r=14ηdpdxr2-Rcore2+C1ηln⁡rRcore-1ητyr-Rcore,whereRcore≤r≤ra.Similar duct modeling analysis is applied to region 3. However, the shear rate in region 3 is γ˙=-duxr/dr. At the center of duct, the absolute shear rates are the same and sign is different according to upper and lower directions. Accordingly, the fluid velocity in region 3 was obtained as follows: (10)ux3r=∫rRcyl-1η12dpdxr+C1r+τydr=14ηdpdxr2-Rcyl2+C1ηln⁡rRcyl-1ητyRcyl-r,whererb≤r≤Rcyl.Since the shear stress in region 2 was less than the yield stress of the MR fluid, the flow velocity gradient was zero. Hence, the shear stress in region 2 can be expressed as follows:(11)ux2r=14ηdpdxra2-Rcore2+C1ηln⁡raRcore-1ητyra-Rcoreor14ηdpdxrb2-Rcyl2+C1ηln⁡rbRcyl-1ητyRcyl-rb,wherera≤r≤rb.In order to obtain integral constant, C1, the free body diagram of MR fluid in Figure 6 can be derived as follows:(12)dpdxπrb2-ra2dx+2πτyrb+radx=dpdxrb-ra+2τy=0.By integrating (8) and (12), C1 can be expressed as follows:(13)C1=rbrarb-ra2τy.Equation (11) can be reexpressed as follows:(14)14dpdxMRRcyl2-Rcore2-rb2+ra2+rbraτyrb-raln⁡RcylraRcorerb+τyRcyl+Rcore-rb-ra0.The total flow rate, Qt, in the annular duct and the flow rate induced from piston movement can be expressed as follows:(15)Qt-Ah-Arvp=0,where vp is the piston velocity. Ah and Ar are the cross-section areas of the piston head and piston rod, respectively. The total flow rate was reexpressed by the sum of the flow volumes of each region: (16)Qt=2π∫Rcoreraux1rrdr+2π∫rarbux2rrdr+2π∫rbRcylux3rrdr.The damping force of the MR damper, Fd, can be expressed as follows:(17)Fd=P1Ah-Ar-P2Ah-Ar.P1 and P2 are the pressures of upper and lower chambers, respectively. Due to pressure drop in annular duct, the pressures of upper and lower chamber are different. Also, damping force of MR damper which consisted of the controllable and viscous damping forces can be reexpressed as follows:(18)Fd=FMR+Fη=dpdxMRL-LpoleAh-Ar+dpdxηLAh-Ar,where dp/dxMR and dp/dxη are the pressure gradients induced from the MR effect and viscous force, respectively; and L is the length of annular duct and Lpole is the sum of the length of 21 magnetic poles. dp/dxMR can be obtained using (14) and (15). In order to obtain dp/dxη, the flow velocity of annular duct without magnetic field can be derived with τrx=ηduxr/dr:(19)uxr=14ηdpdxηr2-Rcore2+C2ηln⁡rRcore,whereC2=-dpdxRcore2-Rcyl24ln⁡Rcore/Rcyl.By substituting (19) for (15), the following equation was obtained:(20)2π∫RcoreRcyluxrrdr-Ah-Arvp=2π∫RcoreRcyl14ηdpdxηr2-Rcore2+C2ηln⁡rRcorerdr-Ah-Arvp=0.dp/dxη was obtained by calculating (20). ra and rb were obtained by applying the two-variable Newton–Raphson method:(21)ra,i+1rb,i+1=ra,irb,i-∂g∂ra∂g∂rb∂w∂ra∂w∂rb-1gra,i,rb,iwra,i,rb,i,where gra,i,rb,i is (15); wra,i,rb,i is (14); and i means the number of iterations. Figure 7 shows the final values as ra=0.19 m and rb=0.1913 m. From the iteration results, it has been revealed that the thickness of regions 1 and 3 is theoretically almost zero. As previously mentioned, the maximum required MR damping force is 818.3 kN at 2 m/s. In order to simplify the design procedure, maximum yield stress of MR fluid is assumed as 40 kPa. Using the above equations and repeated calculations, the damping force can be calculated using (15). The calculated controllable and viscous damping forces of the MR damper (at 2 m/s) are 504.9 kN and 300.9 kN, respectively. As the predicted total damping force was 805.8 kN, the MR damper was manufactured with the design parameters in Table 3. In addition, a magnetic flux analysis was performed to confirm that the magnetic circuit in the MR damper could generate a suitable magnetic field. As shown in Figure 8, the MR fluids created 45 kPa at 4 A current input. Thus, the proposed MR damper can produce the required damping force. Also, it is confirmed that the magnetic field is strongly generated at both sides of the coils. The magnitude of magnetic field at coil is almost zero.Table 3
Geometric specifications of the MR impact damper.
Specification
Symbol
Value
Diameter of piston head
D
h
0.200 m
Diameter of piston rod
D
r
0.140 m
Magnetic core diameter
D
c
o
r
e
0.3800 m
Cylinder diameter
D
c
y
l
0.3826 m
Orifice gap size
h
0.0013 m
Orifice length
L
0.9 m
Total length of magnetic poles
L
p
o
l
e
0.462 m
Length of magnetic pole
/
0.022 mFigure 6
Incompressible viscous axisymmetric annular duct flow model.Figure 7
Optimized design parameters using Newton–Raphson method.Figure 8
FEM result for magnetic circuit of the proposed MR damper.
(a)
Applied current: 2 A
(b)
Applied current: 4 A


## 4. Test Results and Discussions

An experimental apparatus was prepared to investigate the effectiveness of the proposed system, as shown in Figure2. To measure the acceleration of test mass and the damping force of the MR damper, accelerometers (353B02, PCB Piezotronics Inc.) with 250-g measurement range were attached at the test mass and a load cell (Bongshin Loadcell Co.) was placed between test mass and the MR damper. All measured signals were recorded by pulse measurement equipment (type 3038-B, B&K Co.).An impact test was conducted to evaluate the performance of the MR damper. The experimental results are presented in Figure9, specifically, the measured damping force according to the magnitude of applied input currents. From Figure 9, we see that the MR damper met the desired damping properties as set out in Table 2 (i.e., the maximum desired damping force was 805.8 kN at 2 m/s and maximum measured damping force was 775.3 kN). Based on these results, the geometric dimensions of the MR damper were appropriately determined and it is suitable for tuning shock-wave profiles. However, the measured damping force results tended to decrease in fast velocity regions. The primary reason for this was fluid dwell time. According to [23], the total shear stress of MR fluid decreases as the fluid dwell time decreases. The dwell time is reduced when high piston velocity is applied to the MR damper.Figure 9
Damping force versus piston velocity.The measured shock-wave profiles of the test mass are plotted in Figures10–14. Figure 10 showed the experimental results without current input. Particularly, the behavior of MR damper without current input is almost the same as that of hydraulic damper. Figures 11–13 showed experimental results with several current inputs. Accordingly, it is verified that 2nd pulse changed owing to changes in the magnitude of the input current. The maximum accelerations of 2nd pulse from test #1 to test #5 were 141 m/s2, 145 m/s2, 138 m/s2, 148 m/s2, and 145 m/s2, respectively. The duration times of 2nd pulse from test #1 to test #5 were 43 ms, 36.9 ms, 53.8 ms, 56.1 ms, and 58.9 ms, respectively. The total error rates between desired and measured values were 4.8% and 11.6%, respectively. The experimental results without particular cases were reasonable for impact testing. However, the duration time results show a substantial difference between desired and measured values; specifically, the desired duration time in the second test was 54 ms but the measured duration time was 36.9 ms. This was mainly owing to a small damping force in relation to fluid dwell time at the beginning of the 2nd pulse. Since the transmitted force to the test mass was not effectively reduced, the initial gradient of the measured 2nd pulse was larger than that of desired 2nd pulse. It is obvious that the large gradient of the 2nd pulse reduced its duration time. Another substantial difference was observed in Part III of the fifth test. As shown in Figure 14, Part III of the shock-wave profile shows fluctuations with a maximum amplitude higher than 177 m/s2. According to [25], the yield stress induced by the input current produced fluctuations in the shock-wave profile (particularly in Part III); there was no such fluctuation in tests 1–4. It is apparent that the cause of the fluctuations was the large input current, such as 4 A. Since the technical novelty of the proposed method is the tunability of the MR damper, the fluctuations in Part I (owing to the polyurethane spring and velocity generator) were not considered.Figure 10
Comparison test and simulation results (Test #1: 0 A).Figure 11
Comparison test and simulation results (Test #2: 1 A).Figure 12
Comparison test and simulation results (Test #3: 2 A).Figure 13
Comparison test and simulation results (Test #4: 3 A).Figure 14
Comparison test and simulation results (Test #5: 4 A).


## 5. Conclusion

In this work, an impact testing machine system was proposed to experimentally investigate the tunability of dual shock-wave profiles. The proposed testing system comprised a velocity generator, impact mass, test mass, spring, and MR damper. The goal of the proposed system was to generate the desired dual shock-wave profiles. Based on this, the damping force of the MR damper was calculated using an annular duct model and a damping force model. In order to evaluate the tunability of the shock-wave profiles (particularly in Part II), five experimental conditions were chosen. According to current input conditions, desired shock-wave profiles can be altered using the MR damper. From the experimental results, it is known that the error rates in 2nd pulse are 4.8% and 11.7%. Experimental results clearly demonstrated that the MR damper is effective method for impact testing machine for heavy-weight. In future work, the tuning ability of air spring will be analyzed as a second phase of this research.



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