High dynamic range and high dispersion stabilityof a magnetorheological grease damper for semi-active vibration suppression
Authors
Toshihiko SHIRAISHI*, Shuto NAGAMATSU and Hirotaka MISAKI (*corresponding author)
E-mailToshihiko SHIRAISHI (shira ynu.ac.jp)
Affiliation
Graduate School of Environment and Information Sciences, Yokohama National University(79-7 Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, JP)
Abstract
In this study, the dynamic range and dispersion stability of a shear-type magnetorheological (MR) grease damper wereexperimentally investigated as important characteristics affecting the performance of semi-active vibration suppression.
Furthermore, the damper was applied to the semi-active vibration suppression of a small single-degree-of-freedommodel structure subjected to seismic excitation. The performance test results of the damper indicated that it has adynamic range of 157 times, which is much higher than that of conventional MR dampers. Also, because of the highdispersion stability of MR grease, the results showed that its performance can be kept for 9 days longer in comparison
with a similar damper using MR fluid. Moreover, the structural vibration suppression test results demonstrated that thedamper can suppress vibration response over a wide frequency range by applying skyhook-based control, which cantake full advantage of the damper’s high dynamic range.
Keywords
Magnetorheological grease, semi-active vibration suppression, dynamic range, dispersion stability
1.

Introduction
Semi-active vibration suppression is a method of suppressing vibrations by varying the stiffness or damping
characteristics of a damper according to the vibration state of a system. When a controllable damper is used forsemi-active vibration suppression, to generate a desired force and realize high suppression performance, thedamping force should be changed quickly, widely, and continuously. Dampers that use magnetorheological (MR)fluids have been widely studied and applied in many applications, as they satisfy the requirements for controllingthe damping force of dampers. MR fluid is a suspension of micron-sized magnetic particles dispersed in a carrier
fluid, and this method was first reported in a fluid clutch application (Rabinow, 1948). When an external magnetic
field is applied to MR fluid, its rheological characteristics are rapidly changed. The magnetization of the particles
in the fluid results in the formation of chain-like clusters along the applied field. In this phenomenon, which iscalled the MR effect, the MR fluid reversibly changes from a liquid state to a semi-solid state in milliseconds (Dykeet al., 1998). Such change can be activated by an electric current, as the current can be converted into a magnetic
field in an electromagnetic coil. In a semi-active vibration suppression system using an MR damper, variable
damping characteristics are achieved by supplying a variable current to an electromagnetic coil in the damper. As
MR dampers can achieve high-performance suppression by just controlling the current, they have received a lot of
attention and have been frequently studied for many applications, including civil engineering structures and vehicle
suspension systems (Lee et al., 2010; Park et al., 2010; Tu et al., 2011; Yao et al., 2002).
However, conventional MR fluid dampers have two issues to be resolved. The first issue is the sedimentationof their magnetic particles due to the density mismatch between the particles and their carrier oil. In structural
applications, when a damper is exposed to a static state for a long time, sedimentation causes the damping force todiverge from the desired force, resulting in performance degradation. The second issue is the low dynamic range
due to the sliding friction force acting on the sealing elements of dampers, where the dynamic range is the ratio of
the total force to the uncontrollable force, and it is one of the most important parameters for evaluating theperformance of MR dampers (Yang et al., 2002). This sliding friction increases the uncontrollable force and reduces
the dynamic range of MR dampers. However, sealing elements are essential for confining the MR fluid in dampers.
To achieve both high dispersion stability of magnetic particles and high dynamic range of the damper, othermaterials such as elastomer, plastomer, and grease have been proposed as better carriers of magnetic particles (Kintz
et al., 2003; Shen et al., 2004; Xu et al., 2011). While MR elastomer has a problem of low MR effect, MR plastomer
and MR grease can offer both high dispersion stability and high MR effect, thus they have promising usage in thefield of vibration suppression (Ahamed et al., 2018). However, few applications to controllable dampers have beenreported before despite their potential. For high reliability and high suppression performance in practicalapplications, particles should be prevented from settling and the sliding friction force acting on sealing elements
should be eliminated.
To overcome these issues, a shear-type damper using magnetorheological grease (a shear-type MR grease
damper) was developed (Shiraishi and Sugiyama, 2015). It has an oscillating plate moving in a box filled with MR
grease (Figure 1), which is mainly composed of micron-sized magnetic particles and grease as a dispersion medium.
The MR grease used in the damper was developed by researchers including one of the authors of this paper (Kimura
et al., 2013). This damper worked subject to the following mechanism. First, a magnetic field was induced inside
the electromagnetic coil by supplied currents. Then, a closed magnetic circuit was formed through the iron core.
The generated magnetic field passed through the MR grease perpendicular to the plate oscillation. In the gap
between the oscillating plate and the inner wall of the box, magnetic particles of the MR grease formed chain-likeclusters along the field, which resisted shear flows. The damping force value of the shear-type MR grease damper
was obtained as the product of the damper plate area and the shear stress of the MR grease that was determined bya magnetic flux density and shear rate. Therefore, by determining the plate area, the winding number of the coil,and electric currents considering the size of the target structure, the damper can be applied to seismic control.
The combination of the developed MR grease and shear-type damper could solve the abovementioned two
issues. The sedimentation issue could be mitigated by the MR grease, and high dispersion stability was realized.

The low dynamic range issue could be overcome by the shear-type damper without the need for sealing elements.
Unlike MR fluids, semi-solid MR grease does not leak out during operation, so it can be applied to open structures
when the internal pressure is not used to generate forces. As the sliding friction force of shear-type MR grease
dampers is theoretically zero, the dynamic range is high. Furthermore, the absence of sealing elements enhances
the reliability by reducing the number of mechanical parts. Thus, high dispersion stability and high dynamic range
can be achieved for MR dampers using MR grease in combination with shear-type dampers. However, this method
needed to be experimentally and quantitatively verified in detail.
In this study, two experiments using the shear-type MR grease damper were conducted. A performance test of
the damper was performed to investigate its two advantages: high dynamic range of the damper and high dispersion
stability of the MR grease. Then, a structural vibration suppression test was conducted to investigate the semiactive vibration suppression performance of a structure equipped with the damper. In this experiment, thedisplacement response of the upper floor of a small single-degree-of-freedom model structure subjected to actual
seismic excitation was evaluated as a performance indicator.

(a)

(b)

Figure 1. (a) Schematic and (b) photograph of the shear-type MR grease damper.

2.

Magnetorheological grease
Magnetorheological grease is a suspension of magnetic particles in grease, and it was developed to mitigate thesedimentation of MR fluid particles (Kintz et al., 2003; Rankin et al., 1999). After its development, several studies
have focused on its application in controllable devices (Chansheng, 2006; Kavlicoglu et al., 2013) and on measuring
its rheological characteristics (Bahiuddin et al., 2019; Mohamad et al., 2019; Wang et al., 2019). However, to ensurethat MR grease can perform well as a working fluid in controllable dampers, its response time and dispersion
stability should also be experimentally investigated. The characteristics of the MR grease used in this study had
already been examined. It has been experimentally confirmed that the maximum shear stress of the MR grease andthe time constant of the damping force of a prototype damper using the MR grease are as high as those of
conventional dampers using MR fluids (Shiraishi et al., 2011) (Figure 2). Furthermore, the dispersion stability of
the MR grease has been investigated through static observations (Kimura et al., 2013; Shiraishi et al., 2011;
Shiraishi and Misaki, 2016).
In this study, a static observation was performed for 50 days to examine the dispersion stability of conventional
MR fluids and that of the MR grease, and their particle concentrations were found to be 82 and 75 wt%, respectively(Table 1). Also, their dispersion stabilities were quantified using the sedimentation ratio, which was defined in aprevious study as the ratio of the volume of the supernatant liquid in a fluid to that of the entire suspension (Tu etal., 2011).
The developed MR grease remained evenly dispersed. However, the conventional MR fluid was separated into
two layers because of sedimentation (Figure 3). In the static state, the MR fluid produced a visible supernatant witha volume fraction of 20% in total for 50 days (Figure 4). This result indicates that the performance of the dampersthat use MR fluids decreases when they are applied to actual structures, as they are exposed to a static state for along time and then subjected to sudden disturbances. In the MR grease, no visible supernatants were observed, evenafter a 50 day static state (Figure 4). This improvement is attributed to the capacity of grease to suspend particles.
By focusing on the inner structure of the MR grease, the particles may be held using a thickener, which forms threedimensional network structures in grease. These results indicate that dampers using the MR grease can realize highperformance structural vibration suppression for a longer time compared with conventional MR fluid dampers, as
the high dispersion stability of the MR grease allows its dampers to generate desired forces even after a static state.
In the subsequent section, the effect of the high dispersion stability of MR grease on the damping force wasdescribed.

Figure 2. Relationship between the shear stress and shear rate of the developed MR grease in the case of particle
concentration of 75 wt% (Shiraishi et al., 2011).
Table 1. Specifications of conventional MR fluid and the developed MR grease.
MR fluid

MR grease

Magnetic particle

Fe

Carbonyl ironParticle concentration

82 wt%

75 wt%

Base oilHydrocarbon

Mineral oil

Thickener-

Lithium soap

Thickener concentration

-

1 wt%

Figure 3. Static observation of conventional MR fluid (left) and the developed MR grease (right).

Figure 4. Time series of the sedimentation ratio of conventional MR fluid and the developed MR grease.

3.

Performance test of the shear-type MR grease damper
To investigate the dynamic range and dispersion stability of the shear-type MR grease damper, the relationshipbetween its damping force and damper plate displacement was obtained during the excitation of the damper plate(Figure 5). The force and displacement were measured using a load cell and a laser displacement sensor, respectively.
The damping characteristics of the damper were varied by varying an electric current from a bipolar power supply
to the electromagnetic coil in the damper. The force and displacement were measured for each current by setting
the current to a constant value. In this section, the two advantages of the shear-type MR grease damper were
demonstrated. First, the high dynamic range of the MR grease damper was described as a relationship between thedamping force and velocity. Second, the high dispersion stability of the MR grease was described as a time variationof the damping force.

Figure 5. Schematic diagram of the experimental setup for the performance test of the MR damper.

3.1 Dynamic range of the MR grease damper
Methods
To obtain the dynamic range, which is the ratio of the force in the ON-state to that in the OFF-state, the damping
force was measured at the current values of 0, 1, 2, 3, and 4 A (Table 2). The maximum current of 4 A wasdetermined based on the wire temperature. Note that it had been confirmed by simulations and experiments that themagnetic flux density linearly increased with the increase in the supplied current until it reached 0.5 T at 4 A. The
damper plate was excited using a sine wave with a constant displacement amplitude of 5 mm. To examine thevelocity effect on the dynamic ranges, the experiment was conducted at 1, 3, and 5 Hz. The damper plate velocity
was obtained as a time derivative of the displacement measured using the sensor. That velocity was called the“damper velocity” below in this paper. The test temperature of 40℃ was chosen to conduct experiments with aconstant temperature of the MR grease. Considering that the supplied current heated the electromagnetic coil andraised the temperature of the MR grease, it was necessary to set the test temperature to a higher value than the room
temperature.
Table 2. Experimental conditions for measuring the dynamic range of the shear-type MR grease damper.
Current

0, 1, 2, 3, 4 A

Displacement amplitude

5 mm

Frequency

1, 3, 5 Hz

Winding number of the coil1000 turns

Temperature

40°C

Results and discussion
The damping force increased with the increase in the supplied current and damper velocity. The current effect on
the damping force was more dominant than the damper velocity effect (Figure 6). A force–current diagram was
obtained by plotting the maximum damping force at each current. The damping force almost linearly increased
with the increase in the supplied current at all the frequencies (Figure 7). The dynamic range at each frequency wascalculated as the ratio of the maximum force at 4 A to that at 0 A. In the case of 1 Hz, a dynamic range of 157 times
was obtained as the ratio of 94.7 to 0.6 N. This experimental value corresponded with the theoretical one calculatedusing the shear stress of the MR grease. Similarly, dynamic ranges of 125 and 110 times were obtained at 3 and
5 Hz, respectively.
As far as we surveyed, the dynamic range values of the shear-type MR grease damper were much higher thanthose of conventional MR dampers (Table 3). If an MR fluid were applied to a shear-type damper, a high dynamic
range could be obtained due to its low viscosity in the OFF-state. However, in practical applications, an MR fluid
cannot be used as the working fluid of a damper without sealing elements since it may leak out during operation.
On the other hand, the semi-solid MR grease, which allowed the sealing elements to be removed, contributed to an
extremely high dynamic range of over 100 times, which has never been achieved before. Moreover, the eliminationof the sealing elements also improved the damper reliability as the number of mechanical parts decreased.
Compared with a previous study (Shiraishi and Misaki, 2016), the dynamic range of the shear-type MR grease
damper was increased. In this study, the MR grease amount was decreased to a minimum to just cover the crosssection of the iron core in the ON-state based on the attractive force acting on the MR grease by the magnetic field.
In this condition, the damper could keep a large damping force in the ON-state and minimize the damping force in
the OFF-state.
These results indicate that the shear-type MR grease damper has the capability of successfully suppressing
vibrations, as its high dynamic range allows it to generate the required damping forces.

(a)

(b)

(c)
Figure 6. Relationship between the damping force and damper velocity at (a) 1 Hz, (b) 3 Hz, and (c) 5 Hz.

Figure 7. Relationship between the maximum damping force and current. The forces were approximated as linear current
functions (dotted line).
Table 3. Experimental dynamic ranges of conventional MR dampers and the shear-type MR grease damper.
Damper type

Dynamic range

Frictionless MR fluid damper (Machacek et al., 2019)9

Pressure-type MR fluid damper (Yang et al., 2002)

10

Pressure-type MR fluid damper (Tu et al., 2011)18

Shear-type MR grease damper in the present study

157

3.2 Dispersion stability of the MR grease in the damper
Methods
The damping force was measured before the static period and 9 days after it to investigate the dispersion stability
effect on the damping force (Table 4). The developed MR grease and a conventional MR fluid were used as working
fluids for the shear-type damper. It was examined whether the high dispersion stability of the MR grease contributed
to keeping the damper’s damping force for a long time or not by comparing the forces of the shear-type damper inthe cases of using the MR grease and MR fluid. The force was measured when starting to move the damper justafter stirring and 9 days after the static state, as the particle sedimentation would degrade the performance. The
damper plate was excited by a sine wave with a displacement amplitude of 10 mm and a frequency of 1 Hz. The
currents of 0 and 4 A were supplied to the electromagnetic coil, which had 125 turns. These conditions were set
considering a small structure for the vibration suppression tests described below.
Table 4. Experimental conditions for investigating the dispersion stability in the MR damper.
Static period

9 days

Displacement amplitude

10 mm

Frequency

1 Hz

Current

0 A (OFF), 4 A (ON)Winding number of the coil

125 turns

Temperature

40°C

Results and discussion
In the OFF-state with the current of 0 A, the damping force of the MR grease damper was kept for 9 days, whereasthat of the MR fluid damper significantly increased (Figure 8). However, in the ON-state with the current of 4 A,
both the MR grease and fluid dampers kept their damping forces (Figure 9). Regarding the change in the root meansquare (RMS), the damping force was evaluated 10 s after the beginning of excitation. The MR grease damper
generated almost the same damping force in the OFF and ON states, even 9 days after the static state. However, thedamping force of the MR fluid damper increased by 41.6% in the OFF-state (Table 5). The redispersibility of the
MR fluid was so low that it took more than 1 h to return to the desired performance.
Considering the static observation results, the produced supernatants, which had a volume fraction of 10%, led
to the performance degradation of the MR fluid damper 9 days after the static state. In the OFF-state, the dense
particles around the bottom of the box increased the viscosity of the MR fluid and caused an undesired large
damping force. In the ON-state, the damping force of the MR fluid damper could be kept by the magnetic attractive
forces. According to the visual observation of the MR fluid surface, the magnetic particles became supernatants
and fully covered the cross-section of the iron core as soon as the magnetic field was applied. The same experiment
was conducted at the lower current of 1 A to examine the field strength effect, and it was found that the damping
force generated on day 10 corresponded to that generated on day 1 although the attractive force was smaller thanthat at 4 A. These findings indicate that the MR fluid sedimentation affected the performance in the OFF-state
9 days after the static state.
These results also indicate that the high dispersion stability of the MR grease allows its dampers to generate
desired damping forces both in the OFF and ON states. When the damper was applied to the semi-active vibration
suppression of a structure, the vibration response could be successfully suppressed for a long time, even after thestatic state as the damper could generate a force corresponding to the desired force calculated based on the control
law.

(a)

(b)
Figure 8. Time variation of the damping force of (a) the MR fluid damper and (b) MR grease damper in the OFF-statewith the current of 0 A when starting to move, just after stirring, and 9 days after the static state.

(a)

(b)
Figure 9. Time variation of the damping force of (a) the MR fluid damper and (b) MR grease damper in the ON-statewith the current of 4 A when starting to move, just after stirring, and 9 days after the static state.

Table 5. Changes in the RMS damping force of the MR dampers when starting to move on days 1–10 after the staticstate.
MR fluid damper

4.

MR grease damper

Day 1

Day 10ChangeDay 1

Day 10Change

[N]

[N]

[%]

[N]

[N]

[%]

OFF (0 A)

0.09

0.12

41.6

0.96

0.89

−6.8

ON (4 A)

3.08

3.06

−0.7

4.08

4.25

4.2

Structural vibration suppression test
To investigate the semi-active vibration suppression performance of the shear-type MR grease damper, the damper
was applied to the vibration suppression of a structure. A small single-degree-of-freedom model structure equippedwith the damper was subjected to actual seismic excitation (Figure 10). By evaluating the vibration transmissibility
over a frequency range including the natural frequency, it was examined whether the damper could perform wellregarding the vibration suppression of the structure.

Figure 10. Schematic diagram of the small structure equipped with the shear-type MR grease damper.
4.1 Methods
The semi-active vibration suppression experiment comprises three procedures: base excitation, displacement
measurement, and semi-active current control (Figure 11). First, the base floor of the structure was excited by anelectromagnetic exciter using a seismic waveform transmitted from a controller. Then, the displacement response
and input were measured by laser displacement sensors on the target and base floors, respectively. Finally, control
currents calculated in the controller were supplied from a bipolar power supply to the electromagnetic coil in the
MR damper. The current value was determined based on a control law and was updated every 5 ms. To compare
the performance of the semi-active control with that of the passive states, this experiment was also conducted withconstant currents. The currents of 0 and 4 A were supplied in both the OFF and ON states, respectively (Table 6).
The winding number of the coil was determined considering the force magnitude and response time of the damper.
In the case of 125 turns, the damping force in the ON-state was large enough to fix the upper and base floors at anatural frequency of 2.5 Hz. The response time with the coil was fast enough that the supplied current coincidedwith the theoretical value of the controller.
Subsequently, the excitation input of the actual seismic wave and the control law of the skyhook-based control
were described as important conditions to demonstrate that the shear-type MR grease damper can suppress
vibrations by significantly changing the damping characteristics against the base motion of an earthquake.

Figure 11. Schematic diagram of the experimental setup for the structural vibration suppression test.
Table 6. Experimental conditions for the structural vibration suppression test.
Current for passive

0 A (OFF), 4 A (ON)Current for semi-active

0–4 A (Variable)

Excitation input

Actual seismic wave

Displacement amplitudeof the base floor

4.8 mmDominant frequency

2.0–10.0 Hz

Conventional control

ON–OFF control

Proposed control

Skyhook-based control

Winding number of the coil

125 turns

Temperature

40°C

Excitation input
The small structure was subjected to an actual seismic wave. The recorded data during the 2011 earthquake thattook place off the Pacific coast of Tohoku were used. The excitation input was the North–South component of thedisplacement observed in Namie Town, which was severely damaged by that earthquake. Its amplitude was scaled
to a maximum of 4.8 mm considering the limitations of the displacement sensors and exciter (Figure 12). The
frequency was modified so that the frequency of the original data became 10 times higher to examine whether thedamper can provide high suppression performance not only around the natural frequency but also in the highfrequency range. The dominant frequency of the modified displacement was widespread from 2 to 10 Hz (Figure13). Note that the frequency components lower than 2.0 Hz in the input data were removed by a low-pass filter as
the exciter cannot work well at frequencies below 2.0 Hz.
This excitation input was suitable for demonstrating the advantages of semi-active vibration suppression.
Conventional passive vibration suppression methods cannot suppress the vibration caused by such inputs, as they
have a wide range of dominant frequencies. In the passive states, the suppression performance degraded becauseof the fixed damping ratios. For example, when the damping ratio was always high, the response at high frequencies
could not be suppressed. However, the response at low frequencies, including the natural frequency, could be
successfully suppressed. Nevertheless, semi-active vibration methods can provide appropriate damping ratios
according to the dominant frequency. The strategy of realizing the damping ratio variation with the vibration stateis described, as shown below.

Figure 12. Time series of the input displacement measured on the base floor.

Figure 13. Fourier spectrum of the input displacement.

Control law
To achieve high-performance vibration suppression, an appropriate control law for dampers should be used. The
advantages of the shear-type MR grease damper are its high dynamic range and continuous controllability. In thisstudy, a semi-active control law based on the skyhook damper scheme (Karnopp et al., 1974) was applied to makeuse of these advantages. The skyhook damper scheme proposed that the vibrations of single-degree-of-freedom
systems can be successfully suppressed over an entire frequency range by exerting a damping force to reduce theabsolute velocity of the suppression target. The desired damping force 𝐹𝐹𝑑𝑑 can be expressed as
𝐹𝐹𝑑𝑑 = 𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥̇ ,(1)

where 𝑥𝑥̇ denotes the target velocity. However, 𝐹𝐹𝑑𝑑 could not be achieved in this experiment as the damper between
the upper and base floors generated forces to reduce the relative velocity 𝑥𝑥̇ − 𝑥𝑥̇ 0 , where 𝑥𝑥̇ 0 is the base velocity.

In this study, a skyhook-based control was proposed to generate the desired force within the achievable range. By
controlling the current to the electromagnetic coil in the damper, the damping force was varied according to thefollowing equation.
𝐹𝐹(𝐼𝐼 ), 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) > 0
𝐹𝐹 = � 𝑐𝑐
𝐹𝐹𝑂𝑂𝑂𝑂𝑂𝑂 , 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) ≤ 0(2)When the desired and actual forces were in the same direction, the damping force was varied with the vibration
state of the structure by the control current 𝐼𝐼𝑐𝑐 . When they were opposite, the force was minimized by reducing thecurrent to 0 A so as not to degrade the response.

To achieve the desired force 𝐹𝐹𝑑𝑑 , the control current 𝐼𝐼𝑐𝑐 was determined by the target velocity 𝑥𝑥̇ , relative

velocity 𝑥𝑥̇ − 𝑥𝑥̇ 0 , and the parameters representing the relationship between the force, damper velocity, and currentof the shear-type MR grease damper. In the structure, the damper velocity corresponded to the relative velocity

between the floors, 𝑥𝑥̇ − 𝑥𝑥̇ 0 . First, the force was approximated as a linear function of the damper velocity at eachcurrent (Figure 14). The magnitude of the damping force at the current of 𝐼𝐼, 𝐹𝐹(𝐼𝐼), could be expressed as
𝐹𝐹(𝐼𝐼) = 𝑎𝑎(𝐼𝐼)𝑣𝑣 + 𝑏𝑏(𝐼𝐼),(3)where 𝑣𝑣 denotes the relative velocity magnitude, |𝑥𝑥̇ − 𝑥𝑥̇ 0 |. Then, the gradients 𝑎𝑎(𝐼𝐼) and intercepts 𝑏𝑏(𝐼𝐼) wereapproximated as quadratic functions of the current (Figure 15). The force could be expressed as
𝐹𝐹(𝐼𝐼) = (𝑎𝑎2 𝐼𝐼 2 + 𝑎𝑎1 𝐼𝐼 + 𝑎𝑎0 )𝑣𝑣 + (𝑏𝑏2 𝐼𝐼2 + 𝑏𝑏1 𝐼𝐼 + 𝑏𝑏0 ).

(4)

𝐹𝐹(𝐼𝐼) = 𝑐𝑐𝑐𝑐 + 𝑏𝑏0(5)

Using the damping coefficient 𝑐𝑐 (Figure 14), 𝐹𝐹(𝐼𝐼) could be also expressed as
From Eq. (4) and Eq. (5), the control current 𝐼𝐼𝑐𝑐 could be expressed as

𝐼𝐼𝑐𝑐 =−(𝑎𝑎1 𝑣𝑣+𝑏𝑏1 )+�(𝑎𝑎1 𝑣𝑣+𝑏𝑏1 )2 −4(𝑎𝑎2 𝑣𝑣+𝑏𝑏2 )(𝑎𝑎0 −𝑐𝑐)𝑣𝑣
2(𝑎𝑎2 𝑣𝑣+𝑏𝑏2 ).

(6)

By applying Eq. (6) as 𝐼𝐼𝑐𝑐 in Eq. (2), the damping force proportional to the relative velocity could be achieved. In

this study, to generate the desired damping force proportional to the absolute velocity instead of the relative velocity,
the damping coefficient 𝑐𝑐 was updated using an equation expressed as𝑥𝑥̇

𝑐𝑐 = 𝑐𝑐𝑑𝑑 𝑥𝑥̇ −𝑥𝑥̇ ,0(7)

where 𝑐𝑐𝑑𝑑 denotes the proportional gain. Eq. (7) was proposed in a previous study as “continuous skyhook control”

(Liu et al., 2005). By ignoring the intercept 𝑏𝑏0 , the actual damping force brought by Eq. (7) could be expressedusing the following equation.
𝑥𝑥̇

(8)

𝐹𝐹(𝐼𝐼𝑐𝑐 ) ≅ 𝑐𝑐𝑑𝑑 𝑥𝑥̇ −𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) = 𝑐𝑐𝑑𝑑 𝑥𝑥̇0

The damping force proportional to the target velocity could be achieved. Finally, the proposed skyhook-based
control could be formulated as
𝐼𝐼 , 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) > 0.
𝐼𝐼 = � 𝑐𝑐
0 , 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) ≤ 0(9)

In the following subsection, the results of the proposed skyhook-based control were compared with those of
ON–OFF control, which has been widely used in conventional systems (Hu et al., 2017; Liu et al., 2005). A constantcurrent of 4 A was supplied as 𝐼𝐼𝑐𝑐 in the case of the ON–OFF control.

Figure 14. Approximation of the damping force as linear functions of the damper velocity at each current. The currentdependent gradients and intercepts were obtained at each current.

(a)

(b)

Figure 15. (a) Gradient and (b) intercept approximated as quadratic functions of a current.

4.2 Results and discussion
The displacement responses in the OFF and ON states and with the proposed control were different from each other(Figure 16). The proposed skyhook-based control provided the highest performance in terms of reducing the
maximum displacement response, which is defined as half the peak-to-peak amplitude (Figure 17). Its response
reduction rate compared with the OFF-state was 24.1%, whereas the ON-state and conventional ON–OFF control
only achieved 9.1% and 7.5%, respectively. Similarly, the proposed control achieved the highest response reduction
rate of the RMS displacement (Table 7).
In the frequency domain, when the structural vibration was passively suppressed, there was a typical trade-off
between the vibration suppression performance at low frequencies and that at high frequencies (Figure 18). The
high damping ratio in the ON-state was more effective in suppressing vibrations around the natural frequency of
2.5 Hz, whereas the low damping ratio in the OFF-state provided a higher suppression performance in the highfrequency range. Such a trade-off was overcome by varying the damping ratio based on the vibration state using
the proposed skyhook-based control. Around the natural frequency and in the high-frequency range, thedisplacement transmissibility with the proposed control was reduced to the same level as in the ON and OFF states,
respectively.
In conventional studies, the “continuous skyhook control” effect on the vibration suppression performance has
only been demonstrated using numerical simulations. Experimental demonstrations have not been performed as
there were few dampers with a high dynamic range enough to significantly change the damping coefficient.
However, the shear-type MR grease damper in this study succeeded in changing the damping coefficient 𝑐𝑐 over awide range because of its high dynamic range. The current supplied in the experiment almost coincided with thetheoretical one so that the damping coefficient c could be changed to the desired value without time delay (Figure19). The obtained frequency response indicates that the damper could successfully suppress vibrations not onlyaround the natural frequency but also in the high-frequency range by varying the current to update the damping
coefficient 𝑐𝑐 based on absolute and relative velocities.

These results demonstrate that the shear-type MR grease damper can suppress seismic response over a wide

frequency range by applying skyhook-based control, which can take full advantage of high dynamic ranges, evenwhen the excitation input has a wide range of dominant frequencies.

(a)

(b)

(c)
Figure 16. Time series of the displacement responses (a) in the OFF-state, (b) ON-state, and (c) with the proposed
skyhook-based control.

Figure 17. Enlarged time series of the displacement responses around their peaks.

Table 7. Response reduction rates compared with the OFF-state.
Max [%]RMS [%]

OFF

0

0

ON

9.1

25.6

ON–OFF

7.5

15.6

Skyhook-based

24.1

26.7

Figure 18. Frequency response of the structure equipped with the MR grease damper under seismic excitation.

Figure 19. Enlarged time series of the current with the proposed skyhook-based control.

5.

Conclusion

This study experimentally investigated the dynamic range of a shear-type damper using MR grease. Also, the dispersion
stability effect on the damping characteristics and the semi-active vibration suppression performance of a structure were
investigated using the shear-type MR grease damper. First, a performance test of the damper was conducted to obtainits dynamic range as a ratio of the damping force at a current of 4 A in the ON-state to the damping force at 0 A in the
OFF-state. The results indicated that because of the advantage of combining MR grease and a shear-type damper, thedamper showed an extremely high dynamic range reaching up to 157 times, which is much higher than that ofconventional MR dampers. Second, another performance test of the damper was conducted to examine the time changeof its damping characteristics. The results indicated that because of the high dispersion stability of MR grease, thedamper had long-time reliability, thus keeping the damping characteristics in both the ON and OFF states even 9 days
after the static state. However, the same damper type that used a conventional MR fluid changed the damping
characteristics in the OFF-state. Finally, a vibration suppression test of a small single-degree-of-freedom model structure
was conducted under seismic excitation to investigate the semi-active vibration suppression performance when using
the shear-type MR grease damper. The results indicated that because of the high dynamic range of the damper realizing
the desired damping coefficient in skyhook-based control, the damper had high vibration suppression ability that could
ideally reduce the vibration response in a wide frequency range not only around the natural frequency but also in thefrequency range higher than the natural frequency.
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