A simple and novel control strategy for semi-active vibration suppression
by a magnetorheological damper
Authors
Shuto NAGAMATSU* and Toshihiko SHIRAISHI (*corresponding author)
E-mailshutonagamatsu0617@gmail.com; nagamatsu-shuto-df@ynu.jp (Shuto Nagamatsu),
shira@ynu.ac.jp (Toshihiko Shiraishi)
Affiliation
Yokohama National University Graduate School of Environment and Information Sciences
Faculty of Environment and Information Sciences, Artificial Environment(79-7 Tokiwadai, Hodogaya, Yokohama, 240-8501, Japan)
Abstract
Conventional skyhook-based ON–OFF control switches the damping force on a vibration suppression target according
to the sign of the product of the target and relative velocities (which is called the condition function). Here, we propose
a control strategy that uses a novel condition function for improved performance. The proposed strategy is formulated
based on the theory of forced vibration with base excitation. Its effect upon semi-active vibration performance isinvestigated via numerical simulations and experimental tests of the vibration suppression of a small structure equippedwith a magnetorheological (MR) damper. In the simulations, the proposed control strategy can offer high-performancesemi-active vibration suppression, even in the presence of force delays in the damper. The experiments show that thedisplacement response with the proposed control is lower than that with the conventional skyhook-based control overthe entire frequency range; furthermore, the desired performance can be achieved when the proposed condition function
is used with velocity-proportional control. The simplicity and high performance demonstrated by the proposed control
strategy make it applicable to semi-active vibration suppression of practical systems, even in the presence of unavoidable
force delays in controllable dampers.
Keywords
Semi-active vibration suppression, Single-degree-of-freedom, Condition function, ON–OFF control, Skyhook damper
scheme, MR damper

1.

Introduction
Semi-active vibration suppression is a method of suppressing vibration by controlling passive devices withvariable damping or stiffness according to the vibration state. By appropriately designing the control strategy, thedynamic characteristics of the system can be varied as intended without using active devices driven by eternal
power; thus, a high performance can be achieved with a small energy consumption. To successfully suppress thevibration of a single-degree-of-freedom system, a damper must act as a passive ON state around the natural
frequency and as a passive OFF state in the high-frequency range. Unlike passive states that cannot achieve highvibration suppression performance over a wide frequency range because of their constant damping characteristics,
semi-active methods theoretically can achieve it by varying the input signal from the controller to the damper
according to the excitation and/or responses measured by the sensors (Casciati et al., 2012; Saaed et al., 2013;
Symas and Constantinou, 1999).
One of the most successful active control strategies is the skyhook damper scheme proposed for vehicle
suspensions (Karnopp et al., 1974); it has been demonstrated that vibration can be suppressed over the entire
frequency range through the exertion of a damping force to reduce the absolute velocity of the suppression target.
The desired damping force, 𝐹𝐹𝑑𝑑 , is expressed as
𝐹𝐹𝑑𝑑 = −𝑐𝑐sky 𝑥𝑥̇ ,(1)

where 𝑐𝑐sky and 𝑥𝑥̇ denote the proportional gain and the absolute velocity of the target, respectively (Figure 1).

During semi-active vibration suppression by a damper installed between the target and the base, the damper
generates a force proportional to the relative velocity 𝑥𝑥̇ − 𝑥𝑥̇ 0 , where 𝑥𝑥̇ 0 is the base velocity; thus, the desired force
𝐹𝐹𝑑𝑑 cannot be completely achieved. Under these constraints, the damping force is switched according to thefollowing simple equation, which emulates the skyhook damper scheme:
𝐹𝐹 , 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) > 0,
𝐹𝐹 = � 𝑂𝑂𝑂𝑂𝐹𝐹𝑂𝑂𝑂𝑂𝑂𝑂 , 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) ≤ 0.

(2)This equation switches between large damping forces in the ON state (𝐹𝐹𝑂𝑂𝑁𝑁 ) and small ones in the OFF state (𝐹𝐹𝑂𝑂𝑂𝑂𝑂𝑂 )according to the sign of 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ), which is called the condition function (Liu et al., 2005). This switching allows

the damper to generate large forces only when the absolute and relative velocities are in the same direction. Whenthey are opposed, the damping force is minimized so as not to increase the absolute velocity (Figure 2). When 𝐹𝐹𝑂𝑂𝑂𝑂

is provided by variable signals, the continuous skyhook-based control is achieved to exert forces with the required
phase and amplitude. Conversely, to ensure that the force is exerted in the desired direction, when 𝐹𝐹𝑂𝑂𝑂𝑂 is provided

by a single signal, ON–OFF skyhook-based control is achieved. Comparing these two control schemes, ON–OFF

control benefits from a simpler implementation (Liu et al., 2005). The desired performance of the conventional
ON–OFF control is obtained as the relationship between the transmissibility and the frequency ratio. The
transmissibility is defined as the amplitude ratio of the target displacement to the base one, |𝑥𝑥|⁄|𝑥𝑥0 |, whereas thefrequency ratio is defined as the ratio of the excitation frequency to the natural frequency, 𝜔𝜔⁄𝜔𝜔𝑛𝑛 . When thedamping ratios of 1.0 and 0.1 are switched according to the sign of 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) in the simulation, transmissibility

is successfully reduced over the entire frequency range (Figure 3).

However, in experiments, the vibration suppression performance of such conventional ON–OFF control isdegraded because of the delay in the control system and especially that in the force of the damper (Cha et al., 2013;
Qin et al., 2017; Strecker et al., 2015a; Strecker et al., 2015b; Strecker et al., 2018). The delay in switching fromON to OFF when 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) changes from positive to negative induces a degradation of performance, since thelarge damping force is exerted in the opposite direction to that intended. Furthermore, because of the delay inswitching from OFF to ON when 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) changes from negative to positive, the large force required cannot be

exerted to reduce the absolute velocity.

Magnetorheological (MR) dampers have been widely studied as controllable devices with high responses(Dyke et al., 1996; Dyke et al., 1998; Yang et al., 2002; Yao et al., 2002); these are controllable dampers using an
MR fluid composed of micron-sized ferromagnetic particles and base oils. The rheological characteristics of MR
fluids can be varied instantaneously and substantially through the application of an external magnetic field. Anelectric current is converted into a magnetic field by the mounted electromagnet; thus, the damping force of the
MR dampers can be controlled by the supplied currents. The response times of the MR dampers have been reduced
by designing the entire system considering several factors affecting the delay, including the physical dimensions of
the damper, the method of driving the electronics, and the response time of the MR fluid. However, since it isdifficult for a single system to simultaneously satisfy the requirements for physical dimension, electronics, and MR
fluid, the issue of force delays has not been completely overcome. Previous studies have reported that the response
times of the MR dampers are typically several tens of milliseconds (Cha et al., 2013; Strecker et al., 2015b). Thus,
control strategies that can provide high performance, even under delays of several tens of milliseconds in the damper,
should be proposed to successfully suppress vibrations during experiments.
In this study, to achieve high vibration suppression performance in the presence of force delays, control
strategies using a novel condition function are proposed and formulated; furthermore, the advantage of the proposed
control strategy over the conventional one is demonstrated through numerical simulations and experimental testsof the vibration suppression of a small single-degree-of-freedom model structure equipped with an MR damper.

Figure 1. Concept of the conventional skyhook damper scheme.

(a)

(b)

(c)

Figure 2. Emulation of the conventional skyhook damper scheme by ON/OFF switching according to Eq. (2): (a) 𝑥𝑥̇ >0 and 𝑥𝑥̇ − 𝑥𝑥̇ 0 > 0; (b) 𝑥𝑥̇ > 0 and 𝑥𝑥̇ − 𝑥𝑥̇ 0 < 0; (c) 𝑥𝑥̇ < 0 and 𝑥𝑥̇ − 𝑥𝑥̇ 0 < 0. Thick arrows indicate the damping forces.

Figure 3. Desired vibration suppression performance of the conventional skyhook-based ON–OFF control obtained bysimulation. The damping ratios of 1.0 and 0.1 are switched according to the sign of 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ).

2.

Proposed Control Strategy
To achieve a high vibration suppression performance even in the presence of force delays, a simple and novel
control strategy is proposed in this study. It is expressed by the following equation:
𝐹𝐹 , 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) > 0,
𝐹𝐹 = � 𝑂𝑂𝑂𝑂𝐹𝐹𝑂𝑂𝑂𝑂𝑂𝑂 , 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) ≤ 0.

(3)

Eq. (3) that can determine the input signal to the damper by only the target and base velocities is as simple as theabovementioned skyhook-based control. Furthermore, since Eq. (3) does not require the model and parameter
adjustment, it is much simpler than previous model-based controls (Qin et al., 2017; Yoon et al., 2021) and PID
controls (Choi et al., 2016).
Here, the conventional condition function 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) is replaced by 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ). This simple modification

offers a completely different approach to the suppression of vibrations from single-degree-of-freedom systems. The
feasibility of this strategy for semi-active vibration suppression can be demonstrated based on the theory of forced
vibration with base excitation. Considering the theoretical frequency response of the velocity–amplitude ratio 𝑉𝑉

and the phase difference under base excitation (Figure 4), the range of frequency ratios can be divided into threeparts: 0 < 𝜔𝜔⁄𝜔𝜔𝑛𝑛 < 1 , 1 ≤ 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ≤ √2 , and 𝜔𝜔⁄𝜔𝜔𝑛𝑛 > √2 . The sign of 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) can be calculated at eachrange, assuming that the damping ratio of the target system is low. This assumption is valid for practical applications,since the general damping ratios for structures or vehicle suspensions are low enough that the phase difference canbe approximated as 0° at 0 < 𝜔𝜔⁄𝜔𝜔𝑛𝑛 < 1 and 180° at 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ≥ 1. Expressing the base velocity 𝑥𝑥̇ 0 as 𝑣𝑣0 sin 𝜔𝜔𝜔𝜔,
the target velocity 𝑥𝑥̇ can be expressed as follows:𝑥𝑥̇ = �𝑣𝑣

+ � � 𝑥𝑥̇ 0 = +𝑉𝑉𝑣𝑣0 sin 𝜔𝜔𝜔𝜔 ,𝑣𝑣0𝑣𝑣− � � 𝑥𝑥̇ 0 = −𝑉𝑉𝑣𝑣0 sin 𝜔𝜔𝜔𝜔 ,𝑣𝑣0

0 < 𝜔𝜔⁄𝜔𝜔𝑛𝑛 < 1,
1 < 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ,(4)where 𝑣𝑣 denotes the amplitude of the target velocity. The sign of 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) is calculated using these simple

velocity expressions.

The calculation results demonstrate that 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) is 0 or more in 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ≤ √2, whereas it is 0 or less in𝜔𝜔⁄𝜔𝜔𝑛𝑛 > √2 (Table 1); therefore, applying Eq. (3) for the switching of semi-active vibration suppression allows

the damper to act in the ON state when 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ≤ √2 and in the OFF state when 𝜔𝜔⁄𝜔𝜔𝑛𝑛 > √2 (Figure 5). Thisapproach is completely different from the skyhook damper scheme that always dissipates energy during theexcitation. It is interpreted from the viewpoint of ON/OFF switching at 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = √2 on the frequency response
instead of energy transfer and energy dissipation. In terms of physical meaning, the damper is switched to the ON

state when the target and base velocities are in the same direction (Figure 6 (a)). The ON state is also provided
when the absolute value of the target velocity is greater than the base one, even if the velocities are in opposite
directions (Figure 6 (b)). Conversely, the damper is switched to the OFF state when the velocities are in opposite
directions and the absolute value of the base velocity is greater than that of the target (Figure 6 (c)).
This strategy is based on the theory of forced vibration, as opposed to the conventional strategy, which follows
the desired damping force required in the skyhook damper scheme; thus, the performance of the proposed control
can be maintained even in the presence of force delays. Furthermore, the response with the proposed control canbe reduced to the same level as that of the OFF state in the high-frequency range; this theoretically cannot beachieved via the conventional skyhook-based ON–OFF control (Potter et al., 2010).

(a)

(b)
Figure 4. Theoretical frequency responses of (a) the velocity–amplitude ratio and (b) the phase difference at damping
ratios of 0.01 and 0.15.
Table 1. Damping forces provided by the proposed control strategy using 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ). The calculation is performed byexpressing 𝑥𝑥̇ 0 as 𝑣𝑣0 sin 𝜔𝜔𝜔𝜔.

Frequency ratio [-]Base velocity, 𝑥𝑥̇ 0

Target velocity, 𝑥𝑥̇

Amplitude ratio [-]𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 )Sign of 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 )Force by Eq. (3)0 < 𝜔𝜔⁄𝜔𝜔𝑛𝑛 < 1

1 ≤ 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ≤ √2

𝑉𝑉 ≥ 1𝑉𝑉 ≥ 1

𝑣𝑣0 sin 𝜔𝜔𝜔𝜔

+𝑉𝑉𝑣𝑣0 sin 𝜔𝜔𝜔𝜔𝑉𝑉(𝑉𝑉 + 1)𝑣𝑣0 2 sin2 𝜔𝜔𝜔𝜔
0 or more𝐹𝐹𝑂𝑂𝑂𝑂

𝑣𝑣0 sin 𝜔𝜔𝜔𝜔

−𝑉𝑉𝑣𝑣0 sin 𝜔𝜔𝜔𝜔𝑉𝑉(𝑉𝑉 − 1)𝑣𝑣0 2 sin2 𝜔𝜔𝜔𝜔
0 or more𝐹𝐹𝑂𝑂𝑂𝑂√2 < 𝜔𝜔⁄𝜔𝜔𝑛𝑛
𝑣𝑣0 sin 𝜔𝜔𝜔𝜔

−𝑉𝑉𝑣𝑣0 sin 𝜔𝜔𝜔𝜔
0 < 𝑉𝑉 < 1

𝑉𝑉(𝑉𝑉 − 1)𝑣𝑣0 2 sin2 𝜔𝜔𝜔𝜔
0 or less𝐹𝐹𝑂𝑂𝑂𝑂𝑂𝑂

Figure 5. Desired performance of the proposed control strategy using 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) as the condition function.

(a)

(b)

(c)

Figure 6. Relationship between the ON/OFF switching by Eq. (3) and the target and base velocities in the ranges of (a)0 < 𝜔𝜔⁄𝜔𝜔𝑛𝑛 < 1, (b) 1 ≤ 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ≤ √2, and (c) √2 < 𝜔𝜔⁄𝜔𝜔𝑛𝑛 .

3.

Methods
To investigate the effect of the proposed control strategy upon the semi-active vibration suppression
performance, numerical simulations and experimental tests of vibration suppression in a structure equipped withan MR damper were conducted. As a performance indicator, we evaluated the reduction in the absolute
displacement of the upper floor of the structure.
3.1 Target structure
The target of vibration suppression was a small single-degree-of-freedom model structure with a height of 400
mm, equipped with an MR damper (Figure 7). To construct a structure with a low damping ratio, the damper must
generate small damping forces in the OFF state. A shear-type damper using magnetorheological grease (Shiraishi
and Sugiyama, 2015) was used as an appropriate device.
The shear-type MR grease damper was an excellent device, which solved two issues in conventional pressuretype MR fluid dampers. One was the sedimentation of their magnetic particles caused by the density mismatch
between the magnetic particles and their carrier oil, resulting in performance degradation. The other was the low
dynamic range due to the sliding friction force acting on the sealing elements of dampers, where the dynamic range
was the ratio of total force to the uncontrollable force. They were overcome by the MR grease with high dispersion
stability and the shear-type damper without the need for sealing elements, respectively. In practical application,
whereas an MR fluid cannot be used in a damper without sealing elements since it may leak out during operation,
the semi-solid MR grease can be applied to it without leakage. Furthermore, the MR grease, which allowed thesealing elements to be removed, offered very small damping forces in the OFF state and high dynamic range despite
its drawback of higher viscosity than that of MR fluids. A low damping ratio of approximately 0.15 was provided
by the small damping force in the OFF state; in the ON state, the equivalent damping ratio increased significantly
because of the large damping force.
The structure was subjected to a single sine wave excitation with a displacement amplitude of 3 mm. The
excitation frequency ranged from 2.0 to 8.0 Hz, which was determined by considering the frequency limit of theexciter and the natural frequency of 2.5 Hz. This corresponded to the frequency ratio 𝜔𝜔⁄𝜔𝜔𝑛𝑛 from 0.8 to 3.2 (Table2).

To examine the effect of Eq. (3) upon the semi-active vibration suppression performance, the conventional andproposed ON–OFF controls were arranged. Currents of 4 and 0 A were switched according to the sign of 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 )or 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ). Their performances were compared with those of passive states with constant currents. Currents of
0 and 4 A were supplied in the OFF and ON states, respectively.

Figure 7. Schematic diagram of the single-degree-of-freedom structure.
Table 2. Conditions for simulations and experimental tests of structural vibration suppression by an MR damper.
Mass, 𝑚𝑚

1.5 kgDamping ratio in the OFF state

0.15

Currents in the passive states

0 A (OFF state), 4 A (ON state)

Current with the conventional ON–OFF control

4 A 𝑖𝑖𝑖𝑖 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) > 0, 0 A 𝑖𝑖𝑖𝑖 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) ≤ 0

Stiffness, 𝑘𝑘

Current with the proposed ON–OFF control
Waveform of excitation input, 𝑥𝑥0
Input amplitude

Input frequency ratio, 𝜔𝜔⁄𝜔𝜔𝑛𝑛

370 N/m

4 A 𝑖𝑖𝑖𝑖 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) > 0, 0 A 𝑖𝑖𝑖𝑖 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) ≤ 0
Single sine wave
3 mm

0.8–3.2

3.2 Simulation methods
The structure was modeled as a system consisting of a mass, a spring, and a variable damper. If the viscousdamping force is included in the external force 𝐹𝐹MR , the governing equation can be written as follows:𝑚𝑚𝑥𝑥̈ + 𝐹𝐹MR + 𝑘𝑘(𝑥𝑥 − 𝑥𝑥0 ) = 0.

(5)

The simulation for obtaining the displacement response was performed in Microsoft Excel using the fourth-order
Runge–Kutta integration scheme. Some of the simulation results had been verified through comparison with their
exact solutions. The MR damper was installed between the target and base floors; thus, the force 𝐹𝐹MR depended
upon the supplied current 𝐼𝐼 and the relative velocity between floors, 𝑥𝑥̇ − 𝑥𝑥̇ 0 :𝐹𝐹MR = −{𝑎𝑎(𝐼𝐼)|𝑥𝑥̇ − 𝑥𝑥̇ 0 | + 𝑏𝑏(𝐼𝐼)}{1 − exp(− 𝑡𝑡⁄𝜏𝜏)}{1 − exp(−𝛼𝛼|𝑥𝑥̇ − 𝑥𝑥̇ 0 |)}.

(6)

The first and third terms are based on the algebraic model, which has been widely used as a simple model for MR
dampers (Ruangrassamee et al., 2006; Jiang and Christenson, 2011).
The first force-magnitude term means that the force can be calculated using the supplied current 𝐼𝐼 and relative

velocity 𝑥𝑥̇ − 𝑥𝑥̇ 0 . The force magnitude can be expressed as a linear function of the absolute value of the relative

velocity |𝑥𝑥̇ − 𝑥𝑥̇ 0 |, with the gradient and intercept varying with the supplied current 𝐼𝐼. In the first quadrant of theforce–velocity diagram, the force magnitude was approximated as a linear function of the velocity |𝑥𝑥̇ − 𝑥𝑥̇ 0 | ateach current using the least-squares method (Figure 8). The obtained gradient 𝑎𝑎(𝐼𝐼) and intercept 𝑏𝑏(𝐼𝐼) wereapproximated as quadratic functions of the current 𝐼𝐼 according to the least-squares method:𝑎𝑎(𝐼𝐼)|𝑥𝑥̇ − 𝑥𝑥̇ 0 | + 𝑏𝑏(𝐼𝐼) = (𝑎𝑎2 𝐼𝐼 2 + 𝑎𝑎1 𝐼𝐼 + 𝑎𝑎0 )|𝑥𝑥̇ − 𝑥𝑥̇ 0 | + (𝑏𝑏2 𝐼𝐼 2 + 𝑏𝑏1 𝐼𝐼 + 𝑏𝑏0 ).

(7)

Eq. (7) is the detailed form of the first term of Eq. (6). The parameters obtained by the approximations (Table 3)
were substituted into Eq. (7) as fixed parameters in the simulation.
To examine the performance under various delay conditions of the damping force, a second term was added
as a simple exponential form that included the time constant of the force, 𝜏𝜏. A preliminary experiment showed thatthe response time of the shear-type MR grease damper was 60 ms; thus, the time constant was set to four values:
10, 30, 60, and 90 ms.
Finally, the third term, which describes the force–velocity dependence, was added to capture the roll-off at low
velocities. The parameter 𝛼𝛼 was determined such that the simulation results corresponded to the experimental
ones. In the OFF and ON states, the values were set to 10 and 25 s/m, respectively.

Figure 8. Relationship between the damping force, relative velocity, and current of the shear-type MR grease damper.
Table 3. Parameters in Eq. (7) obtained from the approximation of the gradient and intercept as quadratic functions of
the currents.
𝑎𝑎2 [Ns/mA2]
0.48

𝑎𝑎1 [Ns/mA]
0.66

𝑎𝑎0 [Ns/m]
4.77

𝑏𝑏2 [N/A2]
0.14

𝑏𝑏1 [N/A]
0.00

𝑏𝑏0 [N]0.51

3.3 Experimental methods
The semi-active structural vibration suppression experiment comprises four procedures: excitation of the base,measurement of the displacements, calculation of the control current, and supply of the calculated current (Figure9). First, the base floor of the structure was excited by an electromagnetic exciter with a sine wave from a function
generator. Then, during the excitation, the displacements on the target and base floors were measured via laser
displacement sensors. After that, a controller collected and digitalized the measured displacement data every 0.1 ms
and the control current was calculated following the control strategy that determined the current using the floor
velocities. The velocity values were obtained as numerical time derivatives of the measured displacements. They
were moving average values of 50 sampling points; thus, the current values were updated every 5 ms. Finally, thecurrent calculated by the controller was supplied from a bipolar power supply to the MR damper. The displacement
and current data were recorded on a digital oscilloscope.

(a)

(b)
Figure 9. Setup for the structural vibration suppression experiment: (a) photograph and (b) schematic diagram.
4.

Results and Discussion
4.1 Numerical simulation
The proposed ON–OFF control using 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) provides a high vibration suppression performance, which

it maintains even when the response time of the damping force changes from 10 to 90 ms (Figure 10). Around thenatural frequency, the displacement response with the proposed ON–OFF control was always reduced to the samelevel as the desired ON state, whereas that with the conventional ON–OFF control increased as force delays
increased. In the high-frequency range, the proposed ON–OFF control provided superior performance comparedwith the conventional one, regardless of the force response time.
These results demonstrate that the proposed ON–OFF control is more practical, since its performance ismaintained even in the presence of force delays in the damper. As mentioned above, in previous studies, theresponse time typical values of MR dampers have been reported to be several tens of milliseconds. Therefore, whena typical MR damper is applied to the vibration suppression of an actual system, the proposed ON–OFF control
can outperform the conventional one over a wide range of frequencies.

(a)

(b)

(c)

(d)

Figure 10. Simulation results of the frequency response of a structure equipped with an MR damper, having force delaysof (a) 10 ms, (b) 30 ms, (c) 60 ms, and (d) 90 ms upon the application of the conventional and proposed ON–OFF
controls.

4.2 Experiment4.2.1 Proposed ON–OFF control
The proposed ON–OFF control using 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) provided a superior vibration suppression performancecompared with the conventional ON–OFF control using 𝑥𝑥̇ (𝑥𝑥̇ − 𝑥𝑥̇ 0 ) over the entire frequency range. Theirdifference was clear at frequencies higher than 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = √2 (Figure 11); according to the simulation results, theseresults demonstrate that the proposed control strategy is effective not only in the simulation but also in theexperiment with the delay in damping force and that its advantage over the conventional strategy becomessignificant as frequency increases.
Around the natural frequency, the displacement response of the proposed ON–OFF control was reduced to thesame level as in the ON state. The desired performance was achieved by the proposed control. The ratio of the ON
time reached 97% due to 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) being positive for almost the entire period at 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 1.0 (Figure 12 (a)).

In the mid-frequency range, the proposed ON–OFF control outperformed the conventional one in terms ofdisplacement response reduction, although the desired OFF performance was not achieved. The ON time at𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 2.0 reached 67% (Figure 12 (b)); thus, it should be reduced for higher performance. In the high-frequency

range, the proposed ON–OFF control performed well and significantly improved the performance of theconventional ON–OFF control. At 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 3.0, the ratio of the ON time was reduced to only 25% because of𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) being negative for a long period of time (Figure 12 (c)). Although the waveforms of 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) weredistorted due to the control current effect and the velocity calculation of numerical time derivatives and moving
average using 50 sampling points, it did not affect the periodic ON/OFF switching of Eq. (3) and the vibration
suppression performance.
These results demonstrate that the ON–OFF control using 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) can perform as expected according to

the theory of forced vibration with base excitation, especially in the vicinity of the natural frequency and in thehigh-frequency range. For 0 ≤ 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ≤ √2, the velocity–amplitude ratio |𝑥𝑥̇ ⁄𝑥𝑥̇ 0 | is 1 or more, regardless of thedamping ratio; thus, 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) is always 0 or more, as shown by the desired switching (Table 1). This allows the
proposed ON–OFF control to perform as the ON state for a long time and to definitely achieve the desired vibration

suppression performance. In the range of 𝜔𝜔⁄𝜔𝜔𝑛𝑛 > √2, conversely, the performance is degraded by the constantlarge current of 4 A that is supplied when 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) > 0 is achieved. When the control strategy with 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 )was proposed, the damping ratio was assumed to be so low that 𝑥𝑥̇ and 𝑥𝑥̇ 0 were in completely opposite phases.
However, in the experiment, the damping ratio becomes higher than assumed because of the large damping force

exerted at 4 A. It is necessary to adjust the magnitude of the current according to the vibration to improve theperformance of the proposed ON–OFF control in the range of 𝜔𝜔⁄𝜔𝜔𝑛𝑛 > √2, especially in the mid-frequency range

around 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 2.0.

Figure 11. Experimental frequency response of the structure equipped with an MR damper under sine wave excitation
upon applying the conventional and proposed ON–OFF controls.

(a)

(b)

(c)

Figure 12. Time series of the displacement of the target floor (upper), the supplied current (middle), and the condition
function 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) (lower) with the proposed ON–OFF control at (a) 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 1.0 , (b) 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 2.0 , and (c)𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 3.0. The displacements are compared with those with the conventional ON–OFF control.

4.2.2 Proposed velocity-proportional control
This section presents a novel velocity-proportional control scheme by modifying the proposed ON–OFF
control to improve its performance in the mid- and high-frequency range while maintaining a high performance inthe vicinity of the natural frequency. The proposed velocity-proportional control offers a damping force
proportional to the absolute velocity, 𝐹𝐹𝑐𝑐 , when 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) > 0 is achieved:𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) > 0,
𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) ≤ 0.

𝐹𝐹 ,𝐹𝐹 = � 𝑐𝑐
𝐹𝐹𝑂𝑂𝑂𝑂𝑂𝑂 ,(8)This strategy utilizes the velocity-proportional damping force required in the skyhook damper scheme incombination with 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ). The aim of this control strategy is ON/OFF switching at 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = √2, rather thanemulating the forces of the skyhook damper; thus, the name “skyhook” is not included. To achieve the required

force 𝐹𝐹𝑐𝑐 , the supplied current was continuously varied along with the target and base velocities. The damping
coefficient 𝑐𝑐 was updated using the following equation:
𝑐𝑐 = 𝑐𝑐𝑝𝑝 �𝑥𝑥̇𝑥𝑥̇ −𝑥𝑥̇ 0

�,

(9)

where 𝑐𝑐𝑝𝑝 denotes the proportionality constant of 35 Ns/m, which was most effective for experimental response
reduction. When the relative velocity was zero, the control current was switched to 0 A. To facilitate the calculationof Eq. (9), the RMS values of velocity may be applicable though the moving average values of 50 sampling points
were used in this study. By controlling the current according to the velocities to achieve Eq. (9), a damping force
proportional to the absolute velocity can be exerted on the suppression target:|𝐹𝐹𝑐𝑐 | ≅ 𝑐𝑐|𝑥𝑥̇ − 𝑥𝑥̇ 0 | = 𝑐𝑐𝑝𝑝 �𝑥𝑥̇𝑥𝑥̇ −𝑥𝑥̇ 0

� |𝑥𝑥̇ − 𝑥𝑥̇ 0 | = 𝑐𝑐𝑝𝑝 |𝑥𝑥̇ |.

(10)

Although an update cannot be achieved in conventional studies because of their low dynamic ranges, it was realized
in this study by taking full advantage of the high dynamic range of the shear-type MR grease damper.
The proposed velocity-proportional control provided both the desired ON performance around the natural
frequency and the desired OFF performance in the high-frequency range. A clearly improved performance wasachieved compared with that of the ON–OFF control proposed in the previous section, especially in the midfrequency range of 1.8≤ 𝜔𝜔⁄𝜔𝜔𝑛𝑛 ≤2.8 (Figure 13). Around the natural frequency, the displacement response withthe proposed velocity-proportional control was reduced to the same level as in the desired ON state and with theproposed ON–OFF control. The variable current supplied at 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 1.0 was sufficient to suppress the vibration

successfully, although it did not coincide with the constant current of 4 A in the fully ON state (Figure 14 (a)). In
the mid- and high-frequency range, the proposed velocity-proportional control provided the desired OFF
performance in two steps. First, the continuously controlled current allowed the damping force to approach that in
the OFF state; then, since the reduced force lowered the equivalent damping ratio of the target system, theassumption of 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) ≤ 0 in the range of 𝜔𝜔⁄𝜔𝜔𝑛𝑛 > √2 was achieved for a longer time. At 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 2.0, theratio of the ON time was lowered from 67% to 36% by reducing the maximum current from 4.0 to 2.7 A (Figure

14 (b)). Similarly, at 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 3.0 , the ratio of the ON time was lowered from 25% to 17% by reducing the

maximum current from 4.0 to 2.2 A (Figure 14 (c)). These results suggest that the desired vibration suppressionperformance can be achieved by applying 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) in combination with the velocity-proportional control.

Figure 13. Experimental results of the frequency response of the structure equipped with an MR damper under sine
wave excitation upon applying the proposed ON–OFF and velocity-proportional control.

(a)

(b)

(c)

Figure 14. Time series of the displacement of the target floor (upper), the supplied current (middle), and the condition
function 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) (lower) with the proposed velocity-proportional control at (a) 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 1.0, (b) 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 2.0, and(c) 𝜔𝜔⁄𝜔𝜔𝑛𝑛 = 3.0. The displacements and currents are compared with those with the proposed ON–OFF control.

5.

Conclusion
In this study, simple control strategies using a novel condition function 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) were proposed for semi-

active vibration suppression of single-degree-of-freedom systems and formulated based on the theory of forced
vibration with base excitation. Moreover, its effect upon the semi-active vibration suppression performance wasinvestigated using numerical simulations and experiments on a small structure equipped with an MR damper. Ourformulation elucidates that the proposed control strategy can act as an ON state around the natural frequency andas an OFF state in the high-frequency range under the assumption of a low damping ratio of the target system. The
simulation and experimental results demonstrate that ON–OFF control with 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) offers improved vibration

suppression performance compared with that of the conventional control, especially in the high-frequency range.
Furthermore, the velocity-proportional control with 𝑥𝑥̇ (𝑥𝑥̇ + 𝑥𝑥̇ 0 ) achieves the desired performance by exerting small
damping forces upon the suppression target with a continuously controlled current in the mid- to high-frequency

range. These results indicate that the proposed simple control strategy can be applied to practical single-degree-offreedom systems, since it has been shown by both simulation and experiment to provide the desired vibration
suppression performance.
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