1. Introduction
Heavy commercial vehicles, tractors, and construction machinery are obviously excited by vibrations during their operation, due to the poor driving conditions they are in. A vehicle’s tires and suspension system can quickly attenuate the high-frequency vibrational excitations that come from the ground. In contrast, low-frequency vibrational excitations that significantly impact the human body are transmitted from the vehicle floor and seat to the driver or passenger. Therefore, during non-road vehicle driving, the driver must withstand low-frequency and high-intensity vibrations for long periods. These kinds of vibrations can affect the driver’s health, and even cause driving fatigue, leading to safety accidents [
1]. In order to solve this problem, research on reductions in seating system vibrations is important. Seat damping is the simplest, most direct, and most economical method used to meet continually increasing requirements for people’s ride comfort in non-road vehicles. Therefore, research on vehicle seat damping is vital for improving the driving comfort of off-road vehicles.
Currently, the vibration-reduction methods of vehicle seats mainly include passive vibration reduction, semi-active control vibration reduction, and active control vibration reduction. A passive seat suspension has no feedback control system, but instead relies on the spring and damper of the suspension to attenuate vibrational excitation. The differences between semi-active and passive control are that the stiffness or damping parameters of a semi-active control suspension are variable, and that the feedback control of vibrations can be achieved through the output control forces of stiffness or damping elements. An active suspension also achieves strong damping effects through feedback control. The difference between active and semi-active control is that the control forces of an active control suspension are output by a dedicated actuator.
The passive vibration-reduction method is widely used because of its simple structure and low cost. However, the stiffness and damping coefficients of a passive seat cannot change with changing working conditions, and it is difficult to attenuate low-frequency vibrations with a passive vibration-reduction method [
2]. The active control method can effectively suppress low-frequency vibrations and adapt to various working conditions; however, active control devices are expensive, and require highly reliable actuators. The semi-active vibration-reduction method has better adaptability than the passive vibration-reduction method to changing conditions, and has lower costs than the active vibration-reduction method. Therefore, it has strong application prospects [
3].
A magnetorheological fluid (MRF) damper is a new type of semi-active damper that is based on smart materials. This damper is suitable for vehicle suspensions because of its variable damping coefficient and strong vibration-reduction effect. As a result of these advantages, the magnetorheological fluid damper is widely used in vehicle seat vibration-reduction applications. Phu, D. [
4] developed a seat suspension that used an MRF damper combined with a switching adaptive fuzzy controller, and the research results showed that this seat suspension could effectively reduce vibrations transmitted to the driver. Deng [
5] developed a novel compact rotary magnetorheological damper with variable stiffness and variable damping characteristics that could reduce the amplitudes of vibrations effectively. Zhang [
6] developed an innovative seat suspension that operated with an MRF damper combined with a back-stepping sliding mode controller, and the results showed that using this seat suspension could control low-frequency vibrations and achieve better robustness.
For commercial and off-road vehicles, driving behaviors such as cornering, acceleration, and obstacle avoidance, as well as other vibrational excitations resulting from harsh working conditions, cause vibrations in multiple directions. Therefore, it is necessary to carry out multi-degree-of-freedom (MDOF) vibration-reduction research on vehicle seats.
The traditional multi-dimensional vibration-reduction methods are mainly divided into two categories. One type uses elastic materials, such as rubber and sponge, to realize multi-dimensional vibration reduction via the elasticity of flexible materials. However, these materials have shortcomings, such as poor heat dissipation and rapid aging, which may lead to unstable vibration-damping performance and low service life [
7]. Another multi-dimensional vibration-reduction method combines a mechanical structure with a damping element to reduce vibration. For example, Ning developed a two-layer MDOF vehicle seat suspension that can simultaneously control vibration in the vertical direction and two rotational directions [
8]. This suspension can effectively attenuate multi-dimensional vibration. However, it is necessary to adopt a multi-layer structure to realize multi-dimensional vibration reduction, and its structure is complex.
A parallel mechanism can simultaneously move in different directions, and can attenuate vibrations in multiple degrees of freedom. This kind of mechanism has been in development for decades. In 1965, Stewart studied the mechanism that was designed by Gough, and invented the Stewart mechanism as a flight simulator [
9]. In 2007, Preumont et al. [
10] developed a six-axis vibration isolation system that used the Stewart platform, which demonstrated a strong damping effect. Behrouz Afzali-Far [
11] developed a Stewart platform with a symmetric structure, and analyzed the dynamic characteristics of the platform. In 2015, Li [
12] developed a 6 DOF vibration-reducing platform using Stewart’s mechanism, and showed that the platform had a strong damping effect on 6 degrees of freedom. In 2020, He et al. [
13] designed a Stewart platform-based parallel support bumper to alleviate the vibrations of inertially stabilized platforms.
While the Stewart mechanism has many advantages in multi-dimensional vibration reduction, it still has some problems. Movement between the legs of a common Stewart mechanism is coupled, making its kinematic and kinetic analyses complex. In order to solve this problem, Geng first proposed a special-house Stewart mechanism, which is a cubic Stewart mechanism. Any two adjacent legs of this mechanism are perpendicular to each other [
14]. A cubic Stewart platform not only simplifies control algorithms and achieves controller decoupling, but also simplifies calculations of kinematics, and mechanical structural design.
Due to the above-described structural characteristics, the sensors installed along each portion’s axis have orthogonality, as do the vibration signals of each leg. That means that we can transform a multi-input, multi-output (MIMO) vibration control problem into a single-input, single-output control problem [
15,
16].
The authors of this article propose a vehicle seat suspension that is based on a cubic configuration Stewart platform and an MRF damper. The main structure of the paper is described below.
In the first section, we introduce the research background of the multi-dimensional damping of vehicle seats, establish the necessity of this research, and introduce the research scheme adopted in this study. In the second section, we establish a dynamic model of the MRF damper, carry out the kinematics analysis of the cubic configuration Stewart platform, and simplify the control measurement parameters according to the symmetrical characteristics of the platform. In the third section, we establish a virtual prototype model of the damping seat platform, and verify the feasibility of simplifying the control measurement. In the fourth section, we describe our experimental research on a multi-DOF seat damping suspension. In the fifth section, we consider semi-active vibration control, and show that the vibration-reduction effect in multiple directions could be effectively improved. In the final section, we summarize the research of the paper and potential future research directions. The research route of the article is shown in
Figure 1.
5. Semi-Active Control Analysis of the Multi-DOF Seat Damping Suspension
For a passive vibration isolation system, the vibration transmissibility can be shown in the following equation:
In this formula, Td is the vibration transmissibility of the passive vibration isolation system, λ is the ratio of the excitation frequency and natural frequency, and ζ is the damping ratio.
Figure 19 shows that when the excitation frequency was 1.414 times higher than the natural frequency (
λ > 1.414), the vibration transmissibility
Td was less than 1, meaning that the vibration isolation system could effectively reduce vibration. However, when
λ < 1.414,
Td > 1, indicating that the vibration was amplified.
Therefore, to achieve better damping performance, it was necessary to reduce the natural frequency of the seat as much as possible. However, if the natural frequency is too low, the deflection of the vibration isolation system will be too large, so it is not easy to reach a low natural frequency. Therefore, it is necessary to use active or semi-active control methods, in order to attenuate low-frequency vibrational excitation.
5.1. Design of the Semi-Active Controller
5.1.1. Semi-Active Control Algorithm
Since the seat suspension platform was based on the cubic Stewart structure, it could achieve dynamic decoupling between the legs of the structure. This meant that the multi-dimensional vibration-reduction problem could be converted into one-way vibration damping for each leg of the vibration-damping platform problem. The one-way vibration model is shown in
Figure 20 and in Equation (13).
In Formula (13),
x1 is the load displacement,
x2 is the excitation displacement,
m is the load mass one-way vibration model,
k is the spring stiffness of the model, and
c is the damping coefficient. For the legs of a seat suspension platform, the damping element in the model is an MRF damper. Therefore, a semi-active control method can be designed for a single leg. The dynamic formula of a single leg is simplified in the following:
where
m is the load mass acting on each leg of the seat suspension platform,
k is the spring stiffness of the leg, and
fMRF is the output damping force of the MRF damper. In semi-active control, this force includes two parts: the passive damping force
fpassive when the input current is 0A, and the semi-active control force
fcontrol realized by the input control current, calculated by the control algorithm.
5.1.2. Fuzzy Skyhook Control Method
Since an MRF damper was used as the output control force in this study, the output control force had to conform to the output law of the MRF damper. That is, the output control force had to be the opposite to the relative velocity of the MRF damper. As a classical control strategy for semi-active suspensions, the control force output of the skyhook control method conforms to this rule. This method has many advantages, such as simple implementation, few control variables, and easy measurement.
The basis of the skyhook control method is the ideal skyhook control method. The principle of the ideal skyhook control method is shown in
Figure 21.
As shown in
Figure 21, the ideal skyhook control method assumes that a damper (
csky) is installed between the sky inertial system and the platform on a seat. During movement of the seat, the method generates a skyhook damping force with an amplitude of
csky ×
v2 (
v2 is the movement speed of the seat load
m), and the direction is opposite to the direction of the seat load movement; thus, the vibration can be reduced in the seat load. However, in actual use, there is no damper connected with the sky inertial system, and the skyhook damping force needs to be output by the MRF damper. Therefore, in the actual skyhook control strategy, when the output damping force of the MRF damper is opposite to the direction of motion of the seat load, the current is added to output the skyhook damping force in order to generate the damping effect; otherwise, the skyhook damping force will not be output. The skyhook control strategy comprises the following equation:
where
csky is the skyhook damping controlled by the current,
vrel is the relative velocity of the MRF damper (upper rod velocity minus the lower rod), and
vs is the upper rod velocity of the damper. For traditional skyhook control, the damping force is generated by a fixed current; that is, when the motion of the suspension meets the requirements of the output damping force as shown in Equation (15), the current switch is turned on, a fixed current is input, and the damping force is output. Otherwise, the current switch is turned off, and only the minimum damping force is output.
However, skyhook control also has disadvantages: (1) it produces an impact load on the suspension system during damping force switching; (2) the MRF damper can only change between fixed current and no current under skyhook control, so it is unable to fully utilize the ability of the MRF damper to continuously adjust the damping force; as a result, it consumes more electric energy. Therefore, a fuzzy algorithm-based skyhook control was examined, to maximize the MRF damper’s continuously adjustable damping advantage.
The input variables of a fuzzy controller are the upper rod velocity
vs and the relative velocity
vrel, and the output is the damping force factor [
24]. This factor is multiplied by the skyhook damping coefficient to obtain the desired damping force. Through multiple calculations on the suspension model, the skyhook damping coefficient was selected as 3000 Ns/m. This study controlled the damping force by the sign of
vs ×
vrel. The fuzzifications of
vrel and
vs are Negative Big (NB), Negative Small (NS), Zero (ZE), Positive Small (PS), and Positive Big (PB), and the fuzzifications of the desired output are Zero (ZE), Small (S), Middle (M), and Big (B). The fuzzy logic control rules are based on the skyhook strategy, and when the
vs and
vrel are NB, the output is the maximum damping force that corresponds to B in the fuzzy control algorithm, and so forth. The final control rules are shown in
Table 3, and the structure of the fuzzy controller is shown in
Figure 22.
As shown in
Figure 20, the inputs of the fuzzy control were
vrel and
vs, which were then multiplied by the quantization factors
Ka and
Kv, respectively, to obtain the actual fuzzy control inputs. The fuzzy output was set as [0, 1], and then the output was multiplied by the scale factor
Ku, in order to obtain the actual fuzzy control output.
5.1.3. Optimization of the Control Algorithm
The control rules and reasoning mechanism of a designed fuzzy controller cannot be adjusted, but the quantization factors Ka and Kv and the scale factor Ku can be adjusted, which significantly influences the controller’s performance. The experience-based selection of those factors was based on trial and error; it was challenging to obtain an optimum. Therefore, the best combination of the quantization factors Ka and Kv and the scale factor Ku of the fuzzy controller were searched using an optimized algorithm to obtain better control.
The grey wolf optimizer (GWO) algorithm is an intelligent population optimization algorithm that was first proposed by Mirjalili in 2014 [
25]. The GWO algorithm has a simple principle, uses fewer adjustment parameters, and little calculation, and has been applied in face recognition, traffic flow prediction, power system control, and other fields. Therefore, we used the GWO algorithm to quantify factors
Ka and
Kv and the scale factor
Ku, to obtain the best control effect.
The optimization process of the GWO algorithm is described below.
The GWO algorithm defines the best result, second best result, and the third best result as
α wolf,
β wolf, and
δ wolf, respectively, which guide hunting; other candidate results are defined as ω wolf, which follows
α wolf,
β wolf, and
δ wolf. The predation behavior of gray wolves is divided into encircling, hunting, and attacking. Hunting by the gray wolf is expressed as follows [
25]:
where
D is the distance between the wolf and its prey;
X(
t) and
Xp(
t) are the current positions of the wolf and prey, respectively;
t is the current number of iterations; and
A and
C are the coefficient vectors.
The specific definitions of
A and
C are as follows:
where
a is a linearly decreasing vector from 2 to 0.
rand1 and
rand2 are a randomly generated vector assignment in the interval [0, 1].
In these formulas, Dα, Dβ, and Dδ are the distances of α, β, and δ wolf from other gray wolves, respectively; Xα, Xβ, and Xδ are the positions of α, β, and δ wolf, respectively; A1, A2, and A3 are coefficient vectors; C1, C2, and C3 are random vectors; and X is the position of the current gray wolf individual.
Finally, the wolf pack assaults the prey. The value of the convergence factor a in Formula 17 linearly decreases from 2 to 0 with the value of a. If |A|≤ 1, the wolf pack assaults the prey, highlighting the local search; if A > 1, the wolf pack gives up hunting the current prey and spreads out, highlighting the global search.
5.2. Input Signal for the Road Driving Simulation
In order to simulate the vibration-reduction effect of a multi-dimensional damping seat platform when driving on the road, it is necessary to establish a vibrational excitation input channel. Considering that non-road vehicles are often subject to vibration in multiple directions when driving on a bumpy road, the force situation of the seat in this context is shown in
Figure 18. In this figure,
ml,
mr,
mv, and
mb are the masses of the left tire, the right tire, the chassis, and the driver’s body, respectively, and
k and
c are the corresponding stiffness and damping coefficients, respectively. The vehicle chassis is excited by the vertical and roll vibrations of the ground. Vibrations are transmitted to the suspension through the tires, and to the chassis by the suspension. In general, because of the slight roughness of the road, the roll vibration is minimal and can be ignored. In this case, the vertical vibration of the seat is reduced, and the ride comfort is significantly improved. Under terrible road conditions, the roll vibration amplitude also increases with an increase in road roughness, and with the different road excitations on the left and right wheels. At this time, the driver’s body can be regarded as the top mass of an inverted pendulum system [
26]. Therefore, this roll vibration causes high lateral accelerations at high speed, which seriously affects the driver’s riding comfort.
Following the analysis results shown in
Figure 23, to simulate the multi-dimensional vibrational excitation encountered by non-road vehicles during driving, the authors of this paper established a semi-active control simulation model of the multi-DOF seat damping platform, as shown in
Figure 24.
As shown in
Figure 24, a flat plate was built under the lower platform of the seat suspension, in order to simulate the floor of the vehicle’s cab. Random vertical vibration signals were simultaneously applied at both ends of the floor, in order to simulate the vibrational excitation of the cab floor when the vehicle passed over rough road. Random vertical vibrations were generated through the power spectrum density of commercial vehicle seat excitation, as specified in the GB/T8419-2007 national standard [
27]. The random vibration signals are shown in
Figure 25.
5.3. Simulation of the Multi-DOF Seat Damping Suspension
After inputting the vibrational excitation, the optimal fuzzy semi-active control method and the passive vibration-reduction method were used to simulate vibrations of the multi-dimensional damping seat platform. First, the GWO algorithm was applied to optimize the quantization factors
Ka and
Kv, and the scale factor
Ku of the fuzzy semi-active control. According to ISO-2631-1:1997 (E) [
27], the six degrees of freedom of the center position of the upper platform of a damping seat had different weights on the comfort evaluation of the vehicle seat. In this study, the weights of translation along the x-,
y-, and
z-axes were all 1, rotation along the x-axis was 0.63, rotation along the y-axis was 0.4, and rotation along the z-axis was 0.2. Since the vibration input of this simulation involved the vertical, horizontal, and roll directions; that is, the translational direction along the
y- and
z-axes and the rotational direction along the
x-axis, the optimization objective function was set as follows:
where
av is the optimization objective of the fuzzy control,
ay is the translational acceleration along the
y-axis,
az is the translational acceleration along the
z-axis, and
arx is the angular acceleration along the
x-axis. After gray wolf algorithm optimization, the fuzzy control parameters
Ka = 6.88,
Kv = 9.18, and
Ku =9.67 were obtained, and the vibration control simulation was carried out according to the parameters. The simulation output the acceleration in the vertical, horizontal, and roll directions of the upper and lower platforms on the seat damping suspension. The results are shown in
Figure 26,
Figure 27 and
Figure 28.
The blue and red curves in
Figure 26,
Figure 27 and
Figure 28 show the acceleration results of the upper platform of the damping seat under passive damping and semi-active controlled damping, respectively, in three directions: rotation around the
x-axis and translation along the
z-axis and
y-axis. The black curves in
Figure 26,
Figure 27 and
Figure 28 show the vibration acceleration signals of the lower platform of the damping seat in the three directions. The root mean square (RMS) values of the simulation results are shown in
Table 4.
Figure 26,
Figure 27 and
Figure 28 show that under random displacement excitation, due to the low excitation frequency of the vibration, when the passive damping method was used, the maximum values of vibration accelerations of the upper platform rotating around the
x-axis, translating along the
z-axis, and translating along the
y-axis, appeared to be amplified in all three directions. After using the optimized fuzzy skyhook semi-active control method proposed in this paper, the vibration accelerations in all three directions were reduced. As can be seen from
Table 4, when the passive damping method was adopted, the RMS values of the vibration accelerations were amplified in all three directions. After using the semi-active control method proposed in this paper, the angular acceleration, and the accelerations in the three directions of rotation around the x-axis, translation along the
z-axis, and translation along the
y-axis, were reduced by 25.3%, 13.4%, and 20.0%, respectively, compared with the passive damping method. Furthermore, after semi-active control, relative to the lower platform of the seat, the upper platform achieved vibration reductions in all three directions. Rotation around the
x-axis, translation along the
z-axis, and translation along the
y-axis reduced vibrations by 9.4%, 8.9%, and 5.9%, respectively.