Enhanced ride comfort using nonlinear seat suspension with high-static-low-dynamic stiffness

Abstract

To attenuate the low-frequency vibration transmitted to the driver, a nonlinear seat suspension with high-static-low-dynamic stiffness is designed. First, the force and stiffness characteristics are derived. The nonlinear suspension can achieve the quasi-zero stiffness at the static equilibrium position when the structural parameters are properly designed. Then, a car-seat-human coupled model which consists of a quarter car model, a seat suspension, and a 4 degree-of-freedom human model is established to predict the biodynamic response of the driver. Finally, the isolation performance of the high-static-low-dynamic stiffness seat suspension under two typical road excitations is evaluated separately based on the numerical method. The effects of stiffness ratio, damping ratio, and vehicle speed on the ride comfort are investigated. The results showed that the nonlinear seat suspension outperforms the equivalent linear counterpart and can achieve the best ride comfort when the quasi-zero stiffness condition is satisfied.

Keywords

Seat suspension high-static-low-dynamic stiffness ride comfort low-frequency vibration

Introduction

Vehicle drivers usually suffer from fatigue or lumbago due to the whole-body vibration caused by road excitations. In particular, the low-frequency vibration which ranges from 0.5 to 5 Hz is most harmful to human health.¹ Thus, the vehicle suspension and seat suspension play an important role in the vibration attenuation. As is known, the linear suspension is not suitable for low-frequency vibration isolation due to the inherent contradiction between the natural frequency and the static deflection. To further improve the ride comfort, many efforts have been made on the active and semi-active suspensions.^2–4 However, additional energy is required for these controlled suspensions, and the implementations could be complicated. In the past 10 years, the nonlinear vibration isolators with high-static-low-dynamic stiffness (HSLDS) have become the focus of many scholars since they can isolate low-frequency vibration effectively.⁵ The HSLDS vibration isolator can be implemented by combining the linear vibration isolator in parallel with a stiffness corrector, which can provide the negative stiffness in the direction of the load movement. Thus, the positive stiffness can be counteracted by the negative one. Therefore, the HSLDS property can be obtained by designing proper structural parameters, leading to a lower natural frequency. If the stiffness at the static equilibrium position is zero, the quasi-zero stiffness (QZS) is obtained. Therefore, it can be an effective method to attenuate the low-frequency vibration by applying an HSLDS vibration isolator to the seat suspension.

A variety of HSLDS vibration isolators have been proposed. Carrella et al.⁶ developed an HSLDS vibration isolator using two oblique springs as stiffness corrector and analyzed its transmissibility performance analytically. Huang et al.⁷ built a QZS vibration isolator using Euler buckled beams as stiffness corrector and investigated the isolation performance by theoretical and experimental study. Zhou et al.⁸ and Cheng et al.⁹ investigated the dynamic behaviors and the isolation performance of an HSLDS vibration isolator using two cam-roller-spring mechanisms as stiffness corrector. Shan et al.¹⁰ developed an HSLDS vibration isolator which combines two magnetic rings in parallel with the pneumatic spring and verified its effectiveness by experiment. Recently, Yang et al.¹¹ designed a new QZS vibration isolator using connecting rods and tension springs as stiffness corrector. Besides, Su et al.¹² presented a novel HSLDS vibration isolator using two permanent magnets and an electromagnet as stiffness corrector, which can work in the semi-active mode. In addition to isolating translational motion, Zheng et al.¹³ proposed a torsional QZS vibration isolator using a torsion magnetic spring as stiffness corrector. Sun et al.¹⁴ proposed an HSLDS vibration isolator using a parabolic-cam-roller mechanism as stiffness corrector and studied the effects of asymmetric stiffness on the isolation performance. Liu et al.¹⁵ further investigated the effect of Coulomb friction on the displacement transmissibility of a QZS vibration isolator experimentally and found that adding light Coulomb friction is beneficial to the isolation performance.

Besides, Tang and Brennan¹⁶ investigated the isolation performance of an HSLDS vibration isolator under shock excitation and found that it outperforms the equivalent linear counterpart when the excitation magnitude is not too large. Furthermore, Wang et al.¹⁷ analyzed the isolation performance of a QZS vibration isolator under random excitation and demonstrated that the QZS vibration isolator can achieve a lower mean square relative displacement than the equivalent linear one.

It can be concluded from the literature mentioned above that the research is mainly focused on the single degree-of-freedom (SDOF) system. However, less attention is paid to their application on the vehicle seat. Le and Ahn¹⁸ proposed an HSLDS vibration isolator using two symmetric structures as stiffness corrector and applied it to the vehicle seat. But the biodynamic response of a driver was not taken into consideration. Then, Yan et al.¹⁹ proposed an HSLDS seat suspension whose stiffness corrector is similar to the prototype reported in Zhou et al.⁸ A 4-degree-of-freedom (4-DOF) human body model was coupled with the proposed seat suspension. However, the vehicle ride comfort under shock and random excitations induced by uneven road profiles was not investigated. Recently, Zhou et al.²⁰ proposed a QZS suspension and applied it to neonatal transport. But only 2 DOFs were considered. To predict the biodynamic response of a seated human, many lumped parameter models with different degrees of complexity were established and summarized by Liang and Chiang²¹ in detail. An HSLDS seat suspension is developed in this article, which has been proposed in our previous study.²²,²³ Then, a 4-DOF human body model established by Liu et al.²⁴ is considered, which consists of the pelvis, upper torso, viscera, and head. This lumped parameter model can distinguish the responses of the most important human parts. Moreover, a quarter car model is taken into account to predict the biodynamic response more reasonably.

The main contribution of this article is to verify that the proposed HSLDS seat suspension has great isolation performance under shock and random excitations induced by uneven road profiles, which was rarely reported in the literature. The rest of this article is organized as follows. In section “Modeling of an HSLDS seat suspension,” an HSLDS seat suspension with a simple form is developed, and the force and stiffness characteristics are derived. In section “Car-seat-human coupled model,” a car-seat-human coupled model is established, which includes a quarter car model, a seat suspension, and a 4-DOF lumped parameter model of the human body. In section “Isolation performance,” the biodynamic response of driver under shock and random excitations is obtained, respectively, and the ride comfort is analyzed in detail. The main conclusions are drawn in section “Conclusion.”

Modeling of an HSLDS seat suspension

An HSLDS seat suspension consisting of a vertical spring, a stiffness corrector, and a guide mechanism is developed, as shown in Figure 1. The stiffness corrector is mainly composed of horizontal springs, connecting rods, hinge axes, and brackets, as shown in Figure 1(b). The guide mechanism consists of guide rods, rollers, and limiting grooves, which is used to make the loading support only move in the vertical direction, as shown in Figure 1(c). Specially, all the connecting rods have the same length L. Both ends of the connecting rod are hinged with an axis and a bracket, respectively. The horizontal springs are connected with two hinge axes, and their total stiffness is $k_{h}$ . The stiffness corrector is connected with the loading support and the base plate through brackets, while the base plate is fixed to the vehicle floor. Besides, the vertical spring is mainly used to support a load, and its stiffness is $k_{v}$ . When the loading support bears a rated load, it would be balanced in the static equilibrium position where all the connecting rods are on the same horizontal plane. More information can be found in Cheng et al.^22,23

Figure 1.

HSLDS seat suspension model: (a) three-dimensional structural diagram, (b) stiffness corrector, (c) guide mechanism, and (d) physical prototype.

The schematic diagram of the static analysis is shown in Figure 2. $L_{0}$ is the initial length of the horizontal spring and $h_{0}$ denotes the initial distance between points A and B. When the loading support is subjected to a static force F, it will deviate from the initial position by a displacement x, and the length of horizontal spring becomes $L_{1}$ . According to the principle of force balance, the static force is given by

F (x) = f_{v} + f_{1}

(1)

where $f_{v} = k_{v} x$ is the vertical spring force. $f_{1} = f_{h} \tan α$ denotes the force generated by stiffness corrector, and $f_{h} = k_{h} (L_{1} - L_{0})$ represents the horizontal spring force. α is the angle between the connecting rod and the horizontal plane, and $t a n α = (h_{0} - x) / L_{1}$ . Thus, equation (1) can be further written as

F (x) = k_{v} x + k_{h} (h_{0} - x) (1 - \frac{L_{0}}{L_{1}})

(2)

Figure 2.

Schematic diagram of static analysis.

Note that $L_{1} = \sqrt{4 L^{2} - {(h_{0} - x)}^{2}}$ and $h_{0} = \sqrt{4 L^{2} - L_{0}^{2}}$ . Therefore, the restoring force of the seat suspension can be written as

\begin{array}{l} F (x) = k_{v} x + k_{h} (\sqrt{4 L^{2} - L_{0}^{2}} - x) \\ (1 - \frac{L_{0}}{\sqrt{4 L^{2} - {(\sqrt{4 L^{2} - L_{0}^{2}} - x)}^{2}}}) \end{array}

(3)

Equation (3) can be transformed into a nondimensional form

f (u) = u + β (\sqrt{4 - l_{0}^{2}} - u) (1 - \frac{l_{0}}{\sqrt{4 - {(\sqrt{4 - l_{0}^{2}} - u)}^{2}}})

(4)

where $f = F / (k_{v} L), u = x / L, l_{0} = L_{0} / L, and β = k_{h} / k_{v}$ .

The nondimensional stiffness of the seat suspension can be obtained by differentiating equation (4) with respect to u

k (u) = 1 - β (1 - \frac{4 l_{0}}{{(4 - {(\sqrt{4 - l_{0}^{2}} - u)}^{2})}^{3 / 2}})

(5)

It can be easily found that the stiffness has a minimum value at $u = \sqrt{4 - l_{0}^{2}}$ where points A and B coincide, and all the connecting rods are on the same horizontal plane. If the seat suspension bears a rated load, and its static equilibrium position corresponds to the minimum stiffness, the HSLDS property can be achieved by designing proper structural parameters. Besides, the QZS condition can be obtained by setting $k (u = \sqrt{4 - l_{0}^{2}}) = 0$ , which yields

β_{q z s} = \frac{2}{2 - l_{0}}

(6)

By introducing a displacement transformation $q = u - \sqrt{4 - l_{0}^{2}}$ , the restoring force and the stiffness can be rewritten as

f (q) = q - β q (1 - \frac{l_{0}}{\sqrt{4 - q^{2}}}) + \sqrt{4 - l_{0}^{2}}

(7)

k (q) = 1 - β (1 - \frac{4 l_{0}}{{(4 - q^{2})}^{3 / 2}})

(8)

The nondimensional restoring force and stiffness as a function of displacement q for various stiffness ratios are shown in Figure 3. It can be clearly observed that the minimum stiffness is getting smaller as β increases. Note that the minimum stiffness becomes zero when β increases to 4, which means the QZS property is achieved. If β increases further, the minimum stiffness becomes negative, which is undesirable in practice. As a result, when $l_{0}$ is determined, the possible design space for β lies in $(0, β_{q z s}]$ .

Figure 3.

Nondimensional (a) restoring force and (b) stiffness for various β when $l_{0} = 1.5$ .

With the purpose of simplifying the following dynamic analysis, equation (7) can be approximated by a fifth-order Taylor series about $q = 0$ when oscillations are considered

f_{a} (q) = α q + γ_{1} q^{3} + γ_{2} q^{5} + \sqrt{4 - l_{0}^{2}}

(9)

where $α = 1 - β (2 - l_{0}) / 2$ , $γ_{1} = β l_{0} / 16$ , and $γ_{2} = 3 β l_{0} / 256$ .

Then the approximate stiffness of the HSLDS seat suspension is given by

k_{a} (q) = α + 3 γ_{1} q^{2} + 5 γ_{2} q^{4}

(10)

The exact restoring force of the seat suspension and its approximation are shown in Figure 4. It can be observed that the approximate accuracy is dependent on the displacement. Generally, the approximate restoring force matches well with the exact one over the selected displacement range, indicating that it is reasonable to use the approximate expression for the following analysis.

Figure 4.

Exact restoring force and its approximation when $l_{0} = 1.5$ and $β = 4$ .

Car-seat-human coupled model

The car-seat-human coupled model is established as shown in Figure 5. A quarter car model is chosen because of its simplicity and has been widely used for the ride research.²⁵ With the purpose of predicting the biodynamic response of the driver, a 4-DOF lumped parameter model of the human body is introduced. This human model consists of pelvis, torso, viscera, and head, which can distinguish the responses of the most important human parts. Besides the vehicle suspension and the seat suspension, the seat cushion also has the effect of vibration attenuation. Thus, the seat cushion is also considered in the car-seat-human coupled model.

Figure 5.

Car-seat-human coupled model.

As shown in Figure 5, the car-seat-human coupled system has 8 DOFs: x₁, x₂, x₃, x₄, x₅, x₆, x₇, and x₈. It is assumed that the free position of each DOF is the origin of the absolute coordinate system. $x_{0}$ denotes the input from road profile. The tire is modeled by a mass $m_{1}$ , stiffness $k_{1}$ , and damping $c_{1}$ . The vehicle body is represented by a mass $m_{2}$ , stiffness $k_{2}$ , and damping $c_{2}$ . The seat frame is modeled by a mass $m_{3}$ , stiffness $k_{3}$ , and damping $c_{3}$ . The seat cushion is represented by a mass $m_{4}$ , stiffness $k_{4}$ , and damping $c_{4}$ . The human body is modeled by a lumped parameter system comprising mass $m_{i}$ , stiffness $k_{i}$ , and damping $c_{i}$ , for $i = 5, 6, 7, and 8$ .

The tire may depart from the ground occasionally due to uneven road profiles. Therefore, the restoring force $F_{1}$ and the damping force $F_{d 1}$ are given by

F_{1} = {\begin{cases} k_{1} (x_{1} - x_{0}) (x_{0} - x) \geq 0 \\ 0 (x_{0} - x_{1}) < 0 \end{cases}

(11)

F_{d 1} = {\begin{matrix} c_{1} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{0}) (x_{0} - x_{1}) \geq 0 \\ 0 (x_{0} - x_{1}) < 0 \end{matrix}

(12)

Generally, special stoppers are installed in the vehicle suspension with the purpose of avoiding a large stroke and rigid collisions between the tire-wheel assembly and the vehicle body. As a result, the restoring force $F_{2}$ of the vehicle suspension can be modeled by a piecewise function²⁵

F_{2} = {\begin{cases} k_{2 c} (x_{1} - x_{2}) + (k_{2} - k_{2 c}) δ_{2 c} (x_{1} - x_{2}) > δ_{2 c} \\ k_{2} (x_{1} - x_{2}) - δ_{2 e} \leq (x_{1} - x_{2}) \leq δ_{2 c} \\ k_{2 e} (x_{1} - x_{2}) + (k_{2} - k_{2 e}) δ_{2 e} (x_{1} - x_{2}) \leq δ_{2 e} \end{cases}

(13)

where $k_{2 c}$ and $k_{2 e}$ are the stiffness of vehicle suspension stoppers, and $δ_{2 c}$ and $δ_{2 e}$ denote clearance lengths in the vehicle suspension.

In addition, the damping coefficient of the vehicle suspension damper which is placed between the wheel and the vehicle body can take different values in compression and extension processes.²⁶ Thus, the damping coefficient is given by

c_{2} = {\begin{matrix} c_{2 c} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) \geq 0 \\ c_{2 e} ({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{2}) < 0 \end{matrix}

(14)

where $c_{2 c}$ and $c_{2 e}$ denote the damping coefficients in compression and in extension, respectively.

According to the aforementioned static analysis, the restoring force of the HSLDS seat suspension yields the following expression, corresponding to the absolute coordinate system of the car-seat-human model

F_{3} = - k_{v} (\begin{array}{l} α (x_{2} - x_{3} - h_{0}) + \frac{γ_{1}}{L^{2}} {(x_{2} - x_{3} - h_{0})}^{3} \\ + \frac{γ_{2}}{L^{4}} {(x_{2} - x_{3} - h_{0})}^{5} + h_{0} \end{array})

(15)

and the restoring force of equivalent linear seat suspension is given by $F_{3} = k_{v} (x_{3} - x_{2})$ .

For the selected human model, the stiffness of the pelvis is characterized by a nonlinear function

k_{5} = {\begin{cases} 1.6215 \times 10^{8} {(x_{4} - x_{5})}^{2} (x_{4} - x_{5}) \geq 0 \\ 0 (x_{4} - x_{5}) < 0 \end{cases}

(16)

It is worth to note that $k_{5} = 0$ indicates that the pelvis loses contact with the seat cushion. Similarly, the damping force yields

F_{d 5} = {\begin{matrix} c_{5} ({\overset{\cdot}{x}}_{5} - {\overset{\cdot}{x}}_{4}) (x_{4} - x_{5}) \geq 0 \\ 0 (x_{4} - x_{5}) < 0 \end{matrix}

(17)

The stiffness of the torso is also characterized by a nonlinear function

k_{6} = {\begin{cases} 3.7806 \times 10^{6} + 1.09 \times 10^{7} (x_{5} - x_{6}) \\ - 2.6868 \times 10^{7} {(x_{5} - x_{6})}^{2} (x_{5} - x_{6}) \leq 0.04 \\ 77, 043.56 (x_{5} - x_{6}) > 0.04 \end{cases}

(18)

Moreover, part of the damping coefficients are given by

c_{i} = 2 ζ_{j} \sqrt{m_{j} k_{j}}

(19)

where $ζ_{j}$ is the damping ratio corresponding to the jth DOF, and j = 3, 5, 6, 7, 8.

According to Newton’s second law, the equations of motion of the car-seat-human coupled system are governed by

m_{1} {\overset{\cdot\cdot}{x}}_{1} = - F_{1} - F_{d 1} - F_{2} + c_{2} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) - m_{1} g

(20a)

m_{2} {\overset{\cdot\cdot}{x}}_{2} = F_{2} - c_{2} ({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1}) + F_{3} + c_{3} ({\overset{\cdot}{x}}_{3} - {\overset{\cdot}{x}}_{2}) - m_{2} g

(20b)

\begin{array}{l} m_{3} {\overset{\cdot\cdot}{x}}_{3} = - F_{3} - c_{3} ({\overset{\cdot}{x}}_{3} - {\overset{\cdot}{x}}_{2}) + k_{4} (x_{4} - x_{3}) \\ + c_{4} ({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{x}}_{3}) - m_{3} g \end{array}

(20c)

\begin{array}{l} m_{4} {\overset{\cdot\cdot}{x}}_{4} = - k_{4} (x_{4} - x_{3}) - c_{4} ({\overset{\cdot}{x}}_{4} - {\overset{\cdot}{x}}_{3}) \\ + k_{5} (x_{5} - x_{4}) + F_{d 5} - m_{4} g \end{array}

(20d)

\begin{array}{l} m_{5} {\overset{\cdot\cdot}{x}}_{5} = - k_{5} (x_{5} - x_{4}) - F_{d 5} + k_{6} (x_{6} - x_{5}) \\ + c_{6} ({\overset{\cdot}{x}}_{6} - {\overset{\cdot}{x}}_{5}) - m_{5} g \end{array}

(20e)

\begin{array}{l} m_{6} {\overset{\cdot\cdot}{x}}_{6} = - k_{6} (x_{6} - x_{5}) - c_{6} ({\overset{\cdot}{x}}_{6} - {\overset{\cdot}{x}}_{5}) - k_{7} (x_{6} - x_{7}) \\ - c_{7} ({\overset{\cdot}{x}}_{6} - {\overset{\cdot}{x}}_{7}) + k_{8} (x_{8} - x_{6}) + c_{8} ({\overset{\cdot}{x}}_{8} - {\overset{\cdot}{x}}_{6}) - m_{6} g \end{array}

(20f)

m_{7} {\overset{\cdot\cdot}{x}}_{7} = k_{7} (x_{6} - x_{7}) + c_{7} ({\overset{\cdot}{x}}_{6} - {\overset{\cdot}{x}}_{7}) - m_{7} g

(20g)

m_{8} {\overset{\cdot\cdot}{x}}_{8} = - k_{8} (x_{8} - x_{6}) - c_{8} ({\overset{\cdot}{x}}_{8} - {\overset{\cdot}{x}}_{6}) - m_{8} g

(20h)

where g denotes the acceleration of gravity.

Isolation performance

The main parameter values of the car-seat-human coupled model²⁷ are given in Table 1. Note that the total mass of the human model is about 70% of the weight of the driver because the remaining part is supported by the feet.²⁸

Table 1.

Main parameter values of the car-seat-human coupled model.

Degree of freedom	Mass (kg)		Stiffness (kN/m)		Damping coefficient (Ns/m)		Damping ratio
Tire	m ₁	60	k ₁	200	c ₁	7	$ζ_{1}$	–
Vehicle body	m ₂	375	k ₂	15			$ζ_{2}$	–
			k _2c	700	c _2c	475
			k _2e	700	c _2e	1425
Seat frame	m ₃	8	k ₃	16.163	c ₃	–	$ζ_{3}$	0.3
Seat cushion	m ₄	1	k ₄	37.7	c ₄	159	$ζ_{4}$	–
Pelvis	m ₅	29	k ₅	–	c ₅	–	$ζ_{5}$	0.25
Torso	m ₆	21.8	k ₆	–	c ₆	–	$ζ_{6}$	0.11
Viscera	m ₇	6.8	k ₇	2.8318	c ₇	–	$ζ_{7}$	0.5
Head	m ₈	5.5	k ₈	202.27	c ₈	–	$ζ_{8}$	0.1

The numerical simulation based on the fourth-order Runge–Kutta method is used to solve the differential equations of motion. The initial displacement of each DOF is in its static equilibrium position, and the initial velocity is zero. Then the time-domain response can be obtained for analysis.

Performance under shock excitation

The shock excitation is one of the typical road excitations considered in the simulation. When the vehicle passes with a constant speed $v_{0}$ over a bump road profile, the shock excitation can be characterized by the following function

x_{0} (t) = {\begin{cases} \frac{x_{a}}{2} (1 - \cos \frac{2 π v_{0}}{L_{s}} t) 0 \leq t \leq τ \\ 0 t > τ \end{cases}

(21)

where $τ = L_{s} / v_{0}$ is the shock duration. $x_{a}$ and $L_{s}$ denote the height and length of the bump, respectively. $x_{a} = 0.02 m$ , $L_{s} = 2 m$ , and $v_{0} = 30 km/h$ are selected in this simulation.

The shock responses of different human components for various β are shown in Figure 6. For simplicity, the acceleration histories are shown as a function of the normalized time $t / τ$ . It can be observed that the variation trend of peak acceleration for each human component is the same. Generally, increasing β can reduce the peak acceleration effectively. When β increases to 20, the QZS property is achieved. Therefore, the vibration is attenuated significantly, and a very low peak acceleration is obtained. Moreover, the HSLDS seat suspension can achieve a better performance than the equivalent linear counterpart in terms of peak acceleration. In our previous study,²³ we found that increasing the stiffness ratio can improve the vibration isolation performance under harmonic excitation. Thus, this result is consistent with the previous study.

Figure 6.

Acceleration time histories: (a) pelvis, (b) torso, (c) viscera, and (d) head.

More information about the dynamic pelvis load and the seat suspension stroke is shown in Figure 7. It can be observed that applying an HSLDS seat suspension can achieve a lower dynamic load for the pelvis compared to the linear seat suspension. Specifically, increasing the stiffness ratio can lead to a decrease in dynamic pelvis load. However, the stroke of the HSLDS seat suspension is larger than that of the linear counterpart. This is because the dynamic stiffness of the HSLDS seat suspension is smaller than that of the linear one. Thus, there is a sacrifice in terms of seat stroke. A very large seat stroke is adverse to the operation of the driver and safety. Note that the stroke of the HSLDS seat suspension is still within the allowable range (i.e. 0.08 m).²⁹

Figure 7.

Time histories: (a) dynamic pelvis load and (b) seat suspension stroke.

In order to assess the ride comfort, the peak load is selected as the index for the pelvis, torso, and viscera, which is defined as the product of the mass multiplied by the peak value of acceleration history. In particular, the head injury criterion (HIC)²⁷ is chosen as the performance index for the head. These performance indices are expressed as

P_{i} = m_{i} {| {\overset{\cdot\cdot}{x}}_{i} |}_{m a x}

(22)

H I C = m a x {(t_{2} - t_{1}) {(\frac{1}{(t_{2} - t_{1}) g} \int_{t_{1}}^{t_{2}} {\overset{\cdot\cdot}{x}}_{8} (t) d t)}^{2.5}}

(23)

where $i = 5, 6, 7$ and $t_{1} and t_{2}$ denote any two time points of acceleration history. The integration time is limited to 15 ms.³⁰

The peak loads and the HIC as a function of $ζ_{3}$ for various stiffness ratios are shown in Figure 8. For the system with $β = 12$ , the peak loads and the HIC decrease at first and then increase when $ζ_{3}$ increases. However, when β increases to 20, all performance indices keep going up with the increase of $ζ_{3}$ . The QZS seat suspension can achieve the best isolation performance when a light damping is applied. This is because the QZS seat suspension works in the isolation region. Thus, increasing the linear viscous damping can reduce the isolation performance according to the linear vibration theory. In general, the HSLDS seat suspension outperforms the equivalent linear one over the selected range of $ζ_{3}$ . Besides, selecting a greater stiffness ratio is beneficial to ride comfort. It is worth mentioning that the values of these indices are quite small compared to the corresponding critical values.³⁰

Figure 8.

Peak load as a function of $ζ_{3}$ : (a) pelvis, (b) torso, (c) viscera, and (d) head injury criterion.

Figure 9 shows the seat suspension stroke for various stiffness ratios. It can be observed that both strokes of the HSLDS seat suspension and the linear one decrease with the increase in $ζ_{3}$ . The stroke of the HSLDS seat suspension is always greater than that of the linear counterpart regardless of $ζ_{3}$ . In a word, if the nonlinear seat suspension cannot achieve the QZS property, applying proper damping can improve the ride comfort and obtain a relatively small seat stroke.

Figure 9.

Seat suspension stroke as a function of $ζ_{3}$ .

The effects of stiffness ratio β on the peak loads and the HIC for various damping ratios are shown in Figure 10. It can be observed that increasing β can lead to the reduction of peak loads and HIC. This result is consistent with our previous study²³ in which the harmonic base excitation was considered. Thus, applying the nonlinear seat suspension with QZS property can achieve the best ride comfort. When β is close to 20, increasing $ζ_{3}$ could make the ride comfort worse.

Figure 10.

Peak load as a function of β: (a) pelvis, (b) torso, (c) viscera, and (d) head injury criterion.

More information about the seat suspension stroke is shown in Figure 11. It can be observed that the seat suspension stroke increases at first and then decreases with the increase in β. A peak occurs when β is close to 17. Note that the variation of the seat stroke becomes smaller when greater $ζ_{3}$ is applied. Although selecting a greater stiffness ratio leads to a larger seat stroke, it can be controlled by applying appropriate damping.

Figure 11.

Seat suspension stroke as a function of β.

The effects of bump height $x_{a}$ on the peak loads and the HIC are shown in Figure 12. It can be observed that the peak loads and the HIC all keep going up with the increase in $x_{a}$ . Specifically, a critical bump height exists in both the peak loads and the HIC. When $x_{a}$ is not greater than 0.085 m, the HSLDS seat suspension can outperform the equivalent linear counterpart in terms of isolation performance. However, if $x_{a}$ exceeds this critical bump height, the result is just the opposite. This is because that the isolation performance of the HSLDS seat suspension is dependent on the excitation amplitude. Specifically, the isolation performance deteriorates with the increase in excitation amplitude due to the existence of nonlinear stiffness, which was reported in Huang et al.⁷

Figure 12.

Peak load as a function of $x_{a}$ : (a) pelvis, (b) torso, (c) viscera, and (d) head injury criterion.

More information about the seat suspension stroke is given by Figure 13. It can be observed that increasing bump height can lead to an increase in seat suspension stroke. Similarly, when $x_{a}$ is lower than the critical bump height, the stroke of the HSLDS seat suspension is greater than that of the linear one. Therefore, it can be concluded that if $x_{a}$ is not too large, the HSLDS seat suspension can achieve a better shock isolation performance than the linear one.

Figure 13.

Seat suspension stroke as a function of $x_{a}$ .

There is no doubt that the vehicle speed $v_{0}$ has an obvious influence on the ride comfort since it determines the excitation frequency. The simulation results are shown in Figure 14. It can be observed that a valley and a peak exist in both the peak loads and the HIC. When $v_{0}$ is lower than 40 km/h, increase in $v_{0}$ leads to the decrease in peak loads and HIC. However, the situation becomes different when $v_{0}$ exceeds 40 km/h. The peak loads and the HIC increase at first and then decrease as $v_{0}$ goes up. The natural frequencies of the linearized and undamped quarter car model are 0.97 and 9.53 Hz, which are evaluated using the parameters in Table 1. Thus, a peak occurs when $v_{0}$ is close to 80 km/h at which resonance could occur in the wheel subsystem.

Figure 14.

Peak load as a function of $v_{0}$ : (a) pelvis, (b) torso, (c) viscera, and (d) head injury criterion.

The seat suspension stroke is shown in Figure 15. It can be observed that the seat stroke has the same variation trend as the performance indices. The seat stroke also gets a peak value when $v_{0}$ is close to 80 km/h, which is still within the allowable range. Although the vehicle speed can affect the ride comfort, the HSLDS seat suspension still outperforms the equivalent linear counterpart.

Figure 15.

Seat suspension stroke as a function of $v_{0}$ .

Performance under random excitation

A more realistic road profile is considered in this subsection, which generates random excitation. The stochastic road irregularities can be characterized by the following power spectral density

S_{g} (Ω) = {\begin{cases} S_{g} (Ω_{r e f}) {(\frac{Ω}{Ω_{r e f}})}^{- n_{1}} Ω \leq Ω_{r e f} \\ S_{g} (Ω_{r e f}) {(\frac{Ω}{Ω_{r e f}})}^{- n_{2}} Ω \geq Ω_{r e f} \end{cases}

(24)

where Ω is the spatial frequency, and $Ω_{r e f} = 1 / 2 π$ is a reference spatial frequency. The exponents are selected as $n_{1} = 2$ and $n_{2} = 1.5$ . Sg(Ωref) is the road roughness coefficient. Generally, the samples of random road excitation can be generated using the spectral representation method.²⁵ When a vehicle passes through a given road with constant speed $v_{0}$ , the road irregularities can be approximated by a limited series

x_{0} (t) = \sum_{n = 1}^{N} s_{n} \sin (n ω_{0} t + φ_{n})

(25)

where N is the number of harmonic terms considered in the random excitation. $ω_{0} = 2 π v_{0} / L_{r}$ is the fundamental temporal frequency, $L_{r}$ is the length of the road segment considered, and $φ_{n}$ is treated as a random variable and follows a uniform distribution in the interval [0,2π). $s_{n}$ is the amplitude of each harmonic term and is given by

s_{n} = \sqrt{2 S_{g} (n Ω_{0}) Δ Ω}

(26)

where $Δ Ω = Ω_{0} = ω_{0} / v_{0}$ denotes the spatial frequency increment.

The values of the parameters related to the random excitation are chosen as: $L_{r} = 100 m$ , $N = 200$ , and $v_{0} = 60 km / h$ . According to ISO 2631 standards, the road profiles belonging to class C (average quality) with $S_{g} (Ω_{r e f}) = 64 \times 10^{- 6} m^{3}$ are selected. For simplicity, the root-mean-square (RMS) values of the head acceleration and the seat suspension stroke are chosen as the performance indices. The statistical averaging method is used to obtain the expectation of RMS values by conducting the simulation 100 times randomly. The effects of damping ratio $ζ_{3}$ on the head acceleration and the seat suspension stroke are shown in Figure 16. It can be observed that when β is lower than 20, the head acceleration keeps decreasing with the increase in $ζ_{3}$ . However, when β equals to 20, the head acceleration decreases at first and then increases. Compared with the linear seat suspension, applying the nonlinear one with HSLDS property can achieve lower head acceleration and improve the ride comfort. Furthermore, increasing $ζ_{3}$ can suppress the seat stroke, as shown in Figure 16(b). The stroke of the HSLDS seat suspension is larger than that of the linear one but still within the allowable range.

Figure 16.

RMS values as a function of $ζ_{3}$ : (a) head acceleration and (b) seat suspension stroke.

The influences of stiffness ratio β on the head acceleration and the seat suspension stroke are shown in Figure 17. It can be observed that increasing β is beneficial to vibration attenuation. When β increases to 20, the nonlinear seat suspension can achieve the best ride comfort because the QZS property is obtained. The seat suspension stroke keeps going up with the increase of β when $ζ_{3}$ is 0.1. When $ζ_{3}$ is relatively large, the seat suspension stroke increases at first and then decreases. A peak occurs when β is close to 18. Although selecting larger β leads to a greater seat stroke, it can be suppressed by applying proper damping. Moreover, it can be observed that the variation trends of the head acceleration and the seat stroke are similar to the previous case, showing good consistency.

Figure 17.

RMS values as a function of β: (a) head acceleration and (b) seat suspension stroke.

The effects of vehicle speed $v_{0}$ on the head acceleration and the seat suspension stroke are shown in Figure 18. In general, the head acceleration keeps going up with the increase of $v_{0}$ , no matter whether the seat suspension is linear or nonlinear. This is because the increased speed leads to an increase in vehicle body vibration, which has been given in Papalukopoulos and Natsiavas.²⁷ Although increasing $v_{0}$ can reduce the ride comfort, the nonlinear seat suspension still outperforms the equivalent linear counterpart, especially when the QZS condition is satisfied. Besides, the seat suspension stroke increases slowly when $v_{0}$ increases, which means the vehicle speed has a weak effect on the seat suspension stroke.

Figure 18.

RMS values as a function of $v_{0}$ : (a) head acceleration and (b) seat suspension stroke.

Conclusion

The HSLDS seat suspension is designed to attenuate the low-frequency vibration and improve ride comfort. A car-seat-human coupled model is established to predict the biodynamic response of a driver. The numerical method is used to validate the isolation performance of the HSLDS seat suspension under shock and random excitations induced by uneven road profiles. The results show that the HSLDS seat suspension outperforms the equivalent linear counterpart in terms of ride comfort when the bump height is not too large. The HSLDS seat suspension can also perform a better ride comfort under random excitation. Although applying the HSLDS seat suspension could lead to a greater seat stroke, it is still within the allowable range. Besides, applying a proper seat suspension damping can further improve the ride comfort and suppress the seat stroke. Moreover, the nonlinear seat suspension can achieve the best ride comfort when the QZS condition is satisfied. In a word, the application of the HSLDS seat suspension is a simple and effective way to attenuate the undesirable whole-body vibration.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by National Natural Science Foundation of China (Grant No. 51605209), Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 18KJD460003), Natural Science Research Foundation of Jiangsu Normal University (Grant No. 18XLRS009), and Jiangsu Government Scholarship for Studying Abroad.

ORCID iD

Chun Cheng

References

Lee

Goverdovskiy

Temnikov

AI.

Design of springs with “negative” stiffness to improve vehicle driver vibration isolation. J Sound Vib 2007; 302: 865–874.

Ning

Sun

Zhang

, et al. An active seat suspension design for vibration control of heavy-duty vehicles. J Low Freq Noise Vibr Act Control 2016; 35: 264–278.

Alfadhli

Darling

Hillis

AJ.

The control of an active seat with vehicle suspension preview information. J Vib Control 2018; 24: 1412–1426.

, et al. H∞ control for a semi-active scissors linkage seat suspension with magnetorheological damper. J Intell Mater Syst Struct 2019; 30: 708–721.

Ibrahim

RA.

Recent advances in nonlinear passive vibration isolators. J Sound Vib 2008; 314: 371–452.

Carrella

Brennan

Waters

, et al. Force and displacement transmissibility of a nonlinear isolator with high-static-low-dynamic-stiffness. Int J Mech Sci 2012; 55: 22–29.

Huang

Liu

Sun

, et al. Vibration isolation characteristics of a nonlinear isolator using Euler buckled beam as negative stiffness corrector: a theoretical and experimental study. J Sound Vib 2014; 333: 1132–1148.

Zhou

Wang

, et al. Nonlinear dynamic characteristics of a quasi-zero stiffness vibration isolator with cam-roller-spring mechanisms. J Sound Vib 2015; 346: 53–69.

Cheng

Wang

, et al. On the analysis of a high-static-low-dynamic stiffness vibration isolator with time-delayed cubic displacement feedback. J Sound Vib 2016; 378: 76–91.

10.

Shan

Chen

Design of a miniaturized pneumatic vibration isolator with high-static-low-dynamic stiffness. J Vib Acoust 2015; 137 (4): 045001.

11.

Yang

Zheng

, et al. Structural design and isolation characteristic analysis of new quasi-zero-stiffness. J Vib Eng Technol 2020; 8: 47–58.

12.

Liu

, et al. Theoretical design and analysis of a nonlinear electromagnetic vibration isolator with tunable negative stiffness characteristic. J Vib Eng Technol 2020; 8: 85–93.

13.

Zheng

Zhang

Luo

, et al. Analytical study of a quasi-zero stiffness coupling using a torsion magnetic spring with negative stiffness. Mech Syst Sig Process 2018; 100: 135–151.

14.

Sun

Song

, et al. Effect of negative stiffness mechanism in a vibration isolator with asymmetric and high-static-low-dynamic stiffness. Mech Syst Sig Process 2019; 124: 388–407.

15.

Liu

Zhao

Zhang

, et al. An experiment investigation on the effect of Coulomb friction on the displacement transmissibility of a quasi-zero stiffness isolator. J Mech Sci Technol 2019; 33: 121–127.

16.

Tang

Brennan

MJ.

On the shock performance of a nonlinear vibration isolator with high-static-low-dynamic-stiffness. Int J Mech Sci 2014; 81: 207–214.

17.

Wang

Cheng

Investigation on a quasi-zero-stiffness vibration isolator under random excitation. J Theor Appl Mech 2016; 54: 621–632.

18.

Ahn

KK.

A vibration isolation system in low frequency excitation region using negative stiffness structure for vehicle seat. J Sound Vib 2011; 330: 6311–6335.

19.

Yan

Zhu

, et al. Modeling and analysis of static and dynamic characteristics of nonlinear seat suspension for off-road vehicles. Shock Vib 2015; 2015: 938205.

20.

Zhou

Wang

, et al. Vibration isolation in neonatal transport by using a quasi-zero-stiffness isolator. J Vib Control 2018; 24: 3278–3291.

21.

Liang

Chiang

CF.

A study on biodynamic models of seated human subjects exposed to vertical vibration. Int J Ind Ergon 2006; 36: 869–890.

22.

Cheng

Wang

, et al. Force and displacement transmissibility of a quasi-zero stiffness vibration isolator with geometric nonlinear damping. Nonlinear Dyn 2017; 87: 2267–2279.

23.

Cheng

Wang

Modeling and analysis of a high-static-low-dynamic stiffness vibration isolator with experimental investigation. J Vibroeng 2018; 20: 1566–1578.

24.

Liu

Shi

GH.

Biodynamic response and injury estimation of ship personnel to ship shock motion induced by underwater explosion. Proc 69th Shock Vib Symp 1998; 18: 1–18.

25.

Verros

Natsiavas

Papadimitriou

Design optimization of quarter-car models with passive and semi-active suspensions under random road excitation. J Vib Control 2005; 11: 581–606.

26.

Verros

Natsiavas

Stepan

Control and dynamics of quarter-car models with dual-rate damping. J Vib Control 2000; 6: 1045–1063.

27.

Papalukopoulos

Natsiavas

Nonlinear biodynamics of passengers coupled with quarter car models. J Sound Vib 2007; 304: 50–71.

28.

Griffin

MJ.

Handbook of human vibration. San Diego, CA: Academic press, 2012.

29.

Zhao

Gao

Vibration control of seat suspension using H∞ reliable control. J Vib Control 2010; 16: 1859–1879.

30.

Zong

Lam

KY.

Biodynamic response of shipboard sitting subject to ship shock motion. J Biomech 2002; 35: 35–43.