Ride performance of driver’s seat suspension system using various dynamics models

Abstract

The research has proposed three isolation models of the negative-stiffness-element (NSE), the damping-element (DE), and the NSE embedded to the small mass (NSE-SM) that was connected with the seat suspension to ameliorate the ride comfort of the driver. A vibration model of the vehicle and seat built under the random and bumpy excitations has been used for evaluating the isolation performance of the NSE, DE, and NSE-SM. The effect of the NSE, DE, and NSE-SM’s dynamic parameters on their isolation performance is then evaluated via the root-mean-square values of the seat displacement (z_ws) and seat acceleration (a_ws). Based on the genetic algorithm, the NSE, DE, and NSE-SM’s dynamic parameters are optimized to fully reflect their isolation performance. The investigation result indicates that the driver’s ride comfort is slightly affected by the geometric dimension ratios of the NSE, DE, and NSE-SM, while the driver’s ride comfort is greatly affected by the damping and stiffness parameters; and the damping and stiffness ratios of the NSE, DE, and NSE-SM. With the optimized parameters of the NSE, DE, and NSE-SM, the simulation results under the different operating conditions of the vehicle show that the DE slightly improves the driver’s ride comfort. Conversely, both the NSE and NSE-SM greatly improve the driver’s ride comfort, especially the NSE-SM. Therefore, the model of the seat suspension embedded by the NSE-SM should be applied for improving the driver’s ride comfort.

Keywords

vehicle dynamic model seat suspension models ride comfort negative-stiffness-element damping-element negative-stiffness-elemen embedded to the small mass genetic algorithm

Introduction

In order to ameliorate the ride performance of the vehicles, the traditional suspensions of the seat and cab were replaced by using the air spring or semi-active seat suspension with its controlled damping coefficient had been researched.^1,2 The optimal control methods of the genetic algorithm and cultural algorithm-fuzzy controls^3–5 or the combined control methods of the fuzzy-PID control, machine learning algorithm, and neuro-fuzzy^6–8 had been also researched to further enhance the vehicle’s ride comfort as well as the seat suspension’s performance. The study results indicated that the vehicle’s ride performance was remarkably ameliorated by semi-active suspensions. However, compared to passive suspensions, the performance of semi-active suspensions had been only improved by 25%–30%, whereas their structure is extremely complex.

In order to further improve the ride comfort of the driver, the model of the negative-stiffness-element (NSE),^9–11 model of the magnetic-negative-stiffness damper,^12,13 and model of the quasi-zero-stiffness¹⁴ had been studied and embedded into the driver’s seat suspension. The effect of the dynamic parameters, geometrical dimension ratios, and stiffness ratio of the NSE on the ride performance was also analyzed based on the dimensionless characteristics between the restoring force/dynamic stiffness and displacement of the seat suspension.^15,16 The different structures of the NSE including the stiffness elements designed by the steel springs, air pistons, or curved mount spring rollers^10,11,17,18 were simulated and analyzed the performance of the seat suspension. The study results indicated that the seat suspension embedded by the NSE not only remarkably decreased the seat’s displacement and acceleration but also reduced the power spectral density seat’s acceleration at the low frequency. To further enhance the isolation performance of the NSE, the NSE’s parameters were also studied and optimized.¹⁶ However, in the studies of the seat suspension added by the NSE, the effect of the parameters of the NSE on the driver’s ride comfort was mainly assessed based on the dimensionless characteristics of the restoring force/dynamic stiffness and displacement of the seat suspension^8,18; and the driver seat displacement,^19,20 whereas the performance of the seat suspension and the driver’s ride comfort need to be assessed through both the indexes of the displacement and acceleration responses of the driver’s seat.^1,5,21 Furthermore, with the NSE embedded into the driver’s seat suspension, the stiffness elements were mainly designed by the pairs of the steel springs or pneumatic springs in the horizontal direction and symmetrical to obtain the quasi-zero stiffness and decrease the vibration of the driver’s seat.^10,11,17,18 As we all know, in all suspension systems of the vehicle, cab, or seat, the dampers must be added to the suspension systems to generate the damping force and extinguish the vibrations. If the stiffness elements of the two horizontal springs in the NSE are replaced by the damping elements (DE) of the dampers, will the isolation performance of the seat suspension system be better? This issue has not been studied yet. Additionally, a stiff and stable linear oscillator with a negative stiffness element designed by two horizontal springs embedded to the small mass that was connected with the driver’s seat via the seat’s suspension (NSE-SM)^19,20 was also proposed to reduce the vibration of the driver’s seat. The research results indicated that the NSE-SM greatly improved the ride performance of the driver. However, comparative research between the NSE-SM and NSE has also not been performed yet. Moreover, in all the above researches of the seat suspension equipped with the NSE and NSE-SM,^9–16,19,20 a simple model of the seat suspension was mostly applied to evaluate the performance of the NSE and NSE-SM. In the actuality, the seat suspension is installed on the vehicle floor, thus, a dynamics model of the vehicle should also be applied to fully evaluate the isolation performance of the seat suspension models.

To solve the above problems, a dynamics model of the vehicle and the seat suspension is established to evaluate the performance of the NSE, DE, and NSE-SM under the different operating conditions of the vehicle. The effect of the design parameters of the NSE, DE, and NSE-SM on the rider comfort of the driver and the deformation of the seat suspension is evaluated based on the root mean square values of the seat suspension deformation (z_ws) and driver’s seat acceleration (a_ws). These design parameters are then optimized via the genetic algorithm to fully reflect the isolation performance of the NSE, DE, and NSE-SM. The decrease of the deformation of the seat suspension and the increase of the driver’s ride comfort are the goal of this study. The novelty of this study is summarized as follows: (1) the vehicle dynamics model and the seat suspension models are established to evaluate the driver’s ride performance; (2) the DE added into the seat suspension is proposed and researched; (3) based on the optimized parameters of the NSE, DE, and NSE-SM, their isolation performance is then evaluated, respectively.

Mathematical models of vehicle and seat

Vehicle’s dynamic model

In order to evaluate the isolation performance of the NSE, DE, and NSE-SM added into the seat suspension on improving the driver’s ride comfort, a vehicle dynamics model is then established in Figure 1 to calculate the mathematical equations. Herein, {z_s, z_b, z₁, z₂} and {m_s, m_b, m₁, m₂} are the vertical displacement and mass of the seat, vehicle body, front and rear axles, respectively; ϕ_b is the pitch angle of the vehicle body; {c_1,2, c_t1,t2} and {k_1,2, k_t1,t2} are the damping and stiffness coefficients of the front/rear suspension systems and front/rear wheels, respectively; l_s and l_1,2 are the distances of the seat and front/rear wheels; v₀ and q_1,2 are the moving velocity and vibration excitation at the front/rear wheel of the vehicle.

Figure 1.

Mathematical model of the vehicle.

Based on the vehicle dynamics model in Figure 1 and Newton’s second law, the differential equations of the vehicle are written as follows:

\begin{array}{l} m_{s} {\ddot{z}}_{s} = F_{s} \\ m_{b} {\ddot{z}}_{b} = F_{1} + F_{2} - F_{s} \\ I_{b} {\ddot{ϕ}}_{b} = F_{1} l_{1} - F_{2} l_{2} - F_{s} l_{s} \\ m_{1} {\ddot{z}}_{1} = F_{t 1} - F_{1} \\ m_{2} {\ddot{z}}_{2} = F_{t 2} - F_{2} \end{array}

(1)

where F_1,2 and F_t1,t2 are the front/rear force responses of the vehicle suspension systems and wheels determined in equation (2), respectively; F_s is the force response of the seat’s suspension calculated in the seat’s dynamic models.

\begin{array}{l} F_{1} = c_{1} ({\dot{z}}_{1} - {\dot{z}}_{b} + l_{1} {\dot{ϕ}}_{b}) + k_{1} (z_{1} - z_{b} + l_{1} ϕ_{b}) \\ F_{2} = c_{2} ({\dot{z}}_{2} - {\dot{z}}_{b} - l_{2} {\dot{ϕ}}_{b}) + k_{2} (z_{2} - z_{b} - l_{2} ϕ_{b}) \\ F_{t 1} = c_{t 1} ({\dot{q}}_{1} - {\dot{z}}_{1}) + k_{t 1} (q_{1} - z_{1}) \\ F_{t 2} = c_{t 2} ({\dot{q}}_{2} - {\dot{z}}_{2}) + k_{t 2} (q_{2} - z_{2}) \end{array}

(2)

Dynamic models of the seat suspension

The traditional seat suspension (TSS) of the vehicle was designed by a damper and steel spring installed in parallel on the vehicle floor.^1-5 Due to the TSS’s low performance in reducing the seat’s vibration, three vibration models of the NSE, DE, and NSE-SM are then embedded into the TSS to enhance the TSS’s isolation performance and improve the rider comfort of the driver.

Seat suspension added by negative-stiffness-element

The NSE designed by two horizontal springs with the same stiffness k_ns is installed horizontally and symmetrically on the vehicle floor at the seat suspension position. In each horizontal spring, one end of the horizontal spring is fixed to the wall and the other end is attached to a sliding guide block. The sliding block is connected with the seat via a hard bar with a length x₃. The dynamic model of the NSE embedded into the TSS is shown in Figure 2. Where x₁, x₂, and x₄ are the distance from the driver’s seat to the wall in the horizontal direction, the initial length, and the length after deformation of the horizontal spring, respectively; z₀ is the distance between the seat’s initial position and vehicle floor; c_s and k_s are the damping and stiffness values of the TSS; F_ns is the restoring force of the NSE while F_s is the total force of the seat suspension.

Figure 2.

Seat suspension added by NSE.

To calculate both the F_ns and F_s, some assumptions of the model are given following: (1) the vibration of the TSS added by the NSE is only generated in the vertical direction; (2) the mass of the hard bar and sliding block of the NSE is very small and is ignored in the calculation; and (3) the generated frictions at the sliding block and joint of the hard bar is also negligible and is ignored.

Therefore, under the impaction of the static load of the driver and seat, the seat is then deformed by a value z₁ from the initial position z₀, as described in Figure 2. The F_s can be calculated as follows:

F_{s} = c_{s} {\dot{z}}_{1} + k_{s} z_{1} + F_{NSE}

(3)

where F_NSE is the restoring force of the NSE in the vertical direction generated by the restoring force F_ns of two horizontal springs. F_NSE is then determined by:

\begin{array}{l} F_{NSE} = 2 F_{n s} \tan α = 2 k_{n s} (x_{2} - x_{4}) \frac{z_{0} - z_{1}}{x_{1} - x_{4}} \\ = 2 k_{n s} (1 - \frac{x_{1} - x_{2}}{Γ}) (z_{0} - z_{1}) \end{array}

(4)

where

Γ = \sqrt{x_{3}^{2} - {(z_{0} - z_{1})}^{2}}

and

z_{0} = \sqrt{x_{3}^{2} - {(x_{1} - x_{2})}^{2}}

By substituting equation (4) into (3) and transforming equation (3), the F_s is rewritten by:

F_{s} = c_{s} {\dot{z}}_{1} + k_{s} z_{1} + 2 k_{n s} (1 - \frac{x_{1} - x_{2}}{Γ}) (z_{0} - z_{1})

(5)

Let λ₁ = x₃/x₂, λ₂ = x₁/x₂, and γ₁ = k_ns/k_s, equation (5) could be rewritten as follows:

F_{s} = c_{s} {\dot{z}}_{1} + k_{s} z_{1} + 2 γ_{1} k_{s} (1 - \frac{λ_{2} - 1}{\tilde{Γ}}) (z_{0} - z_{1})

(6)

where $\tilde{Γ} = \sqrt{λ_{1}^{2} - {(z_{0} - z_{1})}^{2} / x_{2}^{2}}$ ; λ₁ and λ₂ are defined as the ratios of the geometrical dimensions; γ₁ is defined as the stiffness ratio of the NSE.

Under the vibration excitation z_bs from the vehicle floor at the seat suspension position transmitted to the driver’s seat, the generated vibration of the driver’s seat is z_s (see in the same Figure 2). Thus, the seat’s motion equation in equation (1) added by the NSE is rewritten by:

\begin{array}{l} m_{s} {\ddot{z}}_{s} = c_{s} ({\dot{z}}_{b s} - {\dot{z}}_{s}) + k_{s} (z_{b s} - z_{s}) + . . . \\ + 2 γ_{1} k_{s} (1 - \frac{λ_{2} - 1}{\tilde{Γ}}) (z_{0} - z_{1}) \end{array}

(7)

where z_bs = z_b − l_sϕ_b and z₀ − z₁ = z_bs − z_s.

Seat suspension added by DE

Based on the NSE model using two horizontal springs k_ns in Figure 2, two horizontal dampers with their damping coefficient c_nd are proposed to replace for two horizontal springs. Its dynamic model is shown in Figure 3.Where F_nd is the DE’s damping force and F_s is also the total force of the seat suspension.

Figure 3.

Seat suspension added by DE.

Similarly, under the impaction of the static load of the driver and seat, the seat is also deformed by a value z₁ from the initial position z₀, as described in Figure 3. The F_s can be computed as:

F_{s} = c_{s} {\dot{z}}_{1} + k_{s} z_{1} + F_{DE}

(8)

where F_DE is the damping force of the DE in the vertical direction generated by the damping force F_nd of two horizontal dampers. F_DE is computed by:

F_{DE} = 2 F_{n d} \tan α = 2 c_{n d} ({\dot{x}}_{2} - {\dot{x}}_{4}) \frac{z_{0} - z_{1}}{x_{1} - x_{4}}

(9)

where

x_{4} = x_{1} - \sqrt{x_{3}^{2} - {(z_{0} - z_{1})}^{2}} .

Because the value x₂ is unchanged, therefore, ${\dot{x}}_{2} = 0$ and the DE’s displacement velocity is computed by:

{\dot{x}}_{4} = \frac{d}{d t} (x_{1} - \sqrt{x_{3}^{2} - {(z_{0} - z_{1})}^{2}}) = \frac{- (z_{0} - z_{1})}{\sqrt{x_{3}^{2} - {(z_{0} - z_{1})}^{2}}}

(10)

By substituting equation (10) into (9), the F_DE is then rewritten by:

F_{DE} = 2 c_{n d} \frac{{(z_{0} - z_{1})}^{2}}{x_{3}^{2} - {(z_{0} - z_{1})}^{2}} = 2 c_{n d} \frac{{(z_{0} - z_{1})}^{2}}{Γ^{2}}

(11)

Therefore, the F_s in equation (8) is also rewritten by:

F_{s} = c_{s} {\dot{z}}_{1} + k_{s} z_{1} + 2 γ_{2} c_{s} \frac{{(z_{0} - z_{1})}^{2}}{Γ^{2}}

(12)

where γ₂ = c_nd/c_s is defined as the damping ratio of the DE.

Similarly, under the excitation of z_bs from the vehicle floor at the seat suspension position, the seat’s vibration is generated by z_s (see in the same Figure 3). Thus, the seat’s motion equation added by the DE is rewritten by:

m_{s} {\ddot{z}}_{s} = c_{s} ({\dot{z}}_{b s} - {\dot{z}}_{s}) + k_{s} (z_{b s} - z_{s}) + \frac{2 γ_{2} c_{s}}{Γ} {(z_{0} - z_{1})}^{2}

(13)

Seat suspension added by NSE-SM

Based on the NSE-SM model in Refs.19,20, the NSE-SM is designed by the mass of the driver’s seat (m_s) supported by two parallel springs and a damper. The spring and damper with their stiffness and damping values of k_s and c_s of the seat suspension are connected to a small mass (m_a) while a spring of the seat with its stiffness value k_z is connected to the vehicle floor. The structure of NSE in Figure 2 is then embedded into the small mass m_a to reduce the vibration of the seat suspension. The NSS-SM’s dynamic model added into the seat suspension added is shown in Figure 4.

Where F_a is the vertical force of the seat suspension between the small mass and seat mass, F_NSE-SM is the restoring force of the NSE in the vertical direction generated by the restoring force F_ns of two horizontal springs acting on the small mass m_a, and F_s is the total force of the seat suspension equipped with the NSE-SM.

Figure 4.

Seat suspension added by NSE-SM.

Under the impaction of the static load of the driver and seat, the seat’s mass m_s and small mass m_a are then deformed by a value of z^′₁ and z₁ from the initial position z^′₀ of the seat and z₀ of the small mass, as described in Figure 4. Thus, the F_s can be computed by:

F_{s} = F_{a} + k_{z} z_{1}^{'} = c_{s} \dot{Z} + k_{s} Z + k_{z} z_{1}^{'}

(14)

where Z is the suspension displacement between the small mass and seat mass determined by Z = (z^′₀ − z^′₁) − (z₀ − z₁).

The restoring force F_NSE-SM is also determined by:

F_{NSE - SM} = F_{NSE} = 2 k_{n s} (1 - \frac{x_{1} - x_{2}}{Γ}) (z_{0} - z_{1})

(15)

Equation (15) could be also rewritten by:

F_{NSE - SM} = 2 γ_{1} k_{s} (1 - \frac{λ_{2} - 1}{\tilde{Γ}}) (z_{0} - z_{1})

(16)

where

\tilde{Γ}

, λ₁, and λ₂ have been defined in equation (6).

Under the excitation of z_bs from the vehicle floor at the seat suspension position, the motions of the seat and small mass are generated by z_s and z_a (see in the same Figure 4). The motion equations of the seat and small mass added by the NSE-SM are calculated as follows:

{\begin{cases} m_{s} {\ddot{z}}_{s} = c_{s} ({\dot{z}}_{a} - {\dot{z}}_{s}) + k_{s} (z_{a} - z_{s}) + γ_{3} k_{s} (z_{b s} - z_{s}) \\ m_{a} {\ddot{z}}_{a} = - c_{s} ({\dot{z}}_{a} - {\dot{z}}_{s}) - k_{s} (z_{a} - z_{s}) + F_{NSE - SM} \end{cases}

(17)

where γ₃ = k_z/k_s is the stiffness ratio between k_z and k_s; and F_NSE-SM is calculated in equation (16).

Equation (7) of the NSE, equation (13) of the DE, and equation (17) of the NSE-SM added into the driver’s seat suspension are used to simulate and assess their isolation performance in enhancing the ride comfort of the vehicle.

The vibration of the random road surface

In the actual working condition, the vibration excitation of the vehicle is generated by the random road surface when the vehicle is moving. Therefore, to evaluate the isolation performance of the NSE, DE, and NSE-SM, the random road surface at the front/rear wheels of the vehicle established via the power-spectrum-density (PSD) S(n₀) in ISO-8068²² is used for the simulation. Based on the PSD value of S(n₀), the PSD value of the random road surface in the frequency region S(f) is expressed as:⁵

S (f) = S (n_{0}) {(\frac{n}{n_{0}})}^{- 2} \frac{1}{v_{0}} = S (n_{0}) n_{0}^{2} \frac{v_{0}}{f^{2}}

(18)

where n₀ = 0.1 is the reference spatial frequency, n = f/v₀ is the spatial frequency in a meter length of wave number, and f is the random frequencies from 0.1 to 10 Hz.

Based on the white noise’s random signal W(t) and the PSD value of the random road surface S(f) in equation (18), the random road surface of the front/rear wheels could be expressed as:⁵

{\begin{cases} {\dot{q}}_{1} (t) + 2 π n_{0} v_{0}^{2} q_{1} (t) = 2 π n_{0} \sqrt{S (f) v_{0}} W (t) \\ q_{2} (t) = q_{1} {t + (l_{1} + l_{2}) / v_{0}} \end{cases}

(19)

The excitation of the random road surface in the low frequency range remarkably affected the health and ride comfort of the driver,^21,23 especially with f ≤ 10 Hz. Therefore, based on the PSD value of S (n₀) = 64 × 10⁻⁶ m³ of the random road surface of ISO level B in ISO-8068,²² the vibration excitation of the random road surface at the vehicle moving velocity of v₀ = 20 m/s is simulated and plotted in Figure 5. This excitation is then applied to simulate the system model.

Figure 5.

The random excitations of the road surface.

Effect of design parameters on ride performance

Evaluating index

The isolation performance of suspension systems was mainly evaluated via three indexes of ride quality, suspension deformation, and road friendliness.^1–3,23 With the seat suspension of the vehicles, the suspension deformation was assessed based on the root mean square seat displacement (z_ws), whereas the ride performance of the driver was assessed based on the root mean square seat acceleration (a_ws).^5,23 The values of z_ws and a_ws are expressed by:

z_{w s} = \sqrt{\frac{1}{T} \int_{0}^{T} {(z_{b s} - z_{s})}^{2} d t} a nd a_{w s} = \sqrt{\frac{1}{T} \int_{0}^{T} {({\ddot{z}}_{s})}^{2} d t}

(20)

where z_s and

{\ddot{z}}_{s}

have been determined in equation (1); T is the simulation time.

In order to compare the isolation performance between the NSE, DE, and NSE-SM, the decrease of z_ws and a_ws is chosen as the goal study.

Effect of seat suspension’s parameters

The dynamic parameters of the suspension systems significantly affected the vehicle’s ride comfort.^10,14 Therefore, to evaluate the isolation performance of the NSE, DE, and NSE-SM in improving the driver’s ride comfort, based on the dynamic parameters of the vehicle provided in Table 1³ and design parameters of the NSE, DE, and NSE-SM provided in Table 2,^10,19 the effect of the geometrical dimension ratios {λ₁, λ₂}, the stiffness and damping ratios {γ₁, γ₂, γ₃}, and the stiffness and damping values {k_s and c_s} on the ride comfort of the driver are simulated and analyzed under the random excitation of the vehicle in Figure 5, respectively.

Table 1.

Vehicle’s dynamic parameters.

Parameter	Value	Parameter	Value
m_b/kg	1890	k_t1/kN m⁻¹	290
m_a1/kg	38	k_t2/kN m⁻¹	290
m_a2/kg	38	c_t1/kNs m⁻¹	3.1
k₁/kN m⁻¹	29.35	c_t2/kNs m⁻¹	3.1
k₂/kN m⁻¹	24.73	l_s/m	0.20
c₁/kNs m⁻¹	3.89	l₁/m	1.00
c₂/kNs m⁻¹	2.92	l₂/m	1.43

Table 2.

Design parameters of the seat suspension with the NSE, DE, and NSE-SM.

Parameter	Value	Parameter	Value
m_s/kg	85	k_s/kN m⁻¹	40
m_a/kg	5	c_s/kNs m⁻¹	0.35
x₂/m	0.23	γ ₁	0.65
λ ₁	0.78	γ ₂	0.65
λ ₂	1.20	γ ₃	1.35

Effect of geometrical dimension ratios

The effect of the λ₁ and λ₂ of the NSE, DE, and NSE-SM including λ₁ = [0.5, 0.6, ..., 1.4] and λ₂ = [1, 1.1, ..., 2] on the z_ws and a_ws are plotted in Figure 6(a)–(c), respectively.

Figure 6.

Effect of geometrical dimension ratios on z_ws and a_ws. (a) NSE, (b) DE, and (c) NSE-SM.

With the NSE, Figure 6(a) shows that both the λ₁ and λ₂ significantly affect the z_ws and a_ws. The z_ws reaches the minimum value with 0.8 ≤ λ₁ ≤ 1.3 and 1.1 ≤ λ₂ ≤ 1.5, while the a_ws reaches the minimum value with 0.7 ≤ λ₁ ≤ 1.2 and 1.2 ≤ λ₂ ≤ 1.4. Thus, to optimize the driver’s ride comfort, the values of λ₁ and λ₂ should be designed or optimized in the range of 0.7 ≤ λ₁ ≤ 1.3 and 1.1 ≤ λ₂ ≤ 1.5.

With the DE, Figure 6(b) indicates that the λ₁ significantly affects the z_ws and a_ws. The z_ws is increased while the a_ws is reduced with the increase of λ₁ and vice versa. Conversely, the λ₂ does not affect the z_ws and a_ws. This result is due to the F_SE in equation (11) being not impacted by the λ₂. To optimize the driver’s ride comfort, the values of λ₁ should be designed or optimized in the range of 0.7 ≤ λ₁ ≤ 1.1.

With the NSE-SM, Figure 6(c) shows that both the λ₁ and λ₂ also significantly affect the z_ws and a_ws. With the reduction of λ₁ < 1.0 and increase of λ₂ > 1.4, both the z_ws and a_ws are quickly increased and vice versa. To optimize the driver’s ride comfort, the values of λ₁ and λ₂ should be designed or optimized in the range of 0.9 ≤ λ₁ ≤ 1.4 and 1.0 ≤ λ₂ ≤ 1.5.

Effect of stiffness and damping ratios

With the change of the stiffness and damping values of α₁×k_s and α₂×c_s; stiffness ratio γ₁ of the NSE; damping ratio γ₂ of the DE; and stiffness ratios γ₁ and γ₃ of the NSE-SM including α₁ = α₂ = γ₃ = [0.2, 0.4, ..., 2] and γ₁ = γ₂ = [0.1, 0.2, ..., 1]. The effect of the ratios on the z_ws and a_ws are plotted in Figure 7(a)−(c), respectively.

Figure 7.

Effect of the stiffness and damping ratios on z_ws and a_ws. (a) NSE, (b) DE, and (c) NSE-SM.

With the NSE, Figure 7(a) shows that both the α₁ and γ₁ significantly affect the z_ws and a_ws. The z_ws reaches the minimum value with 1.0 ≤ α₁ ≤ 2.0 and 0.6 ≤ γ₁ ≤ 0.8, while the a_ws reaches the minimum value with 0.2 ≤ α₁ ≤ 1.0 and 0.6 ≤ γ₁ ≤ 0.8. To optimize the driver’s ride comfort, the values of α₁ and γ₁ should be designed or optimized in the range of 0.5 ≤ α₁ ≤ 1.5 and 0.6 ≤ γ₁ ≤ 0.8.

With the DE, Figure 7(b) shows that both the α₂ and γ₂ remarkably affect the z_ws and a_ws, especially with the α₂. Both the z_ws and a_ws are increased with the reduction of the damping ratios of α₂ andγ₂ and vice versa. To optimize the driver’s ride comfort, the values of α₂ and γ₂ should be designed or optimized in the range of 1.0 ≤ α₂ ≤ 2.0 and 0.5 ≤ γ₂ ≤ 1.0.

With the NSE-SM, Figure 7(c) indicates that both the γ₁ and γ₃ lightly affect the z_ws, while these two values remarkably affect the a_ws. Especially with the increase of γ₃, the a_ws is also quickly increased. Similarly, to optimize the ride comfort, the γ₁ and γ₃ should be designed or optimized in the range of 0.4 ≤ γ₁ ≤ 0.7 and 0.2 ≤ γ₃ ≤ 1.0.

Based on the above analysis results, it can conclude that the parameters of the NSE, DE, and NSE-SM remarkably affect the driver’s ride comfort. To compare the isolation performance between the NSE, DE, and NSE-SM in improving the driver’s ride comfort and enhancing their isolation performance, all the NSE, DE, and NSE-SM’s parameters should be optimized.

Optimization of seat suspension parameters

In order to optimize the design parameters of the NSE, DE, and NSE-SM of the driver’s seat suspension, the boundary conditions are given based on the analysis results of the effect of the seat suspension’s design parameters on the ride performance as follows:

{\begin{cases} 0.7 \leq λ_{1} \leq 1.4 \\ 1.1 \leq λ_{2} \leq 1.5 \\ 0.5 \leq α_{1} \leq 1.5 \\ 1.0 \leq α_{2} \leq 2.0 \\ 0.4 \leq γ_{1} \leq 0.8 \\ 0.5 \leq γ_{2} \leq 1.0 \\ 0.2 \leq γ_{3} \leq 1.0 \end{cases} a nd {\begin{cases} x_{2} = 0.23 (m) \\ x_{3} = λ_{1} \times x_{2} \\ x_{1} = λ_{2} \times x_{2} \\ c_{n d} = γ_{2} \times c_{s} \\ c_{s} = α_{2} \times 0.35 (kNs / m) \\ k_{n s} = γ_{1} \times k_{s} \\ k_{s} = α_{1} \times 40 (kN / m) \end{cases}

(21)

Therefore, the initial parameters of the NSE, DE, and NSE-SM should be optimized to reach the minimum (MIN) values of z_ws and a_ws of the NSE, DE, and NSE-SM. To solve this problem, the genetic algorithm (GA)^5,23 has been used to optimize the parameters of NSE, DE, and NSE-SM via the object function of G defined in equation (22).

G = MIN {z_{w s} a nd a_{w s}}

(22)

The GA is defined as finding a vector of decision variables of x = [x₁, x₂, …, x_n]^T satisfying the boundary conditions to obtain the maximum or minimum values of all objective functions of G(x) = [g₁(x), g₂(x), ..., g_n(x)]^T. Its structure includes encoding, population initialization, fitness evaluation, parent selection, genetic operations (crossover and mutate), and termination criterion.^3,4 GA is applied to optimize the NSE, DE, and NSE-SM’s parameters as follows.

Finding the vector of x = [λ₁, λ₂, α₁, α₂, γ₁, γ₂]^T of each NSE, DE, and NSE-SM model to obtain the minimum value of G = [z_ws, a_ws]^T via the boundary conditions in equation (21) and vehicle model in Figure 1. The GA’s model and setting parameters for the optimization are given in Figure 8 and Table 3, respectively.

Figure 8.

The optimization model of the GA and vehicle.

Table 3.

GA’s parameters for the optimization.

Parameter	Value
Initial population	150
Mutation probability	0.05
Termination criterion of \|G _i(x) − G_i+1(x)\|	10⁻³
Crossover probability	0.95
Mutation probability	0.05

Based on the boundary conditions in equation (21), vehicle’s excitation in Figure 5, and GA’s parameters in Table 3, the NSE, DE, and NSE-SM’s parameters are then optimized to obtain the MIN G. The optimization results of z_ws and a_ws using the NSE, DE, and NSE-SM are depicted in Figure 9. The distribution densities of G with the NSE, DE, and NSE-SM indicate that both the z_ws and a_ws have been reduced in the optimization process. The MIN G is obtained by {z_ws = 2.709 and a_ws = 0.1276} with the NSE, {z_ws = 2.902 and a_ws = 0.2975} with the DE, and {z_ws = 2.558 and a_ws = 0.11} with the NSE-SM. At this MIN G, the parameters of the NSE, DE, and NSE-SM optimized have been saved and provided in Table 4. The optimized parameters of each NSE, DE, and NSE-SM are then simulated to compare their isolation performance in improving the driver’s ride comfort, respectively.

Figure 9.

The optimization result of z_ws and a_ws using the NSE, DE, and NSE-SM.

Table 4.

Optimized values of NSE, DE, and NSE-SM.

Parameter	NSE	DE	NSE-SM
λ ₁	0.923	0.972	1.058
λ ₂	1.216		1.205
α ₁	0.788		0.788
α ₂		1.443
γ ₁	0.589		0.616
γ ₂		0.889
γ ₃			0.450

Ride performance under the random excitation

Under the excitation of the random road surface in Figure 5, the performance of the NSE, DE, and NSE-SM with their optimal parameters is then simulated and assessed. The results of the displacement and acceleration responses of the driver’s seat are depicted in Figure 10(a) and (b).

Figure 10.

Result of vibration responses under the random excitation. (a) Seat’s displacement and (b) its acceleration.

The results show that both the displacement and acceleration responses of the driver’s seat with the DE are lower than that of the TSS, while these values with the NSE and NSE-SM are greatly reduced compared to the TSS, particularly with the NSE-SM. Based on the simulation results in Figure 10(a) and (b), the calculation results of the z_ws and a_ws are then provided in Figure 11(a) and (b), respectively.

Figure 11.

The calculated result of z_ws and a_ws. (a) Seat displacement and (b) seat acceleration.

With the seat suspension added by the NSE, the values of z_ws and a_ws are smaller than that of the TSS by 22.8% and 65.4%, especially the a_ws. This result could be due to the generated restoring force F_NSE in equation (4), as shown Figure 12(a), affecting the F_s of the seat suspension in equations (6) and (7). Therefore, the driver’s ride comfort with the NSE is better improved in comparison with the TSS. This result is also similar to the results in Refs.11,12,14

Figure 12.

Result of force responses. (a) With NSE and NSE-SM and (b) with DE.

With the seat suspension added by the DE, the z_ws and a_ws are also smaller than that of the TSS by 15.5% and 14.1%. However, both these values are higher than that of the NSE. Based on the analysis result of the DE in Figures 6(b) and 7(b), both the geometrical dimension ratio and damping ratio of the DE insignificantly affect the z_ws and a_ws. Moreover, the damping force F_DE in equation (11) simulated and calculated in Figure 12(b) shows that the F_DE is very small in comparison with the F_NSE; and the root mean square (RMS) value of F_DE is smaller than the RMS value of F_NSE by 99.8%. This result is the value of (z₀ − z₁)² in equation (11) being very small, thus, the F_s of the seat suspension in equations (12) and (13) is insignificantly affected by the F_DE. This means that the DE improves insignificantly the driver’s ride comfort. Thus, it can be the reason that the DE is not investigated and embedded into the seat suspension in the existing studies.

With the NSE-SM, the results of both the z_ws and a_ws are lower than that of the TSS by 26.7% and 69.2%. Concurrently, these values are also smaller than that of both the NSE and DE. This could be due to the effect of F_s = F_a + k_zz^′₁ in equation (14) and F_NSE-SM in equation (16). Thus, the NSE-SM added into the seat suspension can improve the ride comfort better than the NSE and DE.

The acceleration response of the human body in the low frequency region from 0.5 to 4 Hz greatly affected the driver’s health.²¹ To fully evaluate the isolation performance of the NSE, DE, and NSE-SM on improving the endurance limit of the human body in the frequency domain, based on the fast Fourier transform used to calculate the PSD acceleration of the driver’s seat,¹¹ the PSD acceleration response of the driver’s seat in the frequency region plotted in Figure 13 is also analyzed.

Figure 13.

Result of frequency responses using the different seat suspension models.

The resonant frequency with the TSS and DE appears at the same f = 3.098 Hz (due to the stiffness of the seat suspension being unchanged), while that of the NSE and NSE-SM appears at f = 2.199 and f = 2.5 Hz (due to the stiffness of the seat suspension being changed by adding the NSE and NSE-SM). These frequencies mainly occur at a low frequency region from 0.5 to 4 Hz and they can greatly affect the driver’s health.²¹ However, the maximum PSD acceleration of the driver’s seat with the NSE, DE, and NSE-SM is strongly reduced by 38.5%, 88.3%, and 91% in comparison with the TSS, especially with the NSE-SM. Therefore, with the NSE-SM added to the seat suspension, the PSD acceleration of the driver’s seat always is smaller than that of the NSE and DE under the various frequency excitations.

Based on the analysis results of the vibration response of the driver’s seat under the excitation of the random road surface in both the time and frequency regions, it can be concluded that the DE added to the seat suspension improves insignificantly the driver’s ride comfort while the NSE-SM added in the seat suspension improves the driver’s ride comfort better than the NSE.

Ride performance under different vehicle velocities

In the moving condition of the vehicle on the random road surface, the vehicle velocity could be changed. Thus, the driver’s ride comfort and isolation performance of the seat suspension could be also affected. To fully evaluate the performance of the NSE, DE, and NSE-SM as well as their stability, a moving velocity range of the vehicle from 5 to 35 m s⁻¹ is simulated. The simulation results of the z_ws and a_ws have been provided in Figure 14(a)−(b).

Figure 14.

Effect of the vehicle velocity on the ride performance. (a) With z_ws and (b) with a_ws.

With the seat suspension using the TSS and DE, the result indicates that both the z_ws and a_ws are increased with the increase of the vehicle moving velocity from 5 to 20 m s⁻¹. However, at a moving velocity range from 20 to 35 m s⁻¹, the z_ws is unchanged while the a_ws is quickly increased. This means that the high increase in the vehicle’s moving velocity can reach to reduce the driver’s ride comfort. However, both the z_ws and a_ws with the DE are smaller than that of the TSS under various vehicle velocities. Thus, the driver’s ride comfort is significantly improved by using the DE.

With the seat suspension using the NSE and NSE-SM, both their z_ws and a_ws are strongly reduced in comparison with both the TSS and DE. Especially at the high velocity range of the vehicle moving from 20 to 35 m s⁻¹, the z_ws tends to decrease slightly while the a_ws increases insignificantly. This means that the driver’s ride comfort is greatly improved by using the NSE and NSE-SM, concurrently, their isolation performance is also very stable under the various moving velocities of the vehicle. Besides, the results of the comparison between the NSE and NSE-SM also show that both the z_ws and a_ws with the NSE-SM are lower than that of the NSE when the vehicle is moving on the random road surface at a velocity range from 5 to 30 m s⁻¹, and this is the main velocity range of the vehicle moving. At a velocity range from 30 to 35 m s⁻¹, both the z_ws and a_ws with the NSE-SM are slightly increased in comparison with the NSE, however, their values are small and insignificantly affect the driver’s ride comfort. Therefore, it can conclude that the NSE-SM can improve the driver’s ride comfort better than the TSS, NSE, and DE under various velocities of the vehicle.

Ride performance under the bumpy excitation

The excitation of the bumpy road surface could generate the maximum displacement of the seat suspension and maximum acceleration of the driver’s seat in a very small time. This not only influences the durability of the seat suspension but also influences the ride comfort and health of the driver. Thus, a bumpy road surface with its mathematical equation described in equation (23) is also applied to assess the isolation performance of the NSE, DE, and NSE-SM, respectively.

q_{1} = {\begin{cases} 0.03 \cos (ω t + π / 2) \\ 0 \end{cases} \begin{array}{c} with 0.3 \leq t \leq 0.5 \\ else \end{array}

(23)

where ω = 2πv₀/L in which v₀ is the vehicle velocity and L is the bump width. It is assumed that the vehicle is moving at v₀ = 5 m s⁻¹ on a bumpy road surface with L = 0.4 m, the bumpy excitation is shown in Figure 15(a).

Figure 15.

Result of vibration responses under the bumpy excitation. (a) Seat’s displacement and (b) its acceleration.

Under the excitation of the bumpy road surface in Figure 15(a), the simulation results of the driver’s seat displacement and acceleration responses are plotted in Figure 15(b)−(c). The results also show that the maximum amplitudes of the displacement and acceleration responses of the driver’s seat with the NSE are smaller than that of the DE, whereas the maximum amplitudes of the driver’s seat displacement and acceleration responses with the NSE-SM are the smallest. This implies that the seat suspension system equipped with the NSE-SM is also very effective in isolating the vibration of the driver’s seat under a bumpy excitation.

Conclusions

The driver’s ride comfort is slightly affected by the geometric dimension ratios of the NSE, DE, and NSE-SM, while the driver’s ride comfort is greatly affected by the damping and stiffness parameters and the damping and stiffness ratios of the NSE, DE, and NSE-SM.

With the optimized parameters of the NSE, DE, and NSE-SM, the simulation results under the different operating conditions of the vehicle show that the DE slightly improves the driver’s ride comfort. Conversely, both the NSE and NSE-SM greatly improve the driver’s ride comfort, especially the NSE-SM. Therefore, the model of the seat suspension embedded by the NSE-SM should be applied for improving the driver’s ride comfort.

To further enhance the isolation performance of the seat suspension, the damping parameter c_s of the seat suspension added by the NSE-SM should be controlled. This is also the reason that the NSE and NSE-SM were researched and embedded into the seat suspension,^9-20 concurrently, the damping coefficients of the vehicle suspension systems were independently controlled.^2-8 However, combining both the NSE-SM and the control of the damping coefficient c_s for the seat suspension has not been considered. Thus, this issue needs to be further studied in future work.

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 has been supported by the Key Scientific Research Project of Hubei Polytechnic University (No. 22xjz02A), the Open Fund Project of Hubei Key Laboratory of Intelligent Transportation Technology and Device, Hubei Polytechnic University (No. 2021XY101), and the Teaching and Research Project of Hubei Polytechnic University (Nos. 2020C21, 21xjz04Y).

ORCID iD

Vanliem Nguyen

References

Tang

Zhu

Zhang

, et al. Studies of air spring mathematical model and its performance in cab suspension system of commercial vehicle. SAE Tech Pap 2015; 2015(4): 341–348.

Rao

Sharma

Satyanarayana

, et al. Stochastic optimal preview control response of a quarter-car nonlinear suspension model using spectral decomposition method. Noise Vib Worldwide 2022; 53(4–5): 225–232.

Salman

Majid

Seyed

. Multi-objective optimization of a sports car suspension system using simplified quarter-car models. Mech Ind 2020; 21: 1–12.

Jhon

Mattson

Buckner

. Multi-objective control optimization for semi-active vehicle suspensions. J Sound Vib 2011; 330: 5502–5516.

Nguyen

Jiao

Zhang

. Control performance of damping and air spring of heavy truck air suspension system with optimal fuzzy control. SAE Int J Veh Dyna Stab NVH 2020; 4: 179–194.

Kumar

Singh

. Use of artificial neural network and multi-objective genetic algorithm approach to predict and ascertain stable cutting zone in conventional turning process. Noise Vib Worldwide 2018; 49(5): 191–214.

Pekgökgöz

. Active suspension of cars using fuzzy logic controller optimized by genetic algorithm. Int J Eng App Sci 2010; 2(4): 27–37.

Yuan

Nguyen

Zhou

. Research on semi-active air suspensions of heavy trucks based on a combination of machine learning and optimal fuzzy control. SAE Int J Veh Dyna Stab NVH 2021; 5: 159–172.

Shen

Peng

, et al. Analysis, design and experiment of continuous isolation structure with local quasi-zero-stiffness property by magnetic interaction. Int J Non-linear Mec 2019; 116: 289–301.

10.

Jiao

Van

, et al. Enhancing the ride comfort of off-road vibratory rollers with seat suspension using optimal quasi-zero stiffness. Proc IMechE C: J Mech Eng Sci 2022; 235: 1–15.

11.

Zha

Nguyen

, et al. Performance of the seat suspension system using negative stiffness structure on improving the driver's ride comfort. SAE Int J Veh Dyna Stab NVH 2022; 6(2): 135–146.

12.

Zhou

, et al. Theoretical and experimental analyses of a nonlinear magnetic vibration isolator with quasi-zero-stiffness characteristic. J Sound Vib 2013; 332: 3377–3389.

13.

Jiang

Song

Ding

, et al. Design of magnetic-air hybrid quasi-zero stiffness vibration isolation system. J Sound Vib 2020; 477: 115346.

14.

Wang

Zhou

Wang

, et al. Improving ride comfort in vehicles with seat vibration isolator embedded by different negative stiffness models. SAE Int J Passeng Veh Syst 2022; 15(3): 1–18.

15.

Carrella

Brennan

Waters

. Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic. J Sound Vib 2007; 301: 678–689.

16.

Zha

Nguyen

, et al. Optimizing the geometrical dimensions of the seat suspension equipped with a negative stiffness structure based on a genetic algorithm. SAE Int J Veh Dyna Stab NVH 2022; 6(2): 147–158.

17.

Han

Meng

Sun

. Design and characteristics analysis of a nonlinear isolator using a curved-mount-spring-roller mechanism as negative stiffness element. Math Prob Eng 2018; 2018: 1–15.

18.

Zhang

Wei

Wen

, et al. Design and analysis of a vibration isolation system with cam-roller-spring-rod mechanism. J Vib Ctrl 2021; 28(13–14): 1781–1791.

19.

Antoniadis

Chronopoulos

Spitas

, et al. Hyper-damping properties of a stiff and stable linear oscillator with a negative stiffness element. J Sound Vib 2015; 346: 37–52.

20.

Antoniadis

Kanarachos

Gryllias

, et al. KDamping: A stiffness based vibration absorption concept. J Vib Ctrl 2016; 24(3): 588–606.

21.

International Organization for Standardization . Mechanical Vibration and Shock-Evaluation of Human Exposure to Whole Body Vibration-Part 2: General Requi. Tech. Rep. ISO 2631- 1:1997. Geneva, Switzerland: ISO, 1997.

22.

International Organization for Standardization . Mechanical Vibration-Road Surface Profiles-Reporting of Measured Data. Geneve, Switzerland: ISO, 1995. ISO 8068.

23.

Nguyen

Zhang

, et al. Vibration analysis and modeling of an off-road vibratory roller equipped with three different cab’s isolation mounts. Shock Vib 2018; 2018: 1–17.