An interconnected suspension with adjustable roll and pitch stiffness (IS-ARPS) to enhance anti-roll and anti-dive/squat characteristics

Abstract

The suspension system is critical in maintaining vehicle stability in roll and pitch directions. Hydro-pneumatic interconnected suspension systems, in which the suspension struts are connected through hydraulic hoses, flow control valves, and accumulators, have great potential to further enhance anti-roll and anti-pitch characteristics. To address the existing designs’ shortcomings, this study introduces an interconnected suspension with adjustable roll and pitch stiffness (IS-ARPS). It not only eliminates the conventional anti-roll bars but also enables adjustment of roll stiffness depending on the driving and vehicle load conditions. In the pitch direction, the IS-ARPS provides an additional amount of pitch stiffness to reduce the dive/squat during braking/acceleration. The mathematical modeling of the IS-ARPS is also straightforward, requiring less computational cost and making it more practical for real-time implementations. Overall, the IS-ARPS system represents a significant advancement in the design of roll and pitch coupled interconnected suspensions, offering a more practical and versatile solution. The above features are examined through co-simulation between MATLAB/Simulink and ADAMS/Car.

Keywords

Interconnected suspension hydro-pneumatic suspension adjustable stiffness anti-roll anti-pitch

Introduction

A vehicle suspension system provides a smooth ride over rough roads while ensuring that the wheels remain in contact with the ground and vehicle roll is minimized.¹ A soft vehicle suspension is highly desirable for improving the vehicle ride comfort by attenuating the Whole Body Vibration (WBV) arising from the tire-ground interaction.² However, it tends to reduce the vehicle roll stability where a firm suspension is needed to reduce the vehicle rollover tendency and to improve the handling stability during extreme turning maneuvers. This is mainly due to the strong coupling between the ride and roll of the vehicle that poses complex design challenges associated with conflicting requirements between ride comfort and directional stability, especially in heavy commercial vehicles, which are exposed to wide load variations.³ This conflict needs to be eased by introducing advanced suspension technologies. A shock absorber works well with transient and fast-changing inputs like road disturbances, but it becomes ineffective when dealing with slow-changing motions, such as braking, accelerating, and turning maneuvers. In these situations, higher roll and pitch stiffness is preferred for reducing the vehicle rotational motions. Hence, the hydro-pneumatic interconnected suspension system, which can provide adjustable rotational stiffness and damping, is a good approach for improving vehicle handling stability during steady maneuvers. Meanwhile, the suspension stiffness and damping properties can be easily assigned by designers.

One solution is to use various interconnected structures, such as mechanical,⁴ electrical,⁵ hydraulic,⁶ and pneumatic⁷ connections. The most popular mechanical interconnection is the anti-roll bar, which applies additional roll stiffness to achieve better roll stability. However, it adds extra weight to the vehicle chassis and deteriorates ride comfort to some extent.⁸ To replace anti-roll bars, hydraulic interconnection has received a lot of attention in the past few decades because of its anti-roll/pitch abilities.^9,10 The vehicle’s harsh roll and pitch motions can be improved without compromising the ride comfort. Besides, the suspension roll and pitch characteristics can be varied by interconnecting different suspension struts.

One of the roll-plane interconnected configurations in the literature is the $H_{2}$ arrangement patented by Kinetics Pty Ltd.,¹¹ in which four double-acting hydraulic cylinders and two nitrogen accumulators are diagonally connected through hoses and 10 flow control valves. The valve resistance along the pipelines provides the damping, and the nitrogen gas compression in the accumulators generates the stiffness. Implementing the proposed interconnected suspension system can enhance the roll stability by providing increased roll stiffness and preserving a relatively soft ride at the same time. This configuration has been widely studied in research works,^12–14 and its performance has been evaluated by laboratory and road tests.¹⁵

Other than roll stiffness, pitch stiffness is another critical characteristic that affects vehicle stability and handling performance. Pitch refers to the vehicle’s tendency to rotate about its front-to-back axis, which can be caused by acceleration or braking. Appropriate pitch stiffness helps maintain vehicle stability during acceleration and braking, improves handling, and reduces the risk of accidents. In addition, it can improve the vehicle’s overall performance by reducing weight transfer, which can improve traction and grip. Although anti-roll bars are standard on many vehicles, anti-pitch bars are not as commonly used. Most cars use a combination of shock absorbers, springs, and other suspension components designed to resist pitch and provide a stable ride.¹⁶ In addition, adding an anti-pitch bar could negatively affect vehicle handling and ride quality. It could also make the vehicle more difficult to control in certain situations, such as when driving on slippery roads. However, some vehicles may require higher levels of pitch stiffness, such as high-performance sports cars or race cars.⁸ These vehicles typically have powerful engines designed to accelerate and brake quickly, which can cause significant pitch motions. Some off-road vehicles, such as trucks or SUVs used for heavy-duty towing or hauling, may require higher pitch stiffness to maintain stability and control under heavy loads.² Overall, pitch stiffness is still essential for a vehicle system, but it cannot be maintained by simply adding an anti-pitch bar due to the compact space of the chassis. Thus, some pitch-plane interconnected suspension systems were invented to enhance pitch stiffness without sacrificing ride comfort.

Zhang et al.¹⁷ introduced a bounce and pitch plane hydraulicly interconnected suspension (HIS) for mining vehicles focusing on improving ride quality and pitch stiffness, as shown in Figure 1(a). In detail, a lumped-mass vehicle model with a mechanical-hydraulic coupled suspension system was developed using the free-body diagram method. The impedance of the hydraulic system was modeled according to the transfer matrix method, which was integrated to form the dynamic equations of the mechanical-hydraulic coupled system. The simulation study indicated that the proposed HIS system effectively suppressed the pitch motion to guarantee vehicle handling performance and favorably provided soft vertical stiffness to improve the ride quality. Meanwhile, the force distribution between the front and rear wheels was more reasonable, and the sprung mass vibration decay rate increased somewhat. However, the suspension roll stiffness was not considered by this interconnected configuration. Xu et al.^10,18 developed a roll and pitch independently controlled hydraulically interconnected suspension. As shown in Figure 1(b), the inside four cylinders and fluid circuits formed roll-plane HIS, and the additional four cylinders outside formed pitch-plane HIS. The modal analysis showed the modes-decoupling property of the proposed system. Specifically, the roll-plane HIS increases roll stiffness without affecting other modes. This study demonstrated that the vehicle ride comfort and handling stability could be improved simultaneously by integrating the roll and pitch plane HIS. Meanwhile, it decoupled the four vehicle modes (bounce, roll, pitch, and warp) that could be independently tuned without compromise. Besides, Cao et al.^19,20 developed two novel hydro-pneumatic struts to enhance roll and pitch stiffness, providing a compact design with a considerably large effective working area.

Figure 1.

(a) Bounce and pitch plane HIS²¹ and (b) roll and pitch independently controlled HIS.¹⁸

Compared to the existing designs, a new topology is introduced in this study to realize better anti-roll/pitch characteristics with a less complicated structure. In addition, a stiffness adjustment method is developed, by which the suspension rotational stiffness can be conveniently adjusted to fit various vehicle models, loads, user preferences, driving conditions, etc. The remainder of this paper is structured as follows: Section 2 introduces and models the topology of IS-ARPS and its stiffness characteristics. Section 3 presents and discusses the co-simulation results. In the end, Section 4 summarizes the benefits and highlights of the proposed design.

System modeling

A novel interconnected topology is introduced in this section, as shown in Figure 2. It comprises four double-acting hydraulic cylinders that are connected with each other through hydraulic hoses, four accumulators, and eight flow control valves. The model considers four wheels with vertical degrees of freedom ( $z_{w 1}$ , $z_{w 2}$ , $z_{w 3}$ , $z_{w 4}$ ), and a sprung mass $m_{s}$ with vertical $z_{s}$ , roll $φ_{s}$ , and pitch $θ_{s}$ degrees of freedom. The right-handed Cartesian coordinate system defined in ISO-8855 is placed to describe the vehicle system. The x-axis is substantially horizontal and forward. The y-axis is perpendicular to the vehicle’s longitudinal plane and points to the driver’s left-hand side. The z-axis points upward along the vertical direction.

Figure 2.

Schematic of the tandem IS-ARPS.

In the first hydraulic circuit (solid green line), the upper chamber of the front left cylinder is connected to the lower chamber of the rear right cylinder. In the second hydraulic circuit (solid blue line), the lower chamber of the front left cylinder is connected to the upper chamber of the rear right cylinder. In the third hydraulic circuit (dashed green line), the upper chamber of the front right cylinder is connected to the lower chamber of the rear left cylinder. In the fourth hydraulic circuit (dashed blue line), the lower chamber of the front right cylinder is connected to the upper chamber of the rear left cylinder. Four accumulators are assembled along the four hydraulic circuits to generate enough roll and pitch stiffness. In addition, the adaptive damping functionality can be realized by utilizing eight solenoid flow control valves.

Moreover, a Hydraulic Power Unit (HPU) is connected to the two hydraulic circuits beside the accumulator inlets. It usually contains motor, tank, solenoid valves, and other essential hydraulic components, as shown in Figure 3. The HPU can be used to pump in or drain hydraulic oil from the interconnected suspension system. In this study, the HPU has three operation modes: off (nominal mode), charging (increase stiffness), and discharging (decrease stiffness). Thus, the system pressure can be easily controlled, which results in adjustable roll and pitch stiffness.

Figure 3.

Operation of HPU.

Considering the compressibility of the fluid in each chamber and the continuity of the cylinder movement, the continuity equation in the chamber can be written as

A Δ z - \frac{V Δ P}{β} - \int Q d t = 0

(1)

in which $A$ is the cross-sectional area of the chamber, $Δ z$ is the relative displacement with respect to the equilibrium position, $V$ is the chamber volume, $Δ P$ is the pressure drop in the chamber, $β$ is the bulk modulus, and $Q$ is the corresponding flow rate.

The gas pressure and gas volume in the accumulator always have the following relationship based on the Ideal Gas Law and Boyle’s Law²²:

P_{0} V_{0}^{n} = P_{1} V_{1}^{n}

(2)

where $P_{0}$ and $V_{0}$ are the nominal gas pressure and the nominal gas volume of the accumulator, n is the polytropic coefficient, and $P_{1}$ and $V_{1}$ are the working gas pressure and the corresponding gas volume of the accumulator.

The flow control valves are utilized to realize the damping characteristics. A simplified linear orifice model is used in this study, which assumes that the orifice has negligible fluid volume and involves linear pressure loss. Thus, the orifice equation can be written as

Q = \frac{1}{C} Δ P

(3)

Based on the hydraulic components and their characteristics introduced above, the mathematical modeling of the IS-ARS can be derived. Firstly, the chamber volumes of hydraulic cylinders can be calculated based on the suspension relative displacement. Assuming the strokes of the hydraulic cylinder at its nominal position are $S_{1}$ and $S_{2}$ , the eight chamber volumes can be represented by

V_{1} = - A_{1} Δ z_{1} + A_{1} S_{1}

V_{2} = A_{2} Δ z_{1} + A_{2} S_{2}

V_{3} = A_{2} Δ z_{3} + A_{2} S_{2}

V_{4} = - A_{1} Δ z_{3} + A_{1} S_{1}

\begin{matrix} V_{5} = - A_{1} Δ z_{2} + A_{1} S_{1} \\ V_{6} = A_{2} Δ z_{2} + A_{2} S_{2} \\ V_{7} = A_{2} Δ z_{4} + A_{2} S_{2} \\ V_{8} = - A_{1} Δ z_{4} + A_{1} S_{1} \end{matrix}

(4)

in which the suspension relative displacements at four corners are

\begin{matrix} Δ z_{1} = z_{s} + l_{d} φ_{s} - L_{1} θ_{s} - z_{w 1} \\ Δ z_{2} = z_{s} + l_{d} φ_{s} + L_{2} θ_{s} - z_{w 2} \\ Δ z_{3} = z_{s} - l_{d} φ_{s} - L_{1} θ_{s} - z_{w 3} \\ Δ z_{4} = z_{s} - l_{d} φ_{s} + L_{2} θ_{s} - z_{w 4} \end{matrix}

(5)

where $l_{d}$ represents half the distance between the left and right hydraulic cylinders, $L_{1}$ represents the longitudinal distance from CG location to the front axle, and $L_{2}$ represents the longitudinal distance from the CG location to the rear axle.

Secondly, the time derivative of fluid pressures in the hydraulic cylinders can be calculated according to equations (1) and (3) as follows²³:

{\overset{\cdot}{P}}_{1} = \frac{β}{V_{1}} (A_{1} Δ {\overset{\cdot}{z}}_{1} - Q_{1}), in which Q_{1} = \frac{1}{C_{1}} (P_{1} - P_{9})

{\overset{\cdot}{P}}_{2} = \frac{β}{V_{2}} (- A_{2} Δ {\overset{\cdot}{z}}_{1} - Q_{2}), in which Q_{2} = \frac{1}{C_{2}} (P_{2} - P_{12})

{\overset{\cdot}{P}}_{3} = \frac{β}{V_{3}} (- A_{2} Δ {\overset{\cdot}{z}}_{3} - Q_{3}), in which Q_{3} = \frac{1}{C_{3}} (P_{3} - P_{11})

\begin{matrix} {\overset{\cdot}{P}}_{4} = \frac{β}{V_{4}} (A_{1} Δ {\overset{\cdot}{z}}_{3} - Q_{4}), in which Q_{4} = \frac{1}{C_{4}} (P_{4} - P_{10}) \\ {\overset{\cdot}{P}}_{5} = \frac{β}{V_{5}} (A_{1} Δ {\overset{\cdot}{z}}_{2} - Q_{5}), in which Q_{5} = \frac{1}{C_{5}} (P_{5} - P_{11}) \\ {\overset{\cdot}{P}}_{6} = \frac{β}{V_{6}} (- A_{2} Δ {\overset{\cdot}{z}}_{2} - Q_{6}), in which Q_{6} = \frac{1}{C_{6}} (P_{6} - P_{10}) \\ {\overset{\cdot}{P}}_{7} = \frac{β}{V_{7}} (- A_{2} Δ {\overset{\cdot}{z}}_{4} - Q_{7}), in which Q_{7} = \frac{1}{C_{7}} (P_{7} - P_{9}) \\ {\overset{\cdot}{P}}_{8} = \frac{β}{V_{8}} (A_{1} Δ {\overset{\cdot}{z}}_{4} - Q_{8}), in which Q_{8} = \frac{1}{C_{8}} (P_{8} - P_{12}) \end{matrix}

(6)

in which $C_{1} - C_{8}$ represent the equivalent flow resistance parameters of the hydraulic pipes and valves.

Thirdly, the two accumulator gas volumes can be iteratively calculated based on the nominal gas volume.

\begin{matrix} V_{9} = V_{0} - \int^{(Q_{1} + Q_{7})} dt \\ V_{10} = V_{0} - \int^{(Q_{4} + Q_{6})} dt \\ V_{11} = V_{0} - \int^{(Q_{3} + Q_{5})} dt \\ V_{12} = V_{0} - \int^{(Q_{2} + Q_{8})} dt \end{matrix}

(7)

Fourthly, the time derivative of the gas pressures in the accumulators can be calculated according to equation (2).

\begin{matrix} {\overset{\cdot}{P}}_{9} = - \frac{n P_{0} V_{0}^{n}}{V_{9}^{n + 1}} {\overset{\cdot}{V}}_{9} = \frac{n P_{0} V_{0}^{n}}{V_{9}^{n + 1}} (Q_{1} + Q_{7}) \\ {\overset{\cdot}{P}}_{10} = - \frac{n P_{0} V_{0}^{n}}{V_{10}^{n + 1}} {\overset{\cdot}{V}}_{10} = \frac{n P_{0} V_{0}^{n}}{V_{10}^{n + 1}} (Q_{4} + Q_{6}) \\ {\overset{\cdot}{P}}_{11} = - \frac{n P_{0} V_{0}^{n}}{V_{11}^{n + 1}} {\overset{\cdot}{V}}_{11} = \frac{n P_{0} V_{0}^{n}}{V_{11}^{n + 1}} (Q_{3} + Q_{5}) \\ {\overset{\cdot}{P}}_{12} = - \frac{n P_{0} V_{0}^{n}}{V_{12}^{n + 1}} {\overset{\cdot}{V}}_{12} = \frac{n P_{0} V_{0}^{n}}{V_{12}^{n + 1}} (Q_{2} + Q_{8}) \end{matrix}

(8)

Thus, all the differential equations of the hydraulic system are derived. In this way, the pressures $P_{1}$ to $P_{12}$ along the four hydraulic circuits can be iteratively estimated based on the measured suspension heights and the hydraulic pressures.

Roll stiffness analysis

At the suspension steady status, its roll moment can be written as

\begin{array}{l} M_{θ} = - (k_{1} + k_{2} + k_{3} + k_{4}) l^{2} φ_{s} \\ + (F_{1} + F_{2} - F_{3} - F_{4}) l \\ = - (k_{1} + k_{2} + k_{3} + k_{4}) l^{2} φ_{s} - A_{1} l P_{1} \\ + A_{2} l P_{2} - A_{2} l P_{3} + A_{1} l P_{4} - A_{1} l P_{5} \\ + A_{2} l P_{6} - A_{2} l P_{7} + A_{1} l P_{8} \\ = - (k_{1} + k_{2} + k_{3} + k_{4}) l^{2} φ_{s} - l (A_{1} + A_{2}) \\ (P_{9} - P_{10} + P_{11} - P_{12}) \end{array}

(9)

The roll stiffness can be derived from the restoring roll moment subject to the relative roll deflection as

\begin{matrix} K_{φ} = - \frac{dM}{d φ} = (k_{1} + k_{2} + k_{3} + k_{4}) l^{2} \\ + l (A_{1} + A_{2}) \frac{d}{d φ} (P_{9} - P_{10} + P_{11} - P_{12}) \end{matrix}

(10)

The corresponding derivative of the gas pressures in the accumulators can be calculated as

\begin{matrix} \frac{d P_{9}}{d φ} = \frac{n (A_{1} + A_{2}) l P_{0} V_{0}^{n}}{{V_{9}}^{n + 1}} = \frac{n (A_{1} + A_{2}) l P_{0} V_{0}^{n}}{γ_{5}} \\ \frac{d P_{10}}{d φ} = \frac{- n (A_{1} + A_{2}) l P_{0} V_{0}^{n}}{{V_{10}}^{n + 1}} = \frac{- n (A_{1} + A_{2}) l P_{0} V_{0}^{n}}{γ_{6}} \\ \frac{d P_{11}}{d φ} = \frac{n (A_{1} + A_{2}) l P_{0} V_{0}^{n}}{{V_{11}}^{n + 1}} = \frac{n (A_{1} + A_{2}) l P_{0} V_{0}^{n}}{γ_{7}} \\ \frac{d P_{12}}{d φ} = \frac{- n (A_{1} + A_{2}) l P_{0} V_{0}^{n}}{{V_{12}}^{n + 1}} = \frac{- n (A_{1} + A_{2}) l P_{0} V_{0}^{n}}{γ_{8}} \end{matrix}

(11)

in which

\begin{array}{l} γ_{5} = [V_{0} + (A_{2} - A_{1}) z_{s} - (A_{1} + A_{2}) l φ_{s} \\ + (A_{1} L_{1} + A_{2} L_{2}) θ_{s}]^{n + 1} \\ γ_{6} = [V_{0} + (A_{2} - A_{1}) z_{s} + (A_{1} + A_{2}) l φ_{s} \\ + (A_{1} L_{1} + A_{2} L_{2}) θ_{s}]^{n + 1} \end{array}

\begin{matrix} γ_{7} = [V_{0} + (A_{2} - A_{1}) z_{s} - (A_{1} + A_{2}) l φ_{s} \\ - (A_{1} L_{2} + A_{2} L_{1}) θ_{s}]^{n + 1} \end{matrix}

\begin{matrix} γ_{8} = [V_{0} + (A_{2} - A_{1}) z_{s} + (A_{1} + A_{2}) l φ_{s} \\ - (A_{1} L_{2} + A_{2} L_{1}) θ_{s}]^{n + 1} \end{matrix}

Hence, the roll stiffness of the proposed IS-ARPS can be represented as

\begin{matrix} K_{φ} = (k_{1} + k_{2} + k_{3} + k_{4}) l^{2} + n P_{0} V_{0}^{n} {(A_{1} + A_{2})}^{2} l^{2} \\ [\frac{1}{γ_{5}} + \frac{1}{γ_{6}} + \frac{1}{γ_{7}} + \frac{1}{γ_{8}}] \end{matrix}

(12)

When the Polytropic coefficient $n = 1$ , the roll stiffness formulation can be simplified as

K_{φ} = (k_{1} + k_{2} + k_{3} + k_{4}) l^{2} + 4 {(A_{1} + A_{2})}^{2} l^{2} \frac{P_{0}}{V_{0}}

(13)

Based on equation (12) and the parameters listed in Table 1, the estimated roll stiffness of the IS-ARPS is presented in Figure 4. Specifically, the estimated results of the four example settings are plotted in different colors, that is, $V_{0} = 1.63 L$ & $P_{0} = 245 psi$ , $V_{0} = 1.6 L$ & $P_{0} = 250 psi$ , $V_{0} = 1.57 L$ & $P_{0} = 255 psi$ , and $V_{0} = 1.54 L$ & $P_{0} = 260 psi$ . Noticeably, the roll stiffness can be adjusted by charging or discharging the hydraulic system. In other words, the roll stiffness $K_{φ}$ increases with the system nominal pressure $P_{0}$ . Attributed to the air compression in the accumulator, the roll stiffness tends to increase with the increase of roll angle.

Table 1.

Parameters of the IS-ARPS system.

Symbol	Description	Value	Unit
$k_{1}$ , $k_{3}$	Front air spring stiffness	23,658	N/m
$k_{2}$ , $k_{4}$	Rear air spring stiffness	14,854	N/m
$m_{s}$	Cabin weight	1250	kg
$k_{φ}$	Equivalent roll stiffness of the bushings	155	kN·m/rad
$I_{x}$	Cabin roll moment of inertia	1468	kgm²
$I_{y}$	Cabin pitch moment of inertia	1229	kgm²
$l, l_{d}$	Lateral distance from airbag/cylinder to CG	0.63	m
$L_{1}$	Distance from CG to front suspension	0.936	m
$L_{2}$	Distance from CG to rear suspension	1.055	m
$A_{1}$	Rod side chamber cross-sectional area (rod diameter is 1.5 in)	0.00342	m²
$A_{2}$	Bore side chamber cross-sectional area (bore diameter is 3.5 in)	0.00456	m²
$S_{1}$ , $S_{2}$	Half of the full stroke length	4	inches
$n$	Polytropic coefficient	1.38	-
$β$	Fluid bulk modulus	1400	MPa
$C_{i}$	Resistance coefficient of hydraulic hose ( $i$ = 1, 2, 3, …, 8)	5×10⁷	Pa·s/m³

Figure 4.

Comparison of the dual-axle suspension roll stiffness.

Pitch stiffness analysis

Suspension pitch stiffness measures the tilting resistance about its transverse axis. It determines how much the cabin will tilt forward or backward in response to longitudinal acceleration. A higher pitch stiffness results in less body pitch, which leads to better stability. In conventional air suspension, the pitch stiffness is mainly provided by springs, while it is difficult to be independently adjusted since the springs affect many other suspension design factors. The proposed IS-ARPS adds an additional amount of pitch stiffness to the suspension system. Meanwhile, the rotational stiffness can be easily adjusted for different vehicle models or applications, which is analyzed in the following section.

The pitch moment can be written as

\begin{matrix} M_{θ} = (k_{1} L_{1} - k_{2} L_{2} + k_{3} L_{1} - k_{4} L_{2}) z_{s} \\ - (k_{1} L_{1}^{2} + k_{2} L_{2}^{2} + k_{3} L_{1}^{2} + k_{4} L_{2}^{2}) θ_{s} \\ + (- L_{1} F_{1} + L_{2} F_{2} - L_{1} F_{3} + L_{2} F_{4}) \\ = (k_{1} L_{1} - k_{2} L_{2} + k_{3} L_{1} - k_{4} L_{2}) z_{s} \\ - (k_{1} L_{1}^{2} + k_{2} L_{2}^{2} + k_{3} L_{1}^{2} + k_{4} L_{2}^{2}) θ_{s} \\ - L_{1} (A_{2} P_{2} - A_{1} P_{1}) - L_{1} (A_{2} P_{3} - A_{1} P_{4}) \\ + L_{2} (A_{2} P_{6} - A_{1} P_{5}) + L_{2} (A_{2} P_{7} - A_{1} P_{8}) \\ = (k_{1} L_{1} - k_{2} L_{2} + k_{3} L_{1} - k_{4} L_{2}) z_{s} \\ - (k_{1} L_{1}^{2} + k_{2} L_{2}^{2} + k_{3} L_{1}^{2} + k_{4} L_{2}^{2}) θ_{s} \\ + (A_{1} L_{1} + A_{2} L_{2}) (P_{9} + P_{10}) \\ - (A_{1} L_{2} + A_{2} L_{1}) (P_{11} + P_{12}) \end{matrix}

(14)

The pitch stiffness can be derived from the restoring pitch moment subject to the relative pitch deflection as

\begin{matrix} K_{θ} = - \frac{d M_{θ}}{d θ_{s}} = k_{1} L_{1}^{2} + k_{2} L_{2}^{2} + k_{3} L_{1}^{2} + k_{4} L_{2}^{2} \\ - (A_{1} L_{1} + A_{2} L_{2}) \frac{d}{d θ_{s}} (P_{9} + P_{10}) \\ + (A_{1} L_{2} + A_{2} L_{1}) \frac{d}{d θ_{s}} (P_{11} + P_{12}) \end{matrix}

(15)

The corresponding derivative of the gas pressures in the accumulators can be expressed as

\begin{matrix} \frac{d P_{9}}{d θ_{s}} = \frac{- n (A_{1} L_{1} + A_{2} L_{2}) P_{0} V_{0}^{n}}{{V_{9}}^{n + 1}} \\ = \frac{- n (A_{1} L_{1} + A_{2} L_{2}) P_{0} V_{0}^{n}}{γ_{5}} \end{matrix}

\begin{matrix} \frac{d P_{10}}{d θ_{s}} = \frac{- n (A_{1} L_{1} + A_{2} L_{2}) P_{0} V_{0}^{n}}{{V_{10}}^{n + 1}} \\ = \frac{- n (A_{1} L_{1} + A_{2} L_{2}) P_{0} V_{0}^{n}}{γ_{6}} \end{matrix}

\begin{matrix} \frac{d P_{11}}{d θ_{s}} = \frac{- n (- A_{2} L_{1} - A_{1} L_{2}) P_{0} V_{0}^{n}}{{V_{11}}^{n + 1}} \\ = \frac{n (A_{1} L_{2} + A_{2} L_{1}) P_{0} V_{0}^{n}}{γ_{7}} \end{matrix}

\begin{matrix} \frac{d P_{12}}{d θ_{s}} = \frac{- n (- A_{2} L_{1} - A_{1} L_{2}) P_{0} V_{0}^{n}}{{V_{12}}^{n + 1}} \\ = \frac{n (A_{1} L_{2} + A_{2} L_{1}) P_{0} V_{0}^{n}}{γ_{8}} \end{matrix}

(16)

Hence, the pitch stiffness of the proposed interconnected suspension can be represented as

\begin{matrix} K_{θ} = k_{1} L_{1}^{2} + k_{2} L_{2}^{2} + k_{3} L_{1}^{2} + k_{4} L_{2}^{2} + n P_{0} V_{0}^{n} \\ {(A_{1} L_{1} + A_{2} L_{2})}^{2} [\frac{1}{γ_{5}} + \frac{1}{γ_{6}}] \\ + n P_{0} V_{0}^{n} {(A_{1} L_{2} + A_{2} L_{1})}^{2} [\frac{1}{γ_{7}} + \frac{1}{γ_{8}}] \end{matrix}

(17)

When the Polytropic coefficient $n = 1$ , the pitch stiffness formulation can be simplified as

\begin{matrix} K_{θ} = k_{1} L_{1}^{2} + k_{2} L_{2}^{2} + k_{3} L_{1}^{2} + k_{4} L_{2}^{2} \\ + 2 [{(A_{1} L_{1} + A_{2} L_{2})}^{2} + {(A_{1} L_{2} + A_{2} L_{1})}^{2}] \frac{P_{0}}{V_{0}} \end{matrix}

(18)

As shown in Figure 5, the estimated results of the four settings are plotted in different colors, that is, $V_{0} = 1.63 L$ & $P_{0} = 245 psi$ , $V_{0} = 1.6 L$ & $P_{0} = 250 psi$ , $V_{0} = 1.57 L$ & $P_{0} = 255 psi$ , and $V_{0} = 1.54 L$ & $P_{0} = 260 psi$ . The pitch stiffness changes with the charging or discharging process of the hydraulic system. In other words, the pitch stiffness $K_{θ}$ increases with the system nominal pressure $P_{0}$ . Attributed to the air compression in the accumulators, the pitch stiffness tends to increase with the increase of the pitch angle. Compared to the conventional suspension, the IS-ARPS provides additional pitch stiffness to prevent harsh dive/squat motions. Besides, it is noticeable that the pitch stiffness is not as sensitive as the roll stiffness when the system’s nominal pressure changes.

Figure 5.

Nonlinear pitch stiffness.

Simulation results

The IS-ARPS is examined through the co-simulation between MATLAB/Simulink and ADAMS/Car. A high-fidelity truck model was built in ADAMS/Car based on a 6 × 4 truck, as shown in Figure 6. Three axles are modeled in the truck’s primary suspension, comprising one steering axle at the front and two driving axles at the rear. A cabin equipped with IS-ARPS is installed on the chassis frame. The parameters used in the ADAMS model are listed in Table 1. These two software programs communicated during runtime to simulate the truck’s dynamic responses. In this study, the interconnected suspension system was implemented in MATLAB/Simulink. Therefore, the truck motions were simulated in ADAMS/Car based on the control forces received from MATLAB/Simulink. These two software programs updated their information in each time step, which generated a reliable simulation environment. The simulation results of three testing scenarios are presented in this section: double lane change, steady turn, and emergency braking.

Figure 6.

Co-simulation between MATLAB/Simulink and ADAMS/Car.

Double lane change

Double lane change is a driving maneuver in which a vehicle changes lanes twice in quick succession. It is commonly used in situations where a driver needs to quickly move from one lane to another to avoid an obstacle, merge onto a highway, or get around slower-moving traffic. In this test, the truck was driven on a flat road at 60 km/h, and its trajectory is shown in Figure 7. The responses of conventional air suspension and the IS-ARPS charged at 300 psi are compared.

Figure 7.

Vehicle path (DLC at 60 km/h).

Figure 8 shows the cabin suspension relative roll angles. It can be seen that the maximum roll angle was approximately 6° using the conventional air suspension, which was reduced by IS-ARPS to 4°. Regarding the RMS values, the IS-ARPS improved the anti-roll characteristic by 42%. Meanwhile, the cabin roll rate was reduced by 30%, as shown in Figure 9.

Figure 8.

Cabin suspension relative roll angle (DLC at 60 km/h).

Figure 9.

Cabin roll rate (DLC at 60 km/h).

One of the benefits of the proposed interconnected suspension is the adjustable stiffness, as shown in Figure 10. In other words, a higher nominal pressure results in a smaller suspension relative roll angle. Thus, the roll stiffness characteristics can be conveniently adjusted according to load, user preference, driving condition, etc. Besides, the stiffness range can be further adjusted for different vehicle models by using the proposed topology.

Figure 10.

Cabin suspension relative roll angle at different pressures (DLC at 60 km/h).

Steady turn

A steady turn maneuver means a vehicle makes a controlled turn at a constant speed and radius. This type of turn is commonly used when driving through curved highway ramps or making turns at intersections. Figure 11 shows the vehicle trajectory in this scenario, and the truck turned at 60 km/h.

Figure 11.

Vehicle trajectory (steady turn at 60 km/h).

Figure 12 shows the cabin suspension relative roll angles. It can be seen that the maximum roll angle was approximately 3.5° using the conventional air suspension, which was decreased to 2° by implementing the IS-ARPS. In terms of the RMS values, the IS-ARPS improved the cabin anti-roll characteristic by 45%, and the cabin roll rate was reduced by 42%, as shown in Figure 13.

Figure 12.

Cabin suspension relative roll angle (steady turn at 60 km/h).

Figure 13.

Cabin roll rate (steady turn at 60 km/h).

Emergency brake

On average, a fully loaded truck can have a maximum braking acceleration of around 0.5g to 0.7g. In this scenario, the truck was driven at 80 km/h and braked under non-constant accelerations within 0.45g. The corresponding speed and longitudinal acceleration are plotted in Figure 14.

Figure 14.

Vehicle longitudinal motion (emergency brake).

Figure 15 shows the cabin pitch angles. It can be seen that the maximum pitch angle was 4° using the conventional air suspension. When the nominal pressure equals 250 psi, the maximum pitch angle was reduced to 1.5° by the IS-ARPS. Meanwhile, the cabin pitch rate was reduced by 52% in Figure 16, which significantly improved the anti-pitch characteristic as expected. Moreover, the anti-pitch characteristic could be adjusted by changing the nominal pressure. When the nominal pressure increased to 300 psi, the maximum pitch angle dropped to 1.1°, and the cabin pitch rate was reduced by 59%.

Figure 15.

Cabin suspension relative pitch angle (emergency brake).

Figure 16.

Cabin pitch rate (emergency brake).

Conclusion

A new roll and pitch coupled hydro-pneumatic interconnected suspension named IS-ARPS is introduced in this paper. Its performance was examined through co-simulation between ADAMS/Car and MATLAB/Simulink. Compared to the existing designs, the proposed topology has the following benefits:

Enhanced roll and pitch stability: The IS-ARPS has nonlinear rotational stiffness characteristics, that is, the rotational stiffness increases with the suspension relative roll/pitch angle. Hence, it enhances the suspension’s anti-roll and anti-pitch characteristics.

Adjustable rotational stiffness: The IS-ARPS’s rotational stiffness can be conveniently adjusted according to the driving scenario, road condition, and vehicle load by operating a compact HPU to charge or discharge the hydraulic system.

Feasibility for various applications: The IS-ARPS can be easily tuned and fitted to different car models. Various rotational stiffness levels can be achieved by adjusting the system’s nominal pressure, resulting in one topology for any vehicle suspension. Besides, the proposed topology can be easily extended to multi-axle suspension systems.

As for future work, the IS-ARPS needs to be evaluated in experiments. Besides, a stiffness adjustment algorithm needs to be developed by which the suspension roll/pitch stiffness can be intelligently adjusted according to the vehicle load, speed, and road conditions.

Footnotes

Acknowledgements

The authors gratefully acknowledge the assistance of Dr. Amir Soltani and Dr. Alireza Pazooki.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Yukun Lu

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