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Abstract

Hydraulic interconnected suspension (HIS) systems represent a burgeoning technology in the automotive industry, notably due to their abilities to reconcile the well-known trade-off between ride comfort and handling performance. Numerous design possibilities for HIS systems exist, considering various hydraulic components (such as orifices, accumulators, and pipelines) and their topological connections. At present there is a lack of a systematic design approach for such systems. Traditional design approaches focus on making piecemeal modifications to existing designs, neglecting a substantial catalogue of fundamentally different and transformative schemes. Consequently, significant potential performance enhancements remain untapped. To address this limitation, this paper proposes a novel systematic design approach for HIS systems, which enables a wide range of design possibilities to be explored, and an optimal configuration to be identified. The efficacy of the methodology is demonstrated via a case study. Given a predefined design complexity, a beneficial single-axle HIS system is constructed, which can achieve a 15.6% ride comfort performance improvement while maintaining roll-control performance compared with the traditional system.

Keywords

design methodology hydraulic interconnected suspension network synthesis optimal vibration absorber physical realisation

Introduction

In automotive vehicles, suspension requirements for handling and ride comfort are inherently conflicting.¹ Hydraulic interconnected suspension (HIS) systems have shown superior capability in mitigating this trade-off compared to conventional standalone suspensions.^2–4 An interconnected suspension system, by definition, generates a force at one wheel station in response to a displacement at another wheel station.⁵ HIS systems contain either single or double-acting hydraulic piston cylinders, which replace conventional shock absorbers. Hydraulic circuits interconnect the fluid chambers within these cylinders, incorporating components such as orifices, gas-filled accumulators, and pipelines. These circuits enable tailored force distribution across suspension modes (roll, bounce, pitch, warp), allowing independent optimisation of vehicle dynamics.⁵ Figure 1(a) illustrates a schematic representation of a conventional single-axle HIS system.⁶ Single-axle HIS systems, such as those connecting right and left suspensions, are primarily employed to enhance roll stiffness and bounce mode control, improving stability during cornering without compromising ride comfort. Cross-axle HIS designs interconnect both right/left and front/rear suspension units, enabling simultaneous enhancements in roll and pitch stiffness. This configuration improves handling, braking stability, and overall body control. A well-known example is the Kinetic H2 system, which employs a fully cross-linked hydraulic network to deliver active-like roll and pitch control while remaining a passive system.^2,6 Traditional HIS systems are passive devices, however, more recently, research efforts have been applied to develop advanced HIS systems using active strategies.^7,8 Although active suspension systems are increasingly favoured for their ability to dynamically alter suspension characteristics in real time and further improve ride–handling balance, passive systems remain predominant in production vehicles due to their simpler architecture, lower energy demands, and proven long-term reliability. To this end, alternative passive HIS designs with enhanced performance characteristics are highly valuable to the industry.²

Figure 1.

(a) A conventional single-axle HIS system, defined in the roll mode and (b) an equivalent unified mechanical network for the conventional single-axle HIS system (online version available in colour).

To further the performance of passive HIS systems, efforts have been applied to design the hydraulic circuit for both single-axle HIS configurations and cross-axle HIS designs. For instance, adding accumulators,⁹ integrating air springs,¹⁰ or testing fixed circuit variants.¹¹ While beneficial, these approaches explore only a narrow design space, overlooking fundamentally different configurations that could offer greater performance gains. Rather than focusing on piecemeal modifications to existing devices, network-synthesis-based approaches, such as the structure-immittance approach,¹² offer the opportunity to explore entirely new and distinct designs, in network representation. The structure-immittance approach enables the derivation of all possible series-parallel network configurations for a specified level of complexity (component type and number).¹² By using such techniques, researchers have identified optimal network-represented vibration absorber properties across various engineering fields, including automotive,^13,14 railway,¹⁵ civil engineering,¹⁶ and wind energy sectors¹⁷; these absorber properties showcase significant theoretical performance improvements compared to the best traditional designs. Li et al.¹⁸ extended this concept to multi-domain systems by mapping physical absorbers to equivalent mechanical networks and applying network synthesis to find optimal properties,¹² later realised with physical multidomain components.

Despite a strong potential in performance improvement, it is challenging to apply the multi-domain synthesis approach in reference.¹⁸ for the design of HIS systems. Specifically, the challenge lies in the need to develop an equivalent unifying mechanical network for the system. Naturally, HIS systems form a multi-port hydraulic network, and obtaining a dual equivalent mechanical network¹⁹ which is a prerequisite for reference.¹⁸ is extremely challenging since there is a lack of an existing approach which can obtain the dual of a multiport network.

To address this challenge, this paper introduces a novel design methodology for HIS systems that enables systematic exploration of a broad design space and identification of optimal configurations. Unlike previous approaches that require conversion to the mechanical domain, the proposed methodology models HIS systems directly as multi-port hydraulic networks. Network-synthesis-based techniques (e.g. structure–immittance approach) are then applied to enumerate all feasible topologies for a specified complexity.¹² The optimal network properties derived from this process are then mapped to physical hydraulic components, enabling the realisation of HIS architectures that extend beyond conventional designs. The methodology’s effectiveness is demonstrated through a single-axle HIS case study in this work. It is worth noting that the proposed methodology is also applicable to cross-axle configurations, including systems such as the Kinetic H2.

The remainder of this paper is structured as follows: Section ‘Design methodology overview’ outlines the proposed methodology. A case study demonstrating the effectiveness of this methodology is presented in Sections ‘Design problem formulation’, ‘Benchmark network development’, ‘Optimal network-represented HIS’, and ‘Physical device realisation’, whereby the accumulator branch of a single-axle HIS system is optimally designed. Finally, conclusions are drawn in Section ‘Conclusion’.

Design methodology overview

The existing multi-domain synthesis approach is difficult to implement for the design of HIS systems. Subsection ‘Challenge of developing an equivalent mechanical network’ will discuss the primary challenge associated with its application, while Subsection ‘Methodology overview’ will outline a novel HIS design approach, using a structured framework.

Challenge of developing an equivalent mechanical network

A conventional single-axle HIS system is displayed in Figure 1(a). Fluid flows, instantaneous system pressures, and cross-sectional chamber areas are denoted by the variables $q_{jk}$ , $P_{jk}$ , and $A_{jk},$ respectively. A subscript $j$ corresponds to $l, r$ , $A$ , or $B$ , and a subscript $k$ corresponds to $1$ , $2$ , or $a$ . The notations $l, r$ , $A$ , and $B$ symbolise that the variable belongs to the left piston cylinder, the right piston cylinder, pipeline $A$ , or pipeline $B$ , respectively, and $1$ , $2$ , and $a$ indicate that the variable is a property of the upper piston chamber, the lower piston chamber, or the accumulator branch, respectively. The dynamic force and relative velocity of the left hydraulic piston cylinder are represented by $F_{ld}$ and $Δ v_{l}$ , in that order, while $F_{rd}$ and $Δ v_{r}$ characterise the right hydraulic piston cylinder. Note that $Δ v_{l / r}$ is the relative velocity between the piston and the cylinder of the left/right device. For simplicity of illustration in Figure 1(a), an equivalent $Δ v_{l / r}$ is assumed for the piston motion while the cylinder is assumed to be grounded.

The existing multidomain synthesis approach begins by establishing an equivalent unifying mechanical network for a conventional multidomain absorber.¹⁸ According to Li et al.,¹⁸ the conventional single-axle HIS system, shown in Figure 1(a), must first be translated into an equivalent hydraulic network, and then dually transformed into an equivalent mechanical counterpart.¹⁹ The developed hydraulic network for pipeline A can be found in Figure 2(b), and it can be seen that the network has three-terminals ( $P_{l 1}$ , $P_{r 2}$ , and $P_{Aa 0}$ ). Further details regarding this representation are provided in Section ‘Benchmark network development’.

Figure 2.

(a) A framework summarising the proposed methodology for the design of HIS systems, (b) a three-terminal network representation of pipeline A of a conventional single-axle HIS system, (c) pipeline A of an identified optimal candidate HIS network, and (d) physical device realisation. Note that (b–d) are specific for the case study demonstrated in this paper.

Following Li et al.,¹⁸ an equivalent unified mechanical network for the conventional HIS can be developed as shown in Figure 1(b). For detailed derivation, readers are referred to the Supplementary Material.²⁰ The network consists of two pairs of lever arms and two three-terminal mechanical networks. The use of lever arms is to account for variations between the cross-sectional areas of the chambers, for the left and right single-rod piston cylinders (i.e. $A_{l 1} \neq A_{l 2}$ , and $A_{r 1} \neq A_{r 2}$ ). The solid dark grey bars are rigid, ensuring that points A and B exhibit identical motion with the left suspension velocity $Δ v_{l}$ , and points E and F exhibit identical motion with the right suspension velocity $Δ v_{r}$ . The forces applied at points A and B, representing the resultant forces from the chambers of the left cylinder, are summed to yield the total left dynamic force $F_{ld}$ . Similarly, this analysis applies to the total right dynamic force $F_{rd}$ .

The endpoints C, G, and the ground, are interconnected by a three-terminal mechanical network, which represents pipeline A of the conventional single-axle HIS, and the endpoints D, H, and the ground, are interconnected by another three-terminal mechanical network, which represents pipeline B. Such mechanical network is the dual of its hydraulic counterpart, illustrated in Figure 1(a). The properties $c_{jk}$ and $k_{jk}$ in the mechanical network result from the effects of the hydraulic properties. For example, $k_{Aa}$ and $c_{Aa}$ represent the compliance and inlet damping of the accumulator on pipeline A, while $c_{A 1}$ and $c_{A 2}$ characterise the damping effects within pipeline A.

While the equivalent unified mechanical network for the conventional single-axle HIS can be obtained, the dual transformation process introduces inherent complexities, particularly due to the inclusion of lever arm pairs. These complexities are anticipated to escalate when addressing the mechanical dual of a non-mechanical multi-port network with an increased number of terminals. For instance, in the case of the Kinetic H2 system, a conventional four-wheel HIS configuration, the hydraulic network (representing a single pipeline) features five terminals.²¹ To date, no equivalent unified mechanical network for this system has been identified, primarily because no existing method can reliably derive the dual of a multi-port network.

With the introduction of the inerter, a two-terminal device in which the forces applied at the terminals are proportional to the relative accelerations between them, the mechanical-electrical-hydraulic analogy becomes complete.²² This comprehensive analogy facilitates the application of network-synthesis-based techniques across such domains. As a result, it becomes more practical to develop a network for a complex system and apply network-synthesis-based techniques within the domain that predominates its operation. Accordingly, the design approach proposed in this paper will exclusively utilise the hydraulic domain.

Methodology overview

This section presents an overview of the proposed methodology for the design of HIS systems, outlined as a framework in Figure 2(a). The methodology begins by mathematically formulating a design problem. This involves developing a vehicle model, termed ‘host structure’, and defining its operational scenarios. Based on the identified operational scenarios, an appropriate objective function and constraint(s) are selected to facilitate the optimal design of a HIS. Additionally, a physical model of a conventional HIS system, referred to as the ‘physical benchmark’, is established.

Upon formulating the design problem, a beneficial physical HIS system can be constructed in three key steps, outlined as follows:

(1) Benchmark network development

In this step, the physical benchmark is transformed into an equivalent multi-port hydraulic network configuration. During this transformation, physical components of the HIS system are represented using hydraulic modelling elements, and their topological arrangements are carefully accounted for. These modelling elements capture the essential linear properties of the physical components, in an idealised manner.²³ The resulting hydraulic network serves as a benchmark and is referred to as the ‘network-represented benchmark’. This benchmark network will allow a wide range of network-represented design possibilities to be explored in step 2.

For the case study detailed in this paper, Figure 2(b) illustrates the three-terminal equivalent hydraulic network for pipeline A of the physical benchmark, which is a single-axle HIS system. Linear hydraulic resistance elements ( $R_{A 1}, R_{A 2}$ , and $R_{Aa})$ represent the damping effects of the pipeline and orifice, while a linear hydraulic compliance element ( $C_{Aa})$ represents the compliance effects of the accumulator. Detailed definitions for these variables are provided in Section ‘Benchmark network development’.

(2) Optimal network identification

In this step, an optimal network-represented HIS configuration is determined. Initially, a full set of hydraulic network-represented design possibilities are generated for a pre-determined complexity, using an existing network synthesis technique, such as the structure-immittance approach.¹² The pre-determined complexity is a design choice. Next, a decision is made regarding which part of the network/system will be designed. Once this choice is made, the generated network possibilities will replace the corresponding original section in the network-represented benchmark configuration. These integrated networks are termed ‘candidate HIS networks’. Each candidate HIS network is subsequently incorporated into the host structure model. An optimisation algorithm is then employed to determine the optimal network in terms of topology and element values, assessed against the selected cost function and constraints.

For the case study detailed in this paper, a full set of series-parallel hydraulic network layouts consisting of one inertance, one resistance, and one compliance element are enumerated based on the so-called generic networks shown in Figure 7(a). The focus for this case study is to design the accumulator inlet, that is, modify $R_{Aa}$ and $R_{Ba}$ of the network-represented benchmark configuration. Consequently, each network is integrated in place of $R_{Aa}$ and $R_{Ba}$ . An example optimal network for pipeline A is shown in Figure 2(c).

(3) Physical device realisation

In this step, the optimal network properties identified in step 2 are realised using physical components, such as orifices and pipelines. The topological arrangement of the selected physical components remains consistent with the modelling elements in the optimal network. However, due to inherent non-linearities, the behaviour of the physical device will inevitably differ from its optimal network representation. This necessitates further modelling and optimisation. By integrating the model representing the physical device with the host structure, key design parameters are optimised according to the objective function and constraints, resulting in a physical HIS solution.

For the case study detailed in this paper, different physical components are selected to realise the linear hydraulic inertance, resistance, and compliance modelling elements of the identified optimal network configuration. The selected components are implemented within the multi-physics modelling software AMEsim, as shown in Figure 2(d).²⁴ Parameter definitions for the variables in Figure 2(d), can be found in Section ‘Physical realisation development’.

It should be noted that within this framework, there exists two conditional loops: (1) If step 2 does not yield satisfactory performance improvements, a wider range of network-represented design possibilities should be explored, that is, considering an alternative complexity, and/or alternative design locations; (2) Following step 3, if the performance of the physical HIS system is not satisfactory, a broader selection of physical hydraulic components should be considered. Alternatively, one may revert to step 2. The physical HIS system is deemed ready for prototyping and subsequent performance evaluation only upon completion of these steps.

Design problem formulation

In this section, the design problem for the case study is formulated. Firstly, a linear four-degrees-of-freedom (4-DOF) roll-plane half-car model, compatible with a single-axle HIS system, is developed. Next, an appropriate objective function and constraint(s) is chosen to quantitatively measure the vehicle’s dynamic performance under different operational scenarios. Subsequently, a multiphysics model of a conventional single-axle HIS system is constructed.

Development of a 4DOF half-car model

A linear 4DOF roll-plane half-car model, which exhibits left-right symmetry, is depicted in Figure 3. The model consists of two unsprung masses, $m_{ul}$ and $m_{ur}$ , which represent one wheel set of the vehicle, and a sprung mass, $M$ , which characterises the vehicle body. The sprung mass moment of inertia about the roll axis is given by I. Stiffness coefficients, $K_{tl}$ and $K_{tr}$ , and damping coefficients, $C_{tl}$ and $C_{tr}$ , are used to model the tyre properties. The wheels are connected to the vehicle body by means of a passive HIS system, and a linear coil spring. Stiffness coefficients, $k_{sl}$ and $k_{sr}$ , represent the left and right springs, while reaction forces, $F_{ld}$ and $F_{rd}$ , characterise the left and right dynamic forces exerted by the hydraulic piston cylinders of the HIS. In Figure 3, the vehicle is illustrated in the roll mode, where $F_{ld}$ is positive with extension on the left side, and $F_{rd}$ is positive during compression on the right side. The distances from the centre of the left and the right hydraulic piston cylinders to the centre of gravity of the sprung mass are denoted by $b_{l}$ and $b_{r}$ , respectively. For brevity, the parameter values utilised for the half-car model are not shown but can be obtained in reference.²⁵

Figure 3.

Schematic diagram of a linear 4DOF roll-plane half car model integrated with a single-axle HIS system.

The equations of motion describing the half-car model in the Laplace domain, can be expressed as follows:

[s^{2} M + s C + K + s D_{y}^{T} Z_{m} D_{y}] \tilde{y} = \tilde{F}

(1)

where $M$ , $C$ , and $K$ , are the mass, damping, and stiffness matrices, defined by:

\begin{matrix} M = [\begin{matrix} m_{ul} & 0 & 0 & 0 \\ 0 & m_{ur} & 0 & 0 \\ 0 & 0 & M & 0 \\ 0 & 0 & 0 & I \end{matrix}], C = [\begin{matrix} c_{tl} & 0 & 0 & 0 \\ 0 & c_{tr} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ K = [\begin{matrix} k_{tl} + k_{sl} & 0 & - k_{sl} & b_{l} k_{sl} \\ 0 & K_{tr} + k_{sr} & - k_{sr} & - b_{r} k_{sr} \\ - k_{sl} & - k_{sr} & k_{sl} + k_{sr} & - b_{l} k_{sl} + b_{r} k_{sr} \\ b_{l} k_{sl} & - b_{r} k_{sr} & - b_{l} k_{sl} + b_{r} k_{sr} & b_{l}^{2} k_{sl} + b_{r}^{2} k_{sr} \end{matrix}] \end{matrix}

(2)

The vector $\tilde{F}$ represents the applied excitation force due to the road, defined as $\tilde{F} = B \tilde{ξ}$ . Here, $B$ is the impedance matrix of the tyres, and $\tilde{ξ}$ is the road displacement input applied to the tyres, expressed as $B = [s c_{tl} + k_{tl} 0 0 0; 0 s c_{tr} + k_{tr} 0 0; 0 0 0 0; 0 0 0 0]$ and $\tilde{ξ} = {[\begin{matrix} {\tilde{ξ}}_{l} & {\tilde{ξ}}_{r} & 0 & 0 \end{matrix}]}^{T}$ , respectively. $\tilde{y}$ is a displacement vector which encapsulates the translational motion of the vehicle’s left wheel, right wheel, and body, as well as the rotational motion of the body, defined as $\tilde{y} = {[\begin{matrix} {\tilde{y}}_{ul} & {\tilde{y}}_{ur} & {\tilde{y}}_{s} & \tilde{θ} \end{matrix}]}^{T}$ with positive directions shown in Figure 3. $Z_{m}$ is the mechanical admittance matrix that relates the dynamic hydraulic forces of the HIS system, ${\tilde{F}}_{h}$ , to the relative velocities of the hydraulic piston cylinders, $Δ {\tilde{v}}_{h}$ . The detailed expressions for $Z_{m}$ , ${\tilde{F}}_{h}$ , and $Δ {\tilde{v}}_{h}$ are provided in Section ‘Benchmark network development’. The mathematical relationship between these variables can be represented as follows:

{\tilde{F}}_{h} = Z_{m} Δ {\tilde{v}}_{h} = Z_{m} {\tilde{D}}_{y} \tilde{y} s

(3)

where $D_{y} = [1 0 - 1 b_{l}; 0 - 1 1 b_{r}]$ .

Optimisation cost function and constraints

By implementing the equations of motion for the HIS-integrated half-car model, as detailed in subsection ‘Development of a 4DOF half-car model’, in MATLAB,²⁶ the effects of various HIS systems on vehicle dynamics can be analysed under diverse operational conditions. In this case study, ride-handling dynamics are investigated. The design target is chosen to improve the ride-comfort performance, without compromising the handling performance, of a conventional single-axle HIS system. To enable an analysis of ride-handling dynamics, a single-wheel bump input and a fishhook manoeuvre scenario are selected as the operational conditions for the vehicle, respectively.^27–30

The single-wheel bump input is simulated in MATLAB, characterised by a sinusoidal wave spanning 180°.²⁶ The bump features dimensions $1 m$ in length and $0.8 m$ in height, and it is applied to the left wheel of the vehicle. The vehicle traverses the bump at a speed of $20 km / h$ . To achieve the design target for this work, an indicative cost function, $J_{r}$ , is selected as the total right suspension force of the vehicle under the specified bump input. $J_{r}$ can be expressed as follows:

J_{r} = \max | (F_{rd} + k_{sr} . Δ x_{sr}) |

(4)

where $(F_{rd} + k_{sr} . Δ x_{sr})$ is the total right suspension force of the vehicle, and $Δ x_{sr}$ is the vertical displacement of the right suspension coil spring. Additionally, a constraint is applied to the total left suspension force in this operational scenario, to ensure vehicle balance and comfort. This constraint can be mathematically stated as:

\max | (F_{ld} + k_{sl} . Δ x_{sl}) | \leq \max | {(F_{ld} + k_{sl} . Δ x_{sl})}^{*} |

(5)

where, $(F_{ld} + k_{sl} . Δ x_{sl})$ is the total left suspension force of the suspension under investigation, and ${(F_{ld} + k_{sl} . Δ x_{sl})}^{*}$ is the total left suspension force of the benchmark HIS system. Note that in this case study, jointly decided with the industry collaborator, the total suspension force is selected as the cost function for evaluating ride comfort under a single-wheel bump input, rather than body acceleration. This is because when the vehicle experiences a bump input on the one side, the total suspension force on the other side can be quantified as the vibration transmission across the interconnection, which is an intuitive metric. Furthermore, following equations (1)–(3) the suspension forces directly govern the sprung mass acceleration, such that a reduction in force leads to a proportional reduction in acceleration.

The fishhook manoeuvre is selected to assess the vehicle’s roll control performance. A torque input is generated based on the steering angle input in reference,³¹ and it is applied on the roll DOF of the HIS-integrated half-car model, in MATLAB.²⁶ A constraint is applied to the peak roll angle of the vehicle’s sprung mass during this operational scenario, to ensure vehicle roll control. This constraint can be mathematically expressed as:

max | θ | \leq max | θ^{*} |

(6)

where, $θ^{*}$ is the sprung mass roll angle of the benchmark HIS system. In this work, optimisation of the HIS-integrated half car model is carried out in MATLAB using an embedded function ‘pattern search’.^26,32

Multiphysics HIS model

Establishing a physical benchmark system is essential for defining a reference performance, which in turn is crucial for evaluating new physical designs. In this case study, the physical benchmark system, a conventional single-axle HIS, is developed using the hydraulics library within AMEsim and in accordance with the parameters in References.^24,25 The constructed physical benchmark, termed $P h_{0}$ , is shown in Figure 4, with parameter definitions outlined in Table 1. It is important to highlight that the vehicle model integrated with $P h_{0}$ has been validated using the mathematical models presented in reference.³³

Figure 4.

Multiphysics model of the physical benchmark HIS system, $P h_{0}$ , built in AMEsim.

Table 1.

Properties of $P h_{0}$ .

Parameter	Value	Unit	Description
$ρ$	$870$	$kg m^{- 3}$	Fluid density
$μ$	$0.05$	$Ns m^{- 2}$	Fluid viscosity
$β$	$1400$	$MPa$	Fluid bulk modulus
$d_{pipe}$	$7.1 x 10^{- 3}$	$m$	Pipe diameters
$l_{pipe}$	1	$m$	Pipe length from accumulators to chambers
$V_{Aa}$ , $V_{Ba}$	$2.0 x 10^{- 4}$	$m^{3}$	Accumulator volume
$P_{Aa 0}$ , $P_{Ba 0}$	$0.5$	$MPa$	Accumulator pre-charged pressure
$A_{l 1}, A_{r 1}$	$5.1 x 10^{- 4}$	$m^{2}$	Upper piston chamber area
$A_{l 2}, A_{r 2}$	$4.1 x 10^{- 4}$	$m^{2}$	Lower piston chamber area
$R_{A 1}, R_{A 2}, R_{B 1}, R_{B 2}$	$5.0 x 10^{9}$	$kg s^{- 1} m^{- 4}$	Loss coefficients of cylinder orifices
$d_{Aa}$ , $d_{Ba}$	$3.3 x 10^{- 3}$	$m$	Accumulator inlet orifice diameter

There is a notable difference between the parameters specified in reference²⁵ and those used in this study, which pertains to the accumulator inlet properties. In reference,²⁵ each linear hydraulic resistance element, representing a flow restriction in the accumulator branch, is replaced with an orifice characterised by a realistic physical property, specifically a diameter. This modification aims to achieve a more realistic analysis of the physical benchmark system. Various orifice diameters are examined, using a least squares regression polynomial fitting technique, to determine an outer diameter which can provide equivalent pressure drop and flow rate characteristics to the linear hydraulic resistance described in reference.²⁵ Figure 5 shows the conducted polynomial fitting.³⁴

Figure 5.

(a) Orifice diameter identified as providing equivalent linear properties and (b) relationship between different orifice diameters and their corresponding equivalent linear hydraulic resistance.

The physical benchmark is built to be compatible with MATLAB.²⁶ In Figure 4, the green AMEsim components represent mechanical elements, while the pink AMEsim components are sensors. These sensors facilitate the conversion of hydraulic signals from the HIS into mechanical signals compatible with the host structure model in MATLAB, and vice versa, transforming mechanical signals from the host structure into hydraulic inputs for the HIS system in AMEsim. With the physical benchmark system developed, co-simulation of the HIS-integrated half-car model in MATLAB-AMEsim is performed to establish a benchmark performance.^24,26 This co-simulation is conducted under the operational scenarios defined in Section ‘Optimisation cost function and constraints’. Performance for $P h_{0}$ , is summarised in Table 4, Section ‘Physical parameter identification’.

Benchmark network development

To apply network synthesis techniques for the design of HIS systems, it is essential to represent the physical benchmark system, as shown in Figure 4, as an equivalent hydraulic multi-port network, comprising modelling elements. In Figure 2(b), the three-terminal hydraulic network representation for pipeline A of the physical benchmark system is shown. The network contains hydraulic modelling elements $R_{jk}$ and $C_{jk}$ , which represent linear resistance and compliance elements, respectively, with the indices $j$ and $k$ defined in accordance with Subsection ‘Challenge of developing an equivalent mechanical network’. In addition, the flow and pressure characteristics for the network are also defined in Subsection ‘Challenge of developing an equivalent mechanical network’. It is important to highlight that the variable $P_{Aa 0}$ in the three-terminal hydraulic network represents the accumulator pressure at the initial equilibrium state, distinct from $P_{Aa}$ described in Subsection ‘Challenge of developing an equivalent mechanical network’, which indicates the instantaneous accumulator pressure.

In this case study, the network representation for pipeline B is not provided; however, it would precisely mirror pipeline A as symmetry preservation is fundamental in HIS systems. By leveraging the network representations for both pipelines A and B, the network-represented benchmark system, denoted as $N_{0}$ , can be derived, as shown in Figure 6.

Figure 6.

A schematic diagram of the network-represented benchmark system, $N_{0}$ .

The relationship between the dynamic forces and relative velocities of $N_{0}$ in the Laplace domain can be expressed as:

{\tilde{F}}_{h} = D A Z_{h} A D^{T} Δ {\tilde{v}}_{h}

(7)

Where, ${\tilde{F}}_{h}$ , $Δ {\tilde{v}}_{h}$ , $Z_{h}$ , $D$ , and $A$ are defined as:

\begin{matrix} {\tilde{F}}_{h} = [\begin{matrix} F_{ld} \\ F_{rd} \end{matrix}], Δ {\tilde{v}}_{h} = [\begin{matrix} Δ v_{l} \\ Δ v_{r} \end{matrix}], A = [\begin{matrix} A_{l 1} & 0 & 0 & 0 \\ 0 & A_{r 2} & 0 & 0 \\ 0 & 0 & A_{l 2} & 0 \\ 0 & 0 & 0 & A_{r 1} \end{matrix}], D = [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}], and \\ Z_{h} = \\ [\begin{matrix} R_{A 1} + R_{Aa} + \frac{C_{Aa}}{s} & R_{Aa} + \frac{C_{Aa}}{s} & 0 & 0 \\ R_{Aa} + \frac{C_{Aa}}{s} & R_{A 2} + R_{Aa} + \frac{C_{Aa}}{s} & 0 & 0 \\ 0 & 0 & R_{B 2} + R_{Ba} + \frac{C_{Ba}}{s} & R_{Ba} + \frac{C_{Ba}}{s} \\ 0 & 0 & R_{Ba} + \frac{C_{Ba}}{s} & R_{B 1} + R_{Ba} + \frac{C_{Ba}}{s} \end{matrix}] \end{matrix}

(8)

Here, $A$ is an effective area matrix for the left and right hydraulic piston cylinders, $D$ is a dimension matrix, and $Z_{h}$ is the overall hydraulic impedance matrix of the HIS system. The detailed derivation for this relationship can be obtained in the Supplementary Material.²⁰ The mechanical admittance matrix for the HIS system, corresponding to the hydraulic impedance matrix, can be expressed as follows:

Z_{m} = DA Z_{h} A D^{T}

(9)

The parameter values for $N_{0}$ are outlined in Table 2. These parameters are obtained from reference,²⁵ with the exception of $C_{Aa}$ . Here, $C_{Aa}$ represents a linear approximation of the accumulator compliance effects. This approximation is based on polytropic gas laws, whereby the volume change is directly proportional to the pressure drop across the element.

Table 2.

Properties of $N_{0}$ .

Parameter	Value	Unit	Description
$C_{Aa}, C_{Ba}$	$3.4 x 10^{10}$	$kg m^{- 4} s^{- 2}$	Equivalent linear accumulator compliance
$A_{l 1}, A_{r 1}$	$5.1 x 10^{- 4}$	$m^{2}$	Upper piston chamber area
$A_{l 2}, A_{r 2}$	$4.1 x 10^{- 4}$	$m^{2}$	Lower piston chamber area
$R_{A 1}, R_{A 2}, R_{B 1}, R_{B 2}$	$5.0 x 10^{9}$	$kg s^{- 1} m^{- 4}$	Loss coefficients of cylinder orifices
$R_{Aa}, R_{Ba}$	$3.2 x 10^{9}$	$kg s^{- 1} m^{- 4}$	Loss coefficients of accumulator orifices

By integrating $N_{0}$ , with the host structure model, MATLAB simulations can be conducted to establish a linear benchmark performance.²⁶ These simulations are carried out as described in Subsection ‘Optimisation cost function and constraints’. The system’s performance is detailed in Table 3, Subsection ‘Identification of beneficial network configurations’.

Table 3.

Performance and parameter settings for the benchmark, optimised benchmark, and beneficial candidate network-represented HIS configurations.

Network solution	$J_{r}$ ( $N$ )	Improvement (%)	Parameters*
$N_{0}$	$1730.10$	–	$C_{Aa} = 3.36 x 10^{10} R_{Aa} = 3.20 x 10^{9}$
$N_{0_{optimal}}$	$1047.30$	–	$C_{Aa} = 4.15 x 10^{10} R_{Aa} = 4.04 x 10^{8}$
$N_{6}$	$780.65$	$25.46$	$C_{Aa} = 3.12 x 10^{10} R_{1} = 3.15 x 10^{9}$
			$C_{1} = 1.75 x 10^{10} I_{1} = 7.71 x 10^{7}$
$N_{3}$	$779.60$	$25.56$	$C_{Aa} = 1.00 x 10^{10} R_{1} = 1.01 x 10^{11}$
			$C_{1} = 2.99 x 10^{10} I_{1} = 6.05 x 10^{7}$

Units for hydraulic resistance ( $R_{Aa}$ and $R_{1}$ ), hydraulic compliance ( $C_{Aa}$ and $C_{1}$ ), and hydraulic inertance ( $I_{1}$ ) elements are: $kg s^{- 1} m^{- 4}$ , $kg m^{- 4} s^{- 2}$ , and $kg m^{- 4}$ , respectively.

Optimal network-represented HIS

Based on the benchmark network developed in Section ‘Benchmark network development’, a wide range of candidate HIS networks are explored. Firstly, network-represented candidate design possibilities are proposed using the structure-immittance approach.¹² These design possibilities are then individually integrated into the benchmark network, and in turn the host structure model. By performing optimisation, several beneficial network-represented designs are identified.

Candidate design possibilities

In this case study, the structure-immittance approach is employed to systematically enumerate network-represented design possibilities for the HIS system.¹² Two so-called generic networks, with a given complexity of one hydraulic inertance, $I$ , one hydraulic resistance, $R$ , and one hydraulic compliance, $C$ , elements are derived based on the methodology in reference.¹² These generic networks are illustrated in Figure 7(a). To capture the totality of design possibilities from these so-called generic networks, only one of the four illustrated compliance elements, $C_{1, 2, 3, and 4},$ can be present in each configuration. As a result, each generic network yields four unique possibilities, leading to a total of eight design configurations, denoted as $N_{1 - 8}$ , as depicted in Figure 7(b).

Figure 7.

(a) Two so-called generic networks, $G N_{1}$ and $G N_{2}$ , (b) network-represented candidate design possibilities, $N_{1 - 8}$ , and(c) network representation of the accumulator inlet damping component for pipeline A of the benchmark network ( $N_{0})$ .

The accumulator branch is selected as the design focus for this case study. Specifically, each network possibility will replace the accumulator inlet damping component $R_{Aa}$ in pipeline A, and symmetrically $R_{Ba}$ in pipeline B, of the network-represented benchmark HIS system. Figure 7(c), displays the network configuration for $R_{Aa}$ . Note that the design focus selected in this study is just one example, similarly other components within the benchmark system (such as the pipes) can be designed using network-synthesis based approaches.

Identification of beneficial network configurations

To identify the network configurations that enhance dynamic performance, the network-represented design possibilities proposed in Subsection ‘Candidate design possibilities’ are optimised. Specifically, each network possibility is substituted into the benchmark network-represented system, thereby creating candidate HIS networks. These candidate HIS networks are then integrated within the host structure model. Optimisation is conducted in MATLAB to determine the optimal set of element values that minimise the cost function $J_{r}$ , while satisfying the constraints.²⁶ The design variables for the optimisation are $C_{Aa}, R_{Aa}, C_{1, 2, 3, or 4}, R_{1}$ , and $I_{1}$ . Apart from these variables, all other parameters are maintained constant, in accordance with $N_{0}$ . The constrained performance values are derived from the performance metrics of $N_{0}$ , where $max | θ^{*} | = 2.64 degrees$ and $\max | {(F_{ld} + k_{sl} . Δ x_{sl})}^{*} | = 5822$ $N$ .

In addition to optimising the new designs, it is important to also optimise the benchmark accumulator branch (i.e. $C_{Aa}$ and $R_{Aa}$ ), this is to ensure a fair comparison. In Table 3, the optimised performance and parameter settings for each network-represented HIS is summarised. The performance improvements noted here are in comparison with the optimal network-represented benchmark system $N_{0_{optimal}}$ .

The results indicate that network configurations $N_{3}$ and $N_{6}$ can yield the most significant performance improvements in the $J_{r}$ index, 25.56% and 25.46%, respectively. The time-domain responses for $N_{3}$ , $N_{6}$ , and $N_{0_{optimal}}$ are shown in Figure 8. Although network $N_{3}$ exhibits superior performance in the $J_{r}$ index while meeting optimisation constraints, it leads to a prolonged settling time for the vehicle’s sprung mass roll angle after the fishhook manoeuvre input. This outcome is deemed undesirable in this case study. Given the marginal difference in $J_{r}$ index between the networks $N_{3}$ and $N_{6}$ , $N_{6}$ is considered optimal for further analysis.

Figure 8.

Response for the: (a) total right suspension force when subject to a single-wheel bump input and (b) sprung mass roll angle under a fishhook manoeuvre.

Physical device realisation

In this section, a physical device is constructed to realise the optimal hydraulic network configuration identified in Subsection ‘Identification of beneficial network configurations’. Upon establishment, this device is integrated into the physical benchmark HIS system, replacing each accumulator inlet orifice component. The novel physical HIS system is then unified with the host structure model, and key design parameters are optimised in accordance with the selected objective function and constraints to assess system performance.

Physical realisation development

Having identified the optimal hydraulic network configuration $N_{6}$ , it is necessary to construct its physical realisation, including its modelling elements ( $I_{1}$ , $R_{1}$ , and $C_{1}$ ) and their topological arrangements. A common approach to realise hydraulic resistance is through the application of an orifice. Therefore, an orifice with diameter $d_{A_{orifice}}$ is chosen in this case study. The realisation of linear hydraulic inertance can be accomplished via various methods,^35,36 however a pipe serves as the selected physical component in this case study. The relationship between inertance and physical pipe characteristics, such as length $l_{A_{pipe}}$ and diameter $d_{A_{pipe}}$ , are expressed as follows¹⁸:

I_{1} = \frac{ρ . l_{A_{pipe}}}{A_{pipe}}, where A_{pipe} = \frac{π . d_{A_{pipe}}^{2}}{4}

(10)

Here, $A_{pipe}$ is the cross-sectional area of the pipe, and $ρ$ is the density of the fluid flowing through the pipe. By utilising the linear inertance value determined through network optimisation, and a predetermined pipe diameter, a corresponding pipe length can be calculated. This length can be used to guide the chosen design configuration for the pipe. In contrast to the convenience of realising inertance and resistance elements hydraulically, ‘embedded’ compliance within hydraulic/pneumatic networks is very challenging. Here the term ‘embedded’ means that the compliance element is neither in series nor in parallel with the remainder of the network. A recent solution to this challenge is a component known as the ‘dual spring device’, which incorporates two mechanical springs with identical stiffnesses, $K_{A_{spring}}$ , into a piston cylinder configuration.¹⁸

Through the use of a dual spring device, a straight pipe, and an orifice, the hydraulic network for $N_{6}$ , as depicted in Figure 7, can be physically realised in AMEsim. This physical realisation is denoted as $P h_{6}$ , and it is illustrated in Figure 2(d).

Physical parameter identification

It is essential to identify the parameter values for the physical components within $P h_{6}$ . To achieve this, $P h_{6}$ is integrated in place of each accumulator inlet orifice component within the physical benchmark system in AMEsim.²⁴ AMEsim/MATLAB optimisation is performed to determine the optimal set of parameter values for the components within the accumulator branch, minimising the cost function $J_{r}$ , while satisfying the constraints.^24,26 The design variables for the optimisation are $V_{Aa}, d_{Aa}, K_{A_{spring}}, d_{A_{pipe}}, l_{A_{pipe}},$ and $d_{A_{orifice}}$ . Aside from these variables, all other parameters remain fixed in accordance with $P h_{0}$ . The constrained performance values are derived from the performance metrics of $P h_{0}$ , where $max | θ^{*} | = 2.84$ ° and $\max | {(F_{ld} + k_{sl} . Δ x_{sl})}^{*} | = 4891$ N.

To ensure a fair comparison between the performance of the newly identified physical HIS system and that of the physical benchmark system, it is necessary to also optimise the components within the accumulator branch ( $V_{Aa}$ and $d_{Aa})$ of the physical benchmark system. In Table 4, the identified performance and parameter settings for the physical HIS configurations are summarised. It is important to note that the performance improvement of $P h_{6}$ is compared against the optimal physical benchmark system $P h_{0_{optimal}}$ . The results indicate that $P h_{6}$ can yield a 15.6% performance improvement in the $J_{r}$ index, compared with the best conventional system. The performance response of the total right suspension force during a single-wheel bump input and the sprung mass roll angle during a fishhook manoeuvre for $P h_{0}$ , $P h_{0_{optimal}}$ , and $P h_{6}$ are presented in Figure 9. It is observed that $P h_{6}$ exhibits high-frequency spikes in the suspension force. Under real-world conditions, these spikes would have negligible impact on passenger comfort, as the high-frequency vibrations would typically be attenuated by a top mount, which serves as a buffer between the suspension strut and the chassis.³⁷ However, for simplicity, the top mount has not been modelled in this case study.

Table 4.

Performance and parameter settings for the physical benchmark, optimal physical benchmark, and the newly identified physical HIS system.

Solutions	$J_{r}$ ( $N$ )	Improvement (%)	Parameters*
$P h_{0}$	$1815.88$	–	$V_{Aa} = 2.00 x 10^{- 4} d_{Aa} = 3.33 x 10^{- 3}$
$P h_{0_{optimal}}$	$1117.09$	–	$V_{Aa} = 1.80 x 10^{- 4} d_{Aa} = 6.76 x 10^{- 3}$
$P h_{6}$	$942.99$	$15.59$	$V_{Aa} = 3.20 x 10^{- 4} {d_{A}}_{orifice} = 1.67 x 10^{- 3}$ ${d_{A}}_{pipe} = 9.65 x 10^{- 3} {l_{A}}_{pipe} = 6.40 x 10^{- 1}$ ${K_{A}}_{spring} = 8.72 x 10^{3}$

Units for accumulator volume ( $V_{Aa}),$ spring stiffness ( ${K_{A}}_{spring})$ , pipe diameter ( ${d_{A}}_{pipe})$ , pipe length ( ${l_{A}}_{pipe}$ ), and orifice diameters $({d_{A}}_{orifice}$ and $d_{Aa})$ are: $m^{3}$ , $N / m$ , $m$ , $m$ , and $m$ , respectively.

Figure 9.

Response for the: (a) total right suspension force when subject to a single-wheel bump input and (b) sprung mass roll angle under a fishhook manoeuvre, for the physical benchmark $P h_{0}$ , optimal physical benchmark $P h_{0_{optimal}}$ , and the newly identified physical HIS system $P h_{6}$ .

In addition to the cost function and constraints considered in the optimisation, it is crucial to carry out a comprehensive performance assessment of the obtained beneficial HIS design. Figure 10(a) and (b) depict the suspension travel and tyre–contact force under a single-wheel bump input, while Figure 10(c) and (d) show the corresponding responses during the fishhook manoeuvre. For brevity, only the right side of the vehicle is presented; however, both sides were analysed, and the results are consistent. Under the single-wheel bump input, no compromise in suspension travel or tyre–contact force is observed. During the fishhook manoeuvre, a slight compromise is observed; however, this deviation is minor and remains within an acceptable margin, particularly given that this study serves as a demonstration of the proposed design methodology. To further evaluate the performance of $P h_{6}$ , a rough road input scenario is also assessed. The rough road excitation, consistent with that used in reference,³⁸ is generated using the spectral density matrix

S^{ξ} = [\begin{matrix} S_{D}^{ξ} S_{X}^{ξ} \\ S_{x}^{ξ} S_{D}^{ξ} \end{matrix}]

(11)

where $S_{D}^{ξ}$ is the direct (single side) spectral density, and $S_{X}^{ξ}$ denotes the cross spectral density. Readers can refer to references^25,38 for more details about this excitation. The resulting suspension travel and tyre–contact force responses under this rough road input are shown in Figure 10(e) and (f). The results indicate minimal variation in performance among all physical HIS configurations. This observation is expected, as the primary focus of this case study is on designing the accumulator branch of the system, which predominantly influences the vehicle’s roll dynamics. In contrast, the rough road excitation primarily engages the vehicle’s bounce dynamics, which are largely governed by the damping characteristics of the interconnected pipes and cylinder valves. Since these properties remain unchanged across the evaluated systems, only minor differences in performance are observed under rough road conditions. It is noted that future work could extend the proposed design methodology to include the tuning of pipe properties to further enhance ride comfort on rough road surfaces.

Figure 10.

Further physical HIS system responses for: (a) right suspension travel, (b) right tyre-contact force, over a single-wheel bump input, (c) right suspension travel, (d) right tyre-contact force, during a fishhook manoeuvre input, (e) central body vertical acceleration, and (f) right tyre-contact force, during a rough road input.

Conclusion

This paper proposes a design methodology for HIS systems. The methodology firstly allows a conventional HIS system to be represented as an equivalent hydraulic multi-port network configuration. Subsequently, it enables an optimal network-represented HIS system to be identified, considering the totality of design configurations for a predetermined complexity, utilising existing network synthesis techniques. Finally, the methodology facilitates the physical realisation of a beneficial network-represented HIS system. Compared with traditional HIS design approaches which focus on making piecemeal modifications to existing configurations, the proposed methodology systematically explores a wide range of possibilities. This broadens the design scope for HIS systems, allowing significant performance benefits to be unlocked. The effectiveness of the proposed methodology is demonstrated via a case study, wherein the accumulator branch of a single-axle HIS system is designed. An optimal network represented HIS system is identified, considering a predetermined complexity of one hydraulic inertance, one hydraulic resistance, and one hydraulic compliance elements. This network representation can achieve 25.5% ride comfort performance enhancement over the conventional configuration. The optimal network-represented HIS system is constructed using physical components. Compared with the optimal conventional physical HIS system, the novel physical design can yield a 15.6% performance enhancement in ride comfort under a single wheel bump input, without compromising the roll control performance. A comprehensive performance assessment of the proposed design, compared with the optimal conventional physical HIS system, has also been carried out with no evident compromise identified. It is worth nothing that integrating a multi-objective optimisation function into the design problem formulation could further enhance overall system performance. It is also important to emphasise that this case study represents only one design example. Within the design framework the network complexity can be readily adjusted to consider a wide range of configurations, including variations in the number and types of elements. In addition, the proposed methodology can be directly applied to optimally design other components of the HIS system, such as pipes and piston-cylinder assemblies; and it can be extended to investigate more advanced HIS configurations, including cross-axle interconnection topologies for multi-axle systems and active-passive hybrid configuration.

Supplemental Material

sj-docx-1-pid-10.1177_09544070251395345 – Supplemental material for A design approach for hydraulic interconnected suspension systems

Supplemental material, sj-docx-1-pid-10.1177_09544070251395345 for A design approach for hydraulic interconnected suspension systems by Emily Nar, Duanqi Zhao, Yuan Li, Jason Zheng Jiang, Monzer Al Sakka, Branislav Titurus and Simon Neild in Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering

Footnotes

Appendix

Symbol	Description
$a$	Property associated with the accumulator branch of the HIS system
$A$	Pipeline A of the HIS system
$A$	Effective area matrix for the left and right hydraulic piston cylinders
$A_{jk}$	Cross sectional chamber areas, where a subscript $j$ corresponds to $l, r$ , $A$ , or $B$ , and a subscript $k$ corresponds to $1$ , $2$ , or $a$ .
$A_{l 1}$	Upper cross-sectional area of the left piston cylinder
$A_{l 2}$	Lower cross-sectional area of the left piston cylinder
$A_{pipe}$	Cross-sectional area of a physical pipe
$A_{r 1}$	Upper cross-sectional area of the right piston cylinder
$A_{r 2}$	Lower cross-sectional area of the right piston cylinder
$b_{l}$	Distance from the centre of the left hydraulic piston cylinder to the centre of gravity of the sprung mass
$b_{r}$	Distance from the centre of the right hydraulic piston cylinder to the centre of gravity of the sprung mass
$B$	Pipeline B of the HIS system
$B$	Impedance matrix of the tyres
$c_{A 1}$	Upper mechanical damping coefficient of cylinder orifice for pipeline A
$c_{A 2}$	Lower mechanical damping coefficient of cylinder orifice for pipeline A
$c_{Aa}$	Equivalent linear accumulator compliance for pipeline A of the network-represented HIS systems
$c_{jk}$	General notation for mechanical damping where a subscript $j$ corresponds to $l, r$ , $A$ , or $B$ , and a subscript $k$ corresponds to $1$ , $2$ , or $a$ .
$C_{1, 2, 3, and 4}$	Linear hydraulic compliance elements considered in the network-represented candidate design possibilities
$C_{jk}$	General notation for linear hydraulic compliance elements, where a subscript $j$ corresponds to $l, r$ , $A$ , or $B$ , and a subscript $k$ corresponds to $1$ , $2$ , or $a$ .
$C_{tl}$	Left tyre damping coefficient
$C_{tr}$	Right tyre damping coefficient
$C$	General notation for hydraulic compliance within a network
$d_{A_{orifice}}$	Diameter of the orifice used in the construction of $P h_{6}$
$d_{A_{pipe}}$	Diameter of the physical pipe used in the construction of $P h_{6}$
$d_{Aa}$	Accumulator inlet orifice diameter for pipeline A of the physical HIS system
$d_{Ba}$	Accumulator inlet orifice diameter for pipeline B of the physical HIS system
$d_{pipe}$	Pipe diameter for the physical HIS system
$D$	Dimension matrix
$D_{y}$	Transformation (mapping) matrix
$F_{ld}$	Dynamic force exerted by the left hydraulic piston cylinder of the HIS
$F_{rd}$	Dynamic force exerted by the right hydraulic piston cylinder of the HIS
$\tilde{F}$	Force vector of applied excitation force due to the road
${\tilde{F}}_{h}$	Dynamic hydraulic forces of the HIS system
$G N_{1}$	Generic network 1 for a predetermined complexity of one hydraulic resistance, compliance, and inertance element.
$G N_{2}$	Generic network 2 for a predetermined complexity of one hydraulic resistance, compliance, and inertance element.
$I$	Sprung mass moment of inertia about the roll axis
I	General notation for hydraulic inertance within a network
$J_{r}$	Cost function
$k_{Aa}$	Mechanical stiffness of the accumulator in pipeline A
$k_{jk}$	General notation for mechanical stiffness where a subscript $j$ corresponds to $l, r$ , $A$ , or $B$ , and a subscript $k$ corresponds to $1$ , $2$ , or $a$ .
$k_{sl}$	Left suspension stiffness coefficient
$k_{sr}$	Right suspension stiffness coefficient
$K_{tl}$	Left tyre damping coefficient
$K_{tr}$	Right tyre damping coefficient
$K_{A_{spring}}$	Spring stiffnesses of the dual spring device
$l$	Left hand side
${l_{A}}_{pipe}$	Length of the physical pipe used in the construction of $P h_{6}$
$l_{pipe}$	Pipe length from accumulators to chambers for the physical HIS system
$m_{ul}$	Left unsprung mass
$m_{ur}$	Right unsprung mass
$M$	Sprung mass
$N_{0}$	Network-represented benchmark system
$N_{0_{optimal}}$	Optimised network-represented benchmark system
$N_{1 - 8}$	Network-represented candidate design possibilities
$P_{Aa}$	Instantaneous accumulator pressure for pipeline A of the HIS system
$P_{Aa 0}$	Accumulator pre-charged pressure for pipeline A of the physical HIS system
$P_{Ba 0}$	Accumulator pre-charged pressure for pipeline B of the physical HIS system
$P_{jk}$	Instantaneous system pressures, where a subscript $j$ corresponds to $l, r$ , $A$ , or $B$ , and a subscript $k$ correspondsto $1$ , $2$ , or $a$ .
$P_{l 1}$	Upper left piston cylinder chamber pressure
$P_{r 2}$	Lower right piston cylinder chamber pressure
$P h_{0}$	Physical benchmark HIS system
$P h_{6}$	Physical realisation of the hydraulic network for $N_{6}$
$P h_{0_{optimal}}$	Optimal physical benchmark HIS system
$q_{jk}$	Fluid flows, where a subscript $j$ corresponds to $l, r$ , $A$ , or $B$ , and a subscript $k$ corresponds to $1$ , $2$ , or $a$
$R$	General notation for hydraulic resistance within a network
$R_{A 1}$	Upper loss coefficient of cylinder orifice for pipeline A
$R_{A 2}$	Lower loss coefficient of cylinder orifice for pipeline A
$R_{Aa}$	Loss coefficient of accumulator orifice for pipeline A
$R_{B 1}$	Upper loss coefficient of cylinder orifice for pipeline B
$R_{B 2}$	Lower loss coefficient of cylinder orifice for pipeline B
$R_{Ba}$	Loss coefficient of accumulator orifice for pipeline B
$R_{jk}$	General notation for linear hydraulic resistance elements, where a subscript $j$ corresponds to $l, r$ , $A$ , or $B$ , and a subscript $k$ corresponds to $1$ , $2$ , or $a$ .
$r$	Right hand side
$s$	Laplacian variable
$V_{Aa}$	Accumulator volume for pipeline A of the physical HIS system
$V_{Ba}$	Accumulator volume for pipeline B of the physical HIS system
${\tilde{y}}_{s}$	Translational motion vector of the vehicle’s body
${\tilde{y}}_{ul}$	Translational motion vector of the vehicle’s left wheel
${\tilde{y}}_{ur}$	Translational motion vector of the vehicle’s right wheel
$Z_{h}$	Hydraulic impedance matrix of the HIS system
$Z_{m}$	Mechanical admittance matrix
$Δ v_{l}$	Relative velocity between the piston and the cylinder of the left device
$Δ v_{r}$	Relative velocity between the piston and the cylinder of the right device
$Δ {\tilde{v}}_{h}$	Relative velocities of the hydraulic piston cylinders
$Δ x_{sr}$	Vertical displacement of the right suspension coil spring
$θ$	Sprung mass roll angle
$\tilde{θ}$	Rotational motion vector of the vehicle body
$\tilde{ξ}$	Road displacement vector applied to the tyres
${\tilde{ξ}}_{l}$	Road displacement input applied to the left tyre
${\tilde{ξ}}_{r}$	Road displacement input applied to the right tyre
$β$	Fluid bulk modulus
$μ$	Fluid viscosity
$ρ$	Fluid density

ORCID iD

Jason Zheng Jiang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by Engineering and Physical Science Research Council (Grant no: EP/T016485/1).

Declaration of conflicting interests

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

Supplementary material

Supplementary material for this article is available online.

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Supplementary Material

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