Dynamics analysis and design methodology of roll-resistant hydraulically interconnected suspensions for tri-axle straight trucks

Author’s Accepted Manuscript Dynamics analysis and design methodology of rollresistant hydraulically interconnected suspensions for tri-axle straight trucks Fei Ding, Nong Zhang, Jie Liu, Xu Han www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(16)30291-5 http://dx.doi.org/10.1016/j.jfranklin.2016.08.016 FI2698

To appear in: Journal of the Franklin Institute Received date: 1 December 2015 Revised date: 13 May 2016 Accepted date: 20 August 2016 Cite this article as: Fei Ding, Nong Zhang, Jie Liu and Xu Han, Dynamics analysis and design methodology of roll-resistant hydraulically interconnected suspensions for tri-axle straight trucks, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.08.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Submitted to Journal of The Franklin Institute Engineering and Applied Mathematics, Dec. 2015

Dynamics

analysis

and

design

methodology

of

roll-resistant hydraulically interconnected suspensions for tri-axle straight trucks by

Fei Ding1*, Nong Zhang2, Jie Liu3, Xu Han3 China Automotive Engineering Research Institute Co., Ltd, Chongqing, 401122, People's Republic of China 1

School of Electrical, Mechanical and Mechatronic Engineering, University of Technology, Sydney, NSW 2007, Australia 2

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, People's Republic of China 3

Dr. Fei Ding (Corresponding Author in Submission) Tel: +86-023-6842 8519; Fax: +86-023-6882 1361. E-mail address: [email protected] or [email protected]

Nong Zhang Tel: +61-2-9514 2662; Fax: +61-2-9514 2655. E-mail address: [email protected] (Prof. Nong Zhang)

Jie Liu Tel: +86-0137 8779 2717; Fax: +86-731-8882 2051 E-mail address: [email protected] (Dr. Jie Liu)

Xu Han (Corresponding Author in Publication) Tel: +86-731-8882 3993; Fax: +86-731-8882 2051 E-mail address: [email protected] (Prof. Xu Han)

*Author to whom all further correspondence should be addressed

---------------------------------------------------------------------------------------------------------------------------This paper is submitted for possible publication in Journal of The Franklin Institute Engineering and Applied Mathematics. It has not been previously published, is not currently submitted for review to any other journal, and will not be submitted elsewhere during the peer review.

---------------------------------------------------------------------------------------------------------------------------

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Abstract A new hydraulically interconnected suspension (HIS) system is proposed to enhance the roll dynamics of the tri-axle straight trucks. The impedance of the hydraulic system is derived with impedance transfer matrix method, and integrated to establish the equations of motion of the mechanical and hydraulic coupling system. Based on the obtained equations, the additional mode stiffness/damping of the vehicle body and wheel state forces are explicitly described with the physical parameters of the hydraulic system. The obtained results indicate that the proposed HIS system can be able to independently enhance the mode stiffness/damping, in which the additional bounce/pitch and roll/warp mode stiffness are determined by the difference and summation of the top and the corresponding bottom piston surface area, respectively. The mode damping is caused by the direction and roll damper valves, simultaneously. The later valves alter the mode damping like the accumulators change the mode stiffness. The comparison of dynamic responses between the trucks with the conventional suspension and the HIS system shows that the HIS system can effectively suppress the roll motion of the truck body and favorably reduce the warp mode force for the wheel stations. Finally, the loss coefficients of the damper valves are tuned in terms of dimensionless factors to handle the compromising indices based on the dynamic responses.

Keywords: roll-resistant hydraulically interconnected suspension, mode stiffness/damping, compromise design, tri-axle straight truck

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1. Introduction Vehicle suspensions are designed to isolate the passenger compartment from roadway irregularities, as well as control and distribute the vertical tyre-loads for numerous driving conditions [1, 2]. Therefore, the suspension design involves the selection of optimal spring/damper parameters to provide a better compromise between ride, road holding and handling performances [3-6]. The controlled (semi-active and active) suspensions have been recently developed to achieve better compromise[7-9], while the passive suspensions still predominate in modern society due to inevitable drawbacks involved in these controlled suspensions [10,11]. Therefore, some alternative passive suspension systems are considered desirable to enhance design compromise. Because suspension performances are determined by four fundamental modes [12], e.g. bounce, pitch, roll and articulation, such suspensions should possess the abilities to independently enhance some modes without sacrificing the others. Additionally, full vehicle ride and handling motions are strongly coupled in roll motion induced by manoeuvres, road disturbances and crosswind [6, 13], especially in the case of heavy trucks with high center of gravity (CG) [14]. According to Winlker et al. [15], both higher roll stiffness and roll damping improve the roll stability. Therefore, anti-roll suspensions and similar assist systems are developed to improve the roll stability, such as stabilizer bar (passive) and anti-roll bar (semi-active or active). However, the stabilizer bar, without implementing roll damping, degrades the ride comfort and increases dynamic tire forces [16, 17]. Meanwhile, the wide application of active anti-roll bars is limited as result of power demand, complexity and reliability, as well as weight and cost [10, 11]. Interconnected suspensions (ISs) have been recently reported to be as one of typical alternative solutions to overcome these drawbacks, which is defined as a suspension that a displacement at one wheel station can give rise to forces at other wheel stations [18]. Compared to conventional suspensions, ISs have many advantages and can be realized by either mechanical, hydraulic, air or hydro pneumatic (hydro gas) [19]. Mechanical implementation can be realized using a simple and robust system, but it is difficult to provide individual damping [20, 21]. Air connections put higher requirement on sealing and need more installation space than fluid connections. Hydro pneumatic connections require larger working area and thus greater installation space. In this study, hydraulically interconnected suspensions (HISs) combing with mechanical springs are proposed for heavy trucks to overcome the above mentioned drawbacks, which own both the functionalities of conventional and interconnected suspensions,

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simultaneously. In this system, the single or double acting hydraulic cylinders are interconnected via hydraulic circuits. Accumulators and damper valves are incorporated in these circuits. The mechanical springs share the vehicle load, and HISs provide additional stiffness and damping for specified modes. To the author’s knowledge, so far a limited number of theoretical studies have been carried out for HIS systems [21-33]. For instance, Ortiz [21] conceptually integrated mechanical/hydraulic schematic to provide separate spring and damping for pitch/bounce and roll/articulation. Mace [22] presented a theoretical study for a family of existing passive HIS systems using network theory and system synthesis methodologies. Cao et al. [23] investigated the static and dynamic properties of interconnected hydro-pneumatic suspensions using a generalized analytical model. Zhang et al. [24] derived the impedance of hydraulic subsystem for two-axle sport utility vehicles with transfer impedance matrix method. Our previous work [19] studied the dynamic characteristics of tri-axle trucks with pitch-resistant HIS systems based on established dynamic equations of motion of coupling system. Researchers also evaluated the HIS system with experimental results [20, 25-27]. All obtained results indicated that HIS systems can improve the roll stability without sacrificing ride comfort, and achieve better weight distribution during maneuvers [21-29]. For an up-to-date review of HIS, the readers can refer to two recently published papers by Cao et al. [10] and Smith et. al. [30]. However, it can be seen that the most of above studies assumed ideal interconnections, and thus ignored the pressure loss, so the dynamics of the interconnecting mechanisms were not thoroughly investigated and so it is difficult to qualitatively study the influence of hydraulic physical parameters on the full-vehicle performance. Furthermore, most researchers focused on the performances improvements from HIS system for two-axle vehicles [21-33], especially sport utility vehicles [24-29, 31-33]. In particular there are barely reported studies on the dynamic characteristics of tri-axle straight trucks equipped with HIS system. Different from two-axle vehicles, the typical tri-axle straight trucks are classified into two types, 6 2 and 6 4 straight trucks, which correspond to trucks with dual-steering and dual-driven axles, respectively. It means that there are totally six wheel hops for motions of these wheel stations with respect to rigid frame. Therefore, the conventional method [22, 34], which is based on two-axle vehicles, should be extended to obtain displacement or force modes of suspension systems and wheel stations. This extension is especially important for tri-axle trucks with HIS systems when aiming to improve ride/handling performances with method of individually setting mode stiffness/damping. Additionally, the HIS system used for tri-axle straight truck is generally realized by six interconnected actuators. Compared to four interconnected actuators for two-axle vehicles, the interconnections among six actuators yield more complex

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configurations [32]. As a result, it is required to propose a new configuration of fluidic interconnections for HIS system, in order to independently enhancing roll mode stiffness/damping in this study. In this paper, a new roll-resistant HIS system is proposed to improve performances of tri-axle straight truck. The equations of motion of coupling system are developed by incorporating the hydraulic strut forces into the mechanical subsystem as externally applied forces, which are derived with transfer impedance matrix method. Based on the obtained dynamic equations, the mode forces of vehicle body and wheel stations for tri-axle straight truck are obtained, and the additional mode stiffness/damping yielded by hydraulic system are presented to evaluate the capacities of this HIS system to independently adjust roll mode stiffness/damping. The dynamic responses between trucks with HIS system and conventional suspension are further compared. And a parametric design method is finally employed to tune the stiffness and damping related parameters of hydraulic system.

2. Model development The tri-axle straight truck studied in this paper has one steering axle and two driven axles. The steering axle is connected to rigid frame via leaf spring suspension. While the suspension system for two rear driven axles is inverted semi-elliptic spring centrally pivoted tandem axle bogie suspension [35]. Based on our previous work [36], this bogie suspension can be modeled using a new simplified model, as shown in Fig. 1. In this study, a typical full truck model comprising the lumped sprung mass and unsprung mass is considered. This model has eleven degrees of freedom (DOFs) with linear stiffness and damping parameters. In more details, sprung mass (with mass ms , pitch inertia I P , roll inertia I R ) has three DOFs at center of gravity (CG) of itself, e.g. heave ( z ), pitch (  ) and roll (  ). Each unsprung mass (with mass mui , roll inertia I iR ( i  f , m, r )) has two DOFs at wheel centers, e.g. wheel hops ( zij ( j  l , r )). And each massless working beam of simplified tandem suspension has one DOF at its pivot center, e.g. pitch (  i ( i  L or R )). In this model, suspension system is characterized by linear spring rate k sij and viscous damping csij . The tire is modeled using a simple linear spring with stiffness ktij .

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ksfr

csfr

θ

ϕ x zfr

ksfl

csfl

zfl ktfl

,I fR m uf

ktfr

z

y ms,IP,IR θL ksmr

wfr

ksml

csml

wfl zml

mu

, I mR m

ksrl

θR

csmr

ktmr

zmr ksrr wmr

csrl m

ktml

csrr

,I rR ur

zrr ktrr wrr

zrl

wml ktrl

wrl

Fig. 1. 11 DOFs mathematic model for tri-axle straight truck with tandem rear axle bogie suspension, i.e. three DOFs for vehicle body heave, pitch, roll motion, six DOFs for left and right wheel hops, and two DOFs for pitch motion of left and right walking beams

The proposed HIS system is shown in Fig. 2, in which the actuators, realized by double-acting hydraulic cylinders, are mounted between the sprung mass and unsprung mass. When the vehicle in Fig. 1 turn left, positive roll motion between the sprung mass and unsprung masses occurs, all outer and inner suspensions are compressed and extended, respectively. Therefore, all outer and inner cylinders head move downward and upward relative to the piston rod, respectively. It reduces the volume of outer top and inner bottom cylinder chambers, and simultaneously increases the volume of outer bottom and inner top cylinder chambers, as shown in Fig. 2. Thus, the fluids in the former are squeezed into pipe lines, and the latter accommodate more fluids from pipes lines. The volume difference in circuits I and II will result in outflow and inflow for the corresponding accumulators. As a result, pressures of circuits I and

II are decreased and increased, respectively. It in return causes upward and downward resultant forces for all outer and inner actuators, respectively. Finally, a roll restoring moment is generated to prevent the roll motion of sprung mass relative unsprung mass. As a result, sprung and unsprung masses motions occur due to reaction force provided by the hydraulic system, which in turn alters the hydraulic force. This interaction will continue until a new equilibrium is reached.

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st8 A st9 st1 st7

st2 st10

st3 st20

A' st1'

st11

st10' st2'

st14 st13 st 12 B st15 Circuit I

B'

Circuit II

st19 st16 st4 st26 D st21 st st C 18 17 st23 E

st3'

C'

D' st26' st4' st24'

st25' E' st 27'

st5'

st5 st22

st6

st28' st6'

Fig. 2. Fluid circuit for roll-resistant HIS* *It is noted that the solid and dashed lines represent the high and low pressure lines, respectively, when vehicle turned left. The connection section of different fluid component is marked as sti . sti ' is the symmetrical section for

sti . The T-type junctions are marked using capital letters, such as A, B,...,E, A' , B' ,...,E' .

As described above, the integrated system is mechanical and hydraulic coupling system. Therefore, the investigation of this coupling system involves in developing the equations of motion. In this study, the dynamic equations of mechanical and hydraulic subsystems are separately established, and then integrated together to form the equations of motion via the boundary conditions of the coupling system.

2.1. Equations for mechanical subsystem The equations of motion of full-truck model are developed based on Newton’s second law of motion, assisted by free-body-diagrams of rigid bodies on which the hydraulic forces are considered as external forces:

M Ux  K U x  CU x  FW w  FH

(1)

where M U , C U , K U are the mass, damping and stiffness coefficient matrices for the mechanical



subsystem, respectively. The displacement vector is defined as x111  z



T

zijT , in which  j

  Tj

and zij are column vectors. FW is the coefficient matrix corresponding to the road-disturbance input

 .

column vector w  wij

T

The actuator forces FH  DP P are generated due to the pressures pikj



( k  U , B ) in the cylinder chambers, in which the pressure column vector P  PIT

PIIT



T

, where

8 / 39

PI   piUl

piBr  and PII   piUr T

piBl  are the pressure column vector for circuit I and II, respectively. T

Dp is the coefficient matrix for the pressure column vector P .

2.2. Equations for hydraulic subsystem Equation (1) indicates that the dynamic responses can be obtained if the relationships between FH and x are determined. In fluid dynamics, the flow is related to flow rate with effective flow area. In other words, the relationships between the flow and pressure determine the hydraulic forces FH , which are defined as flow impedance in this paper. In this study, four types of fluid components are included, such as nitrogen-filled diaphragm accumulators, three-way junctions, damper valves and fluid pipelines, as shown in Fig. 2. Along the direction of fluid flow, the connection section of different fluid component is marked as sti . sti is the symmetrical section for sti . A column vectors of pressure and flow ( p and q ) at the in- or out-flow section ( sti or sti 1 ) of the component is referred to as state vectors v i   p

qsti . The transfer matrix T

T j is used to quantitatively relate the two state vectors for the j th component, so that, v i 1  T j v i , where the details of transfer matrices T j for fluid components are given in [19]. The section marked with capital letters, such as A and A , denotes where the flow mutation occurs. Therefore, the transfer matrix between the adjacent flow-mutation sections is defined as connectivity matrix T [19]. Besides, the transfer matrix for the junction BB with bypass accumulator is separately derived. Hence, nine connectivity matrices Tk ( k  1,...,9 ) exist for each circuit. Fig. 2 clearly shows the circuits I and II are symmetrical. Therefore, the impedance derivation for the circuit I is presented as an example in the following sections, which can be also extended to the circuit II . Based on the theory of transfer matrix, the connectivity matrix T can be obtained by production of every individual component transmission matrix. Taking the circuit I as an example, the connectivity matrices TkI can be expressed as follows: 10 pipe v 8 8 T7pipe 7 T1valve  v1  T1I  v1 , and v 2 2 T1valve  v 9  T2Ι  v 9 , 0  T9

(2a-b)

pipe 15 jun 12 pipe pipe 20 valve v16 16 T15  T12  T11  v11  T3Ι  v11 , and v19 19 T20  T3  v 3  T4I  v 3 ,

(2c-d)

pipe 22 valve pipe v 21 21T22  T5  v 5  T5I  v 5 , and v17 17 T18  v18  T6I  v18 , pipe 26 pipe Ι v 24 24 T23  v 23  T7Ι  v 23 , and v 4 4 T2valve 6  T25  v 25  T8  v 25 , 28 pipe Ι v 6 6 T2valve 8  T27  v 27  T9  v 27 ,

(2e-f) (2g-h)

(2i)

9 / 39

where the rear superscript symbol in above matrixes indicates that the component type for the transfer matrix, and the rear subscript and front superscript indicates the upstream and downstream section of this component, respectively. Similarly, the other nine connectivity matrices TkII for the circuit II can be also obtained. It is denoted that each connectivity matrix is a 2x2 matrix. The pressures and flows of flow-mutation sections are related to each other by applying the continuity equation and ignoring pressure loss at the junctions A , C , D and E  , and we have:

pst8  pst9  pst11 , and qst8  qst9  qst11  0 ,

(3a-c)

pst16  pst17  pst23 , and qst16  qst17  qst23  0 ,

(3d-f)

pst18  pst19  pst21 , and qst18  qst19  qst21  0 ,

(3g-i)

pst24  pst25  pst27 , and qst24  qst25  qst27  0 .

(3u-w)

It is denoted that the Equation (2) consists of nine sub-equations, including eighteen state variables. Equation (3) includes twelve equations, which determines the relationships of the twelve state variables. Therefore, the Equation (2) is solvable under the conditions determined by Equation (3) if and only if any six state variables in the vectors v i (e.g. i  1,2,3,4,5,6 ) are given. Then, the impedance matrix Z I for the circuit I can be given as



ZΙ qst1

qst2

qst3

qst4

qst5

qst6

  p T

st1

pst2

pst3

pst4

pst5

pst6

. T

(4)

Similarly, the impedance matrix Z II for the circuit II (Fig. 2) can be shown as:



ZΙI qst1

qst2

qst3

qst4

qst5

qst6

  p T

st1

pst2

pst3

pst4

pst5



T pst6 .

(5)

Equations (4) and (5) can be combined together and expressed as:

 ZΙ 0  66

066   q I   p I    Z1212Q121  P121 , Z ΙI  q II  p II 

(6)

where q I and q II are the flow vectors as defined in Equations (4) and (5), respectively. p I and p II are the pressure vectors corresponding to q I and q II , respectively. Equation (6) describes the impedance equations for the hydraulic subsystem, in which the matrix Z is the impedance matrix. It is denoted that the vectors P and Q consist of the pressure and flow variables at the inlets and outlets of actuator cylinders, respectively. In this study, the pressure losses,

10 / 39

which are caused by sudden expansion/contraction sections at the ports of pipelines connected to cylinders, are incorporated into the damper-valve models. Hence, the vector P can be further used to represent the pressure in the cylinder chambers.

2.3. Boundary conditions The double-acting actuator cylinders form the boundary conditions for the hydraulic subsystem. In more details, the relative motion between the piston head and rod cylinders generates flows at the inlets or outlets of actuators. The relationships between the pressure and flow in expanded and compressed chambers can be given by [16]. Employ these relationships to the six actuator cylinders, and rewritten as:

~ 1 ~  (t)  V P (t)Q(t)  DR P(t)   V 1 (t)DQ x (t) ,

(7)

~

where DQ is the coefficient matrix associated with the relative velocity, V(t) is the time-variant matrix, and D R is the resistance coefficient matrix.

2.4. Integrated equations for coupling system Equations (1) and (6) describe the equations of motion of the mechanical and hydraulic subsystems, respectively. Equation (7) shows the boundary conditions for the coupling system. They can be combined to form the equations of motion of the coupling system and expressed in state-space form as [19]:

sXs  AsXs  BUs ,

(8)

where the system state vector is x341  [xT

x T

P T ]T and the disturbance input vector is u  w 31 ,

the characteristic matrix As  and input coefficient matrix B are given as

 01111  As   M U1K U  01111

I1111 1 U

M CU ~ 1 V (s)DQ

  0113    1  , and B  M U FW . M D    ~  0123  V 1 (s)(Z  DR ) 01112

1 U P 1

(9)

Equation (9) represents the equations of motion of the coupling system, in which the off-diagonal terms

M U1DP describe the amplification factors of pressures in the actuator chambers, and the off-diagonal terms V 1DQ denote the coefficients for the pressure variances in the actuator chambers caused by the

~

mechanical state vectors. The diagonal term V 1 (Z1  DR ) shows the pressure losses due to both the fluid impedance and cross-line leak between up and bottom cylinder chambers. All the above mentioned three terms determine the additional properties provided by the proposed HIS systems.

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3. Suspension characteristics As aforementioned, the suspension modes determine the suspension characteristics. For the vehicle system, the suspensions connect the wheel stations to the vehicle body, and undertake the functions as mentioned above. Therefore, the vehicle-body mode stiffness and damping and tire modal forces are employed to explore the characteristics of the suspensions with the HIS systems. In the following discussions, the full vehicle system is assumed to possess right-left symmetry. All right-side actuators are same as the corresponding left-side actuators, and all cylinder heads are mounted to the vehicle body. The dynamic characteristics of fluid components in the fluid circuits I are same as ones located at the symmetrical positions in the fluid circuit II .

3.1. Mode stiffness and damping for vehicle body The first three diagonal terms in matrices K U and C U present the bounce, roll and pitch mode stiffness and damping of the conventional suspension corresponding to the mode motions at CG of the sprung mass, respectively. Based on the assumptions of impressible fluid and negligible cross-leak, substitutes Equation (6) into the Laplace form of Equation (7) to obtain the pressure vector P , which is the function of the velocity vector sx ( s is the Laplace factor). With the obtained pressure vector P , B

B

the actuator forces FH , which are applied to the vehicle body, can be calculated. The forces FH can be rewritten as FHB  s(K BM s  CBM )z

T   , in which the K BM and CBM are the mode stiffness and

B

B

damping matrices for the truck body, respectively. It is noted that negative K M and CM indicate the HIS system increase the total stiffness and damping of the coupling system. They can be explicitly described as: B

K BM 2sT Acc

MA K  2          Sˆi   Sˆ f a f    Sˆi bi  0   Sˆi         i  f ,m,r    i m,r   i  f ,m,r   2     ˆ    ˆ     , (10) ˆ ˆ ˆ     Si   S f a f    Si bi   S f a f    Si bi  0   i m,r    i m,r    i  f ,m,r    2  ~   ~   0 0  S f l f    Si li     i  m,r   

12 / 39 MAj  C       2  S 2fj a f   Sij2bi 0     Sij i  f ,m,r i  m,r    B Acc CBM  2   R j  S 2fj a f   Sij2bi S 2fj a 2f   Sij2bi2 0    2K MA ZV , j T , B  i  m,r i m,r    0 0 S 2fj l 2f   Sij2li2       i m,r     B

(11)

where T Acc is the impedance of the accumulators ( sT Acc is negative, in which s is the Laplace factor),

ZVAcc is the pressure loss coefficients for damper valves, which connect the bypass accumulators to the main fluid circuits. SiU and SiB ( i  f , m, r ) are the effective cross-section area for the top and bottom

~ cylinders, respectively. Sˆi  SiB  SiU and Si  SiB  SiU are the surface area difference and summation, respectively. a f and bi ( i  m, r ) are the distances from the sprung mass CG center to the front and rear two strut-points of actuators, respectively. 2li ( i  f , m, r ) are the strut-point track of the front and rear two actuators, respectively. It is noted that all top and bottom damper valves correspond to the loss coefficients RT and RB , respectively. The matrices K MA and CBMAj ( j  T,B ) are the area related B

coefficients for the additional mode stiffness and damping, respectively. Equations (10) and (11) clearly show that the HIS system alters the mode stiffness and damping of the vehicle body. Equation (10) indicates that: 1.

The additional stiffness is proportional to T Acc . Furthermore, the additional bounce stiffness is determined by the factor

 Sˆ

i

, as well as the coupling stiffness between the bounce and pitch

~

modes. The additional roll stiffness depends on S i . 2.

B

Both the additional bounce stiffness and bounce-pitch coupling stiffness in the matrix K MA contain the factor

 Sˆ

i

, as shown in Equation (10). Therefore, the HIS system will not increase the bounce

and bounce-pitch coupling stiffness when

 Sˆ

i

 0 , which indicates one of the three (front, middle,

rear) group actuators must be opposite installed. 3.

The pitch and bounce-pitch stiffness if the conventional system will not be changed when





Sˆ f a f  Sˆmbm  Sˆr br  0 , which requires suitable relationships among the area differences Sˆi . 4.

The HIS system can separately increase the roll stiffness when both





 Sˆ

i

0

and

Sˆ f a f  Sˆmbm  Sˆr br  0 , e.g. Sˆ f Sˆr   (bm  br ) (a f  bm ) and Sˆm Sˆr  (br  a f ) (a f  bm ) . Equation (11) shows that the additional mode damping is composed of two terms, the former and latter depend on the pressure loss coefficients of the cylinder-side and accumulator-side damper valves, respectively. The two types of valves are further defined as direction and roll damper valves, respectively.

13 / 39

They are respectively abbreviated as DDVs and RDVs in the following discussions. From Equation (11), it can be found that: 1.

The function of the DDVs behaviors like the dampers in the conventional suspensions, while the RDVs alter the mode damping like the accumulators change the mode stiffness. Therefore, the mode damping caused by the DDVs can not be separately generated with the method of adjusting the surface area Sij and/or installation schematics. The RDVs increase the mode damping.

2.

The bounce damping yielded by DDVs and RDVs depends on the pressure loss coefficients R j and

~ ZVAcc , respectively. The square of surface area ( Sij2 ) and area difference ( Sˆi ) or summation ( S i ) B

determine the elements in the coefficient matrices K MA and CBMAj , respectively. Equations (10) and (11) clearly indicate that the accumulators and damper valves provide additional mode stiffness and damping for the vehicle body, respectively. However, the mechanisms are different.

3.2. Mode forces for wheel stations The

corner

forces

Fij

applied

to

the

wheel

stations

can

be

given

by

Fij  ktij zij  wij   SiU piUj  SiB piBj , which represents the road holding ability. For example, greater positive corner forces indicate the wheels are more heavily pressed on the road. Three fictitious centers ( oCG , oPl and oPr ) for the wheel stations are the sub-points of CG of the sprung mass and rear two pivot centers of the walking beams on the fictitious wheel frame, as shown in Fig. 3. The rear left and right wheel stations can rotation around the left and right fictitious centers, respectively. Hence, the front wheel stations and rear two fictitious pivot centers are located at longitudinal distances a and b from oCG , respectively. The front left and right wheel stations have distances l fl and l fr away from oCG in the transverse direction, respectively. The rear two fictitious oPj ( j  l , r ) pivot centers have track lwl  lwr . The rear-front and -rear axles located at a fixed distance bwm and bwr from oPj , respectively. It is noted that the conventional and fictitious wheel-station corners are marked as ij and j ( i  f , m, r ; j  l , r ), respectively, in which i and j represents the location in longitudinal and transverse directions, respectively.

14 / 39 fr

r

mr

rr

bwm

oCG

lwr

lfr

oPR

bwr

b

lfl

lwl

af

oPL fl

ml

l

rl

Fig. 3. Fictitious frame for mode stiffness/damping study of tri-axle wheel stations (top view)

Therefore, the four master-mode forces, e.g. bounce ( FB ), pitch ( FP ), roll ( FR ) and articulation ( FA ), and two slaver-pitch-modes, e.g. FPl and FPr , with respect to the fictitious centers oCG and oPj can be calculated as: F

TCM FM       1 1 1 1 1   F fl   FB  1  F  a a b b b  b   F fr   P   FR  l fl  l fr lwl  lwr lwl  lwr   Fml  F     TCM 065   l a  l a  l b l b  l b l b F fl f wl wr wl wr A   Fmr      FPl  0 0 bwm 0  bwr 0   Frl       0 0 bwm 0  bwr   Frr   FPr  0

where

F

K T x  K T w  DS P  ,(12)

TCM is the transformation matrix operator for the mode forces from the involved physical states,

K T  diag ([ktij ]) is the stiffness diagonal matrix, DS is the coefficient matrix for the pressure vector P . Equation (12) shows the mode forces are determined by both the stiffness of the conventional suspension and the pressure of the HIS system. In order to study the additional mode stiffness/damping generated by the HIS system, a transformation matrix operator

DV

TCM is used to transfer the displacement from the

corner state zij ( i  f , m, r ; j  l , r ) to the mode state zi ( i  B, P, R, A, Pl , Pr ) at the fictitious centers

oCG and oPj , that is

15 / 39 DV

  M zW     z SW  

TCM    l fl  l fr lwl bwr lwr bwm lwl bwm  lr bwr b ~ a ~ a ~ a ~ a   b l l l l l l b b b b wf wf w w w w   l fl  l fr lwr bwr lwl bwr lwr bwm lwl bwm     zB  ~  ~ ~  ~ z  l l l l l lw b   fl  b b b wf w w w z   wf z  P  1 b  1 b 1 bwr a  1 bwr a 1 bwm a  1 bwm a   fr  ~ ~ ~ ~  zR    1 l lwf lw b lw b lw b lw b   z ml  , (13) wf     z 1 1 bwr 1 bwr 1 bwm 1 bwm  zA  a  b  1   mr     ~ ~ ~ ~  z Pl   lwf   z rl  lwf lw b lw b lw b lw b      ab ab  z Pr  0   z rr  0 0  ~ 0 ~   b b  ab ab  0 0 0  ~ ~ 0  b b  

~

where lwf  l fl  l fr and lw  lwl  lwr is the front and rear wheel track, respectively. b  bwm  bwr is the wheelbase between the rear two axles. It is noted that such transformations must be energy conservation [22,31], e.g.

F

T DV TCM TCM  I 66 , in

which I is the identity matrix. Therefore, the corner state vector can be obtained by premultiplying the inversed matrix of

F

z

TCM , e.g.

fl

z fr

zml

zmr

zrl



1 zB zrr F TCM T

zP

zR

zA

z Pl

T z Pr  .

Hence, Equation (12) can be rewritten as

 FM F TCM  065 

diag 1 1 1 1 1 K T   065 



where the mode sate vector xM  z BM

l

r

zM W

 056  x  K T w  DS P  , 1  M TCM  

(14)

F



z SW , in which z BM  z θ  , the row vector T

S zM W and z W are the master and slave mode displacements for the wheel stations.

Equation (7) shows the pressure P depend on the velocity. Hence, the mode forces defined by Equation (13) are the function of mode displacement and velocity. For simplicity, the master fictitious center is chosen on the right-left symmetry plane, the front and rear wheels have same track, and the slave fictitious centers are equidistant from the middle and rear wheels. It therefore results in lwf  lw ,

l fl  l fr , lwl  lwr and bwm  bwr . With the same methods involved in the study of mode stiffness and W

W

damping for the vehicle body, the additional master mode stiffness K M and damping CM for the wheel stations can be determined by the partitioned matrix corresponding to the transposed vector of z M W and given by:

Λ W  K MA   BP a  b  022

W KM

0.5sT

Acc

022  , Λ RA 

(15)

16 / 39 CW

W CM W  K MA ZVAcc 0.5 a  b 

MAj    Ξ j 0 22    Rj  , 0 Ξ 2  2 j j T , B  

(16)

where the block square matrices Λ BP , Λ RA and Ξ j are given by

Λ BP

      Sˆi  2Sˆ f b    Sˆi a   i  f ,m,r    i  m,r            Sˆ f a    Sˆi b  2Sˆ f b    Sˆi a         i m,r    i m,r 

       Sˆi  2Sˆ f    Sˆi         i  m,r   i  f ,m,r   ,     ˆ    ˆ ˆ ˆ  S f a    Si b  2S f    Si     i  m,r    i m,r  

(17)

Λ RA   ~   ~   ~ ~    S f l f lr    Si  2S f b lr l f    Si a     i  m,r    i  m,r   ~        S f a l f lr    S~i b  2S~f b lr l f    S~i a         i  m,r    i  m,r 

~   ~   ~ ~    S f l f lr    Si  2S f lr l f    Si    i m ,r    i  m,r   , (18)  ~   ~   ~ ~    S f a l f lr    Si b  2S f lr l f    Si    i  m,r    i  m,r   

 2S 2fj b   Sij2 a  2S 2fj   Sij2    i  m ,r i  m ,r Ξ j   .      2S 2fj   Sij2 ab   2S 2fj a   Sij2b   i  m ,r i  m ,r   

(19)

Equations (15) and (16) show that symmetry causes no coupling between the bounce/pitch and roll/warp motions. The diagonal and off-diagonal terms in Equations (15) and (16) describe the main and coupling stiffness/damping factors, respectively. From Equations (15), (17) and (18), the effects of the physical parameters on the additional stiffness W KM can be found:

1.

The additional stiffness depends on T Acc , the additional stiffness for the bounce/pitch and roll/warp

~

modes is a linear function of Sˆi and Si , respectively. For the three fictitious centers, there are same

~ mechanisms for Sˆi to change the bounce/pitch stiffness as Si to change the roll/warp stiffness, as shown in Equations (17,18). 2.

Equation (17) shows that the bounce and bounce/pitch coupling stiffness factor will be unchanged

 Sˆ

when

 Sˆ

i

i

 0 ( i  f , m, r ), the corresponding pitch stiffness factor is 3Sˆ 2f a  b  . When

Sˆ f a b ( i  m, r ), no additional hydraulic pitch torque will not be generated, as shown in

Equation (17). But the pitch motion still cause bounce vibration by the bounce/pitch coupling factor

 Sˆ Sˆ 2  a b , i

f

and the corresponding bounce stiffness factor is

 Sˆ Sˆ a i

f

2



 b 2 b ( i  m, r ).

17 / 39

The additional bounce/pitch forces are independent of the bounce and pitch mode motions when the

 Sˆ

i

3.

 2Sˆ f b a and

 Sˆ

i

 2Sˆ f ( i  m, r ), respectively.

~

The additional roll/warp mode stiffness is explicitly described with Si , as shown in Equation (18). It is indicated that the proposed roll-resistant HIS system inevitably increases the roll mode stiffness. The ~ ~ ~ roll and warp mode forces are determined by the factors  Si ( i  f , m, r ) and S f a   Si b ~ ~ ( i  m, r ), respectively. The roll mode motion will generate same component 2S f b   Si a ( i  m, r ) for the roll and warp mode forces. The components in the roll and warp mode forces, induced by the

~



~

Si ( i  m, r ). Therefore, no warp mode motion, are simultaneously determined by 2S f  i m,r ~ ~ additional warp stiffness can be caused when S f a   Si b  0 ( i  m, r ). Based on the conditions ~ ~ for no warp stiffness, i.e. 2S f   Si  0 ( i  m, r ), it further promotes the roll mode force to be independent of the warp mode motion. It requires the front and rear actuators are installed a and

2b away from the main fictitious center, which is located on the center longitudinal axis of the fictitious wheel frame. Equations (16) and (19) show the mode damping for the wheel stations in terms of physical parameters of the hydraulic subsystem. More in details, the additional mode damping depends on both ZVAcc and

~ R j . The corresponding coefficient matrices are determined by Sˆi and Si , and S ij , respectively. Based on Equations (16) and (19), it is clearly found that 1.

If no RDVs are incorporated, i.e. ZVAcc  0 , the additional damping CM is a quadratic function of Sij . W

The partitioned matrices Ξ j for the bounce/pitch and roll/warp mode damping are the same under the given physical parameter in this study. For the bounce/pitch (or roll/warp) damping, the proposed HIS system reduces and increases the bounce (or roll) and pitch (or warp) damping, respectively. The coupling damping for the bounce/pitch (or roll/warp) mode is negative. Hence, the proposed HIS system reduces the coupling damping of the conventional system for bounce/pitch (or roll/warp). The decrement is proportional to the factor  2S 2fj 

S

i m,r

2 ij

( i  m, r ; j  T , B ). Therefore, no coupling

damping will be generated when the factor is equal to zero. 2.

W If only RDVs are installed, the additional damping is only determined by K MA ZVAcc . Therefore, the

impacts of the piston surface area S ij ( i  f , m, r ; j  T , B ) on the additional damping are the same as the S ij on the additional stiffness. If both RDVs and DDVs are used in the hydraulic system, the total mode damping can be obtained using Equation (16). Generally, the investigations of mode stiffness/damping for the vehicle body and wheel stations reveal

18 / 39

that the proposed HIS system generate additional stiffness/damping to the conventional vehicle system. The bounce/pitch and roll/warp stiffness alteration are determined by the difference and summation of the bottom and the corresponding top piston surface area for each actuator, respectively. The capacitance of the gas-filled accumulators and the pressure loss coefficients of the RDVs are equally enlarged to form the mode stiffness and damping, respectively. The additional mode damping yielded by the DDVs depends on the square of each piston surface area.

4. Responses and design methodology The stiffness/damping alterations of the suspension systems are embodied in the responses of the vehicles subject to random excitations. In this study, the dynamic characteristics are investigated in terms of weighted sprung mass acceleration (SMA), pitch/roll angular displacements (PAD/RAD), suspension working space (SWS) and tire mode force (TMF). Cole [32] stated that modal analysis method can be used as an alternative method to investigate the vibration characteristics. Thus, in this study, the modal analysis method [32] is applied to obtain the evaluation outputs. All outputs can be assembled by

Y  CX  DU , where C  [(HSMA )T X

(20)

(HPAD )T X

(HRAD )T X

(HSWS )T X

(HTMF )T ]T is output coefficient matrix, in which X

H iX ( i  SMA, PAD, RAD, SWS,TMF ) are the transfer functions. D  [0 0 0 0 ( F TCM K T )T ]T is the feed-forward matrix. The symbol T denotes matrix transpose. The root mean square (RMS) values for the corresponding outputs can be given as [33]: U

2   H (ω)S(ω)HT (ω)d ,

(21)

L

where the symbol  denotes the complex conjugate. H  CsI  As  B  D is the transfer function. 1

S(ω) is the input spectral density matrix obtained from the tested gravel road [34]. For PAD, RAD and SWS evaluations, the RMS values are calculated by using Simpson integration. The tire dynamic load coefficient (DLC) Sd  TDF Fst is further adopted to evaluate tire dynamic force. For SMA evaluation, the bounce, pitch and roll RMS values are individually calculated among each one-third octave frequency band between 0.5Hz and 80Hz, and then summed together via the frequency-weighted factors Wki and Wei . A total RMS value aW can therefore be obtained according to ISO 2631 [35]:

aW 

k

b

 (W  ki

  2

bi

)2  k p

 (W  ei

  2

pi

) 2  kr

 (W  ei

ri

)2

, 2

(22)

where k j ( j  b, p, r ) are multiplying factors,  ji are RMS values for the i th one-third octave band.

19 / 39

The study of suspension characteristics indicates that the proposed HIS system provides greater design flexible to independently assign the stiffness/damping to the specified modes and hence conveniently vary the suspension properties, which are achieved by the design of the piston surface area and the fluid components, as well as the installation schematics of the actuators. Equations (10,11,15,16) i

shows that the additional stiffness/damping depends on the area coefficient matrices K MA and CiMAj ( i  B,W ; j  T , B ). Equations (10) and (15) denote the additional stiffness is proportional to the impedance T Acc of the accumulator, which is determined by the pre-charged pressure PP and the corresponding gas volume VP under this pressure, as well as the system operating pressure PW . Equations (11) and (16) indicate that the CiMAj is amplified by the loss coefficients of the damper valves to form the damping coefficient matrix of the coupled system. For practical application, separate enhancement of the roll and/or pitch stiffness are required. Therefore, the areas are firstly tuned to realize both greatest roll/pitch stiffness and softest bounce/warp stiffness. i

These parameters consist of the matrices K MA . The parameters, which determine the capacity

C A  s 1 T Acc of the accumulator, are investigated and manually defined to form the amplification factor i

for the matrix K MA . Based on the work [36], the tuning of the loss coefficients for the damper valves involve in compromising design, which requires the loss coefficients design must be carried out under the given stiffness related parameters in terms of the evaluation indexes. The design methodology is briefly summarized in the block diagram, as shown in Fig. 4. Objective function: Max: Min:

,

Subject to:

Objective function: Min: Subject to:

Capacity of accumulators: Output evaluation indexes

Fig. 4. Design process for the HIS systems

5. Results and discussions In this study, two models named TUCS and THIS are used to represent the trucks with uncoupled conventional suspension and the proposed HIS systems, respectively. All the physical parameters used

20 / 39

for simulation can be obtained from [19,30]. In this study, the dynamic responses are shown to investigate the benefits from the HIS system in terms of the performance indexes defined by Equations (21) and (22). Applying the proposed tuning method, comprehensive discussions are then followed to illustrate the parametric design for the physical parameters of the hydraulic subsystem, such as the design of the state variables, surface areas and loss coefficients for the accumulators, actuators and damper valves, respectively.

5.1 Dynamic responses The performance improvements benefited from the HIS system are investigated in terms of power spectral density (PSD) of the evaluated outputs, defined by Equations (20), (21) and (22), in frequency domain. The simulation was run initially for a constant forward speed of 72Km/h. The road surface spectral density function used throughout this study was that of a gravel road. Fig. 5 shows the PSD responses of the roll angular displacement for the sprung mass at its CG center. Obviously, the proposed HIS system significantly suppresses the resonance peak values corresponding to the oscillation frequencies of the vehicle body and wheel hops. The responses corresponding to other oscillation frequencies are slightly increased under the fixed damping parameters. Compared to the 4

TUCS, the HIS system effectively reduces the RMS value of y RAD over 90% (  RAD  2.65 10 rad and TUCS

6 THIS RAD  7.4110 rad for TUCS and THIS, respectively). Fig. 5 also indicates that the body roll oscillation

frequency is increased from 1.687Hz to 2.735Hz after the truck is fitted with the HIS system, which is obtained with the eigenvalues identification method [19].

10

0

2

PSD response of yRAD [rad /Hz]

TUCS THIS 10

10

10

10

-5

-10

-15

-20

10

0

10 Frequency [Hz]

1

Fig. 5. Roll angular displacement of the sprung mass at its CG center

The displacement PSD responses of the wheel stations for both roll and warp modes are displayed in

21 / 39

Fig. 6. It is noted that the roll mode motion of the wheel stations is heavily restrained, as shown in Fig. 6 (a). With more details, the resonance peak values are greatly reduced, and the values located between the body and wheel hop oscillation frequencies are slightly increased. Furthermore, the oscillation frequency corresponding to the roll mode of the wheel stations is increased. Fig. 6 (b) shows the warp mode displacement of the wheel stations. For TUCS, the oscillation frequency corresponding to the first resonance peak value is the roll natural frequency of the truck body. The dotted curve in Fig. 6 (b) indicates that the roll motion of TUCS induces great warp mode deformation for the wheel stations. However, this case can be improved by the proposed HIS system, as shown the solid curve in Fig. 6 (b). The HIS system effectively suppressed the warp mode deformation and absorbs the shock caused by the roll motion of the truck body, as well as the high frequency excitation (over the oscillation frequency of wheel hop). The corresponding RMS values of z A for TUCS and THIS are 2.0mm and 1.3mm, respectively. Fig. 6 shows that the proposed HIS system is able to significantly reduce the mode deformation of the wheel stations. It is also proved by the comparison analysis of the mode force for the wheel stations, which are omitted here due to the limitation of pages.

10

-4

10

-6

2

PSD response of zR [rad /Hz]

TUCS THIS

10

10

10

-8

-10

-12

10

0

10 Frequency [Hz]

1

(a) 10

-4

10

-6

2

PSD response of zA [m /Hz]

TUCS THIS

10

10

10

10

-8

-10

-12

-14

10

0

10 Frequency [Hz]

1

22 / 39

(b) Fig. 6. Mode displacement for wheel stations: (a) roll mode, (b) warp mode

It is found that ignorable differences of performance exist between the rear two driven wheel stations. Therefore, the comparison investigation of the performances is carried out between the front- and rearrear wheel stations, as listed in Tables 1 and 2. It is denoted that the proposed HIS system increases (by 32.2% and 23.2% for the front left and front right wheels, respectively) and decreases (by 67.9% and 67.7% for the rear left and rear right wheels, respectively) the front and rear suspension deformations, respectively, as shown in Table 1. Furthermore, the HIS system effectively reduces the tire dynamic load coefficients. It also improves the ride comfort (decreasing aW from 21.42 to 1.66), and suppresses the pitch (from 4.361x10-5rad to 1.101x10-5rad) and roll (from 26.492x10-5rad to 0.741x10-5rad) angular displacement, simultaneously. It is noted that the pitch and bounce motions of the truck body are coupled together in this study. Hence, the reduction of pitch angular displacement will indirectly decrease the values of aW . Generally, all the improvements result from the enhancements of stiffness and damping, which are provided by the accumulators and damper valves of the proposed HIS system, respectively.

Table 1. Performance comparisons -3

SWS ×10 (m)

Indexes

aW

 RAD ×10‫־‬5

 PAD ×10‫־‬5

(rad)

(rad)

fl

fr

rl

rr

S DLC

Models

fl

fr

rl

rr

TUCS

21.21

21.72

17.54

17.40

21.42

26.492

4.361

0.274

0.278

2.133

2.198

THIS

28.04

26.75

5.63

5.63

1.66

0.741

1.101

0.025

0.024

0.208

0.232

Table 2. RMS values of mode displacement and force for the wheel stations Master mode×10-3

Indexes Mode Displacement (m or rad)

Models TUCS THIS

Slaver mode×10-3

Bounce

Pitch

Roll

Warp

Pitch@left

Pitch@right

11.73

3.44

5.99

2.03

193.39

193.47

4.68

3.28 8

Force

TUCS

1.68×10

(N or N·m)

THIS

1.73×108

2.95×10

2.14 8

2.94×108

8.33×10

1.35 7

8.87×107

1.81×10

5.89 8

0.89×108

5.90 8

3.07×108

0.54×108

0.54×108

3.07×10

The RMS values of mode displacement and force for the wheel stations are listed in Table 2. For the four master modes, the proposed HIS system significantly decreases the vertical displacement, as well as pitch, roll and warp angular displacement. However, the bounce and pitch mode forces are almost unchanged (increasing and decreasing by 3.0% and 0.3%, respectively). The roll and warp mode forces are enhanced and reduced by 6.5% and 50.8%, respectively. Furthermore, the RMS values of the displacement and force for the slaver modes are significantly reduced, as shown in Table 2. It is indicated

23 / 39

that the roll-resistant HIS system effectively pushes the wheels on the road in the lateral direction, and favorably adjusts the distribution of roll moment between the front and rear wheel stations. The dynamic responses of TUCS and THIS reveal that the proposed roll-resistant HIS system is able to significantly suppress the roll motion of the truck body, effectively improve the tire dynamic load, and greatly reduce the warp mode force for the wheel stations. The additional stiffness and damping provided by the HIS system also enhances the ride performance, as well as increases and reduces the front and rear suspension deformations, respectively.

5.2 Parametric design for accumulators The capacity C A of accumulator is described as CA    1PPVP PW2 , in which  is the ratio of specific heats for the gas, PW is the working pressure of the hydraulic system, VP and PP are the pre-charge volume and the corresponding pressure of the gas in the accumulators, respectively. Obviously, the volume VP is pre-chosen based on the following rules: (1) no negative volume appears during roll movement between the sprung mass and unsprung mass; (2) the maximum and minimum pressures vary among a reasonable range when roll behavior occurs. The former and latter ensure the HIS systems and accumulators are functioning normally and enduringly, respectively. Undoubtedly, the selection of the volume VP depends on the surface areas of the actuators, pre-charged and working pressures of accumulator [36]. The capacity C A of accumulator, as shown in Fig. 7, depends on the pre-charged volume VP and the working pressure PW . When the working pressure PW is fixed, greater pre-charged volume VP causes higher capacity C A . Higher working pressure PW results in larger capacity C A when the pre-charged volume VP is fixed. The contour plot for the capacity C A indicates that each level values of the capacity more heavily depend on the working pressure PW , as shown in Fig. 7. It is noted that the volume-change ratio  V of the gas in the accumulator can be obtained based on the isothermal compression process, i.e. V  1  VW VP 100% , in which VW  PPVP PW is the gas volume corresponding to the working pressure PW . Therefore, taking the pre-charged volume VP =3 x10-3m3 for example, when the volume-change ratio for the gas in Fig. 7 varies from 0 to 50%, the corresponding values for the capacity

C A are -2.143x10-9 and -5.357 x10-10m3/Pa, which changes from 0 to 75% with respect to the capacity under the pre-charged pressure. Fig. 7 also indicates that the capacity for any pre-charged volume almost changes between the same rang, i.e. from 0 to 75%, when the volume-change ratio  V varies from 0 to 50%. That is, the volume-change ratio determines the capacity-change ratio  C A of the accumulator,

24 / 39 P where  C A  1  CW A C A , in which the C A and C A are the capacity corresponds to the pre-charged

P

W

and working pressures, respectively. It is proved by rewriting the capacity-change ratio  C A as





 C A  1  VW2 VP2 100% , i.e.  C  100%  1  V 2 100% . A

Reviewer 1, Q7: x 10

-9

3

CA (Pa/m )

0

-2

-4

x10

-9

-9

4

-3

-1.0

10 .5x

5 x 10

-2 -3. -2. . 0x 10 -9 5x10 -9 0x10 9

-1

-6 6

3

1

1.4

1.2

1.6

2

1.8 x 10

VP (m3)

6

PW (Pa)

Fig. 7. 3D mesh plot with corresponding 2D contour plot for capacity of accumulator C A under pre-charged pressure PP =10x105Pa

The capacity C A is also studied under different pre-charged pressure, as shown in Fig. 8. For comparison, the pre-charged pressure PP is set to 10 x105, 40 x105 and 70 x105Pa, and the maximum volume-change ratio for the gas is up to 50%. Based on the above study, the maximum capacity-change ratio  C A for each pre-charged pressure will be 75%. On the other side, Fig. 8 shows the higher capacity

C A can be obtained when the pre-charged pressure PP is smaller, which is proved by calculating the

C WA in terms of pre-charged volume VP and pressure PP as CWA   1 1  C A VP PP .

x 10

-9

0

3

CA (Pa/m )

-1 -2 -3 -4 -5 6

PP=10x105

PP=40x105

PP=70x105

5 x 10

5 3

VP (m3)

15

10

4

-3

0

x 10

6

PW (Pa)

Fig. 8. Capacity C A of accumulator under different pre-charged pressure PP

In practical application, the volume change of the gas is determined by the structural parameters of the W

trucks, as well as the piston surface areas of the actuator. In other words, the capacity C A only depends on the pre-charged and working pressure after the HIS systems are installed. Fig. 8 indicates that the capacity C A under higher working pressure varies in a wider range than smaller working pressure. It means the additional strut forces provided by the HIS system are generated quickly to suppress the

25 / 39

relative movement between the sprung mass and unsprung mass. But the sharp forces might cause uncomfortable ride. Therefore, for practical application, the design of parameters for the accumulators should compromise the handling and ride performances.

5.3 Area ratio design for actuator cylinders In this study, all front and rear two (middle and rear) actuators are employed with the same cylinders. Three dimensionless factors are further defined when involving in the area design of the actuators, i.e.

 f  S fB S fT and   SiB SiT ( i  m, r ) are the piston surface area ratios of the bottom cylinders with respect to the corresponding top cylinders,   SiT S fT ( i  m, r ) are the area ratios of the middle and rear top cylinders with respect to the front top cylinders. Based on Equation (10), the impact of the piston surface area on the additional mode stiffness is studied. As discussed the above, the additional mode stiffness depends on the piston surface areas, i.e. S fT and  f , as well as  and  . Based on Equation (10), the additional stiffness coefficient matrix K BMA  S fT2 can be described in terms of the dimensionless factors  f ,  and  . From comfort, no additional bounce mode stiffness is expected when involving in enhancing the roll mode stiffness, which requires all the elements in the first row of the coefficient matrix K BMA  S fT2 must be “0”. It provides the design criteria for the dimensionless factors. 5 

4.5

Piston rod Piston rod upward downward

0.8

4 3.5

1



3

6 0.

2.5 2

0.4 0.2 0

1.5 1

2

0.5



3 4

0.5

1 f

1.5

2

2.5 f

3

3.5

4

4.5

5

Fig. 9. Contour plot for the area ratio  satisfying no additional bounce mode stiffness for the vehicle body and wheel stations

Fig. 9 shows the contour values for  when areas are designed so that no additional bounce mode stiffness is generated. It is noted that the values for   1 and   1 (or  f  1 and  f  1 ) correspond to piston rod downward (

) and upward (

) installations, respectively. The values for 

indicate the force distributions among the front and rear actuator. From Fig. 9, it can be found that: 1.

No bounce stiffness requires the front and rear two (middle and rear) actuators are oppositely

26 / 39

installed. All the values for  f to generate smaller (   1 ) and bigger (   1 ) values for  are bigger and smaller than “1”, respectively, as shown in Fig. 9. 2.

The relationships between  f and  , which lead  to generate no bounce stiffness, are linear B (Fig. 9). The common factor for the elements in the first row of the matrix K MA is Sˆi , which can be

expressed by the dimensionless factors as  f  21      . That is, the slope  21    of this line, which

reflects the relationships between  f and  , are determined by the given  .

Furthermore, the slope is numerically equal to the summation of the rear two (middle and rear) piston rod surface areas with respect to the rear top surface area, i.e.  21     2 S B  ST  ST . 3.

The feasible region for  f and  can be roughly determined by   0 , i.e. 2   f  1 , which corresponds to the curve with level value “0” in Fig. 9. It indicates that the summation of rear two (middle and rear) top surface areas must be bigger than the front piston rod surface area.

The additional pitch and roll mode stiffness are determined by the second and third diagonal elements

K BMA 2,2 and K BMA 3,3 , respectively. They can be expressed in terms of the dimensionless factors as





S 2fT  f  1a f    bm  br 

2







and S 2fT 1   f l f     lm  lr  , respectively. If  is designed so 2

as not to generate the additional bounce mode stiffness, the additional pitch and roll mode stiffness are





S 2fT  f  1 a f  0.5bm  br  2

2

and





2 S 2fT 2  lm  lr   1   f l f  0.5lm  0.5lr  , respectively, as

shown in Fig.s 10 and 11. Fig. 10 shows that the additional pitch mode stiffness only depends on  f . The maximum amplification factor is around 16 when the piston rod of the front actuators is downward installed. It is clearly indicated that both the front piston-rod-upward and piston-rod-downward installations provide the same pitch mode stiffness if  f varies between 0 and 2. However, the former is able to generate greater pitch mode stiffness than the latter when  f is larger than 2. It is noted that no additional pitch mode stiffness will be produced when same top and bottom cylinders are employed for the front actuators, i.e.  f  1 , as shown in Fig. 10.

27 / 39 5 KB (2,2)S-2 MA fT

4.5 4 3.5

324

256

196

144

100

64

36

1 f

16

0

0.5

4

4

2.5

16



3

2 1.5 1 0.5 1.5

2

2.5 f

3

3.5

4

4.5

5

Fig. 10. Contour plot for additional pitch mode stiffness of the vehicle body under different area ratio  , which does not generate additional bounce mode stiffness for the vehicle body and wheel stations 5 4.5

KB (3,3)S-2 MA fT

144

4 100

3.5



3

64

2.5 36

2 1.5

16

1 0.5

4 0.5

1 f

1.5

2

2.5 f

3

3.5

4

4.5

5

Fig. 11. Contour plot for additional roll mode stiffness of the vehicle body under different area ratio  , which does not generate additional bounce mode stiffness for the vehicle body and wheel stations

Fig. 11 displays the additional roll mode stiffness corresponding to the specified  . Unlike the additional pitch mode stiffness, any  f is able to generate additional roll mode stiffness, which is a



quadratic function of 1   f



when twice the front strut-point base is not equal to the summation of the

two rear (middle and rear) strut-point bases, i.e. 2l f  lm  lr . Fig. 11 also indicates that the additional roll mode stiffness also depends on  . Larger  is able to generate higher additional roll mode stiffness. Therefore, maximum additional roll mode stiffness requires for a combination of both largest  f and  , as shown in Fig. 11.

28 / 39 350 300

=0.4 =0.6 =0.8

=0.4

fT

-2 KB (2,2)S MA

250 200 150 100 50

0.8 0 0

20

40

60

80 100 -2 KB (3,3)S MA fT

120

140

160

180

Fig. 12. Additional pitch and roll mode stiffness of the vehicle body under different specified area ratio  , which does not generate additional bounce mode stiffness for the vehicle body and wheel stations

The additional pitch and roll mode stiffness corresponding to specified values for the dimensionless factors, in which  f and  are the value sets to reach the specified level values (0.4, 0.6 and 0.8) of  in Fig. 9, are further studied in terms of  , as shown in Fig. 12. It is noted that all the specified values will not produce additional bounce mode stiffness for the vehicle body and wheel stations. Fig. 12 indicates that a combination of specified  f and  subjecting to larger level values of  will produce higher roll mode stiffness than smaller level values of  .

5.4 Loss coefficient design for damper valves The suspension damping has been reported to be able to reduce the tire dynamic load coefficient with ignorable effect on the ride vibration in the ride frequency range [37]. In this study, the additional damping provided by the HIS system is determined by the loss coefficients of the damper valves. The stiffness related parameters are fixed. The pre-charged and working pressure PP and PW are 2x106Pa and 3.5x106Pa, respectively. The forward speed of the truck is 72Km/h. For simplicity, all top and bottom damper valves have the loss coefficients of RT and RB , respectively. The loss coefficient for the roll damper valves is R A . The loss coefficients of direction (or roll) damper valves are fixed when the discussion is focused on the effect from the roll (or direction) damper valves.

29 / 39 1.666 RB=5.7x109

2

aW [m/s ]

1.664

RB=5.8x109

1.662

RB=5.9x109

1.66

RB=6.0x109 RB=6.1x109

1.658

RT = 0.9x109

1.656

RT = 1.0x109

1.654

RB=6.2x109 RB=6.3x109

RT = 1.1x109

1.652 7.395

7.4

7.405

7.41 7.415 RAD [rad]

7.42

7.425

7.43 -6 x 10

Fig. 13. Effect of top ( RT ) and bottom ( RB ) loss coefficients on roll angular displacement  RAD and weight RMS value aW 1.664 RAD 1.662

P1 P2 P3

2

aW [m/s ]

Level values x10-6: P1=7.3997, P2=7.4012, 1.66 P =7.4028, P =7.4043, 3 4

1.658 P6 P7

1.656 1.654

P5 P4

P8 P9 P10

Small 1.652 1.0995 1.1

Large 1.1005

1.101 1.1015 PAD [rad]

Level values x10-6: P5=7.4059, P6=7.4074, P7=7.409, P8=7.4105, P9=7.4121, P10=7.4137 1.102

1.1025

1.103 -5 x 10

Fig.14．Contour plot for roll angular displacement  RAD in terms of pitch angular displacement  PAD and weight RMS value aW

A typical parameter sets for RT and RB are displayed in Fig. 13 when R A is ‘zero’ value. It is indicated that RB can be design to reach minima of  RAD . The corresponding value of RB will be reduced from 6x109kgs-1m-4 to 5.9x109 and 5.8x109kgs-1m-4 when RT increases from 0.9x109kgs-1m-4 to 1.0x109 and 1.1x109kgs-1m-4, respectively. The minima of  RAD is decreased from 7.4136x10-6rad to 7.3981x10-6rad when RT changes from 1.1x109kgs-1m-4 to 0.9x109kgs-1m-4. The minima of  RAD is further obtained using contour plot in terms of pitch angular displacement  PAD and weighted RMS values aW , as shown in Fig. 14. It can be found that the  PAD and aW corresponding to minima (7.3997x10-6rad) of  RAD are 1.10168x10-5rad and 1.66054m/s2, respectively. The values on the ridge line of  RAD show that the parameter sets to generate smaller  RAD will cause higer  PAD and aW under the given range of RT and RB . The dynamic load coefficient S DLC and RMS value of suspension working space  SWS are also investigated in terms of front left tire and suspension, respectively, as shown in Fig. 15. Generally, the effect of RT and RB on S DLC and  SWS of the corresponding suspension is the same. It is found that

30 / 39 FL the value of S DLC are almost unchanged when RT is altered from small to large. For example, the value

corresponding to minima of  RAD from 0.0249 to 0.02496 and 0.025 when RT changes from FL 0.9x109kgs-1m-4 to 1.0x109 and 1.1x109kgs-1m-4, respectively. However, S DLC more heavily depends on

RB than RT , as shown in Fig. 15. Furthermore, the change of RB will cause greater variance of  RAD when RT is higher. -3

0.026

x 10 2.95

SFL DLC

RB=5.7x10

9

RB=5.8x109

0.0255

2.9

RB=6.0x109

FL

SDLC

RT=0.9x10

9

RT=1.0x109

RT=1.1x10

9

RB=6.2x109

0.0245

RB=6.3x109

2.8

2.75

0.024

0.0235 7.395

2.85

RB=6.1x109

FL

0.025

SWS [m]

RB=5.9x109

FL SWS 7.4

7.405

7.41 7.415 RAD [rad]

7.42

7.425

2.7 7.43 -6 x 10

FL Fig.15．Effect of top ( RT ) and bottom ( RB ) loss coefficients on front left tire dynamic load coefficient ( S DLC ) and FL RMS value of suspension working space (  SWS ) 0.24 RB=5.7x109

RL

SDLC

0.216

RB=5.8x109

0.238

0.214

RB=5.9x10

9

0.236

0.212

RB=6.0x109

0.234

RB=6.1x109

0.232

RT=0.9x109

0.21

RT=1.0x109

RT=1.1x109

0.208

RB=6.2x109 RB=6.3x109

0.23 0.228

0.206 0.204 0.202 7.395

RR

SRR DLC

SDLC [m]

0.218

0.226 SRL DLC 7.4

7.405

7.41 7.415 RAD [rad]

7.42

7.425

0.224 7.43 -6 x 10

Fig. 16. Effect of top ( RT ) and bottom ( RB ) loss coefficients on rear left and right tire dynamic load coefficients RL RR S DLC and S DLC

RL RR Fig. 16 shows the variance of dynamic load coefficients S DLC and S DLC for the rear left and right

wheel stations with respect to the RMS value of pitch angular displacement  RAD when the loss coefficients RT and RB change from 0.9x109kgs-1m-4 and 5.7x109kgs-1m-4 to 1.1x109kgs-1m-4 and RL RR 6.3x109kgs-1m-4, respectively. It is noted that there are the same trends for S DLC and S DLC with respect RL to  RAD when RT and RB vary. Higher values for RB are able to generate smaller values of S DLC RR and S DLC . Smaller RT will result in both smaller  RAD and larger tire dynamic load coefficient.

Fig. 17 displays the design of loss coefficients for the direction damper valves by using contour plot of roll angular displacement  RAD in terms of RT and RB . It can be found that minimum  RAD can be

31 / 39

reached by a sets of small RT and medium RB . The minimum  RAD is around 7.4x10-6rad under the given range of RT and RB . The values of  RAD are more heavily dependent on RT than RB when

 RAD is smaller than 7.414x10-6rad, as shown in Fig. 17. It is obvious that the contour plot can be used to find the minima of  RAD . 630 RADx10-6

610

7.414

7.412

7.411

7.409

7.407

7.406

7.404

7.4

7.403

600

7

-1 -4

7.41 7.42 8 7.41 7.417 5

7.401

RBx10 [kgs m ]

620

590

580

570 90

92

94

96

98 100 102 RTx107 [kgs-1m-4]

104

106

108

110

Fig. 17. Loss coefficient design for the direction damper valves

The above study has investigated the effect of RT and RB on body responses ( aW ,  PAD and  RAD ) and suspension working space (  SWS ), as well as the tire dynamic load coefficient (  DLC ). It is indicated that the design of RT and RB involves in compromising between the ride comfort ( aW ) and handling performance (  RAD ,  DLC ). For the fixed loss coefficient R A of the roll damper valves, the tuned parameters of RT and RB can be obtained using the method as shown in Fig. 17. Based on the founded parameters, other performance indexes can be calculated. Before tuning the parameters of RT ,

RB and R A in terms of the performance indexes, the impact of R A on the performance indexes is studied in the following content. The values for RT and RB are 1x109kgs-1m-4 and 6x109kgs-1m-4, respectively. R A is changed from “0” to 10x109kgs-1m-4. The body responses and tire dynamic load coefficients are shown in Fig.s 18 and 19, respectively. Intuitively, the increase of R A will cause higher values of aW ,  PAD and  RAD . The corresponding increments are 38.6%, 0.8% and 129.7%, respectively. It is indicated that R A is able to greatly alter the vibration characteristics of the truck body. Fig. 19 shows the tire dynamic load coefficients of the front wheels when R A is altered. It is noted that R A (1.6x109kgs-1m-4) can be design to reach FL FR minimum front S DLC , as shown in Fig. 19. The corresponding decrements for S DLC and S DLC are 3.5%

RL RR and 4.6%, respectively. However, the rear dynamic load coefficient S DLC and S DLC go straight down by

2.3% and 2.1%, respectively, when R A is altered from ‘0’ to 10x109kgs-1m-4, which is omitted due to the length of this paper.

32 / 39 -5

x 10 1.12

2.5 PAD

PAD [rad]

2

aW [m/s ]

aW

2

1.11 RA

1.5 0.6

0.8

1

1.2 RAD [rad]

1.4

1.6

1.1 1.8 -5 x 10

Fig. 18. Body responses when R A varies from ‘0’ to 10x109kgs-1m-4 0.025

0.025

0.0245

0.024

FL

FR

SDLC

SDLC

RA

0.024

0.023

SFL DLC 0.0235 0.6

SFR DLC 0.8

1

1.2 RAD [rad]

1.4

1.6

0.022 1.8 -5 x 10

Fig. 19. Tire dynamic load coefficients for the front wheels when R A varies from ‘0’ to 10x109kgs-1m-4

The above study indicates that the damping tuning of the proposed HIS system involves in handling the compromise between the ride comfort and handling performance. Generally, the optimized parameters can be found if the upper limit of some index is given, which determines the feasible region for the damping design. For example, if the upper limit for  RAD is  RAD , then the feasible region of the loss





coefficients for the damping tuning is   R RAD (R)  RAD , where R is the design vector and composed of RT , RB and RA . As stated the above, all the performances corresponding to R i   can be obtained using Equations (21) and (22). In this study, the radar-type chart is employed to describe the normalized conflictual indexes and Fig. out the optimal parameters. More details, each performances, e.g.

 RAD , are normalized and expressed with the dimensionless factor (i) , which are defined by the ratio between

the

range

and

distance

max(~RAD )  ~RAD  max(~RAD )  min(~RAD ) ,

with

respect

to

the

maximum,

i.e.

( RAD ) =

~ in which  RAD is the column vector composed of  RAD (R i ) .

~ , are the maxima or minima. When (i ) reachs 0 or 1, it indicates that the evaluation indexes, i.e.  RAD

33 / 39

(FR ) SWS (RL ) SWS

(FL ) SWS1 0.8 0.6 0.4 0.2

(RAD)

(FL ) SWS1 0.8 0.6 0.4 0.2

(FR ) SWS (PAD)

(RL ) SWS

(RAD) (PAD)

(aW )

(aW )

(RR ) SWS

(RR ) SWS

(SRR ) d

(SRR ) d (SFL ) d (SFR ) d

(SFL ) d

(SRL ) d

P = 1.5x106, R = 0.2x109, R = 1x109, R = 6x109

R = 0x109, R = 1x109, R = 6x109 A

T

(SRL ) d

(SFR ) d W

B

A

T

B

6

9

9

9

6

9

9

9

9

9

9

PW = 1.0x10 , RA = 0.4x10 , RT = 1x10 , RB = 6x10

9

9

9

PW = 0.5x10 , RA = 1.0x10 , RT = 1x10 , RB = 5x10

RA = 0.2x10 , RT = 1x10 , RB = 6x10 RA = 0.4x10 , RT = 1x10 , RB = 6x10

Fig. 20. Dimensionless factor (i) when PW =2.0Mpa

Fig. 21. Optimal damping under different PW

The effect of R A on the performance indexes is displayed in terms of the dimensionless factor  when RT and RB are fixed simultaneously, as shown in Fig. 20. It is indicated that the proposed HIS system without roll damper valves is able to achieve better compromise between the ride comfort and handling performance. Higher loss coefficient for the roll damper valves will generate greater RMS values of roll angular displacement, as shown the curve with the diamond symbol in Fig. 20. When the pre-charged pressure PP is changed (from 2Mpa to 1.5, 1.0, 0.5Mpa), the above method can be employed to find the optimal loss coefficient for the damper valves, as shown in Fig. 21. The optimal damping also can be obtained when the forward speed of the truck is changed, which is omitted due to the length of this study.

6. Conclusions and future work This paper has proposed a new roll-resistant HIS system to improve the roll dynamics of the truck body. The dynamic equations for the HIS system have been derived with impedance transfer matrix method. Based on the derived equations, the additional mode stiffness/damping rates provided by the HIS system are quantitatively described in terms of the physical parameters of the hydraulic system. It can be found that (1) the difference and summation of the top and the corresponding bottom piston surface area determine the additional bounce/pitch and roll/warp mode stiffness, respectively. (2) The capacitance of the gas-filled accumulators and the pressure loss coefficients of the roll damper valves are equally enlarged to form the mode stiffness and damping, respectively. (3) The additional mode damping yielded

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by the DDVs depends on the square of each piston surface area. (4) When symmetry circuits are incorporated in the HIS system, the opposite installation between the front and rear actuators will not generate additional bounce stiffness. The obtained results confirm the truth that the proposed HIS system can be designed to independently improve the roll stiffness. Based on modal analysis, the proposed HIS system is able to (1) effectively reduce the RMS value of body roll response, (2) greatly improve the vibration characteristics, (3) significantly enhance the roll mode force, and (4) favorably reduce the warp mode force for the wheel stations. The damping parametric analysis indicates that the damping tuning involves in dealing with the compromise between ride comfort and handling performance. Hence, the radar-type chart is proposed to tune the loss coefficients for the damper valves. For future work, one of our ongoing researches is to apply this proposed HIS system to further investigate the benefits from separately controlled roll mode with respect to heave and pitch modes in terms of transient responses and handling performances. These main control requirements are now achieved by using hydraulic pump and adjustable damping valves. The comparison between different control schemes is being considered to suppress vehicle body motion and wheel hops, simultaneously, according to transient body and wheel station modes estimated with the proposed method in this study. Another task for our future work is to validate the methodology experimentally, including both fluid components and complete system-level field testing.

Acknowledgements This research was partly supported by the China National Science Funds for Distinguished Young Scholars (10725208) and International Science & Technology Cooperation Program of China (2015DFA13060), and also by the National Natural Science Foundation (51175157).

Nomenclature CG

center of gravity

DDV

direction damper valve

DOF

degree-of-freedom

DLC

dynamic load coefficient

HIS

hydraulically interconnected suspension

PAD,

pitch-angular displacement of the vehicle body

PSD

power spectrum density

RAD

roll-angular displacement of the vehicle body

RDV

roll damper valve

35 / 39 SMA

sprung mass acceleration

SWS

suspension working space

TDF

tire dynamic force

TMF

tire mode force

TUCS

truck with uncoupled conventional suspension

THIS

truck with hydraulically interconnected suspension



pitch angle displacement of vehicle body around CG of itself

i 

pitch angle displacement of rear-left and -right walking beams ( i  l , r )



area ratio between rear and front top cylinders

i

area ratios between piston rod and head sides surface areas of front and rear cylinders ( i  f , r )



ratio of specific heats for the gas

i

ratios of volume- and capacity-change for accumulator ( i  V , C A )



root mean square value of output evaluations



dimensionless coefficients defined by ratio between range and distance relative to maximum

a, b

longitudinal distance from CG of vehicle body to the front wheel stations and rear pivot centers, respectively

af

longitudinal distance from CG of vehicle body to front actuators

aW

total weight root mean squ

bwi

longitudinal distances from rear pivot centers to rear-front and rear-rear axles, respectively

ki

weight factors for bounce, pitch, roll vibration proposed in ISO 2631 ( i  b, p, r )

k sij

suspension linear stiffness coefficients for UCS systems ( i  f,m,r ; j  l , r )

k tij

tire vertical stiffness for UCS systems ( i  f,m,r ; j  l , r )

lij

transverse distance of vehicle body from CG of vehicle body to wheel stations ( i  f,w ; j  l , r )

ms

mass of vehicle body

mui

mass of unsprung mass ( i  f,m,r )

oi

fictitious centers of vehicle body’s CG and walking beams’ two pivot centers projected onto fictitious wheel-center

roll angle displacement of vehicle body around CG of itself

level frame ( i  CG, PL, PR )

p

pressure

q

volume flow rate

sti

connection sections of different fluid components at fluid circuits I and II

wij

road disturbance inputs at wheel-road contact points ( i  f,m,r ; j  l , r )

z

bounce displacement of vehicle body

zi

mode displacement of unsprung masses ( i  B, P, R, A, Pl , Pr )

z ij

vertical displacement of unsprung masses ( i  f , m, r ; j  l , r )

CA

capacity of the accumulator

CW A

capacity of the accumulator under working status

Fi

mode forces applied to wheel stations ( i  B, P, R, A, Pl , Pr )

Fij

corner forces applied to wheel stations ( i  f , m, r ; j  l , r )

Ii

inertia for vehicle body ( i  P, R )

I iR

roll inertia for unsprung mass ( i  f,m,r )

Pi

pressure of accumulator under pre-charged and working status ( i  P,W )

Ri

loss coefficients for direction damper valves located at output ports of actuators ( i  T , B )

S ij ~ Sˆi , S i

effective cross-section area for top and bottom cylinders, respectively ( i  f , m, r ; j  U , B ) area difference and summation of each actuator between its bottom cylinders and top cylinders ( i  f , m, r )

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T Acc

impedance of accumulators

Vi

volume of accumulator under pre-charge and working status ( i  P,W )

Wki , Wei

weighted factors for the i th one-third octave

ZVAcc

pressure loss coefficients for roll damper valves



sets of feasible pressure loss coefficient

pi

pressure for fluid circuits ( i  I, II )

qi

flow for fluid circuits ( i  I, II )

u v w

disturbance input vector

x

displacement vector of mechanical system

x

state vector for mechanical and hydraulical coupled system

z iW

mode displacement vector of wheel stations ( i  M , S )

Λi , Ξ j

area dependent square matrices of master mode stiffness for wheel stations of vehicle with HIS system

state vector composed of pressure and flow at different section displacement vector of road-disturbance input

( i  BP, RA ; j  T , B )

A

characteristic matrix for mechanical and hydraulical coupled system

B

input coefficient matrix for mechanical and hydraulical coupled system

C

output coefficient matrix

C iM

coefficient matrices of master mode damping for truck body and wheel stations of vehicle with HIS system ( i  B,W )

C BMAj

area dependent coefficient matric of mode damping for truck body of vehicle with HIS system ( j  T,B )

CU

damping coefficient matrix for the vehicles with UCS systems

D

feed-forward matrix

DQ

coefficient matrix of associated with deformation velocity of suspension

DR

coefficient matrix of associated with cylinder chamber pressure

FH

hydraulic strut force vector

FHB

mode force vector applied to truck body

FM

mode force vector of wheel stations

FW

coefficient matrix corresponding to road-disturbance input

H

assembled transfer functions for all evaluation outputs of the SMA, PAD, RAD, SWS and TDF

H iX K iM

transfer functions for evaluation outputs ( i  SMA,PAD,RAD,SWS,TMF ) coefficient matric of master mode stiffness for truck body and wheel stations of vehicle with HIS system ( i  B,W )

K BMA

area dependent coefficient matric of mode stiffness for truck body of vehicle with HIS system

KT

stiffness diagonal matrix composed of tire vertical stiffness

KU

stiffness coefficient matrices for vehicles with UCS systems

MU

mass coefficient matrix for vehicles

F

transformation matrix operator for mode forces from involved physical states

TCM

DV

TCM

transformation matrix operator for mode displacement/velocity from corners connectivity matrix for fluid circuit j ( k  1,...,9 ; j  I, II )

Tkj j k Ti

the k

P

pressure vector of hydraulic system

Q

flow vector of hydraulic system

R

design vector composed of pressure loss coefficients

th

transfer matrix for fluid component k from up-stream i to down-stream j

37 / 39

S

road input spectrum matrix

U

Laplace transfer of disturbance input vector

V

volume matric composed of cylinder chamber

X

Laplace transfer of state vector x

Y

Laplace transfer of evaluation outputs

Zi

impedance matrices for fluid circuits ( i  I, II )

Z

impedance matrices for hydraulic system

u

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Highlights 

A new hydraulically interconnected suspension system for tri-axle truck is proposed.



Transfer impedance matrix method is used to model multi-body dynamic systems.



Additional mode stiffness/damping for vehicle body and wheel station has been derived.



A parametric tuning method is applied to design key parameters of hydraulic systems.



Valve loss coefficients are used to monitor conflicts between ride and handling.