# Simulation of the Vibration Characteristics for Agricultural Wheeled Tractor with Implement and Front Axle Hydropneumatic Suspension

Authors: Enlai Zheng; Song Cui; Yuanzhao Yang; Jinlin Xue; Yue Zhu; Xiangze Lin
Journal: Shock and Vibration
 (2019)

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## Abstract

The dynamic model is significant for the analysis of the vibrational characteristics of the wheeled tractor system with implement and front axle hydropneumatic suspension. In this work, the nonlinear stiffness and damping equations are derived first. The dynamic coupling relationship among cabin, three-point hitch structure, and implement are figured out, and the dynamic model of the half agricultural wheeled tractor/implement system is presented considering the effects of three-point hitch structure, passive silent blocks of cabin, and front axle hydropneumatic suspension together. To validate the model, the power spectral densities (PSDs) of the driver seat, cabin, chassis, and implement acquired from numerical simulations are compared with those from experiments, respectively. Under different forward speeds, two groups of results match well. Based on the model, the influences of passive cabin suspension, implement, and front axle hydropneumatic suspension on the dynamical characteristics of the tractor system are investigated. Results indicate that the front axle hydropneumatic suspension will deteriorate the ride comfort of the driver but improve the handing stability. The passive cabin suspension reduces the operational stability while improving much more ride comfort than the front axle hydropneumatic suspension does. The driver’s comfortableness will be increased due to implement; meanwhile, handling stability will be compromised. Besides, the impacts of initial nitrogen volume, pressure of accumulators, and orifice diameter of throttle valves on the vibration characteristics of the tractor system are also inspected.


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## Body

## 1. Introduction

The low-frequency vibration experienced by an agricultural tractor comes from the interaction between the tractor and the rough terrain [1–4]. It is severe when primary tractors only have the tires as the elastic component between the road and the tractor [5, 6], for the reason that tires are unable to provide proper suspension characteristics required to absorb these vibrations. According to the International Standard (ISO) 2631-1:1997 [7], daily exposure to whole body vibration (WBV) among tractor drivers may lead to adverse effects on health such as musculoskeletal disorder and low back pain [8]. The primary contact point for transmission of vertical WBV to drivers is through the tractor seat (amplifies or attenuates vibration at the base). It should be noted that the human-sensitive frequency range for vertical vibration is from 4 to 8 Hz. Seat effective amplitude transmissibility (SEAT) value is also used to represent the efficiency of seat isolation.Moreover, the tractor vibration leads to the tires’ dynamic load on the ground, which will not only damage the road and intensify farmland soil compaction [9–13] but also affect the handling stability of the tractor. With the development of technology, the front axle of a modern tractor has been equipped with the hydropneumatic suspension. Therefore, it is essential to develop a dynamic model of the tractor/implement system with front axle suspension to predict its vibration characteristics and evaluate its performances, which provides a theoretical basis for structural design and parameter optimization of both the suspension and tractor system.The main feature of this work is that it considers the hydropneumatic suspension on the front axle, the three-point hitch structure between implement and tractor frame, and excitations from ground surface roughness, which leads to strong nonlinear dynamics. There are six degrees of freedom (DOF) related to the tractor body motions including three-dimensional and three rotational degrees of freedom. Thus, it is challenging to establish an integrated dynamic model of the wheeled tractor/implement system with the hydropneumatic suspension on the front axle.In order to investigate the vibration characteristics of agricultural wheeled tractors, there have been two main approaches. One is the experimental approach, and the other is the computer simulation method. In the aspect of the experimental approach, Clijmans et al. [14] used the experimental modal analysis technique to predict the structural behaviour of the machine under a set of excitation conditions. When studying the effects of vibration on drivers, Cutini et al. [15] operated three agricultural tractors on six test tracks at different forward speeds to evaluate the whole body vibration of the agricultural tractor’s operator. Adam and Jalil [16] performed measurements on a healthy male tractor driver to determine the vertical suspension seat transmissibility and SEAT values. In addition, some researchers measured the vertical vibration accelerations of the front axles, rear axles, and cabin for the two-wheel-drive tractor under different road conditions and forward speeds [17–19]. Also, Nguyen and Inaba [20] measured the vertical wheel load of the left and right rear wheels and the roll, bounce, and pitch accelerations of the rear axle center on an asphalt road and a sandy loam field. Although the experimental method can provide more precise and reliable results, high expenses, insecurity, and inconvenience limit its development. Moreover, Yang et al. [21] pointed out that the results from an experiment are based on the specified test tractor and field conditions, which did not suit other test conditions.Compared with the experimental approach, the computer simulation method employs the mathematical models or multibody dynamic models constructed by commercial software to extrapolate the experimental results over the range of test conditions [22, 23]. The quarter-vehicle model, half-vehicle model, and full-vehicle body models have been developed to predict the dynamic behaviour of the tractor and to carry out the vibration control. Especially the one-dimensional quarter-vehicle models with single or two DOF are commonly used to study the heave motion of the tractor body [24]. Cuong et al. [4] modeled tire-soil system as an equivalent system by the mechanism of a linear parallel spring and damper in vertical the direction with one end connected to the tire axle and the other end connected to the hard layer. To calculate the structural parameters of the hydropneumatic spring, Yuan et al. [25] established a vibration model of the front axle suspension tractor and the stiffness and damping nonlinear mathematical model of the hydropneumatic spring.The two-dimensional half-vehicle model represents the tractor’s longitudinal dynamics using the heave and pitch motion of the tractor body and the vertical motion of the front and rear wheels [26–28]. In order to study the influence of implement’s mass on the vibration characteristics of the tractor/implement system, Zhu et al. [29] established a two-degree-of-freedom vibration model of the tractor with a rear implement. On this basis, Zheng et al. [30] developed a complete nonlinear dynamic model of the wheeled tractor with a suspended driver seat including air spring and MR damper, and the effects of nonlinear stiffness for scissors linkage seat, the air spring with auxiliary chamber as well as MR damper, and dynamic characteristics of real tire are also considered in this model. The presented half-vehicle model can also be used to describe the lateral dynamics, with the roll and heave motion of the tractor body and the vertical motion of the left and right wheels.The three-dimensional full-tractor model has various number of DOFs, which starts from six DOFs up to hundreds. The concrete number depends on the number of components being modeled, the model versatility, and the simplifying assumptions being used [31]. For example, in a multibody model of a tractor, Melzi et al. [32] used not only the spring-damper elements to represent the suspension system but also the rigid body to represent vehicle chassis, cabin, and the seat with different DOFs (yaw, pitch, and roll). Considering that the tractor consists of a rotatable front end (anterior part) and the main body (posterior part), Li et al. [33] analysed the pitch motion of the front part of the tractor as well as the roll motion of the rear part of the tractor and predicted tractor behaviour when a tire is off the ground. In vehicle driving simulations, Sim et al. [34] established three-dimensional vehicle dynamic models containing the seat, cabin, body, and tires with a 14-DOF model which describes the bounce, pitch, roll, longitudinal response, and lateral response of the tractor’s seat, cabin, and tire.In addition to the cabin and driver seat suspension [35–37], to improve the driver’s comfort and the steering response, front axle suspension is always the focus of this research. For the tractor without implement, Martelli et al. [38] pointed out the front axle elastic suspension had limited effect on the vertical vibration and driving safety depending on the driving conditions, while its effect on the pitch and lateral vibration is obvious [28]. Furthermore, Mazhei et al. [39] found that when the implement is mounted in front of a tractor, the effect of the front axle elastic suspension relies on the damping element, but it was less effective than that without front implement. Apart from the linear elastic front axle suspension, the researchers also designed the hydropneumatic nonlinear suspension as the front axle suspension of the tractor.When studying the output force of hydropneumatic suspension, Theron and Els [40] considered mathematical modelling of a suspension unit that comprises a hydraulic cylinder connecting the vehicle body to the unsprung mass, two nitrogen-filled accumulator springs, and two damper ports. van der Westhuizen and Schalk Els [41] developed an accurate suspension model to obtain realistic vehicle dynamics simulation results when considering different suspension characteristics or control algorithms. Considering polytropic change in the gas state and seal friction, the gas-oil emulsion flows through orifices and valves. Yin et al. [42] formulated an analytical model of the hydropneumatic suspension, which considers one and two bleed orifices configurations of the strut. Eventually, the validity of these mathematical models is proved by the experimental rig in the lab. Unlike the conventional linear elastic front axle suspension, Yilidaer et al. [43] found that the hydropneumatic nonlinear suspension plays an important role in reducing the vertical vibration acceleration and pitching vibration angular acceleration of the tractor.As far as we know, few attempts have been made to study the effect of the front axle hydropneumatic suspension along with the passive cabin suspension on the vibration characteristics of the agricultural tractor/implement system including the structure of three-point hitch. In this work, the dynamic model of the wheeled tractor/implement system with hydropneumatic suspension on the front axle is established and the vibration characteristics in both time and frequency domains are analysed. Furthermore, the effect of forward speed, nitrogen volume, and pressure of accumulators and orifice area of proportional and throttle valves on the vibration characteristics of the tractor system are also investigated.


## 2. Tractor/Implement Description

As shown in Figure1, the rodless chamber 1 in the front axle hydropneumatic suspension system is connected with the accumulator A using the proportional valve 5 and the small throttle valve 6. While the rod chamber in the cylinder 1 is connected in series with the accumulator B via the proportional valve 7 and the small throttle valve 8. The large throttle valve 10 is installed between the accumulators A and C to adjust the oil pressure between the rod and rodless chambers of cylinder 1, and the membrane-coated cylinders of the accumulators A, B, and C can isolate the nitrogen through valve block 3 and ball valve 4.Figure 1
Physical structure of the front axle hydropneumatic suspension system.The piston rod 2 is hinged to the front axle, and the cylinder 1 of the hydropneumatic suspension is hinged to the chassis. Running on rough ground, the tractor’s piston rod reciprocates in the cylinder due to the relative motion between the front axle and the chassis. Once the front axle comes close to the chassis and the throttle valve 9 is closed, the hydropneumatic suspension is compressed and the volume of the rodless chamber will be reduced. As a consequence of the oil pressure increase in the rodless chamber 1, the oil flows to the accumulator A through the proportional valve 5 and the small throttle valve 6. Meanwhile, the oil pressure of the rod chamber is lessened, making the oil in the accumulator B flow to the pole room. The proportional valve 7 and the small throttle valve 8 are used to adjust the oil pressure. The flow direction of oil will be opposite to the above direction if the front axle is separated from the chassis. When the throttle valve 9 is opened, part of oil in the rodless and rod chambers will form a closed oil circuit.As can be seen in Figure2(a), the wheeled tractor with implements consists of the chassis, front and rear axles, tires, cabin, driver seat, and implement. The implement is connected to the tractor body via the three-point hitch structure shown in Figure 2(b). The three-point suspension is composed of a lifting shaft, a lifting arm, an upper connecting rod, a lifting link, a lower connecting rod, and a plow mounting bracket. Once the lift shaft starts to rotate, the lift link and the upper link will get the plow mounting bracket off the ground. Between the cabin and the chassis, passive silent blocks are set up to improve the ride comfort, as shown in Figure 2(c).Figure 2
Physical structure of the tractor system: (a) tractor/implement system; (b) three-point hitch structure; (c) silent blocks of cabin.
(a)
(b)
(c)


## 3. Dynamic Model of Tractor/Implement System with Front Axle Hydropneumatic Suspension

The upper and lower ends of the cylinder for the hydropneumatic suspension are hinged to the front axle and the chassis, respectively. The tractor body structure is represented by mass and moment of inertia relative to a central axis perpendicular to the symmetry plane. A dynamic model of the wheeled tractor with the implement and front axle hydropneumatic suspension system was developed, as shown in Figure3. The front and rear tires of the tractor system are subjected to the displacement excitations qfzt and qrzt, respectively, since there is a difference between the amplitudes qfzt and qrzt and the angular displacement with respect to the mass center arises.Figure 3
Dynamic model of the wheeled tractor system with the implement and front axle hydropneumatic suspension.The ground excitation relation between the front and rear axles of the tractor/implement system at the instanti by considering the time lag τ between qfzit and qrzit can be expressed by(1)qfzit=qrzit+τ,where qfzit and qrzit are the displacement excitations at front and rear axles, respectively, τ is the time lag and can be calculated as lbf+lbr/v, lbf is the distance of the mass center between the chassis and the front axle, lbr is the distance of the mass center between the chassis and the rear axle, and v is the velocity of the tractor/implement system.The governing equations of motion for the front axle suspension and cabin as well as driver seat can be given by(2a)mez¨e=Fp−kfze−qf−cfz˙e−q˙f,(2b)mcz¨c=kcrzb+lbrφb−zc−lcrφc+ccrz˙b+lbrφ˙b−z˙c−lcrφ˙c+kcfzb−zc+lcfφc−lbfφb+ccfz˙b−z˙c+lcfφ˙c−lbfφ˙b−kdzc+ldφc−zd−cdz˙c+ldφ˙c−z˙d,(2c)Icφ¨c=kcrzb+lbrφb−zc−lcrφc+ccrz˙b+lbrφ˙b−z˙c−lcrφ˙clcr−kcfzb−zc+lcfφc−lbfφb+ccfz˙b−z˙c+lcfφ˙c−lbfφ˙blcf−kdzc+ldφc−zd+cdz˙c+ldφ˙c−z˙dld,(2d)mdz¨d=kdzc+ldφc−zd+cdz˙c+ldφ˙c−z˙d.


### 3.1. Model of Front Axle Hydropneumatic Suspension

The large throttle valve 10 is usually closed and can be used to control the volume of the accumulator A. The influences of the accumulator C can be neglected. The model of the front axle hydrosuspension is developed, as shown in Figure4. Compared with the nitrogen, the oil in the accumulator can be perceived as an incompressible flow. Therefore, the force produced by the piston rod can be given by(3)Fh=P1A1−P2A2,where P1 and P2 are the transient pressures of the rod cavity and of the rodless cavity, respectively, A1 is the area of the piston with A1=πD2/4, D denotes the diameter of the piston, A2 represents the effective area of the rod cavity with A2=πD2−d2/4, and d stands for the diameter of the piston rod.Figure 4
Model of front axle hydropneumatic suspension.The orifice size of the front axle hydropneumatic suspension is very small, so is that of the proportional valve. Based on this, the oil flow can be calculated as(4)q=CdA2ΔPρ,where ρ is the density of oil, A is the flow area of the hole that mounts the throttle and proportional valves, Cd denotes the flow coefficient of the hole, and ΔP represents the pressure difference between the oil cavity and the accumulator.The compression on the hydropneumatic suspension will cause the oil in the rodless chamber 1 to flow into the accumulator A via the throttle valve 6 and the proportional valve 5; meanwhile, the oil in the accumulator B passes through the throttle valve 8 and the proportional valve 7 and enters the rod cavity 2. When the hydropneumatic suspension rebounds, the flow direction of the oil will be reversed. The numerical relationship between the flow rate and the speed of the piston rod can be described by(5a)A1x˙=sgnx˙CdA92ΔP21ρ+CdA5+A62ΔPA1ρ,(5b)A2x˙=sgnx˙CdA92ΔP21ρ+CdA7+A82ΔP2Bρ,where A6 and A8 denote the flow area of the throttle valves 6 and 8, respectively, A5 and A7 are the flow area of the proportional valves 5 and 7, A9 is the flow area of the throttle valve 9, ΔP1A is the absolute value of the oil pressure difference between the rodless cavity 1 and the accumulator A, ΔP2B is the absolute value of the oil pressure difference between the rod cavity 2 and the accumulator B, sgnx˙ is the symbolic function, and x˙ is the relative velocity between the chassis and front axle with x˙=z˙b−lfφ˙b−z˙e.The symbolic function is(6)sgnx˙=1,x˙>0,0,x˙=0,−1,x˙<0.The velocity with respect to the volume change of the accumulators A and B can be expressed as(7)sgnx˙CdA5+A62ΔPA1ρdt=V˙A,sgnx˙CdA7+A82ΔP2Bρdt=−V˙B.It is assumed that the gas in accumulators is ideal and its temperature is considered to be invariant. Thus, the state equation of this ideal gas will be(8)PAVA=PA0VA0,PBVB=PB0VB0,where PA represents the transient pressure of the accumulator A; VA is the transient volume of the accumulator A; likewise, PB and VB denote the transient pressure and the transient of the accumulator B; PA0 and VA0 are the initial pressure and the initial volume of the accumulator A, and PB0 and VB0 are the initial pressure and the initial volume of the accumulator B.According to the state equation of the ideal gas, the oil pressure of the accumulators A and B can be expressed as(9)PA=PA0VA0VA0−A1x,PB=PB0VB0VB0+A2x,where x is the relative displacement between the chassis and the front axle, x=zb−lfφb−ze.Based on equation (8), the derivatives of the volume change of the accumulators A and B with respect to time are(10)V˙A=−PA0VA0P˙APA2,V˙B=−PB0VB0P˙BPB2.By solving equation (5), the oil pressure can be given by(11)P1=PA−ρPA02VA02P˙A/PA2sgnx˙2Cd2A5+A62,P2=PB+ρPB02VB02P˙B/PB2sgnx˙2Cd2A7+A82.Friction force between the cylinder and the piston can be(12)Ffx˙=sgnx˙Fc+Fs−Fce−x˙/x˙s2+fvx˙,where Fc is the Coulomb friction force, Fs is static friction force, fv is the coefficient of the viscous friction force, and x˙s is the Stribeck velocity.The total force acting on the piston rod with the damping force of the orifice and the friction force between cylinder and piston considered can be given by(13)Fp=Fh−Ff.The throttle valve 9 will be either closed or fully opened. When the throttle valve 9 is in the closed state, the force of the hydropneumatic suspension can be calculated as(14)Fp=ρA13x˙2signx˙2Cd2Aj+Ac2+PA0VA0rA1VA0−A1xr−signx˙Fc+Fs−Fce−x˙/x˙s2+Fvx˙−PB0VB0rA2VB0+A2xr+ρA23x˙2signx˙2Cd2Az+Ad2.Otherwise, the force of the hydropneumatic suspension can be expressed as(15)Fp=PA0VA0+VB0A1−A2PA0VA0+VB0A1−A2/Mg−A1−A2x+ρA1−A23x˙24Cd2A5+A62+ρA1−A23x˙24Cd2A7+A82−Fc−Fs−Fce−x˙/x˙s2−Fvx˙signx˙.


### 3.2. Model of Three-point Hitch Structure

The model of the three-point hitch structure is shown in Figure5. Separating the upper link CD from the three-point hitch structure, its force balance relationship can be given by(16)Fdx−Fcx−m2a2x=0,Fcz−Fdz−m2g−m2a2z=0,Fdxlcd−lcmsinφ2+Fdzlcd−lcmcosφ2+Fcxlcmsinφ2+Fczlcmcosφ2=I2φ¨2.Figure 5
Model of three-point hitch structure.Similarly, the force balance relation of the implement can be expressed as(17)Fbx−Fdx=0,Fdz−mag−maz¨a−Fbz=0,Fbxl4+Fbzl3+Fdxl5−Fdzl6=Iaφ¨a.Separating the lower link AB from the three-point hitch structure, its force balance relation can be expressed as(18)Fax−Fbx−m1a1x+Fecosγ−φ1=0,Fbz−Fesinγ−φ1−Faz−m1g−m1a1z=0,−Fbzl1+l2−lamcosφ1−Fazlamcosφ1−Fesinγlam−l1−Fbxl1+l2−lamsinφ1−Faxlamsinφ1=I1φ¨1.Similarly, the force balance relation of the chassis can be given by(19a)mbx¨b=Fax−Fcx+Fecosγ−φ1+Ffx+Frx,(19b)mbz¨b=−Fp+krqr−zb−lrφb+crq˙r−z˙b−lrφ˙b−kcrzb−zc−lcrφc+lbrφb−ccrz˙b−z˙c−lcrφ˙c+lbrφ˙b−kcfzb−zc+lcfφc−lbfφb−ccfz˙b−z˙c+lcfφ˙c−lbfφ˙b+Fcz−Fesinγ−φ1−Faz,(19c)Ibφ¨b=Fplf+krqr−zb−lrφb+crq˙r−z˙b−lrφ˙blr−kcrzb−zc−lcrφc+lbrφb+ccrz˙b−z˙c−lcrφ˙c+lbrφ˙blbr+kcfzb−zc+lcfφc−lbfφb+ccfz˙b−z˙c+lcfφ˙c−lbfφ˙blbf+Fcxlbc+Fczlb−Fecosγ−φ1lag−lba−Fesinγ−φ1lb+Faxlba−Fazlb−Ffxzb−qf−Frxzb−qr.The force balance relation of the implement can be expressed as(20)Fbx=Fdx,Fdz−Fbz−mag=maz¨a,Fbxl4+Fbzl3+Fdxl5−Fdzl6=Iaφ¨a.The angular velocity and acceleration of link CD can be described as(21)φ¨2=adxsinφ2−adzcosφ2−acxsinφ2+aczcosφ2lcd,φ˙2=adzsinφ2−adxcosφ2−aczsinφ2+acxcosφ2lcd1/2,where adz=z¨a−l6φ¨a−l5φ˙a2, adx=−l5φ¨a+l6φ˙a2, acz=z¨b+lbφ¨b−lbcφ˙b2, and acx=−lbcφ¨b−lbφ˙b2.The acceleration at the mass center of link CD in the horizontal and vertical directions can be expressed as(22)a2x=acx+lcmφ¨2sinφ2+lcmφ˙22cosφ2,a2z=acz−lcmφ¨2cosφ2+lcmφ˙22sinφ2.Similarly, the angular velocity and acceleration of link AB can also be given by(23)φ¨1=−z¨acosφ1+l3cosφ1φ¨a+l4sinφ1φ¨a−l4cosφ1φ˙a2+l3sinφ1φ˙a2+z¨bcosφ1+lbcosφ1φ¨b−lbasinφ1φ¨b+lbacosφ1φ˙b2+Lbsinφ1φ˙b2lab,φ˙1=−z¨bsinφ1−lbsinφ1φ¨b−lbacosφ1φ¨b−lbasinφ1φ˙b2+lbcosφ2φ˙b2+z¨asinφ1−l3sinφ1φ¨a+l4cosφ1φ¨a+l4sinφ1φ¨a+l3cosφ1φ˙a2lab1/2,where abx=l4φ¨a+l3φ˙a2, abz=z¨a−l3φ¨a+l4φ˙a2, aax=lbaφ¨b−lbφ˙b2, and aaz=z¨b+lbφ¨b+lbaφ˙b2.The acceleration at the mass center of link AB in horizontal and vertical directions can be given by(24)a1x=aax+lamφ¨1sinφ1+lamφ˙12cosφ1,a1z=aaz−lamφ¨1cosφ1+lamφ˙12sinφ1.A set of state variables selected for analysis are listed below:(25a)u1=za,u2=φa,u3=xb,u4=zb,u5=φb,u6=zc,u7=φc,u8=zd,u9=ze,(25b)u10=z˙a,u11=φ˙a,u12=x˙b,u13=z˙b,u14=φ˙b,u15=z˙c,u16=φ˙c,u17=z˙d,u18=z˙e.By substituting equations (25a) and (25b) into equations (2a)–(2d) and equations (19a)–(19c), the second-order differential equations can be reduced to the first order, which makes it easy to be handled mathematically.


## 3.1. Model of Front Axle Hydropneumatic Suspension

The large throttle valve 10 is usually closed and can be used to control the volume of the accumulator A. The influences of the accumulator C can be neglected. The model of the front axle hydrosuspension is developed, as shown in Figure4. Compared with the nitrogen, the oil in the accumulator can be perceived as an incompressible flow. Therefore, the force produced by the piston rod can be given by(3)Fh=P1A1−P2A2,where P1 and P2 are the transient pressures of the rod cavity and of the rodless cavity, respectively, A1 is the area of the piston with A1=πD2/4, D denotes the diameter of the piston, A2 represents the effective area of the rod cavity with A2=πD2−d2/4, and d stands for the diameter of the piston rod.Figure 4
Model of front axle hydropneumatic suspension.The orifice size of the front axle hydropneumatic suspension is very small, so is that of the proportional valve. Based on this, the oil flow can be calculated as(4)q=CdA2ΔPρ,where ρ is the density of oil, A is the flow area of the hole that mounts the throttle and proportional valves, Cd denotes the flow coefficient of the hole, and ΔP represents the pressure difference between the oil cavity and the accumulator.The compression on the hydropneumatic suspension will cause the oil in the rodless chamber 1 to flow into the accumulator A via the throttle valve 6 and the proportional valve 5; meanwhile, the oil in the accumulator B passes through the throttle valve 8 and the proportional valve 7 and enters the rod cavity 2. When the hydropneumatic suspension rebounds, the flow direction of the oil will be reversed. The numerical relationship between the flow rate and the speed of the piston rod can be described by(5a)A1x˙=sgnx˙CdA92ΔP21ρ+CdA5+A62ΔPA1ρ,(5b)A2x˙=sgnx˙CdA92ΔP21ρ+CdA7+A82ΔP2Bρ,where A6 and A8 denote the flow area of the throttle valves 6 and 8, respectively, A5 and A7 are the flow area of the proportional valves 5 and 7, A9 is the flow area of the throttle valve 9, ΔP1A is the absolute value of the oil pressure difference between the rodless cavity 1 and the accumulator A, ΔP2B is the absolute value of the oil pressure difference between the rod cavity 2 and the accumulator B, sgnx˙ is the symbolic function, and x˙ is the relative velocity between the chassis and front axle with x˙=z˙b−lfφ˙b−z˙e.The symbolic function is(6)sgnx˙=1,x˙>0,0,x˙=0,−1,x˙<0.The velocity with respect to the volume change of the accumulators A and B can be expressed as(7)sgnx˙CdA5+A62ΔPA1ρdt=V˙A,sgnx˙CdA7+A82ΔP2Bρdt=−V˙B.It is assumed that the gas in accumulators is ideal and its temperature is considered to be invariant. Thus, the state equation of this ideal gas will be(8)PAVA=PA0VA0,PBVB=PB0VB0,where PA represents the transient pressure of the accumulator A; VA is the transient volume of the accumulator A; likewise, PB and VB denote the transient pressure and the transient of the accumulator B; PA0 and VA0 are the initial pressure and the initial volume of the accumulator A, and PB0 and VB0 are the initial pressure and the initial volume of the accumulator B.According to the state equation of the ideal gas, the oil pressure of the accumulators A and B can be expressed as(9)PA=PA0VA0VA0−A1x,PB=PB0VB0VB0+A2x,where x is the relative displacement between the chassis and the front axle, x=zb−lfφb−ze.Based on equation (8), the derivatives of the volume change of the accumulators A and B with respect to time are(10)V˙A=−PA0VA0P˙APA2,V˙B=−PB0VB0P˙BPB2.By solving equation (5), the oil pressure can be given by(11)P1=PA−ρPA02VA02P˙A/PA2sgnx˙2Cd2A5+A62,P2=PB+ρPB02VB02P˙B/PB2sgnx˙2Cd2A7+A82.Friction force between the cylinder and the piston can be(12)Ffx˙=sgnx˙Fc+Fs−Fce−x˙/x˙s2+fvx˙,where Fc is the Coulomb friction force, Fs is static friction force, fv is the coefficient of the viscous friction force, and x˙s is the Stribeck velocity.The total force acting on the piston rod with the damping force of the orifice and the friction force between cylinder and piston considered can be given by(13)Fp=Fh−Ff.The throttle valve 9 will be either closed or fully opened. When the throttle valve 9 is in the closed state, the force of the hydropneumatic suspension can be calculated as(14)Fp=ρA13x˙2signx˙2Cd2Aj+Ac2+PA0VA0rA1VA0−A1xr−signx˙Fc+Fs−Fce−x˙/x˙s2+Fvx˙−PB0VB0rA2VB0+A2xr+ρA23x˙2signx˙2Cd2Az+Ad2.Otherwise, the force of the hydropneumatic suspension can be expressed as(15)Fp=PA0VA0+VB0A1−A2PA0VA0+VB0A1−A2/Mg−A1−A2x+ρA1−A23x˙24Cd2A5+A62+ρA1−A23x˙24Cd2A7+A82−Fc−Fs−Fce−x˙/x˙s2−Fvx˙signx˙.


## 3.2. Model of Three-point Hitch Structure

The model of the three-point hitch structure is shown in Figure5. Separating the upper link CD from the three-point hitch structure, its force balance relationship can be given by(16)Fdx−Fcx−m2a2x=0,Fcz−Fdz−m2g−m2a2z=0,Fdxlcd−lcmsinφ2+Fdzlcd−lcmcosφ2+Fcxlcmsinφ2+Fczlcmcosφ2=I2φ¨2.Figure 5
Model of three-point hitch structure.Similarly, the force balance relation of the implement can be expressed as(17)Fbx−Fdx=0,Fdz−mag−maz¨a−Fbz=0,Fbxl4+Fbzl3+Fdxl5−Fdzl6=Iaφ¨a.Separating the lower link AB from the three-point hitch structure, its force balance relation can be expressed as(18)Fax−Fbx−m1a1x+Fecosγ−φ1=0,Fbz−Fesinγ−φ1−Faz−m1g−m1a1z=0,−Fbzl1+l2−lamcosφ1−Fazlamcosφ1−Fesinγlam−l1−Fbxl1+l2−lamsinφ1−Faxlamsinφ1=I1φ¨1.Similarly, the force balance relation of the chassis can be given by(19a)mbx¨b=Fax−Fcx+Fecosγ−φ1+Ffx+Frx,(19b)mbz¨b=−Fp+krqr−zb−lrφb+crq˙r−z˙b−lrφ˙b−kcrzb−zc−lcrφc+lbrφb−ccrz˙b−z˙c−lcrφ˙c+lbrφ˙b−kcfzb−zc+lcfφc−lbfφb−ccfz˙b−z˙c+lcfφ˙c−lbfφ˙b+Fcz−Fesinγ−φ1−Faz,(19c)Ibφ¨b=Fplf+krqr−zb−lrφb+crq˙r−z˙b−lrφ˙blr−kcrzb−zc−lcrφc+lbrφb+ccrz˙b−z˙c−lcrφ˙c+lbrφ˙blbr+kcfzb−zc+lcfφc−lbfφb+ccfz˙b−z˙c+lcfφ˙c−lbfφ˙blbf+Fcxlbc+Fczlb−Fecosγ−φ1lag−lba−Fesinγ−φ1lb+Faxlba−Fazlb−Ffxzb−qf−Frxzb−qr.The force balance relation of the implement can be expressed as(20)Fbx=Fdx,Fdz−Fbz−mag=maz¨a,Fbxl4+Fbzl3+Fdxl5−Fdzl6=Iaφ¨a.The angular velocity and acceleration of link CD can be described as(21)φ¨2=adxsinφ2−adzcosφ2−acxsinφ2+aczcosφ2lcd,φ˙2=adzsinφ2−adxcosφ2−aczsinφ2+acxcosφ2lcd1/2,where adz=z¨a−l6φ¨a−l5φ˙a2, adx=−l5φ¨a+l6φ˙a2, acz=z¨b+lbφ¨b−lbcφ˙b2, and acx=−lbcφ¨b−lbφ˙b2.The acceleration at the mass center of link CD in the horizontal and vertical directions can be expressed as(22)a2x=acx+lcmφ¨2sinφ2+lcmφ˙22cosφ2,a2z=acz−lcmφ¨2cosφ2+lcmφ˙22sinφ2.Similarly, the angular velocity and acceleration of link AB can also be given by(23)φ¨1=−z¨acosφ1+l3cosφ1φ¨a+l4sinφ1φ¨a−l4cosφ1φ˙a2+l3sinφ1φ˙a2+z¨bcosφ1+lbcosφ1φ¨b−lbasinφ1φ¨b+lbacosφ1φ˙b2+Lbsinφ1φ˙b2lab,φ˙1=−z¨bsinφ1−lbsinφ1φ¨b−lbacosφ1φ¨b−lbasinφ1φ˙b2+lbcosφ2φ˙b2+z¨asinφ1−l3sinφ1φ¨a+l4cosφ1φ¨a+l4sinφ1φ¨a+l3cosφ1φ˙a2lab1/2,where abx=l4φ¨a+l3φ˙a2, abz=z¨a−l3φ¨a+l4φ˙a2, aax=lbaφ¨b−lbφ˙b2, and aaz=z¨b+lbφ¨b+lbaφ˙b2.The acceleration at the mass center of link AB in horizontal and vertical directions can be given by(24)a1x=aax+lamφ¨1sinφ1+lamφ˙12cosφ1,a1z=aaz−lamφ¨1cosφ1+lamφ˙12sinφ1.A set of state variables selected for analysis are listed below:(25a)u1=za,u2=φa,u3=xb,u4=zb,u5=φb,u6=zc,u7=φc,u8=zd,u9=ze,(25b)u10=z˙a,u11=φ˙a,u12=x˙b,u13=z˙b,u14=φ˙b,u15=z˙c,u16=φ˙c,u17=z˙d,u18=z˙e.By substituting equations (25a) and (25b) into equations (2a)–(2d) and equations (19a)–(19c), the second-order differential equations can be reduced to the first order, which makes it easy to be handled mathematically.


## 4. Derivation Procedure of Model Parameters

In order to acquire the vertical stiffness and damping of tire, we simplify the tire to be a single-DOF dynamical system. By using the logarithmic decrement method of free vibration, the vertical stiffness and damping of front and rear tires can be gained. The site map and the test rig are demonstrated in Figure6. The test rig system includes support frame (1), acceleration sensor (2), weight plate (3), rings (4), tire (5), rope (6), beam (7), linear bearing beam (8), displacement sensor (9), holder of sensor (10), suction plate (11), suspension bracket of magnetic lifter (12), magnetic lifter (13), ground (14), tire connection plate (15), linear bearing (16), and guide rods (17).Figure 6
Test device of stiffness and damping for tire: (a) site map; (b) structure principle.
(a)
(b)As can be seen in Figure6, the under-test tire is supported by the beam. The weight plates are installed in the same way to inflict the load on the tire. The guide rod connects with the beam pass through the linear bearing. One end of the rope passes through the rings fixed to the lower side of the upper frame to suspend the under-test tire, and the other end is fastened to the suction plate. The cup of the suction plate is absorbed on the lower surface of the magnetic elevator that is mounted to the suspension bracket. For the purpose of eliminating the influence of gravity on the under-test tire, the magnetic lift is adjusted by operators depending on the situation. By regulating the height of the linear bearing beam, tires with different diameters can be installed by the test bed. The acceleration signal of the tire is acquired and saved by the data acquisition card. The test software is programmed with Labview code [44].When the tire is lifted above the ground with a heighth, the lower beam with the guide rod will fall along the guide sleeve due to gravity. After the tire hits the ground block, the vibration in time domain begins to attenuate freely and is collected using an acceleration sensor fixed to the beam. The log reduction can be calculated as(26)δ=lnAi/Ajj−i=lnz¨i/z¨jj−i.The damping ratio of the system can be obtained based on the logarithmic decrement and is given by(27)ξ=δδ2+2π2.Then, the vertical stiffness and damping of front and rear tires can be given by(28)k=2π/T2m1−ξ2,c=2ξkm.With the first lag introduced, the dynamic response of the tested tire in the longitudinal direction can be given by(29)σγF˙xv+Fx=Cγγ,where γ is the slip of tire in the longitudinal direction, γ=rθ˙−x˙/v, r is the tire radius, x˙ is the longitudinal tire speed, θ˙ represents the tire’s roll speed, and v is the forward speed of tractor.The longitudinal stiffness and damping of tire can be expressed as(30)kx=Cγσγ,cx=Cγv.Similarly, the site map of stiffness and damping test rig for rubber rings between the cabin and the chassis is shown in Figure7, and the vertical stiffness and damping of the driver seat can also be obtained in the same way (Figure 8).Figure 7
Site map of stiffness and damping test rig for the rubber suspension.Figure 8
Site map of stiffness and damping test rig for the driver seat.The stiffness and damping of silent blocks of the cabin and driver seat can be given by(31)kc=mω2α2−αcosφα2−2αcosφ+1,cc=mωαsinφα2−2αcosφ+1,where α is the acceleration amplitude ratio between output and input signals, ω is the excitation frequency, m is the mass of the driver seat, and φ is the phase difference of output acceleration with respect to input acceleration.The mean value of the measured modal parameters between 4 and 8 Hz, which is very sensitive to the health of the driver, can be used as the stiffness and damping parameters of simulation.In order to obtain the mass and moment of inertia of wheeled tractor, its three-dimensional physical model is established in Pro/E software [45]. When the material property of each part for wheeled tractor is defined, the parameter of mass and the moment of inertia can be calculated easily.


## 5. Results and Discussion

The CF700 wheeled tractor system equipped with the implement and front axle hydropneumatic suspension as well as passive rubber cabin suspension is modeled in the numerical example. The corresponding parameters are listed in Table1. A standard artificial test track with a length of 100 m, as shown in Figure 9, is employed to produce road excitations, which are applied to both front and rear tires of the tractor system.Table 1
Parameters of the wheeled tractor/implement with front axle hydropneumatic suspension.
l
f
1348 mm
l
r
840 mm
l
bf
360 mm
l
br
840 mm
k
f
357177 N/m
c
r
2516.8 N·s/m
k
cr
1012383 N/m
c
d
1473 N·s/m
I
a
166.67 kg·m2
l
bc
0.14 m
l
1
0.42 m
l
cd
0.604 m
l
5
0.343 m
l
cm
0.302 m
C
d
0.7
A
9
0 m2
P
B0
6 MPa
F
s
220 N
l
f
1348 mm
l
r
840 mm
l
cf
600 mm
l
cr
290 mm
l
d
290 mm
m
b
2694 kg
c
f
1789.2 N·s/m
k
cf
1012383 N/m
c
cr
5093 N·s/m
m
a
300 kg
γ
1.4775 rad
l
ag
0.576 m
l
2
0.42 m
l
3
0.125 m
l
6
0.287 m
D
0.05 m
ρ
900 kg/m3
P
A0
6 MPa
V
B0
0.5 L
f
v
2
l
cf
600 mm
l
cr
290 mm
m
c
375 kg
m
d
75 kg
I
b
3249.3 kg·m2
I
c
218.8 kg·m2
k
r
359304 N/m
c
cf
5093 N·s/m
A
7
+
A
8
4.52 × 10−6 m2
m
e
146 kg
l
ba
0.21 m
l
b
0.94 m
l
ab
0.84 m
l
4
0.338 m
l
am
0.42 m
d
0.032 m
A
5 + A6
4.52 × 10−6 m2
V
A0
0.5 L
F
c
100 N
x
˙
s
0.003 m/s
m
c
375 kg
m
d
75 kgFigure 9
Road roughness of the standard artificial test track.When the forward speed of the tractor/implement system is 2 m/s and the nitrogen pressure and volume of the accumulators A and B are 6 MPa and 0.5 L, the measured and predicted acceleration values of the driver seat, cabin, chassis, and implement and their corresponding power spectral density (PSD) with the orifice diameter 2.4 mm of throttle valves 6 and 8 and 0 mm of throttle valve 9 (closed) for front axle hydropneumatic suspension are shown in Figures10 and 11.Figure 10
Vibration acceleration of the tractor/implement system: (a) driver seat; (b) cabin; (c) chassis; (d) implement.
(a)
(b)
(c)
(d)Figure 11
PSD of acceleration for the tractor/implement system: (a) driver seat; (b) cabin; (c) chassis; (d) implement.
(a)
(b)
(c)
(d)As can be seen from Figures10 and 11, there are slight quantitative variations between the simulation time series and the measured values generally. The vibration responses from the proposed model are in good agreement with experimental results, and the validity of the presented model in this work is verified. A pronounced resonant frequency was observed around 2–4 Hz for the tractor/implement system. To prove the proposed model further, the experimental and the predicted root mean square (RMS) values of acceleration for the tractor under different forward speeds are compared (Figure 12). It is demonstrated that the presented model is also effective in predicting the vibration characteristics of the tractor under different forward speeds. With the increase of the forward speed, the RMS values of acceleration for the driver seat, cabin, chassis, and implement increase dramatically and the RMS of cabin’s acceleration is less than that of acceleration for the driver seat, chassis, and implement due to the passive rubber suspension between the cabin and chassis.Figure 12
RMS of vibration responses for the tractor system under different forward speeds: (a) driver seat; (b) cabin; (c) chassis; (d) implement.
(a)
(b)
(c)
(d)To evaluate the driver comfort of the tractor system, the frequency-weighted acceleration in the time domain is used to describe the exposure of the seated driver to the vertical vibration. Two typical weightings are often applied in practice:Wk is defined in the International Standard ISO 2631 [7] and Wb from British Standard, BS 6841 [8]. The RMS vibration magnitudes were determined for the vertical vibration simulated on the cabin floor and surface of the seat based on International Standard ISO 2631:(32)RMS=∫0Ta2tdtT1/2,where at is the frequency-weighted acceleration in time domain and T is the measurement period.The SEAT values were calculated as the ratio of the frequency-weighted acceleration on the seat, RMSseat, to the frequency-weight acceleration on the tractor floor, RMSfloor:(33)SEATRMS%=RMSseatRMSfloor%.The SEAT value is used to describe the effectiveness of the driver seat, and the SEAT value below 100% implies that the seat is isolating vibration on the floor and thus reducing discomfort from the tractor vibration.To make a comparative study on the effect of the passive cabin suspension, implement, and front axle hydropneumatic suspension on the vibration characteristics and driver comfort of the tractor system, the dynamic models of tractor system under such four cases as (I) without cabin, front axle suspension, and implement, (II) with cabin suspension and without front axle suspension and implement, (III) with cabin suspension and implement and without front axle suspension, and (IV) with cabin, front axle suspension, and implement are developed. The corresponding RMS values of vibration responses and SEAT values for the tractor/implement system under different settings are listed in Table2.Table 2
RMS of vibration responses and SEAT values for the tractor/implement system under different settings.
Settings
z
¨
d (m/s2)
x
¨
b (m/s2)
z
¨
b (m/s2)
φ
¨
b (rad/s2)
F
fdyn (N)
F
rdyn (N)
SEAT
Case I
2.3942
2.0311
2.1048
0.9355
4335.2
2669.3
1.184
Case II
2.1700
1.5341
1.6914
1.1003
5342.8
4082.6
1.416
Case III
1.5697
1.40231
1.3742
0.8580
12080
7201.8
1.293
Case IV
2.6330
1.2562
1.5372
0.5240
4342.5
3484.5
1.851Table2 shows that the front axle hydropneumatic suspension increases the RMS of vertical vibration acceleration for the driver seat under the case with cabin and front axle suspension as well as implement compared with the case without front axle hydropneumatic suspension; meanwhile, the front axle hydropneumatic suspension decreases the RMS of longitudinal, vertical, and pitch vibrations for the chassis as well as dynamic load for front and rear tires dramatically. The passive cabin suspension can reduce the RMS of vertical vibration of the driver seat and longitudinal and vertical vibrations of the chassis, while increasing the RMS of pitch vibration for the chassis, dynamic loads for front and rear tires, and SEAT value tremendously compared with the case without cabin, front axle suspension, and implement. The RMS values of the vertical vibration for the driver seat, longitudinal, vertical, and pitch vibrations for the chassis, and SEAT value under the case II is larger than those under the case III, but the RMS values of dynamic loads for front and rear tires are on the contrary. Therefore, the front axle hydropneumatic suspension will deteriorate the ride comfort of the driver but will improve the handing stability obviously. The passive cabin suspension plays a more important role in improving the ride comfort of the driver than the front axle hydropneumatic suspension does, while taking the expense of operational stability to some extent. The existence of the implement will raise the comfortableness of the driver and worsen the corresponding handing stability at the same time dramatically. The calculated SEAT values indicated that the driver seat increases the vibration transmitted to the driver.


### 5.1. Effect of Nitrogen Volume of Accumulators

When the forward speed of the tractor system is 1.21 m/s and the nitrogen pressure of accumulators A and B is 6 MPa, the RMS values of vibration responses and SEAT values for the tractor/implement system with the orifice diameter 2.4 mm of throttle valves 6 and 8 and 0 or 5 mm of throttle valve 9 (closed or fully opened) for front axle hydropneumatic suspension under different initial nitrogen volumes of accumulators A and B are shown in Figure13.Figure 13
RMS of vibration responses of the tractor system under different initial nitrogen volumes of accumulators: (a) vertical acceleration of the driver seat; (b) longitudinal acceleration of the chassis; (c) vertical acceleration of the chassis; (d) pitch acceleration of the chassis; (e) dynamic load of the front tire; (f) dynamic load of the rear tire; (g) dynamic deflection of the hydropneumatic suspension; (h) SEAT value of the driver seat.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure13 shows that when the throttle valve 9 is closed, the RMS values of vertical vibration for the driver seat, longitudinal, vertical, and pitch vibration for the chassis, dynamic load for front and rear tires, dynamic deflection for suspension, and SEAT values of the driver seat almost remain constant with the increase in nitrogen volume of the accumulators A and B. When the throttle valve 9 is fully opened, the RMS values of vertical vibration for driver seat and chassis, pitch vibration for the chassis, and SEAT values of the driver seat decrease dramatically with the increase in nitrogen volume of the accumulators A and B, while RMS values of dynamic loads for front and rear tires and dynamic deflection for suspension also remain constant. The RMS of longitudinal acceleration for the chassis increases and then decreases with the increase in nitrogen volume of the accumulators A and B. Therefore, the RMS values of vertical vibration for the driver seat, longitudinal, vertical, and pitch vibrations for the chassis, and SEAT values of the driver seat with the throttle valve fully opened are more sensitive to the change of initial nitrogen volume of the accumulators A and B than those with the throttle valve closed, while other RMS values of vibration responses for the tractor system are insensitive to the change of initial nitrogen volume of accumulators and almost remain constant regardless of the throttle valve being closed or opened. The optimized initial nitrogen volume for the front axle hydropneumatic suspension is 0.7 L when the throttle valve 9 is fully opened.


### 5.2. Effect of Nitrogen Pressure of Accumulators

When the forward speed of the tractor system is 1.21 m/s and the initial nitrogen volumes of the accumulators A and B is 0.5 L, the RMS values of vibration responses and SEAT values for the tractor/implement system with a 2.4 mm orifice diameter of throttle valves 6 and 8 for front axle hydropneumatic suspension under different nitrogen pressures of the accumulators A and B are shown in Figure14.Figure 14
RMS of vibration responses of tractor system under different nitrogen pressure of the accumulators: (a) vertical acceleration of the driver seat; (b) longitudinal acceleration of the chassis; (c) vertical acceleration of the chassis; (d) pitch acceleration of the chassis; (e) dynamic load of the front tire; (f) dynamic load of the rear tire; (g) dynamic deflection of the hydropneumatic suspension; (h) SEAT value of the driver seat.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)It can be seen that when the throttle valve 9 is fully opened, the RMS values of vertical vibration for the driver seat and longitudinal, vertical, and pitch vibrations for the chassis decrease first and then increase with the initial nitrogen pressure ranging from 5.8 MPa to 6.2 MPa and continue to decrease with the increase of initial nitrogen pressure in the interval above 6.2 MPa, while RMS values of dynamic loads for front and rear tires, dynamic deflection for suspension, and SEAT values of the driver seat decrease. When the throttle valve 9 is closed, RMS of vibration responses (except longitudinal acceleration of the chassis) for the tractor system also remains constant with the increase of initial nitrogen pressure for the accumulators A and B. Therefore, only the RMS values of vertical vibration for the driver seat, longitudinal, vertical, and pitch vibrations of the chassis, and SEAT value of the driver seat when the throttle valve is fully opened are sensitive to the change of initial nitrogen pressure of the accumulators A and B. The optimized initial nitrogen pressure for the front axle hydropneumatic suspension when the throttle valve 9 is fully opened is 6.5 MPa.


### 5.3. Effect of Orifice Diameter of Throttle Valves 6 and 8

In this section, the effect of orifice diameter of throttle valves 6 and 8 is studied. When the forward speed of the tractor system is 1.21 m/s and the nitrogen pressure of the accumulators A and B is 6 MPa, the RMS values of vibration responses and SEAT values of the tractor/implement system with a 0.5 L initial nitrogen volume of the accumulators A and B under different orifice diameters of the throttle valves 6 and 8 are shown in Figure15.Figure 15
RMS of vibration responses of tractor system under different orifice diameters of throttle valves: (a) vertical acceleration of the driver seat; (b) longitudinal acceleration of the chassis; (c) vertical acceleration of the chassis; (d) pitch acceleration of the chassis; (e) dynamic load of the front tire; (f) dynamic load of the rear tire; (g) dynamic deflection of the hydropneumatic suspension; (h) SEAT value of the driver seat.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure15 shows that when the throttle valve 9 is closed, the RMS values of vertical vibration for the driver seat and chassis, dynamic loads for front and rear tires, and SEAT values of driver seat decrease with the orifice diameter of the throttle valves 6 and 8 changing between 1 mm and 2 mm and continue to increase with the increase of orifice diameter of throttle valves 6 and 8 in the interval above 2 mm, while the RMS of longitudinal and pitch vibrations for the chassis rises obviously with the increase of orifice diameter of throttle valves 6, 8. When the throttle valve 9 is fully opened, the RMS values of vibration responses (except the dynamic load of the rear tire and dynamic deflection of the suspension) and SEAT values for the tractor system increase with an increase in the orifice diameter of throttle valves 6 and 8. Therefore, the recommended orifice diameter of throttle valves 6 and 8 is 2 mm when the throttle is closed to obtain the optimized ride comfort. When the throttle valve 9 is fully opened, the smaller the orifice diameter of the throttle valves 6 and 8 is, the better the ride comfort would be.


## 5.1. Effect of Nitrogen Volume of Accumulators

When the forward speed of the tractor system is 1.21 m/s and the nitrogen pressure of accumulators A and B is 6 MPa, the RMS values of vibration responses and SEAT values for the tractor/implement system with the orifice diameter 2.4 mm of throttle valves 6 and 8 and 0 or 5 mm of throttle valve 9 (closed or fully opened) for front axle hydropneumatic suspension under different initial nitrogen volumes of accumulators A and B are shown in Figure13.Figure 13
RMS of vibration responses of the tractor system under different initial nitrogen volumes of accumulators: (a) vertical acceleration of the driver seat; (b) longitudinal acceleration of the chassis; (c) vertical acceleration of the chassis; (d) pitch acceleration of the chassis; (e) dynamic load of the front tire; (f) dynamic load of the rear tire; (g) dynamic deflection of the hydropneumatic suspension; (h) SEAT value of the driver seat.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure13 shows that when the throttle valve 9 is closed, the RMS values of vertical vibration for the driver seat, longitudinal, vertical, and pitch vibration for the chassis, dynamic load for front and rear tires, dynamic deflection for suspension, and SEAT values of the driver seat almost remain constant with the increase in nitrogen volume of the accumulators A and B. When the throttle valve 9 is fully opened, the RMS values of vertical vibration for driver seat and chassis, pitch vibration for the chassis, and SEAT values of the driver seat decrease dramatically with the increase in nitrogen volume of the accumulators A and B, while RMS values of dynamic loads for front and rear tires and dynamic deflection for suspension also remain constant. The RMS of longitudinal acceleration for the chassis increases and then decreases with the increase in nitrogen volume of the accumulators A and B. Therefore, the RMS values of vertical vibration for the driver seat, longitudinal, vertical, and pitch vibrations for the chassis, and SEAT values of the driver seat with the throttle valve fully opened are more sensitive to the change of initial nitrogen volume of the accumulators A and B than those with the throttle valve closed, while other RMS values of vibration responses for the tractor system are insensitive to the change of initial nitrogen volume of accumulators and almost remain constant regardless of the throttle valve being closed or opened. The optimized initial nitrogen volume for the front axle hydropneumatic suspension is 0.7 L when the throttle valve 9 is fully opened.


## 5.2. Effect of Nitrogen Pressure of Accumulators

When the forward speed of the tractor system is 1.21 m/s and the initial nitrogen volumes of the accumulators A and B is 0.5 L, the RMS values of vibration responses and SEAT values for the tractor/implement system with a 2.4 mm orifice diameter of throttle valves 6 and 8 for front axle hydropneumatic suspension under different nitrogen pressures of the accumulators A and B are shown in Figure14.Figure 14
RMS of vibration responses of tractor system under different nitrogen pressure of the accumulators: (a) vertical acceleration of the driver seat; (b) longitudinal acceleration of the chassis; (c) vertical acceleration of the chassis; (d) pitch acceleration of the chassis; (e) dynamic load of the front tire; (f) dynamic load of the rear tire; (g) dynamic deflection of the hydropneumatic suspension; (h) SEAT value of the driver seat.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)It can be seen that when the throttle valve 9 is fully opened, the RMS values of vertical vibration for the driver seat and longitudinal, vertical, and pitch vibrations for the chassis decrease first and then increase with the initial nitrogen pressure ranging from 5.8 MPa to 6.2 MPa and continue to decrease with the increase of initial nitrogen pressure in the interval above 6.2 MPa, while RMS values of dynamic loads for front and rear tires, dynamic deflection for suspension, and SEAT values of the driver seat decrease. When the throttle valve 9 is closed, RMS of vibration responses (except longitudinal acceleration of the chassis) for the tractor system also remains constant with the increase of initial nitrogen pressure for the accumulators A and B. Therefore, only the RMS values of vertical vibration for the driver seat, longitudinal, vertical, and pitch vibrations of the chassis, and SEAT value of the driver seat when the throttle valve is fully opened are sensitive to the change of initial nitrogen pressure of the accumulators A and B. The optimized initial nitrogen pressure for the front axle hydropneumatic suspension when the throttle valve 9 is fully opened is 6.5 MPa.


## 5.3. Effect of Orifice Diameter of Throttle Valves 6 and 8

In this section, the effect of orifice diameter of throttle valves 6 and 8 is studied. When the forward speed of the tractor system is 1.21 m/s and the nitrogen pressure of the accumulators A and B is 6 MPa, the RMS values of vibration responses and SEAT values of the tractor/implement system with a 0.5 L initial nitrogen volume of the accumulators A and B under different orifice diameters of the throttle valves 6 and 8 are shown in Figure15.Figure 15
RMS of vibration responses of tractor system under different orifice diameters of throttle valves: (a) vertical acceleration of the driver seat; (b) longitudinal acceleration of the chassis; (c) vertical acceleration of the chassis; (d) pitch acceleration of the chassis; (e) dynamic load of the front tire; (f) dynamic load of the rear tire; (g) dynamic deflection of the hydropneumatic suspension; (h) SEAT value of the driver seat.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure15 shows that when the throttle valve 9 is closed, the RMS values of vertical vibration for the driver seat and chassis, dynamic loads for front and rear tires, and SEAT values of driver seat decrease with the orifice diameter of the throttle valves 6 and 8 changing between 1 mm and 2 mm and continue to increase with the increase of orifice diameter of throttle valves 6 and 8 in the interval above 2 mm, while the RMS of longitudinal and pitch vibrations for the chassis rises obviously with the increase of orifice diameter of throttle valves 6, 8. When the throttle valve 9 is fully opened, the RMS values of vibration responses (except the dynamic load of the rear tire and dynamic deflection of the suspension) and SEAT values for the tractor system increase with an increase in the orifice diameter of throttle valves 6 and 8. Therefore, the recommended orifice diameter of throttle valves 6 and 8 is 2 mm when the throttle is closed to obtain the optimized ride comfort. When the throttle valve 9 is fully opened, the smaller the orifice diameter of the throttle valves 6 and 8 is, the better the ride comfort would be.


## 6. Conclusion

In this paper, a complete dynamic model of the wheeled tractor/implement system with front axle hydraulic-pneumatic suspension is established with the three-point suspension structure and the nonlinear effect of the front axle hydraulic suspension considered. Numerical and experimental results show that the model can be used to predict the vibration characteristics of real structures, and a pronounced resonant frequency was observed around 2–4 Hz for the tractor/implement system. The influences of the passive cabin suspension, the implement, and the hydraulic suspension of the front axle on the dynamic behaviour of the tractor system are studied and compared in four cases. The numerical example manifests that the front axle hydraulic suspension does not improve the driver’s ride comfort but it significantly improves the steering stability. The passive cabin suspension contributes more to enhancing the ride comfort than the front axle oil and gas suspension do, while sacrificing operational stability to some extent. Because of the implement, the driver’s comfort is improved, but the steering stability is lessened. The calculated SEAT values indicate that the driver seat increases the vibration transmitted to a driver. When the throttle valve 9 is fully opened, the initial nitrogen volume is 0.7 L and the optimized pressure of the front axle hydraulic suspension is 6.5 MPa. The dynamic load of front and rear tires and dynamic deflection of front axle hydraulic-pneumatic suspension are not sensitive to the change of the initial nitrogen volume and pressure of accumulators whether the throttle is closed or open. When the throttle is closed, the diameter of the recommended orifices of the throttles 6 and 8 should be 2 mm for the optimum ride comfort. When the throttle valve 9 is fully opened, the smaller the diameter of the holes of the throttle valves 6 and 8 is, the better the ride comfort will be.



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*Source: 9135412-2019-07-07.xml*