# Modeling and Analysis of the Kinetic Influence of Liquid Sloshing Characteristics on High-Clearance Sprayers

Authors: Weijun Wang; Jing’an Feng; Wenkang Wan; Peng Zhang; Songlin Yang
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

To research the influence of liquid sloshing on the driving stability of high-clearance sprayers, this paper builds an equivalent liquid sloshing mechanical model and obtains the stochastic acceleration excitation of the rectangular spray tank using the Adams kinetic model, thus obtaining the relationship between the impact force, moment, and the stochastic acceleration using Fluent numerical simulation analysis. This paper makes further calculations with MATLAB/Simulink system models, and the result from comparing these two calculations shows that the equivalent strategy proposed in this paper has a better consistency. Based on the consideration of the acting forces of the additional moment due to lateral movement of the center of mass of the liquid and the dynamic pressure due to liquid sloshing in the tank, this paper builds a mathematical model of the sprayer and researches the influence of the filling ratio and vehicle velocity on the vehicle stability through stochastic acceleration excitation. The results show that, in the case of different speeds, the liquid sloshing has a small influence on the overall roll angle; in the case of different filling ratios, the liquid sloshing has a big influence on the overall roll angle, the slip angle of the center of mass, and the yaw angular velocity; the filling ratiok = 0.85 and the speed u = 1 m/s−2 m/s are safe operation parameters of the sprayer. This research provides reference solutions for the stability control and optimization problems of the high-clearance sprayer and semitrailer.


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## Body

## 1. Introduction

During the pesticide application process, multiple factors affect the stability of high-clearance sprayers. High-clearance sprayers are likely to roll over due to instability during the pesticide application process, which could in turn trigger casualties and economic losses; therefore, researching and analyzing the stability of working high-clearance sprayers is increasingly important. High-clearance sprayers transport a large amount of liquid, while in motion, the large-amplitude sloshing of liquid will change the load distribution on the axle of the high-clearance sprayer and produce additional force and moment, thus reducing the stability of the high-clearance sprayer. Liquid sloshing is an important factor that causes instability in high-clearance sprayers and is of great significance to the analysis and resolution of the working stability problems of the high-clearance sprayer by building an equivalent mechanical model of the liquid sloshing of the high-clearance sprayer and researching the liquid sloshing behaviors [1–5].Since the 1950s, the problem of liquid sloshing of mobile containers has received more attention [6]. Liquid sloshing is a common problem in engineering, and extensive researches into the transportation problems of tank trucks and the application of such symmetrical structures in engineering have been conducted [7–10]. However, the liquid sloshing in truck tanks is not a symmetrical problem, and the coupling of the liquid sloshing and the tank truck body makes such problems more complicated.In recent years, domestic and overseas scholars have conducted numerous researches on liquid sloshing in truck tanks. Feng Ran from Jilin University built an equivalent model of liquid sloshing in truck tanks to solve associated parameters of the equivalent model by utilizing MATLAB ident system identification [11]. Pan and Ma have built an equivalent pendulum model of liquid sloshing using a rectangular open-moon pool as an example. Liu Chunyuan et al. took a rectangular full-open moon pool as an example, established a liquid sloshing equivalent simple pendulum model, used Lagrangian equations to establish the kinematics and dynamics equations of the simple pendulum, and then used Ibrahim to solve the moon pool based on linear potential flow theory. Kinematics, dynamics, and related mechanics equations through simultaneous equations solve the relevant parameter values of the equivalent model [12]. Hu Xiaoming et al. built an equivalent mechanical model of liquid forced sloshing and solved associated parameters by taking the liquid tank semitrailer as an example. Afterward, they built a semitrailer kinetic model, added the forces and moments obtained from the equivalent model to the semitrailer kinetic model, and analyzed the influence of liquid sloshing on the stability of the semitrailer [13].To sum up, many domestic and overseas scholars have conducted multiple research on the liquid sloshing characteristics; however, they only take into account the acting force of the lateral movement of the center of mass of liquid in the tank. In practical traffic and transport, the dynamic pressure generated from the lateral sloshing of the liquid cannot be ignored. Therefore, it is necessary to conduct further research on the lateral sloshing of the liquid in the high-clearance sprayer. This paper establishes a coupling relationship between the liquid sloshing and the high-clearance sprayer by building an equivalent mechanical model of the pendulum mass damping to make a synthetic analysis of the influence of liquid sloshing on the driving stability of the sprayer through simulation. The research in this paper will provide references for the modeling of the chassis system kinetic model and the stability control strategies of the high-clearance sprayer.


## 2. Designing the Simulation Model

### 2.1. 3D High-Clearance Sprayer Model

An overall 3D model is built for the high-clearance sprayer using SolidWorks, as shown in Figure1. The machine is mainly composed of high-clearance chassis, steering mechanism, driver cab, spray tank, spray lance, and so on, and the main parameters are shown in Table 1.Figure 1
Overall 3D model.Table 1
Main parameters of Adams overall simulation.
ParameterValueParameterValueWheelbase3.0 mDistance between the sprung mass center and roll center0.3 mWheel track2.0 mRotational inertia of sprung mass surrounding roll center308.4 kg m2Ground clearance1.5 mEquivalent roll stiffness of suspension4000.0 N • s • rad−1Total mass1000 kgEquivalent roll damping coefficient of suspension20.0 N • s • rad−1Sprung mass976 kgFigure2 is the simulation flowchart of reliability verification of the liquid sloshing equivalent mechanical model in this paper. The equivalent mechanical model of liquid forced sloshing is built based on the equivalence principle, the stochastic acceleration excitation suffered by the spray tank is output based on the Adams virtual prototype simulation model, and the model of liquid forced sloshing of the spray tank is built using Fluent to conduct numerical simulation of liquid sloshing and obtain the relationship between the impact force and impact moment and the stochastic acceleration; relevant parameters of the equivalent mechanical model are obtained through the mechanical analysis of liquid sloshing in the rectangular-section spray tank to build the MATLAB/Simulink system simulation model, then the reliability of the system with the filling ratio of 0.5 is verified through Fluent, and later the coupling with the constructed kinetic model of the high-clearance sprayer is conducted to analyze the influence control factors of liquid sloshing on the stability of the high-clearance sprayer.Figure 2
Simulation flowchart.


### 2.2. Adams Kinetic Model

To obtain the stochastic acceleration excitation suffered by the spray tank and dynamically to load it into the Fluent environment in real time, this paper builds a multibody kinetic simulation model of a high-clearance sprayer, converts the 3D high-clearance sprayer model equipped in the SolidWorks software into a Parasolid.xt format, and imports it into the Adams software. Besides, the model adds a rotating set to the four tires of the high-clearance sprayer separately and adds a rotating drive to the rear rotating sets separately to make the high-clearance sprayer move in a forward direction, and each component is set with parameters like material attributes, and so on [14]. The operating speed is 1 m/s, and the stochastic acceleration suffered by the spray tank in three directions is dynamically loaded into the Fluent model of liquid forced sloshing of the spray tank for the numerical simulation of the sloshing of liquid in the spray tank.


#### 2.2.1. Tire Model

The tire is an important component of the high-clearance sprayer and is an important contact part between the high-clearance sprayer and the pavement. It bears the weight of the sprayer and the external acting force applied by the pavement to the sprayer, playing a role in force conduction between the sprayer and ground.The Adams software has multiple tire models, with different tire models that apply to diverse working environments. According to the characteristics and requirements of a working high-clearance in the field, this paper selects the Fiala tire model, and its parameters are shown in Table2.Table 2
Tire model parameters.
ParameterValueParameterValueTire mass20 kgWheel hub width0.17 mFree radius of the tire0.38 mVertical stiffness of the tire190000 N • m−1Tire width0.21 mVertical damping coefficient of the tire50 N • s • m−1Flatness ratio of the tire0.55Static friction coefficient of the tire0.8Radius of the wheel hub0.2 mDynamic friction coefficient of the tire1.1


#### 2.2.2. Pavement Model

The paper mainly researches the liquid sloshing caused by uneven field pavement and does not take into account the influence of the pavement gradient on liquid sloshing; the paper establishes the requirements that meet Adams simulation pavement excitation and finally generates a GradeE three-dimensional pavement surface spectrum file of 40 m long and 20 m wide in MATLAB software according to the three-dimensional pavement node generation algorithm and based on the pavement unevenness expression and pavement unevenness classification standard established according to the harmony superposition principle, and the pavement file is shown in Figure 3.Figure 3
Grade e pavement.


### 2.3. Fluent Numerical Simulation

Two states occur during the pesticide application process of the high-clearance sprayer, namely, the gas phase and liquid phase in the spray tank at all times, as shown in Figure4. During the impact process, the total volume of the gas phase and liquid phase remains constant, and only the shape and position change.Figure 4
Schematic diagram of the two-phase flow in the spray tank.Therefore, the liquid impact on the spray tank is a multiphase flow problem and can be analyzed through simulation by utilizing the VOF model. Fluent can be used for single-phase, multiphase, and complicated problems in the numerical model, for example, multiphase flow problem, laminar flow problem, and turbulence problem. Table3 shows the setting of the key parameters.Table 3
Fluent simulation setting.
ParameterOptionParameterOptionCalculation pattern3D double precisionUnit conditionApply stochastic acceleration on the spray tank in three directionsSolverPressure solverSpeed and pressure couplingPISOSolving modelVOF modelUnit gradient interpolation methodGreen-gauss cell-basedViscosity modelStandardk-ε turbulence modelDispersion scheme of the pressure modification coefficientBody force weightedFlow modelTransient flowMonitoring parameterImpact force on the sidewall (the curved surface) of the spray tank; impact moment on the bottom center of the spray tankDispersion scheme of convection termsFirst-order upwind schemeInitializationMixed initialization; path liquid phaseMaterialWater (main phase); air (second phase)Timestep0.01 sDuring the numerical simulation, Fluent detects the impact force of liquid sloshing to the wall surface of the spray tank and the moment of liquid sloshing to the bottom center of the spray tank.


### 2.4. Analysis of Simulation Results

The paper prepares the GradeE stochastic pavement model required by the sprayer operating conditions through MATLAB software, then generates the stochastic pavement features suitable for Adams software simulation, and obtains the stochastic acceleration excitation output of the spray tank with Adams software simulation as the input of the Fluent software of liquid sloshing and applies C language to compile and expand the program codes of Fluent, which are then dynamically loaded to Fluent environment to establish the forced sloshing model of liquid in the spray tank of a high-clearance sprayer, when the filling ratio in numerical simulation is k = 0.5; the simulation results of the additional force and moment generated by liquid sloshing to the sidewall of the spray tank are shown in Figures 5 and 6.Figure 5
Impact force on the spray tank wall surface inZ direction.Figure 6
Central moment at the spray tank bottom surface inZ direction.As can be seen from Figures5 and 6, under the influence of the external stochastic acceleration, the additional force and moment generated by liquid sloshing in the spray tank gradually become larger, then tend to be stable, and finally increase, where the maximum impact force and moment suffered by the spray tank wall surface are 19.6 N and 0.6 N • m, respectively, and the average impact force and moment are 0.47 N and 0.04 N • m, respectively.To research the relationship between the stochastic acceleration excitation and the impact force and moment suffered by the wall surface of the spray tank, we apply a stochastic acceleration to the fluid region, taking the stochastic acceleration suffered by the spray tank in theZ direction as an example, detect the impact force and moment suffered by the wall surface of the spray tank, and obtain the relationship between the stochastic acceleration and the impact force and moment suffered by the wall surface in the Z direction, as shown in Figures 7 and 8.Figure 7
Relationship between stochastic acceleration and impact force.Figure 8
Relationship between stochastic acceleration and impact moment.As can be seen from Figures7 and 8, the variation trend of the stochastic acceleration applied to the spray tank in the Z direction mirrors that of the impact force and moment suffered by the wall surface in the Z direction, where the impact force and moment have hysteresis phenomena, which are consistent with real-life phenomena. The results show that the impact force and moment suffered by the wall surface of the spray tank are affected by the stochastic acceleration; namely, when the acceleration increases gradually, the impact force and moment suffered by the wall surface will also increase, and the variation trend is the same, where the average absolute value of the acceleration is 0.06 m/s2 and the maximum value is 1.5 m/s2.


## 2.1. 3D High-Clearance Sprayer Model

An overall 3D model is built for the high-clearance sprayer using SolidWorks, as shown in Figure1. The machine is mainly composed of high-clearance chassis, steering mechanism, driver cab, spray tank, spray lance, and so on, and the main parameters are shown in Table 1.Figure 1
Overall 3D model.Table 1
Main parameters of Adams overall simulation.
ParameterValueParameterValueWheelbase3.0 mDistance between the sprung mass center and roll center0.3 mWheel track2.0 mRotational inertia of sprung mass surrounding roll center308.4 kg m2Ground clearance1.5 mEquivalent roll stiffness of suspension4000.0 N • s • rad−1Total mass1000 kgEquivalent roll damping coefficient of suspension20.0 N • s • rad−1Sprung mass976 kgFigure2 is the simulation flowchart of reliability verification of the liquid sloshing equivalent mechanical model in this paper. The equivalent mechanical model of liquid forced sloshing is built based on the equivalence principle, the stochastic acceleration excitation suffered by the spray tank is output based on the Adams virtual prototype simulation model, and the model of liquid forced sloshing of the spray tank is built using Fluent to conduct numerical simulation of liquid sloshing and obtain the relationship between the impact force and impact moment and the stochastic acceleration; relevant parameters of the equivalent mechanical model are obtained through the mechanical analysis of liquid sloshing in the rectangular-section spray tank to build the MATLAB/Simulink system simulation model, then the reliability of the system with the filling ratio of 0.5 is verified through Fluent, and later the coupling with the constructed kinetic model of the high-clearance sprayer is conducted to analyze the influence control factors of liquid sloshing on the stability of the high-clearance sprayer.Figure 2
Simulation flowchart.


## 2.2. Adams Kinetic Model

To obtain the stochastic acceleration excitation suffered by the spray tank and dynamically to load it into the Fluent environment in real time, this paper builds a multibody kinetic simulation model of a high-clearance sprayer, converts the 3D high-clearance sprayer model equipped in the SolidWorks software into a Parasolid.xt format, and imports it into the Adams software. Besides, the model adds a rotating set to the four tires of the high-clearance sprayer separately and adds a rotating drive to the rear rotating sets separately to make the high-clearance sprayer move in a forward direction, and each component is set with parameters like material attributes, and so on [14]. The operating speed is 1 m/s, and the stochastic acceleration suffered by the spray tank in three directions is dynamically loaded into the Fluent model of liquid forced sloshing of the spray tank for the numerical simulation of the sloshing of liquid in the spray tank.


### 2.2.1. Tire Model

The tire is an important component of the high-clearance sprayer and is an important contact part between the high-clearance sprayer and the pavement. It bears the weight of the sprayer and the external acting force applied by the pavement to the sprayer, playing a role in force conduction between the sprayer and ground.The Adams software has multiple tire models, with different tire models that apply to diverse working environments. According to the characteristics and requirements of a working high-clearance in the field, this paper selects the Fiala tire model, and its parameters are shown in Table2.Table 2
Tire model parameters.
ParameterValueParameterValueTire mass20 kgWheel hub width0.17 mFree radius of the tire0.38 mVertical stiffness of the tire190000 N • m−1Tire width0.21 mVertical damping coefficient of the tire50 N • s • m−1Flatness ratio of the tire0.55Static friction coefficient of the tire0.8Radius of the wheel hub0.2 mDynamic friction coefficient of the tire1.1


### 2.2.2. Pavement Model

The paper mainly researches the liquid sloshing caused by uneven field pavement and does not take into account the influence of the pavement gradient on liquid sloshing; the paper establishes the requirements that meet Adams simulation pavement excitation and finally generates a GradeE three-dimensional pavement surface spectrum file of 40 m long and 20 m wide in MATLAB software according to the three-dimensional pavement node generation algorithm and based on the pavement unevenness expression and pavement unevenness classification standard established according to the harmony superposition principle, and the pavement file is shown in Figure 3.Figure 3
Grade e pavement.


## 2.2.1. Tire Model

The tire is an important component of the high-clearance sprayer and is an important contact part between the high-clearance sprayer and the pavement. It bears the weight of the sprayer and the external acting force applied by the pavement to the sprayer, playing a role in force conduction between the sprayer and ground.The Adams software has multiple tire models, with different tire models that apply to diverse working environments. According to the characteristics and requirements of a working high-clearance in the field, this paper selects the Fiala tire model, and its parameters are shown in Table2.Table 2
Tire model parameters.
ParameterValueParameterValueTire mass20 kgWheel hub width0.17 mFree radius of the tire0.38 mVertical stiffness of the tire190000 N • m−1Tire width0.21 mVertical damping coefficient of the tire50 N • s • m−1Flatness ratio of the tire0.55Static friction coefficient of the tire0.8Radius of the wheel hub0.2 mDynamic friction coefficient of the tire1.1


## 2.2.2. Pavement Model

The paper mainly researches the liquid sloshing caused by uneven field pavement and does not take into account the influence of the pavement gradient on liquid sloshing; the paper establishes the requirements that meet Adams simulation pavement excitation and finally generates a GradeE three-dimensional pavement surface spectrum file of 40 m long and 20 m wide in MATLAB software according to the three-dimensional pavement node generation algorithm and based on the pavement unevenness expression and pavement unevenness classification standard established according to the harmony superposition principle, and the pavement file is shown in Figure 3.Figure 3
Grade e pavement.


## 2.3. Fluent Numerical Simulation

Two states occur during the pesticide application process of the high-clearance sprayer, namely, the gas phase and liquid phase in the spray tank at all times, as shown in Figure4. During the impact process, the total volume of the gas phase and liquid phase remains constant, and only the shape and position change.Figure 4
Schematic diagram of the two-phase flow in the spray tank.Therefore, the liquid impact on the spray tank is a multiphase flow problem and can be analyzed through simulation by utilizing the VOF model. Fluent can be used for single-phase, multiphase, and complicated problems in the numerical model, for example, multiphase flow problem, laminar flow problem, and turbulence problem. Table3 shows the setting of the key parameters.Table 3
Fluent simulation setting.
ParameterOptionParameterOptionCalculation pattern3D double precisionUnit conditionApply stochastic acceleration on the spray tank in three directionsSolverPressure solverSpeed and pressure couplingPISOSolving modelVOF modelUnit gradient interpolation methodGreen-gauss cell-basedViscosity modelStandardk-ε turbulence modelDispersion scheme of the pressure modification coefficientBody force weightedFlow modelTransient flowMonitoring parameterImpact force on the sidewall (the curved surface) of the spray tank; impact moment on the bottom center of the spray tankDispersion scheme of convection termsFirst-order upwind schemeInitializationMixed initialization; path liquid phaseMaterialWater (main phase); air (second phase)Timestep0.01 sDuring the numerical simulation, Fluent detects the impact force of liquid sloshing to the wall surface of the spray tank and the moment of liquid sloshing to the bottom center of the spray tank.


## 2.4. Analysis of Simulation Results

The paper prepares the GradeE stochastic pavement model required by the sprayer operating conditions through MATLAB software, then generates the stochastic pavement features suitable for Adams software simulation, and obtains the stochastic acceleration excitation output of the spray tank with Adams software simulation as the input of the Fluent software of liquid sloshing and applies C language to compile and expand the program codes of Fluent, which are then dynamically loaded to Fluent environment to establish the forced sloshing model of liquid in the spray tank of a high-clearance sprayer, when the filling ratio in numerical simulation is k = 0.5; the simulation results of the additional force and moment generated by liquid sloshing to the sidewall of the spray tank are shown in Figures 5 and 6.Figure 5
Impact force on the spray tank wall surface inZ direction.Figure 6
Central moment at the spray tank bottom surface inZ direction.As can be seen from Figures5 and 6, under the influence of the external stochastic acceleration, the additional force and moment generated by liquid sloshing in the spray tank gradually become larger, then tend to be stable, and finally increase, where the maximum impact force and moment suffered by the spray tank wall surface are 19.6 N and 0.6 N • m, respectively, and the average impact force and moment are 0.47 N and 0.04 N • m, respectively.To research the relationship between the stochastic acceleration excitation and the impact force and moment suffered by the wall surface of the spray tank, we apply a stochastic acceleration to the fluid region, taking the stochastic acceleration suffered by the spray tank in theZ direction as an example, detect the impact force and moment suffered by the wall surface of the spray tank, and obtain the relationship between the stochastic acceleration and the impact force and moment suffered by the wall surface in the Z direction, as shown in Figures 7 and 8.Figure 7
Relationship between stochastic acceleration and impact force.Figure 8
Relationship between stochastic acceleration and impact moment.As can be seen from Figures7 and 8, the variation trend of the stochastic acceleration applied to the spray tank in the Z direction mirrors that of the impact force and moment suffered by the wall surface in the Z direction, where the impact force and moment have hysteresis phenomena, which are consistent with real-life phenomena. The results show that the impact force and moment suffered by the wall surface of the spray tank are affected by the stochastic acceleration; namely, when the acceleration increases gradually, the impact force and moment suffered by the wall surface will also increase, and the variation trend is the same, where the average absolute value of the acceleration is 0.06 m/s2 and the maximum value is 1.5 m/s2.


## 3. Building a Parallel Liquid Sloshing Mechanical Model

To facilitate the research, liquid sloshing problems in engineering have always been designed to parallel mechanical models. This paper designs a spray tank filling system that mirrors the pendulum-mass-damping system as shown in Figure6 and adopts the rigid system kinetics to research the response of such system under the action of external excitations.The building of the equivalent mechanical model of the spray tank has to meet the following 4 conditions:(1)
The equivalent model must have the same mass as the liquid in the spray tank.(2)
The equivalent model must have the same sloshing frequency and damping force as the liquid in the spray tank.(3)
The center of mass of the liquid in the spray tank remains consistent with the center of mass of the system in the equivalent model.(4)
After the application of lateral excitation, the equivalent model and the liquid in the spray tank generate the same acting force and moment to the tank.According to the equivalent mechanical model shown in Figure9, we establish the pendulum model kinetic equation of liquid sloshing as shown in the following:(1)mklkθ¨+cklkθ˙+mkgθ=mka.Figure 9
Schematic diagram of the liquid sloshing equipment model.The acting force of the lateral sloshing of the pendulum model to the spray tank is(2)F=mklkθ¨−mla.The roll sloshing moment of liquid sloshing on the bottom center of the spray tank center is(3)M=−hkmklkθ¨−mla+h0m0a+mklkgθ.The physical meanings of parameters in the above-mentioned equivalent pendulum model are as shown in Table4.Table 4
Physical meanings of parameters in the liquid sloshing pendulum model.
ParameterPhysical meaning of parameterParameterPhysical meaning of parametermkEquivalent pendulum mass of the liquid sloshingm0Fixed mass of the liquid sloshingmlTotal mass of the liquid in the spray tankθPendulum angle of the equivalent pendulum of liquid sloshingckDamping coefficient of the equivalent modellkLength of the equivalent pendulumhkHeight of the center of mass of the pendulum from the bottom of the spray tankh0Height of the center of mass of fixed mass from the bottom of the spray tankLWidth of the spray tankhbHeight of the liquid in the spray tankBLength of the spray tankhWave height of the liquid sloshinghlHeight of the center of mass of liquid to the bottom of the spray tankaLateral acceleration of the spray tankThe specific calculation formulas for parameters in the equivalent model of liquid sloshing are shown as follows:(4)ml=ρhblb,(5)mk=mlb3.87hbtanh3.14hbb,(6)m0=ml-mk,(7)hk=1-mk2mlhb,(8)h0=m02mlhb.The pendulum mode of liquid sloshing built in this paper can be approximately deemed as at the center point of the spray tank; namely, all rotations are deemed as near such point. Therefore, the distance between the center of the spray tank and the center of mass of the pendulum can be approximately deemed as the equivalent pendulum length 1k.(9)lk=gwij2.The data in Table5 are derived according to formulas (1)–(10) and used in the following calculations [15]:(10)wij2=πi2b2+j2l2tanhπhbi2b2+j2l2g,where i and j are, respectively, the number of half-waves in two orthogonal directions within the cross section.Table 5
Free sloshing frequency of liquid in the rectangular spray tank.
Frequency order of liquid sloshingValue (s−1)w000w101.36w010.8702w112.23w205.439w023.481w124.841w216.309w228.92The damping ratio is calculated with the logarithmic decrement method according to [16–18](11)ξ=12πnlnh1hn+1,where h1 and hn+1, respectively, represent the wave heights of the 1st and the n+1th vibration cycles.The damping of liquid sloshing in the rigid spray tank is small, and the influence of the shape of the spray tank on the damping is not sensitive; therefore, the approximate calculation formula of the damping of small damping system is(12)ck=2πξ1−ξ2.Table6 shows the parameters of the equivalent pendulum model of liquid sloshing in the wide spray tank calculated from the above formulas.Table 6
Parameters of equivalent pendulum model.
ParameterValuemk0.3651 kgml10 kgck0.4063hk0.04909 ml0.4 mb0.5 mhl−0.02494 mm09.635 kglk1.97068 mh00.0240875 mhb0.05 mh0.045 mThe above calculations are parameters of the equivalent pendulum model with the spray tank 0.5 m long∗ 0.4 m wide, and according to the similarity principle of fluid mechanics, the relevant parameters in the above pendulum model have to be magnified accordingly to obtain parameters of the spray tank of 2 m long ∗ 1 m wide in this paper, as shown in Table 7.Table 7
Relevant parameters of the pendulum model of the 2 m long∗ 1 m wide spray tank.
ParameterValuemk0.54765 kgck0.60945hk0.073635 ml2 mb1 mlk2.95602 m


## 4. Verification of Equivalent Mechanical Model

We designed an equivalent model in MATLAB/Simulink based on the parameters obtained from the above calculations to calculate the impact force and moment of liquid sloshing and verify the accuracy of the equivalent pendulum model by comparing the calculation results obtained from Fluent numerical simulation and the equivalent model of liquid sloshing in the spray tank. The comparison results are as shown in Figures10 and 11. The maximum impact force and average to the spray tank solved from the equivalent mechanical model are 19.6 N and 0.47 N, respectively, and the maximum moment and average moment on the bottom center of the spray tank are 0.6 N/m and 0.04 N/m, respectively.Figure 10
Impact force of liquid sloshing.Figure 11
Liquid sloshing moment.As can be seen from Figures10 and 11, under the excitation of the stochastic acceleration, the impact force of liquid sloshing oscillates constantly with time and the variation trend is the same as that of the external acceleration.The amplitude of oscillation of the impact force increases gradually, with the moment first increasing and then decreasing. The simulation results of the equivalent model of liquid sloshing built with MATLAB/Simulink are consistent with the numerical simulation results from using Fluent, where the maximum error in impact force is 2.2% and the maximum error at the moment is 15%. Thus, the built equivalent model can better describe the sloshing behavior of liquid in the spray tank. Therefore, the research in this paper will provide references for the modeling of the chassis system kinetic model and the stability control strategies of the high-clearance sprayer.


## 5. Building the High-Clearance Sprayer Kinetic Model

The high-clearance sprayer model was designed based on the equivalent mechanical model of the pendulum mass damping of liquid sloshing built in the above section in combination with the overall high-clearance sprayer rigid model [19, 20], as shown in Figure 12. When researching the roll and driving stability of a high-clearance sprayer on a level road, we neglected the pitching and vertical motion and assumed the steering angle of the front wheel to be small; namely, the lateral force of the tire can be expressed by the linear tire slip model.Figure 12
Roll model of the high-clearance sprayer.


### 5.1. Kinetic Model

Lateral movement of the high-clearance sprayer is as follows:(13)Mudβdt+w−Mlhld2φδt2−hk+lk+Z0+Z1mkd2φδt2−mklk=∑i=14Fyi.Yawing movement of the high-clearance sprayer is as follows:(14)Izdwdt=Fy1+Fy2b1−Fy3+Fy4b2.Roll movement of the high-clearance sprayer is as follows:(15)Ixd2φ∂t2−Mlhl+hk+lk+Z0+Z1mkudβdt+w−hk+lk+Z0+Z1lkd2cosθdt2=−Df+Drdφdt−Cφf+Cφrφ+Mlhlgφ.


### 5.2. Tire Model

Assume the steering angle of the front wheel of the high-clearance sprayer (δ, rad) and use the linear tire slip model; the slip angles of tires from the first axle to the fourth axle can be separately described as [21](16)a1=β1+ω1ua1−δ,a2=β1−ω1ub1,a3=β1−ω1ub2,ai=β1−lp+a2+bi−1ω1u−a2−bi−1dθ/dtu−θ,i=4,5,6.The lateral force of each tire can be separately expressed as(17)Fy1=Fy2=Cyfa1,Fy3=Fy4=Cyra2.Meanings and values of parameters in the model system are shown in Table8.Table 8
System parameters.
ParameterMeaningValueMTotal mass of the high-clearance sprayer, including the liquid (kg)1546MlSprung mass of the high-clearance sprayer with no load (kg)976hlDistance from the center of the sprung mass of the high-clearance sprayer with no load to the roll axis (m)1φRoll angle of the sprung mass of the high-clearance sprayer (rad)IzYaw rotational inertia of the high-clearance sprayer including liquid (kg • m2)IXRoll rotational inertia of the high-clearance sprayer including liquid (kg • m2)wYaw angular velocity of the high-clearance sprayer (rad • s−1)βSlip angle of the center of mass of the high-clearance sprayer (rad)b1Distance from the center of mass of the high-clearance sprayer to the front axle (m)1.6b2Distance from the center of mass of the high-clearance sprayer to the rear axle (m)1.4DfRoll damping of the front suspension of the high-clearance sprayer (N • s • m • rad−1)400DrRoll damping of the rear suspension of the high-clearance sprayer (N • s • m • rad−1)400CφfRoll angle stiffness of the front suspension of the high-clearance sprayer (N • s • m • rad−1)4000CφrRoll angle stiffness of the rear suspension of the high-clearance sprayer (N • s • m • rad−1)4000CyfLateral stiffness of the front tire of the high-clearance sprayer (KN • rad−1)15CyrLateral stiffness of the rear tire of the high-clearance sprayer (KN • rad−1)25FyiLateral force of the tires of the high-clearance sprayer (N)uDriving the speed of the high-clearance sprayer (m • s)Z0Distance between the roll axes at the bottom of the spray tank (m)0.8Note. The parameters M, Iz,, and IX are related to the filling ratio k, and the definitions of other parameters are consistent with that of the equivalent mechanical model system of the pendulum mass damping.


### 5.3. Simulation Test and Analysis

#### 5.3.1. Influence of the Driving Speed on the Stability of the Sprayer

Considering that it is dangerous to test drive at high speed, this paper selects the simulation test method for analysis. The filling ratiok is defined as the ratio between the liquid depth hband the spray tank heighth, namely, k = hb/H. To research and analyze the influence of the driving speed on the stability of a high-clearance sprayer with a certain filling ratio, we set the filling ratio k as 0.5 and the driving speed u as 1m/s, 2 m/s, and 3 m/s, respectively, during the simulation test. The simulation analysis results are as shown in Figure 13, and the input external stochastic acceleration is as shown in Figure 14.Figure 13
Simulation results of different speeds. (a) Slip angle of the center of mass of the high-clearance sprayer; (b) yaw angular velocity of the sprayer; (c) roll angle of sprung mass of the sprayer.
(a)(b)(c)Figure 14
Variation curve of stochastic acceleration.As can be seen from Figure13, in the case of the filling ratio k = 0.5, stochastic acceleration, and different speed conditions, the liquid sloshing has a small influence on the roll angle of the high-clearance sprayer but has a big influence on the slip angle of the center of mass and the yaw angular velocity. This is attributable to the liquid flowing outwards under the action of centrifugal force and increasing the lateral displacement of the center of mass of the liquid. On the other hand, the dynamic pressure produced by liquid sloshing due to the increased speed also increases the slip angle of the center of mass and the yaw angular velocity. When the driving speed is increased to 3 m/s from 1 m/s, the slip angle of the center of mass increases by 46% and 40%, respectively, the yaw angular velocity increases by 20% and 13%, respectively, and the roll angle of the sprung mass increases by 8.5% and 2.6%, respectively. As can be seen from the above data, when the driving speed of the high-clearance sprayer is increased to 3 m/s from 2 m/s, the liquid sloshing is violent; as a result, we can conclude that it is safe to maintain the driving speed of the high-clearance sprayer between 1 m/s and 2 m/s when working in the field.


#### 5.3.2. Influence of the Filling Ratio on the Stability of Sprayer

To research and analyze the influence of different filling ratios on the stability of high-clearance sprayers, we setu = 2 m/s and set a as the stochastic acceleration in the simulation test. Figure 15 shows the simulation results.Figure 15
Simulation results of different filling ratio. (a) Variation curve of the slip angle of the center of mass. (b) Variation curve of yaw angular velocity. (c) Variation curve of mass roll angle.
(a)(b)(c)As can be seen from Figure15, when u = 2 m/s and at stochastic acceleration, the filling ratio greatly influences the roll angle, the slip angle of the center of mass, and the yaw angular velocity of the high-clearance sprayer. This is because, with the increase of the filling ratio, the liquid sloshing produces additional moment and acting force in the spray tank, increasing the roll angle, slip angle of the center of mass, and yaw angular velocity of the high-clearance sprayer. When the filling ratio increases to 0.95 from 0.75, the roll angle separately increases by 32% and 75%, the slip angle of the center of mass separately increases by 54% and 67%, and the yaw angular velocity separately increases by 43% and 69%. As can be seen from the above data, when the filling ratio of the high-clearance sprayer increases to 0.95 from 0.85, the liquid sloshing is violent. Consequently, we can conclude that it is safe to maintain the filling ratio of liquid in the spray tank at about 0.85 when the high-clearance sprayer is working.As can be seen from Figure16, when the driving speed is 2 m/s and a is the stochastic acceleration, the liquid sloshing hugely influences the lateral acceleration of the high-clearance sprayer, and with the increase of the filling ratio, the lateral acceleration will increase randomly. When the filling ratio increases to 0.95 from 0.75, the lateral acceleration will increase by 12.4 and 26.4, respectively. u, as depicted in the above data, improves the operating efficiency and the driving stability of the high-clearance sprayer while working, and the best filling ratio is 0.85.Figure 16
Lateral acceleration curves of the high-clearance sprayer.


## 5.1. Kinetic Model

Lateral movement of the high-clearance sprayer is as follows:(13)Mudβdt+w−Mlhld2φδt2−hk+lk+Z0+Z1mkd2φδt2−mklk=∑i=14Fyi.Yawing movement of the high-clearance sprayer is as follows:(14)Izdwdt=Fy1+Fy2b1−Fy3+Fy4b2.Roll movement of the high-clearance sprayer is as follows:(15)Ixd2φ∂t2−Mlhl+hk+lk+Z0+Z1mkudβdt+w−hk+lk+Z0+Z1lkd2cosθdt2=−Df+Drdφdt−Cφf+Cφrφ+Mlhlgφ.


## 5.2. Tire Model

Assume the steering angle of the front wheel of the high-clearance sprayer (δ, rad) and use the linear tire slip model; the slip angles of tires from the first axle to the fourth axle can be separately described as [21](16)a1=β1+ω1ua1−δ,a2=β1−ω1ub1,a3=β1−ω1ub2,ai=β1−lp+a2+bi−1ω1u−a2−bi−1dθ/dtu−θ,i=4,5,6.The lateral force of each tire can be separately expressed as(17)Fy1=Fy2=Cyfa1,Fy3=Fy4=Cyra2.Meanings and values of parameters in the model system are shown in Table8.Table 8
System parameters.
ParameterMeaningValueMTotal mass of the high-clearance sprayer, including the liquid (kg)1546MlSprung mass of the high-clearance sprayer with no load (kg)976hlDistance from the center of the sprung mass of the high-clearance sprayer with no load to the roll axis (m)1φRoll angle of the sprung mass of the high-clearance sprayer (rad)IzYaw rotational inertia of the high-clearance sprayer including liquid (kg • m2)IXRoll rotational inertia of the high-clearance sprayer including liquid (kg • m2)wYaw angular velocity of the high-clearance sprayer (rad • s−1)βSlip angle of the center of mass of the high-clearance sprayer (rad)b1Distance from the center of mass of the high-clearance sprayer to the front axle (m)1.6b2Distance from the center of mass of the high-clearance sprayer to the rear axle (m)1.4DfRoll damping of the front suspension of the high-clearance sprayer (N • s • m • rad−1)400DrRoll damping of the rear suspension of the high-clearance sprayer (N • s • m • rad−1)400CφfRoll angle stiffness of the front suspension of the high-clearance sprayer (N • s • m • rad−1)4000CφrRoll angle stiffness of the rear suspension of the high-clearance sprayer (N • s • m • rad−1)4000CyfLateral stiffness of the front tire of the high-clearance sprayer (KN • rad−1)15CyrLateral stiffness of the rear tire of the high-clearance sprayer (KN • rad−1)25FyiLateral force of the tires of the high-clearance sprayer (N)uDriving the speed of the high-clearance sprayer (m • s)Z0Distance between the roll axes at the bottom of the spray tank (m)0.8Note. The parameters M, Iz,, and IX are related to the filling ratio k, and the definitions of other parameters are consistent with that of the equivalent mechanical model system of the pendulum mass damping.


## 5.3. Simulation Test and Analysis

### 5.3.1. Influence of the Driving Speed on the Stability of the Sprayer

Considering that it is dangerous to test drive at high speed, this paper selects the simulation test method for analysis. The filling ratiok is defined as the ratio between the liquid depth hband the spray tank heighth, namely, k = hb/H. To research and analyze the influence of the driving speed on the stability of a high-clearance sprayer with a certain filling ratio, we set the filling ratio k as 0.5 and the driving speed u as 1m/s, 2 m/s, and 3 m/s, respectively, during the simulation test. The simulation analysis results are as shown in Figure 13, and the input external stochastic acceleration is as shown in Figure 14.Figure 13
Simulation results of different speeds. (a) Slip angle of the center of mass of the high-clearance sprayer; (b) yaw angular velocity of the sprayer; (c) roll angle of sprung mass of the sprayer.
(a)(b)(c)Figure 14
Variation curve of stochastic acceleration.As can be seen from Figure13, in the case of the filling ratio k = 0.5, stochastic acceleration, and different speed conditions, the liquid sloshing has a small influence on the roll angle of the high-clearance sprayer but has a big influence on the slip angle of the center of mass and the yaw angular velocity. This is attributable to the liquid flowing outwards under the action of centrifugal force and increasing the lateral displacement of the center of mass of the liquid. On the other hand, the dynamic pressure produced by liquid sloshing due to the increased speed also increases the slip angle of the center of mass and the yaw angular velocity. When the driving speed is increased to 3 m/s from 1 m/s, the slip angle of the center of mass increases by 46% and 40%, respectively, the yaw angular velocity increases by 20% and 13%, respectively, and the roll angle of the sprung mass increases by 8.5% and 2.6%, respectively. As can be seen from the above data, when the driving speed of the high-clearance sprayer is increased to 3 m/s from 2 m/s, the liquid sloshing is violent; as a result, we can conclude that it is safe to maintain the driving speed of the high-clearance sprayer between 1 m/s and 2 m/s when working in the field.


### 5.3.2. Influence of the Filling Ratio on the Stability of Sprayer

To research and analyze the influence of different filling ratios on the stability of high-clearance sprayers, we setu = 2 m/s and set a as the stochastic acceleration in the simulation test. Figure 15 shows the simulation results.Figure 15
Simulation results of different filling ratio. (a) Variation curve of the slip angle of the center of mass. (b) Variation curve of yaw angular velocity. (c) Variation curve of mass roll angle.
(a)(b)(c)As can be seen from Figure15, when u = 2 m/s and at stochastic acceleration, the filling ratio greatly influences the roll angle, the slip angle of the center of mass, and the yaw angular velocity of the high-clearance sprayer. This is because, with the increase of the filling ratio, the liquid sloshing produces additional moment and acting force in the spray tank, increasing the roll angle, slip angle of the center of mass, and yaw angular velocity of the high-clearance sprayer. When the filling ratio increases to 0.95 from 0.75, the roll angle separately increases by 32% and 75%, the slip angle of the center of mass separately increases by 54% and 67%, and the yaw angular velocity separately increases by 43% and 69%. As can be seen from the above data, when the filling ratio of the high-clearance sprayer increases to 0.95 from 0.85, the liquid sloshing is violent. Consequently, we can conclude that it is safe to maintain the filling ratio of liquid in the spray tank at about 0.85 when the high-clearance sprayer is working.As can be seen from Figure16, when the driving speed is 2 m/s and a is the stochastic acceleration, the liquid sloshing hugely influences the lateral acceleration of the high-clearance sprayer, and with the increase of the filling ratio, the lateral acceleration will increase randomly. When the filling ratio increases to 0.95 from 0.75, the lateral acceleration will increase by 12.4 and 26.4, respectively. u, as depicted in the above data, improves the operating efficiency and the driving stability of the high-clearance sprayer while working, and the best filling ratio is 0.85.Figure 16
Lateral acceleration curves of the high-clearance sprayer.


## 5.3.1. Influence of the Driving Speed on the Stability of the Sprayer

Considering that it is dangerous to test drive at high speed, this paper selects the simulation test method for analysis. The filling ratiok is defined as the ratio between the liquid depth hband the spray tank heighth, namely, k = hb/H. To research and analyze the influence of the driving speed on the stability of a high-clearance sprayer with a certain filling ratio, we set the filling ratio k as 0.5 and the driving speed u as 1m/s, 2 m/s, and 3 m/s, respectively, during the simulation test. The simulation analysis results are as shown in Figure 13, and the input external stochastic acceleration is as shown in Figure 14.Figure 13
Simulation results of different speeds. (a) Slip angle of the center of mass of the high-clearance sprayer; (b) yaw angular velocity of the sprayer; (c) roll angle of sprung mass of the sprayer.
(a)(b)(c)Figure 14
Variation curve of stochastic acceleration.As can be seen from Figure13, in the case of the filling ratio k = 0.5, stochastic acceleration, and different speed conditions, the liquid sloshing has a small influence on the roll angle of the high-clearance sprayer but has a big influence on the slip angle of the center of mass and the yaw angular velocity. This is attributable to the liquid flowing outwards under the action of centrifugal force and increasing the lateral displacement of the center of mass of the liquid. On the other hand, the dynamic pressure produced by liquid sloshing due to the increased speed also increases the slip angle of the center of mass and the yaw angular velocity. When the driving speed is increased to 3 m/s from 1 m/s, the slip angle of the center of mass increases by 46% and 40%, respectively, the yaw angular velocity increases by 20% and 13%, respectively, and the roll angle of the sprung mass increases by 8.5% and 2.6%, respectively. As can be seen from the above data, when the driving speed of the high-clearance sprayer is increased to 3 m/s from 2 m/s, the liquid sloshing is violent; as a result, we can conclude that it is safe to maintain the driving speed of the high-clearance sprayer between 1 m/s and 2 m/s when working in the field.


## 5.3.2. Influence of the Filling Ratio on the Stability of Sprayer

To research and analyze the influence of different filling ratios on the stability of high-clearance sprayers, we setu = 2 m/s and set a as the stochastic acceleration in the simulation test. Figure 15 shows the simulation results.Figure 15
Simulation results of different filling ratio. (a) Variation curve of the slip angle of the center of mass. (b) Variation curve of yaw angular velocity. (c) Variation curve of mass roll angle.
(a)(b)(c)As can be seen from Figure15, when u = 2 m/s and at stochastic acceleration, the filling ratio greatly influences the roll angle, the slip angle of the center of mass, and the yaw angular velocity of the high-clearance sprayer. This is because, with the increase of the filling ratio, the liquid sloshing produces additional moment and acting force in the spray tank, increasing the roll angle, slip angle of the center of mass, and yaw angular velocity of the high-clearance sprayer. When the filling ratio increases to 0.95 from 0.75, the roll angle separately increases by 32% and 75%, the slip angle of the center of mass separately increases by 54% and 67%, and the yaw angular velocity separately increases by 43% and 69%. As can be seen from the above data, when the filling ratio of the high-clearance sprayer increases to 0.95 from 0.85, the liquid sloshing is violent. Consequently, we can conclude that it is safe to maintain the filling ratio of liquid in the spray tank at about 0.85 when the high-clearance sprayer is working.As can be seen from Figure16, when the driving speed is 2 m/s and a is the stochastic acceleration, the liquid sloshing hugely influences the lateral acceleration of the high-clearance sprayer, and with the increase of the filling ratio, the lateral acceleration will increase randomly. When the filling ratio increases to 0.95 from 0.75, the lateral acceleration will increase by 12.4 and 26.4, respectively. u, as depicted in the above data, improves the operating efficiency and the driving stability of the high-clearance sprayer while working, and the best filling ratio is 0.85.Figure 16
Lateral acceleration curves of the high-clearance sprayer.


## 6. Conclusions

(1)
By researching the liquid sloshing in the spray tank, we were able to design a kinetic model of the high-clearance sprayer and simulate two working conditions, namely, a particular filling ratio and different driving speeds. Also, certain driving speeds, different filling ratios, and simulation results show that liquid sloshing has a different influence on the driving parameters of a high-clearance sprayer under these two working conditions. Under the working condition of certain filling ratio and different driving speeds, liquid sloshing has a small influence on the roll angle of the high-clearance sprayer, while under the working condition of certain driving speeds and different filling ratios, liquid sloshing has a big influence on the driving parameters of the high-clearance sprayer.(2)
When the driving speed is increased to 3 m/s from 1 m/s, the liquid flows outwards under the action of centrifugal force and increases the lateral displacement of the center of mass of liquid. Meanwhile, the dynamic pressure produced by liquid sloshing due to the increased speed also increases the slip angle of the center of mass and the yaw angular velocity. When the driving speed is increased to 3 m/s from 2 m/s, the liquid sloshing becomes violent, so we can conclude from the analysis that it is appropriate to maintain the driving speed between 1 m/s and 2 m/s when the sprayer is working.(3)
When the filling ratio increases to 0.95 from 0.75, the liquid mass in the spray tank increases, and the sloshing force and inertia force produced by liquid jointly act on the sprayer to aggravate the yawing motion and lateral acceleration. When the filling ratio increases to 0.95 from 0.85, the liquid mass participating in sloshing gradually increases, and the produced sloshing force has a big influence on the yawing motion and lateral acceleration of the sprayer. Therefore, considering the working efficiency of a high-clearance sprayer in the field, when the filling ratio K is about 0.85, the liquid sloshing in the spray tank has a small influence on the driving stability of high-clearance sprayer, so such filling ratio is a preferential filling ratio for high-clearance sprayers in operation.



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*Source: 9926962-2021-11-12.xml*