To investigate the driving stability of tank trucks, an equivalent trammel pendulum was utilized to approximately demonstrate the dynamic characteristics of liquid sloshing in a partially filled tank. The oscillation movement of the trammel pendulum in the tank was described under the tankfixed coordinate system and its motion equation under the noninertia coordinate system was derived using a Lagrangian function. The motion of the pendulum that expresses the fluid cargo dynamic behavior and that of the solid truck was coupled with each other by the tank. Therefore, a tank truck dynamic model was established using Newton’s first law and the angular momentum. A typical tank truck was selected and used to study its driving stability under steering angle step test. The study on tankers driving stability is of great importance for evaluating tankers driving safety, investing the main impact factor aspecting tankers driving stability, and developing active/passive roll control systems for them.
About 80% of global chemical and petroleum products are delivered by road tank vehicles. The transportation freight has already reached 4 billion tons per year. In America, tankers make up more than 55% of all freight trucks. Tankers are hugely convenient for fluid material exchange and have a positive effect on boosting the national economic development. However, they also create severe traffic safety problems which would result in huge people injury and property damage. Statistical data collected by Statistique Canada has shown that 83% of lorry rollover accidents on highways are caused by tank vehicles [
Much works have been carried out on the characteristics of tanker accidents, in an attempt to investigate the primary accident type. It was found by Treichel et al. that rollover is the most frequent accident type for tankers [
To explore the influence of transient liquid sloshing on tanker roll stability, many research works have been explored and some important conclusions have been drawn. Considering the complexity of tank vehicle dynamic analysis—as tank vehicles are fluidsolid coupling multibody system—some important simplifications have to be made before the analysis. Strandberg et al. assumed that liquid free surface is a tilted straight line whose gradient is a function of vehicle’s lateral acceleration and tank’s roll angle [
To investigate the accuracy of the QS method, some researchers studied the relationship between the evaluated liquid sloshing effect obtained by calculating the movement of the CG of the liquid bulk and the observed effect. They have found that the mean values of sloshing forces and the coordinates of the CG of the liquid bulk are quite close between the estimated and observed cases [
However, the QS method cannot reflect the dynamic characteristics of liquid sloshing. Therefore, an equivalent mechanical model which can precisely demonstrate the dynamic characteristic of liquid sloshing in a partially filled tank must be modeled first [
Therefore, the purpose of this paper is to establish a tank vehicle dynamic model based on the equivalent mechanical model for transient liquid sloshing in a partially filled tank. The trammel pendulum was used to describe liquid sloshing, its motion was analyzed, and the motion equation was derived under the tankfixed coordinate. Then, the coupling condition for the fluid cargo and the solid vehicle was studied, and a tanker dynamic model was established based on this. Finally, MATLAB was used to simulate the roll stability of a typical tank truck under steering angle step test, and some important conclusions were drawn. The research achievement is of great significance for commanding and evaluating tank trucks roll stability performance, judging the conditions under which a tanker will lose its driving stability and designing roll stability control systems for tankers to improve their driving safety.
Theoretical analysis and experimental studies had revealed that the firstorder sloshing mode, which can be described by the oscillation of liquidfree surface, is the most important mode of liquid sloshing in partially filled tanks [
Schematic diagram for the motion of the trammel pendulum.
In Figure
The Lagrangian function was used to derive the kinetic equation of the trammel pendulum system. The motion equation of the trammel pendulum that oscillates under tank fixed inertia coordinate system whose origin locates at the center of the tank (as shown in Figure
Analytical diagram for the motion of the trammel pendulum under an inertial coordinate system.
The pendulum parameters in (
The detailed derivation process for (
According to Figure
From (
As the mass of liquid bulk can easily be acquired while the tank’s size and liquid density are given, the pendulum mass and the fixed liquid mass can be obtained using (
During the driving process, the vehicle’s driving state changes regularly under the influence of a variety of factors. It leads to the movement of the tank, and in turn the coordinate system is being fixed relative to the tank. Therefore, motion equations for a trammel pendulum under a noninertial coordinate system (translation and rotation are included) should be derived in order to study the motion characteristics of liquid bulk when tankers are being driven on curved sections of road, avoiding obstacles or changing lanes.
The schematic diagram for the motion of the tank when vehicle turns right is presented in Figure
Analytical diagram for the motion of the trammel pendulum under the noninertial coordinate system.
With reference to Figure
The velocity of the pendulum mass ball can be obtained by solving the firstorder derivative of (
Similarly, the acceleration of the pendulum mass ball can be acquired by solving the differential of (
The zero of potential energy is defined as the
According to (
The motion of the trammel pendulum system can be expressed by
According to the motion analysis of the trammel pendulum,
The differential of (
The differential of (
Finally, the differential of (
Substituting (
The differential of (
The differential of (
According to (
Substituting (
Equations (
The roll stability of a tank truck is greatly influenced by transient liquid sloshing in partially filled tanks. Due to the flow characteristic of liquid bulk, transient liquid sloshing is produced when the vehicle’s driving state changes, which leads to the oscillation of the pendulum mass ball. To simplify the analysis of the tank truck’s roll stability, some assumptions are made and listed as follows.
The sprung mass is the vehicle mass that lies over the suspension system, in which the mass of the liquid cargo is not included. In this paper, vehicle and liquid bulk, as the solid and fluid parts respectively, are not mixed with each other.
In the transverse direction, the CG of the sprung mass is located at the bottom of the tank. Moreover, the sprung mass is symmetrical about the longitudinal axis of the vehicle.
Transient liquid sloshing is merely produced along the transverse direction; longitudinal liquid sloshing is not taken into consideration. The movement and mass distribution of the liquid bulk are completely the same at different tank cross sections, which means that the CG of liquid bulk is located in the middle of the tank along the longitudinal direction.
The liquidfree surface is not broken during the vehicle’s driving process.
Based on the assumptions given in Section
The coordinate system being fixed relative to the vehicle was chosen as the reference coordinate system for the dynamic analysis of the tank truck and the establishment of the vehicle’s dynamic model. The origin of the coordinate system is the point where the vertical line that goes through the CG of the vehicle when it is static intersects with the vehicle’s roll axis.
The coordinate system being fixed relative to the vehicle’s sprung mass was defined as the
The vehicle’s unsprung mass is assumed not to produce roll movement.
The top view and the back view of the vehicle’s dynamic analysis are plotted in Figures
Top view of vehicle dynamic analysis.
Back view of vehicle dynamic analysis.
Generally, the fixed liquid mass does not stay static but slides slowly with the aid of gravity while the tank tilts. However, the shift of the fixed liquid mass is quite small compared to that of the pendulum mass ball and its shift amount drops greatly with the increase in the liquid fill percentage. Therefore, the fixed liquid mass is assumed to be static and its position is assumed to stay constant as the tank rolls over. Based on this assumption, the fixed liquid mass can be seen as the solid part and its lateral acceleration is equal to that of the vehicle’s sprung mass.
According to the dynamic analysis of the vehicle, the lateral acceleration of the sprung mass and unsprung mass can be presented by
According to (
The force analysis in the roll plane is plotted in Figure
The schematic diagram for the pendulum rotating around the roll axis.
According to Figure
Therefore, the absolute roll rate and roll angular acceleration can be obtained as follows based on (
On the basis of (
In general, the angular momentum of a rotating object is defined as the product of a body’s rotational inertia tensor and rotational velocities about a particular coordinate.
According to the definition of angular momentum, that for the sprung mass and fixed liquid mass can be expressed as follows:
According to the assumptions given in Section
As the fixed liquid mass does not move while the tank tilts, the distribution of the fixed liquid mass is symmetrical about the
Therefore, based on (
The expanded form of (
The change in the vehicle fixed coordinate after it has been driving for time
The change of the
The firstorder derivatives of (
According to the moment balance between the vehicle inertia moment and the external moment, the vehicle’s roll and yaw moment balance equation can be expressed as follows:
Based on the parallel axis theorem of the moment of inertia, the following equation can be obtained:
The dynamic model of the tank truck is composed of (
The tire cornering force in the dynamic model can be obtained using the magic formula for tires, which is given by
The tire sideslip angle in (
For the convenience of vehicle dynamic simulation, the tank truck dynamic model is translated into the following form:
According to the Cramer rule, the variables in (
Based on (
Before solving the differential equations of the vehicle’s dynamic model, the inertia tensor for the fixed liquid mass and the pendulum mass are needed.
It is known that the liquidfree surface can be approximately described by a tilted straight line when transient liquid sloshing occurs. This means that the inertia tensor for the liquid bulk can be obtained. However, the obtained values are the sum of the pendulum part and the fixed liquid part. As it is quite difficult to divide liquid bulk physically into the pendulum part and the fixed liquid part, the fixed liquid part is assumed to locate at the bottom of the liquid bulk (as shown in Figure
Schematic diagram for the division into the pendulum part and the fixed liquid part.
According to a market survey, the crosssectional area of cylindrical tanks is usually close to 2.4 m^{2}.
The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 0.2.







 

0°  1831  18574  0  597  0  215  4088  101 
10°  1926  18613  117  607  338  215  4088  101 
20°  2208  18711  230  637  684  215  4088  101 
30°  2670  18871  336  687  1044  215  4088  101 
40°  3298  19075  432  754  1420  215  4088  101 
50°  4071  19283  515  837  1806  215  4088  101 
60°  4967  19485  582  933  2192  215  4088  101 
70°  5959  19650  632  1039  2558  215  4088  101 
80°  7020  19770  662  1153  2883  215  4088  101 
90°  8101  19783  672  1268  3135  215  4088  101 
The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 0.5.







 

0°  10837  45002  0  2285  0  3998  35642  1212 
10°  11023  45002  228  2305  1063  3998  35642  1212 
20°  11575  45002  449  2364  2094  3998  35642  1212 
30°  12477  45002  656  2461  3062  3998  35642  1212 
40°  13702  45002  843  2592  3936  3998  35642  1212 
50°  15212  45002  1005  2754  4691  3998  35642  1212 
60°  16960  45002  1136  2941  5303  3998  35642  1212 
70°  18895  45002  1233  3149  5754  3998  35642  1212 
80°  20957  45002  1292  3370  6031  3998  35642  1212 
90°  23084  45002  1312  3598  6124  3998  35642  1212 
The inertia tensors of the pendulum mass and the fixed liquid mass when the liquid fill level is 0.8.







 

0°  16269  34190  0  2570  0  23308  104440  5008 
10°  16363  34150  117  2579  751  23308  104440  5008 
20°  16644  34040  230  2610  1461  23308  104440  5008 
30°  17107  33880  336  2659  2091  23308  104440  5008 
40°  17732  33680  432  2726  2611  23308  104440  5008 
50°  18506  33470  515  2809  2998  23308  104440  5008 
60°  19400  33270  582  2905  3240  23308  104440  5008 
70°  20387  33100  632  3010  3337  23308  104440  5008 
80°  21438  32980  662  3122  3298  23308  104440  5008 
90°  22539  32980  672  3241  3135  23308  104440  5008 
As the fixed liquid part is assumed to be static as the tank tilts, the inertia tensor of the fixed liquid mass is always constant.
Polynomials are fitted for the data points listed in Tables
A MATLAB simulation is carried out for the steering angle step test of the
Truck driving stability when liquid fill level is 0.5.
Curves of slip angle
Curves of yaw rate
Curves of roll angle
Curves of roll rate
Curves of pendulum amplitude
Curves of pendulum angular velocity
Driving stability of normal truck with 50% laden situation.
Curves of slip angle
Curves of yaw rate
Curves of roll angle
Curves of roll rate
It is quite obvious that the vehicle’s roll stability is greatly influenced by transient liquid sloshing in a partially filled tank. While the curves of the driving variables for the normal truck quickly return to the steady state, those for the tank truck fluctuate up and down for about 40 s before becoming steady, and the overshoot is quite large. However, the overshoot dampens quickly with the passage of time, and the damping frequency is close to the pendulum oscillation frequency. It can also be seen that the bigger the steering angle is, the more significant the effect on vehicle roll stability given by transient liquid sloshing is. As can be seen in Figure
When the steering angle is 0.05 rad, the pendulum’s amplitude changes from its original value of 4.71 rad (270 degrees) to 5.09 rad (291.7 degrees) and stays steady with the aid of lateral acceleration and gravity. The movement of the pendulum mass ball increases the vehicle’s rollover torque and results in a bigger roll angle for the tank truck (Figure
As the maximum roll rate of the vehicle is already quite large even though the steering angle is small, it seems that the vehicle may encounter rollover. However, as seen in Figure
From the comparison of Figures
To investigate the impact of the liquid fill level on the vehicle’s roll stability, tank vehicles with liquid fill levels of 0.2 and 0.8, with a steering angle of 0.02 rad, and with a driving speed of 90 km/h are studied in a steering angle step test. The simulation results for the roll angle and roll rate are plotted in Figures
Tank truck driving stability when liquid fill level is 0.2.
Curves of roll angle
Curves of roll rate
At the same time, the roll stability of normal trucks with the same steering angle, parameter values, load situations, and driving speed are also investigated to determine the influence of transient liquid sloshing in a partially filled tank on vehicle roll stability. The simulation results for the roll angle and roll rate for the normal trucks are presented in Figures
Driving stability of normal truck that is 20% laden.
Curves of roll angle
Curves of roll rate
Tank truck driving stability when liquid fill level is 0.8.
Curves of roll angle
Curves of roll rate
Driving stability of normal truck that is 80% laden.
Curves of roll angle
Curves of roll rate
As can be seen from Figure
Above all, the tank vehicle’s roll stability is greatly influenced when the liquid fill level is near to 0.5, in which case the sloshing liquid mass occupies a large proportion of the liquid bulk and the mass of the liquid bulk is quite large. In this situation, the tank vehicle’s roll stability is much worse than that of normal cars. However, when the liquid fill level is either fairly small or fairly large, the tank vehicle’s roll stability is only slightly influenced by transient liquid sloshing. Therefore, a liquid fill level of 0.4–0.6 is the worst load situation for tankers and should be avoided if possible.
To investigate the driving stability of tank trucks, the paper used an equivalent trammel pendulum for liquid sloshing in a partially filled tank. The trammel pendulum oscillated in the tank and was described in reference to tankfixed coordinates. To couple the motion of the trammel pendulum with that of the tank truck, the motion equation of the pendulum under a noninertial coordinate system was analyzed using a Lagrangian function. Based on this, a dynamic model for a tank truck was established using Newton’s first law and the angular momentum. A typical tank truck was selected and its driving stability is studied under a steering angle step test. Studying tankers’ driving stability and the main influencing factors is of great importance for evaluating tankers’ driving safety, as well as for developing active/passive roll control systems for them.
The following important discoveries were found from the driving stability simulations.
The curves of the tank truck driving variables fluctuate until they return to the steady state. The fluctuation frequency descends with an increase in the liquid fill level while holding the testing situation constant. The fluctuation frequency is close to the pendulum’s oscillation frequency. The overshoots of the driving variables depend on the pendulum mass, which is equal to the liquid mass that participates in transient sloshing.
The pendulum mass occupies a large proportion of the mass of liquid bulk when the liquid fill level is in the range of 0.3–0.7. Also, the mass of liquid bulk is quite large in such cases, the overshooting of the tanker driving variables is quite large, and the tanker’s driving stability is greatly impacted by transient liquid sloshing. Therefore, overshoot control for tankers’ driving variables must be considered to support their driving safety. Moreover, the situation of a liquid fill level near to 0.5 is the worst state in terms of load and should be avoided.
The driving stability of a tanker depends on the duration of the driving variables to a large degree.
Since we made the assumption in deriving the inertia tensors for the pendulum mass and the fixed liquid mass that the fixed mass is located at the bottom of the tank and is always static when the tank tilts, the inertia tensor value differs from the real situation, and the simulation results for the tank truck’s driving stability will also have been slightly influenced. Thus, a theory and method for physically distinguishing the pendulum’s mass and the fixed liquid mass from the liquid bulk will be proposed in a future study.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by Research on Evaluation and Detection Technology of Handling and Driving Stability for Commercial Vehicles (no. 2009BAG13A04) and the National Natural Science Foundation of China (Grant nos. 51375200 and 51208225).