A Simple Model for the Determination of Jackknifing

Jackknifing is uncontrolled braking of a truck-trailer combination which causes the trailer rotates relative to tractor and as a result extends to adjacent lanes, causing in many cases sever accidents. In this paper a simple two dimensional model is presented. The model includes parameters such as truck’s and trailer’s dimensions, coefficients of friction, masses, location of the load on the trailer etc. Based on this model simulation program, which allow the user to determine whether or not jackknifing will occur, was written. Simulations results for few cases are presented. Nomenclature L1 Distance between the tractor’s front axle and its GC L2 Distance between the tractor’s rear axle and its GC L3 Distance between the tractor’s rear axle and the trailer’s GC L4 Distance between the trailer’s GC and its rear axle Wc Tractor’s weight Wt Trailer’s weight (including load) w Tractor/trailer width b Lateral location of the trailer’s center of gravity μf Coefficient of friction on the tractor’s front tires μm Coefficient of friction on the tractor’s rear tires μr Coefficient of friction on the trailer’s rear tires μrr Coefficient of friction on the trailer’s right rear tires μrl Coefficient of friction on the trailer’s left rear tires It Mass moment of inertia of the trailer about the Z-axis F Horizontal force acting on the kingpin P Vertical force acting on the kingpin Nf Normal force on the tractor’s front axial Nm Normal force at the tractor’s rear axial Nr Normal force at the trailer’s rear axial Nrr Normal force at the trailer’s right rear wheels Nrl Normal force at the trailer’s left rear wheels Introduction At the present, the truck transport is one of the cornerstones of the economy of the United States. Trucking is vital to the US economy. The American Trucking Association states, “Trucks move most of America’s freight and are the transportation backbone of the American economy...the trucking industry accounts for 81% of the country’s freight revenue.” [1] This massive flow of products and materials can be easily interrupted, however, by a variety of potential accidents especially semi-trailer accidents that often occur as the truck and the trailer jackknife. Hence, in 2000, 457,000 large trucks that vehicles which weight rating are greater than 10,000 pounds; were involved in traffic crashes in the United States; 4,930 were involved in fatal crashes. A total of 5,211 people died, 12 percent of all the traffic fatalities reported in 2000, and an additional 140,000 were injured in those crashes. Such accidents can close highways for hours, to say nothing of the threat to the lives of truck drivers and car passengers. Jackknifing might cause a very severe accident due to the fact that the trailer, which rotates about the tractor, extends to adjacent lanes blocking the incoming or following traffic. A full understanding of this phenomenon is therefore critical for accident reconstruction and braking system design. Previous work on trailer/truck jackknifing include: Timothy V. Fossum and Gilbert N. Lewis presenting a differential equation [2] which is a model for the position of a trailer relatively to the cab which is pulling it. Chieh Chen and Masayoshi Tomizuka from the University of California, Berkeley [3] proposed a modeling approach designed to prevent jackknifing and furthermore reduce tracking errors of the trailer. Some work was done in the University of Windsor, Ontario, determining the equations of motion of tractor/trailer combination. The linearized equations were written in terms of the lateral speed of the truck, the yaw rate of the truck, and the sway angle of the trailer. Finally, the stability of the system had been evaluated by performing an eigen value analysis on the matrix yielded from the previous equations [4]. A general method of deriving dynamic models for any configuration of heavy duty vehicles, including vehicles with N units, is presented in appendix [5]. By the way, commercial heavy vehicle research at the California Partners for Advanced Transit and Highways (PATH) program has focused on the development of controllers for partially to fully autonomous driving environments. A multi-body dynamic model of a tractor semi-trailer has been developed using a commercially available dynamic analysis software program to predict any one of the many failure modes such as rollover or jackknifing that are common to articulated heavy vehicles [6]. A worstcase scenario generation algorithm able to find the input combination which is most detrimental for truck rollover or jackknife when the steering and braking inputs of a truck are limited below a certain threshold value, was also presented [7]. Many truck brake producers such as Radlinski & Associates, Inc working on the same topic also developed anti-jackknife devices [8]. The aim of those works has been to figure out a dynamical model able to determine whether the articulated truck would jackknife or not, which required the knowledge of a lot of very specific parameters. This paper presents a simple two dimensional model, which is an extension of a one dimensional stick model. The model accommodates variables, which could not be treated with one dimensional mode, and therefore, through simulations, the effect of these variables on the jackknifing phenomena can be studied. These variables include tractor/trailer lateral dimensions, location of the center of gravity of the trailer (in order to treat load position on the trailer), coefficient of friction for each on the tire and other. As a result, more realistic analysis and can be performed while maintaining the simplicity of the model. Two-dimensional model The model assumes no motion in the vertical direction (Z-direction). Thus, the sum of the forces in this direction and the sum of the moments about the Y-axis are zero. Using the dimensions of the tractor-trailer combination shown in Figure 1 and the vertical forces acting on the tractor and the trailer are shown in Figures 2 and 3 respectfully, the reactions at the wheels and the force at the king pin can be determined. Referring to Figure 2 the static equilibrium is given by: 0 = + − = ∑ P W N F t r z (1) 0 3 4 = − = ∑ PL L N M r y (2) Solving equations 1 and 2 yields the normal forces acting on the trailer’s tires and the vertical force on the king pin: ) ( 3 4 4 L L L W P t + = (3) ) ( 4 3 3 L L L W N t r + = Once Nr and P are known, Nm and Nf, the reaction forces on the tractor, can be easily found using the same methodology. Summation of the forces acting on the tractor in the Z-direction and the moments about the Y direction (refer to Figure 3): 0 = − − + = ∑ P W N N F c m f z (4) 0 1 2 2 = − − = ∑ L N PL L N M f m y (5) Solving equations 4 and 5 yields the forces acting on the front and rear tires of the tractor: