TH E515 This is to certify that the thesis entitled The Lateral Stability of the Michigan Double Tanker presented by Martin John Vanderploeg has been accepted towards fulfillment of the requirements for Masters degreein Mech. Engr. gfi/watw/ Major professor Date I/ / IL / L 0-7639 LIBRARY Michigan Stat: , .. " .p 2.; OVERDUE FINES: 2.“; _ _‘ '_ 25¢ per day per item t": 1;“.“3. j RETURNING LIBRARY MATERIALS: ,5? ,;§;.‘,”g' ‘ 1 Place in book return to remove r ‘N ’ ‘ 4 charge from circulation records THE LATERAL STABILITY OF THE MICHIGAN DOUBLE TANKER BY Martin John Vanderploeg A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1980 l “k. w __ J lief o” b -. ABSTRACT THE LATERAL STABILITY OF THE MICHIGAN DOUBLE TANKER By Martin John Vanderploeg This thesis uses linear mathematics to analyze the Michigan tanker. Several mathematical models are used, from a two degree of freedom model of the dolly-pup trailer combination to higher degree of freedom models of entire vehicles. I The results of the modeling and analysis show that the Michigan double tanker cannot be improved by simple design changes involving easily changed items such as tongue length, axle spread or tire stiffness. More basic changes are therefore analyzed, including the so-called Canadian double, the four point hitch, and a con- straint linkage between the semitrailer and the pup trailer. The Canadian double is shown to be the best idea, offering signifi- cantly improved dynamic properties over the Michigan double tanker. ACKNOWLEDGEMENTS I would like to thank my major professor and good friend, Dr. James E. Bernard, for his assistance and constant encourage- ment during my graduate career. I also wish to thank Ms. Jan Swift for typing this disser- tation and for her organizational assistance. Finally, I would like to thank my parents, Marvin and Joann Vanderploeg for their loving support throughout my graduate ‘ work. ii TABLE OF CONTENTS LIST OF TABLES ........................................... iv LIST OF FIGURES .......................................... v Section 1.0 INTRODUCTION ...................................... 1 2.0 LITERATURE SURVEY ........ . ....... . ................ 4 3.0 MODELS FOR MULTIPLY ARTICULATED VEHICLES .......... 8 4.0 INFORMATION FROM THE PUP TRAILER MODEL ............ 17 4.1 Eigenvalue Analysis .......................... 17 4.2 Transient Analysis ........................... 23 5.0 METHODS OF INTRODUCING DYNAMICS COUPLING .......... 30 5.1 Eigenvalue Analysis .......................... 34 5.2 Transient Analysis ........................... 37 6.0 OFFTRACKING ....................................... 40 7.0 CONCLUSIONS ....................................... 43 APPENDICES ............................................... 45 REFERENCES ............................................... 53 LIST OF TABLES List of Nomenclature Used in Derivation of the Equations of Motion ................................ 2 List of Parameters for the Baseline Dolly Pup Trailer ............................................ 3 Peak Lateral Acceleration Gain for Several Parameter Variations ................................... 28 Source of Equations and Parameters for Design Modifications .......................................... 35 Frequency and Damping Ratio for Several Design Modifications .......................................... 35 Acceleration Gain for Several Design Modifications ..... 39 Vehicle Components Offtracking (inches) at low speed in a 50 ft. Radius Turn .......................... 41 Vehicle Components Offtracking (inches) at 50 mph in a 600 ft. Radius Turn ............................... 42 iv 10. ll. 12. l3. T4. 15. 16. LIST OF FIGURES Diagram of Various Vehicle Configurations ............. 2 Yaw Plane Model of Michigan Double Bottom Tanker ...... 9 Diagram of Tractor-Semitrailer and Dolly-Pup Trailer Models .................................. . ...... lO Eigenvalues of the Full Model and the Simplified Models ................................................ 15 Lateral Acceleration of the Pup Trailer Mass Center in a Lane Change Maneuver ...................... 16 Effect of Tongue Length on Eigenvalues ................ 19 Effect of Stiffening Dolly Tires on Eigenvalues ....... 20 Effect of Stiffening Pup Trailer Tires on Eigenvalues ........................................... 21 Effect of Stiffening Dolly and Pup Trailer Tires on Eigenvalues .................................. 22 Effect of Changing Pup Trailer Wheelbase on Eigenvalues ............................... . ........... 24 Effect of Changing Pup Trailer Axle Spread on Eigenvalues ........................................ 25 Lateral Accelerations of Vehicle Components of the Simplified Model .................................. 27 Effect of Parameter Changes on Lateral Acceleration of the Pup Trailer ....................... 29 Diagram of the Canadian Double ........................ 31 Diagram of the Four Point Hitch ....................... 32 Diagram of the Constraint Linkage... .................. 33 17. 18. Al. Bl. Eigenvalues of the Modified Designs ................... 36 Lateral Acceleration in a Lane Change for the Modified Designs .................................. 38 Free Body Diagram of Dolly Pup Trailer Model .......... 46 Constraint Linkage in a Deformed Configuration ........ 51 vi 1.0 INTRODUCTION The transport of commercial cargo via trucks takes various forms in the United States and elsewhere depending on local rules. For example, it is clearly economically advantageous to use vehicles which are as long as local length limits will allow, thus carrying as much cargo as possible with each commercial vehicle. Local length limits are usually lenient enough to lead to the desire to articulate the vehicle at least once and often more than once so that good low speed maneuvering is provided. Thus in various localities it is common to see simply articulated tractor semitrailers; doubly articulated Canadian doubles, or so-called B-trains; triply articulated vehicles such as the Michigan double tanker. A sketch of each of these vehicles is shown in Figure l, which is reproduced from Reference 2. These vehicles have been the subject of analysis by various researchers over the years, starting as early as 1951 when Williams considered "Snaking of Commercial Vehicles" [13]. Williams was followed by several others, as will be indicated in the literature survey in this thesis. This survey clearly indicates an escalation in the complexity of the mathematical models as digital computers become more powerful and more convenient to use, culminating in 1979 with [5], in which a 128 degree of freedom model is pre- sented to simulate the directional response of triples. l. Tractor Semitrailer Tractor First Semitrailer Second Semitrailer 2. Canadian Double Tractor First Semitrailer Dolly Pup Trailer .3. Michigan Double Bottom Tanker Figure 1. Diagram of Various Vehicle Configurations The main subject of this thesis is the so-called Michigan double tanker shown in Figure 1. Since this vehicle was often used to haul volatile fuel, and since it received significant notoriety due to several spectacular accidents during the middle and late 1970's, it is an appropriate subject for thoughtful ana- lysis. This thesis will show that significant conclusions con- cerning the stability of the Michigan double can be reached using straightforward mathematics and simple models. The thesis is divided into several sections. Section 2 presents a survey of the literature concerning the directional performance of multiply articulated vehicles. Section 3 presents the standard form of the linear model for the Michigan double and shows that the model can conveniently be broken into two simple models. Section 4 shows that these simple models yield the in- formation that the inherent problems of the Michigan double are not amenable to simple solution via changes in pup trailer parameters. Section 5 introduces the design changes now commonly used to improve the performance of the Michigan double and Section 6 considers the penalty for these changes, degraded low speed Offtracking performance. Conclusions are presented in Section 7. 2.0 LITERATURE SURVEY Analysis of the dynamic stability of multiply articulated vehicles first appeared in the literature in 1951. These first efforts were concerned with linear analysis, a procedure based on the assumption of constant speed and small angles. Later, as available computing capacity increased, researchers used non- linear models as well. In 1951, Williams [13] used a linear model to determine stability criteria for a tractor-semitrailer. His results delt primarily with trailer mass center location. These results were extended to a tractor-dolly-pup trailer. His results indicated that snaking was inherent in the dolly-pup trailer configuration. The first well known linear analysis was done by Jindra [8] in 1965. He obtained numerical solutions for the 6th order characteristic equation of a tractor-dolly-pup trailer. Ex- tension of this to a tractor-semitrailer-pup trailer configura- tion, with an 8th order characteristic equation, was considered too large a problem for existing digital capabilities. Hazemoto [6] numerically solved the 8th order equation in 1973, and showed that at normal highway operating speeds, the tractor-semitrailer-pup trailer configuration has a natural mode at around 0.8 hz with less than 20% critical damping. Frequency response calculations also showed that, at frequencies near 0.8 hz, the peak yaw rate of the pup trailer is significantly higher than the peak yaw rate of the tractor. Although many parameter variations were studied, no practical design change was suggested to significantly reduce the gain of the pup vehicle. Hales [7], using the same techniques in 1975, also found a lightly damped mode at normal highway speeds. Again, no practical solution was offered. Eshleman [2] developed AVDS, the first nonlinear digital simulation of multiply articulated vehicles, in 1973. AVDS was noteworthy because of its inverse formulation, i.e., the trajec- tory of the tractor is input and the steer angle and braking, as well as tractor sideslip and all the articulation angles, are output. The model was used in an attempt to judge multiply articulated vehicle stability. Eshleman concluded that "the double articulated vehicle is almost as stable as the single" [2], [3]. This conclusion, which is now generally regarded as incorrect, was based on steady turn results of a vehicle with a low center of gravity. In 1974, Standberg and Nordstrom [11] presented an inverse model similar to the model developed by Eshleman. In 1973, a roll degree of freedom was added far each vehicle [12]. The dynamic behavior of the model was studied through simulation of a double lane change maneuver. Vehicle stability was determined by comparing mass center accelerations up to the rollover limit for each vehicle. They concluded that, in a lane change, the rearmost vehicle experienced the highest accelerations. In addition, they showed that an increased number of articulations among comparable vehicle combinations led to higher accelera- tions. In 1978, Mallikarjunarao and Fancher [9] developed and used a linear model to study one particular vehicle, the Michigan double tanker. They computed eigenvalues, finding a lightly damped mode near 0.75 hz. No practical parameter changes were found that significantly added to the damping of this mode. They also noted that certain eigenvalues of the doubles combination closely matched eigenvalues of the tractor semitrailer without the pup trailer. This supported the idea that the dynamic coup- ling between the tractor semitrailer and the dolly pup trailer was weak. ‘ Mallikarjunarao and Fancher also studied a transient lane change, paying close attention to the lateral accelerations of the mass centers of each vehicle. A large acceleration gain between the tractor and the pup trailer was found at frequencies near 0.75 hertz. Their suggested solution to this problem re- configured the coupling between the semitrailer and the pup, effectively changing the vehicle to a tractor-semitrailer-semi- trailer. This configuration offers better dynamic characteristics than the traditional tractor-semitrailer-dolly-pup trailer, with the penalty of degraded low speed maneuvering and tire wear. In 1979, Gillespie et. a1. [5] developed a multidegree of freedom simulation for vehicle configurations with a tractor, semitrailer and up to two dollies and full trailers. The model includes tandem axles for each vehicle and antiskid brakes. The trend of the literature to date indicates escalation in the complexity of the models with time, an apparent indication of increasing availability of computational power. In the next section, an attempt will be made to reverse this trend by considering the information available from very simple models. 3.0 MODELS FOR MULTIPLY ARTICULATED VEHICLES The previous section showed that research in the area of multiply articulated vehicles has been making use of increasingly complex models. As an example, consider Figure 2, a yaw plane model of the Michigan double tanker reproduced from Reference 9. The linearized differential equations which describe the motion of this vehicle are based on four assumptions, namely, 1) lateral forces are a linear function of tire slip angles, 2) artic- ulation angles are small, 3) forward speed is a constant and 4) all motion takes place in the yaw plane. This section will show that additional information may be gained using much simpler models. For example, consider Figure 3, which presents a model of a tractor-semitrailer and a dolly pup trailer (a dolly pup trailer is often referred to as a full trailer). The rationale for the use of simplified models in place of the more complete model of Figure 2 rests on the following obser- vations: 1) Since the pup trailer kingpin is approximately vertically over the dolly's suspension center, only small lateral forces can be applied to the semitrailer by the dolly, and 2) since there is only lateral force coupling between pup trailer and semitrailer (linear analysis assumes constant speed), the pup trailer has little influence on the trajectory of the semi- trailer. Thus, tractor-semitrailer calculations, which involve w> . whzmwm _ anon cmcwgupz 0 one: mzm 3 N emxcm Ecuaom @— .flms 241m. . O \ L l 215/ . "Mn ,3 . A .a . «n . o----_ a. .A-TII:||. .aw . .m——— ¢e+ I . 5% p n. y «i 0 —x :x ‘6‘ 3 529m 3.. 205230 .232: «1 10 \, iv. Tractor Semitrailer \Yz Dolly Pup Trailer (Full Trailer) Figure 3. Diagram of Tractor-Semitrailer and Dolly Pup Trailer Models . 11 only three degrees of freedom, can be made without regard to the pup trailer, and the computed time-varying position of the pintle hook, point * in Figure 3, can be used as input to the pup trailer model. And the separate eigenvalues of each model should match the eigenvalues of the complete system. The advantage of this procedure is the relative simplicity of the analysis which will lead to clear and concise conclusions concerning pup trailer design. The pup trailer model has two degrees of freedom, w] and oz, which are the dolly and pup trailer articulation angles. Following methodology developed in [1], the dolly articulation angle is measured from the velocity vector of the hitch point at the rear of the semitrailer. It is assumed that this point has a constant speed U with variable direction. The rotation rate of the velocity vector is given by ru. The pup trailer articulation angle is measured with respect to the center line of the dolly. There was not an opportunity in the course of this thesis to design a test program for the purpose of testing the validity of the pup trailer model. Thus, results computed using the simple models were compared to the calculations and test results presented in Reference 3, which considered the full five degree of freedom model. Parameters which correspond to the Figures and the nomen- clature in Table 1 are listed in Table 2. This vehicle configuration, which will be referred to as the baseline vehicle, models the Michigan double tanker [9]. 12 TABLE 1 LIST OF NOMENCLATURE USED IN DERIVATION OF THE EQUATIONS OF MOTION :p yi "fl'fl'fl'flfi lateral acceleration of the ith vehicle. combined cornering stiffness at the jth axle of the ith vehicle. combined lateral tire force of the jth axle on the ith vehicle. longitudinal force at the hitch. lateral force at the hitch. longitudinal force at dolly kingpin. lateral force at dolly kingpin. yaw inertia of the ith vehicle. distance from the dolly c.g. to the hitch. distance from the dolly c.g. to the kingpin. distance from the puptrailer c.g. to the kingpin. mass of the ith vehicle. combined aligning moment of the jth axle on the ith vehicle. combined aligning moment coefficient of the jth axle of the ith vehicle. yaw rate of the ith vehicle. rotation rate of pintle hook point velocity vector. magnitude of the pintle hook velocity vector. forward velocity of the ith vehicle c.g. lateral velocity of the ith vehicle, c.g. distance of axle ij from the mass center of the ith vehicle. tire slip angle at the jth axle of the ith vehicle. articulation angle of the ith vehicle. 13 TABLE 2 LIST OF PARAMETERS FOR THE BASELINE DOLLY PUP TRAILER M1 = 4525.0 1b. x12 = 21.0 in. M2 = 59975.0 lb. x2] = 2.0 in. I1 = 21627. lb.*in.*sec.*sec x22 = 44.0 in. 12 = 782079. lb.*in.*sec.*sec x23 = 86.0 in. ij = 1673 lb./deg. £1 = 70.0 in. ij = 248 ft. lb./deg. 22 = 0.0 in. x]] = 21.0 in. 23 = 81.0 in. Verification that the eigenvalues of the simpler models are close to the eigenvalues of the tractor-semitrailer-dolly-pup trailer [9] is presented in Figure 4. The figure indicates that the eigenvalues of the full trailer model closely match two of the eigenvalues of the more complete model. Figure 4 also indicates that two modes of the 8th order system can be predicted very accurately by a tractor-semitrailer model, thus illustrating the weak dynamic coupling between the tractor-semitrailer and the full trailer. Transient response of the pup trailer model was compared to the lane change test and calculations presented in Reference 9. In this case, the input to the pup trailer simulation was the velocity vector of the pintle hook calculated using the tractor- semitrailer model. Figure 5 shows that the calculated acceler- ation of the pup mass center matches very closely the results from Reference 9. 14 The results of this section indicate that uncoupling the semitrailer from the full trailer for purposes of eigenvalue or transient analysis is a useful simplification. This point of view will be used in the next section wherein the pup trailer model will be utilized to illustrate dynamical phenomena which are peculiar to this configuration and to study potential pup trailer design changes. 15 I3 Baseline Full Model 6 0 Dolly-Pup Trailer ‘1' Tractor-Semitrailer 0 5 I: ‘C’ C] a E H 4" S .. 3 Y C R .1 X 2 1 S i(rad/sec) - I 1 i ‘1 1 1 o -5 -4 '3 -2 '1 0 Real. mus Figure 5. Lateral Acceleration of the Pup Trailer lass Center in a Lane Change Maneuver 4.0 INFORMATION FROM THE PUP TRAILER MODEL This section will be devoted to the study of the baseline full trailer model to determine feasible design improvements. This will be done via eigenvalue analysis and via examination of vehicle performance in lane change maneuvers. 4.1 Eigenvalue Analysis Previous studies of articulated vehicles have used eigenvalues to study vehicle stability [9], [11]. Consider the eigenvalue A = a + ib. The value of a is, for most vehicles, always negative. Researchers in vehicle dynamics commonly consider the damping ratio, cos {tan'1(3%)}, as an indicator of the amount of time for a disturbance to damp out. The worst case, positive a, occurs only for oversteer vehicles driven above their critical speed. The b term has often been considered to be a benign indicator of characteristic frequencies. However, since emergency lane change maneuvers of commercial vehicles typically entail frequencies which closely match the imaginary parts of one of the eigenvalues of tractor-semitrailer-full trailers, it's important to keep in mind both the imaginary and real parts of the eigenvalue. The eigenvalues of the full trailer model change with design parameters. Since the frequency of an emergency lane change is near 0.5 hz [9] and the natural frequency of the baseline dolly- pup trailer is near 0.75 hz, it is desirable to change the design 17 to increase the frequency of the full trailer roots, moving them further from the input frequencies. In addition, any resulting- increase in the damping ratio is desirable because it decreases the peak response. The full trailer model was used to compute eigenvalues corresponding to several feasible changes from the baseline configuration, including changes in dolly tongue length, the cornering stiffness of the tires, and the wheelbase. Special attention was given to the affect of these changes on the lightly 4 damped mode. Figure 6 compares the eigenvalues for several different tongue lengths. Lengthening the tongue from the baseline con- figuration leads to degraded performance as indicated by lower frequencies and lower damping. Shortening of the tongue led to favorable results, but this has limited applicability due to interference of the leading pup trailer axle with the trailing semitrailer axle in sharp turns. Figures 7 through 9 present the eigenvalues for several different tire stiffness configurations. Small increases in damping ratio and frequency occur for some configurations. Simultaneous stiffening of dolly and pup tires gave the best results of the combinations modeled. A 20% increase in the stiffness of all tires resulted in a 10% increase in damping ratio and frequency of the lightly damped mode. Decreasing the tire stiffnesses reduces damping ratio and frequency in all cases modeled. 19 ‘1 6 "‘ 5 I H 2 N A R ’ fl Tongue Length .. 2 X o 50 Inches g [3 70 Inches (Baseline) 4' 90 Inches (rad/sec) * 120 Inches -l I L_, . j l I 1, o -5 '4 ‘3 '2 -I 0 Ram. mus (sec") Figure 6. Effect of Tongue Length on Eigenvalues 20 "' 6 {\- x H 2 4—1 4 I N A R -3 Y R . -2 X G 40% Increase é a 20% Increase + Baseline (rad/ sec) * 20% Decrease - I l I l_ 1 1A 0 '5 '4 -3 -2 -l a REAL AXIS (sec-1) Figure 7. Effect of Stiffening Dolly Tires on Eigenvalues 21 ‘16 M d 5 I H 2 N A R ‘3 Y A ..2 X (J 40% Increase % a 20% Increase 4? Baseline (radlsec) * 20% Decrease _. 1 1 1 f 1 1 1 a '5 -4 -3 -2 -I 0 REAL axis (sec-l) Figure 8. Effect of Stiffening Pup Trailer Tires on Eigenvalues 22 ‘116 ‘5 I H 2 .1 I I H A R -3 Y A 1.112 X C) 40% Increase g I: 20% Increase (rad/sec) +Baseline , it 20% Decrease - I l I l, l I a -5 -4 -3 -2 -I 0 REAL AXIS