Citation
Modeling the side slipping of an articulated vehicle for speed determination

Material Information

Title:
Modeling the side slipping of an articulated vehicle for speed determination
Creator:
Bloxsom, William Allan
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
Physical Description:
156 leaves : illustrations ; 29 cm

Subjects

Subjects / Keywords:
Motor vehicles -- Skidding -- Mathematical models ( lcsh )
Motor vehicles -- Skidding -- Mathematical models ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references.
Thesis:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Mechanical Engineering
Statement of Responsibility:
by William Allan Bloxsom.

Record Information

Source Institution:
|University of Colorado Denver
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
25485451 ( OCLC )
ocm25485451
Classification:
LD1190.E55 1991m .B56 ( lcc )

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MODELING THE SIDE SLIPPING OF AN ARTICULATED VEHICLE FOR SPEED DETERMINATION by William Allan Bloxsom B.S., Eastern Michigan University, 1972 B.S., University of Colorado, 1985 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Engineering May 1991

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This thesis for the Master of Science degree by William Allan Bloxsom has been approved for the Department of Mechanical Engineering by John A. Trapp Date

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Bloxsom, William Allan (M.S. Mechanical Engineering) Modeling the Side Slipping of an Articulated Vehicle for Speed Determination Thesis directed by Professor John A. Trapp Traffic investigators are able to quantify the speed of a single unit vehicle which generates side-slip scuff marks in a high speed turning maneuver by equating the radius of the described arc to a velocity by means of a permutation of the definition of the scalar component of the normal acceleration of a particle on a circular path. That simple approach is not valid for articulated vehicles. Using Newtonian Physics a single vehicle model is designed and the tire forces identified. Constraint equations are used to describe the motion with lateral adhesion. The motions and forces are evaluated numerically. The iterations continue after the loss of lateral traction so that the radius of the curved path can be determined; The computed radius is compared to the desired radius. The velocity which was adjusted by the algorithm to achieve the desired radius is the velocity determined by the far simpler algebraic formula. A two-body articulated vehicle model is devised with the tire forces at each of the three axles as well as the forces at the pin coupling identified. Equations of constraint describe the required non-slip motion as well as the relationship of the centers-of-mass which are linked by the pin connector. The motion and forces are evaluated by computer algorithm in a manner similar to

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the single vehicle model. Because of the nature of the braking systems of tractor/trailer combinations as well as the vertical height of the centers-of-mass, the articulated model addresses those parameters. The weight transfer is computed using Euler angles. The braking is limited to values less than one hundred percent. The single vehicle model verifies the validity of the more basic algebraic formula. The difficulty in ascertaining all of the needed articulated vehicle input values diminishes that model's applicability. The values obtained do show a correlation between the speed and radius values that were typical for the single vehicle model. The form and content of this abstract are approved. I recommend its publication.

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TABLE OF CONTENTS INTRODUCTION ................................. Page 1 PART I Single Unit Vehicle ........................... Page 10 PARTll Computer Program For Single Unit Model ......... Page 23 PART III Two Unit Articulated Vehicle ................... Page PARTlY Articulated Vehicle With Weight Shift and Braking Page 69 APPENDIX ...................................... Page 81 BmLIOGRAPHY

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Traffic accident reconstructionists have for some time used centrifugal scuffmarks or yaw marks to quantify the speed of single unit vehicle which generated the marks in a turning maneuver. The algebraic equation used, = where S speed miles per hour R radius of the center-of-mass of the vehicle f coefficient of friction between tires and roadway, has its roots in the scalar equation for the normal component of acceleration of a particle a curved path. The scalar nonnal component of y2 n

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is metamorphized by substituting = 3,600 = where S speed in miles per hour and scalar velocity in feet per second and = where g is the acceleration due to gravity and is the coefficient of friction. The equation now takes on the form: = (1.466 8)2 = 2.149 82 = = 15 = The traffic equation has validity because: There is a particle analogy with the typical treatment of the body as a point representation at the center-of-mass.

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2. The magnitude of the total normal force at the road surface-tire interface is preserved even though the distribution on each of the tires may change. 3. The maximum available force is exceeded as evidenced by the transverse side-slipping. With the maximum available acceleration, and therefore force, toward the geometric center of the circular path known, the prescribed ratio between the scalar velocity squared and the radius of the circular path is also maximized. That ratio determines the maximum scalar velocity or speed at a given radius. When the maximum speed at a given radius is exceeded traction is lost, lateral sideslip is initiated and the center-of-mass will attempt to track at a radius which corresponds to the ratio representing maximum acceleration at the existing velocity. It is that sideslip that produces a distinctive tire mark on the roadway. The generated tiremarks are characterized by a narrow width and diagonal striations. The striations must represent a: sideslip parallel to the axle while the tire has continued to rotate. Since the center-of-mass compelled to travel a circular path with a larger radius and since the rear axle generally has a negative weight transfer making it the "lighter" axle, the rear tires. will yaw and track outside of their respective front tires. Because all of the tiremarks are easily associated with the individual tire which generated it, those marks can be documented .and reproduced on a scale drawing. By careful utilization of an

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appropriate scale model of the involved vehicle on the scale drawing, the track of the center-of-mass can be plotted. The radius of the described center-of-mass is not consistent, but will be larger at the onset of the marks and will decrease as speed is scrubbed off during the continued side-slip. The speed at the initiation of the side-slip is most closely approximated by the speed associated with the earliest discernable radius. The speed determination from yaw marks is a combination of both science and art. The science is the application of physical laws to make the quantitative assessments. The art is in the skill of the investigator assessing the tire marks in the roadway. To guarantee that there are no forces at the tire/road surface interface which would correspond to the vehicle braking or vehicle acceleration, the investigator must see evidence that the striations in the yaw marks are indicative of side-slipping parallel to the axle. The two linear measurements used to determine the radius described must be made with accuracy and care. The choice of placement and length of the arbitrary chord across the arc has significance. The measurement of the middle ordinate, the perpendicular distance between the center of the chord and the curve, is very important. The radius calculation is very sensitive to the much shorter middle ordinate measurement. The relationship used to calculate the radius of the arc,

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2 where R radius in feet C chord length in feet M = middle ordinate in feet is easily evolved by geometric proof. The third factor, and perhaps the most artful, is the assessment of the kinetic coefficient of friction between the tire and the roadway. There have been numerous articles devoted to that topic. There are several methods of testing to determine that value. Some methods have more apparent validity than others. The range of values used for a coefficient of friction is left to the investigator and is defensible independent of the validity of the general methodology employed in the speed determination. Having viewed the physics used in this speed determination, is necessary to see where this process starts from the investigators perspective: 1. The investigator has a reason to believe that there was a high-speed steering maneuver attempted. That belief can be based on the character of the roadway or an avoidance maneuver by the involved vehicle.

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2. There must be tiremarks on the roadway which both describe an arc and have the characteristic diagonal striations. 3. A linear chord and middle ordinate measurement must be made of one or more of the yaw marks. 4. Coupled with the assessment of the coefficient of friction, the investigator calculates the radius of the measured segment of the arc and then the speed associated with that mark. The mathematics and physics currently used are appropriate if used properly .. The physics, however, limits the application of the algebraic formulas to single, rigid bodies where all of the external forces are applied at the roadway/tire interface. There is an additional requirement that all of the friction force at the road surface be available to lateral traction and not be exceeded either by longitudinal slipping due to acceleration or braking forces. In order to include a larger class of vehicles, that is the articulated, combination vehicle, into this manner of speed determination, a number of additional factors must be considered. The effect of these factors on the

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applicable physics must be included in any calculation. Among the factors to be considered are: 1. The presence of a force at the pin connection between the two vehicles. 2. The effect of a very high center-of-mass of each of the bodies involved. 3. The effect of independent braking at the various axles. Newtonian mechanics have been used to develop the mathematics to deal with the dynamics involved. The decision was based on: 1. The nature of the time spans and velocities do not necessitate the evaluation of any relativistic effects. 2. The motion anticipated is not very complex. 3. The model lends itself to the use of rectangular coordinates on a plane surface. 4. Constraint equations can be established. 5. The nature of the equations remains apparent through the calculations. The use of Lagrange's equations or any other energy-related method has not been used.

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The vector mathematics involved in this discussion will be established a coordinate system typically utilized in the transportation industry. The longitudinal axis of the vehicle shall be defined as the x-axis, with the front of the vehicle toward the positive direction. The y-axis shall correspond to the trans verse axis of the car, with the positive direction toward the right side, as viewed from above, of the vehicle. In order to maintain a "right-handed" system, the vertical z-axis is positive into the ground. The origin of the system is at the center-of-mass. Each of the individual bodies which are linked in this articulated vehicle shall be described by their own "local" coordinate systems. Overall translation and rotation are described in relation to a global or fixed coordinate system. All angles will be measured positively, clockwise from the positive x-axis of the global system to the positive x-axis of the local system. The global system shall be identified by capital letters X and Y. The local system of the first unit the articulated group shall be designated by Xl and YI. The angle between the global X-axis and the local x-axis of the first unit shall be designated by the greek letter theta. The second unit will have local designation x and the angle denoted by the greek letter phi. -8-

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All angles are described in radians unless otherwise designated. The English system of units has been used. Distances, velocities and forces have units of feet, feet per second, and pounds, respectively. -9-

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The most apparent method to approach this problem was to develop the necessary force and constraint equations for a single-unit vehicle and correlate the results of that analysis to the results of the simplified equation based on the normal acceleration and scalar velocity on a curved path. the single vehicle model, there will be no friction forces used in the longitudinal direction. This is accommodated by allowing the vehicle to "free wheel", i.e., the vehicle is neither braked nor accelerated. This is a requirement for the simplified approach with the normal acceleration equation. This assumption has a basis in reality. In a turning maneuver where traction is lost, there is little likelihood of the operator either accelerating the vehicle or being able to modulate the braking to avoid skidding. The model will also trace the path of the vehicle from a straight-line motion through incremental steering changes. With no acceleration to increase speed, it is anticipated that during the change in steering angle, given a sufficient

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scalar velocity while the turning radius is continuing to diminish, that the veloci ty /radius relationship will cause loss of lateral adhesion. The necessary physical parameters associated with the vehicle are: QALI x Figure 1.1; Single Vehicle Input Parameters x y

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WB1 = wheelbase D1 = the distance from the center-of-gravity to the front axle FO = front overhang T = the track width OAL1 = overall length It is also necessary to define the forces present the free body diagram. The transverse forces at the tire/road surface interface are related to the orientation of tires. The change in vector velocity is related to the global coordinate system. The steer angle will cause the front wheels to be oriented with regard to the local coordinate system differently than the rear wheels. The steer angle is measured from the local. coordinate system. Figure 1.2; Single Vehicle Forces -12-

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oc the steer angle a the angle between the global X-axis and the local x-axis There are four forces (F F F F shown, one at each of the four tires. Those forces are the friction forces at the tire/road contact area which oppose lateral motion. The forces are oriented to be parallel to the axis of rotation of the tire. The maximum value for the four forces is defined as: where u the static coefficient of friction Ni the normal force at each tire The value of the force at each tire prior to side-slipping is not known. -13-

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With the free body diagram for the X-Y plane complete, the basic Newtonian equations of motion can be written: (1.1) Ml Xl = (6 90) -cos (6 90) (6 oc: -90) (6 -90) (1.2) Mt t = sin (6 oc: -90) -sin (6 90) sin (6 oc: -90) -sin (6 90) (1.3) The three equations of motion contain the seven following unknowns: The three equations are not sufficient to solve for the seven unknowns. To augment the three equations of motion, there must be equations of constraint developed. -14-

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The first apparent restraint deals the pairs of forces at each axle. the only weight shift considered is about the roll (longitudinal or x-axis) axis, then the total weight distribution on each axle does not change even though it might be re-allocated between the two wheels. Mathematically, this can be written as: (1.4) (1.5) (1.6) Additionally, the location of the wheels on anyone axle are constrained in relation to each other by the length of the axle. Additionally, each pair of forces, Fl and .F2 as well as F3 and F4 are constrained by the geometry of the vehicle to be parallel. The total frictional force on any.axle which prohibits that axle from side-slipping is therefore a combination of the frictional forces at each tire on the axle. Each of the pairs of forces can. therefore be replaced by one composite force per axle which relates physically as follows: = F2 (1.8) F uN -15-

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Since the appearance of FI and F2 have the same coefficient in the translational equations, the substitution of FF is simple. The same may be said of FR for F3 and F4 The torque equation has, however, different coefficients for FI and F3.The mathematical treatment as follows: (1.11) (1.12) long as FI arid F2 are parallel: therefore (1.14) which results in (1.15) (0) -16-

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The three equations of motion may now be written in terms of five unknowns as follows: (1.16) m Xl cos (0 oc -90) -cos (0 90) (1.17) m1 Yl = sin (0 oc -90)-sin (e 90) = cos oc The next simplification possible is to combine the angles used in the trigonometric functions by using the trigonometric addition formulas. (1.19) (1.20) (1.21) (1.22) (1.23) (1.24) (1.25) (1.26) (1.27) (1.28) sin(u -v) == sin cos v cos sin v v) = cos cos sin sin since + ex -90) = sineS + oc) cos( +90) cos(O + oc) sin(9O) cos(O oc) sin(6 90) == sin(6) cos(9O) cos(6) sin(9O) cos(6 + ex -90) = cos(6 +oc) cos(90) + since + oc) sin(90) = (6 oc) cos(e 90) = cos 6 cos(90) sin e sin(9O) sin 6 -17-

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Therefore, the three equations are: or (1.29) (1.30) (1.31) (1.32) (1.33) (1.34) ml Xl = sin (e 0<) -sin e l Y1 = (e 0<) e e = 0< ml Xl sin (e 0<) sin e = 0 Y (6 0<) = 0 11 e 0< 0 The rigid body has four tires that relate the vehicle to the roadway. Two additional equations of constraint are needed. Since the two tires on the front would have similar equations as would the two tires on the rear, the two needed equations should come from a relationship formulated for a front wheel and for a rear wheel. Each of the wheels is physically constrained by the rigid body to their constant relationship to the center-of-mass. The velocity of the wheels must therefore be equal to the velocity of the center-of-mass translation plus a velocity component, which is the cross product of the angular rotation vector (in -18-

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this case a k) and the vector distance between the center-of-mass and the tire. That distance now needs some discussion. The forces at each wheel on an axle have already been combined to produce a single friction force equivalent at each axle. The analogy will be carried one step further. Instead of using either wheel on the front axle or rear axle, the point chosen will be where the longitudinal axis crosses the axle. That point, as well, will have constraints imposed by the rigid body. That point is at a fixed distance from the center-of-mass. The point on the axle is also bound to move in the same direction as the front wheels. The vector distance from the center-of-mass to the front axle is: (1.35) The vector distance from the center-of-mass to the rear axle is: -19-

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The vector cross-products of the rotation vector and the distance vector are: (1.37) (1.38) (1.39) (1.40) 9 cos 8 sin 8 cos Dl sin 8 Al:le = [ 1 cos sin = cos sin The velocity of the non-centroidal point in relation to the global coordinate system is: (1.41) (1.42) (1.43) (1.44) (1.45) (1.46) Dl cos Dl sin e Dl sin 8) 9 cos 8) kMAl:le = cos 8 sir e = 9 1 cos The direction of the velocity at the axle in a non-slip condition is the direction of roll of the parallel wheels on each end of the axle. That direction is defined by the angle between the local and global X-axes. The relationship between the icomponent and the j-component of that velocity is defined by the

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tangent of the aforementioned angle. (In the case of the front axle, the steer angle must also be added). For the front axle, the ratio is written as: (1.47) (1.48) tan(S oc) = Vz Y e cos S tan(S ex) = sin 8 which may be written as: (1.49) Xl tan (8 oc) e sin 8 tan (8 oc) = 1\ e cos 8 or (1.50) il[tan (8 oc)] e [Dl(sin S (6 oc) cos 6)] = 0 The velocity components of the rear axle are related as: (1.51) (1.52) l 6(WBl cos 8 8 = Xl 6(WBl sin 8 -21-

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which may be rewritten as: (1.53) Xl tan 6 (WBI -D I ) sin 6 tan 6 = (WB1 cos 6 (1.54) Xl [tan a] Y [(WB D1)(sin a tan a cos a)] = 0 The five equations can now be written as: (1.55) [sin (0 oc)] = 0 [cos (a ce)] -(6)] 0 (1.57) cos [WI;' DI] 0 (1.58) [tan (6 oc)] y-a a sin a tan (6 oc] = 0 (1.59) iItan 6] DI)(sin 6 tan 6 cos an = 0 The five are X(t), Y(t), 9(t), FF and FR. The combination of the first and second time derivatives, coupled with the number of equations and the number of iterative solutions required, presuppose a numerical solution.

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PART II COMPUTER PROGRAM; SINGLE VEHICLE There are five equations with five unknowns. There are time-dependent unknowns in the equations which appear as algebraic values, as first derivatives and as second derivatives. In order to facilitate the analysis by computer and to solve for the zeroth, first and second time derivative at the same time, the time derivatives are discretized. Because of the second derivative, three time levels must be employed. These are: n present time step values n-l = last time step values n 1 = next time step value Of course, with each newly completed time step, n-l is no longer needed, n becomes the new n-l, n 1 becomes the new n, and n 1 is the new goal for the next time step.

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or The first time derivative, or the "velocity", is defined as: (2.1) The second time derivative or the "acceleration" is defined as: (2.2) = ___ (2.3) 1) 1)], (2.4) =

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The five equations may now be written in the following form where: sin(6 oc) sin(8) = 0 hl y" -(2.6) cos (6 oc) -cos(8) = 0 oc)] "[tan(8 ex)] y" y" 8" --[D(cos 8 sin 8 tan(8 oc] 8" 8 sin 8 tan (8 oc] = 0 + l[tan 8] X"[tan 8] (sin tan cos 0 -25-

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The equations can now be written one more time with the unknowns and their coefficients on the left and the quantifiable terms on the right. (2.10) x" ., [( sin(O [( sin 0]-2X" -X" J (2.U) (0 [( 0] --, (2.12) 0"" l cos 0] D'l] 2 0" -8" (2.13) l[tan(O oc)] 6 (0 oc)] = 6 sin 0 tan(O oc 1 [tan I [1] I = 0] = 6 tan cos 6) The left sides of the five equations are now in a form where the coefficients may be placed in a five-by-five matrix. In this matrix, each column represents an unknown. The sequence of the unknowns from left to right are Xl' Y1 a, F and Each row of the matrix represents one of the five equations. The problem is now a form which is compatible to a computer solution. However, order to find the first "n 1" level there must exist values for "n" and "n-l" levels. -26-

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The initial conditions of the problem are input into the computer by the operator. Without operator control, the coordinates of the center-of-mass are established at (0,0), with the velocity directed along the X-axis. Initially, the angular orientation is zero degrees. The operator establishes the track width and the wheelbase of the vehicle. The center-of-mass is established by the operator by specifying either the percentage of weight on the front axle or its distance from the front axle. The operator inputs the weight of the vehicle. Additionally, the operator inputs the axle that produced the observed yaw mark as well as the measured radius of that yaw mark. The computer uses that radius to establish an initial velocity (X) by means of: (1.466)(15 Where R is the operator input radius and f is the operator input coefficient of friction. The mass of the vehicle is computed by: (2.16) 32.2

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The mass moment of inertia about the vertical (Z) axis is computed from input values of mass (as calculated from the input value of weight), track width (T) and the wheelbase of the vehicle (WB from the following definition of mass moment of inertia of a rectangular prism: (2.17) = % 12 The value of I is calculated as: (2.18) This value is approximate because of the non-uniform distribution of the mass over the length. The length value is also an approximation based on the wheelbase length. A value of 1.8 is used to convert the wheel base dimension into a length dimension for the inertia value. Realistically, a value of 1.8 would convert wheelbase to actual overall length as demonstrated by the following table: -28-

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WB 1984 Cadillac Seville 71 114 205 205 1986 Chevy Sprint 61 88.4 142 159 1989 Dodge Shadow 68 97 172 175 1989 Ford Probe 68 99 177 178 1983 Porsche 928 73 98.4 176 177 1986 Celica 67 99.4 172 179 Table 2.1: Dimensions of Typical Vehicles Trantech who provides a computer program to assist in traffic accident reconstruction, used moment of inertia to resolve post collision spin trajectories of involved vehicles. Their default moment of inertia values are for various size cars based on a range of wheelbase lengths. -29-

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RANGE WHEELBASE (INCHES) K2(IN 2 ) I lb' 2 ) 1 90-95 2006 952 2 95-101.6 2951 1942 3 101.6-110.4 3324 2543 4 110.4-117.5 3741 3428 5 117.5-123.2 4040 4238 6 123.2-150 4229 4842 Table 2.2: Mass Moment of Inertial of Size Groups of Cars By using a multiplier of 1.8 on the wheelbase dimension from: CAR RANGE COMPUTED K2 1984 Cadillac Seville 4 3929 1986 Chevy Sprint 1 2420 1989 Dodge Shadow 2 2925 1989 Ford Probe 2 3031 1983 Porsche 928 2 3058 1986 Toyota Celica 2 3042 Table 2.3: Calculated Radii of Gyration of Typical Vehicles SLAM Program Trantech Corp., 1229 Cornwall Avenue, Bellingham, Washington. -30-

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Each of the computer radii of gyration fall within an acceptable range for the calculations to be done. There is no computed radius of gyration that would fall within a different size range of vehicle from the published data. Because the motion is prescribed as being completely along the longitudinal axis of the vehicle and because at time zero the steer angle is set at zero degrees, there are no lateral forces. Consequently, at T = 0, FF and FR are both set at zero. At time zero, all of the variable values have been established either by operator input or by program design. The computer algorithm that is used for the iterations requires that two previous time steps be available to create a new time step. Since only one time step is available, an alternative method must be devised to generate the values at "0 h." That method is based on the applied definitions of velocity and acceleration, namely: (2.19) (2.20) -31-

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or by solving for time level lin (2.21) (2.22) Equation (2.14) can be rearranged and made equal to Y. By using the velocity/displacement relationship one equation for time step two may be written as: (2.23) (2.24) tan tan cos Similarly, equation (2.10) may be stated in terms of the values that equal xn. That substitution may be made into the acceleration equation to yield value xn 1. With equation (2.12) and the acceleration equation the value for 1 may also be determined. yn+l, a displacement, may now be used in the velocity relationship to generate a value for (This argument requires an assumption that, over a

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small increment of time from an initial condition of stability with no acceleration, any accelerations present will be very small.) That relationship is: (2.25) Y" Y" By a similar argument, the two velocities at this increment, and may be used to establish displacement values by: (2.26) (2.27) X" = X" X" 811 1 = 811 e" 1 With three velocities known at time steps "0" and "0 h" the accelerations may be written from, for example: (2.28) 1 1 X" The initial conditions dictate unaccelerated, straight line motion which preclude the existence of any side forces between the tires and the ground. At time "0 h" and subsequent time steps, the steer angle is increased at the rate of 0.229 degrees or .004 radians per time step until a maximum of 6.875 degrees or 0.12 radians of steer angle is achieved. The steer angle is always to the right, creating a positive angle and is measured from the local coordinate system.

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For this model, at time step "0+ h", the very small steer angle is assumed to cause very small lateral tire forces. At this time step, the tire forces, FF and FR are simply evaluated as zero. Subsequent iterations are completed using Gaussian elimination to resolve the 5x5 matrix. The resolution of the matrix yields values for yo FF1 and FR1. The values of the first derivatives are established using the appropriate finite difference formula, for example: (2.29) in 1 1 X, The values of the second derivatives are generated by, for example: (2.30) 1 1 -1) X :: Therefore, at each time step all of the unknowns are evaluated. The goal of the program is to correlate a vehicle's speed at loss of adhesion to the radius of the side scuff tire marks described on the roadway. Sideslipping begins when the lateral acceleration forces exceed the maximum available lateral frictional forces. the vehicle is to slip, then the computed

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forces, Fp and FR' must be compared at every step to the maximum force available. The maximum friction force available is a function of the normal force available to the vehicle at that axle and the available coefficient of friction. The normal force at the axle is determined by the weight of the vehicle and positioning of the center-of-massbetween the two axles. At each time step, the computed frictional forces are compared to the maximum available frictional forces. the computed value for either Fp or FR meet or exceed the maximum available force, then several things occur. The maximum value is substituted for the computed value in the output for that time step. 2. The matrix is reduced from a 5x5 to a 4x4 with the elimination of either equation (2.13) (if Fp is at maximum) or equation (2.14) (if FR is at maximum). 3. In the remaining equation containing the maximum frictional force that value is made a coristant and moved to the non-variable column. 4. The new smaller matrix is solved as before.

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When and if the remaining frictional force reached a maximum value the maximum value, replaces the computed value for that time step. Subsequent iterations are computed using a 3x3 matrix, with the remaining constraint equation eliminated. From the onset of slipping, the program will continue with 30 additional iterations. From those values attained during the sliding, three points are identified. Those points are at time (1) sliding onset plus 5 time steps, (2) sliding onset plus 15 time steps and (3) sliding onset plus 25 time steps. By using the global coordinates of the three points in pairs (Le., one and two, two and three), the slopes of the two chords that connect the points in sequence can be quantified: (2.31), (2.32) Additionally, the coordinates of the mid-points of the two chord distances can be stated:

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The perpendicular bisector of a chord of an arc is a line that passes through the center of that arc. The two chords each representing arcs which, although contiguous, may not be identical in radius will have perpendicular bisectors which will cross. The point where the two bisectors cross may not be at the center of either arc; but, because of the small interval of time between the selected points, will not be too different than the respective arc centers. The point where the two bisectors intersect is identified with the coordinates (XR' Y The slope of each bisector may be identified in two ways. The bisector slope because it is perpendicular to the chord may be written as: (2.32), (2.33) (2.34), (2.35) =1

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Additionally, the slope may be written in terms of the coordinates of two of the points on the bisector itself. (2.36), (2.37) Y, ; Y11 Y. ; Y, 1 M = G (At ; 1 ; 1 YI Y2 Y2 Y3 Xl Xl Yl Y3 (2.38), (2.39) By equating the two equations for each slope: (2.40) Xl Xl YI Y2 = Yl YI Xl Xl Xl Y2 Y3 Y, Yl Xl (2.41) There are now two equations with two unknowns (XR and Y 0 which can be solved algebraically. The radius of the path of the center-of-mass of the vehicle can be computed using the coordinates of the center of the arc and the coordinates of the first point selected. -38-

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-X 1)2 Yl)2]2 The radius of the center-of-mass oithe vehicle is not what the investigator sees described by tire marks in the road. The investigator is able to identify marks the road. The investigator is able to identify marks left by the outside tite on the slipping axle or axles. In order to generate a program output which is compatible with the investigator's observations, the radius of any tiremark is computed based on the selection the operator made on the data input screens as to which tire path was measured. The procedure used to calculate that radius is very similar to the method described for the center-of-mass.The same three time increments are identified, but instead of proceeding with the coordinates of the center-of-mass, new coordinates are identified which represent the position of the appropriate tire at those points in time based on the geometry of the car and the angle the vehicle is at with reference to the global system. The program then checks the computed radius of the selected tire against the measured radius of the investigator. the difference is greater than one thousandth of the measured radius value, the entire process is reiterated from time zero with a new initial velocity (X) until the two values (computed and

PAGE 45

measured) come within the allowed tolerance. The speed of the vehicle when the radius of the tire paths match is the goal of the investigator. That speed is determined by computing the square root of the sum of the squares of the X velocity and the Y velocity. That value which is in feet per second is converted into miles per hour (by dividing by 1.466) for output. As the steering angle is increased through its range, traction will be lost if the velocity is high enough. In this program, the speed is not affected by increased power to the wheels or by longitudinal friction. This assumption is valid because of the normal driver reaction of "lifting his foot" from the accelerator pedal and the short period of time used .. Any hard braking by the driver would preclude the vehicle from generating true yaw marks on the roadway. The steering angle is slowly increased to permit a reasonable rate of increase of lateral friction at the wheels. An instantaneous steering angle of sufficient magnitude could lead to the computation of values associated with a completely different phenomena. For the single vehicle model the output was noted to do two things: 1. With an input velocity below the threshold of traction loss, the vehicle, when permitted enough time, described a complete circle.

PAGE 46

2. The computed radius of the vehicle at a given speed compared very favorably with the speed calculated by the normal acceleration formula. The single vehicle modeled with the bicycle analogy can be rewritten with the incorporation of the geometry associated with having a wheel on each end of an axle of some finite length. The three equations of motion are not effected, but the two geometry equations are. This model allows the velocity of the mid point of each axle to be calculated as the average of the velocities of each wheel. The first constraint equation used in the initial test was: (1.59) X[tan(6)] 8[(Wl D1)(cos6 sin 6 tan 6)] = 0 The geometry used assumes that the distance between the center-of-mass and the "wheel" is the distance along the longitudinal axis to the center of the axle is the appropriate distance and that there is no angle or other coordinates in the local system needed to define the rectangular coordinates of the rotational portion of the motion of the wheels. In fact, this new model needs to take these two additional factors into account. -41-

PAGE 47

x Figure 2.1: Geometry Associated With Single Vehicle Model

PAGE 48

= (W, sin(8 (W, cos(8 11)]/, = (W, D,? sin(8 (W, D,? cos(8 11)]/, T2 2 (2.45) (i)aJCle XD (W, D,? cos 8 cos

PAGE 49

Equation (1.59) now is derived in a similar manner as before to become: (2.47) i[tan OJ 0 (cos P)(sin 8 8 cos 8)] = 0 A constraint equation. for the front axle will be derived a similar manner. The new equation (2.48) is: (2:48) oc)] (cos A)(sin 8 (8 oc) cos 8)] = 0 A second computer run at a speed insufficient to induce side slipping and for a long enough time for the angleS to progress through more that 3600 has the center-of-mass described a circular path. At a higher speed, the computed radius for a sideslip path again compares very favorably with the radius of the less complicated formula based on the normal acceleration. In fact, both the bicycle analogy and the second generation model produce the same speed for a given radius.

PAGE 50

The bicycle model and this model are mathematically identical. The coefficients of the first time derivative of theta are the same in both applicable equations. -45-

PAGE 51

The articulated vehicle model involves a tractor unit with two axles and a pin receptacle and a trailer unit with a single axle and a pin. The model allows a portion of the weight of the trailer to be supported vertically by the pin connection with the tractor. Most tractor-trailer combinations actually have five axles; one steering axle, tandem drive axles on the tractor, and tandem axles on the trailer. Since, however, the wheels in a set of tandem axles operate in concert and are linked to the chassis at a pivoted mid-point, the tandem axles with eight wheels and tires is easily modeled by a single axle with a single wheel and tire at each end. The input quantities and the unknown variables used in this model are as follows: Tractor WI Wheelbase (feet) T Track width (feet) D1 = Distance from axle to center-of-mass (along longitudinal axis) (feet)

PAGE 52

D2 = Distance from center-of-mass to pin connection (along longitudinal axis) (feet) = Steer angle (angle between orientation of from wheel and longitudinal axis of tractor) e = Angle between global X-axis and longitudinal axis of tractor (Xt,Y t ) = Coordinates of center-of-mass of tractor Trailer W 2 Distance from trailer axle to pin (along longitudinal axle) (feet) D3 Distance from center-of-mass to pin (along longitudinal axle) (feet) T = Track width (feet) Angle between the global X-axis and the longitudinal axis of trailer-(X2 ,Y 2 ) -Coordinates of center-of-mass of trailer As shown, there are eight identified forces; six at the tires and two at the pin. The six forces at'the tires are oriented so that they are always perpendicular to the orientation of the tire to which they are assigned. The two forces at the pin remain coincident with the global X and Y axes.

PAGE 53

y Figure 3.1: Input Values of Articulated Vehicle Model -48-

PAGE 54

For each body, there are two equations of translation and one of rotation from Newton's second law for a total of six equations. These equations are: (3.1) m1 = cos(S ex -90) cos(S ex -90) cos(S 90) -(3.2) m1 = sin(S oc -90) F2sin(S oc -90) sin(S 90) (3.3) 1 8 = cos oc -sin ex) (Dcos oc sin oc) 2 2 Cos(S 90) sin(S 90 X2 = cos(4)> 90) -sin (4)> 90) = sin 4 cos 4 There are six equations and fourteen unknowns. In a manner similar to that of the single vehicle, three forces can be eliminated. One of the forces on each of the axles can be eliminated by combining its effective value with the remaining force on the axle. In this model Fl and F2 when combined at the left front wheel, shall be identified as F1 ; F3 and F4 when combined at the left rear tractor wheel, shall be identified as F3 ; F7 and Fa when combined at the left rear trailer wheel shall be identified as F 7 With that reduction of unknowns to eleven, there remain five equations of constraint to be written. -49-

PAGE 55

It is appropriate to develop a constraint equation for each of the axles (or wheels with assigned forces) as well as the pin which is the point of articulation. The constraint equations, which deal with the pin, have to do with the geometric constraints imposed by the pin on the two centers-of-mass. the pin is given the coordinates (XA' Y A) then the centers-of-mass of the two bodies have a definable relationship to that point. (X2,Y2) D3 D2 Figure 3.2: Geometric Relationship Between Centers-Of-Mass and Pivot Point

PAGE 56

For the trailer: = X2 cos(l80 4) (3.S) = Y2 sin(180 4) For the tractor: Xl cos(1S0 6) = Dl sin(180 With two equations for XA and two equations for Y A' the equations may be combined to eliminate the coordinates of the pivot point. The combined equations are: (3.11) (3.12) Xl cos(lSO 4) = Xl cos(1S0 6) sin(lSO 4) = sin(180 Eliminating the double angles, the equations may be written as: (3.13) (3.14) 4 = Xl cos 6 Ysin 4 Y sin 6 -51-

PAGE 57

or as: (3.15) Xl Xl Dl cos COS 0 (3.16) 1 -2 -sin e -sin In the single vehicle matrix, the five columns represented the two involved forces and the velocities of the three position variable. The matrix for the articulated vehicle model will have a size of llxl1. The eleven columns will represent the five forces identified and the velocities of the six position variables. order to put the above equations into that form, each of the variables in those equations must be differentiated with respect to time. The result is: (3.17) (3.18) 2 e sin c;, sin = 2 e cP e The front axle constraint equation is similar to the equation from the single vehicle model. Figure 3.3: Geometry in Front Axle Constraint -52-

PAGE 58

(3.19) (3.20) (3.21) (3.22) = DI (} sin 6) I DI (} cos 6) Dl (} cos 6 tan(e oc) =. x 6 sin 6 tan(6 oc) (} DI sin 6 tan(6 oc) cos 6 X[tan(6 oc)] S[DI sin 6 tan (6 oc) cos 61 = 0 Using the same basic approach, the constraint equation for the drive wheels of the tractor is developed as follows: Wl-Dl Figure 3.4: Geometry in Rear Tractor Axle Constraint (3.23) (3.24) (3.25) (3.26) = [i S(W sin 61 S(W I cos 6 I S(W l cos6 tan 6 = ------6 S (W sin e e = (W cos e XI[tan 61 6[(Wl Dl)(sin 6 tan 6 cos e)l 0 -53-

PAGE 59

The remaining axle is that of the rear wheels on the trailer. Once again, the translational velocities of the center-of-mass are combined with the rotational components, which are a function of the rate of rotation about the center-of-mass and the distance from that center-of-mass. with the previous two derivations, the model is simplified to a bicycle analogy. <1>] <1>] cos (3.28). tan = ------z sin (3.29) X2 sin tan = cos I (3.30) 2[tan -D3)(sin I cos = 0 y W2-D3 Figure 3.5: Geometry in Trailer Wheel Constraint

PAGE 60

x = X, -1 -sln(4) P) (4) P) = X, -sln(4) -P) = t cj, P) 2 2

PAGE 61

Rewriting the x-component of the velocity of both wheels using the appropriate trigonometric addition formulas yields: (3.35) (3.36) Since the x-component of the velocity of the axle is somewhere between the velocity of the two wheels and since the wheels are physically constrained to each other, it will be appropriate to average the two velocities. (3.37) (3.38) The same philosophy produces a y-component of velocity for the axle of:

PAGE 62

These components may be carried out as before. (3.40) tan = (3.41) X2 Y2 (sin cos P cos cos = 0 This equation is very close to equation (3.30) if the quantity (W2 -D3 ) is much larger than the quantity of T /2. With no control over that ratio (especially in a trailer where 75 percent of the weight may be load and the placement of that load may vary), the more desirable approximation will be achieved with the more accurate model. -57-

PAGE 63

The model of the tractor with two wheels on the axle equation (3.26) may be rewritten as: . rlT'2 (3.42) Xl [tan 5] -Y1 e 4"" (WI (cos (sin e tan e cos e)] = [ 1 (W and equation (3.22) becomes: (3.43) C)(sin 0 tan(0 cos 0)] 0 There are now eleven equations and eleven unknowns. As with the single unit model, the coefficients of each of the variables of each of the equations will be arranged in a matrix which by Gaussian elimination algorithm, will be solved. The eleven equations with the eleven unknowns need to be manipulated so that the time-dependent kinematic variables all represent displacement values. -58-

PAGE 64

The relationship between those displacements and the velocity value and the acceleration value is as follows (with h equal to the time increment). (3.44) ... -= (3.45) 1 ... = -59-

PAGE 65

(3.47) cos(O' r; ,[ cos 0'] .,[ 1"'-' (3.48) 0" cos r; 2 1 cos 2 O' 8" -, (3.49) 4>-] (3.50) ,[ !:]cos 4>"] 4>"" sin 4>"] cos 4>'] D)] = -7 2 3 2

PAGE 66

1 = xt [D3 2n [D2 an] [D3 1 r; <1>" (cos p) (sin tan <1] X;[tan 4>1 y; tan l[tan(6 I 6" (COS C)(cos 8 sin 8 tan (8 )] X; l[tan BI I 6" 0{ D.)'(cos P)(OO9. 8 sin B tan B)]

PAGE 67

The coefficients of the unknowns are arranged in an eleven-by-eleven matrix. All of the coefficients of each of the eleven equations are arranged so that they are the following order. For example, equation (3.48) has matrix values of: = 2) 0 3) = 1 4) 0 = = 7) sin 11 8) 11 9) = --oc 10) = (WI 11) = 0 12) 2611 1

PAGE 68

And the matrix values for equation (3.56) are: (11, 1) tan a (11, = (11, 3) -)2 (cos j})(cos 6 sin tan 6) (11, 4) 0 (11, 5) 0 (11, = (11, 0 (11, 8) = 0 (11, 9) 0 (11, 10) = 0 (11, 11) = 0 (11, 12) a) yt' (11, 3)] As with the single vehicle model, the articulated vehicle utilizes the values of the position coordinates for time increments "n" and "n_l" to produce new values for time increment "n+ 1". This is possible for the third and subsequent time increments where the values for the two immediately prior time increments are available. At time zero, the initial conditions some of the needed values and allow the remaining values to be determined so that at time zero, all eleven unknowns are established. In order to approximate the values at time step "0+ h", the definitions of velocity and acceleration are again employed. From the definitions: -63-

PAGE 69

(3.57) and (3.58) i" = The method of determining the values at time "0 h" is from: (3.59) = Equation (3.52) is rewritten so that Xl is isolated as: That expression, evaluated at time zero, is used to quantify Xl at time "0+ h". YI is evaluated from equation (3.55) in a similar manner while Y2 has its root in equation (3.53). Theta and Phi at time "0+ h" are related to equations (3.56) and (3.54), respectively. The velocity X2 is evaluated from acceleration equation (3.49) and that value is used to establish the value of X2 at time "0+ h". -64-

PAGE 70

With all of the position values, the velocity values are approximated with: 1 = Those values are used to approximate the acceleration values with: = The five forces are now computed using permutations of equations (3.46 through (3.51). With a complete array of values for the first two time increments, the 11x11 matrix can be iteratively resolved using Gaussian elimination. For each iteration, three values besides the variables are monitored. The time increment is added to a total cumulative time from time zero, and that cumulative value is stored association with the values for the eleven variables. 2. The steering angle (alpha) is incremented from zero to a maximum value which is equal to about seven degrees. When that maximum value of alpha is reached, it remains unchanged. Regardless of its value at any

PAGE 71

time increment, it is captured as a value and stored in conjunction with the cumulative time and variable values. 3. The computed values of the roadway-tire friction forces (Fl' F F are compared at each time increment with the maximum allowable force values permitted. That maximum value is determined as a function of the coefficient of friction and the weight on the axle. the computed value of either Fl F3 or F7 exceeds the respective maximum, then several other things occur: a. Since the maximum value cannot be exceeded, the higher comput ed value is replaced by the maximum allowable value and is stored as such. b. The constraint equation which relates to that axle is eliminated from the matrix and the maximized force is no longer computed as a variable, but rather is treated as a constant. c. Because the forces are maximized prior to reaching the maximum value of alpha, the increases in alpha will not cause the side forces to decrease. a result, once the maximu111, value of any road friction force is computed, it is no longer checked in subsequent iterations to see if the computed force has fallen below the maximum value.

PAGE 72

d. The program takes note of the fact that lateral adhesion is lost at that cumulative time on that particular axle. As with the single vehicle model, this articulated vehicle was iterated around a circle at a low speed. The articulated vehicle behaved as suspected. A circle was described. 2. The rear trailer wheels tracked inside the tractor wheels. 3. The value of phi (trailer angle to the global axis) lagged behind the computed value of theta (the tractor's angle to the global axis). shorter duration sequences at higher speeds, there was an indication of lateral traction loss. The trailer wheels and the rear tractor wheels did track outside of the steering axle wheels. The lagging relationship of angle phi to angle theta was no longer observed. Once again, after the onset of sideslip by the axle of interest, there are an additional 30 iterations completed. three iterations are isolated from the 30 and their values used to determine three points of the outside tire's path so that radius information can be calculated. That radius information is compared with the operator's input value of the magnitude of the radius observed. The computer program compares the magnitude of the two values. there is a

PAGE 73

sufficient difference in the magnitudes, the initial velocity is again modified and the process is repeated. The goal is to simulate the operator observed radius with the algorithm and then use the initial velocity of the algorithm as an approximation of the actual speed needed for the actual yaw marks. -68-

PAGE 74

The tractor-trailer combination has two additional factors which do not affect the single unit vehicle. 1. The total gross weight of the articulated vehicle is frequently 40 times that of the four wheel car. The center-of-mass of the units in the com1?ination vehicle may also be four or five times higher than that of the single vehicle. The magnitude of the mass and the height of the center-of-mass make any weight transfer a significant factor. 2. The nature of air brakes and the varied braking systems available to the driver make possible for some axles or wheels to be braked while others are not subjected to any action. Because of that type of braking, there may be a situation where an axle can leave the classic yaw marks while another axle may have a significant braking force. The primary rotation of the two vehicles is about the vertical z-axis through the center-of-mass. That rotation, both as a rate and as a cumulative

PAGE 75

displacement, is computed at every time step. Because the center-of-mass is, assumed as the center of rotation, the Euler equations apply. The equations, where subscripts .1, 2 and 3 represent local axes x, y, and z which are fixed to and rotate with the body, are: (4.1) II WI = Nt = WI = where the "I's" represent the moments of inertia and the "N's" are the applied torques about the designated axis. The model does not differentiate between the varying loads on the separate wheels of-an axle. The wheels on the axle are physically restrained from separating and contribute to the total lateral force on that axle regardless of the normal force exerted by anyone wheel, ,so long as the total normal force remains the same. As a result, there is no effect of any small angular rotation (roll) about the longitudinal x-axis. The concern is to approximate the change in normal force from axle to axle due to rotation about the transverse y-axis (pitch). This increase and decrease normal force will cause the maximum lateral frictional force available

PAGE 76

at each axle to change and, consequently, will alter: the onset of loss of lateral traction. the rotation about the z-axis is denoted by 6, 6 and 6 and the small rotation about the y-axis is denoted by the a, a, and then the angular velocity components on each principal, fixed axis of each veh!c1e become: and therefore: (4.7) (4.8) (4.9) = sin = = cos 2 e cos' e sin -71-

PAGE 77

The situation is simplified by assuming that the small rotation about the y-axis is small and relatively invariant. It follows, that although "a" has a real value, "a" and "a" are zero The three equations become: (4.10) (4.11) 2 (4.12) These values are substituted into the Euler equations to yield: (4.13) (4.14) '2 e sin cos = (4.15) The torques on the body fixed axes related to the global coordinate system as follows: Nt cos sin = (4.18) Nz = sin cos

PAGE 78

y, 2 1 3 z Figure 4,1: Positions of Axes Mter Rotation About Therefore: (4.19) = (I} 6 sin cos (13 6 cos sin (4.20) '2 a sin cos (I} -13 ) (4.21) = I} e sin2 13 e cosl

PAGE 79

Which can be rewritten as: (4.22) NJ: = sin cos (4.23) '2 = e sin cos (4.24) z ::; sin The torques that apply to each of the two bodies must be identified. In addition to the lateral and normal forces at the wheels and the forces at the pin, a braking force must be considered. In order to maintain a true yaw mark (sideslipping parallel to the axle without braking) on at least one axle, the braking will not be applied uniformly to all axles. Because of the nature of the airbrake systems on articulated vehicles, the braking force will be applied at the rear tractor The front tractor brakes are small in size, have a variable proportioning value iII. the air supply and in fact may even be turned off. The trailer brakes are more worn than the tractor brakes and consequently may not operate well. The available braking force will be limited to 75 percent of the maximum available. By limiting the braking to less than maximum, the orientation of that braking force is parallel to the orientation of the wheels. The braking force

PAGE 80

reduces the lateral traction available at the wheel as the vector combination of the two must not exceed the maximum friction force available. On the trailer, the identified forces are: x Figure 4.2: Forces on Trailer in Three Dimensions -75-

PAGE 81

kl 11, 0 --2 2 where: = sm(4) 2 3 4 2 2' R4J: = sin e -3.5 -76-

PAGE 82

The components of Mo are therefore: = = Substituting back into the Euler equations: (4.28) (4.29) cp sm cos = (4.30) e sin2 cos2 = Equations (4.28) and (4.29) can each be in terms of "sin a cos a". With "sin a cos a" isolated on one side of each equation, the two equations can be equated and the terms involving angle "a" eliminated, producing: (4.31)


PAGE 83

(4.32) The process for the tractor is identical and evolves from the following diagram identifying the forces and their locations. x Figure 4.3: Forces on Tractor in Three Dimensions

PAGE 84

Which has a composite torque of: The algebraic manipulation in addition to thd relationship: allows the equation to be solved for N2 : (4.35) By computing the values of Nt, N2 and N3 at each time step, a more accurate assessment of the maximum available lateral force may be made. As before, the calculated lateral force required at ach time step for non-slip conditions is compared to the calculated maximum -79-

PAGE 85

CONCLUSION The more complex the combination of vehicles becomes, the more geometry-related data is required. Every time a value is inserted as a given, there is the possibility of introducing a small error. The re:quirement of eight or more such values each with a varying effect on the sensitivity of the computed solution is sure to induce some uncertainty into the solution.! The artiCulated vehicle model is designed in a parallel manner to the single vehicle model. The single vehicle model generates a solution which is favorably compared to the simpler, traditional, manual solution. Because of the genuine sparsity of accurate tire mark measurement and alternate speed detennination in the same situation, the articulated model was not tested against a known scenario.

PAGE 86

A CIRCULAR PATH OF BASIC SINGLE VEillCLE BICYCLE MODEL CIRCULAR PATH OF SINGLE VEillCLE ENHANCE MODEL OUTPUT OF BASIC SINGLE VEillCLEMODEL D. OUTPUT OF ENHANCE SINGLE VEillCLE MODEL E. COMPUTER PROGRAM LISTING! OF SINGLE VEHICLE MODEL F. OUTPUT OF BASIC ARTICULATED MODEL G. OUTPUT OF ARTICULATED MOPEL WHICH INCLUDES WEIGHT SHIFT AND BRAKING H. COMPUTER PROGRAM LISTING OF ARTICULATED VEHICLE MODEL

PAGE 87

APPENDIX A 82

PAGE 88

Xl X1DOT X1-DD VI V1DOT VI-DD 19.8215 0 0 0 .25 4.9548 19.8149 -.0399 .0364 .3249 1.6399 .5 9.9061 19.7929 -.1199 .1764 .829'7 2.3999 .75 14.8488 19.7409 -.2799 .4673 1.5319 3.1999 1 19.7727 19.6379 -.5599 .9579 2.4269 3.9599 1.25 24.6617 19.4569 -.9199 1. 6961 3.5119 4.7199 1.5 29.4'i25 Iq.172t;> -1.15'79 4.7:109 4.6199 1.75 34.2464 18.8459 -1.4399 4.0558 5.8689 4.4399 39.9103 18.4539 -1.6799 5.6605 : 6.9649 4 .. 2.25 43.4683 17.9989 -1.9599 7.5363 8.0359 4.2399 2.5 47.9047 17.480'7 -2.1999 9.6761 9.0769 4.1199 2.75 52.204 16.9029 -2.4399 12.0719. 10.0839 3.9999 3 56.3515 16.2679 -2.6399 14.7152 1 1.0549 3.79'79 3.25 60.3333 15.5759 -2.8799 17.5964: 11.9869 3.6399 3.5 64.1353 14.8309 -3.0799 20.705J' 12.8749 3.479'7 3.75 67.7448 14.0359 -3.27'79 24.0304: 13.7179 3.27'79 4 71.14'77 13.1949 -3.47'7'7 27.5604, 14.5129 3.07'79 4.25 74.3387 12.3089 -3.6399 31.282'7' 1 2579 2.8799 4.5 77.301 11.3839 -3.75'79 35.1847: 15.9449 2.5'7'79 4.75 80.0279 10.4239 -3.9199 39. 2506i 16.5719 2.3999 5 82.5105 9.4309 -4.0399 43.465.9: 17.1399 2.1599 5.25 84.7407 8.4059 -4.1599 47.8158! 17.6479 1.9199 86.7114 7.3549 -4.27'7'7 52.2855 18.0979 1.6799 5.75 88.4162 6.2789 -4.3599 56.8595: 18.4839 1.4399 6 89.8497 5.1849 -4.4399 61.52251 18.8089 1.1599 6.25 91.0075 -4.4799 66.2586: 19.0689 .9199 6.5 91.8858 2.9509 -4.51'79 71.0516 19.2619 .6399 6.75 92.4821 1.8179 -4.5599 75.8848 19.3919 .3999 7 92.7945 .6809 -4.5199 80.7422: 19.4549 .1199 7.25 92.8225 -.4579 -4.5599 85.6071 19.4519 -.1599 7.5 92.560 -1.5929 -4.5199 90.4628 19.3819 -.3999 7.75 92.0265 -2.7219 -4.4799 95.2927 19.2449 -.6799 8 9J.2059 -3.8379 -4.3999 100.0803 1'i.0449 -.9199 8.25 90.11)92 -4.9329 -4.3599 104.8099 18.7799 -1.1999 8.5 88.7408 -6.0099 -4.2399 109.4653 18.4519 -1.4399 8.75 87.1057 -7.0659 -4.1599 114.031 :: 18.0639 -1.6799 9 85.2096 -8.0979 -4.0799 118.4926 17.6159 -1.'i199 9.25 83 .,59 -9. 1 ()09 -3.9599 122.8346 11.1099 -2.1599 9.5 80.6614 -111.0729 -3.7999 127.0426' 16.5449 -2.3599 "11.01)99 -3.6799 131.1028 15.9269 -2.5999 1 75.1595 I I .9079 -3.5199 -2.7999 83

PAGE 89

rIME XI XIDOT XI-DD Yl YIDOT YI-DD 10.25 72.07<11 -3. 1<1.5309 -2.9999 10.5 68.7791 -13.5839 -::'.19<;19 142.2635 13.7619 -::':.1999 10.75 145. 6lY:'. I:!. 945." -3.:.5590;; 1 I 61.6065 -15.(1739 -::.799<; 1 4B. -15.743<;1 lSi .. 64::;[.,1 1 1 189'" -:;.6799 !:':, 738b -16. ;:'599 154.3253 11.1.2549 -;c.7999 7':. 5768 -16.924'1 '2.1199 156.7689 9.2879 -3.9199 I;: 45. :::. -17.4299 -1.9199 158.9668 8.2889 -4.0399 ]2.25 41l.8667 -17.8759 -1.6799 7.2619 -4.1599 12.5 36.3483 -18.2619 -1.4399 162.5962 6.2119 -4.2399 12.75 31.7408 -1. 1599 164.0159 13 27.0598 -18.8509 -.9199 165.1661 4.0559 -4.3999 13.25 22.321)6 -.6799 166.0429 2.9559 13.5 17.5392 -19.1879 -.3999 166.6434 1.8469 -4.4399 13.75 12.7318 -19.2599 -.1599 166.9656 -4.4399 14 7.9144 -19.2679 .0799 167.0088 -.3859 -4.4799 3.103 -19.2109 .3599 166.7725 -1.5029 -4.4399 14.5 -1.6859 -19.0889 .5999 166.2578 -2.6129 -4.4399 14.75 -6.4364 -18.9039 .8799 165.4674 -3.7069 -4.3599 15 -11.1328 -18.6559 1. 1199 164.4055 -4.7849 -4.2799 15.25 -15.7597 -18.3479 1.3599 163.0758 -5.8489 -4.1999 15.5 -20.3019 -17.9779 1.5999 161.4826 -6.8929 -4.1199 -24.744:'> -17.5499 1.8399 159.6313. -7:9129 -4.0399 16 -29.0723 -17.(1649 2.0399 157.528 ; -8.9079 -3.9199 16.25 -16.5229 2.2799 155.1797 -9.8729 -3.7999 16.5 -37.3296 -15.9279 2.4799 152.5941 -10.8049 -3.6399 16.75 -41.231"4 -15.2789 2.6799 149.78 -11.7019 -3.5199 17 -44.965 -14.5799 2.8799 146.7464 -12.5599 -3.3599 17.25 -48.518 -13.8349 3.0799 143.5033 -13.3769 -3.1599 17.5 -51.8788 -13.0419 3.'2799 140.0615: -14.1489 -2.9999 17.75 -55.0358 -12.'2059 3.4399 136.4325 -14.8739 -2.7999 -57.9788 -11.3299 3.5999 132.6285 -15.5489 -2.5999 18.25 -60.698 -10.4169 3.7199 128.662 -16.1739 -2.3999 18.5 -63.1847 -9.4689 3.8799 124.5462 -16.7419 -2.1599 18.75 -65.4303 -8.4909 3.9599 120.2952 -17.2569 -1.9599 -67.4279 -7.4839 4.0799 115.9223: -17.7149 -1.7199 19.25 -69.1707 -6.4529 4.1999 111.4424 -18.1129 -1.4799 19.5 -70.6527 -5.3999 4.2399 106. 8705! -18.4509 -1.2399 19.75 -71.8696 -4.3309 4.2799 102.2216 -18.7289 -.9999 20 -72.8172 -3.2469 4.3599 97.5111 -18.9439 -.7199 84

PAGE 90

t xlDOr XI-DO VI VIDOT YI-OD 20.25 -73.4923 -2.1519 4.3999 92.7547 -19.0959 -.4799 -73.8928 -1.0519 4.3999 87.9683 -19.1839 -.2399 20.75 -74.0177 .0529 4.4399 83.1679 -19.2089 .0399 2] -73.8665 1.1559 4.3999 78.3689 -19. ]709 .2799 21.25 -73.44 2.2559 4.3999 73.5875 -19.0689 .5199 21.5 -72.7395 3.3469 4.3599 68.8394 -18.9039 .7999 21.75 -71.7675 4.4269 4.2799 64.1405 -18.6759 1.0399 -70.5273 5.4909 4.1999 59.5064; -18.3859 ].2799 22.25 -69.0231 6.5369 4.1199 54.9522 -18.0359 1.5199 -67.2605 7.5609 4.0399 5C1.4932, -17.6239 1.7599 -65.2451 8.5579 3.9199 46.1445: -17.1559 1.9999 -62.9838 9.5269 3.7999 41.92 -16.6309 2.1999 23.25 -60.4842 10.4639 3.6799 37.8337' -16.0499 2.3999 -57.7549 11.3639 3.5199 33.9992, -15.4159 2.6399 23.75 -54.9052 12.2259 3.3599 30.1297' -14.7309 2.8399 2"1 -51.6451 13.0479 3.1999 26.5374: -13.9999 3.0399 24.25 -49.2848 13.8259 3.0399 23.1337, -13.2219 3.1999 -44.7359 14.5569 2.8399 19.93 -12.3999 3.3599 24.75 -41.0099 15.2419 2.6399 16.9366: -11. 5399 3.5199 25 -::::7.1192 15.874GJ 2.4399 14.163 -10.6419 3.6799 25.25 -33.0769 16.4529 2.1999 11.6184 -9.7089 3.7999 -28.8GJ7 16. 974GJ 1.9599 9.3112 -8.7449 3.8799 25.75 -24.5941 17.4369 1.7199 7.2475 -7:7SGJ9 3.9999 26 -20.1831 17.8409 1.4799 5.4332 -6.7499 4.0799 26.25 -15.6786 18.1859 1.2799 3.874 -5.7199 4.1599 26.5 -11.0951 18.4719 1.0399 2.5746 -4.6719 4.2399 26.75 -6.4476 18.6969 .7599 1.5392 -3.6089 4.2799 27 -1.7516 18.8609 .5199 .7708 -2.5369 4.2799 27.25 2.9778 18.9629 .2799 .2713 -1.4579 4.3199 27.5 7.7252 19.0049 .0399 .0423 -.3739 4.3199 27.75 12.4754 18.9849 -.1999 .0841 .7079 4.3199 28 17.2129 18.9029 -.4399 .3961 1.7879 4.3199 28.25 21.9222 18.7609 -.6799 .9774 2.8609 4.2799 28.5 26.5882 lB.5559 -.9599 1.8254 3.9219 4.2399 28.75 31.1952 18.2909 -1.1599 2.9373 4.9699 4.1599 29 35.7287 17.9669 -1.3999 4.3091 5.9999 4.0799 29.25 40.1741 11.5849 -1.6399 5.9358 7.0089 3.9599 29.5 44.5171 17.1499 -1.8399 7.BI06 7.9849 3.8799 29.75 48.7448 16.6599 -2.0799 9.9263 8.9349 3.7199 30 16.1159 -2.2399 12.2757 9.8539 3.5999 85

PAGE 91

Xl XIDOT XI-DD VI VIDOT VI-DD 30.25 56.7988 15.5229 -2.4399 14.8508 10.7409 3.4799 30.5 60.6005 14.8809 -2.6799 17.6432 11.5919 3.3599 31).75 64.236 14.1949 -2.8399 20.6438 12.4059 3.1999 31 67.6942 13.4619 -3.0399 23.84311 13.1789 2.9999 31.25 70.9638 12.6879 -3.1999 27.2301 13.9089 2.8399 31. 5 74.0349 11.8749 -3.3199 30.7941 14.5939 2.6399 31.75 76.8981 11. 0249 -3.4799 34.5234: 15.2319 2.4799 32 79.5446 10.1399 -3.5999 38.4062 15.8199 2.2399 32.25 81.9657 9.2229 -3.7199 42.4292, 16.3549 2.0399 32.5 84.1542 8.2799 -3.8399 46.5797' 16.8399 1.8399 32.75 86.1046 7.3189 -3.8799 50.8442' 17.2659 1.5999 33 87.812 6.3369 -3.9599 55.2082 17.6359 1.3599 89.2714 5.3349 -4.0399 59.65771 17.9509 1.1599 33.5 90.4786 4.3199 -4.0799 64.1786 18.2069 .9199 33.75 91.4301 3.2909 -4.1199 68.7571 18.4109 .6799 34 92.1236 -4.1599 73.3792' 18.5549 .4399 34.25 92.557 1.2119 -4.1599 78.0297 18.6399 .2399 34.5 92.7294 .1669 -4.1999 82.6946: 18.6679 0 34.75 92.6403 -.8779 -4.1599 87.3593 18.6379 -.2399 86

PAGE 92

TIME ALPHA THETA THETA-D THETA-DO, FI F2 0 0 0 0 0 0 .25 .0199 .0053 .0419 .1599 139.2531 43.1439 .5 .0399 .0208 .0819 .1599 194.3475 90.8288 .75 .0599 .0463 .1219 .1599 245.4086 135.2007 .0799 .0818 .1619 .1599 298.0997 180.7018 1.25 .0999 .1273 .2019 .1599 351.4777 226.6445 1.5 .1199 .1825 .2359 410.8787 262.6848 t.75 .1199 .2416 .2359 0 326.7529 254.1904 2 .1199 .3006 .2359 0 326.6052 255.4555 2.25 .1199 .3596 .2359 0 324.1333 254.5754 2.5 .1199 .4186 2359 324. 145J 255.7293 2.75 .1199 .4776 .2359 0 321.8626 255.2796 3 .1199 .5365 .2359 0 321.9674 256.5905 3.25 .1199 .5956 .2359 0 321.5059 257.3752 3.5 .1199 .6546 .2359 319.1933 256.7925 3.75 .1199 .7136 .2359 0 318.9141 257.8161 4 .1199 .7726 .2359 0 317.3273 257.615 4.25 .1199 .8316 .2359 c) 317.0939 258. 360b 4.5 .1199 .8904 .2339 0 321.5482 248.3929 4.75 .1199 .9489 .2339 0 320.4255 248.6529 5 .1199 1.0074 .2339 0 319.2042 248.9378 5.25 .1199 1.0658 .2339 0 318.3558 249.6204 5.5 .-1199 1.1244 .2339 0 319.0453 251.255 .1199 1.1829 .2339 0 316.4876 250.5169 6 .1199 1.2414 .2339 (1 316.715 251.667 6.25 .1199 1.2999 .2339 (> 316.3549 252.3174 6.5 .1199 1.3584 .2339 313.2013 250.7936 6.75 .1199 1.4169 .2339 0 312.6737 251.3723 7 .1199 4754 .2339 0 312.1992 251.787 7.25 .1199 1.5339 .2339 0 312.2233 252.73 7.5 .1199 1.5924 .2339 0 310.109 251.8819 7.75 .1199 1.6509 .2339 0 311.5497 253.9117 8 .1199 1.7092 .2319 0 '316.6167 244.2429 8.25 .1199 1.7672 .2319 0 315.0989 243.9605 8.5 .1199 1.8252 .2319 313.2111 243.2566 8.75 .1199 1.8832 .2319 0 313.6938 244.5175 9 .1]99 1.9412 .2319 0 312.4359 244.2481 9.25 .1199 1.9992 .2319 0 311.7933 244.5614 9.5 .1199 2.0571 .2319 310.1562 244.0687 9.75 .1199 2.1152 .2319 0 310.0345 244.6921 1(1 .1199 2.1732 .2319 309.0598 244.8425

PAGE 93

TIME ALPHA THETA THETA-O THETA-DO Fl F2 10.25 .1199 2.2312 .2319 310.9794 247.1147 10.5 .1199 2.2892 .2319 246.3141 10.75 .1199 2.3472 .2319 308.4869 246.3314 11 .1199 2.4052 .2319 0 309.1172 247.6047 11.25 .1199 2.4632 .2319 0 308.6062 247.7343 11.5 .1199 2.5212 .2319 0 307.5804 247.3877 11.75 .1199 2.5792 .2319 0 307.1829 247.4557 12 .1199 2.6372 .2319 0 307.2805 12.25 .1199 2.6952 .2319 0 307.5613 248.724 12.5 .1199 2.7532 .2319 (I 306.4323 248.3143 12.75 .1199 2.8111 .2319 0 306.6848 248.8743 .1199 2.8691 .2319 0 306.7536 249.2652 13.25 .1199 2.9272 .2319 0 307.0295 249.7163 13.5 .1199 2.9852 .2319 0 305.6204 248.8115 13.75 .1199 3.0432 .2319 0 305.608 248.995 14 .1199 3.1012 .2319 0 304.6005 248.4785 14.25 .1199 3.1592 .2319 305.9677 249.643 14.5 .1199 3.2172 .2309 -.0399 303.848 248. 14.75 .1199 3.2747 .2299 0 310.5225 239.2782 15 .119<;1 3.3322 .2299 310.0782 239.128 15.25 .1199 3.3897 .2299 0 310.8363 239.996 15.5 .1199 3.4472 .2299 0 310.2436 239.8398 15.75 .1199 3.5047 .2299 0 310.4572 240.2025 16 .119<;1 3.5622 .2299 310.319 240.4294 .1199 3.6197 .2299 308.8557 239.2536 .1199 3.6772 .2299 0 309.3028 239.785 16.75 .1199 3.7347 .2299 311).057 240.6937 .1199 3.7922 .2299 311.3941 241.7559 17.25 .1199 3.8497 .2299 0 311.0211 241.4755 .1199 3.9072 .2299 311.6439 242.0066 17.75 .1199 3.9647 .2299 309.9443 240.7687 18 .1199 .2299 0 310.5342 241.4312 18.25 .1199 4.1)797 .2299 0 310.664 241.6483 18.5 .1199 4.1372 .2299 0 310.5359 241.7734 18.75 .1199 4.1947 .2299 0 311.0021 242.4641 19 .1199 4.2522 .2299 3ICI. 991 9 242.5206 19.25 .1199 4.3097 .2299 0 311.3091 243.0206 19.5 .1199 4.3672 .2299 0 309.1992 241.6274 19.75 .119<;1 4.4247 .2299 0 310.264 242.68 20 .1199 4.-4822 .2299 0 310.2197 242.8715 88

PAGE 94

TIME ALPHA THETA THETA-D THETA-DO F1 F2 20.25 .1199 4.5397 .2299 0 309.4426 242.4462 20.5 .1199 4.5971 .2299 0 308.7011 242.2994 20.75 .1199 4.6546 .2291;1 307.9912 242.0527 21 .111;11;1 4.7121 .2299 0 308.141;1 242.3721 21.25 .1199 4.7697 .2299 0 301;1.1492 243.6274 21.5 .1199 4.8272 .2299 0 309.1862 243.994 21. 75 .1199 4.8847 .2299 0 308.881 244.0255 22 .1199 4.9422 .2299 0 307.7845 243.5861 22.25 .1199 4.1;191;17 .2299 0 307.5686 243.8024 J 199 5.(1572 .2299 0 307.2678 244.1528 22.75 .1199 5.1147 .2299 306.3686 243.9593 .1191;1 5.1722 .2299 0 306.11;102 244.2765 .1199 5.2297 .2299 0 306.4274 245.0484 .1199 5.2872 .2299 0 305.697 245.0685 23.75 .1199 5.3447 .2299 304.8686 245.1048 24 .1199 5.4022 .2299 304.5526 245.4768 24.25 .1199 5.4597 .221;19 303.8261 245.4423 24.5 .1199 .2299 0 305.2745 247.3374 .1199 5".5747 .2299 302.5042 245.586 .1199 .2299 304.0418 247.3296 25.25 .1199 5.6897 .2291;1 302.4726 246.6492 .1199 5.747 .2279 0 307.8447 237.0858 ."1199 5.804 .2279 C) 305.6746 236.0933 26 .1199 5.861 .2279 306.6777 237.8969 .1199 5.918 .2279 304.6217 237.2991 26.5 .1199 5.975 .2279 305.2949 238.7508 26.75 .1199 6.032 .2279 304.685 239.1738 27 .1199 6.089 .2279 (J 303.6846 239.4062 27.25 .1199 6.146 .2279 0 302.5598 239.4131 .1199 6.203 .2271;1 0 302.2624 240.1123 27.75 .1199 6.26 .2271;1 300.6255 239.7588 28 .1199 6.3161;1 .2279 0 300.9997 241.065 28.25 .111;19 6.374 .2279 0 21;18.9385 240.2464 28.5 .1199 6.431 .2279 300.0767 242.1664 28.75 .1199 6.488 .2279 296.6103 240.436 29 .1199 6.545 .2279 (J 295.7549 240.7292 .1199 6.6019 .2259 r) 295.4111 241.4339 .1199 6.6584 .2259 0 301.8976 29.75 .1199 6.7149 .225<;1 299.6884 232.4103 .11 <;19 6.7714 .2259 297.6417 232.1881 89

PAGE 95

TIME ALPHA THETA THETA-O THETA-DO FI F2 -------------30.25 .1199 6.8279 .2259 0 297.8937 233.5932 30.S .1199 6.8844 .2259 Cl 298.4662 235.1446 30.75 .1199 6.9409 .2259 0 297.0494 235.0616 31 .1199 6.997.11 .2259 0 296.8414 235.9027 31.25 .1199 7.0539 .2259 0 293.0527 234.01)39 31.5 .1199 7.110.11 .2259 292.6814 23.11.826.11 31.75 .1199 7.1669 (. '.;"/:'..4 .. ;8.'. 2:50.4245 ,1 J99 .2259 :9;/.4584 :::36.6129 .1199 .', .?799 /.0141 .11"''' -.1.('.99 2A9.91c'" 236.9988 -::? .119<;1 7.39:><1 0 295.192 227.6077 33 .119<;1 7.4484 .2239 0 294.8619 228.5635 33.25 .1199 7.5044 .2239 0 293.1435 228.2714 33.5 .1199 7.5604 .2239 0 292.3069 228.8305 33.75 .1199 7.6164 .2239 0 292.1643 229.6441 34 .1199 7.6724 .2239 289.4719 228.443 34.25 .1199 7.7284 .2239 0 290.7555 230.5833 34.5 .1199 7.784.11 .223<;1 288.4903 229.652 34.75 .1199 7.8404 .2239 286.7992 229.1157 90

PAGE 96

,c 91

PAGE 97

TIME Xl X1DOT Xl-DO Yl Y1D01 Vl-DD 0 19.8215 0 0 0 4.9548 19.8149 -.0399 .0352 .3119 1.5199 .5 19.7929 -.1199 .1684 .7839 2.2399 .75 14.8493 19.7459 -.2399 .441 1.4279 2.9599 19.77cl 19.c579 -.4799 89c5 2.2459 3.5999 1.25 24.673 19.5029 -.7599 1.5775 3.2299 4.2399 1.5 29.5209 19.2659 -.9599 4.3529 4.0399 1.75 34.3054 19.0009 -1.1599 3.7383 5.3549 3.9599 2 39.017 18.6819 -1.3599 5.2006 6.3409 3.8799 43.6423 18.3109 -1.5599 6.9071 7.3069 3.7999 48.1687 17.8919 -1.7599 8.8526 8.2529 3.7199 2.75 52.5839 17.4209 -1.9599 11.0316 9.1739 3.5999 3 56.8759 16.9059 -2.1599 13.4377 10.0699 3.5199 '.3.25 61.0331 16.3439 -2.3199 16.0642: 10.9369 3.3999 3.5 c5.0443 15.7369 -2.5199 18.9035 11.7719 3.2799 3.75 68.8985 15.0889 -2.6799 21.9474 12.5729 3.1599 4 72.5855 14.3989 -2.8399 25.1875 13.3399 2.9599 4.25 76.0949 13.6689 -2.9999 28.6142 ; 14.0669 2.8399 4.5 79.4174 12.9039 -3.1199 32.218 14.7559 2.6399 4.75 82.5442 12.1039 -3.2799 35.9886 15.4029 2.5199 5 85.4673 11.2759 -3.3599 39.9151 16.0009 2.2799 5.25 88.1804 10.4229 -3.4799 43.9856 16.5549 2.1199 5.5 90.6765 9.5409 -3.5599 48.1887 : 17.0599 9199 5.75 92.9492 8.6359 -3.6799 52.5127 17.5239 1.7599 6 94.9929 7.7099 -3.7599 56.9467 17.9389 1.5599 6.25 96.81)23 6.7619 -3.8399 61.4782 : 18.3039 1.3599 6.5 98.3727 5.7989 -3.8799 66.0947 : 18.6189 1.1599 6.75 99.7002 4.8199 -3.9199 70.7837 18.8839 .9599 7 lu".7817 3.8299 -3.9999 75.5327 19.0989 .7599 1 01.614:::2.8299 -1.9999 B(i.3292 19.2639 .5599 102.1961 1.8249 -4.0399 85. 160:! 19.3739 .3199 7.75 102.5262 .8149 -4.0399 90.0119 19.4289 .1199 8 1(12.6033 -.1969 -4.0399 94.8709 19.4339 -.0799 8.25 102.4278 -1.2059 -3.9999 99.7249 19.3889 -.2799 102.0005 -2.2119 -3.9999 104.5614 19.2939 -.4799 8.75 11)1.3226 -3.2089 -3.9599 109.3669, 19.1399 -.7199 9 100.397 -4.1919 -3.9199 114.1274, 18.9349 -.9199 9.25 99.2279 -5.1579 -3.8399 118.8304' 18.0799 -1. 1199 9.5 97.8192 -0.1089 -3.7999 123.4641; 18.3809 -1.3199 9.75 96.1746 -7.0449 -3.7199 128.0171 18.0349 -1.4799 III 94.2984 -7.9599 -3.5999 132.4776: 17.6399 -1.6799 92

PAGE 98

Xl XIDOT Xl-DO Vl VIDOT VI-DD 10.25 92.1961 -3.5199 136.8334 17.1999 -1.8399 10.5 89.8734 -9.7239 -3.4399 141.0738 16.7149 -2.0399 10.75 87.3364 -10.5669 -3.3199 145.1873 16.1849 -2.1999 11 84.5923 -11.3819 -3.1999 149.163 15.6129 -2.3599 11.25 81.6481 -12.1649 -3.0799 152.9907 15.0009 -2.5199 11.5 78.5122 -12.9169 -2.9199 156.6604 14.3489 -2.6799 11.75 75.1926 -13.6339 -2.7999 160.1625 13.6609 -2.8399 12 71.6983 -14.3139 -2.6399 163.488 .12.9379 -2.9599 12.25 68.0383 -14.9579 -2.4799 166.6286 12.1809 -3.0799 12.5 64.2223 -15.5619 -2.3199 169.576: 11.3919 -3.1999 12.75 60.2607 -16.1239 -2.1599 172.3222 10.5729 -3.3199 13 56.1639 -16.6429 -1. 9599 174.8603 9.7269 -3.3999 13.25 51.9429 -17.1169 -1.7999 177.1839 9.8549 -3.5599 13.5 47.6(194 -17.5429 -1 . 6399 179.286 7.9589 -3.6399 13.75 43.1744 -17.9279 -1.4399 181.1619 7.0449 -3.7199 14 38.6494 -18.2629 -1.2399 182.8068 6.1119 -3.7599 14.25 0469 -18.5479 -1.0399 184.2163 5.1619 -3.9399 14.5 29.3794 -18.7829 -.8399 185.3866 4.198<; -3.8799 14.75 24.6594 -18.9679 -.6399 196.3149 3.2259 -3.9199 15 19.8994 -19.1029 -.4399 186.9987 2.2439 -3.9199 15.25 15.1119 -19.1879 -.2399 197.4365 1.2569 -3.9599 15.5 10.3093 -19.2229 -.0399 197.627' .2669 -3.9599 15.75 5.5044 -19.2079 .1599 197.57 -.7219 -3.9199 16 .7094 -19.1429 .3599 197.2659 -1.7099 -3.9199 16.25 -4.0631 -19.0279 .5599 196.7155 -2;6919 -3.9199 16.S -8.8006 -18.9629 .7599 195.9204 -3.6679 -3.9199 16.75 -13.4906 -19.6479 .9599 184.983:2 -4.6249 -3.7999 17 -18.1206 -18.3829 1.1599 193.6098 -5.5679 -3.7599 17.25 -22.6792 -18.0709 1.3199 192.1005 .-6.4959 -3.6799 17.5 -27.1527 -17.7159 1.5199 180.3619 -7.4089 -3.6399 17.75 -31.5322 -17.3109 1.7199 178.3977 -9.2999 -3.5199 18 -35.8048 -16.9629 1.8799 176.2134 -9.1699 -3.4399 18.25 -39.9601 -16.3729 2.0399 173.8143 -10.0179 -3.3599 18.5 -43.9877 -15.8399 2.2399 171.2064 -10.8389 -3.2399 18.75 -47.8766 -15.2639 2.3999 168.3972 -11.6299 -3.1199 19 -51.6167 -14.6489 2.5199 -12.3909 -2.9999 19.25 -55.1983 -13.9969 2.6799 162.2041. -13.1209 -2.8399 19.5 -58.6121 -13.3069 2.8399 158.8365 -13.8139 -2.1199 19.75 -61.8492 -12.5839 2.9599 155.2998. -14.4719 -2.5599 2(1 -64.9(113 -11.8269 3.0799 151.6035-15.0919 -2.3999 93

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TIME XIDDT XI-DO VI VIDOT YI-DO 20.25 -67.7604 3.1999 147.7572 -15.6719 -2.2399 20.5 -70.4193 -10.22JJ9 3.3199 143.7708 -16.2119 -2.0799 :::0.75 -72.8711 -9.3849 3.3999 -16.7099 -1.9199 21 -75.1098 -8.5199 3.519'7 135.4192 -17.1649 -1. 7199 21.25 -77.12'72 -7.6319 3.5999 131.0761 -17.5729 -1.5599 21. 5 -78.9244 -6.7259 3.6799 126.6361 -17.9379 -1. 3599 21.75 -80.4906 -5.8009 3.7199 t 22. 1111 -18.2529 -1.1599 22 -81.824 -4.8629 3.799'7 117.5132 -18.5229 -.9999 22.25 -82.9212 -3.9119 3.8399 112.9532 -18.7479 -.7999 22.5 -83.7795 -2.9519 3.83'7'7 108.1432 -18.9229 -.5999 22.75 -84.3965 -1.9829 3.8799 11)3.3951 -19.0479 -.3999 -84.7708 -1.0099 3.9199 98.6232 -19.1229 -.1999 23.25 -84.9013 -.033'7 3.919'7 93.8382 -19.1479 0 -84.7876 .9419 3.9199 89.0532 -19.1229 .1999 23.75 -84.4304 1.9149 3.8799 84.2807 -19.0479 .3999 24 -83.8303 ::.8839 3.9399 79.5332 -18.'722'7 .59'79 24.25 -82.9892 3.8439 3.8399 74.8232 -18.7479 .7999 24.5 -81.9092 4.7'739 ::;.759'7 70.1632 -18.5229 .'7999 24.75 -80.5"'35 5.7289 3.7199 65.5657 : -18.247<;1 1.1999 25 -79.0456 6.6509 3.6399 61.0428 : -17.9279 1.3599 25.25 -77.2693 7.5549 3.559'7 56.6053 -17.5629 1.5599 -75.2698 8.4379 3.5199 -17.1499 1.7599 -73.0521 9.2989 3.3999 48.0336 -16.6939 1.9199 26 -70.6224 10.1339 43.922 -16.1919 2.0799 -67.9871 10.'7429 3.159'7 39.9409 -15.6499 2.2399 26.5 -65.153 11.7239 -15.0679 2.3999 -62.1277 12.4729 2.9199 -14.4459 2.5599 -58.9191 1: . 1899 2.7999 28.8801 -13.7879 2.7199 -55.5355 13.8719 2.6399 25.519 -13.0949 2.8399 .,... c -51.986 14.5159 2.4799 -12.3659 2.9599 27.75 -48.2805 15.1209 2.3599 19.3388 -11.6039 3.1199 28 -44. 428L1 15.6889 2.1999 16.5357 -10.8159 3.1999 28.25 -40.4394 16.2139 1.9999 13.9331 -9.9989 3.3199 28.5 -36.3244 16.6979 1.8399 11.5382 -'7.1559 3.4399 28.75 -.12.0942 17.1349 1.6399 9.3571 -8.2889 3.4799 29 -27.7603 17.5289 1.4799 7.3953 -7.4019 3.5999 29.25 -23.3342 17.8719 1.2799 5.6569 -6.5039 3.5999 29.5 -18.8282 1.0799 4.1447 -5.5899 3.67'79 29.75 -14.2547 18.4129 .9199 2.8628 -4.6629 3.7199 .31'1 -9.6241 18.6179 .7199 1.814 -3.7259 3.7599 9 4

PAGE 100

Xl Xl DDT XI-DD Yl YIDDT YI-DD 30.25 -4.9497 18.7729 .5199 1.0007 -2.7789 3. 79'i''i' 30.5 -.2422 18.877'i' .3199 .4247 -1. 8279 3.8399 30.75 4.4852 18.9329 .1199 .0872 -.871'i' 3. 83'i'9 31 'i'.2203 18.'i'379 -.0799 -.0107 .0849 3.7999 31.25 13.9313 18.9019 -.2399 .1301 1.0409 3.7'i'99 31.5 18.6673 18.8169 -.4399 .5094 1.9929 3.7999 31.7S 23.3558 18.6819 -.6399 1.126 2.9379 3.7S99 32 28.0043 18.4969 -.8399 1.9779 3.8749 3.7199 32.25 32.6006 18. 266fj1 -.fjl999 3.0626 4.7'i'fjl9 3.6799 32.5 37.1341 17.9919 -1.1999 4.3773 5.7139 3.5'i'99 32.75 41.5927 17. 669fj1 -1.35fj19 5.9182 6.6109 3. 55'i'9 33 45.9657 17.304'i' -1.559fj1 7.6814 7.4909 3.4799 33.25 50.2413 16. 892fj1 -1.7199 9.6618 8.3489 3.3999 33.5 54.4091 16.4429 -1.8799 11.8536, 9.1799 3. 27fj19 33.75 58.4595 15.fjl52fj1 -2.03fj19 14. 24fj16 9.9829 3. 15fj1fj1 34 62.3826 15.4249 -2.1999 16.8434 10.7619 3.039fj1 34.25 66.1687 14. 856fj1 -2. 35fj19 19.6282 11. 5109 2.9199 34.5 69.8082 14.2539 -2.4799 22.5964 12.2309 2.8399 34.75 73.2929 13.6169 -2.5999 25.7413 12.9209 2. 67'i'9 95

PAGE 101

lIME rHErA THErA-V I HE'I A-DJ) F 1 .,c' ._'-' .1)199 .0048 .0379 .1599 126.9191 .0:':9<; .0187 .1199 177.8794 82.6108 .75 .0599 .0414 .1089 .1199 124.5336 .0799 .073 .1439 .1599 270.4874 168.5694 1. 25 .11999 .1134 .1789 .1199 320.6556 199.9661 1. .1199 .1625 .2099 366.9391 242.4497 1.7::, .1199 .215 .2099 0 291.0196 225.7716 .1199 .2675 .2099 291.995::-.1199 .32 .2099 289.2848 226.2849 2.5 .1199 .3725 .209" 226.60494 2.75 .1199 .425 .2099 269.2409 226.2934 .119<:; .4775 .2099 287.8887 228.1471 .1199 .53 .2099 286.7199 226.115::' 3.5 .1199 .5825 .2099 266.2371 .1199 .635 .2099 (. 285.7742 229.3557 4 .1199 .6875 .2099 (J 285.617:-.1199 .74 .2099 281.9108 228.1715 .1199 .7925 .2099 282.2676 229.4689 .1199 .845 .2099 282.622 230.8376 5 .1199 .8971 .2079 287.145:: 220.9016 .1199 .9491 .21)79 284.7518 219.9373 .119<;' 1.0011 .2079 (, 1.0531 .2079 283.297::' 220.8991 6 .1199 1. 1051 .2079 () 282.9756 221.4554 .1199 1.1571 .2079 281.4572 221.1822 .119<;' 1.2091 .2079 282.5362 6.75 .1199 1.2611 .2079 279.6188 .119'7 1.3131 .2079 280.3499 223.0033 7 .,'" .1199 1.3651 .2079 277.9938 221.8069 .1199 1.417J .2079 277.7345 222.282 7.75 .1199 1.4691 .2079 278.7499 8 .1199 5211 .2079 276.4997 8.25 .1199 1.5731 .2079 276.0156 .1199 1.6251 276.804::' 224.8471 .11<;19 1.6771 .2079 274.050::;, 223.4308 .1199 1.7288 .2059 280.046:!. 7141 .7803 .2059 278.36'" :>14.47:52 9. .119':: 83IEI :77.827<;' 214.896< I.B83::': .2059 278.0437 1.9348 {, 96

PAGE 102

M; FHA THETA THETA-O THETA-DOl Fl F2 4.0387 .2039 274.602 213.4095 .1199 4.0898 .2039 (J 275.3521 214.2493 20.75 .1199 4.1408 .2039 274.9028 214.1759 21 .1199 4.1918 .2039 273.5111 213.075 21.25 .1199 4.2429 .2039 273.5267 213.5571 21.5 .1199 4.2937 .2039 273.5937 213.606 21.75 .1199 4.3448 .2039 272.6595 213.1692 .1199 4.3959 .2039 0 273.8442 214.4606 .1199 4.4468 .2039 272.1297 212.9578 .1199 4.4978 .2039 274.8473 215.1119 .1199 4.5488 .2039 0 272.8269 213.5467 .1199 4.5998 .2039 0 273.3574 214.0699 23.25 .1199 4.6508 .2039 272.864 213.7992 23.5 .1199 4.7018 .2039 214.272 23.75 .1199 4.7528 .2039 273.8567 214.892 24 .1.199 4.8038 .2039 273.7385 214.9867 24.25 .1199 4.9548 .2039 272.0703 213.8626 24.5 .1199 4.9058 .2039 0 273.2152 215.144 24.75 .1199 .2039 273.3281 215.7333 25 .1199 5.0078 .2039 0 272.1894 215.3185 .1199 5.0588 .2039 272.236 215.4897 .1199 5.1098 .2039 270.749 214.73 25.75 .1199 5.1608 .2039 0 271.3841 215.5214 26 .-1199 5.2119 .2039 272.6973 217.1125 26.25 .1199 5.2629 .2039 0 269.9645 215.4418 26.5 .1199 5.3138 .2039 269.1716 215.1375 26.75 .1199 5.3648 .2039 269.5773 215.9432 27 .1199 5.4157 .2039 271.2949 217.701 .1199 5.4668 .2039 0 269.0917 216.2709 .1199 5.5178 .2039 269.0182 216.6657 27.75 .1199 5.5688 .2039 0 268.9227 217.2849 28 .1199 5.6198 .2039 0 270.1626 218.7997 28.25 .1199 5.6707 .2039 269.7432 218.7945 28.5 .1199 5.7218 .2039 0 269.4982 218.295 28.75 .1199 5.7728 .2039 0 268.5746 219.0721 29 .1199 5.8238 .2029 -.0399 217.995 29.25 .1199 5.8743 .2019 271.076 207.7004 29.5 .1199 5.9248 .2019 271. 6436 208.9074 29.75 .1199 5.9752 .2019 2"10.66:51 209.176 30 .1199 6.0258 .2019 209.4654 97

PAGE 103

ALPHA THETA THETA-D THETA-DD FI F2 1.9863 .2059 0 276.5964 215.9517 to.5 199 2.0378 .2059 0 216.3852 10.75 .1199 2.0893 .2059 0 276.4474 217.077. 6 I 1 .1199 :2.1408 .2059 276.2353 217.4187 11.25 .1199 2.1923 .2059 275.4608 217.33 11.5 .1199 2.2438 .2059 0 276.3232 218.6764 11.75 .1199 .2059 0 275.0312 218.1501 12 .1199 2.3468 .2059 0 273.9927 217.7997 12.25 .1199 2.3983 .2059 274.0101 218.076 12.5 .1199 2.4498 .2059 0 272.992 217.6111 12.75 .1199 2.5013 .2059 0 273.1856 218.1618 13 .1199 2.5528 .2059 0 273.282 218.5919 13.25 .1199 2.6043 .2059 0 273.5038 219.1213 13.::> .1199 2.6558 .2059 0 272.3844 218.8298 13.75 .1199 :2.7073 .2059 0 219.6304 14 .1199 2.7387 .2059 0 273.0414 219.8355 14.25 .1199 2.8103 .2059 0 219.1602 14.5 .1199 2.8618 .2059 0 270.432 218.6391 14.75 .1199 2.9133 .2059 271).9644 219.5591 15 .1199 2.9648 .2059 269.6753 218.991 15.25 .1199 3.0163 .2059 0 271.2418 15.5 .1199 3 .0678 .2059 0 269.4833 219.6115 15.75 .1199 3.119;" .2059 269.2249 219.6803 16 1199 3.1708 269.8(116 220.3587 16.25 .1199 3.2223 .:!059 0 270.531 Z21.0972 16.5 .119<;' 3.2738 .2049 -.0399 270.675 221.3181 16.75 .1199 3.3248 .203Q 274.903 1 .1199 3.3758 .2039 0 276.329:;: .1199 3.4268 .2039 211).2967 11.5 J 199 3 .4778 .2039 274.3251 210.6522 17.75 .1199 3.5288 .2039 0 274.384 210.9586 18 .119C! ;!,.5198 .2039 0 274.8359 211.6751 .1199 3.6308 .2039 274.9705 211.9327 18.5 .1199 3.6818 .2039 274.0S44 211. 2204 .1199 .3.7329 .2039 275.3898 Z12.4037 19 .1199 3.783 8 .::039 ( .119<;-:":. ; :;:7,:'. ::':: :::11. ::111 1 co . : .n.qc: 1;'. 2u: ''<'> ''::11.2885 0 274.647:' 213.1329 98

PAGE 104

lIME ALPHA THETA THETA-D THETA-DO F1 F2 30.25 .1199 6.0763 .2019 269.0561 209.2049 .1199 6.1268 .2019 269.3660 210.3094 30.75 .1199 6.1773 .2019 267.3916 209.705 31 .1199 6.2278 .2019 0 267.646 211.036 .1199 6.2783 .2019 267.0977 211.3797 31.5 .1199 6.3288 .2019 265.6799 210.8326 31.75 .1199 6.3793 .2019 .9973 32 .1199 6.4298 .2019 (J 263.7414 210.9927 .1199 6.4802 .2019 0 264.3477 212.5415 .1199 6.5308 .2019 263.2357 212.3033 32.75 .1199 6.5813 .2019 261.652 212.0193 33 .1199 6.6318 .2019 Cl 261.1766 212.2298 33.25 .1199 6.6823 .2019 0 260.386 212.7087 .1199 6.7324 .1999 265.7134 203.2458 .33.75 .1199 6.7824 .1999 0 263.9303 202.7471 .1199 6.8324 .1999 264.1652 203.802 34.25 .1199 6.8824 .1999 0 263.7632 204.4781 .1199 6.9324 .1999 0 263.0098 204.9925 34.75 .1199 6.9824 .1999 262.3049 205.1926 99

PAGE 105

100

PAGE 106

VELOCITY IS 85 FEET PER SECOND SPEED IS 58 MILES PER HOUR DRAG FACTOR USED IS .65 RADIUS OF AXLE 1 355.0928649902344 RADIUS OF AXLE 2 348.7281494140625 SLIPPING BEGINS AT TIME .3000000044703484 AND .3500000052154064 VELOCITY IS 85 FEET PER SECOND SPEED IS 58 MILES PER HOUR DR AS FACTOR USED IS .65 RADIUS OF AXLE 1 355.5943603515625 RADIUS OF AXLE 2 349.2236938476563 SLIPPING BEGINS AT TIME .3000000044703484 AND 35q00000521 54064 VELOCITY IS 85 FEET PER SECOND SPEED IS 58 MILES PER HOUR DRAG FACTOR USED IS .65 RADIUS OF AXLE 1 355.9066772460938 RADIUS OF AXLE 2 349.5313110351563 SLIPPING BEGINS AT TIME" .3000000044703484 AND .350;0000052154064 VELOCITY IS 85 FEET PER SECOND SPEED IS 58 MILES PER HOUR DRAG FACTOR USED IS .65 RADIUS OF AXLE 1 356.0843505859375 RADIUS OF AXLE 2 349.7060241699219 SLIPPING BEGINS AT TIME .3000000044703484 AND 101

PAGE 107

102

PAGE 108

VELOCITY IS 85 FEET PER SECOND SPEED IS 58 MILES PER HOUR DRAG FACTOR USED IS .65 RADIUS OF AXLE 1 354.8875427246094 RADIUS OF AXLE 2 348.3240661621094 SLIPPING BEBINS AT TIME .3500000052154064 AND .4000000059604645 VELOCITY IS 85 FEET PER SECOND SPEED IS 58 MILES PER HOUR DRAB FACTOR USED IS .65 RADIUS OF AXLE 1 355.5525512695313 RADIUS OF AXLE 2 348.980224609375 SLIPPING BEBINS AT TIME .3500000052154064 AND .4000000059604645 VELOCITY IS 85 FEET PER SECOND SPEED IS 58 MILES PER HOUR DRAG FACTOR USED IS .65 RADIUS OF AXLE 1 355.9640808105469 RADIUS OF AXLE 2 349.3860168457031 SLIPPING BEGINS AT TIME .3500000052154064 AND .4000000059604645 VELOCITY IS 85 FEET PER SECOND IS 58 MILES PER HOUR DRAG FACTOR USED IS .65 RADIUS OF AXLE 1 356.2079467773438 RADIUS OF AXLE 2 349.6265258789063 SLIPPING BEGINS A"r TIME .3500000052154064 AND .4000000059604645 VELOCITY IS 85 FEET PER SECOND SPEED 15 58 MILES PER HOUR DRAG FACTOR USED IS .65 RADIUS OF AXLE 1 356.35302734375 RADIUS OF AXLE 2 349.7695922851563 SLIPPING BEGINS AT TIME .3500000052154064 AND .4000000059604645 103

PAGE 109

APPENDIX E COMPUTER PROGRAM LISTING OF SINGLE VEHICLE MODEL 104

PAGE 110

tt) 20 30 40 50 60 70 80 90 101) 111) 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 331) 340 350 360 370 380 390 400 410 420 430 440 450 460 47(' 4BI.l 490 51)1) 510 520 530 540 550 560 REM PROGRAM TESTSIN.BAS REM : SINGLE VEHICLE YAW PROGRAM IN mCROSOFT BASIC REM ************.* REM DEFDBL A-Z KEY OFF REM CHOSE SINGLE OR ARTICULATED VEHICLE .** ***.* *****" 'ALL VARIABLES DOUBLE PRECISION 'DISABLE THE VIDEO DESCRIPTORS OF FUNCTIONN KEYS REM ***.* REDEFINE THE FUNCTION KEYS REM KEY 1. "" : KEY 2. "SCR 1 "+CHR$ (13) I KEY 3, KEY 4, "SCR 2"+CHR$(13) KEY 5, KEY 6. "SCR 3": KEY 7, "" KEY 8, "SCR 4" : KEY 9, "" : KEY 10, "" KEY (2) ON KEY (4) ON : KEY (6) ON KEY (8) ON REM CLS REM REM ** CHOSE SINGLE OR ARTICULATED VEHICLE REM JUNK = 'VARIABLE DEFINED FOR NEXT QUI l=SINGLE VEH : 2=ARTICULATED VEH LOCATE 10, 5: INPUT;" IS THIS FOR AN ARTICULATED VEHICLE (Y or 0$ CLS IF QS = "Y" OR CIS = "v" THEN JUNK = 2 : LOAD "TESTART.BAS",R REM REM ************ SEQUENCING SUBROUTINES REM ** SINGLE VEHICLE REM **.** *.** INPUT INITIAL VALUES REI'! REM REM REM REM REM REM GOTO 930 GOSLIB 2640 GOSUB 640 INPUT INITIAL VALUES STORE INITIAL VALUES STEP TIME COUNTER ************ BOSUB 2860 GOSUB 640 JTERATION FOR STEP TWO COMPUTE LEVEL TWO STEP COUNTER ************ ALL SUBSEQUENT ITERATIONS Mcr = 1 WHILE TM (= TMMAX GOSUe 3200 GOSUB 4750 GOSUe 760 WHILE MCT I GUSUe 2640 Mel WEND MC"I =' I GOSUe 640 WEND BOSUB 760 GOSUB 2640 COMPUTE VALUES AT BASE LEVEL COMPUTE FIRS1 AND SECOND" DERIVATIVES ROTATE VALUES IN SHORT TERM STORAGE STORE COMPUTED VALUES STEP COUNTER ROTATE VALUES STORE CmlPUTED VALUES REM *** ...... *****.** PRINT OUT FROM SlORAGE REM GIJSI.lB 5031.1 ******.*** 57(. 'F REDO 1 THEN GOTO 2290 580 REI'! .... *** ... ******* END OF PROGRAM 591) REM 6(l(, END 610 REM 105

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620 630 640 650 660 67() 680 690 700 710 72(1 730 740 750 760 770 780 790 BOO 810 820 830 B40 850 860 870 880 890 900 910 920 930 940 950 960 970 980 990 REM ********************************************************************* REM REM REM ****************** STEP TIME INCREMENT AND ALPHA ***************** REM TM TM H ALPHA = ALPHA + H (.08) IF ALPHA .12 THEN ALPHA K(LIMIT-l.3) TM K(LIMIT.3) = ALPHA REM RETURN REM .12 REM ... *********** .. ********************************************* REM ******************* ROTATE VALUES REM REM COUNTERS USED N = I WHILE N LIMIT = WHILE L <= 2 L N K(N,L) K(N.L+I) RIN.L) RIN.L+l) SIN.L) S(N.L+l) = + WEND N = N + I WEND REM RETURN REM REM *********** REM INPUT VALUES FOR SINGLE VEHICLE CASE ******************* REM ************************* REM INPUT INITIAL VALUES ******************** REM INITIALIZE DATA STORAGE REM 1 (1)0 0 I M A (t 3. 13). M 1 13, 13), K ( 13. 13). R (7,3), S 1010 OPEN "DATA. OAT" AS til LEN 104 1020 FIELD til. 8 AS AU.8 AS A2S.B AS A3S,8 AS Bl$,8 AS AS G2S,B AS G3S,S AS HIS,S AS Jl$,S AS PlS.8 AS P3$ 1030 REM 11">40 REM 1050 REM 1060 REM 1070 REM A IS XI. e IS YI. G IS THETA. H IS 1M .J AL P J 5 FORCE III IS DISTANCE FROM CG TO FRONT AXLE 10Ell) 01 = 4.5 11)90 REM WI IS THE WHEELBASE (FEET! WI '" 10 REM T IS THE TRACK WIDTH T 5.1 T(13) B2S.8 AS B3S.8 AS 61$,8 100 11 () 1130 REM WTl IS THE WEIGHT, HTI IS THE HEIGHT OF THE CENTER OF MASS 1 1 41) WT 1 4000 : Hl I 1. 75 1150 REM 1160 REM KEY 2 ACIVATES THE FIRST INPUT SCREEN 1170 ON KEY(2) GOSUB 1280 1180 REM .(EY 4 ACIVATES THE SECOND INPUT SCREEN 1 1 90 ON KEY (4) GOSUB 1660 121)0 REM KEY 6 AC!VATES THE THIRD INPUT SCREEN 1<.1:0 122Q REM KEY II.l INITIATES THE PROGRAM 1230 ON t
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126(1 1270 1280 1290 13(1) 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 14:2(1 144(1 1450 1460 1471) 148(1 1'190 1500 1510 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1631) 1640 1650 1660 1670 REM F1Rsr TNF"IJ't SCREEN REM CLS LOCATE 1,5 : PRINT "THIS IS SCREEN 1 (VEHICLE )" LOCATE 3,10: PRINT "SCREEN 2 IS ROADWAY PARAMETERS ( F 4 RETURN )" KEY(4) ON LOCATE 4,IU: PRINT "SCREEN 3 IS INITIAL CONDITIONS (6) ON LOCATE 6.5 : PRINT "VEHICLE WHEELBASE LOCATE 6,601 PRINT LOCATE 6.60: PRINT 11.11 (FEET> LOCATE e .51 PRINT TRAGK WIDTH LOCATE 8 ,601 PRINT LOCATE 8 ,6U: PRINT LOCATE 10,5 : PRINT "WEIGHT OF VEHICLE (POUNDS)" LOCATE 10,60: PRINT LOCATE '10,6u: PRINT WTl F 6 RETURN )" LOCATE 12.5 : PRINT "HEIGHT OF C G (IF VEHICLE' (FEETl" LOCATE 12,6UI PRlNT HTI LOCATE 14,5 : PRINT "YOU CAN INPUT THE LONGITUDINAL LOCATIUN OF THE C G" L.OCATE 15,10: PRIN'! "1. BY INPLlHNG THE DISTANCE FROM THE FRONT AXLE 01"" LOCATE PRINT "2. BY INPUTING THE WEIGHT PERCENTAGE UN FRONT AXLE" LOCATE 17-,15: PRI Nr "YOUR CHOtCE IS" LOCATE 18,5 : PRINT "DlSTANCE FROM FRONT AXLE" LOCATE PRINT" LOCATE 16,6U: PRINT Dl I_OCATE I PR l NT "WE I GHT PERCENTAGE ON THE FRONT A LE" LOCATE 19,60: PRINT" LOCATE 19,60. PRINT (WI-Ol)/WI) LOCATE 6,60 : lNPLIT" ", BES : IF GES THEN WI GES GES'" LOCATE 8 ,61): lNPUT ", GES : IF BES <> THEN Ti = GES : GES LOCATE 10,6tJ: INPLIT ", BES : IF GES THEN WTl = GES: GES 0 HI = II = HI T*T + 1.8 1.8 11.11 "11.11 ) LOCATE 12,601 INPUT" ", GES tF GES (> THEN Hil = GES. GES 0 LOCATE 17,46: INPUT" ", CHI IF CHI = 2 THEN BOTO 1630 LOCATE I B,61 NPUT ", GES : IF GES THEN 01 GES: GES 1..0CA1'E 19.60: PRINT (WI-Oil/WI) IF CHI = 1 THEN CHI = GOTO 1650 LOCATE 19,6'-': INPI,JT ", GES : IF GES THEN 01 LOCATE 18,6u: PRINI DJ GOTO 154(1 CL5 : 17(1;) REM 107

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1080 1690 1700 1710 1720 1730 1740 1750 1760 177(1 1780 1790 1800 1810 1820 IB30 IB40 IB60 1870 IBBO lB90 19()O 1910 1920 1930 REM SECOND INPUT SCREEN REM LOCATE 1,5 PRINT "THIS IS SCREEN 2 ( LOCATE 3,10: PRINT "SCREEN 1 IS VEHICLE ROADWAY:PARAMETERS )" PARAMETERS ( F RETURN )" KEY(2) ON LOCATE 4,10: PRINT "SCREEN 3 IS INITIAL KEY(6) ON CONDITIONS F 0 RETURN )" LOCATE 7,5 : PRINT "GRADE OF ROADWAY (PERCENT> ": LOCATE 7,6(1: PRINT" LOCATE 7,601 PRINT SR LOCATE ,5: PRINT "SUPER-ELEVATION OF ROADWAY (PERCENT) LOCATE 9 ,601 PRINT LOCATE ,601 PRINT SE LOCATE II,S ; PRINT "LEVEL DRAG FACTOR BETWEEN RPAD AND TIRES" LOCATE 11,601 PRINT" LOCATE 11,601 PRINT F LOCATE 13,5 ; PRINT "RADIUS OF CURVATURE OF OUTSIDE EDGE OF" LOCATE 14,101 PRINT "LANE OF TRAVEL" LOCATE 14,601 PRINT" LOCATE 14,60: PRINT RRD LOCATE 7,6UI INPUT" ", GES; IF LOCATE 9,00; INPUT GES; IF LOCATE 11,6u: INPUT ,GES; IF LOCATE 14-,60; INPUT" ", GES: IF = SE GR GOTO CLS : RETURN 1960 RE"M GES GES < GES GES .: 108 THEN o THEN THEN o THEN GR GE5 SE GES F! = GES RRD= GES GES GES GES GES

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*.*.*.***.** 2 ": 7,11); (1),0)" VI = 0" "1. 10 1 = 0 = IF =1 = 109

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2210 2220 2230 2240 2250 2260 2270 2280 2300 2310 2320 2330 2340 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 245(1 2460 247(1 2480 2490 2500 2510 2520 2530 2540 2550 2560 2570 2590 2600 2610 2620 2630 2640 REM ********** POSITION VALUES REM REM REM REM REM REM o ISPLACEMENT VELOCITY, ACCELERAT IONa FORCES: XI, VI, THETA XIDOT, VIOOT, THETAO )ClOD, V1DD, THETADD FI, F3 XIDOT 1.466 SQRC 15 GOALRAD) K(I,ll .. Xl K(2,l) .. YI 1(3,1) THETA Fl K(5,1) F3 Xl = VI .. 0 THETA = 0 Fl .. R(l,t) XIDOT = VIDOT ,. VtDD .. 0 THETAD = THETADD '" .(J REM SIl,l ) R(2,l ) S(3,ll VIDOT :z VIDO THETAD THETADO REM *********** SET TIME AND INCREMENT LIMITS REM H .05 TM .. TM 'rMMAx ALPHA ,", 0 REM REM ************ REM Cl I REM PRINTER CONTROL REM ************* BLANK COUNTERS REM J K '" :a 11 I REM ALPHA REM ********** SET LIMtT ON ARRAV VALUES REM LIM!l' .. 7 REM GOTO REM 110

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2650 2660 2670 2680 2690 27(10 2710 2720 2730 2740 2750 2760 2770 2780 2790 2800 2810 2820 2830 2840 2850 2860 '2870 2880 2890 2900 2910 2920 2930 2940 2950 2960 2970 REM ************************ STORE COMPUTED VALUES SINGLE VEH ******** REM REM COUNTERS USED M REM LSET AIS= MKD$CCFIXCKC1.1)*10000/l0000) LSET A2$= MKDSCCFIXCRI1,l)*10000/10000) L5ET MKDSCCFIXCSC1,l)*10000/l0000) LSET 815= MKD$ICFIXCKC2.t)*10000/l00001 L5ET 825= MKD$ C IFJ X CR C2, 1) *10(00) ) 110000) L5ET 835= MI
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3210 3230 3240 3250 3260 3270 3280 3290 3300 3310 3320 3330 3340 3350 3360 3370 3380 3390 3400 3410 3420 3430 3440 3450 3460 3470 3480 3490 3501) 3510 3520 3530 3540 3550 3560 3570 3580 3590 3600 361fJ 3620 3630 3640 3650 3660 3670 3680 3690 3700 REM THIRD AND SUBSEQUENT (SING VEHi REM REM ESTABLISH MAXIMUM VALUE OF FRICTION FORCES FOR NO-SLIP REM F!MAX F IIH-D!) WI) WTt F3MAlC = F C I WI-Dil Wll ) WTI REM REM BLANK COEFFICIENT STORAGE OF MATRIX REM I 1 WHILE I < 6 WHILE J 7 ACI,J) 0 J + 1 WEND I = J 1 WEND REM REM INPUT COEFFICIENTS FOR SOLUTION AT TIME STEP' REM ACI,ll A (1,4) ACI,5) A (1,6) REM A(2,2) AC2,4) AC2,5l AC2,6) REM A(3,3) A(3,4) A(3,5) A(3,6) REM A(4,ll AC4,2) AC4,3) AC4,6) REM A(S,! ) A(S,2) A(S,3) A(S,6) REM NOTY MEMI=1 NTEST REM (HA2*SINCK(3,2) + ALPHA MI MI '2*K ( I 2) -K 1) ALPHA MI MI 2*K(2,2) -K(2,1) -CH*H*Dl*COSCALPHA II (H*H*CW1-Dl II 2*K(3,2) -K(3,1) -SQRT*T/4) + DI*Dl)*
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3710 REM TEST FOR Fl FORCE AT MAXIMUM 3720 REM 3730 IF (ABS(Ft ( F1MAX THEN NGlTV = NGlTV + t ELSE NTEST .. 3740 WHILE NTEST t 3750 MEM! = 0 3760 t 3770 WHILE 6 3780 A(I,6) A(I,6l -A(I,4l FIMAX 3790 A(I,4) A(I,S) 3800 J 8 + 3810 WEND 3820 NTEST = 3830 WEND 3840 REM 3850 REM TEST FOR FORCE AT MAXIMUM 3860 REM 3870 IF (ABS(F3 F3MAX THEN NGlTY = NGITY ELSE Nl:EST 3880 WHILE NTESl 3890 MEM3 3900 391() WHILE 6 3920 AH,6) = A(I,6l A(I,S) F3MAX 3930 AtI,S) 0 3940 J J 3950 WEND 3960 NTESl 3970 WEND 3980 IF MEM! THEN GOTO 40S0 3990 4000 WHiLE 1 6 4010 A(I,4) A(I,S) 4020 A(I,S) '" A(I,6) 4030 I 4040 WEND 4050 IF MEM3 THEN GOTD 4150 4060 I = 1 4070 WHILE I ( 6 4080 A ( I ,4) A ( I 5 ) 4090 A(I,S) = AII,6) 4100 I = I + 1 4110 WEND 4120 REM 113

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4130 REM BESIN GAUSSIAN ELIMTNATION 4140 REM 4150 I 1 4160 WHILE I NQTY P I WHILE P (NCITY + 1) IF A(P,I) <> THEN = P P P 1 WEND IF N THEN GOTO 4310 K = J ,WHILE K (NCITY + 2) PIVOT = A(I,K) A A(I,I> K c: I WHILE K (NGITY + 2) (NCITY + 4170 4180 4190 4200 4210 4220 4230 4240 4250 4260 4270 4280 4291) 4300 4310 4320 4330 4340 4350 4360 4370 4380 4390 A(J,IO = AeJ,K) -M(J,I) A(I,K) K = K 4400 4410 WEND J = WEND I = I 1 WEND T (NIHY) = A (NtITY, (NOTY+1) ) (NGITY ,NClTY) REM REM CHECK FOR MAXIMUM FORCES AT CURRENT TIME STEP REM NOTE TIME OF SLIPPAGE FOR EACH AXLE REM 4420 4430 4440 4450 4460 4470 4480 IF (N(HY 5 AND MEM3 = 1 AND ABSeT(5 :, F3MAX THEN T(5) = F3MAXo TM TMLCK3 4490 IF (NOTY,'" TMLCK3 4 AND MEM!= AND ABS(Te4 F3MAX ) THEN T(4) F3MAX; TM 4500 IF (NOTV,= 4 AND MEM1= TMLCIO 4510 = NOTY 4520 WHILE 4530 + 454n GlTV = 4550 WHILE J { TM (NDlY 1) AND ABS 4) ) 4560 QTY 4570 DTY + A(I,J)*T(Jl + 1 4580 WEND FIMAX ) : THEN T(4) 4590 T (I> ( A (I, (NOTY+! CITY ) A (I. II 4600 IF (NQTY TMLCKI 4610 I = I -1 4620 WEND 5 AND MEM!= AND ABSeT(4 ) FIMAX ) THEN T(4) = rM 114 FIMAl(; FIMAX;

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4630 K 3 ) T 1 1) I K (2,3) T 12 ) 4640 K(3,3) = T(3) K14,3) = T(4) 1,3) 4650 IF MEMI c THEN K(4,3) ... FIMAlC 4660 IF (MEl'll AND MEM3 .. 1) THEN K(5,3) 4670 IF (MEM3 0) THEN K(S.3) F3MAlC 4680 FIMAX F 1 (WI-01) WI) WTl 4690 F3MAlC a F I WI-OI) WI) ) WTl 4700 FI = K(4,3) 4710 F3 = K(S,3) 4720 1,3) = TM 4730 RETURN 4740 REM 4750 REM K(7,3) ALPHA = T(S) T(4) 4760 REM COMPUTE FIRST AND SECOND DERIVATIVES VALUES 4770 REM 4780 REM 4790 REM 4800 ( 1 1 ) 4810 (1,2) 4820 K
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***************** *************** = Q #1, I< = K = I K (3' 1 I = Q 116

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REM REM ************ DETERMINE RADIUS ********************** REM REDO = 0 J = IF BOALA)( IF BOALA)( WHILE I < 4 1 THEN NMB 2 THEN NME! TMLCKI H TMLCK3 H IF] 1 THEN NN1 = NM8 5 IF I 2 THEN NN1 NM8 + 15 IF I 3 THEN NNJ = NM8 25 BET #1, NN1 Xl CVDIAlS) VI CVDI81S) IF THEN XVEL CVOIA2S) IF 1 'HEN VVEL CVO (82$) IF = THEN VEL = SGiR 1 XVEL XVEL + VVEt. YVEL ) THETA CVOIB1S) IF JUNK 2 THEN X2 CVDIC1S) IF JUNK 2 THEN V2 = CVDCD1$) IF JUNK 2 THEN PHI = CVDIE1$) XXII.I) XI 01 COS (THETA) + YYII,J) YI + 01 SINITHETA) IT/2) COSITHETA) 5320 5330 534n 5350 5360 5371) 5380 539() 5400 5410 5430 5440 5450 5460 5470 5480 5490 550C) 5510 5520 5530 5550 556(1 5570 5580 5590 IF JUNK 2 THEN XXCI,3) = X2-(W2-D3).COSCPHI) IT/2).SINCPHI) IF JUNK 2 THEN YYII,3) = IT/2)*COSIPHI) XX 11.,2) XI (WI-DlI COSCTHETAI .. SINITHETA) VYII,2) YI (WI-OJ) SINITHETA) COSITHETA) 1 1 5600 WEND 5610 REM LPRINT TABCI01"VELOCITY IS "I TAB(25) INT/VELl; TAB(32) "FEET PER SECOND" 5630 LPRINT TAB 1 10) "SPEED IS"; TAB (25) INT CVELll. 466); TAB (32) "MILES PER HOUR" 564() LPRINT TAB(10) "DRAB FACTOR USED IS ", TA8(32) F 565() I 1 5661) IF JUNK 2 THEN BORP 4 ELSE BORP = 3 5670 WHILE I BORP 5681) 5690 GlT2 XXC3,I)A2 + YYC2,I)A2 -YY (2, I) XXC1,I)A2 + YYC2,IIA2 YY (1, I) ) ) 5700 5710 5720 OT3 XX(I,I) XXC2,II I YYC2,1) -VY(l.l) GlT4 XXI2,Il XX(3,I> ) YY(3,I> YY(2,1l XR = IGiTI QT2) GlT3 GlT4 ) 573(1 YR OT3 GlT2 574(1 RAD(1l = ( :':R)A2 (VV(I,I) YR)'') 5750 PRINT "RADIUS "RAD(I) LPRlt.JT "RADIUS OF AXLE "J" "RAD (J ; 5770 = 1 1 5780 WEND 5791' PRINT "XIDOT XIDOT 5800 LPRINT "SLIPPING BEBINS AT TIME "TMLCKI" AND "TMLCK3 581(l LPRINT ": LPRINT .. 5820 DIFF = BOALRAD RADCBOALAX) IF DIFF THEN REDO I 5840 IF DTFF ;. BDALRADI THEN REDO = ELSE BOTO 600 5850 )(1001 = XIDOI ... .2*DTFF)/BOALRAP) 586() PRINT "OIFF "OIFF 5870 PRINT "XIDOT XIDOT 588n REM 5890 RETURN 591)0 REM THIS IS THE END OF THE CALCULATIONS 12*(YV (3, I) 12*(YY(2,l) 5910 REM 117

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118

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rIME Xl00T ;'(1-0D .,.1 Y1DD1 yt-Dll .34.7692 1.7384 34.7691 -.0(149 .002 .1073 1.9107 .1 3.4769 34.7689 -.0068 .1)099 2.1525 34.7685 -.0112 .0231 2.3945 6.9537 34.7677 -.0183 .1)423 .4483 2.6365 8.6921 34.7666 -.0261 .0681 .5862 2.8786 10.4304 34.765 -.0401 1011 .7362 12.1686 34.7628 -.0542 .1418 .8983 .4 13.9066 34.7599 -.0687 .191 1.1)725 3.6053 15.6446 34.7561 -.0861 .2492 1.2588 3.847B .5 17.3822 -.11 .3171 1.4573 4.0904 19.119-; -.1339 1.6679 4.3331 .6 20.8568 34.7379 .484 1.891)6 4.5761 22.59361 34.7291 4.B19 24.3297 34.7185 -.2317 .6967 2.3725 5.0621 ... 26.065:::: 3.q.706 .... 270B .8218 2. .8 27.8002 34.6913 -.3142 .961)1 2.9031 5.549 34.67.111 -.3664 3.1866 .9 34.6544 -.4225 1.2789 3.4823 32.9998 3'1.6:.H8 -.4S3=' 1.4606 ::::.7902 6.2792 34.606 -.5508 1.658 4. I 11)'2 6.5226 1.05 :;/1.5767 -.6219 1.8718 -1.44211 ':1.766;;' 1.1 38.1884 34 ... 5436 -.699=' 2.1024 4.786B "/. (1)98 15 .. '9.9147 34.5065 -.786 2.3506 5.1.113.11 7.252<; 4 .6-::") 34.4640 -.8754 2.61.,9 5.5121 7.4963 1.2=. i3.3611 :.54.4185 -.9769 2.90:' 5.893 7.7387 45.0808 .34.367 -1.0837 6.286 7.981 46.7977 34.31 -1.1963 6.6911 8.2232 1. 48.5117 34.2471 -1.3207 3.8757 7.1083 8.4641 50.2224 3.11.1779 -1.4483 4.2417 7.5375 8.7053 51.929/1 34.1056 -1.2967 4.6296 7.9578 7.2698 1.55 53.6331 34.0389 -1.3686 5.0365 7.2654 1.6 55.333:. .33.9686 -1.444 5.4617 8.6843 7.2595 1. 65 57.02<;19 33.8946 -1.5165 5.905 9.0471 7.2537 58.7227 -1.594'; 6.3664 9.4096 7.2456 1. 60.4115 -1.6685 6.845<; 9.7717 7.2381 1.8 33.6498 "'1.745:: 10.1334 7.:229 1.85 33.5606 -1.8234 7.8593 10.4946 7.2188 t;' 33.4675 -1.8G17 B.39::. 10.8553 7. I. 67.1237 33.3706 -1.977[. 11.21::i4 7. 68.789:; 33.2698 -2.055:. .'.1841 33.1651 1(1. 10:;: ::. 11.9336 7.1711 1 :;:':'.0565 10.71)8 12.292 7. 15614 :2 .. 32.Q44 -2.2897 12.649::, 7.141 J 32.8276 -2.3667 11.9729 13.01.16:2 l.1255 -2.446<1 12.6321 13.362 7.1078 -0 32.5831 -::.5256 13.3()91 13.717 7.1)898 <, 80.2967 32.455 -2.601:? 14.0036 14.0711 2.4 81.916"2 3':.3229 -2.6921 14."162 14.4242 7.051.3 83.5289 3:'.1869 -2.7566 15.446;-1.11.7763 85.1348 32.047 -2.8388 16.1938 7.0096 2.5:' 66. -2.9151 16.9589 6.9881 '2.6 88.3251 31.7554 ... 2.9929 17.7415 15.8261 6.9652 89.9()91 31.6038 -3.0721 18.5415 16.1737 6.94.14 91. 31.4483 -3.1499 19.3589 16.52(11 6.9155 :? 31.2888 16.8653 6.8898 2.8 9'1.6147. 31.1255 17.2091 6.862::: 30.958:' :'1.914' 17.5515 6.8339 2. 97.7; 3(1.7972 -.::.4607 22.8(1)6 17.892:> 6.8064 119

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PAGE 126

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PAGE 127

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PAGE 128

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PAGE 129

6.25 161. 1143 16.0915 -7.1014 94.399 32.7657 3.9815 6.3 161.91 15.7351 -7.1519 96.0323 32.9629 3.9063 6 . 35 162.6978 15.3761 -7.2059 97.6853 33.1563 3.8329 6.4 163.4476 15.0145 -7.2583 99.3478 33.346 3.756 6.45 164.1892 14.6503 -7.3073 101.0198 33.5318 3.6769 6.5 164.9126 14.2836 -7.3572 102.701 33.7139 3.6011 6.55 165.6175 13.9145 -7.4056 104.3912 33.892 3.5235 6.6 166.304 13.5429 -7.4587 106.0901 34.0662 3.4475 6.65 166.9718 13.1689 -7.504 107.7977 34.2365 3.3673 6.7 167.6209 12.7926 -7.5478 109.5137 34.4028 3.2814 6.75 168.251 12.4139 -7.5946 111. 238 34.5651 3.2032 6.8 168.8622 -7.6391 112.9702 34.7233 3.1246 6.85 169.4543 11.6498 -7.6864 114.7102 34.8774 3.0418 6.9 170.0272 11. 2645 -7.7303 116.4579 35.0274 2.9584 6.95 170.5807 10.877 -7.77 118.2129 35.1733 2.874 7 171.1149 10.4874 -7.8102 119.9752 35.315 2.7879 7.05 171.6295 10.0958 -7.952 121.7444 35.4524 2.7078 7.1 172.1244 9.7021 -7.8895 123.5204 35.5856 2.6176 7.15 172.5997 9.3065 -7.9326 125.3029 35.7146 2.5367 7.2 173.0551 9.9089 -7.9691 127.0918 35.8392 2.4487 7.25 173.4905 -8.0047 128.8868 35.9595 2.3628 7.3 173.906 8.1082 -8.0431 130.6977 36.0754 2.2721 174.3013 7.7052 -8.0782 132.4943 36.1869 2.185 7.4 J74.6765 7.3004 -8.1126 134.3063 36.294 2.0968 7.45 175.0313 6.8939 -8.1442 136.1236 36.3967 2.0075 7.5 175.3659 137.9459 0 0 124

PAGE 130

THETA rHE1'A-D THETA-DO PHI PHI-DO" PHI-DO t) 0 0 0 .05 .0001 .0086 .139 0 .0009 .0198 1 .0007 .0156 .139 0 .002 .0262 .15 .0017 .0226 .139 .0002 .0035 .0323 .2 .003 .029:5 .1391 .0004 .0052 .038 .25 .0046 .0365 .1391 .0007 .0073 .0434 .3 .0066 .0434 .1391 .0011 .0096 .0485 .35 .009 .0504 .1391 .0017 .0121 .0534 .4 .0117 .0573 .1392 .0023 .OI4'i' .059 .45 .0147 .1393 .0032 .0179 .0624 .5 .0181 .0713 .1393 .0041 .0211 .0665 .55 .0219 .0782 13'i'4 .0053 .0704 .6 .026 .0852 .1395 .0066 .0282 .0741 .65 .0304 .0'i'22 .1396 .0081 .032 .0776 .7 .0352 .0'i'92 .1397 .0098 .036 .081 .75 .0403 .1062 .1398 .0117 .0401 .8 .0458 .1132 .14 .0138 .0444 .0872 .85 .0516 .1401 .0162 .0488 .09 .9 .0578 .1272 .1403 .0187 .0534 .0927 .95 .0644 .1342 .1405 .0215 .0581 .0953 1 .0712 .1412 1407 .1)629 .0978 1.05 .0785 .1483 .1409 .0679 .1002 1.1 .0861 .1553 .1412 .1024 15 .094 .1624 .1414 .0351 .0781 .1046 1.2 .1023 .1695 .1417 .0391 .0834 .1066 25 .111 .1766 .142 .0435 .0888 .1086 1.3 .12 .1837 .1424 .048 .0943 .1105 .1293 .1908 .1427 .0529 .0998 .1123 1.4 .1391 .198 .1431 .058 lOSS .1141 1.45 .1492 .2051 .1435 .0634 .1112 .1158 1.5 .1596 .2105 .0023 .1169 .1043 1.55 .1701 .2107 .0023 .0751 .122 .0989 1.6 .1807 .2108 .0023 .0814 .1268 .0938 1.65 .1912 .2109 .0024 .0878 .1314 .089 1.7 .2018 .211 .0024 .0945 .1357 .0844 1.75 .2123 :111 .0024 .1014 .1398 .0801 1.8 .2229 .2113 .0024 .1085 .1437 .076 85 .2334 .2114 .0024 .1158 .1474 .0721 1.9 .244 .2115 .0025 .1232 .151 .0684 1. 95 .2546 .2116 .0025 .1309 .1543 .065 2 .2652 .2118 .0025 .1387 .1575 .0617 2.05 .2758 .2119 .0025 .1466 .1605 .0585 2.1 .2864 .212 .0025 .1547 .0556 2.15 .297 .2122 .0025 .1629 .166 .0528 2.2 .3076 .2123 .0025 .1713 .1686 .0501 2.25 .3182 .2124 .0025 .1798 .1711 .1)476 .3289 .2125 .0025 .1884 .1734 .0452 2.35 .3395 .2127 .0025 .1971 .1756 .043 2.4 .351)1 .2128 .0025 .206 .1777 .0408 :Z.45 .3606 .212'" .0025 .2149 .1797 .0388 .3714 .213 .0025 .224 .1816 .0369 .382J .2132 .0025 .1834 2.6 .3927 .1)025 .2423 .185t .0334 .21;:;4 .0025 .2510 .1867 .0317 ::.7 .4141 .2136 .0025 .261 .1883 .0302 .4248 .2137 .0025 .2704 .1897 .0287 .4355 .0025 .2799 .1911 '4462 .213<;1 .0025 .2895 .026 2.Q .456<;1 .2141 nr)25 .299:.1938 .0248 214::

PAGE 131

-.4783 .2143 .0025 .3187 .1961 .0225 3.05 .489 .2145 .0025 .3285 .197'2 .0214 3.1 .4997 .2146 .0025 .3384 .1983 .0205 3.15 .5105 .2147 .0025 .3484 .1993 .0195 ..:..-.5212 .2148 .0025 .3584 .2002 .0186 3.25 .532 .215 .0025 .3684 .2011 .0177 .3.3 .:5427 .2151 .0025 .3785 .202 .0169 3.35 .5535 .2152 .0025 .3886 .2028 .0162 3.4 .5642 .2154 .01)25 .3988 .2036 .0155 3.45 .575 .2155 .0025 .409 .2044 .0148 .5858 .2156 .0025 .4192 .2051 .0141 ::.55 .5966 .2158 .0026 .4295 .2058 .0135 3.6 .6074 .2159 .0025 .4398 :2065 .0129 3.65 .6182 .216 .0025 .4501 .2071 .0124 3.7 .629 .2161 .0025 .4605 .2077 .0118 3.75 .6398 .2163 .0025 .4709 .2093 .0113 3.8 .6506 .2164 .0025 .4813 .2089 .0109 3.85 .6614 .2165 .0025 .4918 .2094 (1104 3.9 .6723 .2167 .0025 .2099 .01 ::.95 .6831 .2168 .0026 .5128 .21(14 .0090 4 .694 .2169 .0025 .21 ()9 .0092 4.05 .7048 .217 .0026 .5339 .2113 .0089 4. J .7157 .2172 .0026 .5445 .2118 .0086 .2173 .0025 .2122 (1(182 4.2 .73711 .2174 .0026 .5657 .2126 .0079 4.25 .748:.'. .217t> .0026 .5763 .213 .0076 4.3 .7592 .2177 .t)026 .587 .2134 .01)74 4.35 .7701 .2170 .0026 .5977 .2137 .0071 4.4 .781 .218 .0026 .6094 .2141 .0069 4.45 .7919 .2181 .0026 .0191 .2144 .0066 4.5 .8028 .2182 .0026 .6298 .2148 .0064 4.55 .8137 .2184 .0026 .6406 .2151 .0062 4.6 .8246 .2185 .01)26 .651:: .2154 .1)1)0 4.65 .8355 ;"186 .06::1 .0059 4. 7 .8465 .2187 .672:" .0057 -. ;'184 .. 'JI)'2r-.68:-.' . .0055 .-;. 1.;. ..,945 .2165 .0054 ,,c: 19: .(1):<'6 .7054 .:2168 .0052 .J. .890:;' .2193 .0026 .7162 .2171 .0051 4.95 .9013 .2194 .0026 .7271 .2173 .0049 .2195 t)026 .738 .2176 .0048 .2197 .0026 .7488 .2178 .0047 5.1 .9342 .2198 .0026 .7597 .218 .0046 .9452 .2199 .0026 .7707 .2183 .0045 5.:? .9562 .2201 .0026 .78t6 .2185 .0044 .9672 .2202 .0(126 .7925 .2187 0043 5.3 .9782 .2203 .0026 .8035 .2189 .0042 .9893 .2205 .0026 .8144 .2191 .0041 5.4 1.0003 .2200 .0026 .8254 .2193 .1)04 5.45 1.011:0. .2207 .0026 .8363 .2195 .004 CO' 1.0':'::24 .2'2(.'" ')026 .8473 .21'H .0039 5.55 1.0334 .. 221 .0026 85El3 .2199 .0038 5.6 1.0445 .2211 .0026 .8693 .2201 .0038 ::5.65 1.0555 .(1)26 .8803 .22(1:; .0037 5.7 1.0666 .2214 .0026 .8914 .2205 .0036 1.0777 (11)26 .9024 .2207 .0036 S.B 1.0888 .2217 .1)026 .9134 .2209 5.85 J .0999 .2216 .n026 .. 9245 .Z21 .0035 5.9 I. 111)9 .2219 .0026 .9355 .:2212 .0035 CO' oCO' ....... .... 1.1221 .. :!221 .0027 .9466 .2214 .0034 126

PAGE 132

..... 6 1.1332 .0027 .9577 .2216 .0034 6.05 1.1443 .0026 .9688 .2217 .0033 6.1 1.1554 .2225 .0027 .979'9 .2219 .0033 6.15 1.1665 .0027 .991 .2221 .0033 6.2 1.1777 .2228 .0027 1.nn21 .2222 .0032 6.25 1. 1888 .2229 .0027 1. 0132 .2224 (1(132 6.3 1.2 .223 .0027 1.0243 .2226 .0032 6.35 1.2111 .2232 .0027 1.0355 .2227 .0032 6.4 1.2223 .2233 .0027 1.0466 .222'7 .0031 6.45 1.2335 .2234 .0027 1.0578 .223 .0031 6.5 1.2446 .2236 .0027 1.0689 .2232 .0031 6.55 1.2558 .2237 .0027 1. 0801 .2234 .0031 6.6 1.267 .2238 .0027 1. 0913 .2235 .003 b.65 1.2782 .224 .0027 1. 1024 .2237 .003 6.7 1.2894 .2241 .0027 1.1136 .2238 .003 6.75 1.3006 .2243 .0027 1. 1248 .2211 6.8 1.3118 .2244 .0027 1.136 .2241 .003 6.85 1.3231 .22115 .0027 1.1472 .2243 .003 6.'7 1.3343 .2247 .0027 1.1585 .2244 .003 6.'95 1.3455 .2248 .(1027 1.1697 .2246 .003 7 1.3568 .2249 .0027 1.1809 .22117 .003 7.05 1.368 .2251 .0027 1.1922 .2249 .0029 7.1 .2252 .0027 1.2034 .225 .0029 7.15 1.3'9(16 .2254 .0027 1.2147 .2252 .0029 7.2 1.11018 .2255 .0027 1.226 .2253 .0029 7 1.4131 .2256 .0027 1.2372 .2255 .0029 7.3 1.4244 .2258 .0027 1.2485 .2256 .0029 7.35 1.4357 .2259 .0028 1.2598 .2258 .0029 7.4 1.447 .2261 .0027 1.2711 .2259 .01)29 7.45 1.4583 .2262 .0028 1.2824 .2261 .0029 1.4696 0 1.2937 0

PAGE 133

ALPHA F5 Fe. FI F3 F7 (J 0 -1 0 0 .05 .0039 -6 -756 1669 951 208 .1 .0079 -8 -717 1007 844 225 .15 .0119 -12 -946 1122 1093 318 .0159 -18 -1178 1238 1345 420 .25 .0199 -27 -1413 1355 529 .3 .0239 -37 -1651 1472 1855 646 .35 .0279 -52 -1990 1590 2112 770 .4 -65 -2132 1707 2372 900 .45 .0359 -84 -2376 1826 2634 1037 .... -105 -2621 1945 2999 1179 55 -13l) -2969 2064 3163 1327 .6 .0479 -158 -3118 2184 3430 1480 .0519 -191 -3370 2304 3699 1639 .7 .0559 -228 -3622 2425 3969 1902 .75 .0599 -270 -3976 2547 4241 1970 .8 .0639 -315 -4132 2669 4515 2142 .85 .0679 -366 -4388 2791 4790 2318 .9 .0719 -426 -4646 2914 5067 2498 .95 .Q759 -490 -4905 5346 2681 1 .0799 -559 -5165 3163 5627 2869 1.05 .1)839 -635 -5426 3288 5909 3059 1.1 .0879 -719 -5688 3414 6193 3253 1.15 .0919 -808 -5950 3541 6479 3450 .0959 -907 -6213 3668 6767 3650 1. 25 .0999 -1012 -6476 7056 3853 1.3 .1039 -1126 3926 7348 4058 1.35 .1079 -1250 -7003 4056 7643 4267 1.4 .1119 -1380 -7266 4188 7938 4477 1. 45 .1159 -1521 -7528 4320 8237 4690 1.5 .1199 -1670 -7791 4453 8537 4906 1.55 .1199 -1636 -7569 3668 8222 4979 1.6 .1199 -1729 -7605 3679 8271 5115 1.65 .1-199 -1822 -7637 3690 8317 5243 1.7 .1199 -1916 -7666 3700 8362 5365 75 .1199 -2009 -7691 3711 8404 5481 1.8 .1199 -2101 -7713 3720 8444 5590 1.85 .1199 -2195 3730 8484 5695 1.9 .1199 -2288 -7750 3739 8522 5794 1.95 .1199 -2381 -7764 3749 8558 5889 .1199 -2478 -7777 3758 8594 5979 2.05 .1'199 -2569 -7786 3766 8627 6065 2.1 .1199 -2663 -7793 3775 866(. 2.15 1-199 -2758 -7797 3783 8693 6225 .119<;1 -2e51 -7800 3792 6299 .1199 -2945 -7798 3799 8752 6368 .1199 -3040 -7798 3808 8183 6438 .1199 -3132 -77Cj12 3815 8809 6500 ::'.'1 J '199 -3225 -7795 8935 6562 2.45 .1199 -3318 3831) 8863 6622 .1199 -3410 -7766 3837 8886 6676 2.55 .1199 -3504 3844 8911 6730 .1199 -3596 -7740 3851 8935 6782 .1199 -3689 -7724 3858 8958 6930 .1199 -3782 -7705 3864 9981 6877 .1199 -3871 -7683 3870 8999 6919 .1199 -3965 -7664 3977 9024 6964 .1199 -4058 -7641 3884 9045 7005 9 9Jb4 7n013 128

PAGE 134

.119<;1 -4330 -7562 3903 9104 7119 3.05 .1199 -4419 -7533 3909 9123 7154 3.1 .! 199 -4509 -7500 3914 9140 7186 3.15 .1199 -4599 -7469 3920 9159 7219 .1199 -4689 -7435 3926 9176 7250 .3.25 .1199 -4779 -7400 3932 9195 7281 .1199 -4867 -7363 3938 9212 7310 3.35 .1199 -4955 -7324 3944 9227 7336 3.4 .1199 -5(144 -7285 3950 9245 7364 3.45 .1199 -5132 -7245 3955 9262 7390 3.5 .1199 -5218 -7202 3961 9277 7415 3.55 .1199 -5305 -7159 3966 9293 7439 3.6 .1199 -5390 -7114 3972 9308 7462 3.65 .1199 -547G' -7069 3978 9326 7487 3.7 .119G' -5565 -7022 3983 9341 7509 3.75 .1199 -5648 -6971 3988 9354 7528 3.8 .1199 -5731 -6921 3994 9368 7548 3.85 .1199 -5817 -6871 3999 9384 7570 3.9 .1199 -5898 -6819 4004 9397 7588 3.95 .1199 -5984 -6766 4010 9414 7609 4 .1199 -6066 -6712 4016 9429 7628 4.05 .1199 -6143 -6653 4020 9438 7642 4.1 .1199 -6227 -6598 4026 9454 7662 4.15 .1199 -6309 -6540 4031 9469 7679 4.2 .1199 -6388 4036 9481 7695 .1199 -6467 -6420 4041 9494 7711 4.:c. .119<;1 -6544 -6357 4046 9506 7725 4.35 .1199 -6625 -6296 4052 9522 7743 4.4 .1199 -6233 4057 9535 7759 4.45 .1199 -6780 -6169 4062 9549 7774 4.5 .1199 -6856 4067 7788 4.55 .1199 -6930 -6036 4072 9573 7802 4.6 .1199 -7005 -5967 4077 9586 7816 4.65 .1199 -7081 -5900 4083 7831 4.7 .1199 -7154 -5830 4088 9612 7844 4.75 .1199 -7227 -5759 4093 9625 7858 4.8 .1199 -7300 -5688 4098 9638 7872 4.85 .1199 -737.":-5615 4103 9652 7885 4.9 .1199 -7443 -5540 4108 9663 7897 4.95 .11G'9 -7515 -5469 4114 9679 7913 5 .1199 -7583 -5391 4119 9689 7923 5.05 .1199 -7654 -5316 4124 9704 7939 .1199 -77'20 4129 9714 7949 5.15 .1199 -7788 -5160 413011 9728 7962 ., .... .1199 -7955 -5081 4139 9741 7975 .1199 -7919 -5000 4144 9752 7985 5.3 .1199 -7983 -4918 4149 9764 7996 5.35 .1199 -8049 -4838 4155 9778 801'0 5.4 .1199 -8110 -4753 4159 9788 8020 5.45 .1199 -8176 -4671 4165 9804 8034 5.5 .1199 -8233 -4583 4169 9811 8041 5.55 -8301) -4502 4176 9831 8059 5.6 J 199 -4412 4180 9837 8064 5.65 .1199 -8415 -4326 4185 9851 8077 .... .1.199 -8475 -4238 4191 9865 8091) .1199 -8533 -4151 4196 9879 8103 .1199 -8588 -4061 4201 9891 81 J 3 5.85 .1199 -8641 -3970 4206 990(' 8122 5.Q .119c -8695 -3878 4211 991::, 8133 5.95 .1199 -8749 -3785 4216 9925 8143 129

PAGE 135

6 -8801 -3693 4221 9937 9154 6.05 .1199 -8854 -3600 4227 9951 8166 6.1 -8906 -3506 4232 9964 8179 6.15 .1199 -8957 -3411 4238 9978 9190 6.2 .1199 -9004 -3316 4243 9989 8200 6.25 t -9050 -3219 4247 10000 9210 6.3 .1199 -9096 -3121 4253 10011 8219 6.35 .1199 -9145 -3024 4259 10027 8233 6.4 .llQ9 -9186 -2924 4263 10036 9241 6.45 .1199 -9229 -2926 4268 10049 8252 6.5 .1199 -9274 -2727 4274 10064 9265 6.55 .1199 -9314 -2625 4279 10074 8273 6.6 .1199 -9356 -2524 4284 10088 8285 6.65 -9391 -2421 4289 10097 9292 6.7 .1199 -9433 -2319 4295 10112 8306 6.75 .1199 -9471 -2218 4300 10127 8319 6.8 .1199 -9505 -2113 4305 10137 8327 6.85 .1199 -9541 -2008 4311 10150 8338 6.9 -9573 -1904 4316 10161 9347 6.95 .1199 -9606 4321 10174 8358 7 .1199 -9638 -1693 4327 10188 8370 7. ()5 .1199 -9671 -1587 4332 10202 8382 7.1 .1199 -96<;>9 -1478 4337 10213 8391 7.15 .1199 -9728 4343 10227 8404 .1199 -9751 -1262 4348 10237 8410 7.25 .1199 -9780 -1155 4354 10252 8424 7.::. .119<;> -9804 -1045 4359 10264 8434 7.35 .1199 -9827 -937 4364 10276 8444 7.4 .1199 -9848 -826 4369 10289 8454 7.45 .1199 -9869 -715 4375 10301 8465 7.5 .1199 -9892 -604 4381 10316 8478 130

PAGE 136

RCD 5 CKRAD 2 XPT( I -8.520647854320146 -3.498499373851717 VELYAW = 34.76952743530273 RCD 15 Ct
PAGE 137

APPENDIX G OUTPUT OF ARTICULATED MODEL WHICH INCLUDES WEIGHT SHIFT AND BRAKING 132

PAGE 138

RCD = YPT ( 1 VELYALoi RCD = 17 -1.403328538365476 -3.313893755097687 70.72492218017578 CKRAD 2 XPT( 2 ) YPT( 2 ) VEL YAW = RCD = 27 16.22464831666201 -2.056613475215435 70.72492218017578 CKRAD = 2 XPT( 3) YPT( 3 .3034424406647682 VELYAW 70.72492218017578 QT1 OT2 101 217535967876,4 OT3 -14.02072166174307 OT4 -7.453248167454503 XR -12.84875274069669 YR 281.3663218457431 RADVALoi = 284.9102172851563 REFERENCE AXLE 2 RADIUS IS 284.910217285156 lTME OF SLIP 5.0()l)uuOQ74:;U l::lu6l.l-{) VELOCITy SPEEI, AI SI.IF" is :3 .. :n4679 J X I ()[ll : 0'. 8':,'<;171)'1-$,.,"74,,,,87:' = CI
PAGE 139

RCD = -, CKRAD 2 .0515714616345242 VPT( 1 -3.313993755097687 IJELYAW 80.41'906738281:25 RCD '" 17 CKRAD = :2 XPT( 2 ) VPT ( 2 ) VEL YAW Reo = 27 20.10588812197447 80.4190673828125 CKRAD = 2 XPT ( 3 ) 41),.12638274800181 VPT( 3 ) .274091010427475 IJELYAW 80.4190673828125 QTl 256.9343084453917 QT2 159.0649413333062 QT3 -16.04906222836246 QT4 -8.561175763306516 XR = -13.07035938229259 YR 368.8319524067808 RADYAW = 372.3771667480469 REFERENCE AXLE 2 RADIUS IS 37:2.3771667480469 TIME OF SLIP 5.0000000745058060-02 VELOCITY AT SLIP 15 (fps) 80.4190673828125 SPEED AT SLIP 15 54.85611836815977 OIFF 27.62283325195313 HDOT"= 82.08511931546021 CKRAD 2 XPT( 1 .3057714616345242 VPl ( 1 -3.314093755097687 VELVAW 92.113525390625 RCD = 17 CKRAT> = 2 XPT( 2) 20.7943230749324 VPT( -2.066134900343418 VELVAW = 92.113525390625 RCD CKRAD = 2 XPT( 41.22953364977092 YPT( VELVAW QTI QT2 G!T3 QT'l .2688635686039925 82.113525390625 270.5975691507827 170.3502746561725 -16.40963685243471 -8.75598457418046 XR' -13.09796824444656 VR 385.2831770522623 RADYAW = 398.8282470703125 REFERENCE AXLE 2 RADIUS IS 388.8282470703125 re:. 'F: (f r" ) SPEEU SLIF' IS UI195605779585 XIDDl = 82.80153943135148 134

PAGE 140

RCI) c 7 CKRAO 2 XPTC 1 .4132714616345241 YPT( 1 -3.314093755097687 VEL YAW 92.82962036132813 RCD = 17 CKRAD = 2 XPT( 2 ) 21.07125903114995 YPT( 2 ) = -2.067539201015234 VELYAW = 92.82962036132813 RCD = 27 CKRAD = 2 XPT( 3 41.69564989525229 YPT ( 3 .2669542350649834 VEL YAW 82.82962036132813 OTI 276.3609577245802 OT2 175.3307227488011 OT3 -16.5720677862519 OT4 -8.834632621084672 XR -13.05732879424981 YR 391.7176606344878 RADVAW = 395.2613830566406 REFERENCE AXLE 2 RADIUS IS 395.2613830566406 TIME OF SLIP 5.0000000745058060-02 VELOCITY AT SLIP IS (fps) 82.92962036132813 SPEED AT SLIP 15 (mph) 56.50042467294085 OIFF = 4.7;>8616943359375 X1DOT = 83.12399375486763 RCD 0: -:' CI(RAO 2 XPT( 1) .4616714616345241 VPTC 1) -3.314093755097687 VELYAW 83.15187835693359 RCO 17 Cf(RAO XPT( 2 ::1.20025803114995 VPT( 2) -2.067639201015234 VEL YAW S:;.15187835693359 RCD 27 CI
PAGE 141

APPENDIXH COMPUTER PROGRAM LISTING OF ARTICULATED VEHICLE MODEL

PAGE 142

10 REM PROGRAM TESTART.BAS 20 REM : ARTICULATED VEH YAW PROGRAM IN MICROSOFT BASIC 30 REM ********************************************************************** 40 REM 50 KEY OFF 60 REM 'DISABLE VIDEO DESCRIPTORS OF FUNCTION KEYS 70 REM ************ REDEFINE THE FUNCTION KEYS **********+************** 90 REM 90 KEY 1. "" : KEY 2. "SCR 1"+CHR$(13) : KEY 3. KEY 5. "" 100 KEY 6. "SCR 3": KEY 7. ". : KEY 9. SCR 4" KEY 9. 110 KEY (2) ON : KEY (4) ON : KEY (6) ON : KEY (8) ON 1221 REM 130 CLS 1421 REM "SCR 2"+CHR$(13': : KEY 10. "" 150 REM ************ SEQUENCING SUBROUTINES ** ARTICULATED ************* 160 REM ************ INPUT INITIAL VALUES 170 REM REM GOTO 600 GOSUB 5330 GOSUB 5660 1821 210 220 230 240 250 2621 270 REM *******+**** REM 280 REM GOSue 3440 G.OSUB 5660 INPUT INITIAL VALUES STORE INITIAL VALUES STEP TIME COUNTER ITERATION FOR STEP TWO COMPUTE LEVEL TWO VALUES STEP TIME COUNTER 2921 REM ************ ALL SUBSEQUENT ITERATIONS 300 REM 310 320 3421 3521 3621 370 380 390 400 410 420 430 440 451Z1 460 REM MCT = 1 WHILE TM TMMAX GOSUB 5770 GOSUB 80210 eOSuB 7830 WHILE MCT 1 GOSUB 5330 MCT .. 0 WEND MCT :: 1 GOSUe 566(!l WEND GOsue 7830 eosue 5330 COMPUTE VALUES AT BASE LEVEL COMPUTE FIRST AND SECOND DERIVATIVES ROTATE VALUES IN SHORT TERM STORAGE STORE COMPUTED VALUES STEP TIME COUNTER ROTATE VALUES STORE COMPUTED VALUES 470 REM ****+*+****** PRINT OUT FROM STORAGE 480 REM 490 500 5121 REM 520 530 REM IF REDO = THEN GOSue 4380 'PRINT ALL CALCULATED DATA GO sue 4870 'CALCULATE AND PRINT RADIUS INFO IF REDO 1 THEN GOTO 2680 540 REM *****+******* END OF PROGRAM 550 REM 560 END 570 REM 580 REM **********+****+***************************************************** 137

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590 REM 600 REM ************************* 610 REM INITIALIZE DATA STORAGE 6213 REM DEFINE VARIABLES AND ARRAVS ************* 630 DEFDBL A-Z 640 REM 'ALL VARIABLES DOUBLE PRECISION 650 D 1M A ( 13, 13), M ( 13, 13), K ( 13, 3), R ( 13 ,3), S ( 13 3), T ( 13) 660 REM 67121 OPEN "DATA.DAT" AS #1 LEN = 2111111 680 FIELD _I, 8 AS AlS,8 AS A2s,8 AS A3S,8 AS 91$,8 AS 92$,9 AS B3$,8 AS Cl$,8 AS C2$,8 AS C3$,8 AS 01$,8 AS D2$,8 AS D3$,8 AS El$,8 AS E2$,8 AS E3$,8 AS Gl$,8 AS G2$,8 AS G3S.9 AS HIS,8 AS Jl$.8 AS AS P3$,8 AS P5$,8 AS P7$ 69121 REM A IS Xl, B IS VI, C IS X2. 0 IS Y2, E IS PHI, G IS THETA, H IS TM 700 REM J IS ALPHA, P IS FORCE 710 REM 720 REM ************************* 73121 REM INPUT INITIAL VALUES ******************** 740 REM *********** COEFFICIENT OF FRICTION (DIMENSIONLESS) 750 760 REM 770 REM *********** SUPER-ELEVATION OF ROADWAY (RADIANS) 780 REM 790 SE" 0 800 REM 810 REM *********** TRACTOR INPUT CONSTANTS 820 Dl 830 D2 840 WI 850 OALI 860 T 870 WTl 98121 Ml 890 900 REM 8 9 20 = 25 18000 HTI 5! WTl / 32.2 (1/12)*Ml*(T*T + OALl*OALI) 910 REM *************** TRAILER INPUT CONSTANTS 920 REM 930 03 940 W2 950 OAL2 96111 WT2 970 M2 980 12 990 REM 15 32 = 40 70000! : HT2 = 6.5 WT2 + OAL2*OAL2) REM *********** PROGRAM FUNCTION KEYS 1010 REI" 1020 ON KEV(2) SOSUB 11121111 1030 ON KEY(4) GOSUB 1540 1040 ON KEY(6) GOSUB 1970 1050 ON KEY(8) GOSUB 2280 1060 ON KEY(10) GOSUB 2680 REM 138

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11218121 11219121 11121121 111111 112121 113121 1140 1150 116111 1170 118121 119121 1200 1210 1220 1230 1240 125121 1260 1270 1280 129121 131Z11Z1 131121 132121 133121 1340 135111 1360 137121 1380 139121 141210 141121 142111 143121 144111 1450 146121 1470 1480 149121 1500 1510 152121 153121 154121 1550 REM *********** FIRST INPUT SCREEN ****.**********.**.***.*** ***.**** REM CLS LOCATE 1.5 ; PRINT "THIS IS SCREEN 1 (TRACTOR PARAMETERS )" LOCATE 3.1111: PRINT "SCREEN 2 IS TRAILER PARAMETERS ( F 4 RETURN )" KEY(4) ON LOCATE 4.11Z1: PRINT "SCREEN 3 IS ROADWAY PARAMETERS F 6 + RETURN )" KEV(6) ON LOCATE 5.1111: PRINT "SCREEN 4 IS INITIAL CONDITIONS F 8 + RETURN ) ": KEY(8) ON LOCATE 7.5 : PRINT "TRACTOR WHEELBASE (FEET!" LOCATE 7.6121: PRINT" LOCATE 7.6111: PRINT WI LOCATE 9 .5: PRINT "TRACTOR TRACK WIDTH (FEET) 11 __ LOCATE 9 .6121 PRINT T LOCATE 11 PRINT "DISTANCE FRONT AXLE TO FIFTH WHEEL (FT)" LOCATE 11.60 PRINT" LOCATE 11.6121: PRINT (01+02) LOCATE 13.5 : PRINT "WEIGHT OF TRACTOR (POUNDS)" LOCATE 13.6121: PRINT __ LOCATE 13.60: PRINT WTl LOCATE 15.5 : PRINT "HEIGHT OF C G OF TRACTOR TRAILER (FT!" LOCATE 15.60: PRINT HTI LOCATE 17.5 : PRINT "YOU CAN INPUT THE LONGITUDINAL LOCATION OF THE C G" LOCATE 18.10: PRINT "1. BY INPUTING THE DISTANCE FROM THE FRONT AXLE 0,.." LOCATE 19.1121: PRINT "2. BY INPUTING THE WEIGHT PERCENTAGE ON FRONT AXLE" LOCATE 2121.15: PRINT "YOUR CHOICE IS" LOCATE 21.5 : PRINT "DISTANCE FROM FRONT AXLE" LOCATE 21.61111 PRINT" LOCATE 21 . 6121: PRINT 01---LOCATE 22.5 PRINT "WEIGHT PERCENTAGE ON THE FRONT AXLE" LOCATE 22.6121: PRINT" LOCATE 22.6121: PRINT WI-01)/Wl) LOCATE 7.6121 : INPUT"" GES IF GES <> 121 THEN WI = GES : GES .. 121 LOCATE 9 .6121: INPUT"" GES; IF GES <> 121 THEN T = GES 1 GES 0 LOCATE 11.6121: INPUT GES IF GES <> 121 THEN (01+02) GES : GES c 121 LOCATE 13.6121: INPUT"" GES IF GES 121 THEN WTl = GES: GES 121 Ml .. LOCATE 15.6121: INPUT"" GES IF GES <> 121 THEN HTl = GES: GES .. 121 LOCATE 2121.46: INPUT "" CHI IF CHI = 2 THEN GOTO 151121 LOCATE 21.6121: INPUT"" GES IF GES <> 121 THEN 01 GES: GES 121 LOCATE 22.60: PRINT WI-0l'/Wl) + OAL1*OALl) IF CHI 1 THEN CHI 121: GO TO 153121 LOCATE 22.60: INPUT GES : IF GES <> 121 THEN 01 '" WI GES"'WI LOCATE 21.6121: PRINT 01 GOTO 141121 CLS :RETURN 158121 REM 139

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1560 1570 1580 1590 1600 1610 REM SECOND INPUT SCREEN REM LOCATE 1.5 : PRINT "THIS IS SCREEN 2 TRAILER PARAMETERS )" LOCATE 3.10: PRINT "SCREEN 1 IS TRAILER PARAMETERS (. F 2 RETURN )": KEY(2) ON LOCATE 4.10: PRINT "SCREEN 3 IS ROADWAY PARAMETERS F 6 + RETURN )" KEY(6) ON LOCATE 5.10: PRINT "SCREEN 4 IS INITIAL CONDITIONS F 8 RETURN )" KEY(8) ON 1620 LOCATE 7.5 : PRINT "DISTANCE FROM PIN TO REAR AXLE (FEET>" 1630 LOCATE 7,60: PRINT" 1640 LOCATE 7.60: PRINT W2 1650 LOCATE 9 .5: PRINT "TRAILER TRACK WIDTH (FEET> 1660 LOCATE 9 .60: PRINT __ 1670 LOCATE 9 .60: PRINT T 1680 LOCATE 11.5 : PRINT "WEIGHT OF LOADED TRAILER (POUNDS) 1690 LOCATE 11,60: PRINT" 1700 LOCATE 11.60: PRINT WT2 1710 LOCATE 13.5 : PRINT "HEIGHT OF TRAILER C G (FEETl" 1720 LOCATE 13.60: PRINT ___ 1730 LOCATE 13.601 PRINT HT2 1740 LOCATE 15.5 : PRINT "YOU CAN INPUT THE LONGITUDINAL LOCATION OF THE C 13" 1750 LOCATE 16.10: PRINT "I. BY INPUTING THE DISTANCE FROM THE FIFTH WHEEL or" 1760 LOCATE 17.10: PRINT "2. BY INPUTING THE WEIGHT PERCENTAGE ON FIFTH WHEEL" 1770 LOCATE 18.15: PRINT "YOUR CHOICE IS" 1790 LOCATE 18.45: PRINT" 1790 LOCATE 19.5 : PRINT "DISTANCE FROM FIFTH WHEEL" 1800 LOCATE 19.60: PRINT 181121 LOCATE PRINT 03 1820 LOCATE 20.5 : PRINT "WEIGHT PERCENTAGE ON THE FIFTH WHEEL" 1830 LOCATE 20,60: PRINT 184121 LOCATE 20.60: PRINT W2-D3)/W2) 195121 LOCATE 7.60 : INPUT"" GES IF GES <> 0 THEN W2 GES : GES co 0 1960 LOCATE 9 .60: INPUT"" GES IF GES <> 0 THEN GES : I3ES .. 1870 LOCATE 11.60: INPUT"" GES IF I3ES <> 121 THEN WT2 GES : GES 0 1890 1990 1900 1910 1920 1930 1940 195121 1960 1970 1990 1'12 WT2 LOCATE 13.60: INPUT"" GES IF I3ES <> 0 THEN HT2 IS GES: I3ES .. 0 LOCATE 19.46: INPUT "" CHI IF CHI .. 2 THEN GOTO 1930 LOCATE 19.60: INPUT"" GES IF GES <> 111 THEN 0.3 = GES: GES .. 0 LOCATE 20.60: PRINT W2-D3)/W2) IF CHI = 1 THEN CHI = 0: GO TO 1960 LOCATE 20.60: INPUT" ". GES : IF GES 0 THEN 03 = W2 GES.W2 LOCATE 19.60: PRINT OAL2+0AL2) GOTO 1850 CLS : RETURN 2010 REM 140

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1990 REM THIRD INPUT SCREEN 21210121 REM 212110 LOCATE 1.5 : PRINT "THIS IS SCREEN 3 (ROADWAY PARAMETERS )" 21212121 LOCATE 3,1121: PRINT "SCREEN 1 IS TRACTOR PARAMETERS ( F 2 RETURN )" : KEY(2) ON 212130 LOCATE 4.10: PRINT "SCREEN 2 IS TRAILER PARAMETERS KEY(4) ON 21214121 LOCATE 5.1121: PRINT "SCREEN 4 IS INITIAL CONDITIONS KEV(S) ON 21215121 LOCATE 7,5 : PRINT "GRADE OF ROADWAY (PERCENT>" 2060 LOCATE 7,60r PRINT" 212170 LOCATE 7.6121: PRINT GR F 4 + RETURN F 8 RETURN 2080 LOCATE 9 .5: PRINT "SUPER-ELEVATION OF ROADWAY (PERCENT> 212190 LOCATE 9 ,60: PRINT 2100 LOCATE 9 ,60: 'PRINT SE 211121 LOCATE 11.5 : PRINT "LEVEL DRAG FACTOR BETWEEN ROAD AND TIRES" 2120 LOCATE 11,60: PRINT" 2130 LOCATE 11.6121: PRINT F 2140 LOCATE 13.5 : PRINT "RADIUS OF CURVATURE OF OUTSIDE EDGE OF" 2150 LOCATE 14.10: PRINT "LANE TRAVEL" 2160 LOCATE 14.60: PRINT " 217121 LOCATE 14.60: PRINT RRO 2180 LOCATE 16. 5: PRINT "PERCENT ( 0 to 75 ) BRAKING ON DRIVE AXLE" 219121 LOCATE 16,60: PRINT __ GR GES SE ... GE5 : F GES : RRD= GES : GES 121 GES = 121 GES = 121 GES = III 2200 LOCATE 7.60: INPUT" ". GES: IF GES <> 0 THEN 221121 LOCATE 9.6121: INPUT" ". GES: IF GES 121 THEN 2220 LOCATE 11.60: INPUT" ". GES: IF GES <> 0 THEN 223121 LOCATE 14.6121: INPUT" ". GES: IF GES <> 121 THEN 224121 LOCATE 16.6121: INPUT" ". GES: IF GES <> 0 THEN 2250 IF ORVBK ) .75 THEN LOCATE 16,6121: PRINT" ORVBK= GES ": GOTO 224121 GES 121 226121 F F -GR + SE 227121 GOTO 22121121 2280 CLS : RETURN 2320 229121 REM 141

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23121121 REM *********** FOURTH INPUT SCREEN ************************************ 2310 REM 232121 LOCATE 1.5 : PRINT "THIS IS SCREEN 4 (INITIAL CONDITIONS)" 2330 LOCATE 3.10: PRINT "SCREEN 1 IS TRAILER PARAMETERS ( F 2 + RETURN )": KEV(2) ON 234121 LOCATE 4.1121: PRINT "SCREEN 2 IS TRAILER PARAMETERS F 4 + RETURN )" KEY(4) ON 2350 LOCATE 5.1121: PRINT "SCREEN 3 IS ROADWAV PARAMETERS F 6 + RETURN )" ON 236121 LOCATE 7.1121: PRINT "THIS PROGRAM WILL START WITH AN INITIALLV STABLE." 2370 LOCATE 8.5 : PRINT "STRAIGHT LINE MOTION ALONG THE POSITIVE X-AXIS WITH" 2380 LOCATE 9.5 : PRINT "THE TRACTOR CENTER OF GRAVITY AT COORDINATES (121.121)" 239121 LOCATE 11.5: PRINT "TRACTOR CENTER OF GRAVITY )(1 0" 24121121 LOCATE 12.5 PRINT" YI 121" 241121 LOCATE 13.5 PRINT "TRACTOR ANGLE TO X-AXIS: THETA 0" 242121 LOCATE 14.5 PRINT "TRAILER CENTER OF GRAVITY X2 "C-D2-D3) 2430 LOCATE 15.5 PRINT" Y2 0" 2440 LOCATE 16.5 PRINT "TRAILER ANGLE TO X-AXIS: PHI 121" 2450 LOCATE 18.5 PRINT "WHICH AXLE LEFT IDENTIFIABLE YAW MARKS 2460 LOCATE 18.6121: PRINT" 247121 LOCATE 19.1121: PRINT "I. STEERING AXLE" 2480 LOCATE 2121.1121: PRINT "2. TRACTOR DRIVE AXLE" 2490 LOCATE PRINT "3. TRAILER AXLE" 251110 LOCATE 22.5 : PRINT "WHAT IS THE MEASURED RADIUS" 2510 LOCATE 22,60: PRINT 252111 LOCATE 22.6121: PRINT GOALRAD 253111 LOCATE 18.6121: INPUT" ". GES: GOALAX = GES : GES 0 2540 LOCATE 22.6121: INPUT GES: IF GES <> 121 THEN GOALRAD GES: GES 0 2550 REM 2560 REM ************* START COMPUTATION PROCESS 257121 REM 2580 LOCATE 23.5 : INPUT" HIT '1' TO START COMPUTATION CK2 2590 IF CK2 1 THEN GOTO 261210 ELSE GOTO 2530: CK2 = 121 260121 REM 142

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2610 2620 2630 2640 2650 266111 2670 2675 2680 2690 2710 2720 273111 2740 275121 2760 2770 2780 2790 2800 2810 2820 2830 2840 2850 2860 2870 2880 2890 290111 2910 292111 293111 294111 295121 2960 297111 298111 2990 3000 3010 31112111 3030 3040 3050 306111 312170 3080 31119111 3100 311111 3120 3130 3140 3150 3160 REM ************** VARIABLE INITIAL VALUES DISPLACEMENT: Xl, V1, THETA, X2, V2, PHI REM REM REM REM REM REM VELOCITV: ACCELERATI ON: FORCES XIDOT. VIDOT, THETAO. X2DOT. Y2DOT. PHID XIOD. VIDD. THETADD. X2DD. V2DD. PHIDD Fl. F3. F5. Fe, F7, F9 XIDOT 1.466 .5 SQR( 15 GOALRAD F ) Xl .. Vl 0 THETA = X2 = (-D2-D3) : Y2 = 0 : PHI 0 X2DOT XIDOT VIDOT = 0 THETAD V2DOT = PHID = 0 XIDD = 0 VIDD = 0 X2DD .. 0 V2DD = 111 V2DD = 0 REM ALPHA .. 13 1) ALPHA REM .. Xl VI THETA X2 THETADO .. 111 PHIDD R(1.1) XIDOT R (2.1> VIDOT R(3.1) THETAO R(4.1l X2DOT 5(1.1> 5(2.1) 5(3.1) 5 (4.1> XIDD YIDD THETADD X2DD K(2.1) K(3.1) K(5.1> (6.1) REM Y2 PHI R(5.l> Y2DOT 5(5.1> Y2DD REM ********** REM N7 WT2. ( 03 N4 WT2 -N7 INITIAL W2 R(6.1> PHID 5(6.1> NORMAL FORCES NJ = WTl WI 01 WI + N4 (WI -01 -02 ) / WI N3 = WTI N4 NI REM REM ********** INITIAL BRAKING FORCE REM F9 = N3 F DRVBK REM REM ********** TIME INCREMENT REM H .025 TM = 121 K( 12.1) TM TMMAX 50 H REM REM ********** ACCEPTIBLE ERROR VALUE REM 5TD .0005 REM REM ********** BLANK COUNTERS REM 0 M 1 REM 143 PHIDD

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3170 3180 3190 321110 3210 3220 3230 324111 325111 3260 3270 328111 REM PRINT CONTROL REM = REM REM SET NUMBER OF TERMS REM REM REM COMPUTE FORCE VALUES REM ANSI COS CTHETA+ALPHA) CW2-D3) + D3*COSCPHI)*COS(PHI,*COSCTHETA+ ALPHA) + D3*COSCPHI)*SINCPHI)*9INCTHETA+ALPHA) ANS2 .. Dl*COSCALPHA) + D2*COSCTHETA) *COS CTHETA+ALPHA) + D2*SINCTHETA'* SINCTHETA+ALPHA) ANS3 C (W2 D3)*COSCTHETA) + D3*COS(THETA)*COS(PHI)*COSCPHI) + 03* SINCTHETA)*SIN(PHI)*COS(PHI / (WI-D2-Dl) 3290 Fl = ( 12*PHIDD*COSCPHI) +CM2*Y2DD + MI*Y1DD)* CW2-D3) + M1*Y10D*D3*CO SCPHl)*COS(PHI) Ml*XlDD*D3*SINCPHI'*COSCPHI)+ ( II*THETADD + Ml*YlDO*D2*COSCTH ETA) Ml*XIDO*02*SINCTHETA) )*ANS3 '* C 1 / (ANSl+(ANS2*ANS3) ) ) 33111121 F3 = C 11*THETADD + MI*Y1DO*D2*COS(THETA) M1*X10D*02*SINCTHETA) 331111 3320 3330 334111 3350 336111 3370 338111 3390 341110 3410 342111 3430 ANS2*Fl ) (01 + 02 WI ) F5 = -MI*XIDD F3*SINCTHETA) -Fl SINCTHETA + ALPHA) F6 .. Ml*YIDD -* COSCTHETA) -Fl COSCTHETA + ALPHA) F7 = ( M2*Y2DD + Ml*YIDD F3*COS(THETA) Fl*COSCTHETA+ALPHA) ) / COS CPHIl Xl VI THETA X2 Y2 KCl.l ) KC2.1) KC3.1l Ke4.1l KC5.1l K C6.1l .. PHI GO TO 21111 REM REM VALUES OUT: REM W C 1.1) WC2.1) WC3.1l W(4.1) WC5.1l WC6,l) XIDOT YIDOT = THETAD = X2DOT .. Y2DOT = PHID KC7.1I = F5 KC8.1l F6 K(9,ll Fl K( 1121,1>= F3 KCll,ll= F7 Dl 02 D3 WI W2 I'll 1'12 K(l.l' RCJ.l) SeJ.l) 1=1.13 11,I2 144

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3440 345111 3460 347111 3480 3490 3500 3510 3520 REM ITERATION THRU FIRST TIME STEP REM REM COUNTERS USED TEST REM K <12.2) TM K ( 13. 2) ALPHA REM COMPUTE NEW VALUES REM Xl 3530 REM 3540 3550 REM 3560 357111 REM 3580 3590 REM 3600 3610 REM 3620 Kl Xl + H (X2DOT 02 THE TAO SINCTHETA) 03 PHIO SIN(PHIl ) VI K2 VI + H (Xl TANCTHETA + ALPHA) 01 *THETAO CCOSCTHETA) + SINCTHETA)*TANCTHETA + ALPHA) X2DOT R4 = X200T + H C F5 -F7 SINCPHI) ) M2 V2 K5 = V2 + H C VIDOT 02*THETAD*COS(THETA) D3*PHID*COSCPHI) ) THETA K3 THETA + H C C VI XI*TAN(THETA) ) (WI-Ol) ( SINCTHETA)* TANCTHETA) + COSCTHETA) ) ) ) PHI 1(6 PHI H ( C V2 X2*TAN (PHI) ) CW2-03) ( SIN CPHI). TANCPHI) + COSCPHI) ) ) ) 36311.1 REM X2 3640 K4 .. R4 H + X2 365111 GOT a 4280 3660 3670 3680 3690 3700 3710 3720 3730 3740 3750 3760 3770 3780 ::790 3800 3810 3820 3830 38413 3850 3860 3870 REM REM KK4 .. Kl 02 COSCK3) KK5 K2 02 SIN(K3) K4 .. (K4 + KK4) 2 K5 (K5 + KK5) 2 KK2 K5 + 02 SINCK3) KKI K4 + 02 COSCK3) Kl CKKI + Kl) K2 .. CKK2 + K2) 2 Rl (Kl Xl> H R2 CK2 VI> H R3 CK3 THETA) R4 (K4 X2) H R5 CK5 V2) H R6 (K6 -PHI) H SI (Rt XI00Tl H 52 (R2 VIDOT) H S3 (R3 THETAO) 54 (R4 X200Tl H 55 (R5 Y200Tl H 56 CR6 PHIO) H 03 COSCK6) 03 SINCK6) + 03 SINCK6) + 03 COSCK6) H H 145

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3880 3890 3900 3910 3920 3930 3940 3950 REM ERROR REM CHECK TEST = 1 : DUMB 1 REM WHILE TEST ANSI AN52 ANS3 75 COSCTHETA+ALPHA)*CW2-D3) + D3*COSCPHI)*COSCPHI)*COSCTHETA+ ALPHA) + D3*COS(PHI)*SIN(PHI)*SINCTHETA+ALPHA) Dl*COSCALPHA) + D2*COSCTHETA) *COS CTHETA+ALPHA) + D2*SINCTHETA)* SIN CTHETA+ALPHA) ( (W2 D3)*COSCTHETA) + D3*COSCTHETA)*COSCPHI)*COSCPHI) + D3* SIN(THETA)*SINCPHI)*COSCPHI (Wl-D2-Dl) 3960 Fl = C I2*PHIDD*COS(PHI) +CM2*Y2DD + Hl*YIDD)* CW2-D3) + Ml*YIDD*D3*CO SCPHI)*COSCPHI) Ml*XIDD*D3*SINCPHI)*COS(PHI)+ ( Il*THETADD + Ml*YIDD*D2*COSCTH ETA) Ml*XlDD*D2*SINCTHETA) )*ANS3 )* C 1 CANSl+CANS2*ANS3) ) ) 3970 F3 = ( Il*THETAOD + Ml*YIDD*D2*COSCTHETA) Ml*XlDD*02*SINCTHETA) 3980 3990 4000 4010 4020 4030 4040 4050 4060 4070 4080 4090 4100 4110 4120 4130 4140 4150 4160 4170 4180 4190 4200 4220 4230 4240 ANS2*FI ) COl + 02 WI ) F5 = -Ml*XlDD F3*SINCTHETA) -Fl SINCTHETA + ALPHA) = Ml*YlOD,F3 COSCTHETA) -Fl COS(THETA + ALPHA) F7 C M2*Y2DD + Ml*YlDD F3*COSCTHETA) Fl*COSCTHETA+ALPHA) ) COS (PHI) RRI R4 02 R3 SINCK3) 03* R6 SINCK6) ) RR2 R4 02 R3 COSCK3) 03 R6 COSCK6) S54 CF5 -F7 SIN(K6 M2 RR5 (R2 D2*R3*COSCK3) 03*R6*COSCK6) ) RR3 C C K2 -Kl*TAN(K3) ) C (WI-0l) C 5INCK3)* TANCK3) + C05CK3) ) ) ) RRb =C C K5 K4*TANCK6) ) CW2-D3) ( SINCK6)* TANCK6) + COSCK6) ) ) ) CI(l ABS CRI -RRt) CK2 ABSCR2 RR2) CK3 ABSCR3 RR3) CK4 AB9(S4 9S4) CK5 ABSCR5 RRS) CK6 = ABS(R6 RR6) IF CKI ( CK2 THEN CKI IF CKI CK3 THEN CKI IF CKI < CK4 THEN CKI IF CKI CK5 THEN CKI IF CKI CK6 THEN CKI IF CKI < STO THEN Rl RRI Kl R2 RR2 K2 54 SS4 R4 R5 RR5 K5 CK2 CK3 CK4 .. CK6 .. 75 Rl H + Xl .. R2 H V1 54 H + X2DOT R5 H V2 R3 RR3 K3 = R3 H + THETA R6 H PHI Rt. RR6 K6 51 52 .. K4 .. 55 53 S6 CRI XIDOT) H CR2 VIDOT) H (R4 H) X2 (R5 -V2DOT) H CR3 -THETAD) H (Re. PHID) H 4250 TEST = TEST + DUMB DUMB 1 4260 WEND 4270 REM 4280 REM SET NEW VALUES FOR NEXT ITERATION 4290 REM RETURN X2 1<4 Y2 K5 Xl Kl Y1 K2 PHI = K6 THETA K3 4300 4310 4320 4330 4340 4350 4360 4370 4390 REM VALUES OUT: REM R4 R5 X2DOT V2DOT XIDOT YIDOT PHID THETAD= R2 R6 R3 X2DD = 54 Y2DD = 95 X1DD .. 91 YIDO S2 PHIDD= 56 THETADD 1. 13 146 KC4.2)=K4 (5.2)=K5 KC1.2)=Kl KC2.2)=K2 KC6.2)=K6 93: KC3.2)=K3 KC7.2) F5 K(8.2) = F6 KC9.2) .. FI KCI0.2) F3 K C 11.2) = F7

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4390 REM PRINT SUBROUTINE 4400 REM 4410 REM COUNTERS USED K 4420 REM 4430 STPT M 4440 K = 1 4450 LPRINT TAB(3) "TIME": TAB(13)"XI": TAB(24) "XIDOT": TABC3S) "XI-DD": TAB(46) "VI": TAB(57) "VIDOT": TAB(68)"VI-DD" 4460 LPRINT TAB(3) "----": TAB(13)"--": TAB(24) "-----": TAB(35) "-----"; TAB(46) 4470 4480 4490 4500 4510 4520 4530 4540 4550 4560 4570 4580 4590 4600 4610 4620 4630 4640 4650 4660 4670 4680 4690 4700 4710 TAB(57) "-----": TAB(68)"-----" WHILE K STPT GET #1, LPRINT TAB(3)CVDCH1S): TABCI3)CVDCA1S); TAB(24)CVDCA2S):TABC35)CVDCA3S); TAB(46)CVDCB1S); TAB(57)CVDCB2S) : TAB(68)CVDCB3S) K = K IF CI( 40 OR K = 80 DR K 120 OR I( 160 OR K 200 OR K = 240 OR K 280) THEN LPRINT CHRSCI2) : GOTO 4450 WEND LPRINT CHRS C 12) LPRINT TAB(3) "TII'1E": TA8(13)"X2"; TAB(24) "X200T": TABc3S) "X2-0D": TA8(46) "V2"; TAB(57)"V200T": TA8(68)"Y2-00" LPRINT TAB(3) "----"; TABCI3)"--": TAB(24) "-----"; TAB(35) "-----"; TAB(46) "--":, TAB (57) "-----": TAB (68) "-----" WHILE K STPT GET #1, LPRINT TAB(3)CVOCH1S): TABCI3)CVOCCls): TAB(24)CVOCC2S):TABC35)CVOCC3S)1 = f( IF (K WEND TAB(46)CVOCOIS)rTAB(57)CVDCD2S): TAB(68)CVOC03S) + 40 OR = 80 OR = 120 OR tc: 160, OR 200 OR = 280) THEN LPRINT CHRsCI2) : GOTO 4550 LPRINT CHRs(12) K 1 240 OR LPRINT TAB(3) "TIME": TAB(13) "THETA"; TAB(24) "THETA-D": TABC3S) "THETA-OO" TAB(46) "PHI": TABCS7)"PHI-OOT"; TAB(68)"PHI-DD" LPRINT TAB(3) "----"; TAB(13)"--": TAB(24) "-----": TABC3S) "-----": TAB(46) "--"; TAB(57)"----:..,,; TAB(68)"-----" WHILE K STPT GET #1, K LPRINT TAB(3)CVOeH1S): TAB(13)CVOeGlS): TAB(24)CVOCG2S):TABC3S)CVOeG3s): IF TAB(46)CVDcElS) TABCS7)CVOCE2s): TAB(68)CVDCE3S) 40 DR K 80 OR K = 120 OR K = 160 OR K = 200 OR K 240 OR 280) THEN LPRINT CHRs(12) : GOTO 46S0 4720 WENO 4730 LPRINT CHRS(12) 4740 = 4750 LPRINT TAB(3) "TIME"; TAB(13) "ALPHA"; TAB(24) "FS"; TABC3S) "F6"; TAB(46)"Fl": TAB(57) "F3": TAB(68)"F7" 4760 LPRINT TAB(3) "----": TAB(13)"--": TAB(24) "-----": TABC3S) "-----"; TAB(46) "--": TAB(57) "-----": TAB(68)"-----" K STPT 4780 GET #1. K 4790 LPRINT TAB(3)CVO(H1S): TABCI3)CVOeJls): TAB(24)INTCCVDepls:TABC35)INTC CVDep3$: TAB(46) INT(CVO(P5$:TABe57) INT(CVDep6S: TAB(68)INTCCVDep7S 4800 = + Q 4BHl IF 40 OR 80 OR = 120 OR = 160 OR 200 OR 240 OR K 290) THEN LPRINT CHRS(12) : GOTO 4750 4820 WEND 4830 LPRINT CHRS(12) 4840 REM 147

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4850 RETURN 4860 REM 4870 REM ********** RADIUS CALCULATION 4880 REM 4890 IF GOALAlC 4900 IF GOALAX 4910 IF GOALAX 492111 K 1 THEN I 5 2 THEN I 5 3 THEN I 5 + 4930 WHILE K < 4 4940 IF = THEN RCD = T5LIPl / H TSLIP2 H TSLIP3 / H LPRINT "RCD "RCD 4950 IF K = 2 THEN RCD 4960 IF K 3 THEN RCD + 10 LPRINT "RCD "RCD 20 : LPRINT "RCD ="ReD 4965 GET _1. RCD 4970 CKRAD = GOALAK :LPRINT "CKRAD "CKRAD 4980 WHILE CKRAD 4990 XPTCK) = CVDCAts) 01 COSCCVDCG1S C ) SINCCVDCGlS 501110 VPTCK) = CVDCBlS) 01 9INCCVDCG1S -( ) COSCCVD(Gl$ 5010 CKRAD 0 : IF K = 1 THEN VELVAW SQR( CVDCB2$)A2) 502111 WEND 5030 WHILE CKRAD 2 5040 XPTCK) CVDCA1S)-CWI-Dl)* COSCCVDCGlS + ( T/2 ) SINCCVDCGls 5050 VPTCK) = CVDCB1$)-CWI-Dl). SINCCVDCGls C ) COSCCVDCG1S 5060 CKRAD = 0 : IF K = I THEN VELVAW SQR( CVDCA2S)A2 + CVDCB2$)A2) 5062 LPRINT "XPTC"K") = XPTCK) 5064 LPRINT "VPTC"K") VPTCK) 5066 LPRINT "VELVAW VELVAW 5070 WEND 5080 WHILE CKRAD = 3 5090 XPTCK) = CVDCCIS)-CW2-D3)* COSCCVD(EI$ + ( T/2 ) SINCCVDCElS 5100 VPT(K) CVDIDl$)-(W2-D3)* SINCCVDCElS C COS(CVD(Els 5110 CKRAD = 0 : IF K = 1 THEN VELVAW = SQRC CVDCC2S)A2 + CVD(D2S)A2) 5120 WEND 5130 K = K + 1 5140 WEND 5150 QTI = CXPT(3)A2 XPT(2)A2 + VPT(3)A2 VPT(2)A2) / C2*CVPT(3) VPT(2) 5160 QT2 = CXPT(2)A2 XPT(I)A2 VPT(2)A2 VPT(1)A2) (2*(YPTC2) VPT(I) 5170 QT3 = (XPT(l) XPT(2 / CVPT(2) VPT(I 5180 QT4 (XPT(2) XPT(3 / (VPTC3) VPT(2 5190 XR (QTl / (QT3 QT4) 5200 VR QT3 OT2 5201 LPRINT "QTI = 5202 LPRINT "QT2 "QT2 5203 LPRINT "QT3 = QT3 5204 LPRINT "QT4 "QT4 5205 LPRINT "XR "lCR 5206 LPRINT "VR "VR 5210 RADVAW = ( (XPT(l) XR)A2 + (VPT(l) VR)A2 5211 LPRINT "RADVAW RADVAW 5220 LPRINT TAB(20) "REFERENCE AXLE "GOALAX 5230 LPRINT TAB(25) "RADlUS IS "RADYAW 5240 LPRINT TAB (25) "TIME OF SLIP "( C 1-5) *H) 5250 LPRINT TAB(25) "VELOCITV AT SLIP IS "VELYAW 5260 LPRINT TAB(25) "SPEED AT SLIP IS Cmch) uCVELVAW/l.466) 5270 LPRINT ": LPRINT " 5280 DIFF GOALRAD RADVAW 5282 LPRINT "DIFF = DIFF 5290 IF AB5CDIFF) > (.001 GOALRAD) THEN REDO I ELSE REDO 5300 VELYAW = VELVAW + VELVAW DIFF) / GOALRAD ) 5302 XI00T = VELVAW 5305 LPRINT "XI00T 5306 TSLIPl 0 5307 TSLIP2 .. '" 5308 TSLIP3 = (!) 5310 RETURN 5320 REM "XI00T 148 o

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REM STORE COMPUTED VALUES .................. 5340 REM REM COUNTERS USED M REM 5370 LSET AlS= 5380 LSET A2S= 5390 LSET A3S= MKDSCCFrXCSCI.l).10000/10000) 5400 LSET MKDSCCFIXCKC2.1).10000/10000) 5410 LSET 82$= MKDSCCFIXCRC2.1).10000/10000) 5420 LSET 93s= MKDSCCFIXCSC2.1)+10000/10000) 5430 LSET CIS= MKDSCCFIXCKC4.1)+10000/10000) 5440 LSET C2S= MKDSCCFIXCRC4.1).10000')/10000' 5450 LSET C3$= 5460 LSET 01$= MKD$(CFIX(KC5.1).10000/10000) 5470 LSET D2Sm MKDSCCFIXCRCS,I)+100001)/10000) 5480 LSET D3$= MKD$CCFIXCSCS,l'*10000/10000) 5490 LSET El$= MKO$CCFIXCKC6,1'*100001)/10000) 5500 LSET E2S= MKDSCCFIXCRC6,1)*10000)1/10000) 5510 LSET E3$= MKDSCCFIXCSC6.1).10000/10000) 5520 LSET MKDSCCFIXCKC3,1).10000/10000) 5530 LSET MKDSCCFIXCRC3.1)*100001)/10000) 5540 LSET MKDSCCFIXCSC3,1).10000/10000) 5550 LSET H1S= MKDSCCFIXCKC12.1)*10000/10000) 5560 LSET Jl$= MKDSCCFIXCKC13.1).10000/10000) LSET Pl$=' MKD$CCFIXCKC7.1).10000"/10000) 5580 LSET P3$= LSET P5S= MKDSCCFIXCKC9.1)*1000011/10000) LSET P6$= 5610 LSET MKD$CCFIXCKC11.1)+10000/10000) PUT #1. M 5630 M = M + 1 5640 RETURN 5650 REM 5660 REM 5670 REM .......... ******.* STEP TIME INCREMENT AND ALPHA **** *.** 5680 REM 5690 TM = TM + H 5700 ALPHA = ALPHA + H (.08) 5710 IF ALPHA => .12 THEN ALPHA s .12 KCI2.3) TM 5730 KCI3.3) = ALPHA 5740 REM 5750 RETURN 5760 REM 5770 REM .*******. BLANK COEFFICIENT MATRIX 5790 REM 5790 = 5800 WHILE < 12 5810 J = 1 5820 WHILE J < 13 5830 ACI.J) 0 = J WEND I = I + 1 5870 WEND 5880 REM 149

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5890 REM LOAD COEFFICIENTS INTO MATRIK 5900 REM 5910 AC1.l) = 1 5920 AC1.9) = CHA2*SINCKC3.2) + ALPHA Ml 5930 AC1.10) = Ml 5940 AC1.7) = CHA2) Ml 5950 AC1.12)= 2*K(I.2) KC1.l) + C H*H*F9*COSCK(3.2) Ml CMl 32.2 F COSCKC3.2 ORVBK N3 ) C WTl + WT2) 5960 REM 5970 AC2.2) 5980 AC2.9) + ALPHA Ml 5990 = -1*CHA2*COSCKC3.2) Ml 6000 A(2.8) = Ml 6010 AC2.12)= KC2.1) + C F9 H H SINCKC3.2) HI CMl 32.2 F DRVBK N3 SINCKC3.2) CWTl + WT2) 6020 REM 611130 AC3.3) = 1 604111 AC3.9) = -1*CHA2*OI*COSCALPHA 11 611150 AC3.10) = (HA2*(Wl-Dl 6060 AC3.7) = 6070 AC3.8). 6080 AC3.12)= 2*KC3.2) KC3.1) 6090 AC4.4) = 6100 AC4.7) = -1*HA2 M2 6110 A(4.11)= (H*H*SINCKC6.2) M2 6120 AC4.12)= 2*kC4.2) KC4.1) 6130 AC5.5) = 1 6140 ACS.8)= M2 6150 A(5.11)= -1*CHft2*COSCKC6.2) M2 6160 AC5.12)= 2*KCS.2) KC5.1) 6170 AC6.6) = 6180 AC6.7) = CHA2*D3*SINCKC6.2) 12 6190 AC6.8)= CHft2*D3*COSCKC6.2) 12 6200 AC6.11)= CH2*CW2-D3 12 6210 AC6.12)= 2*KC6.2) -K[6.1) 6220 AC7.1) 6230 AC7,3) = 02 SINCKC3.2 6240 AC7.4) = 6250 AC7.6) = 03*SINCKC6.2 6260 A(7.12)= AC7.1)*K(1.2) AC7.3)*KC3.2) + AC7.6)*KC6,2) 6270 AC8.2) = 6280 A(8.3) -1*02+COSCKC3.2 6290 AC8.5) .:.. 6300 AC8.6) -1*D3*COSCKC6.2 6310 AC8.12)= ACS.2)+KC2.2) ACS.5)*KCS.2) + ACB.3)+KC3.2) + ACS.6)*K(6.2) 6320 AC9.4) TANCK(6.2 6330 AC9.5) = 6340 AC9.6) = CW2-D3)*CSINCKC6.2*TANCKC6.2 + COSCKC6.2) 6350 AC9.12)= AC9.4)*KC4.2) + AC9.S).KCS.2) AC9.6)*KC6.2) 6360 A 'Cl0.1J= TANCKC3.2)+ ALPHA) 6370 ACl0.2)= -I 6380 AC10.3)= -1*01*CCOSCKC3.2 + SIN(KC3.2 TAN(KC3.2) + ALPHA) AC10.12) = + AC10.2)*K(2.2) + AC10.3)*K(3.2) 6400 AC1I.I) = TANCK(3.2 6410 A ( 11 .2) = -I 6420 A(11.3) = (WI-DI, \ CSINCKC3.2 TANCKC3.2 + COSCf(C3.2) 6430 ACll.12)= AClt.I)* KC1.2) + ACII.2)*K(2.2) A(11.3).KC3.2) 6440 REM 150

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6450 REM TEST FOR MAXIMUM LATERAL FORCE 6460 REM 6470 MEl'll 6480 MEM2 6490 MEM3 .. 6500 NGITY = 8 6510 NTEST 111 6520 F9 .. DRVBK F N3 6530 FIMAX .. F Nl 6540 F3MAX SQR C (F F9 F9 ) 6550 F7MAX = F N7 6560 IF NQUIZl = 1 THEN NTEST 1 : GO TO 6580 6570 IF CABSCF1 FIMAX THEN NQTY NQTY + ELSE NTEST 6580 WHILE NTEST = 1 6590 MEl'll E 0 6600 IF TSLIPI = 0 THEN TSLIPI TM TMMAX TM + 30 H 6610 NQUIZI = 1 6620 I 1 6630 WHILE 12 6640 ACI.12) .. ACI.12) ACI.9).FIMAX 6650 ACI.9) = 0 6660 T(9) FIMAX 6670 I = I 1 6680 WEND 6690 NTEST = 0 6700 WEND 6710 IF NQUIZ3 = THEN NTEST = 1 : GOTO 6730 6720 IF ABS/CF3 F3MAX THEN NQTY NQTY ELSE NTEST 6730 WHILE NTEST 6740 MEM2 0 6750 IF TSLIP2 = 0 THEN TSLIP2 TM TMMAX = TM + 30 H 6760 NGlUIZ3 .. 6770 1 6780 WHILE < 12 6790 ACI.12) ACI.12) ACI.10).F3MAX 6800 ACI.10) 0 6810 T(10) F3MAX 6820 I I 1 6930 WEND 6940 NTEST = 111 6950 WEND 6960 IF NQUIZ7 = THEN NTEST 1 : GOTO 6880 6870 fF ABS/CF7 ( F7MAX THEN NQTY = NQTY + 1 ELSE NTEST 6980 WHILE NTEST = 1 6890 MEM3 0 6900 IF TSLIP3 0 THEN TSLIP3 TM TMMAX 6910 NQUIZ7. 1 692111 I 1 6930 WHILE 12 694. 0 A C 12) 6950 ACI.ll) 6960 T(ll) 6970 T .. T 6990 WEND 6990 NTEST = 0 7000 WEND A( I .12) o F7MAX ACI.l1).F7MAX 151 TM 3111 H

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7010 lei 7020 IF MEM3 1 THEN GOTO 7120 7030 I 1 7040 WHILE I ( 12 7050 A(I,II) c ACI.12) 7060 A(I,12) 0 7070 = + 1 7080 WEND 709/11 REM 7100 REM ********** FOR EACH MAX FORCE REDUCE MATRIX SIZE BY ONE 711/11 REM 7120 IF MEM2 = 1 THEN GO TO 7200 713121 = 1 7140 WHILE I < 12 715/11 ACI.II21) .. ACI.ll> 716/11 ACI.l1) ACI.12) 7170 ACI,12). 0 718121 I I + 1 7190 WEND 721210 IF MEl'll = 1 THEN GO TO 7320 7210 I .. 1 7220 WHILE I < 12 7230 A( I .9) 7240 ACI.10) 725121 A C 11 ) 7260 A ( I 12) 727121 I = + 7280 WEND 7290 REM -ACI.10) A 0 THEN N P P .. CNQTY+l) 7380 P P + 1 7390 WEND 7400 IF N == THEN GOTO 7480 7410 K = 1 7420 WHILE < CNQTY+2) 7430 PIVOT ACI.K) 7440 ACI.K) ACN,K) 7450 ACN.K) PIVOT 7460 K K + 1 7470 WEND 7480 J = I + 1 7490 WHILE J < CNQTY+l) 7500 1'1 1> = A (J, 1> A (I. I) 751"0 K .. I 7520 WHILE (NQTY+2) 7530 ACJ.K) ACJ,K) MCJ,I) ACI.K) 7540 K K + 1 755121 WEND 7560 J = J 7570 WEND 7580 I J + 1 7590 WEND 152

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7600 7610 7620 7630 7640 7650 7660 767111 7690 7690 770111 7710 772111 7730 774111 775111 7752 7754 7760 7770 7790 7790 7B00 7910 7820 7830 7840 7950 786121 7970 7890 7990 7900 7910 7920 7930 7940 7950 796 7970 7990 7990 8000 TCNQTY) = ACNQTY. CNQTY+l/ACNQTY.NQTY) I = NQTY-l WHILE > 0 J I + 1 ClTY WHILE J < CNQTY+l) QTY QTY + ACI.J)*TeJ) J = J 1 WEND TCI) A(I, (N(;ITY+I (;ITY ) ACI,I) IF I 9) AND (TC9) FIMAX THEN T(9) FIMAX : TRAP = 1 IF I 10) AND (T(10) F3MAX THEN T(10) F3MAX : TRAP = IF I 11) AND (T(11) F7MAX THEN T(ll) F7MAX cTRAP = 1 IF (TRAP 1 AND UNTRP = 0) THEN TMLCKI TM : UNTRP = 1 = I -1 WEND T(4) T(l) -D2 T(S) = T(2) -D2 K(l,3) T(ll KC3.3) T(3) KC5.3) TeS) K(7,3) T(7) KC9,3) T(9)' K(11.3) COS(T(3 D3 COSCT(6 -D3 SIN(TC6 SIN (3 KC2,3) K(4.3) KC6,3) K(8.3) KC10.3) = RETURN REM REM REM REM COUNTERS USED N = 1 WHILE N 13 = WHILE L 2 L N K(N,L) K(N,L+l) R(N,L) RCN.L+l) SCN,L) = SCN,L+l) = + WEND N N + 1 WEND RETURN REM REM T(2) T(4) T(6) T(10) TM ROTATE VALUES 153 : K(13.3) ALPHA

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8010 REM *.***.***** COMPUTE FIRST AND SECOND DERIVATIVES *. 812!2m REM 803121 REM VALUES IN: K(I.l). K(I,2), K(I,3) I 1 6 804121 REM 80512! GOTO 8240 8060 K (1,1> 807121 K(I,2) 808I21K(l,3) 909121 K(2.1> 810121 K(2.2) 8110 K(2.3) 8120 K(3.1) 813111 K(3.2) 9140 K(3.3) 915121 K (4,1) 8160 K(4.2) 817111 K(4.3) 8180 K (5,1) 919121 K(5.2) 821210 K(5.3) 821111 K(6.1) 8220 1.2) 823121 K<6.3) 8240 R(I.2) 825111 R(2.2) 8260 R(3.2) 827121 R(4.2) 828111 R(5.2) 829121 8300 REM 8310 5(1.2) 832121 9(2.2) 8330 5(3.2) 834121 5(4.2) 9350 5(5.2) 936121 5(6.2) 837111 REM tNT INT INT FIX FIX FIX FIX FIX FIX INT INT INT FIX FIX FIX ( FIX FIX K(I.3) K(2.3) K(3.3) K(4.3) K(5.3) K(6.3) K(I.3) K(2.3) K(3.3) K(4,3) K(:5.3) K(6,3) 9380 REM VALUES OUT: 9390 REM 8400 RETURN 841121 REM K(I.1)+.00001 10000 K(I.2)+.00001 1000121 K(I.3)+.00001 1000121 K(2,1)+.0001211 10000 K(2.2'+.0001211 If1l000 K(2.3)+.00001 10000 K(3, 1'+.0012101 1000121 K(3,2)+.00001 10000 K(3,3)+.0001211 If1l000 K(4,l)+.000f1l1 If1l0f1l0 K(4.2)+.12I12112101 112101210 K(4.3)+.000f1l1 1000121 K(5.1)+.12I001211 10f1l00 K(5.2)+.00001 10000 K(5.3)+.00f1l01 100121121 K(6.1)+.0001211 1012100 K(6.2)+.00001 1001210 K(6,3)+.00001 ) 10121121121 -K ( 1 1) ) H*2) -K(2,1) ) (H.2) KC3,1) (H*2) -K(4.1) ) (H*2) -K (5.1) ) CH.2) -K(6,1) ) (H*2) 10000 100f1l121 U!1000 11211210111 10000 1121000 100121121 112100121 100121121 / 10000 1000121 10000 / 1011100 112112100 1121000 / 10000 2.KC1.2) + KC1.1) 2.KC2,2) + K(2.1) -2*K(3,2) + K(3.1) -2*K(4.2) + K(4.1) (H.H) CH*H) (H.H) (H.H) -2*K(5,2) + K(5.1) -2*K(6,2) + K(6.1) RCI.I).5CI.2) 154 CH*H) / (H*H) I = 1 6

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8420 REM ******* COMPUTE NORMAL AND FRICTIONAL FORCES ******* 8430 REM 8440 IF CRC3.1) <> 0 AND RC6.1) <> 0 ) GOTO 8460 ELSE GOTO 8980 8450 IF CKC9.3) >= FIMAX OR K(10,3) F3MAX OR K(11.3) >= F7MAX) THEN GO TO 8980 8460 REM 8470 Rll X = 01 8480 RIIV -1 T*.5 8490 R13X -1 CWI 01) 8500 R13V -1 T*.5 8510 Rll Z HTl 8520 R13Z '" HTl 8530 R152 '" HT1 -3.5 8540 R15X 02 8550 112 11 8560 III = Mt CHTt*HT1*4 + OALl*OALl) 9570 REM 9590 R23X (W2-03) 8590 R23V = -1 T*.5 8600 R24X .. 03 8610 HT2 -3.5 8620 R23Z ". HT2 8630 121 = (1/12)*M2*(4*HT2*HT2 + OAL2*OAL2) 8640 122 ". 12 8650 Fl = K(9.3) 8660 F5 K(7.3) 8670 REM F3 = KCll21.3) F6 .. KC8.3) : F7 = K C 11 ,3) 8690 F51X F5 COSCKC3.3 : F51Y '" +1*F5*SINCKC3,3 8690 F61X +1 F6 SINCKC3.3 : F61Y = +1 F6*COSCKC3,3 8700 F52X = -1 F5 COSCKC6,3; F52Y'" F5 SINCKC6,3 8710 F62X = F6 SINCKC6.3: F62Y". F6 COS(KC6.3 8720 IF NQUIZ7 = 1 THEN GOTO 8790 8730 TRLI C-R24X'* WT2 R24Z F52X + R24Z F62X) (R(6,lA2 8740 TRL2 C-R24Z *F62Y R24Z F52Y R23Z*F7) S(6,l) 8750 TRL3 = (R23Y + R24Y)1 SC6,l) 8760 TRL4 C R23X R24X ) (RC6,l) RC6,l 8770 N7 (+TRL1 + TRL2) (TRL4) : N7LST = N7 8780 N4 .. WT2 -N7 8790 IF NQUIZ3 = 1 OR NQUIZ1 = 1 THEN GOTO 8860 8800 TCTI (+RI1X*WT1 + RIIX*N4 + R13Z*F9 + R15Z*F5X + R15X*N4 R15Z*F6X) C R C 3, 1 .... 2 8810 TCT2 = (RIIZ*F1 R13Z*F3 +F15Z*F51Y +R15Z*F61Y) SC3.1) 8820 TCT3 = (R13Y RIIY) S(3.1) 8830 TCT4 = -1 (R13X + RIIX)' (RC3.1A2 8840 N3 C+TCTI + TCT2) (TCT4): N3LST N3 8850 Nt = WTl + N4 -N3 886121 Fl = K(9.3) 887121 F3 KCtl2l.3) 888121 F7 = K(11.3) 889121 REM ********** ON SCREEN ITERATIONS FOR DEBUGGING 8900 GOTO 8980 8910 PRINT "Nt = "N1 8920 PRINT" N3 "N3 8930 PRINT" N4 = "N4 8940 PRINT N7 = "N7 895121 LPRINT TABC2'TM: TAB