DYNAMIC BEHAVIOR OF THE FULL-CAR MODEL IN THE J-TURN MANEUVER BY CONSIDERING THE ENGINE GYROSCOPIC EFFECTS

Resume This study presents a new dynamic modeling of a vehicle by considering the engine dynamics. By selecting the vehicle coordinate system as the reference frame, all the force-torque equations of the sprung mass and unsprung masses are derived in this coordinate system by using the Newton’s equations of motion. Unlike the previous researches, in this work the sprung mass of the vehicle is not considered as a rigid body. The dynamics of the sprung mass components, such as gyroscopic effects of the engine crankshaft, is considered. In order to study the vehicle's dynamic behavior, in the J-turn maneuver, the governing equations of the full-car model are evaluated and validated by the numerical simulation method and ADAMS/ Car software. Based on the results, the maximum roll angle and roll rate of a vehicle reach about 8 degrees and 40 degrees per second, respectively.

and its lateral stability by experimental validations, [6]. Phanomchoeng and Rajamani [7] developed a new rollover index that can detect both tripped and untripped rollovers by experimental and simulation examinations. The purpose of computer simulations is to reveal the effect of systems and components on the dynamic behavior of a vehicle as much as possible. By using the computer simulations, this purpose can be reached much earlier in the targeting and initial design stages of a vehicle than for the actual prototype. Chen et al. [8] simulated the rollover dynamics, such as rollover speed thresholds of a vehicle in roundabouts. Loktev et al. [9] determined the geometric, kinematic and dynamic characteristics of a vehicle and its state parameters on the road by computer vision algorithms. Phalke and Mitra [10] simulated a quarter-car model to investigate the effect of damping coefficient, stiffness, sprung mass and velocity on ride comfort and road holding. Saga et al. [11] investigated possibilities of a fuzzy technique in a vehicle dynamic analysis by computer simulations. Kazemian et al. [12][13] presented the rollover index and new dynamics of suspension system in order to study the vehicle's dynamic behavior by computer simulations. Rajamani [14] and Gillespie [15] studied the vehicle dynamics and its subsystems such as tire in order to study on the vehicle's dynamic behavior, such as roll and yaw dynamics, by considering the dynamics' basics [16]. Pacejka [17][18][19] examined the tire characteristics, such

Introduction
Safety assessment is one of the most important issues in the automotive industry. The safety of vehicles can be examined from two perspectives: the pre-crash safety and post-crash safety. Stability analysis is an important factor in the pre-crash safety and vehicle instability plays a major role in fatal crashes. Understanding the dynamic behavior of a vehicle in standard maneuvers can be effective in assessing its stability.
A set of experimental examinations was performed to evaluate the actual dynamic behavior of a vehicle in various standard maneuvers [1], from which the phase IV rollover tests of the National Highway Traffic Safety Administration (NHTSA) [2] can be pointed out. Cooperrider et al. [3] performed experiments to investigate tripped rollover, the results of which included five types of the curb-tripped rollover tests. In their researches, acceptable information was obtained through the minimum required speed for the rollover occurring, as well as the characteristics of the rollover phenomenon at different speeds. Labuda et al. [4] studied the simulation of wheeled vehicle dynamic regimes in laboratory conditions by considering the congruent courses of driving speeds. Sindha et al. [5] developed an experimental prototype along with its simulation to study the steer system on the vehicle stability improvement. Zhang et al. investigated the vehicle maneuverability located on the roll axis, which is the most important coordinate system and all variables are expressed in that system, is located at a point on the roll axis and below the center of gravity of the vehicle mass (VCG). It is also assumed that this coordinate system rotates only around the Z axis. To obtain the equations of wheels motion in the vertical direction and around their rotation axis for each wheel, a coordinate system is used at the center of rotation. According to coordinate systems and using the Newton's equations of motion, all equations of motion for the sprung and unsprung masses are obtained in the vehicle coordinate system. Therefore, for the sprung mass, the following Equations can be written: , cos cos sin s in . cos sin cos Similarly, for the unsprung masses, next Equations are obtained: By placing velocities of Equations (1)-(3) in the Newton's linear momentum equation of the sprung mass, the force vector of the sprung mass in the vehicle coordinate system is obtained: as its longitudinal and lateral forces and obtained the data, by mathematical expressions based on a formula. In vehicles, knowing the dynamic behavior of the engine elements, such as crankshaft, can be useful for obtaining the dynamic behavior of the vehicle [20][21]. The effect of static misalignment on the dynamic behavior of a main crankshaft bearing is examined by Lahmar et al. [22]. Ahmadabadi [23] proposed an application of lightweight vibration control strategy known as nonlinear energy sink (NES) to mitigate the undesired vibrations in engine crankshaft systems. Huang et al. [24] proposed a method for dynamic balance measurement and imbalance compensation of crankshaft assemblies. Moreover, basic researches of dynamic modeling have been reported in [25][26][27][28] This study, by presenting a 15-DOF model of the vehicle dynamics, considers reducing the complexity of the model to the extent that it would be acceptable for the dynamic behavior of the vehicle studying and modeling the necessary subsystems, such as tire and engine, with sufficient accuracy. The tire is modeled with the Pacejka 89 model, which calculates the tire forces using the longitudinal and lateral slips. The moments due to the gyroscopic effect of engine rotation are taken into account in the equations and the final equations of motion of the vehicle were derived. The dynamic behavior of the 15-DOF presented model is validated by the ADAMS/Car software. Dynamic behavior and stability of a vehicle in the J-turn maneuver is simulated by the Newmark numerical method under the supervision of the NHTSA, [25].

Modeling and equations
In this research, the vehicle is considered as a set of lumped masses including the sprung mass and four unsprung masses as a set of wheels and tires. The unsprung masses are connected to the sprung mass by the spring and a damper. Each tire is assumed equivalent to a spring and a damper, parallel in the vertical direction. The number of degrees of freedom that are considered for the vehicle model is 15, out of which 6 DOFs are related to translation and rotation of the sprung mass. The next 4 DOFs are for vertical movement of the unsprung masses, which indicate the vertical movement of the suspension systems, while the following 4 DOFs are related to rotation of the wheels around their axes and 1 DOF is considered for steerability of the front wheels. For this set of masses, separate coordinate systems, such as fixed inertial coordinate system (O), sprung mass coordinate system (S), roll axis coordinate system (vehicle coordinates (V)) and wheel coordinate system (US), are considered. The direction of the coordinate systems, as shown in Figure 1, is in accordance with the SAE (Society of Automotive Engineers) standard [15]. In Figure 1 the sprung mass coordinate system is located at its center of gravity (CGS). The coordinate system According to Equation (13), the total forces acting on the vehicle include the forces acting on all the sprung and unsprung masses: where m m m m 2 2 s u sf usr = + + is the total mass of a vehicle. According to the Newton's angular momentum equation, the resultant of the torques, acting on the body, is equal to change of the body's angular momentum. If one writes this relation for the center of the vehicle's coordinate system, the distance of which to the center of the sprung mass coordinate system remains constant, Equation is obtained: where RS V is the matrix of coordinate transformation from the sprung mass coordinates to the vehicle coordinates (see Appendix 1). Assuming that values of inertia moments remain constant, with small rotation of the roll and pitch of a vehicle, on obtains: where: cos sin sin sin cos cos Similarly, for the unsprung masses the following Equations are obtained: cos sin sin cos Considering that the coordinate center of a vehicle is a point that differs from the center of mass of the sprung and unsprung masses, the total torque applied to the vehicle can be calculated as [16]: .
Equation (22) shows the angular torque around the desired point Q. According to the toques principle in which the sum of the torques of all the external forces of a system around the point Q must be equal to resultant torque around Q and the following Equation is obtained: By placing the vector of forces and torques in Equation (23) for the proposed vehicle model, one obtains: By entering the force-torque equations (Equations (7), (9)-(12), (18) and (21)) in Equation (24) Since the xy plane is a symmetry plane of the vehicle with a good approximation, so I I 0 xy yz = = and with ignoring the value of Ixz , the matrix of the sprung mass inertia moments is presented by: By placing , S V S Ṽõ and IS V , respectively, from Equations (2), (16) and (17) in Equation (15) To derive the equation of the unsprung mass angular momentum, similar to e equation of the sprung mass angular momentum (15), Equations (19)-(21) are obtained as follows:

Equations of external forces
By writing the resultant of forces and torques applied to the vehicle, next two Equations are obtained: where n shows the unsprung masses.
In Equations (28) and (29) Z n is the displacement of a point of the sprung mass in the vertical direction above the unsprung masses, which is calculated according to Equation The left-hand side of Equations (13) and (25)- (27), which are the main equations of motion of the vehicle, includes the forces and torques of the external forces applied to the vehicle and an example of them is shown in Figures 2 and 3. Note that by considering the directions of the coordinate systems, these forces and torques are entered into the equations of motion of the vehicle. engine and Iae is the crankshaft inertia moment around its rotation axis. In this study, the gyroscopic moment of the crankshaft is considered like other external torques and is added directly to the torque vector of external forces.

Wheels equations of motion
The wheel's equations of motion in the vertical direction are determined according to Equations (35)-(38). With writing the resultant of forces for each of the unsprung masses (Figure 4), one obtains:

The tire modeling
In this research, the model of Pacejka 89 (Magic Formula), which has the ability to estimate the lateral and longitudinal forces of the tire under the lateral and longitudinal slips, is considered [17][18][19]. This model receives variables such as vertical force of the wheel and longitudinal and lateral slips as input, while its output are the longitudinal and lateral forces of a tire. The longitudinal slip is considered based on Equations (39) and (40): The longitudinal slip during the acceleration is: while the longitudinal slip during the braking is:

Equations of engine rotating
Generally, in deriving equations of a vehicle, the sprung mass is considered as rigid. Now, if a part of the sprung mass has a rotation, relative to the vehicle coordinate system (such as the engine crankshaft), then it is necessary to enter the effect of that rotation in the force-torque equations that have been calculated so far. In this study, it is assumed that the rotating components are symmetric, so the product of inertia multiplications is zero. In this case, the rotation of these components will not create any force and therefore the force equation remains stable. However, if the rotational velocity of the rotating components or their moment of inertia are significant, then the gyroscopic moments, due to the angular momentum of the rotating component, are considerable and their effect must be considered in the torque equation. The gyroscopic moment is obtained according to: In Equation (31) H is the angular momentum vector of the engine and the vector of the angular velocity of the engine in the vehicle coordinate system e Ṽ h is considered according to: In this study, the crankshaft coordinate axes are considered to correspond to the vehicle coordinate axes, in which case the product of the crankshaft inertia multiplications in the vehicle coordinates is also equal to zero. Thus: In Equation (34) where, C is the inertia matrix and C and G are functions of the displacement and velocity, K is the vector of external excitations, E0 is the initial displacement, E0 o is the initial velocity and a0 is the initial acceleration. Displacement and velocity can be predicted by: arctan

Validation and simulation results
Several different test programs in the phase IV of the light vehicle rollover research program were reviewed and evaluated to select the most appropriate maneuver to investigate the rollover. According to Table 1, it can be seen that one of the best maneuvers is the J-turn test, which has obtained the highest score. Due to the fact that the vehicle is assumed to be perfectly symmetrical with respect to its xz plane, direction of the steering wheel angle (clockwise or counterclockwise) of Figure 9 has no effect on the maneuver results. The initial speed of the vehicle is shown in Figures 10 and 11 and the engine speed is 5000 rpms. During the J-turn maneuver, the passed trajectory by the vehicle is shown in Figure 12. By applying the steering wheel angle according to Figure  9, the Newmark numerical method and parameters of Table 2, the dynamic behavior of the vehicle is compared by ADAMS/Car software according to Figures 13 and 14. Based on the validation results, it is found that the 15-DOF presented model in this research simulates the vehicle's dynamic behavior with a good accuracy. According to the numerical results (Figures 13b, 14b, 15 and 16), by applying the steering input, the lateral acceleration reaches about -0.8 g and finally decreases with decreasing longitudinal velocity of a vehicle.
The maximum roll angle and roll rate of a vehicle reach about 8 º and 40 º/s. Moreover, the yow rate of the vehicle reaches a maximum of about -41 º/s and then decreases to about -15 º/s and finally reaches about -35 º/s.
The process of the Newmark method is shown in Figure 8.  Figure 9 Schematics of the steering wheel angle of a vehicle in: (a) NHTSA J-turn maneuver [2] and (b) this study

Conclusions
This study presents the dynamics of a 15-DOF model of a vehicle by performing simulations to investigate the vehicle's dynamic behavior in the J-turn maneuver under the supervision of the phase IV of NHTSA's light vehicle rollover research program. Using the Newton's equations of motion, the equations of motion for the sprung and unsprung masses are all written in the vehicle coordinate system. In order to study the engine dynamics, the crankshaft coordinate axes are considered to correspond to the vehicle coordinate axes. Finally, the gyroscopic moment of the crankshaft is added directly to the torque vector of external forces and the governing equations are evaluated by numerical The transformation relation between the coordinate systems of the sprung mass and a vehicle, which results from the pitch and roll rotation, is obtained as follows: