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Technical Paper

State Space Formulation by Bond Graph Models for Vehicle System Dynamics

Modeling and simulation of dynamic systems is not always a simple task. In this paper, the mathematical model of a 4 Degree Of Freedom (DOF) ride model is presented using a bond-graph technique with state energy variables. We believe that for the physical model as described in this research, the use of a bond-graph approach is the only feasible solution. Any attempt to use classical methods such as Lagrange equations or Newton's second law, will create tremendous difficulties in the transformation of a set of second order linear differential equations to a set of first order differential equations without violating the existence and the uniqueness of the solution of the differential equations, the only approach is the elimination of the damping of the tires, which makes the model unrealistic. The bond-graph model is transformed to a mathematical model. Matlab is used for writing a computer script that solves the engineering problem.
Technical Paper

Mathematical Modeling and Computer Simulation of a Passenger Vehicle Steering System

Standard analysis of vehicle dynamics often treats the steering system as a rigid system. This paper examines the response of a compliant steering system via a mathematical model and computer simulation using MATLAB. This model represents a valuable tool for examination of steering system component stiffness effects on vehicle response. The vehicle chosen for analysis is a typical passenger sedan. The input represents a situation experienced during standard vehicle operation as well as emergency avoidance maneuvers. The findings of this paper will benefit those engineers interested in the steering performance of typical passenger vehicles. Any problems in vehicle performance relating to the steering system, such as steering precision, response, and gain, to name a few, can be made apparent through equation derivation and simulation such as was done in this paper.
Technical Paper

Effect of Chassis Design Factors (CDF) on the Ride Quality Using a Seven Degree of Freedom Vehicle Model

The kinematics and kinetics of a seven degree of freedom vehicle ride model with independent front and rear suspension are developed. Lagrange's equation is used to obtain the mathematical model of the vehicle. The equations of motion are transformed to state space equations in Linear Time Invariant (LTI) form. The effect of Chassis Design Factors (CDF) such as stabilizer bars, stiffness', Dynamic Index in Pitch (DIP) and mass ratio on the vehicle ride quality are investigated. The ride quality of the 3 dimensional vehicle that includes bounce, pitch, roll and unsprung masses motion is demonstrated in time domain response. The vehicle is considered as a Multi-Input-Multi-Output System (MIMO) subjected to deterministic ground inputs. Outputs of interest for the ride quality investigation are vertical and angular displacement and vertical accelerations. Numerical computer simulation analysis is performed using MATLAB® software.
Technical Paper

On Body Roll Angle During Transient Response Maneuver of a 3-D Model

This paper addresses the issues of mathematical modeling of a three-dimensional vehicle with 3 Degree-Of-Freedom (DOF), yaw rate r, roll rate p and side-slip angle β and computer simulation using “Matlab”. The research investigates the case of an untripped vehicle body roll angle, induced rollover on road involving a single vehicle. The untripped Body Roll Angle (BRA) and ROLLOVER risks are associated with not only the Vehicle Design Factors (VDF) but also with the driver inputs, namely forward speed u and steering wheel angle - δSW. The lateral acceleration ay is reaching its peak value during transient response due to abrupt maneuvers by the driver. The magnitude of the lateral acceleration may lead to a rollover situation. LaGrange equation for a holonomic system is used to determine the equations of motion. A unique State-Space Formulation with 4 state variables is proposed in order to obtain the Linear Time Invariant (LTI) equations.
Technical Paper

4-DOF Vehicle Ride Model

Ride quality is one of the most important criteria by which people judge the design of a car. At the most basic level, ride isolation properties are investigated using a quarter vehicle model. But the input from road roughness would excite not only bounce motions, but also pitch motions. Understanding the pitch and bounce motions is essential because it is their combination that determines the vertical and longitudinal vibrations at any point on the vehicle [2]. In this paper, a 4-degree-of-freedom (4-DOF) Vehicle Ride Model, which is shown in Figure 1, is used to investigate the effect on the ride quality of the dynamic index in pitch, mass ratio, weight distribution and flat ride tuning. A Lagrange equation is used to derive the equations of motion. A state-space formulation is obtained by using state variables. From these, the characteristic equation, natural frequency and damping ratio are obtained.
Technical Paper

Application of Bond Graph Technique and Computer Simulation to the Design of Passenger Car Steering System

Vehicle Dynamics play an important role in responsiveness of a vehicle. The performance of a vehicle depends on its ride and handling characteristics [1]. Handling is a measure of the directional response of a vehicle and one of the important characteristics from the vehicle dynamics point of view. The directional response of a vehicle depends on the dynamics of the steering system. A good steering control provides an accurate feedback about how the vehicle reacts to the road. In this paper, the powerful techniques of Bond graphs and state equations [2] are used to design and analyze the dynamics of a manual rack and pinion steering system. The author obtains the transfer function between the Angle of rotation of front tire and the Angle of rotation of steering wheel. The overall steering ratio of the bond graph modeled steering system is compared with the overall ratio of a similar vehicle to validate the model.