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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.

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.

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.

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.

It is a challenge to design vehicle for good ride characteristics to the engineer, given the many design constraints. The vibration environment is a very important criterion by which people judge the design and construction “ride quality” of a car. It needs a quarter - vehicle mathematical model to describe the dynamic motions of a vehicle on the road. In this paper, LaGrange Equation is used to derive the equations of motion. A state - space formulation is obtained by using state variables. And the characteristic equation, natural frequency and damping ratio are achieved. Through the analytical and numerical study on the ride quality of a quarter - vehicle (2 - DOF, 0 - dimension) model, the dynamic behavior of vehicle is represented by simulating the input, which is single-input-single-output (SISO) - output relationship. [1]* The effects of mass ratio, stiffness ratio and damping coefficient ratio on the vibration of the sprung mass are researched.

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.

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.

This paper reviews the state-of-the-art of semi-active and active suspensions for use in ground transportation vehicles and compares their performances to conventional suspension. The evolution of active and semi-active suspensions from theoretical and analytical consideration to application is presented. The primary focus of the paper is on ride quality control. The section on theoretical considerations summarizes the results of a multi-degree of freedom (MDOF) vehicle models. The paper investigates analytically, the use of active and semi-active suspensions for ride quality. It is shown that the separate control structures using different measurements and actuator are effective in controlling ride. Generally speaking, an active suspension system is required to optimize between ride comfort and road handling, between soft and firm ride. It is shown in this paper that improved performance is coupled with increased hardware complexity and higher costs.

Design specifications for suspensions of passenger cars for ride and handling in the past half century were amazingly associated with some “Magic Numbers.” These parameter values are needed for mathematical models which describe the dynamic motions of a vehicle on the road, help researchers to validate the overall performance of their simulations. Moreover, these numbers should be viewed as a guide for evaluation of vehicle design practice for moderate payload and driving conditions. Number 1, for instance, is the magic number associated with bounce resonant frequency of the sprung mass, but also related to the desired value for the dynamic index for ride quality and directional control. Number 10, on the other hand, is the magic number associated with wheel hop resonant frequency and the desired total mass/-unsprung masses ratio.

A comprehensive analytical and numerical study on the design of passive, adjustable and automatic suspensions is presented. The fundamental limitations inherent in a passive system which are incompatible with the active and semi-active systems are demonstrated. The mathematical models of the vehicle are linear time-invariant lumped parameter systems with fixed configurations. The vibrations of the vehicles which are assumed to move forward at constant velocity are induced by deterministic ground roughness. With the above models the influence of active, semi-active damping on ride control is discussed. Computer simulation results of single-input-single-output (SISO) and multiple-input-multiple-output (MIMO) automobile models, show that the semi-active-damper (SAD) significantly improves the vibration isolation capabilities of today's vehicles, and consequently represents an attractive alternative for the automobile industry.