URI Mechanical Engineering Projects -- Dynamics

Engineering Academy of Southern New England
University of Rhode Island -- Mechanical Engineering Projects

Traffic Light Design Problem

Objectives:
Apply the dynamic simulation package Working Model to perform a preliminary kinematic analysis of automobiles approaching an intersection controlled by a traffic light. Determine the yellow light duration based on typical but conservative parameters describing the driver and the vehicles performances.

Instructional Objectives:
Learn how to use the features of working model to perform a kinematic analysis. Create object of interest, assign it the necessary properties, establish an acceleration time profile. Learn how to use meters to report out acceleration, velocity, position and time. Learn how to use the tape player to examine the details of a simulation.

Background:
Working Model is a software package which is designed to analyze and animate the motion of a rigid body based on Newton's second law. This is expressed by the famous equation

F = ma
                                                          (1)
Here, F represents the force applied to the rigid body, m is the objects mass and a is its acceleration. In this exercise we will only be concerned with the scalar version of this equation since the motions of concern all occur along a straight line.

Remember that this is really a differential equation since if one looks at the basic relations of kinematics one finds that 

                                                          (2)
That is acceleration is the time derivative of the velocity and velocity is itself the time derivative of the objects position. Do not be concerned if you do not know anything about the various methods of solution of differential equations. All that is necessary here is the idea of integration.

Basically Working model performs its task by obtaining information about acceleration at some particular time by solving equation (1). It then takes this and uses equations (2) to calculate a new value of velocity and a new value of position at a time dt later. This is a process of numerical integration. Visualizing this information at each time step constitutes the simulation of motion.

Performing a Kinematics Analysis:
Kinematics is the study of motion with out concern for the forces that must be present to create the motion. Therefore we must fool Working Model into ignoring "forces" and working with just accelerations. This is easily accomplished by setting the mass of the object under consideration to unity (m=1). With this specification, from equation (1) force becomes directly equal to acceleration and it is easy to specify forces in working model. Thus the procedure for performing a kinematic analysis is to set the mass to one and then specify an appropriate force history for the object. This will be taken as an acceleration history and subsequently integrated to provide values for velocity and position.

Problem Statement:
You are to determine the appropriate duration for the yellow light at a highway intersection. Assume that cars will be approaching the intersection traveling at a maximum of 65 mi/hr, that drivers reaction times can be as high as 0.5 seconds, and that typical cars can safely achieve a deceleration of at least 0.4 g.

Suppose the scenario is changed as far as the drivers strategy is concerned. If instead of putting on the brakes with maximum effect and continuing to apply them with constant force until the car stops suppose that initially the brakes are applied to produce the deceleration of 0.4g but then as the car begins to slow the driver decreases his force on the pedal proportional to the cars actual velocity until at zero speed the breaking force is also zero. That is suppose the deceleration is given by

What effect does this have on your results and conclusions?