MEI Mechanics-1 Maths course-work An investigation into automatic Traffic Lights Daniel Baird The Edinburgh Academy Autumn 1996 Table of contents Table of contents 2 Introduction 3 Question 3 Aims 3 Simplifications and Assumptions 4 Road 4 Traffic Lights 4 Vehicles 4 General assumptions. 4 SEQUENCE OF THE TRAFFIC LIGHT 5 MODEL 1 5 Maximum distance travelled in order for a vehicle to clear the system 5 Period of one cycle 6 Period of green light 6 Dead Time 6 Period of red light 7 MODEL 2 8 Data collection 9 Real Situation 9 List of other factors 9 CONCLUSION 10 Introduction In this brief report, my aim is to attempt to determine the factors that affect the pattern of operation of temporary traffic lights in operation while roadworks or another obstruction is blocking one lane of a two-way, single carriageway road. Once this has been accomplished, I shall attempt to create a simple mathematical model and attempt to apply it to real life. Question When major roadworks take place on a two-way single carriageway road, contractors frequently regulate a one-way flow of traffic in alternating directions by use of automatically controlled traffic lights. What considerations affect the timings of such lights Aims 1.
Analyse problem. 2. Propose a formula. 3. suggest errors involved. 4. look into ways where timings from formulae could be modified other than simplest case. Simplifications and Assumptions + Road n The road is two-way, single carriageway. n The road is straight, there are no obstructions. n Road surface is constant, friction is constant. + Traffic Lights n Drivers have infinite patience and always obey traffic lights perfectly n All sets of lights are perfectly co-ordinated, and their pattern constant. + Vehicles n Vehicle size is constant. n Vehicle acceleration / deceleration is constant. n Distance between vehicles is constant. n All vehicles are perfect. n Distance travelled by cars between traffic lights is constant.
General assumptions. n Weather is constant, clear and dry. n The time that the amber light is displayed for is quasi-instantaneous and therefore can be ignored. N.B. These assumptions have been made in order to allow my models to function properly. The theoretical values that the models formulate are to be taken with a pinch of salt and therefore to take account of other variables, error bounds should be applied to reduce these discrepancies. SEQUENCE OF THE TRAFFIC LIGHT 1. Red 2. Red / Amber 3.
Green 4. Amber 5. Red MODEL 1 Distance between traffic signals. 34 m Mean speed of traffic. 5 ms-1 Time for traffic to clear the system (34 m) at 5 ms-1 6.8 sec Mean length of one vehicle. 4 m Mean inter-vehicle gap (dynamic).
5 m Mean inter-vehicle gap (static). 0.5 m Number of vehicles per flow. 8 Time delay between cars starting at first set of signals. 1 sec.
Duration of amber light. 3 sec's. In this model, it is assumed that every cycle of the lights, no mre or less than eight vehicles pass through the system in one direction. The minimum distance a car has to travel to clear the sytem is 34 m. On the lights turning green, the first car instantly reaches a speed of 5 ms-1 one second later the second car does the same and so on.
Maximum distance travelled in order for a vehicle to clear the system 34+ (70.5) + (84) = 69.5 m The last vehicle in the flow has to travel the 34 m between the traffic lights plus the distance between itself and the first set of lights. This latter distance is the length of all eight cars plus the distance of seven static inter-car gaps. Period of one cycle Time = Distance / speed = 69.5 m / 5 ms-1 = 13.9 sec's This is the time taken for the first vehicle to clear the system, however the last vehicle has to wait 7 seconds before it can start to move. 13.9+7 = 20.9 sec's The time from the first car starting until the last car clearing the last set of lights is therefore 20.9 seconds. Period of green light = 35.5 m / 5 m /'s = 7.1 sec's This is the time for the last vehicle to clear the first set of lights, however, we have to consider the time delay between cars starting. 7.1+7 = 14.1 sec's Therefore the minimum time the green light must be illuminated is 14.1 seconds.
Dead Time Time = period of one cycle - period of green light. = 20.9's - 14.1's = 6.8's Therefore, the system cannot be clear until 6.8 seconds after the green light has been extinguished. Period of red light To find the duration of the red light, I shall map out the two sets of lights and fill in the blanks! Light one Period / seconds Light two Green 14.1 Red Amber 3 Red Red 6.8 Red Red 3 Red / Amber Red 14.1 Green Red 3 Amber Red 6.8 Red Red / Amber 3 Red Therefore the period of the red light is (6.8+3+14.1+3+6.8) = 33.7 seconds MODEL 2 In order to produce an equation for these traffic lights, I shall introduce some unknowns. Variable Symbol Distance between traffic signals. Dsig Mean speed of traffic. vs. Mean length of one vehicle.
L Mean inter-vehicle gap (static). Gs Number of vehicles per flow. n Time delay between cars starting at first set of signals. t Duration of amber light. A Maximum distance. Dmax Maximum time. Tmax Dead Time.
Tdead Period of green light. Tgreen Period of red light. Tred Dmax = Dsig+ [ (n-1) Gs]+nL Dmax = Dsig+nGs+nL-Gs Tmax = (Dmax / v.) + (n-1) t Tmax = Dmaxv-1+nt-t Tgreen = (Dmax - Dsig) /vs. + t (n-1) Tgreen = Dmaxv-1 - Dsig-1 + tn - t Tdead = Tmax - Tgreen Tred = 2 Tdead +Tgreen + 2 A Data collection The data I have used was collected in Broughton, Edinburgh. A construction site near the road is blocking one lane of this two-way, single carriageway road. The data was collected by the students in the Mechanics A-level maths set.
Stop watches were used to time the traffic signals, and a wheel ruler to measure the dimensions of the road. The weather was clear and windy. Real Situation In a real situation it is even more complicated as not only would we have to bear in mind the acceleration and deceleration of the vehicles, but even these would not be constant. Many other factors also effect the results, such as if the traffic is already moving at the first set of signals, also factors such as weather conditions effect the style of driving. Factor Error bounds Car Length x 0.75 m Inter-car gap 2.6 mx 0.2 m Time when system is empty 5 sec 2.5 sec Speed of flow 4.26 ms-10.75 ms-1 List of other factors + Weather + Time of day + Presence of special vehicles, e.g. Articulated lorries, bicycles, horses etc. + Reckless drivers jumping lights.
CONCLUSION This project has given me an interesting insight into the outwardly simple maths of everyday systems. I would consider these models to be anything but perfect as there are far to many random factors involved and this makes it too difficult to model accurately without chaos theory. 336.