01 - Introduction to trajectories Analysis - OMT

🕒 17:30: Problem solved in class (see iPad)

Imagine three friends who want to take a long trip using a tandem bicycle for two persons. Because the bike riders travel at 20 km/h, independent of the number of riders, and all three persons walk at 4 km/h, they proceed as follows:

To start the journey, friends “A” and “B” ride the bicycle and friend “C” walks.
After a while, friend “A” drops off friend “B” who starts walking and “A” rides the bicycle alone in reverse direction.
When “A” and “C” meet, they turn the bicycle and ride forward until they catch up with “B”.
At that moment, the 3 friends have completed a basic cycle of their strategy, which they then repeat a number of times until they reach their destination. Could you determine their average travel speed?

Solution: $v_{m} = 10 \frac{k m}{h}$

Introduction to trajectories Analysis 2024-10-01 18.15.47.excalidraw.png
%%🖋 Edit in Excalidraw%%

Trajectories of a traffic stream on a time-space diagram

When we work with time and space we use coordinated axis, with time ( $t$ ) on the horizontal axis and space ( $x)$ on the vertical axis.

This visualization allows to have all the main parameters readily available:

Position
Speed/velocity
Acceleration
Time

Velocity:

v = \frac{\partial x}{\partial t} = \dot{x}

Acceleration:

a = \frac{\partial^{2} x}{\partial t^{2}} = \ddot{x}

The graph shows a possible trajectory of an elevator that stops at floor 3 and 5.

Scheduling

We want to figure out the capacity of a rail line (single rail)

We have a train traveling from A to B at constant speed $V_{t e c h}$ . We neglect accelerations.

Capacity is a flow: $q_{m a x} [\frac{v e h}{h}]$

flow

It's the number of passengers or vehicles or expeditions per hour
$\underset{―}{q}$

Dwell time is the time taken for passengers to board and un-board.
Headway: ( $h$ )

\underset{―}{q} = \frac{1}{\overset{―}{h}}

then:

{\underset{―}{q}}_{m a x} = \frac{1}{h_{m i n}}

We can increase capacity of the diagram above in different ways:

Add a way for trains to pass each other in the middle: side track
We could add more sidetracks but it becomes a problem of controlling the switches. If any train is delayed, then every train gets delayed
Faster trains would allow for shorter headways
Have more tracks at the ends so we can have train ready to leave as soon as one arrives