Resonant cycle length

[tradephoric]

The concept of a resonant cycle is a cycle length that achieves good two-way progression.  A resonant cycle allows both directions of travel along a 2-way road to receive a "green wave"  which limit the number of red lights drivers get stopped at. 

Below is a graph that shows the resonant cycle at varying signal spacing and speed limits:


While a resonant cycle is an ideal cycle to shoot for, it's often not practical to achieve.  As a real life example, consider Beach Boulevard near Long Beach, California.   The corridor has a speed limit of 50 mph with major signals spaced roughly 2600 feet apart.  Referring to the chart, a resonant cycle along Beach Boulevard would be achieved at a 70 second cycle.   A 4-phase signal (which are common along Beach Blvd.) running a 70 second cycle would lead to a very high lost time.  Since a 70 second cycle length isn't practical, there is no way to achieve a resonant cycle for Beach Boulevard (drivers will get stopped at red lights in at least one direction of travel, regardless of how well the signals are timed). 

[Jardine]

IIRC, Rockford Illinois at one time (maybe still?) had the lights sync'd east of downtown for 5 MPH less than the speed limit.  For safety.  So there  can be other purposes of timing than just getting the maximum throughput.

LOL, if I owned a convenience store along a very well "sync'd" street, I would look for a pliable city council person to bribe to put the signal on my corner 180 degrees out of phase with the rest of them.

Ka-ching!!!

[tradephoric]

IIRC, Rockford Illinois at one time (maybe still?) had the lights sync'd east of downtown for 5 MPH less than the speed limit.  For safety.  So there  can be other purposes of timing than just getting the maximum throughput.

This can be done with a resonant cycle.  Just find the resonant cycle that is 5 MPH under the posted speed limit.  That way, BOTH directions will be timed for 5 MPH under the speed limit.

[Duke87]

So am I correct in concluding here that the mathematical derivation of these lines is cycle length = 2 x distance/speed?

2 x 2600 ft / 50 mph (x 3600 s/h / 5280 ft/mi) = 70.9 seconds. Seems to roughly work out for another couple points I tried it with as well. And makes sense because if we assume roughly half of each cycle is green (35 seconds in this case), then when that matches the time it takes to drive from one intersection to the next, you do indeed get a "resonance" that allows both directions to flow freely.


Unfortunately you are right about alternatives: if 2d/s is too short a cycle length, 3d/s, 4d/s, etc. do not achieve the same desired effect. A "higher order" resonance is not possible.


What this underscores is the need to avoid having signals too close together on high speed roadways!

[vtk]

A half-mile grid can have well-timed signals, if each street on that grid is optimized for one direction.  Parallel streets would alternate their direction of optimization, and the cycle period would be the same as for a full-mile grid with two-way resonance.  Then each street could be designed with more capacity in the favored direction. Drivers would adapt, learning to use the streets as one-way pairs over long distances, with contraflow usage still available and practical for shorter distances.

Intermediate signals on the half-mile streets need only be synchronized to the favored direction.  If they run on double-length periods and with shorter green time for the minor cross street, degradation of throughput in the disfavored direction on the half-mile streets by these intermediate signals can be abated.

I should make an animation of this. That would be nifty.

[tradephoric]

So am I correct in concluding here that the mathematical derivation of these lines is cycle length = 2 x distance/speed?

2 x 2600 ft / 50 mph (x 3600 s/h / 5280 ft/mi) = 70.9 seconds. Seems to roughly work out for another couple points I tried it with as well. And makes sense because if we assume roughly half of each cycle is green (35 seconds in this case), then when that matches the time it takes to drive from one intersection to the next, you do indeed get a "resonance" that allows both directions to flow freely.

Yep, that looks perfect.  Turns out to be double the travel time.

Really what the chart is showing is the "critical"  resonant cycle.  If the required cycle length along a corridor is higher than the "critical"  resonant cycle, then perfect dual progression isn't possible (where: critical resonant cycle = 2 x travel time).   Too often along major corridors, the required cycle length is higher than the critical resonant cycle making dual progression impossible.

I find it easiest to visualize on a time-distance diagram.  Here is the time-distance for Beach Boulevard running its critical resonant cycle of roughly 70 seconds.  If the cycle length increases to higher than 70 seconds, than perfect dual progression is lost.