The document describes an algorithm to determine the common or synchronization period (Tc) of two independent asynchronous events with periods of ta and tb. The algorithm involves 4 steps: 1) Determine the maximum (t2) and minimum (t1) periods. 2) Calculate the difference in periods (Δt). 3) Determine the number of t2 periods in Δt (N2). 4) The synchronization period is Tc = N2 * t1. Two examples applying the algorithm are also provided.

1. Mathcad - Synchronicity Algorithm.xmcd Page 1 of 6
ASynchronicity Algorithm
by Julio C. Banks
The synchronicity time or common time of two (2) independent and asynchronous events can be readily
completed in a four (4) steps algorithm to be described in this article.Additionally, two (2) illustrative
examples are also provided for completeness of presentation of the Synchronicity Algorithm.
SynchronicityAlgorithm ProblemStatement
Create an algorithm (i.e., a procedure) which allows the determination of a synchronicity (or common)
period of two (2) independent and asynchronous events. Each asynchronous event has a period ta
and tb where tb ta
 . Name the resulting synchronicity period, Tc .
1.0 SynchronicityAlgorithm Development
Step1: Choose t
2
t
1
>
t
2
max ta tb
,
( )
= (maximum period) 1
( )
t
1
min ta tb
,
( )
= (Minimum period) 2
( )
Step2: Calculate Δt t
2
t
1
-
= 3
( )
Step3: Calculate the number of time-difference, Δt, in cycles 1 and 2 describing the largest period, t
2
Ν
2
t
2
Δt
= 4
( )
Step4: Finally the synchronization or common period is
Tc Ν
2
t
1

= 5
( )
2.0 SynchronicityAlgorithm
Tc ta tb
,
( ) t
2
max ta tb
,
( )

t
1
min ta tb
,
( )

Δt t
2
t
1
-

Ν
2
t
2
Δt

Tc
t
2
Δt

=

2. Mathcad - Synchronicity Algorithm.xmcd Page 2 of 6
3.0 SynchronicityAlgorithm Simplification
Tc Ν
2
t
1

=
t
2
Δt






t
1

=
Tc
t
1
t
2

t
2
t
1
-
= 6
( )
This is recognized as having the form of the beating period in vibration. This phenomenon occurs when
the forcing frequency is close but not equal to the natural frequency of a vibrating system. The equation
for the beating period is given as follows
Tb
2 π

ωforcing ωnatural
-
( )
= 7
( )
Equation 6 may be manipulated to better resemble Eq. 7 as follows
Tc
1
1
t
2






1
t
1






-
= 8
( )
Equation 7 may be manipulated to better resemble Eq. 8 as follows
Tb
1
fforcing fnatural
-
( )
= 9
( )
Comparison of equations 8 and 9 leads us to conclude that the synchronicity (common) period is found
by inverting the difference in frequencies of events.
The most effective method of calculating the common period is by the use of Eq. 6 as follows:
Tc ta tb
,
( ) t
2
max ta tb
,
( )

t
1
min ta tb
,
( )

Tsynchronicity
t
1
t
2

t
2
t
1
-

:=
It should be noted that as the difference in time in the denominator of Eq. 6 is diminished, the
synchronicity time increases at a faster rate.

3. Mathcad - Synchronicity Algorithm.xmcd Page 3 of 6
Rewire Eq. 6 in terms of the minimum time period, t1, and the ratio of time, 0 < t1/t2<1
Tc
1
1
t
1
t
2








-










t
1

= 10
( )
Let α
t
1
t
2








= 11
( )
and β α
( )
1
1 α
-
:= 12
( )
Tc α
( ) β α
( ) t
1

= 13
( )
For a given minimum period, t1, the amplification factor increases as α approaches unity. Plot Eq. 12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Synchronicity Factor Versus Period Ratio
β α
( )
α
Equation factor is the synchronicity amplification factor. This function increases as the fastest event
period, t2, decreases to approach the slower event, t1 which is assumed to be fixed for illustrative
purpose. The greatest change occurs at α = 0.5 and larger (i.e., α 0.5
 ).

4. Mathcad - Synchronicity Algorithm.xmcd Page 4 of 6
The Synchronicity Factor increases at such a high rate that it is best to plot in semi-log scale as follows
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
10
100
1 10
3

Synchronicity Factor Versus Period Ratio
β α
( )
α

5. Mathcad - Synchronicity Algorithm.xmcd Page 5 of 6
In the following two (2) synchronicity examples, the common time, Tc ta tb
,
( ), as well as the Synchronicity Factor,
β ta tb
,
( )are readily calculated.
Equation 6 may be written as
β ta tb
,
( ) "Synchronicity Factor"
t
2
max ta tb
,
( )

t
1
min ta tb
,
( )

Alpha
t
1
t
2

Beta
1
1 Alpha
-

:=
Synchronicity Example 1
Casey's Watch beepsevery 15 minutes. Her brother's watch beeps every10 minutes.The last time both
watches beeped was at 5:30 P
.M. What time will be the next time both watches beep simultaneously ?
Explain how you know.
Solution
ta 15 min

:= tb 10 min

:=
The synchronicity time: Tc ta tb
,
( ) 30 min

=
The synchronicity Factor: β ta tb
,
( ) 3
= (for reference only)
The time at which both beeping events with periods of 10 and 15 minutes is every 30 minutes.
The synchronicity algorithm explains how we can predict the next time both events will occur
simultaneously.

6. Mathcad - Synchronicity Algorithm.xmcd Page 6 of 6
Synchronicity Example 2
Train Aarrives at Central Station on the hour and 12 minutes. TrainB arrives at Central Station on the
hour and 15 minutes. Predict when both trains will arrive at the same time at Central Station?
Explain how you know how to choose the correct answer.
A. On the hour and 30 minutes past the hour
B. On the hour and 15 minutes to the hour
C. On the hour and 27 minutes past the hour
D. On the hour only
Solution
ta 1 hr
 12 min

+ 72 min

=
:= tb 1 hr
 15 min

+ 75 min

=
:=
The synchronicity time: Tc ta tb
,
( ) 30 hr

=
The synchronicity Factor: β ta tb
,
( ) 25
= (for reference only)
The time at which both trains with periods of 72 and 75 minutes will simultaneously arrive at Central
Station is in 30 hours. Since Tc is 30 hours the answer is D. On the hour only. The synchronicity
algorithm explains how we can predict the next time both events will occur simultaneously.
Conclusion
Asynchronicity algorithm has been developed and compared to the Beating Period Phenomenon of
vibrating system in which the forcing function is close to the natural frequency of the vibrating system.
Asynchronicity factor, β ta tb
,
( ), has also been developed and it indicates the number of t1
which will
need to be multiplied to arrive at the synchronicity (common) time, Tc ta tb
,
( )