Gear Freq. Using Euclidean Algorithm

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Gear Frequencies Using

The Euclidean Algorithm


William C. Foiles

Consulting Services Dept.

Saudi Aramco

Chairman, Saudi Arabian Chapter

of The Vibration Institute


Reference 1 (Euclid),which every educated person used to read (in Greek or Latin, at one time), describes a technique by which the greatest common divisor (GCD) of two integers (or line segments of different integral lengths) can be obtained. More modern treatments ,such as reference 2, extend this to Euclidean rings and Principal Ideal Domains (Rings); this includes polynomials. This method is commonly called the Euclidean Algorithm.

The Euclidean Algorithm is found in Theorems 1 and 2 of Book 7 of Elements by Euclid who lived in Alexandria (not Virginia, at the time there was really only one Alexandria) around 300 BC.. This 2300 year old text has truly endured the test of time; Bertrand Russell in his classic, A History of Western Philosophy, says about the Elements, "when I was young, [it was] the sole acknowledged text-book of geometry for boys." (So, what about girls?)

It is interesting to note that even the word algorithm has it origins many years ago. Algorithm derives from the name of the Arab mathematician, Al Khawarizmi (825 AD), who lived during the Abbasi period (750 -1258 AD) when the arts and sciences flourished in the Muslim world.

The problem of finding the number of assembly phases and the tooth repeat frequency for a gear set is directly related to finding the GCD of the number of teeth in both elements. The gears will have a "hunting tooth" combination, where each tooth on the input gear contacts every tooth on the output gear, if and only if the GCD of the number of teeth on the two gears is 1 (i.e. They are relatively prime.). The number of assembly phases (ways the gear set can be assembled) is equal to this GCD of the gear teeth.

|[pic] |

|Figure 1: Two Gears whose mesh combinations and assembly phases are given in tables 1 and 2 below. |

Table 1: Assembly Phase 1 - Tooth Mesh for figure 1

|G1/G2 |Tooth 1 |Tooth 2 |Tooth 3 |Tooth 4 |Tooth 5 |Tooth 6 |

|Tooth 1 |mesh | |mesh | |mesh | |

|Tooth 2 | |mesh | |mesh | |mesh |

|Tooth 3 |mesh | |mesh | |mesh | |

|Tooth 4 | |mesh | |mesh | |mesh |

|Tooth 5 |mesh | |mesh | |mesh | |

|Tooth 6 | |mesh | |mesh | |mesh |

|Tooth 7 |mesh | |mesh | |mesh | |

|Tooth 8 | |mesh | |mesh | |mesh |

Table 2: Assembly Phase 2 - Tooth Mesh for figure 1

|G1/G2 |Tooth 1 |Tooth 2 |Tooth 3 |Tooth 4 |Tooth 5 |Tooth 6 |

|Tooth 1 | |mesh | |mesh | |mesh |

|Tooth 2 |mesh | |mesh | |mesh | |

|Tooth 3 | |mesh | |mesh | |mesh |

|Tooth 4 |mesh | |mesh | |mesh | |

|Tooth 5 | |mesh | |mesh | |mesh |

|Tooth 6 |mesh | |mesh | |mesh | |

|Tooth 7 | |mesh | |mesh | |mesh |

|Tooth 8 |mesh | |mesh | |mesh | |

Tables 1 and 2 show that for the gears in figure 1 each tooth on one gear meshes with half the teeth on the opposing gear. The GCD of 6 and 8 is of course 2, the number of assembly phases for this gear set.


The frequencies to be looked at are :




F1 = Angular Velocity of Gear 1 (cpm, Hz., Rad/sec)

F2 = Angular Velocity of Gear 2 (cpm, Hz., Rad/sec)

T1 = Number of teeth on Gear 1

T2 = Number of teeth on Gear 2

Ap = Number of Assembly Phases

TRP = Tooth Repeat Frequency

The number of teeth on gear 2 which any one tooth on gear 1 contacts is T2/GCD, and similarly the number of teeth on gear 1 that a single tooth on gear 2 contacts is T1/GCD.

The usual method that I have encountered engineers and technicians using to compute the GCD involves the prime factorization of the numbers of gear teeth. Prime factorization, in general, is not trivial, nor is it efficient. The following examples illustrates both methods.


T1 = 30 - number of teeth on the first gear element.

T2 = 75 - number of teeth on the second gear element.

The prime factorizations are simple for this case (Why else would I choose it?) and are given below.

T1 = 5×3×2

T2 = 52×3

The greatest common divisor can now easily be found to be:

GCD = 5×3 = 15 = Number of assembly phases

TRF = F1×15/75 = F1/5 = F2 /2

Each tooth on gear 1 mates with 5 teeth on gear 2; while each tooth on gear 2 mates with only 2 teeth on gear 1. Such a simple combination has 15 different ways of assembly and the resulting wear patterns.

Euclidean Algorithm

The Euclidean Algorithm uses the modulo or remainder function of integer division; this is available on most calculators and in most computer programming languages. This method provides an efficient solution to this type of problem. It proceeds as follows:

For two integers a, b; there GCD, gcd(a,b) can be found by taking successive remainders until a remainder of zero is found as follows (assume a=gear2teeth){




else {




/* Euclidean Algorithm Start */

do {

gcd= a%b; /* Modulo function - part of "C" */




while (gcd != 0);

gcd = a ; /* gcd is the greatest common divisor of the two gear teeth */

/* End of Euclidean Algorithm */

toothrepeatfrq =inputspeed/gear2teeth*gcd;

/* Print Results */

printf("tooth repeat freq. = %8.3f\n",toothrepeatfrq);

printf("number of assembly phases = %d\n",gcd);

printf("gearmesh = %9.2f\n",inputspeed*gear1teeth);

printf("output speed = %9.2f\n",outputspeed);

return 0; /* No error codes to return */



1. Euclid, Elements, circa 300 BC.

2. van der Waerden, B. L., Translation of the seventh edition by Fred Blum and John R. Schulenberger, Algebra, Volume 1, Frederick Ungar Publishing Co., 1970.


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