Toothed wheels

Each toothed wheel is identified by two unique and simple paramenters:

- the number of teeth "n"

- the modulus "m"

where m=diameter/n

The modulus is the most important parameter because two wheels needs the same modulus to transfer the rotation properlly. Furthermore this ratio gives indications on the strongness of the tooth; in fact when "m" increases, also the wheel diameter increases, if the teeth number is still the same. Practically speaking, in order to make a tooothed wheel with "n" between 10 and 100, the following formula is used to calculate the wheel diameter:

Diameter( mm )=m(n+1)

For the pinions the following formulas have to be used:

6 teeth pinions: diameter=m(n+2)

8 teeth pinions: diameter=m(n+1,5)

For wooden wheels is better to use "m" values between 1,25 and 2 depending on the kind of wood used.

Coupled toothed wheels

If you couple two weels, that is put into gear the A wheel with the B one, with the A driving B, the B wheel will rotate in the opposite way and the KB axis will make a number of rounds given by the ratio of the teeth of the A wheel and the teeth of the B one or: KB=nA/nB. If for example nA is 60 and nB is 30, KB will make two rounds for each round of KA.

The same is valid if you move the B wheel; in this case B has to turn two times in order to obtain the complete rotation of A.

Time gearing mechanism calculation

In order to calculate the number of wheels necessary to move the escapement, first of all has to be decided the time of the escapement wheel for a complete turn and then how many rounds has to make the driving wheel in one hour time. We will indicate as:

A = Main wheel (Driving wheel)

B = Mean wheel

C = Escapement wheel (CATERINA)

B1 = Mean wheel pinion

B2 = Caterina wheel pinion

P = Weight

P1 = Counterweight

T = Rope drum

If C will make one round in one minute and A one round in two hours, let's continue with the wheels calculation.

We preferred impose one round every two hours to A in order to have more charge autonomy for the mechanism; the rope where the weight is hanging is winding on the drum T linked to the A wheel: so the working time wil be:

Tf = L*Hg/3,14*DA where:

L = clock height from the groung

DA= Drum A diameter

Hg= Hours for each A round

If the clock is fixed on the wall at about 180 cm from the ground and id the T drum diameter is about 4 cm thetime will be:

Tf= 180x2/3.14x4= 28 hours about

Coming back to the wheels calculation, C has to make 120 rounds because it takes 1 minute for a complete round and A makes one round every two hours that are 120 minutes. Now we have to find two numbers that when miltiplied will give 120. These numbers are the gearing ratio between A and B1 coupled with B and C1. For example 12 and 10 can be an optimum solution.

If we take: nA = 72 ( wheel A teeth number)

nB1 = 6 ( pinion B1 teeth number)

72:6= 12

If we take. nB = 60 ( wheel B teeth number) nC1 = 6 ( pinion C1 teeth number) 60/6 = 10 Anyway other ratios that can give the same results could be accettable. Let's verify what we choose. The 72 teeth wheel gears with B1 that has 6 teeth, so for one A round, B1 makes 12 rounds; but B wheel (60 teeth) gears with C1 of 6 teeth so C1 will make 10 rounds for one round of B, but B is linked with B1 so 12*10=120. Considering the prrevious formulas, diameter=m(n+1) and if m=1.5

diameter A= 1,5(72+1)=110 mm( 72 teeth)

diameter B= 1,5(60+1)=92 mm( 60 teeth)

diameter B1= 1,5(6+2)=12 mm( 6 teeth)

diameter C1= 1,5(6+2)=12 mm( 6 teeth)

The thickness of the wheels can change depending on the material, anyway in case of wood it can be from 6 up to 10 mm.

Escapement calculation

N.B.: the calculus of the escapement will be made exclusively on the type with pendulum, ratchet wheel and pallet to recoil, because it is of simple construction of notable precision and reliability. The basic element in the construction of an escapement is the regulator of the time is the component that for its physical characteristics completes a periodic movement (oscillation). This means that the interval of time employed from this element (pendulum) to complete a cycle is always constant. The pendulum is a regulator excellent because his period is independent from the mass "M" (pendulum bob) and it depends exclusively from his length "l" and from the strength of gravity. In fact:

T= 2x3,14x square root of l/ g

where:

T= time that employs the pendulum for complete an express oscillation in seconds.

g= acceleration of gravity- Italy (Rome)= 9.804 m/ [secq]

l= length of the pendulum expressed in meters

Since the wheel of escapement "C" will do 1 turn in 60 seconds, we could establish the number of necessary teeth to give a likewise impulses to the pendulum for maintain it in oscillation, and accordingly know the period "T" and calculate the length "l." If we set for instance [nC]= 40 (number of the ratchet wheel of the C wheel of escapement) and we call TC the time employed from this to terminate a complete rotation it will be:

T= TC/ [nC]= 60/ 40= 1.5 [sec]

This value represents the necessary time to our pendulum for complete a complete oscillation. Replaced in the formula:

T= 2x3,14x square root of l/ g

the value of l is drawn, that is the length that will have the pendulum. Resolving is had:

l= g x T to the square/ 3.14 to the square= 9,804x2,25/ 39,4384= around 0.56 meters

This is the theoretical calculus. In practice "l" real it is the found value +10% and it is gotten a perfect adjustment moving the mass "M" (pendulum bob).

Calculus of the indicative device

The indicative device is a system of one or more wheels that commanded from the movement they allow, by means of specific indexes, of read the hour. Returning to our model and remembers us that the wheel "A" make 1 turn each 2 hours; the wheel "D" will be driven from the pinion "A" and, firm staying all that told already on the cogwheels, it will have a number of teeth [nD]= 12x [nA']/ Hg.

Small glossary

CATERINA: wheel of the escapement to rocker shaft with ratchet wheel that it gear in the shovels of the rocker shaft. The number of teeth is odd, but varied according to the models, 13 or 15, sometimes more numerous.

CENTRE ARBOR: cylindrical revolving element, generally of steel, that it drags wheels and pinions.

CHARGE AUTONOMY: operation with which they rise the weights or they burden the rubber bands that furnish the power.

DRUM: Is the cylinder of grooved metal or of wood, around the which is entwines the rope of the weight.

ESCAPEMENT: mechanism that check power and transmits to the regulator so much energy how much the same regulator loses any during the oscillations.

FOLIOT: ancient French term for designate a balance wheel to bar.

GEAR: together of cogwheels and pinions that has the function of transmit the movement.

IMPULSE: push imparted through the escapement to the balance wheel or to the pendulum.

ISOCHRONISM: ownership of a regulator of repeat an oscillation in a period of time always equal.

WEIGHT and COUNTERWEIGHT: there are the weight motor and a smaller counterweight that allow the charge because, throwing the counterweight, it get up again the weight motor.

WING: each tooth of the pinion. It is also an of the thin foils of a handwheel that they brake for resistance of the air.

LIFE: interval of time during which a clock works without need of loaded being.

MOTRICE RUBBER BAND: generally foil of moderate steel, it is rolled up and contained in a barrel. It moves the train of the time and that of the ringing. It comes loaded during the position and it entwines on if same.

MOVEMENT: mechanical complex of a clock in its completeness, except quadrant, hands and box. It constitute the train of wheels and pinions, the escapement and the regulator.

PALLET: fundamental element of any types of escapement to recoil, to rest and free.

PALLET STAFF: they are births that receive the impulse of the wheel of escapement and they regulate the movement of it. Those is fixes on the rocker shaft.There are pallet staff also the terminal parts of the pallet.

PENDULUM: organ regulator with characteristics of isochronism, for the measurement of the time because it completes oscillations of the same duration in particulary conditions.

PENDULUM BOB:weight to form of disk, that constitutes the inferior extremity of the pendulum, thin to the edge, thicker to the center for avoid the resistance of the air. Usually it is fully grown you from two caps of brass join between them and full of lead.

PIVOT: conic or cylindrical element that wheel in a specific center. Generally it is the terminal part of a centre arbor.

PINION: organ to cylindrical form that gears with a wheel that has a diameter and a number of greater teeth. The teeth are also said wings.

POWER: in the mechanical clocks, it has given from a weight or from a rubber band; the weight descends for strength of gravity, it makes rotate a drum by means of a rope and it operate the train of the time or that of the ringing. The rubber band comes charged and it is used like power the effort that does in the stretch and return in the position of rest.

RECOIL: escapements in which while a tooth of the wheel of escapement gives the impulse to a shovel, the other shovel halts the fall of the same wheel and makes it to demote before it could give a new impulse.

ROCKER SHAFT: it is the centre arbor with the pallet staff that they come to use in the escapement to rod. It can being vertical, for support the balance wheel, or horizontal if it is joined with the pole of a short pendulum.

TEETH: peripheral elements of the wheels of gears of different form; triangular, rounded.

TRAIN:it is the succession of theet wheels and pinions that transmits the power. There are trains with different functions; the train of the time operates the escapement and the regulator, but there are trains for the ringing, for the carillon, for the astronomic movements.