MESA PROJECT Lesson of Mechanics and Machines done in the 5th A-M, 2012-2013 by the teacher Pietro Calicchio. THE GEARS To transmit high power are usually used gear wheels. In this case, the transmission of motion is not caused by the friction between two surfaces in contact, as occurs in the friction wheels, but by the force that is transmitted between the teeth, appropriately shaped, which characterize the gear wheels. The profile of the teeth must be designed in such a way to ensure the continuity of motion and the constancy of the transmission ratio. The profile that best meets the needs described above, is the involute of a circle. CYLINDRICAL STRAIGHT TEETH GEARS Looking at the illustration below, a gear (a transmission formed by a pair of gear wheels) can be considered as a connection between two friction wheels whose outer surfaces have been replaced by a series of protrusions and recesses of appropriate form. pitch circle pitch circle (driven wheel) (drive wheel) Commission. 1
Each cylindrical gear (formed by a pair of cylindrical toothed wheels) forms a kinematic pair whose motion is defined by the mutual rolling, without slipping, of two cylinders, tangent to each other, called primitive cylinders. They represent the contour of the cylindrical friction wheels having the same transmission ratio of the gear and, as these, during operation, they roll on each other without slipping. Intersecting the cylinders primitive with a plane perpendicular to their axes are obtained two circumferences called primitive circles whose diameters, pitch diameters, are indicated in the picture with d 1 for the drive wheel and d 2 for the driven wheel. The C point of tangency between the two pitch circles is the "instantaneous rotation center. Examining from the geometrical point of view a cylindrical toothed wheel with straight tooting, the teeth of which are developed in length in the direction of the axis of the wheel, we can see that the tooting is between two circumference: the circumference of foot (with radius r p ) internally and the circumference of head (with radius r t ) which limits it outside. The radial height (h) of the teeth will therefore be the difference between the radii (r t and r p ) of the above circumferences: h = r t - r p The part of the tooth between the circumference of the head and the primitive is called "tooth head", and its radial height (h a ), called the addendum, that is: h a = r t - r The portion of the tooth between the pitch circle and the circumference of foot is called foot of the tooth : its radial height h f (dedendum) is: h f = r - r p It is defined pitch (p) the length of the arc of the pitch circle between the axes of two successive teeth. The lateral surfaces of the teeth are called sides, their profiles are a part of an involute of circle. The profiles of the teeth will have to be "conjugated", i.e. such that the normal (that is the perpendicular straight line) common to the two surfaces in contact (straight line n), conducted at the point where they are touching (A), always passes through the center of instantaneous rotation (C), that is the point of tangency of the two pitch circles. Commission. 2
circumference of foot Pitch circle circumference of head circumference of head Pitch circle circumference of foot It is defined step (p) the length of the pitch circle divided by the number of the teeth Z ; It is defined module (m) the step divided by π: m = p/π = d/z It is defined straight line of action (n) the normal common at the profiles of the teeth in their point of contact, according to which direction they exchange forces during meshing. The straight line of action is also tangent to the two circumferences of the base; it is defined pressure angle (α) the angle between the straight line of action and the tangent common to the two pitch circles (usually it is assumed α = 20 ). Proportioning (method Reuleaux) Considering the tooth as a shelf stuck at one end and impressed at the free end by a concentrated load F, of length equal to the height of the tooth. Are known the power P to be transmitted and the numbers of revolutions per minute (n) of the drive wheel and the driven wheel. Determined the torque, the equation of stability in flexion is obtained: m = ³ (10.5 / λ) ³ (Mt / K ' z) The proportionality factor λ (λ = b / m) is normally 10 (can vary from 8 to 12 according to the peripheral speed). The module thus calculated, expressed in mm, is not generally an integer; must be rounded to the next higher value in the series of modules unified. Commission. 3
The dynamic coefficient of safety flexural K' varies with the speed of the device. For the safety load wheels are distinguished wheels of strength and wheels of work. Said wheels of strenght those used with peripheral speeds very low (less than 1 m / s). Said wheels of work those used with peripheral speeds higher. For the first assumes K equal to the load static safety; for the second must be taken of the dynamic stresses due to the peripheral speed and K' is determined by the formulas: - Gears slow and poorly accurate: K '= K [3 / (3 + v)]; - Gear fast and accurate: K '= K [6 / (6 + v)]; - Gears very fast and very precise: K '= K [5.6 / (5.6 + v)]; Made a tentative value for the peripheral speed and calculated the module is obtained the pitch radius: r = m z / 2 it recalculates the value of the peripheral speed v = 2 π r n / 60: if it is equal to or less than the value assumed the module defined it can be considered correct; if the calculated value is higher than that assumed, you must repeat the calculation of the module, whereas a peripheral speed higher. Check for wear Since the wear of the teeth greatly affects the peripheral speed, the check is carried out on the wheel of smaller diameter, with the greater angular velocity, regardless of whether it is conducted or conductive. The contact pressure P CON [dan/mm 2 ], for Θ = 20, is calculated with the formula: P CON = C [2 Mt (1 + e) / b d 2 ] where: C = numerical coefficient that depends on the nature of the materials; steel / steel C = 151 steel / cast iron C = 123 cast iron / cast iron C = 107 Mt = torque [danmm] b = face width [mm] d = pitch diameter of the lesser wheel [mm] e = ratio between the number of teeth of the wheel and the smaller number of teeth of the large wheel. The maximum allowable pressure is determined with the formula: p max = 2.5 HB / 6 (n h) where: HB = Brinell hardness of the material; n = number of revolutions per minute of the lower wheel; h = hours of operation provided; you can think of the following terms: - 130,000 150,000 hours for continuous operation; Commission. 4
- 10,000 to 30,000 hours for discontinuous operations; The Brinell hardness can be calculated from the table: - Steels HB 200 to 250 - Gray cast iron HB 180 to 200 - Special cast iron HB 180 240 If it is not satisfied with the condition: P CON p max it must reduce P CON (increasing the width of the wheel or the pitch diameter) or increase p max (increasing the hardness of the material). Performance of spur gears The power available on the driven shaft is less than that found on the crankshaft. Performance is expressed by the formula: η = 1 / {1 + f π [(1 / z 1 ) + (1 / z 2 )]} The value of (f) normally ranges from 0.15 to 0.25. The performance of the gears is high; can be improved with effective lubrication between the teeth in mesh. Commission. 5