Chapter 13 - equations

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Mechanical Engineering

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Oct 30, 2023

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Chapter 13 – Gears (General) Nomenclature Terms Term Definition Pitch Circle Circle that dictates all calculations for a gear (“active circle”), the pitch circles of two mated gears are tangent to one another. Pitch Diameter Diameter of a pitch circle. Pitch Point Point of intersection of pitch diameters Pinion Smaller of two mated gears, the other is called the gear Circular Pitch The distance, measured on the pitch circle, from a point on one tooth to a corresponding point on an adjacent tooth. Thus, the circular pitch is equal to the sum of the tooth thickness and the width of space. Module Ratio of pitch diameter to number of teeth (typical unit: mm/tooth) Diametral Pitch Ratio of number of teeth to the pitch (Reciprocal of the module). Only used in U.S units, so unit is teeth per inch. Addendum Radial distance between top land (tip of the tooth) and pitch circle Dedendum Radial distance from the bottom land to the pitch circle. Whole Depth Sum of Addendum and Dedendum Clearance Circle Circle tangent to addendum circle of mated gear Clearance Distance between the dedendum and clearance circle Backlash Amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circles. Pressure angle Angle between pressure line and a line perpendicular to the two centers of rotation. Pressure Line Also called generating line or line of action, the pressure line is an imaginary line drawn tangent to two mated gears base circles. Base Circle Where involute profile starts

Variables Variable Meaning Units/Notes P Diametral Pitch Teeth/In N Number of teeth Teeth d Pitch Diameter In or mm m Module Mm p Circular Pitch In or mm r bi Base Radius of Gear i Distance ϕ Pressure angle Degrees or radians V Pitch Line Velocity m/s q a Arc of approach q r Arc of Recess q t Arc of action When this equals p continuous motion is met, when it exceeds p then multiple teeth may be in contact at once m c Contact Ratio “average number of teeth in contact”, should be greater than 1.2 to reduce errors L ab Length of line of action Distance between the two points of intersection of the line of action and the gears. p b Base Pitch p c Circular pitch N p Smallest number of teeth Without interference k Constant for Np K=1 for full depth teeth and 0.8 for stub teeth m g ∨ m Gear ratio n Revolutions RPM e Train Value Positive if last gear rotates in same direction as first n L Speed of last gear n F Speed of first gear H Power Transmitted T Torque ω Angular velocity Rad/s W t Transmitted Load Defining equations: P = N d m = d N

p = πd N = πm pP = π m g = N G N P Fundamentals: V = | r 1 ω 1 | = | r 2 ω 2 | | ω 1 ω 2 | = r 2 r 1 r bi = r i cos ϕ p b = p c cos ϕ m c = q t p = L ab p cos ϕ Smallest number of teeth Gear ratio 1:1 N p = 2 k 3sin 2 ϕ ( 1 + √ 1 + 3sin 2 ϕ ) Other Gear ratios N p = 2 k ( 1 + 2 m ) sin 2 ϕ ( m + √ m 2 + ( 1 + 2 m ) sin 2 ϕ ) Largest number of teeth (specified pinion) interference free N ( ¿¿ P 2 sin 2 ϕ − 4 k 2 )/( 4 k − 2 N P sin 2 ϕ ) N G = ¿ Smallest spur pinion that will operate with a rack N P = 2 k sin 2 ϕ Gear Trains

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n 3 = | N 2 N 3 n 2 | = | d 2 d 3 n 2 | e = ± Product of drivingtoothnumbers Product of driventoothnumbers n L = en F Desired Gear ration Number of gear trains m < 10:1 None, one pair of gears will suffice 10:1 ≤m≤ 100:1 Two stage compound gear train 100:1 < m Multi-stage compound gear train Force Analysis H = Tω = ( W t d 2 ) ω V = πdn 12 W t = 33000 ∗ H V = 60000 H πdn