# Part VII: Gear Systems: Analysis

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1 Part VII: Gear Systems: Analysis This section will review standard gear systems and will provide the basic tools to perform analysis on these systems. The areas covered in this section are: 1) Gears 101: The geometric details about standard gears (involute) 2) Salient features of involute gears 3) Gear-tooth geometry equations 4) Gear train systems: fixed-axis and Planetary 5) Types of Gears Gears 101: Fundamental Law of Gearing ME 3610 Course Notes - Outline Part VII -1

2 Gears 102: Details about the involute gear profile: Gears were created to transmit constant-velocity rotating motion between shafts relying on kinematic contact (not friction) to transmit forces. Recall that in order to have constant velocity, the line of action and line of centers must intersect at a constant location: Different types (that do not work) ME 3610 Course Notes - Outline Part VII -2

3 Animation (of what is required): The point of contact (and direction) defines the tooth. Any number of arbitrary, conjugate shapes could be defined to complete this task, however two profiles of significance work: involute and cycloidal profiles. The involute profile is the standard for gear teeth, o the involute is conjugate to itself o The involute is easy to manufacture and o does not depend on distance between gear centers. ME 3610 Course Notes - Outline Part VII -3

4 Based on this involute geometry of gear teeth, the geometry of a gear can be standardized and named, as in the following figures. The nature of tooth contact is described as well on these figures. ME 3610 Course Notes - Outline Part VII -4

5 ME 3610 Course Notes - Outline Part VII -5

6 The velocity ratio between gears is given as: Conditions of Interchangeability (For Standard Gears) ME 3610 Course Notes - Outline Part VII -6

7 Salient Features of Involute Gears: 1. Velocity ratio of gears in mesh is constant and independent of center distance (just like belts and pulleys). 2. Line of Action: tangent to the base circles, all tooth contact takes place on this line. 3. Interference points: points where line of action meets the base circles. 4. If contact occurs outside the interference points, interference will occur between the gears. 5. A gear that can mesh with a rack with out undercutting can mesh with any other gear of equal or greater size without undercutting. 6. Standard gears with the same diametral pitch and same pressure angle are interchangeable. 7. Contact ratio, mp, is the average number of teeth in contact (in design, 1.2 is a minimum, the larger the better). 8. A rack is a gear with infinite diameter. On a rack, the involute becomes a straight line. 9. Common gear cutters: Hobs, and Shapers. Hobs act like a bit in a milling machine. Shapers have a translating cutting stroke. 10. The type of cutter determines the minimum number of teeth possible with out undercutting. The hob represents the worst case, since it is like a rack. For a gear cut with a shaper, the minimum number of teeth may be given by the shaper, if N of the shaper is known. 11. Salient: (sa' li-ent) adj. Standing out from the rest; noticeable; conspicuous; prominent. (Webster's, College Ed.) ME 3610 Course Notes - Outline Part VII -7

8 Details of Involute Gears 1. Equations not found in Norton: (Refer to Fig. 1) Base pitch (distance between one tooth set measured along base circle): Length of action: (1) (2) Contact ratio (average number of teeth in contact): (3) Diametral Pitch (number of teeth per inch): (4) Module (mm per tooth): Minimum number of teeth to avoid interference: (k=1 for full depth teeth) a) for a rack: (5) (6) ME 3610 Course Notes - Outline Part VII -8

9 b) for two gears in mesh: (7) Center distance: (8) "operating" center distance and pressure angle: Backlash resulting from an increased operating center distance: (9) Tooth thickness: (requires the tooth thickness at some radius to be known, generally at the pitch circle): (10) Radius and angle at various points along the involute: (11) (12) ME 3610 Course Notes - Outline Part VII -9

10 Gear-Train Systems: Gears are used in combinations to create a desired torque/velocity ratio. Combinations of gears can be divided into two classes: Fixed-axis gear trains, and planetary gear trains. Fixed-axis gear trains: The sign change occurs for external gears. The diameters listed are the pitch diameters. Now consider a series of gears in mesh: In this arrangement, the intermediate gears do not affect the overall velocity ratio, and therefore should be replaced with a more cost effective means of power transmission. Only the outer two gears are useful in achieving the desired velocity ratio. Since the velocity ratio of a single gear set is practically limited to 10:1 or less (actually, more like 5:1 greater), compound gears (two gears constrained to have the same angular velocity) are used in gears trains to achieve larger velocity ratios: ME 3610 Course Notes - Outline Part VII -10

11 Compound Gears: Gears from a clock, reverted train. Select gears assuming 24 pd and find nominal center distance. If the center distance tolerance is +/-.006 in., what is the maximum backlash (9.33 Norton) ME 3610 Course Notes - Outline Part VII -11

12 Given 3-speed shiftable transmission (this is not how transmissions work btw). Select appropriate gears to give output speeds of splined shaft of 150, 350 and 550 rpm for input speed of 450 rpm. Select pd and tooth numbers. (9-34 Norton). What is efficiency assuming Eo = 0.95? ME 3610 Course Notes - Outline Part VII -12

13 Planetary Gear Trains: A Planetary gear train (see Fig. below) results when certain gears in the train (called the planet gears) have moving axes. The arm, while not a gear, is an essential part of the planetary because it defines the motion of the moving planet gear axes. The planetary is also unique to a standard gear train in that it requires two inputs to define one output (verify this using mobility). A good example is your car's differential, which has two inputs: one the drive-shaft, and the second a constraint between the two driven wheels provided by whatever you are driving on (e.g. dry pavement, one wheel on ice, etc.) The planetary gear train consists of three parts: ME 3610 Course Notes - Outline Part VII -13

14 Planetary Gear Equation: The planetary gear train equation must be used to solve the angular velocities of elements in the planetary. The equation is: where: f, and l identify two gears in the planetary (call them first and last), a represents the arm, ω l, ω f, ω a,. ω la, ω fa, ω la / ω fa, Example: Planetary Gear Trains Given this gear train, find the speed and direction of the drum. What is train efficiency for Eo = 0.97 (Norton 9-35). ME 3610 Course Notes - Outline Part VII -14

15 Example: Planetary Gear Train Given the planetary gear train above with inputs, what is the velocity of ring gear D? Given the planetary gear train above with inputs: Arm CCW at 50 rpm and gear A fixed to ground, find the speed of gear D. (Norton 9-37), ME 3610 Course Notes - Outline Part VII -15

16 A-36 E - 18 Given the differential above, assume Gear E is driven at 500 RPM, Gear B is rigidly attached to gear A, and the right output shaft is held fixed. What is the speed of the left output shaft (gear D?) (Norton 9-38) ME 3610 Course Notes - Outline Part VII -16

17 Choosing f, l, and a: Choosing elements for the first, last, and arm is the first step in solving a planetary. Solving will fall into one of the three following scenarios (remember that you must know two pieces of information to solve the planetary equation). case i You want to find the arm velocity, (wa is not known) knowing the velocity of two gears: Choose f and l as the two known gears, and the arm as a, an unknown. Solve for wa. case ii You want to find the velocity of a gear, and you know the velocity of the arm and one other gear: Choose l as the desired unknown gear, choose f as the known gear and a as the known arm. Solve for wl. case iii You want to find the velocity of a gear, and you know the velocity of two gears but not the arm. First, choose f and l as the known gears and solve the arm velocity, wa. Then go to case ii. Mixed Gear trains: A general gear train can include both fixed axis and planetary gear trains, or multiple planetaries. Solving systems like these requires using the procedures outlined above and looking for elements that share the same angular velocity between the mixed gear trains. ME 3610 Course Notes - Outline Part VII -17

18 Gear Types: A gear train consists of one or more gear sets intended to give a specific velocity ratio, or change direction of motion. Gear and gear train types can be grouped based on their application and tooth geometry. Table I: Gear Types Grouped According to Shaft Arrangement Parallel Axes Intersecting Axes Non-Intersecting (Non-parallel) Axes Rotary to Translation Spur gears (Fig. 1): Spur gears connect parallel shafts, have involute teeth that are parallel to the shafts, and can have either internal or external teeth. Notes: ME 3610 Course Notes - Outline Part VII -18

19 Helical gears (Fig. 2): Helical gears also connect parallel shafts, but the involute teeth are cut at an angle (called the helix angle) to the axis of rotation. Note that two mating helical gears must have equal helix angle but opposite hand. These are found in automotive transmissions, and any application requiring high speed rotation and good performance. Notes: Herringbone gears (Fig. 3): To avoid axial thrust, two helical gears of opposite hand can be mounted side by side, to cancel resulting thrust forces. These are called double helical or herringbone gears ME 3610 Course Notes - Outline Part VII -19

20 Bevel gears (Fig. 4): Bevel gears connect intersecting axes, and come in several types (listed below). For bevel gears, the pitch surface is a cone, (it was a cylinder in spur and helical gears) and mating spiral gears can be modeled as two cones in rolling contact. Types of bevel gears: 1. Straight bevel: These are like spur gears, the teeth have no helix angle. Straight bevel gears can be a. Miter gears, equal size gears with a 90 degree shaft angle, b. Angular bevel gears, shaft angle other than 90 degrees, or c. Crown gears, one gear is flat, has a pitch angle of 90 degree. 2. Spiral bevel gears(fig. 4a): Teeth have a spiral angle which gives performance improvements much like helical gears 3. Zerol bevel gears (Fig. 4b): Teeth are crowned, so that tooth contact takes place first at the tooth center. ME 3610 Course Notes - Outline Part VII -20

21 Hypoid gears (Fig. 5): Similar to spiral bevel gears, but connect non-parallel shafts that do not intersect. The pitch surface of a hypoid gear is a hyperboloid of revolution (rather than a cone, the pitch surface in bevel gears), hence the name. Crossed helical gears (Fig. 6): Helical gears that connect skew shafts. The teeth have sliding motion and therefore lower efficiency. One application is connecting distributer to cam shaft in pre-electronic ignition vehicles. Worm Gears (Fig. 7): The driving gear is called a worm, and typically has 1, 2, or four teeth. The low number of teeth on the worm can result in a very large velocity ratio. These can also be designed to be non-backdriveable, and can carry high loads. Because of sliding action, efficiency is low. ME 3610 Course Notes - Outline Part VII -21

22 Rack and Pinion (Fig. 8): These transmit rotary motion (from the pinion) to translational motion (of the rack). The rack is a gear with infinite radius; its teeth, although flat sided, are involute. The rack and pinion is commonly used in steering units and jacks. ME 3610 Course Notes - Outline Part VII -22

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