Wk 10 - A & B


  • Applications

    High velocity ratios with moderate size gears in a comparatively lesser space
    For example:
    – Back gears of lathes
    – Differential gears of automobiles
    – Hoists
    – Pulley blocks
    – Wrist watches
    – Etc.


    Epicyclic Gears

    Introduction
    – Applications
    Velocity Ratio
    Compound Gear Trains
    Solved Problems

    Introduction

    Arm C Fixed
    – Gear A or Gear B is driver of other gear
    Gear A Fixed
    – Gear B rotates upon (epi) and around (cyclic) Gear A

  • Compound Gear Train

    📷   Compound Gear Train cross section - enlarge above image

    Co-axial shafts S1 and S2
    Annulus Gear A
    – Internal Teeth
    Compound Gear (Planet) B-C
    – Carried by Arm and revolves freely on pin
    Sun Gear D
    – Co-axial with A and H but independent
    Arm H
    – When A is fixed, D provides the drive
    – When D is fixed, A provides the drive
    – H acts as a follower for both cases

    Continues on next tab.


    Velocity Ratio

    zA = number of teeth on Gear A
    zB = number of teeth on Gear B
    Assume Arm C, in the diagram above, is fixed
    – Therefore axes of arms fixed relative to each other
    Assume Arm C is fixed
    – Therefore axes of arms fixed relative to each other, and:$$ \large \frac{N_B}{N_A} \; = \; \frac{z_A}{z_B} \quad Eqn.(5.1)$$
    – When A makes one revolution anticlockwise, B makes zA/zB revolutions clockwise
    Assume anticlockwise is positive rotation and clockwise is negative rotation
    For Gear A making +1 revolutions, Gear B makes: $ \bigg( - \Large \frac{z_A}{z_B} \bigg) $ revolutions.
    📷   Velocity Ratio Table
  • The equations above can be used to generate the magnitude of the revolution value, but the directions must be taken from the following rules:
    – Gears that mesh external teeth to external teeth rotate in opposite directions.
    – Gears that mesh external teeth to internal teeth rotate in the same direction.

    Velocity Ratio Table - Run Through Video

    Worked example 1 - Epicyclic Gears Video
    Download worked example



    Compound Gear Train (cont.)


    📷   Compound Gear Train cross section - enlarge above image

    zA, zB, zC and zD number of teeth for each gear
    NA, NB, NC and ND speed of each gear
    When H is fixed, D is turned anti-clockwise, B-C and A rotate in clockwise direction
    From Equation (5.1) for ND = 1: $$ \large N_c = \frac{z_D}{z_C} \quad Eqn.(5.2) $$
    As B-C is a compound gear, NB = NC, and the next speed ratio to consider is NA/NB. Therefore; $$ \large \frac{N_A}{N_B} \; = \; \frac{z_B}{z_A} \quad Eqn.(5.3) $$
    Re-arranging for NA:
    $$ \large N_A = N_B \times \frac{z_B}{z_A} \; \quad Eqn.(5.4) $$
    Substituting the value for NB from Equation (5.2) (equal to NC):
    $$ \large N_A = \frac{z_D}{z_C} \times \frac{z_B}{z_A} \quad Eqn.(5.5) $$
  • Slope of thread($ \theta $)
    - It is the inclination of the thread with the horizontal

        Mathematically,
    $$ \text{tan} \; \theta = \frac{Lead of screw}{Circumference of screw} = \frac{l}{ \pi D _{ms}} $$
    $ l $ =lead of the screw = $ np $

    $ p $ = pitch of the screw

    $ D _{ms} $ = Mean diameter of the screw, and

    $ n $ = number of start threads

    Derivation of the Angle of a Thread Equation Video


    Design of Power Screws

    Introduction
    - Nomenclature
    Screw Jacks
    - Torque Required
    - Efficiency
    Overhauling and self-locking screws

    Introduction

    Screws
    - Made by cutting continuous helical groove on a cylindrical surface
    - External Threads
    Cut on outer surface of a solid rod
    - Internal Threads
    Cut on the inner surface of a hollow rod
    - V-Threads
    Stronger and higher friction resistance than square threads
    Prevent nut from slackening
    Used for bolts and nuts
    - Square Threads
    Used for screw jacks, vice screws, etc.

    Nomenclature

    Helix
    - Curve traced by particle moving along screw thread
    Pitch
    - Distance between screw threads
    Lead
    - Distance screw thread advances axially in one turn
    Depth of Thread
    - Distance between top and bottom surfaces of thread
    Single-Threaded Screw
    - Lead of the screw is equal to the pitch
    Multi-Threaded Screw
    - More than one thread is cut in one lead distance of a screw, e.g. double threaded screw has two threads cut in one length
    Lead = Pitch $ \times $ Number of Threads
  • (Cont.)
    Or with no collar bearing: $$ T = W \; tan \; ( \phi + \theta ) \frac{D _{ms}}{2} \quad \small \textbf{Eqn(6.3)} $$
    And to calculate the force P at the end of a lever length L to apply this torque: $$ T = W \; tan \; ( \phi + \theta ) \frac{D _{ms}}{2} + W \mu _c \frac{D _{mc}}{2} = P \times L $$

    Screw Diameters

    Nominal Screw Diameter (DO)
    - Also known as outside diameter or major diameter
    Core Diameter (DC)
    - Also known as outside diameter or major diameter
    Mean Diameter (DM)
    - If the nominal and/or the core diameter are known:
    $$ D _{m} = \frac{D_o + D_c}{2} = D _o - \frac{p}{2} = D _c + \frac{p}{2} $$
    Working Out the Mean Diameter of the Screw Video



    Screw Jack

    Screw jack with thrust collar
    Used to lift heavy loads by applying comparatively small effort at the handle
    Works on principle similar to inclined plane
    Load to be raised or lowered is placed on head of square threaded rod
    Square threaded rod rotated by application of force at end of the lever

    Required Torque

    Torque required to lift the weight
    Where:
    $ p $ = pitch of the screw
    $ D _{ms} $ = Mean diameter of the screw
    $ D _{mc} $= Mean diameter of the collar
    $ \theta $ = helix angle
    $ W $ = load to be lifted
    $ \mu _{s}$ = coefficient of friction of screw thread, i.e., between the screw and nut = tan $ \phi $ , where $ \phi $ is the friction angle
    $ \mu _{c}$ = coefficient of friction of collar bearing

  • Therefore
    - A screw will always be self locking if:
    the friction angle is greater than the helix angle, or
    the coefficient of friction is greater than the tangent of the helix angle

    Comparison of Thread Angle and Angle of Friction Video

    Worked Example 2 Introduction Video

    Power Screws Nomenclature
    Power Screws Worked Examples


    Efficiency of a Screw Jack

    Efficiency defined as:
    - Ratio between the ideal effort (neglecting friction) to the actual effort (taking friction into account
    Remember:
    $$ T = W \; tan \; ( \phi + \theta ) \frac{D _{ms}}{2} + W \mu _c \frac{D _{mc}}{2} \quad \small \textbf{Eqn(6.6)} $$
    For no friction, $ \mu _s $ = 0, therefore:
    $$ T _0 = W \; \text{tan} \; \theta \frac{D _{ms}}{2} $$
    Therefore, efficiency equation defined as: $$ \mu = \frac{T_0}{T} = \frac{W \; \text{tan} \; \theta \; \frac{\large D _{ms}}{2}}{W \; tan \; ( \phi + \theta ) \frac{\large D _{ms}}{2} + W \mu _c \frac{\large D _{mc}}{2}} $$
    or for no collar bearing: $$ \eta = \frac{tan \; \theta}{tan \; (\theta + \phi)} $$

    Overhauling and Self-Locking Screws

    Torque required to lower the load: $$ T = W \; tan \; ( \phi - \theta ) \frac{\large D _{ms}}{2} + W \mu _c \frac{\large D _{mc}}{2} $$
    If $ \phi $ < $ \theta $
    - Torque required to lower the load may be negative
    Load moves down without application of torque
    Known as overhauling
    If $ \phi $ > $ \theta $
    - Torque required to lower the load will always be positive
    Effort is required to lower the load
    Known as self-locking

  • Tutorials

    Epicylic Gears
    Power Screws