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

z_A = number of teeth on Gear A

z_B = 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 z_A/z_B 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

z_A, z_B, z_C and z_D number of teeth for each gear

N_A, N_B, N_C and N_D 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 N_D = 1: $$ \large N_c = \frac{z_D}{z_C} \quad Eqn.(5.2) $$

As B-C is a compound gear, N_B = N_C, and the next speed ratio to consider is N_A/N_B. Therefore; $$ \large \frac{N_A}{N_B} \; = \; \frac{z_B}{z_A} \quad Eqn.(5.3) $$

Re-arranging for N_A:

$$ \large N_A = N_B \times \frac{z_B}{z_A} \; \quad Eqn.(5.4) $$

Substituting the value for N_B from Equation (5.2) (equal to N_C):

$$ \large N_A = \frac{z_D}{z_C} \times \frac{z_B}{z_A} \quad Eqn.(5.5) $$

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

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 (D_O)

- Also known as outside diameter or major diameter

Core Diameter (D_C)

- Also known as outside diameter or major diameter

Mean Diameter (D_M)

- 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

Summary

⬇ Epicyclic Gear and Power Screw Summary

Tutorials

⬇ Epicylic Gears

⬇ Power Screws