Arithmetic & Geometric Progression: Example 1

Question:

A lathe has to have five spindle speeds. It is to cater for work ranging from 25mm to 200mm in diameter. Allowing for a cutting speed of 25m/min, find the spindle speeds using:

a) Arithmetic Progression

b) Geometric Progression

Answer:

a) Firstly utilise the formula for Feed Rate:

Calculate spindle speed for the 1st and 5th term in the series as we have the diameters for these terms:

1st term: Diameter = 200mm,

5th term: Diameter = 25mm,

The formula for Arithmetic Progression is as follows. Use Arithmetic formula and re-arrange to find the common difference (d):

Where,

a = First term

d = Common difference

n = nth term

Now use formula to find remaining spindle speeds:

Therefore according to Arithmetic Progression the 1st to 5th spindle speeds are

[40, 110, 179, 249, 318 rev/min].

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b) We know from answer (a) that the 1st and 5th spindle speeds are:

The formula for Geometric Progression is as follows.

Where,

a = First term

r = Common ratio

n = nth term

Use Geometric formula and re-arrange the 5th term to find the common ratio (r):

Now use Geometric formula to find the remaining spindle speeds:

Therefore according to Geometric Progression the 1st to 5th spindle speeds are

[40, 67, 113, 189, 318 rev/min].