Calculating Lathe Machine Spindle Step Ratio

The question asks for the step ratio of the headstock spindle for a lathe machine given its maximum speed, minimum speed, and the total number of speeds available. The spindle speeds in a lathe machine are typically arranged in a geometric progression. The common ratio of this geometric progression is called the step ratio.

We are given the following data:

• Maximum spindle speed, \(N_{\text{max}}\) = 2100 rev/min
• Minimum spindle speed, \(N_{\text{min}}\) = 70 rev/min
• Number of available speeds, \(Z\) = 21

Let the step ratio be denoted by \(\phi\). In a geometric progression of speeds, if the minimum speed is \(N_{\text{min}}\) and there are \(Z\) speeds, the speeds are \(N_{\text{min}}\), \(N_{\text{min}}\phi\), \(N_{\text{min}}\phi^{2}\), ..., \(N_{\text{min}}\phi^{Z-1}\).

The maximum speed will be the last term in this series, which is \(N_{\text{max}} = N_{\text{min}} \times \phi^{Z-1}\).

We can rearrange this formula to solve for the step ratio, \(\phi\):

\[\phi^{Z-1} = \frac{N_{\text{max}}}{N_{\text{min}}}\]

\[\phi = \left(\frac{N_{\text{max}}}{N_{\text{min}}}\right)^{\frac{1}{Z-1}}\]

Now, substitute the given values into the formula:

• \(N_{\text{max}}\) = 2100
• \(N_{\text{min}}\) = 70
• \(Z\) = 21

\[\phi = \left(\frac{2100}{70}\right)^{\frac{1}{21-1}}\]

\[\phi = \left(30\right)^{\frac{1}{20}}\]

Thus, the step ratio of the headstock spindle for the given lathe machine data is \(\left(30\right)^{1/20}\).

Let's look at the provided options:

• Option 1: \(\left(30\right)^{1/9}\)
• Option 2: \(\left(30\right)^{1/20}\)
• Option 3: \(\left(30\right)^{1/21}\)
• Option 4: \(\left(30\right)^{1/22}\)

Comparing our calculated step ratio with the options, we find that Option 2 matches our result.

The step ratio is a crucial parameter in designing the gearbox of a lathe machine's headstock, as it determines the progression of spindle speeds available for different machining operations.