Optimum Spindle Speed given Tool Changing Cost Solution

STEP 0: Pre-Calculation Summary
Formula Used
Rotational Frequency of Spindle = (Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece))*((((1+Taylor's Tool Life Exponent)*Cost of a Tool*Reference Tool Life*(1-Workpiece Radius Ratio))/((1-Taylor's Tool Life Exponent)*(Cost of changing each Tool+Cost of a Tool)*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))))^Taylor's Tool Life Exponent)
ns = (Vref/(2*pi*ro))*((((1+n)*Ct*Tref*(1-ar))/((1-n)*(CCT+Ct)*(1-(ar^((1+n)/n)))))^n)
This formula uses 1 Constants, 8 Variables
Constants Used
pi - Archimedes' constant Value Taken As 3.14159265358979323846264338327950288
Variables Used
Rotational Frequency of Spindle - (Measured in Hertz) - Rotational Frequency of Spindle is the number of turns made by the spindle of the Machine for cutting in one second.
Reference Cutting Velocity - (Measured in Meter per Second) - Reference Cutting Velocity is the Cutting Velocity of the tool used in the reference Machining Condition.
Outside Radius of the Workpiece - (Measured in Meter) - Outside Radius of the Workpiece is the radius of the outermost surface of the workpiece, away from the machining tool.
Taylor's Tool Life Exponent - Taylor's Tool Life Exponent is an experimental exponent that helps in quantifying the rate of Tool Wear.
Cost of a Tool - The Cost of a Tool is simply the cost of one tool being used for machining.
Reference Tool Life - (Measured in Second) - Reference Tool Life is the Tool Life of the tool obtained in the reference Machining Condition.
Workpiece Radius Ratio - Workpiece Radius Ratio is the ratio of the inner radius of the workpiece to its outer radius.
Cost of changing each Tool - The cost of changing each Tool is the cost that arises due to the time taken by the operator to change one tool when he is paid by the hour.
STEP 1: Convert Input(s) to Base Unit
Reference Cutting Velocity: 5000 Millimeter per Minute --> 0.0833333333333333 Meter per Second (Check conversion here)
Outside Radius of the Workpiece: 1000 Millimeter --> 1 Meter (Check conversion here)
Taylor's Tool Life Exponent: 0.5 --> No Conversion Required
Cost of a Tool: 70 --> No Conversion Required
Reference Tool Life: 5 Minute --> 300 Second (Check conversion here)
Workpiece Radius Ratio: 0.45 --> No Conversion Required
Cost of changing each Tool: 25 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
ns = (Vref/(2*pi*ro))*((((1+n)*Ct*Tref*(1-ar))/((1-n)*(CCT+Ct)*(1-(ar^((1+n)/n)))))^n) --> (0.0833333333333333/(2*pi*1))*((((1+0.5)*70*300*(1-0.45))/((1-0.5)*(25+70)*(1-(0.45^((1+0.5)/0.5)))))^0.5)
Evaluating ... ...
ns = 0.265690865532833
STEP 3: Convert Result to Output's Unit
0.265690865532833 Hertz --> No Conversion Required
FINAL ANSWER
0.265690865532833 0.265691 Hertz <-- Rotational Frequency of Spindle
(Calculation completed in 00.004 seconds)

Credits

Created by Kumar Siddhant
Indian Institute of Information Technology, Design and Manufacturing (IIITDM), Jabalpur
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National Institute of Technology (NIT), Srinagar
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19 Facing Operation Calculators

Optimum Spindle Speed
Go Rotational Frequency of Spindle = (Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece))*((((1+Taylor's Tool Life Exponent)*Cost of a Tool*Reference Tool Life*(1-Workpiece Radius Ratio))/((1-Taylor's Tool Life Exponent)*(Cost of a Tool*Time to Change One Tool+Cost of a Tool)*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))))^Taylor's Tool Life Exponent)
Machining and Operating Rate given Optimum Spindle Speed
Go Machining and Operating Rate = (Cost of a Tool/(((((((Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece)))/Rotational Frequency of Spindle)^(1/Taylor's Tool Life Exponent))*((((1+Taylor's Tool Life Exponent)/(1-Taylor's Tool Life Exponent)))*((1-Workpiece Radius Ratio)/(1-((Workpiece Radius Ratio)^((Taylor's Tool Life Exponent+1)/Taylor's Tool Life Exponent))))*Reference Tool Life))))-Time to Change One Tool)
Cost of 1 Tool given Optimum Spindle Speed
Go Cost of a Tool = (Machining and Operating Rate*(((((((Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece)))/Rotational Frequency of Spindle)^(1/Taylor's Tool Life Exponent))*((((1+Taylor's Tool Life Exponent)/(1-Taylor's Tool Life Exponent)))*((1-Workpiece Radius Ratio)/(1-((Workpiece Radius Ratio)^((Taylor's Tool Life Exponent+1)/Taylor's Tool Life Exponent))))*Maximum Tool Life))))-Time to Change One Tool)
Optimum Spindle Speed given Tool Changing Cost
Go Rotational Frequency of Spindle = (Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece))*((((1+Taylor's Tool Life Exponent)*Cost of a Tool*Reference Tool Life*(1-Workpiece Radius Ratio))/((1-Taylor's Tool Life Exponent)*(Cost of changing each Tool+Cost of a Tool)*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))))^Taylor's Tool Life Exponent)
Tool Changing Time given Optimum Spindle Speed
Go Time to Change One Tool = Reference Tool Life/((Rotational Frequency of Spindle*2*pi*Outer Radius of Workpiece/Reference Cutting Velocity)^(1/Taylor's Tool Life Exponent)*(1-Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent))*(1-Taylor's Tool Life Exponent)/((1+Taylor's Tool Life Exponent)*(1-Workpiece Radius Ratio)))-Cost of a Tool/Machining and Operating Rate
Tool Changing Cost given Optimum Spindle Speed
Go Cost of changing each Tool = (Cost of a Tool*Maximum Tool Life/(((Rotational Frequency of Spindle*2*pi*Outside Radius of the Workpiece/Reference Cutting Velocity)^(1/Taylor's Tool Life Exponent))*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))*(1-Taylor's Tool Life Exponent)/((1+Taylor's Tool Life Exponent)*(1-Workpiece Radius Ratio))))-Cost of a Tool
Machining Time given Rate of Increase of Wear-Land Width
Go Machining Time = Tool Life/(Rate of Increase of Wear Land Width*Reference Tool Life*((Reference Cutting Velocity/Cutting Velocity)^(1/Taylor's Tool Life Exponent))/Increase in Wear Land Width per Component)
Taylor's Exponent given Cutting Speed for Constant-Cutting-Speed Operation
Go Taylor's Tool Life Exponent = ln(Cutting Velocity/Reference Cutting Velocity)/ln(Maximum Tool Life/(Tool Life*Time Proportion of Cutting Edge Engagement))
Time for Facing given Instantaneous Cutting Speed
Go Process Time = (Outside Radius of the Workpiece-(Cutting Velocity/(2*pi*Rotational Frequency of Spindle)))/(Rotational Frequency of Spindle*Feed)
Feed given Instantaneous Cutting Speed
Go Feed = (Outside Radius of the Workpiece-(Cutting Velocity/(2*pi*Rotational Frequency of Spindle)))/(Rotational Frequency of Spindle*Process Time)
Time Proportion of Edge Engagement given Cutting Speed for Constant-Cutting-Speed Operation
Go Time Proportion of Cutting Edge Engagement = Reference Tool Life*((Reference Cutting Velocity/Cutting Velocity)^(1/Taylor's Tool Life Exponent))/Tool Life
Feed of Workpiece given Machining Time for Facing
Go Feed = (Outside Radius of the Workpiece-Inner Radius of Workpiece)/(Rotational Frequency of Spindle*Machining Time)
Total Machining Time for single Facing Operation
Go Machining Time = (Outside Radius of the Workpiece-Inner Radius of Workpiece)/(Rotational Frequency of Spindle*Feed)
Feed given Instantaneous Radius for Cut
Go Feed = (Outside Radius of the Workpiece-Instantaneous Radius for Cut)/(Rotational Frequency of Spindle*Process Time)
Time for Facing
Go Process Time = (Outside Radius of the Workpiece-Instantaneous Radius for Cut)/(Rotational Frequency of Spindle*Feed)
Inner Radius of Workpiece given Machining Time for Facing
Go Inner Radius of Workpiece = Outside Radius of the Workpiece-Rotational Frequency of Spindle*Feed*Machining Time
Machining Time given Maximum Wear-Land Width
Go Machining Time = Increase in Wear Land Width per Component*Tool Life/Maximum Wear Land Width
Inside Radius given Workpiece Radius Ratio
Go Inner Radius of Workpiece = Workpiece Radius Ratio*Outside Radius of the Workpiece
Workpiece Radius Ratio
Go Workpiece Radius Ratio = Inner Radius of Workpiece/Outside Radius of the Workpiece

Optimum Spindle Speed given Tool Changing Cost Formula

Rotational Frequency of Spindle = (Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece))*((((1+Taylor's Tool Life Exponent)*Cost of a Tool*Reference Tool Life*(1-Workpiece Radius Ratio))/((1-Taylor's Tool Life Exponent)*(Cost of changing each Tool+Cost of a Tool)*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))))^Taylor's Tool Life Exponent)
ns = (Vref/(2*pi*ro))*((((1+n)*Ct*Tref*(1-ar))/((1-n)*(CCT+Ct)*(1-(ar^((1+n)/n)))))^n)

Production Cost for Facing

The Total Production Cost is defined as the cumulative of all the costs of different processes in Facing Operation. This includes the cost of loading/unloading the tool, the cost of setup of the workpiece, the machining cost, and the cost of tools used.

How to Calculate Optimum Spindle Speed given Tool Changing Cost?

Optimum Spindle Speed given Tool Changing Cost calculator uses Rotational Frequency of Spindle = (Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece))*((((1+Taylor's Tool Life Exponent)*Cost of a Tool*Reference Tool Life*(1-Workpiece Radius Ratio))/((1-Taylor's Tool Life Exponent)*(Cost of changing each Tool+Cost of a Tool)*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))))^Taylor's Tool Life Exponent) to calculate the Rotational Frequency of Spindle, The Optimum Spindle Speed given Tool Changing Cost is used to determine the rotational frequency of the spindle of the working Tool that results in the minimum Production Cost of Facing when Tool Changing Cost is known. Rotational Frequency of Spindle is denoted by ns symbol.

How to calculate Optimum Spindle Speed given Tool Changing Cost using this online calculator? To use this online calculator for Optimum Spindle Speed given Tool Changing Cost, enter Reference Cutting Velocity (Vref), Outside Radius of the Workpiece (ro), Taylor's Tool Life Exponent (n), Cost of a Tool (Ct), Reference Tool Life (Tref), Workpiece Radius Ratio (ar) & Cost of changing each Tool (CCT) and hit the calculate button. Here is how the Optimum Spindle Speed given Tool Changing Cost calculation can be explained with given input values -> 0.265691 = (0.0833333333333333/(2*pi*1))*((((1+0.5)*70*300*(1-0.45))/((1-0.5)*(25+70)*(1-(0.45^((1+0.5)/0.5)))))^0.5) .

FAQ

What is Optimum Spindle Speed given Tool Changing Cost?
The Optimum Spindle Speed given Tool Changing Cost is used to determine the rotational frequency of the spindle of the working Tool that results in the minimum Production Cost of Facing when Tool Changing Cost is known and is represented as ns = (Vref/(2*pi*ro))*((((1+n)*Ct*Tref*(1-ar))/((1-n)*(CCT+Ct)*(1-(ar^((1+n)/n)))))^n) or Rotational Frequency of Spindle = (Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece))*((((1+Taylor's Tool Life Exponent)*Cost of a Tool*Reference Tool Life*(1-Workpiece Radius Ratio))/((1-Taylor's Tool Life Exponent)*(Cost of changing each Tool+Cost of a Tool)*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))))^Taylor's Tool Life Exponent). Reference Cutting Velocity is the Cutting Velocity of the tool used in the reference Machining Condition, Outside Radius of the Workpiece is the radius of the outermost surface of the workpiece, away from the machining tool, Taylor's Tool Life Exponent is an experimental exponent that helps in quantifying the rate of Tool Wear, The Cost of a Tool is simply the cost of one tool being used for machining, Reference Tool Life is the Tool Life of the tool obtained in the reference Machining Condition, Workpiece Radius Ratio is the ratio of the inner radius of the workpiece to its outer radius & The cost of changing each Tool is the cost that arises due to the time taken by the operator to change one tool when he is paid by the hour.
How to calculate Optimum Spindle Speed given Tool Changing Cost?
The Optimum Spindle Speed given Tool Changing Cost is used to determine the rotational frequency of the spindle of the working Tool that results in the minimum Production Cost of Facing when Tool Changing Cost is known is calculated using Rotational Frequency of Spindle = (Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece))*((((1+Taylor's Tool Life Exponent)*Cost of a Tool*Reference Tool Life*(1-Workpiece Radius Ratio))/((1-Taylor's Tool Life Exponent)*(Cost of changing each Tool+Cost of a Tool)*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))))^Taylor's Tool Life Exponent). To calculate Optimum Spindle Speed given Tool Changing Cost, you need Reference Cutting Velocity (Vref), Outside Radius of the Workpiece (ro), Taylor's Tool Life Exponent (n), Cost of a Tool (Ct), Reference Tool Life (Tref), Workpiece Radius Ratio (ar) & Cost of changing each Tool (CCT). With our tool, you need to enter the respective value for Reference Cutting Velocity, Outside Radius of the Workpiece, Taylor's Tool Life Exponent, Cost of a Tool, Reference Tool Life, Workpiece Radius Ratio & Cost of changing each Tool and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Rotational Frequency of Spindle?
In this formula, Rotational Frequency of Spindle uses Reference Cutting Velocity, Outside Radius of the Workpiece, Taylor's Tool Life Exponent, Cost of a Tool, Reference Tool Life, Workpiece Radius Ratio & Cost of changing each Tool. We can use 1 other way(s) to calculate the same, which is/are as follows -
  • Rotational Frequency of Spindle = (Reference Cutting Velocity/(2*pi*Outside Radius of the Workpiece))*((((1+Taylor's Tool Life Exponent)*Cost of a Tool*Reference Tool Life*(1-Workpiece Radius Ratio))/((1-Taylor's Tool Life Exponent)*(Cost of a Tool*Time to Change One Tool+Cost of a Tool)*(1-(Workpiece Radius Ratio^((1+Taylor's Tool Life Exponent)/Taylor's Tool Life Exponent)))))^Taylor's Tool Life Exponent)
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