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Length of Path of Contact Formula

GoMaximum Length of Path of Contact Formula

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Path of Contact is the path traced by the contact point of a pair of tooth profiles. Check FAQs

P = \sqrt{{R a}^{2} - {R w}^{2} \cdot {(\cos (Φ g))}^{2}} + \sqrt{{r a}^{2} - r^{2} \cdot {(\cos (Φ g))}^{2}} - (R w + r) \cdot \sin (Φ g)

P - Path of Contact?R_a - Radius of Addendum Circle of Wheel?R_w - Radius of Pitch Circle of Wheel?Φ_g - Pressure Angle of Gear?r_a - Radius of Addendum Circle of Pinion?r - Radius of Pitch Circle of Pinion?

Length of Path of Contact Example

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Here is how the Length of Path of Contact equation looks like with Values.

Here is how the Length of Path of Contact equation looks like with Units.

Here is how the Length of Path of Contact equation looks like.

16.8077 = \sqrt{{18.63}^{2} - {12.4}^{2} \cdot {(\cos (32))}^{2}} + \sqrt{{15.954}^{2} - {10.2}^{2} \cdot {(\cos (32))}^{2}} - (12.4 + 10.2) \cdot \sin (32)

16.8077mm = \sqrt{{18.63mm}^{2} - {12.4mm}^{2} \cdot {(\cos (32°))}^{2}} + \sqrt{{15.954mm}^{2} - {10.2mm}^{2} \cdot {(\cos (32°))}^{2}} - (12.4mm + 10.2mm) \cdot \sin (32°)

16.8077 = \sqrt{{18.63}^{2} - {12.4}^{2} \cdot {(\cos (32))}^{2}} + \sqrt{{15.954}^{2} - {10.2}^{2} \cdot {(\cos (32))}^{2}} - (12.4 + 10.2) \cdot \sin (32)

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Length of Path of Contact Solution

Follow our step by step solution on how to calculate Length of Path of Contact?

FIRST Step Consider the formula

P = \sqrt{{R a}^{2} - {R w}^{2} \cdot {(\cos (Φ g))}^{2}} + \sqrt{{r a}^{2} - r^{2} \cdot {(\cos (Φ g))}^{2}} - (R w + r) \cdot \sin (Φ g)

Next Step Substitute values of Variables

∴

P = \sqrt{{18.63mm}^{2} - {12.4mm}^{2} \cdot {(\cos (32°))}^{2}} + \sqrt{{15.954mm}^{2} - {10.2mm}^{2} \cdot {(\cos (32°))}^{2}} - (12.4mm + 10.2mm) \cdot \sin (32°)

Next Step Convert Units

∴

P = \sqrt{{0.0186m}^{2} - {0.0124m}^{2} \cdot {(\cos (0.5585rad))}^{2}} + \sqrt{{0.016m}^{2} - {0.0102m}^{2} \cdot {(\cos (0.5585rad))}^{2}} - (0.0124m + 0.0102m) \cdot \sin (0.5585rad)

Next Step Prepare to Evaluate

∴

P = \sqrt{{0.0186}^{2} - {0.0124}^{2} \cdot {(\cos (0.5585))}^{2}} + \sqrt{{0.016}^{2} - {0.0102}^{2} \cdot {(\cos (0.5585))}^{2}} - (0.0124 + 0.0102) \cdot \sin (0.5585)

Next Step Evaluate

∴

P = 0.0168076646441216m

Next Step Convert to Output's Unit

∴

P = 16.8076646441216mm

LAST Step Rounding Answer

∴

P = 16.8077mm

Length of Path of Contact Formula Elements

Variables

Functions

Path of Contact

Path of Contact is the path traced by the contact point of a pair of tooth profiles.

Symbol: P

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Path of Contact

Radius of Addendum Circle of Wheel

Radius of Addendum Circle of Wheel is radial distance between the pitch circle and the root circle.

Symbol: R_a

Measurement: LengthUnit: mm

Note: Value can be positive or negative.

Formulas to find Radius of Addendum Circle of Wheel

Radius of Pitch Circle of Wheel

Radius of Pitch Circle of Wheel is radial distance of the tooth measuring from the pitch circle to the bottom of the tooth space.

Symbol: R_w

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Radius of Pitch Circle of Wheel

Pressure Angle of Gear

Pressure Angle of Gear also known as the angle of obliquity is the angle between the tooth face and the gear wheel tangent.

Symbol: Φ_g

Measurement: AngleUnit: °

Note: Value should be greater than 0.

Formulas to find Pressure Angle of Gear

Radius of Addendum Circle of Pinion

Radius of Addendum Circle of Pinion is radial distance between the pitch circle and the root circle.

Symbol: r_a

Measurement: LengthUnit: mm

Note: Value can be positive or negative.

Formulas to find Radius of Addendum Circle of Pinion

Radius of Pitch Circle of Pinion

Radius of Pitch Circle of Pinion is the radial distance of the tooth measuring from the pitch circle to the bottom of the tooth space.

Symbol: r

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Radius of Pitch Circle of Pinion

sin

Sine is a trigonometric function that describes the ratio of the length of the opposite side of a right triangle to the length of the hypotenuse.

Syntax: sin(Angle)

cos

Cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse of the triangle.

Syntax: cos(Angle)

sqrt

A square root function is a function that takes a non-negative number as an input and returns the square root of the given input number.

Syntax: sqrt(Number)

Other Formulas to find Path of Contact

Go Maximum Length of Path of Contact

P = (R w + r) \cdot \sin (Φ g)

Other formulas in Length category

Go Length of Arc of Contact

L = \frac{P}{\cos (Φ g)}

Go Length of Path of Approach

P 1 = \sqrt{{R a}^{2} - {R w}^{2} \cdot {(\cos (Φ g))}^{2}} - R w \cdot \sin (Φ g)

Go Length of Path of Recess

P 2 = \sqrt{{r a}^{2} - r^{2} \cdot {(\cos (Φ g))}^{2}} - r \cdot \sin (Φ g)

Go Maximum Length of Path of Approach

P 1 = r \cdot \sin (Φ g)

How to Evaluate Length of Path of Contact?

Length of Path of Contact evaluator uses Path of Contact = sqrt(Radius of Addendum Circle of Wheel^2-Radius of Pitch Circle of Wheel^2*(cos(Pressure Angle of Gear))^2)+sqrt(Radius of Addendum Circle of Pinion^2-Radius of Pitch Circle of Pinion^2*(cos(Pressure Angle of Gear))^2)-(Radius of Pitch Circle of Wheel+Radius of Pitch Circle of Pinion)*sin(Pressure Angle of Gear) to evaluate the Path of Contact, Length of Path of Contact formula is defined as the distance traveled by the point of contact between two interacting gears or wheels during a specific period, providing insight into the mechanical interaction and motion of the system. Path of Contact is denoted by P symbol.

How to evaluate Length of Path of Contact using this online evaluator? To use this online evaluator for Length of Path of Contact, enter Radius of Addendum Circle of Wheel (R_a), Radius of Pitch Circle of Wheel (R_w), Pressure Angle of Gear (Φ_g), Radius of Addendum Circle of Pinion (r_a) & Radius of Pitch Circle of Pinion (r) and hit the calculate button.

FAQs on Length of Path of Contact

What is the formula to find Length of Path of Contact?

The formula of Length of Path of Contact is expressed as Path of Contact = sqrt(Radius of Addendum Circle of Wheel^2-Radius of Pitch Circle of Wheel^2*(cos(Pressure Angle of Gear))^2)+sqrt(Radius of Addendum Circle of Pinion^2-Radius of Pitch Circle of Pinion^2*(cos(Pressure Angle of Gear))^2)-(Radius of Pitch Circle of Wheel+Radius of Pitch Circle of Pinion)*sin(Pressure Angle of Gear). Here is an example- 16807.66 = sqrt(0.01863^2-0.0124^2*(cos(0.55850536063808))^2)+sqrt(0.015954^2-0.0102^2*(cos(0.55850536063808))^2)-(0.0124+0.0102)*sin(0.55850536063808).

How to calculate Length of Path of Contact?

With Radius of Addendum Circle of Wheel (R_a), Radius of Pitch Circle of Wheel (R_w), Pressure Angle of Gear (Φ_g), Radius of Addendum Circle of Pinion (r_a) & Radius of Pitch Circle of Pinion (r) we can find Length of Path of Contact using the formula - Path of Contact = sqrt(Radius of Addendum Circle of Wheel^2-Radius of Pitch Circle of Wheel^2*(cos(Pressure Angle of Gear))^2)+sqrt(Radius of Addendum Circle of Pinion^2-Radius of Pitch Circle of Pinion^2*(cos(Pressure Angle of Gear))^2)-(Radius of Pitch Circle of Wheel+Radius of Pitch Circle of Pinion)*sin(Pressure Angle of Gear). This formula also uses Sine (sin)Cosine (cos), Square Root (sqrt) function(s).

What are the other ways to Calculate Path of Contact?

Here are the different ways to Calculate Path of Contact-

Path of Contact=(Radius of Pitch Circle of Wheel+Radius of Pitch Circle of Pinion)*sin(Pressure Angle of Gear)

Can the Length of Path of Contact be negative?

Yes, the Length of Path of Contact, measured in Length can be negative.

Which unit is used to measure Length of Path of Contact?

Length of Path of Contact is usually measured using the Millimeter[mm] for Length. Meter[mm], Kilometer[mm], Decimeter[mm] are the few other units in which Length of Path of Contact can be measured.

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Length of Path of Contact Formula

​GoMaximum Length of Path of Contact Formula

Length of Path of Contact Example

Length of Path of Contact Solution

Length of Path of Contact Formula Elements

Other Formulas to find Path of Contact

Other formulas in Length category

How to Evaluate Length of Path of Contact?

FAQs on Length of Path of Contact

Variable Name

GoMaximum Length of Path of Contact Formula