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Length of Path of Recess Formula

GoMaximum Length of Path of Recess Formula

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Path of recess is the portion of the path of contact from the pitch point to the end of contact. Check FAQs

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

P₂ - Path of Recess?r_a - Radius of Addendum Circle of Pinion?r - Radius of Pitch Circle of Pinion?Φ_g - Pressure Angle of Gear?

Length of Path of Recess Example

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

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

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

8.0003 = \sqrt{{15.954}^{2} - {10.2}^{2} \cdot {(\cos (32))}^{2}} - 10.2 \cdot \sin (32)

8.0003mm = \sqrt{{15.954mm}^{2} - {10.2mm}^{2} \cdot {(\cos (32°))}^{2}} - 10.2mm \cdot \sin (32°)

8.0003 = \sqrt{{15.954}^{2} - {10.2}^{2} \cdot {(\cos (32))}^{2}} - 10.2 \cdot \sin (32)

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Length of Path of Recess Solution

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

FIRST Step Consider the formula

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

Next Step Substitute values of Variables

∴

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

Next Step Convert Units

∴

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

Next Step Prepare to Evaluate

∴

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

Next Step Evaluate

∴

P 2 = 0.00800027198460171m

Next Step Convert to Output's Unit

∴

P 2 = 8.00027198460171mm

LAST Step Rounding Answer

∴

P 2 = 8.0003mm

Length of Path of Recess Formula Elements

Variables

Functions

Path of Recess

Path of recess is the portion of the path of contact from the pitch point to the end of contact.

Symbol: P₂

Measurement: LengthUnit: mm

Note: Value should be greater than 0.

Formulas to find Path of Recess

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

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

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 Recess

Go Maximum Length of Path of Recess

P 2 = R w \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 Contact

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)

Go Maximum Length of Path of Approach

P 1 = r \cdot \sin (Φ g)

How to Evaluate Length of Path of Recess?

Length of Path of Recess evaluator uses Path of Recess = 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 Pinion*sin(Pressure Angle of Gear) to evaluate the Path of Recess, Length of Path of Recess formula is defined as the distance along the path of recess of a gear tooth, which is a critical parameter in gear design, influencing the gear's strength, durability, and overall performance, and is used to determine the optimal gear tooth shape and size. Path of Recess is denoted by P₂ symbol.

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

FAQs on Length of Path of Recess

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

The formula of Length of Path of Recess is expressed as Path of Recess = 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 Pinion*sin(Pressure Angle of Gear). Here is an example- 12627.46 = sqrt(0.015954^2-0.0102^2*(cos(0.55850536063808))^2)-0.0102*sin(0.55850536063808).

How to calculate Length of Path of Recess?

With Radius of Addendum Circle of Pinion (r_a), Radius of Pitch Circle of Pinion (r) & Pressure Angle of Gear (Φ_g) we can find Length of Path of Recess using the formula - Path of Recess = 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 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 Recess?

Here are the different ways to Calculate Path of Recess-

Path of Recess=Radius of Pitch Circle of Wheel*sin(Pressure Angle of Gear)

Can the Length of Path of Recess be negative?

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

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

Length of Path of Recess 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 Recess can be measured.

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Length of Path of Recess Formula

​GoMaximum Length of Path of Recess Formula

Length of Path of Recess Example

Length of Path of Recess Solution

Length of Path of Recess Formula Elements

Other Formulas to find Path of Recess

Other formulas in Length category

How to Evaluate Length of Path of Recess?

FAQs on Length of Path of Recess

Variable Name

GoMaximum Length of Path of Recess Formula