Outer Radius Of Disc Given Maximum Radial Stress In Solid Disc Calculator

Formula Used:

\[ r_{outer} = \sqrt{\frac{8 \times \sigma_r}{\rho \times \omega^2 \times (3 + \nu)}} \]

Radial Stress (σr):

Pascal

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

(unitless)

Outer Radius Disc (router):

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1. What Is The Outer Radius Of Disc Given Maximum Radial Stress In Solid Disc?

This calculation determines the outer radius of a solid disc based on the maximum radial stress, material density, angular velocity, and Poisson's ratio. It's essential for designing rotating mechanical components to ensure they can withstand operational stresses.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ r_{outer} = \sqrt{\frac{8 \times \sigma_r}{\rho \times \omega^2 \times (3 + \nu)}} \]

Where:

\( r_{outer} \) — Outer radius of the disc (m)
\( \sigma_r \) — Radial stress (Pascal)
\( \rho \) — Density of the disc material (kg/m³)
\( \omega \) — Angular velocity (rad/s)
\( \nu \) — Poisson's ratio (unitless)

Explanation: This formula calculates the maximum outer radius that a solid disc can have while maintaining the specified maximum radial stress under given rotational conditions.

3. Importance Of Outer Radius Calculation

Details: Accurate calculation of the outer radius is crucial for designing rotating machinery components like flywheels, turbine discs, and gears to ensure they operate within safe stress limits and prevent mechanical failure.

4. Using The Calculator

Tips: Enter radial stress in Pascals, density in kg/m³, angular velocity in rad/s, and Poisson's ratio (typically between 0.1-0.5 for metals). All values must be positive and within valid ranges.

5. Frequently Asked Questions (FAQ)

Q1: What is radial stress in a rotating disc?
A: Radial stress is the stress component acting perpendicular to the radius of the disc, caused by centrifugal forces during rotation.

Q2: Why is Poisson's ratio important in this calculation?
A: Poisson's ratio accounts for the material's tendency to expand or contract in directions perpendicular to the applied stress, affecting the stress distribution.

Q3: What are typical values for Poisson's ratio?
A: For most metals and alloys, Poisson's ratio ranges between 0.25-0.35. Rubber-like materials can have values close to 0.5.

Q4: How does angular velocity affect the outer radius?
A: Higher angular velocities create greater centrifugal forces, requiring smaller outer radii to maintain the same maximum radial stress level.

Q5: Can this formula be used for hollow discs?
A: No, this specific formula is derived for solid discs. Hollow discs have different stress distributions and require different calculations.