Outer Radius Of Disc Given Constant At Boundary Condition For Circular Disc Calculator

Formula Used:

\[ r_{outer} = \sqrt{\frac{8 \times C_1}{\rho \times \omega^2 \times (3 + \nu)}} \]

Constant at boundary condition (C₁):

N/m²

Density Of Disc (ρ):

kg/m³

Angular Velocity (ω):

rad/s

Poisson's Ratio (ν):

Outer Radius Disc (rₒᵤₜₑᵣ):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Outer Radius Calculation?

The outer radius calculation determines the radius of a circular disc based on boundary conditions, material properties, and rotational dynamics. It's derived from stress analysis in rotating discs under specific boundary constraints.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{outer} = \sqrt{\frac{8 \times C_1}{\rho \times \omega^2 \times (3 + \nu)}} \]

Where:

\( C_1 \) — Constant at boundary condition (N/m²)
\( \rho \) — Density of disc material (kg/m³)
\( \omega \) — Angular velocity (rad/s)
\( \nu \) — Poisson's ratio (dimensionless)

Explanation: This formula calculates the outer radius of a solid circular disc rotating at constant angular velocity, considering material properties and boundary conditions.

3. Importance of Outer Radius Calculation

Details: Accurate radius calculation is crucial for mechanical design, stress analysis, and ensuring structural integrity in rotating disc applications such as turbines, flywheels, and rotating machinery components.

4. Using the Calculator

Tips: Enter constant at boundary condition in N/m², density in kg/m³, angular velocity in rad/s, and Poisson's ratio (typically between 0.1-0.5). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the constant at boundary condition?
A: It's a stress-related constant determined from the boundary conditions of the disc, typically obtained from stress analysis equations.

Q2: Why is Poisson's ratio important in this calculation?
A: Poisson's ratio accounts for the material's lateral contraction when stretched, affecting stress distribution in the rotating disc.

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接近 0.5.

Q4: How does angular velocity affect the outer radius?
A: Higher angular velocities create greater centrifugal forces, which influence the stress distribution and thus the calculated outer radius.

Q5: What are the limitations of this formula?
A: This formula assumes homogeneous material, constant angular velocity, and specific boundary conditions. It may not apply to complex geometries or non-uniform materials.