Disc springs, also known as Belleville washers, are mechanical components used in applications requiring high force in a small space. A disc spring calculator helps in determining the force exerted by the spring and its deflection under load.
Purpose and Functionality
The main purpose of a disc spring calculator is to predict how a disc spring will behave under certain conditions. This involves calculating:
- The force the spring can exert.
- The deflection (or compression) of the spring when a load is applied.
- The stress in the spring material.
Formulae for Disc Spring Calculation
Force Calculation (F)
The force exerted by the disc spring can be calculated using the formula:
F=4Et3(s−y)0.75Do2−1.33Di2F = \frac{4Et^3 (s - y)}{0.75D_o^2 - 1.33D_i^2}F=0.75Do2−1.33Di24Et3(s−y)
Where:
- FFF is the force exerted by the spring (N).
- EEE is the modulus of elasticity of the material (Pa).
- ttt is the thickness of the spring (m).
- sss is the deflection (m).
- yyy is the preload deflection (m).
- DoD_oDo is the outer diameter of the spring (m).
- DiD_iDi is the inner diameter of the spring (m).
Deflection Calculation (s)
The total deflection under load can be calculated as:
s=F(0.75Do2−1.33Di2)4Et3+ys = \frac{F (0.75D_o^2 - 1.33D_i^2)}{4Et^3} + ys=4Et3F(0.75Do2−1.33Di2)+y
Where:
- sss is the total deflection under load (m).
Stress Calculation (σ)
The stress in the spring material can be calculated using:
σ=8Ft(0.5Do−0.5Di)π(Do4−Di4)\sigma = \frac{8Ft(0.5D_o - 0.5D_i)}{\pi (D_o^4 - D_i^4)}σ=π(Do4−Di4)8Ft(0.5Do−0.5Di)
Where:
- σ\sigmaσ is the stress in the spring material (Pa).
Inputs Needed
To use the disc spring calculator, you need the following inputs:
- EEE: Modulus of elasticity of the material (e.g., steel, stainless steel).
- ttt: Thickness of the spring.
- DoD_oDo: Outer diameter of the spring.
- DiD_iDi: Inner diameter of the spring.
- sss: Desired deflection.
- yyy: Preload deflection.
Example Calculation
Let's perform a sample calculation with hypothetical values:
- E=207×109E = 207 \times 10^9E=207×109 Pa (for steel).
- t=0.003t = 0.003t=0.003 m.
- Do=0.050D_o = 0.050Do=0.050 m.
- Di=0.025D_i = 0.025Di=0.025 m.
- y=0.0005y = 0.0005y=0.0005 m.
- s=0.001s = 0.001s=0.001 m (desired deflection).
Step-by-Step Calculation
- Force Calculation (F): F=4×207×109×(0.003)3×(0.001−0.0005)0.75×(0.050)2−1.33×(0.025)2F = \frac{4 \times 207 \times 10^9 \times (0.003)^3 \times (0.001 - 0.0005)}{0.75 \times (0.050)^2 - 1.33 \times (0.025)^2}F=0.75×(0.050)2−1.33×(0.025)24×207×109×(0.003)3×(0.001−0.0005) F=4×207×109×2.7×10−11×0.00050.001875−0.00083125F = \frac{4 \times 207 \times 10^9 \times 2.7 \times 10^{-11} \times 0.0005}{0.001875 - 0.00083125}F=0.001875−0.000831254×207×109×2.7×10−11×0.0005 F≈1.12×10−10.00104375F \approx \frac{1.12 \times 10^{-1}}{0.00104375}F≈0.001043751.12×10−1 F≈107.32 NF \approx 107.32 \, \text{N}F≈107.32N
- Deflection Calculation (s): s=107.32×(0.001875−0.00083125)4×207×109×(0.003)3+0.0005s = \frac{107.32 \times (0.001875 - 0.00083125)}{4 \times 207 \times 10^9 \times (0.003)^3} + 0.0005s=4×207×109×(0.003)3107.32×(0.001875−0.00083125)+0.0005 s≈0.001 ms \approx 0.001 \, \text{m}s≈0.001m
- Stress Calculation (σ): σ=8×107.32×0.003×(0.025)π×(0.0504−0.0254)\sigma = \frac{8 \times 107.32 \times 0.003 \times (0.025)}{\pi \times (0.050^4 - 0.025^4)}σ=π×(0.0504−0.0254)8×107.32×0.003×(0.025) σ≈2.68×106 Pa\sigma \approx 2.68 \times 10^6 \, \text{Pa}σ≈2.68×106Pa
Relevant Data Table
Here is a table summarizing the example calculation:
| Parameter | Value |
|---|---|
| EEE | 207×109207 \times 10^9207×109 Pa |
| ttt | 0.003 m |
| DoD_oDo | 0.050 m |
| DiD_iDi | 0.025 m |
| yyy | 0.0005 m |
| sss | 0.001 m |
| FFF | 107.32 N |
| σ\sigmaσ | 2.68×1062.68 \times 10^62.68×106 Pa |
Conclusion
A disc spring calculator is an essential tool for engineers and designers working with disc springs. By understanding the formulas and inputs required, you can predict the behavior of the spring under load, ensuring it functions correctly in your application. This helps in designing efficient and reliable mechanical systems.