Belleville Spring (Coned-Disk) Calculator
What is a Belleville Spring?
A Belleville spring, also known as a coned-disk spring or Belleville washer, is a type of spring shaped like a frustum of a cone. This unique geometry allows the spring to provide very high loads in restricted axial spaces with relatively small deflections. They are widely used in industrial applications such as bolted joints, clutch mechanisms, and pressure relief valves.
The Almen-Laszlo Equations
The calculation of Belleville spring forces is primarily based on the Almen-Laszlo equations. Unlike standard coil springs, the rate of a Belleville spring is non-linear. The relationship between load and deflection depends heavily on the ratio of the internal height (h) to the thickness (t).
- Linear: If h/t < 0.4, the spring is almost linear.
- Regressive: If 0.4 < h/t < 1.414, the spring rate decreases as deflection increases.
- Constant Load: If h/t = 1.414, the spring reaches a plateau where the load remains constant for a portion of the deflection.
Calculation Example
Suppose you have a steel Belleville washer (E = 206,000 N/mm²) with an Outer Diameter of 50mm, Inner Diameter of 25.4mm, thickness of 2mm, and a total height of 3.5mm. If you compress it by 0.75mm:
- Internal Height (h): 3.5mm – 2mm = 1.5mm.
- Diameter Ratio (δ): 50 / 25.4 = 1.968.
- Force Calculation: Using the constants K1, K2, K3 derived from δ, the load is calculated based on the third-degree polynomial of deflection.
- Result: For these parameters, the force exerted would be approximately 4,280 Newtons.
Applications of Coned-Disk Springs
Belleville springs are favored in high-tension applications. In bolted joints, they maintain preload even when thermal expansion or vibration occurs. In clutches, they provide the necessary clamping force in a very compact package. They can also be "stacked" in series to increase deflection or in parallel to increase load capacity.