The waves or features around the tolerance ring compress like springs. But how does this actually create the forces to hold everything together? Well, this is down to simple spring theory!

Hooke’s Law of elasticity gives a basic understanding of how tolerance rings work and is summarized by the equation on the left. It essentially says that the force needed to extend or compress a spring is directly proportional to the distance displaced (it actually says this for all materials, but we are talking springs here).

When we look at the tolerance ring, the main part of this formula is **K**, or what we call the spring constant. This is essentially the stiffness of each of the waves added together – the greater the stiffness, the more force is needed to compress the tolerance ring the same amount.

The stiffness of the tolerance ring can be changed in a number of ways including

- Young’s modulus of the material
- Material thickness
- Shape of the waves

For a very simple tolerance ring the spring constant can be approximated using these factors in this equation:

*K = 4.8 E w (t/p) ^{3}*

Where:

**E**is the Elastic Modulus for the material [kN/mm²]**w**is the width of the wave [mm]**t**is the material thickness [mm]**p**is the wave pitch [mm]

In reality this equation is too simplistic and doesn’t account for many factors. At Saint-Gobain we use sophisticated predictive design tools are used to calculate the performance of the tolerance ring. To take a look at how our engineers take you through the design process look at **How are RENCOL ^{®} Tolerance Rings designed?**.

Applying the above spring theory to the design of a tolerance ring gives the ability to tune the spring stiffness for a wide range of applications and performance requirements. For example; a stiff wave geometry can be developed for applications with high radial load or torque requirements. Alternatively using a gentler wave geometry will generate a lower stiffness for applications with low loading requirements.

This design flexibility allows the tolerance rings to be specifically designed for each application by varying a combination of; the complex ring geometry, material thickness, hardness and operating compression range to create an appropriate spring constant and hence a pre-determined retention forces and/or slip torques.

In order to work out the true spring rate in practice, a compression test is normally done. A tolerance ring would be put into a tensile tester and compressed. The output shows what force the ring generates at certain amounts of compression. This data can then be used to validate predictions and also calculate other performance criteria (such as torque and slip).

The above equation gives you the radial force **F _{R}**. This is useful to know to understand how much force the tolerance ring is putting onto mating components, but usually the axial force

Again, in theory these are fairly simple calculations. The axial force

**F _{A} = F_{R} * µ**

And the torque **T** by simply multiplying the axial force **F _{A}** by half the ring diameter

**T = F _{A} * r**

Again, these formulas will give a rough indication of performance, but in reality there are many factors that affect these such as material deformation, ‘ploughing’ effect of the waves on component and others. Because of this our engineers have developed sophisticated design tools that take into account these factors and allow us to predict the performance much more accurately.

If you would like to find out if a RENCOL^{®} Tolerance Ring is suitable for you, **contact us**.