What are linear springs and how do they work?

Springs that follow Hooke’s Law are often referred to as “linear springs” because they have a linear relationship between load and deflection.

A linear spring has the same diameter along its entire length, and this uniform diameter gives it a constant spring rate. In other words, the spring rate doesn’t change regardless of the load acting on the spring, and the spring’s deflection (or displacement) will be proportional to the applied force.

Spring equations for linear compression springs:

F = applied load (force) (N)

k = spring constant (N/m)

x = deflection (displacement) from spring’s neutral position (m)

U = potential (strain) energy (Nm, J)

Linear springs are helical coil springs and can be compression, extension, or torsion designs. Although these types of springs experience a constant deflection per unit of force (or exert a constant rate of force per deflected length), their load vs. deflection curve can be adjusted by changing the diameter of the coils or by adjusting the pitch (also referred to as rise angle) of the coils. It’s important to note, however, that if the pitch is adjusted so that it varies along the spring’s length, the spring will no longer have a linear load vs. deflection relationship.

Unlike compression springs, which resist forces that attempt to compress, or shorten them, extension springs resist forces that try to pull them apart. The primary difference between linear extension and linear compression spring designs is that an extension spring’s coils have no pitch in their neutral state. (Extension springs are also designed with an initial tension, or pre-tension.) When an extension spring is lengthened, it creates a pulling force as it tries to return to its original length.

When a linear extension spring is lengthened, it provides a “pulling” force that’s proportional to the change in length.
Image credit: Lee Spring

Torsion springs — which provide resistance to twisting forces — are helically wound designs with arms on the ends that rotate about the spring’s central axis. A linear torsional spring is simply a torsion spring whose travel varies proportionally with the load applied to the springs’ legs.

This conical spring is a type of non-linear spring.

The opposite of a linear spring is a non-linear spring. As the name suggests, these springs have a variable (non-linear) relationship between the applied load and the spring’s deflection.

Non-linear springs can be either compression or extension styles and come in a variety of designs, including conical, variable pitch, or tapered material — each with different load vs. deflection curves.

Note that the term “linear spring” can also refer to a linear wave spring, which is made from spring-tempered wire length that is formed into a wave. These springs produce an axial force when the waves are deformed by a radial force, and they can produce a radial force when the spring is deflected in the axial direction. Like “traditional” linear springs, linear wave springs also have a linear relationship between load and deflection.