8.2 Springs: Types, Design, and Applications

Spring Types

Coiled and Leaf Springs

Helical springs are wires coiled into a helical shape, and they're the most common spring type you'll encounter. They come in two main varieties:

• Compression springs resist compressive forces and store mechanical energy. You'll find these in mattresses, ballpoint pens, and valve assemblies. They're open-coiled, meaning there's space between the coils for them to compress.
• Extension springs resist tensile (pulling) forces. Unlike compression springs, they start with coils touching and stretch apart under load. Trampolines and garage door counterbalance systems use these.

Leaf springs are made of flat plates (leaves) stacked and clamped together. They serve a dual purpose in vehicle suspensions: they absorb shock and act as a structural linkage that locates the axle and transfers forces to the chassis. Truck rear suspensions are the classic example. The friction between the stacked leaves also provides some built-in damping.

Torsion Springs

Torsion springs apply a torque (rotary force) when twisted about their axis. Think clothespins, mousetraps, and wind-up toys. The key relationship is that the restoring torque is proportional to the angular deflection, just like how a linear spring's force is proportional to its linear deflection.

These springs can be made from round or rectangular cross-section wire. Round wire is far more common, but rectangular bars handle higher loads and are used in applications like vehicle stabilizer bars.

Spring Design Parameters

Spring Rate and Natural Frequency

Spring rate ($k$) defines how stiff a spring is. It's the change in applied force divided by the change in deflection:

$k = \frac{\Delta F}{\Delta x}$

Units are force per length (N/m or lbf/in). A stiffer spring has a higher $k$ and requires more force for the same deflection.

For a helical compression spring, the spring rate depends on material and geometry:

$k = \frac{Gd^4}{8D^3 N_a}$

where $G$ is the shear modulus, $d$ is wire diameter, $D$ is mean coil diameter, and $N_a$ is the number of active coils.

Natural frequency ($f_n$) is the frequency at which a spring-mass system vibrates freely after being disturbed:

$f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

This matters because if an external excitation frequency gets close to $f_n$, you get resonance, which can destroy the spring. Higher $k$ or lower mass $m$ pushes the natural frequency up.

Spring Index and Stress

Spring index ($C$) is the ratio of mean coil diameter to wire diameter:

$C = \frac{D}{d}$

Typical values range from about 4 to 12. A low $C$ (fat wire, tight coils) means the spring is hard to manufacture and has high bending stresses during coiling. A high $C$ (thin wire, wide coils) is easier to make but produces a softer spring with a lower spring rate. Most practical designs target $C$ between 6 and 10.

Stress in a helical spring is primarily torsional shear stress, with an additional transverse shear component. The maximum stress occurs at the inner surface of the coil, where curvature effects concentrate the stress. The Wahl correction factor ($K_W$) accounts for this curvature and direct shear:

$K_W = \frac{4C - 1}{4C - 4} + \frac{0.615}{C}$

The corrected shear stress is then:

$\tau = K_W \frac{8FD}{\pi d^3}$

You can reduce peak stress by increasing $C$ (within practical limits) or selecting materials with higher shear strength, such as music wire (ASTM A228) or chrome-silicon steel (ASTM A401).

Spring Failure Modes

Fatigue and Stress Relaxation

Fatigue is the most common failure mode for springs under cyclic loading. A spring might survive any single load application just fine, but after thousands or millions of cycles, cracks nucleate at stress concentrations (usually the inner coil surface) and propagate until fracture.

Fatigue life depends on:

• Stress amplitude relative to the material's endurance limit
• Material quality and surface finish
• Manufacturing processes like shot peening, which introduces beneficial compressive residual stresses on the surface, closing down potential crack initiation sites

Stress relaxation is a different problem: the spring gradually loses force while held at a constant deflection. This happens because the material slowly creeps at the microstructural level, especially at elevated temperatures. Engine valve springs are a classic case where stress relaxation matters. You can minimize it by choosing thermally stable alloys and keeping operating temperatures within the material's rated range.

Buckling and Surging

Buckling affects compression springs the same way it affects columns. When a compression spring is too slender (long relative to its diameter), it can suddenly bow sideways under axial load instead of compressing straight.

The critical factor is the free length-to-diameter ratio ($L_f / D$). Springs with $L_f / D$ greater than about 4 (depending on end conditions) are at risk. You can prevent buckling by:

1. Reducing the free length or increasing the coil diameter to lower the $L_f / D$ ratio
2. Using a guide rod or housing bore to constrain lateral movement
3. Choosing squared-and-ground ends for better load alignment

Surging is a resonance phenomenon. When the frequency of the applied load matches (or is a harmonic of) the spring's natural frequency, standing waves develop along the coils. This causes coils to clash into each other, producing impact stresses far beyond normal design levels and leading to rapid fatigue failure.

To prevent surging:

• Change the spring's natural frequency by adjusting $k$ or the spring mass (variable-pitch springs are one approach)
• Add damping to the system
• Ensure the operating frequency stays well below the spring's fundamental natural frequency (a common guideline is to keep $f_n$ at least 15 to 20 times higher than the excitation frequency)