Springs Taxonomy — Family Index
A spring is an elastic element that stores energy in a controlled, reversible deflection and releases it as force or torque. This note maps the major mechanical-spring families, their governing equations, the wire/strip materials used to make them, and the heuristics for picking one over another.
1. At a glance
Major categories (by geometry / loading mode):
- Helical — round/square wire wound on a cylinder
- Compression (axial push)
- Extension (axial pull, with initial tension)
- Torsion (angular, legs reacting to applied torque)
- Conical / volute — variable-pitch helical for variable rate and low solid height
- Belleville / disc — conical washer, very high load in short stroke
- Leaf — bending of a strip or stacked strips
- Flat / clock / spiral — strip wound in a planar spiral, torque output
- Wave — wave-form washer stack, low stack height
- Constant-force / Negator — pre-stressed strip rolled tight, nearly flat F–x curve
- Garter — closed-loop helical, clamps radially
- Gas spring — sealed N₂-charged cylinder
- Hydropneumatic — gas spring plus inline hydraulic damper
- Magnetic spring — repelling permanent magnets, non-contact
- Elastomeric — rubber or polyurethane block in shear/compression
Two-pillar selection axis:
- Deflection vs. load envelope — how much travel is needed for what force? Belleville stacks dominate the short-stroke / very-high-load corner; Negator strips dominate the long-stroke / near-constant-force corner; helical compression fills the broad middle.
- Space envelope — solid height, ID/OD limits, mounting style, weight. Wave springs and Belleville stacks beat helical when axial space is tight. Conical/volute springs allow pancaking to wire diameter.
Auxiliary axes: environment (corrosion, temperature), fatigue life (static / cyclic / infinite life), magnetic permissibility, electrical conductivity, cost.
2. Helical compression
The workhorse spring. A round (most common) or square wire is wound on a mandrel into a cylindrical, conical, hourglass, or barrel form.
Forms:
- Straight cylindrical — constant pitch, constant coil diameter; linear rate
- Variable-pitch — close-wound section bottoms out first → progressive rate
- Conical / tapered — coil diameter tapers; allows nesting at full compression to ≈ wire diameter
- Hourglass / barrel — diameter varies along axis for tuned natural frequency and side-load suppression
End conditions (affect active turns N_a and solid height):
- Plain (P) — wire cut, ends remain helical
- Plain-ground (PG) — ends ground flat but still helical
- Squared / closed (C) — last turn deflected to perpendicular, not ground
- Squared-and-ground (CG) — closed and ground flat; preferred for stability
Geometry index C = D/d, where D = mean coil diameter, d = wire diameter.
- Practical range C ≈ 4 to 12
- C ≈ 8 is the sweet spot (manufacturable, low stress concentration)
- C < 4 → manufacturing difficulty, high surface stress
- C > 12 → tangles, buckles, very flexible
Linear rate (Hooke’s law):
F = k · x
k = G · d⁴ / (8 · D³ · N_a)
where G is the shear modulus of the wire material (≈ 79.3 GPa for steel music wire, 77 GPa for stainless 302). Active turns N_a depends on end style: N_a = N_t for plain ends, N_a = N_t − 2 for squared-and-ground.
Shear stress (uncorrected) τ = 8·F·D / (π·d³).
Wahl correction for curvature plus direct shear:
K_w ≈ (4C − 1) / (4C − 4) + 0.615 / C
τ_max = K_w · 8·F·D / (π·d³)
K_w drops from ≈ 1.40 at C = 4 to ≈ 1.10 at C = 12.
Solid height L_s ≈ N_t · d (squared-and-ground); add wire dia for plain ends. Always design with operating deflection ≤ 80% of (free length − solid length) to leave clash margin.
Buckling check — slenderness L_f / D > 4 with one end free → check Euler-type buckling per SMI handbook nomogram.
3. Helical extension
Wound tight with the coils touching (or nearly so) under no external load — this is the initial tension F_i. External force must first overcome F_i before deflection begins.
F = F_i + k · x for F > F_i
k = G · d⁴ / (8 · D³ · N)
Free length L_f = wire-stack length + 2 × hook reach. Hooks are the fatigue-critical feature — stress concentrates at the bend where the hook turns up out of the body coils.
Hook / end styles:
- Machine half-hook / full loop — last coil bent up 90° or 180°, simplest and weakest
- Crossover loop — loop centered on coil axis
- Side loop — loop offset to one side
- Extended hook — loop pulled axially out from body, long lever arm
- Threaded insert — screw-in eye for high-force applications (no bend stress)
- Custom swivel / clevis — for dynamic applications
Hook stress can be 2–3× body stress, so derate accordingly; for cyclic service, use the threaded insert or a torsion spring instead.
4. Helical torsion
Wound like a compression spring but loaded angularly: legs project radially, tangentially, or axially and react to an applied torque. The body coils are loaded in bending, not shear — different governing equation.
M = k_T · θ
k_T = E · d⁴ / (64 · D · N_a) (radians; torque in N·m if SI)
Note k_T is proportional to d⁴ and inversely to D and N — same scaling as compression for d, but bending modulus E, not shear modulus G.
Leg styles — radial (most common), tangential (closer to compact), straight-axial (specialty), hinged with formed end.
Wind direction matters — torsion springs are deflected in the direction that closes the coil (reduces D and increases N) to keep stress benign. Loading them open is allowed for short cycles but is fatigue-bad.
Applications — mousetraps, clothespins, hinges, garage-door counterbalance, relay return springs, clipboard clips, retractable badge holders.
5. Conical / volute springs
A helical compression spring with progressively decreasing coil diameter. Properties:
- Variable rate — large coils bottom against each other first as load increases (effective N decreases), so stiffness rises with deflection
- Low solid height — when fully compressed, coils nest inside each other, bringing solid height to ≈ wire diameter
- Good lateral stability — flared base resists tipping
Volute springs are conical springs made from rectangular strip wound on edge — historically used in tank suspension and heavy machinery.
Use cases — battery contacts (low solid height under door), recoil mechanisms, vehicle bumpers, packaging.
6. Belleville / disc springs
A conical washer that flattens under axial load. Standardized by DIN 2093 (spring discs) and DIN 6796 (high-load conical lock washers). ASME B18.21 covers Belleville lock washers used with bolts.
Geometry parameters: outer diameter D_o, inner diameter D_i, free height h₀ (of cone), thickness t. The h₀/t ratio drives the F–x character:
- h₀/t < 0.4 → near-linear (small deflection)
- h₀/t ≈ 1.4 → flat segment of curve crosses (constant force over a range)
- h₀/t ≈ 2.0 → bistable snap-through (negative-stiffness region)
- h₀/t > 2.83 → snap-through with hysteresis
Stacking:
- Series (same-direction cones nested cup-to-cone) — n discs → n × stroke, same force
- Parallel (same-direction, base-to-base or cup-to-cup) — n discs → same stroke, n × force
- Alternating series/parallel groups — independent tuning of force and stroke
Use cases:
- Bolt-preload maintenance — Belleville stack under a fastener compensates for gasket creep and thermal expansion, keeping clamp force stable
- Mechanical seal face-loading — provides preload that follows face wear
- Valve actuator springs — Belleville packs in pneumatic spring-return actuators
- Press / clutch packs — sintered-friction clutch engagement
- Vibration isolators — non-linear F–x can be tuned to natural frequency
7. Leaf springs
A beam in bending — a flat strip (or stack of strips) anchored at both ends or cantilevered.
Forms:
- Cantilever leaf — one end fixed, other free; simple gauge / contact spring
- Single full-elliptic — two semi-elliptics back-to-back (vintage carriage suspension)
- Semi-elliptic multi-leaf — main leaf plus progressively shorter helper leaves; the classic truck rear suspension. Interleaf friction provides inherent damping
- Parabolic — single tapered leaf or 2–3 leaves with parabolic thickness profile, lower weight, less friction (modern OEM trucks, light commercial)
- Mono-leaf transverse — single leaf mounted across vehicle (e.g. Corvette composite leaf — fiberglass)
Materials — typically silico-manganese spring steel:
- SAE 5160 (Cr) — common automotive
- SAE 9260 (Si-Mn) — fatigue-grade truck leaves
- SAE 51CrV4 / 50CrV4 — European spec (DIN 17221), chromium-vanadium
- Composite leaves — GFRP unidirectional (corvette, Volvo, Mercedes monoleaf) — 60% mass reduction
Applications — automotive rear suspension (live axles), rail rolling-stock bolster springs, structural / architectural cantilevers, large industrial shock-absorption.
8. Flat / clock / spiral springs
A flat strip wound in a planar spiral, loaded in bending. Torque output.
- Power spring (motor spring) — flat strip in a drum, drives wind-up toys, clocks, mechanical watches (mainspring). Approximately constant torque over the working range
- Spiral torsion spring — short coiled strip used in instruments, gauges and pointers
- Hairspring (balance spring) — fine flat spiral in mechanical watches, oscillates the balance wheel — Nivarox / Elinvar / Parachrom (low-thermal- coefficient alloys)
9. Wave springs
Crest-to-crest stacked wave-form washers (Smalley, Rotor Clip). Provide the same deflection and force as a coil spring of similar size while occupying 50–70% less axial stack height.
Variants:
- Single-turn (open) — one ring, snap-in
- Multi-turn — continuous coil of wave-form flat wire (Smalley Crest-to-Crest)
- Nested — wave springs stacked in parallel for higher load
Use cases — bearing preload (axial take-up), valve seat preload, transmission clutch packs, mechanical seals where coil height is forbidden, medical-device latches.
Wave-form wire is edge-wound flat stainless or carbon spring steel; outer and inner diameters fit retaining-ring grooves directly.
10. Constant-force springs (Negator)
A pre-stressed flat strip rolled tightly onto itself — when unwound, it tries to re-roll, exerting a force that is nearly constant over a long stroke (typically within ±5% over the working range). Originally developed and patented as “Negator” by Hunter Spring Co.; “Neg’ator” is the SDP/SI trademark.
Variants:
- Standard flat strip — work mechanism is the un-rolling, force is the pull-out tension
- Spring motor (back-to-back drums) — Negator on one drum drives a take-up drum, used as a low-speed constant-torque source
- Carbon-fiber / composite tape — high-stroke deployable booms in aerospace
Use cases — retractable seat belts, counterbalance for hospital monitor arms, retractable badge reels, brushes in motor commutators (constant brush-to-commutator force as the brush wears), elevator door counterbalance.
11. Garter springs
A small-diameter helical extension spring whose ends are joined to form a closed loop. The loop circumferentially clamps onto a shaft or seal lip.
Use cases:
- Radial-lip oil seals — the garter spring inside the rubber lip clamps it onto the rotating shaft to maintain sealing force as the lip wears (Simrit / SKF / Freudenberg standard)
- Brush-seal retention in jet engines and turbomachinery
- Mechanical packing in pumps and valves
12. Gas springs
A sealed steel cylinder with a piston and rod, pre-charged with nitrogen gas (typically 30–180 bar) and a small charge of damping oil. The gas acts as the spring; the oil dampens the rod motion.
Major manufacturer references: Stabilus (Lift-O-Mat, Bloc-O-Lift), SUSPA, ACE Stoßdämpfer, Bansbach.
Characteristic curve — force is roughly constant over the stroke (typically within ±30%), with a small force rise as the piston compresses the gas. The “P1 / P2” force callout in catalogs gives extended (lowest) and retracted (highest) force.
Variants:
- Standard (push-type) — extends to push (hatchback, hood, office-chair back tilt)
- Tension (pull-type) — extends to pull (specialty)
- Locking gas spring — internal valve locked by an external trigger (rigid-back office chairs, hospital beds, machine guards)
- Adjustable / dampers-only — pure shock absorber, no preload (door closers, washing-machine drum dampers)
Use cases — hatchback / tailgate / hood props, office chair height adjustment, monitor arms, machine guards, exercise equipment, aviation seats, hospital beds.
13. Hydropneumatic / disc-pack systems
A gas spring with an explicit hydraulic damper inline — the gas charge provides the spring force, the hydraulic flow through orifices provides the damping curve.
- Automotive struts (MacPherson) — gas-pressurized monotube or twin-tube damper; the gas suppresses oil cavitation more than it provides spring force
- Aircraft landing gear oleo strut — gas spring with metering pin providing progressive hydraulic damping during touchdown; sized for energy absorption per FAR Part 25
- Citroën hydropneumatic suspension — sphere-and-membrane gas accumulator with hydraulic interconnect (1955–2017 production)
14. Magnetic springs
Two permanent magnets oriented to repel — no contact, no wear, no fatigue. The restoring force varies nonlinearly with gap (≈ 1/r² to 1/r⁴ depending on geometry), but can be linearized over a small range.
Use cases:
- MEMS accelerometers and energy-harvesting transducers
- Magnetic levitation (passive support + active feedback)
- Mechanical-watch escapements (Slim d’Hermès, Frederique Constant Monolithic)
- Linear oscillators (free-piston Stirling cryocoolers — Sunpower / Qdrive)
Limit — repelling-magnet equilibrium is unstable in all three axes (Earnshaw’s theorem), so magnetic springs need a mechanical or active constraint in at least one direction.
15. Elastomeric / rubber springs
Bulk rubber or polyurethane in compression or shear. The bulk modulus of rubber is high (≈ 2 GPa) but the shear modulus is low (≈ 1 MPa), so geometry — specifically the shape factor S = loaded area / bulged free area — drives the effective stiffness.
For a simple block in compression:
k ≈ E_c · A / t where E_c = E · (1 + 2·S²) — apparent modulus
For shear:
k ≈ G · A / t
Materials:
- Natural rubber (NR) — best fatigue, low damping, –30 to +80 °C
- Styrene-butadiene (SBR) — cost
- Polychloroprene (CR / Neoprene) — oil and weather resistance
- EPDM — high temperature, ozone, –40 to +130 °C
- Polyurethane (PU, MPM) — high load, abrasion resistance, narrower temp range
Use cases:
- Engine mounts (rubber-bonded-to-metal hydromounts — fluid-damped rubber isolator, Trelleborg / Continental / Vibracoustic)
- Rail pads (under-tie elastomer, Pandrol)
- Building base-isolation bearings (high-damping rubber, lead-rubber bearings — Bridgestone, Dynamic Isolation Systems)
- Machine snubbers, conveyor return rollers
- Earphone gaskets, vibration feet on lab equipment
16. Wire and strip materials
Spring properties hinge on the wire/strip. The big four for steel springs are music wire, hard-drawn, oil-tempered, and chrome-silicon. Stainless and exotic alloys cover environment, temperature, and corrosion.
- Music wire (ASTM A228) — cold-drawn high-carbon (0.70–1.00% C). UTS up to ≈ 2700 MPa for small diameters. Industry workhorse for d < 4 mm precision springs. Surface defects minimal. Service to ≈ 120 °C.
- Hard-drawn (ASTM A227) — cold-drawn medium-high carbon. Lower UTS, cheaper. General-purpose static or low-cycle applications.
- Oil-tempered MB (ASTM A229) — quenched and tempered carbon wire. Used for d > 4 mm where music wire is no longer available; auto suspension.
- Chrome-silicon (ASTM A401) — Cr-Si alloyed, oil-tempered. Excellent fatigue and ≈ 220 °C service. Engine valve springs, fuel-injection.
- Chrome-vanadium (ASTM A232) — Cr-V oil-tempered. Shock and impact, 200 °C service.
- Stainless 302 (ASTM A313 Type 302 / AISI 302) — austenitic, mildly ferromagnetic when cold-worked, corrosion-resistant. Most common stainless spring wire.
- 17-7PH (ASTM A313 Type 631) — precipitation-hardening stainless, hardened post-forming, excellent fatigue, service to ≈ 320 °C.
- Inconel X-750 (UNS N07750, AMS 5698/5699) — Ni-Cr-Fe age-hardenable; 600 °C service, oxidation resistance. Turbine engine springs.
- Inconel 718 (UNS N07718) — similar service envelope, higher yield.
- Elgiloy / Phynox (Co-Cr-Ni-Mo, UNS R30003) — high strength, corrosion resistance, non-magnetic. Medical implants, intravascular.
- MP35N (UNS R30035) — Co-Ni-Cr-Mo, very high strength, non-magnetic, corrosion resistance. Medical and marine.
- Beryllium-copper (C17200, UNS C17200) — non-magnetic, electrically conductive, good fatigue. Electrical contacts, RF springs.
- Phosphor bronze (C51000) — copper-tin, mildly conductive, modest fatigue.
- Brass C26000 (cartridge brass) — low-stress decorative or electrical.
- Hastelloy C-276 — corrosion (chloride / acid).
- Titanium Beta-C (Ti-3Al-8V-6Cr-4Mo-4Zr) — ≈ 40% mass reduction vs steel, used in MotoGP / F1 valve springs and aerospace.
- Nivarox / Parachrom / Elinvar — low thermal coefficient Fe-Ni-Cr-Be alloys for watch hairsprings.
17. Property table
| Material | Spec | UTS (MPa) | T_max service (°C) | Corrosion | Comments |
|---|---|---|---|---|---|
| Music wire | ASTM A228 | 1700–2700 (d-dependent) | 120 | Poor | Workhorse for d < 4 mm precision springs |
| Hard-drawn | ASTM A227 | 1100–1700 | 120 | Poor | Cost-driven static or low-cycle |
| Oil-tempered MB | ASTM A229 | 1200–1800 | 150 | Poor | d > 4 mm, auto suspension |
| Chrome-silicon | ASTM A401 | 1700–2100 | 220 | Poor | Engine valve, shot-peened |
| Chrome-vanadium | ASTM A232 | 1300–1700 | 200 | Poor | Shock / impact |
| 302 stainless | ASTM A313-302 | 1400–1900 | 290 | Good | General-purpose corrosion |
| 17-7PH | ASTM A313-631 | 1700–2100 | 320 | Good | Precipitation-hardened, fatigue |
| Inconel X-750 | AMS 5698/5699 | 1100–1400 | 600 | Excellent | Turbine engine, age-hardened |
| Inconel 718 | UNS N07718 | 1200–1500 | 650 | Excellent | Higher yield than X-750 |
| Elgiloy / Phynox | UNS R30003 | 1700–2300 | 400 | Excellent | Medical, non-magnetic |
| MP35N | UNS R30035 | 1800–2400 | 425 | Excellent | Marine / medical, non-magnetic |
| Beryllium-copper | UNS C17200 | 1100–1450 | 200 | Good | Non-magnetic, conductive |
| Phosphor bronze | UNS C51000 | 700–1000 | 150 | Good | Conductive, modest fatigue |
| Brass C26000 | UNS C26000 | 500–700 | 100 | Fair | Low-stress, decorative / electrical |
| Ti Beta-C | UNS R58640 | 1300–1500 | 315 | Excellent | 40% lighter than steel, motorsport |
18. Selection heuristics
- Small (d < 4 mm) precision compression / extension / torsion → music wire ASTM A228; if corrosion is a factor, 302 stainless ASTM A313.
- Auto suspension coil or leaf → SAE 9260 / 5160 oil-tempered, shot-peened; composite GFRP mono-leaf where mass matters.
- Engine valve spring → chrome-silicon ASTM A401, oil-tempered, shot-peened; Ti Beta-C in motorsport.
- Hatchback / hood prop / chair lift → off-the-shelf gas spring (Stabilus Lift-O-Mat, SUSPA, Bansbach); sized by extended force (P1) and stroke.
- High-temperature turbomachinery → Inconel X-750 or 718 helical; Belleville stack in heat-resistant alloy for bolt preload at hot interfaces.
- Corrosive / wet / chloride → 17-7PH or 316 stainless or BeCu (also non-magnetic) or MP35N for the worst environments.
- Saving stack height in axial direction → wave spring (Smalley Crest-to-Crest) or Belleville stack — both replace coil at 50–70% less height.
- Bolt preload maintenance over temperature / gasket creep → Belleville stack under the fastener, sized so working range sits in the flat region of the F–x curve.
- Constant retraction force over long stroke → Negator constant-force strip (seat belt, retractable cord, balance arm).
- Angular return spring (mousetrap, hinge, latch) → helical torsion; always load in the closing direction.
- Vibration isolation under a machine or vehicle → elastomeric mount (rubber-bonded-to-metal hydromount, polyurethane snubber).
- Radial-lip seal clamping → garter spring inside the elastomer lip.
- Non-magnetic / non-sparking / electrically conductive → BeCu C17200, or phosphor bronze for lower stress.
- Medical implant / endovascular → MP35N or Elgiloy / Phynox.
- Watch hairspring / instrument → Nivarox / Parachrom / Elinvar.
19. Cross-references
- steel-grades — wire-source carbon and alloy steels (SAE 1060, 5160, 9260, 51CrV4) used for leaf and large-d coil springs
- stainless-steels — austenitic and precipitation- hardening grades used as spring wire (302, 316, 17-7PH)
- copper-alloys — phosphor bronze, beryllium-copper, brass for conductive / non-magnetic springs
- titanium-alloys — Ti Beta-C for lightweight valve springs
- fasteners-taxonomy — Belleville lock-washers (ASME B18.21, DIN 6796) used under bolted joints
- bearings-taxonomy — axial preload from wave springs and Belleville stacks
- fatigue-analysis — S–N curves, Goodman / Soderberg lines, shot-peening, surface-finish effects critical to spring fatigue life
20. Citations
- Spring Manufacturers Institute, Encyclopedia of Spring Design, 4th edition, 2018.
- Shigley & Mischke, Mechanical Engineering Design, 11th ed., McGraw-Hill, 2020 — Chapter 10 (Mechanical Springs).
- ASTM A228 / A227 / A229 / A232 / A313 / A401 — wire specifications.
- AMS 5698 / 5699 — Inconel X-750 spring wire.
- DIN 2089 (cylindrical helical compression springs).
- DIN 2092 (calculation of cylindrical helical compression springs).
- DIN 2093 (disc / Belleville springs).
- DIN 6796 (conical spring washers for bolted connections).
- ASME B18.21 — Belleville lock-washer dimensions.
- Stabilus Lift-O-Mat technical catalog (gas-spring sizing).
- Smalley Crest-to-Crest Wave Spring engineering manual.