Coulomb damping¶

Model mechanical energy loss by an approximately constant-magnitude dry-friction force opposing sliding velocity and producing piecewise motion with possible sticking.

Version: v1 · 2026-08-30 · History
Domain-specific #: 1570
Origin domain: mechanical engineering
Subdomain: dry friction vibration
Aliases: Dry-friction damping, Constant-friction damping

Core Idea¶

Coulomb damping is vibration attenuation produced by sliding dry friction idealized as a force of nearly constant magnitude \(F_c=\mu N\) directed opposite instantaneous velocity. For a single-degree-of-freedom oscillator the moving phases may be written \(m\ddot x+kx=-F_c\,\operatorname{sgn}(\dot x)\), with separate conditions at zero velocity. The force law is discontinuous and motion is solved piecewise between reversals.^[1]

During each sliding half-cycle, friction removes work equal to its magnitude times distance traveled. Under the ideal constant-force, linear-spring model, successive peak amplitudes decrease by an approximately constant amount rather than the exponential envelope characteristic of viscous damping. As energy falls, static friction can hold the body before the spring returns it to nominal equilibrium; the final rest position can therefore lie in a sticking band.^[2]

The Coulomb model is an idealization, not a universal friction law. Breakaway force, velocity dependence, surface state, normal-load changes, Stribeck behavior, and microslip can invalidate constant magnitude. Equivalent viscous damping may match energy loss at one amplitude and frequency but does not preserve the piecewise waveform or constant decrement. The account remains analytical and descriptive rather than instructions for manipulating machinery.^[3]

Structural Signature¶

Sliding interface. Contact between surfaces supplies dry-friction energy loss.
Normal load. Contact loading scales the ideal friction magnitude.
Opposing force. Force direction reverses with sliding velocity.
Restoring dynamics. Elastic and inertial terms produce oscillatory reversals.
Piecewise phases. A fixed force sign applies between consecutive zero-velocity events.
Energy dissipation. Frictional work reduces mechanical energy over each traveled segment.
Amplitude decrement. Ideal free vibration loses approximately equal peak amplitude per cycle.
Sticking condition. Static friction can terminate motion inside a force-balance interval.

What It Is Not¶

Not viscous damping. Viscous force scales with velocity and usually yields an exponential envelope.
Not hysteretic damping. Material hysteresis follows a force-displacement loop rather than simple sliding sign reversal.
Not all friction. Real contacts can depend on velocity, temperature, wear, and state.
Not a smooth linear model. The sign change and zero-velocity set make dynamics piecewise and nonsmooth.
Not constant decay in time. The ideal result concerns peak decrements over cycles.
Not guaranteed return to zero. Static friction can leave an offset final position.

Scope of Application¶

The abstraction is literal wherever practitioners can identify the same constitutive roles, apply the same boundary tests, and obtain the same kind of output. The following habitats are uses of Coulomb damping itself, not metaphors based only on resemblance.

Vibration analysis. Predicting free-decay envelopes under sliding friction.
Mechanical joints. Approximating dissipation where interfaces undergo gross slip.
Isolation devices. Comparing frictional and viscous attenuation models.
Parameter estimation. Inferring effective friction magnitude from peak decrement.
Simulation. Handling sign changes and stick conditions with event-aware dynamics.
Model validation. Checking amplitude and velocity dependence against observed decay.

Clarity¶

A clear account of Coulomb damping must preserve the recognition invariant stated in the Core Idea rather than rely on the title alone. State whether static and kinetic friction are distinguished. Declare the normal-load and constant-magnitude assumptions. Solve between velocity reversals rather than smoothing the sign change silently. Test the predicted constant peak decrement across more than one amplitude range. These declarations are not editorial extras: each changes what observations count, which transformations are licensed, and what conclusion can be drawn. A reader should be able to reconstruct the input, the operative rule, the output, and at least one defeater from the account without consulting an implementation or guessing an unstated convention.

Manages Complexity¶

Coulomb damping manages complexity by replacing a diffuse field of observations or possible operations with a bounded role structure: sliding interface supplies contact between surfaces supplies dry-friction energy loss.; normal load supplies contact loading scales the ideal friction magnitude.; opposing force supplies force direction reverses with sliding velocity.; restoring dynamics supplies elastic and inertial terms produce oscillatory reversals.; piecewise phases supplies a fixed force sign applies between consecutive zero-velocity events.. The compression is useful because it localizes disagreement. One can ask whether the input was properly formed, whether a constitutive relation held, whether an alternative explanation defeats the inference, or whether the output was overinterpreted. The same compression can mislead when its discarded detail is exactly what the decision requires. A reference-grade use therefore reports both the invariant retained and the information intentionally lost.

Abstract Reasoning¶

Draw the free-body model and identify the sliding interface.
Estimate friction magnitude and direction for each moving phase.
Integrate the linear dynamics until the next zero-velocity event.
Update force sign or test the static-friction sticking condition.
Track frictional work and successive peak amplitudes.
Compare predictions against viscous and state-dependent friction alternatives.
Retain the law only over the range where its diagnostics hold.
Test the candidate interpretation against the nearest named confusable rather than accepting a shared surface feature.
State the conclusion at the same scope as the source conditions, and retain uncertainty or nonuniqueness where the construct does not remove it.

Knowledge Transfer¶

The strict upward abstraction is Damping. Coulomb Damping instantiates Damping because dry-friction work irreversibly removes mechanical energy from motion. Within dry friction vibration, the full mechanism transfers literally when the same roles and boundary tests recur. Beyond that domain, only the parent-level skeleton should travel. Reusing the label Coulomb damping after removing its constitutive vocabulary would hide a change of mechanism behind an analogy. The honest transfer rule is therefore two-stage: recognize the domain-specific pattern first, then lift only the parent relation that remains invariant under a substrate change.

Examples¶

Canonical¶

A mass attached to a linear spring slides on a dry surface. If ideal friction magnitude is \(F_c\), energy balance gives peak reduction of approximately \(4F_c/k\) per full cycle while sliding. When spring force at a reversal no longer exceeds static friction, the mass sticks rather than completing another symmetric cycle.

Mapped back: input and conventions → constitutive role test → bounded output → explicit interpretation and defeater check.

Applied / In Practice¶

Measured peaks fall by nearly equal displacements for several cycles but depart at small amplitude. An analyst fits Coulomb damping only to the gross-slip interval, reports normal-load uncertainty, and uses a separate sticking or microslip description near rest. One viscous ratio would hide waveform and termination behavior.

Mapped back: field observation or problem → candidate recognition → confusable and limit checks → appropriately scoped conclusion.

Structural Tensions¶

T1: Simple force law versus real friction. Surface state and velocity can change force magnitude. Diagnostic: Plot inferred friction against velocity and amplitude.
T2: Sliding solution versus sticking. The sign law alone is undefined at rest. Diagnostic: Test static-friction balance at every reversal.
T3: Constant decrement versus noisy peaks. Measurement and nonlinear stiffness can obscure the envelope. Diagnostic: Fit multiple cycles and compare rival residual patterns.
T4: Energy equivalence versus dynamic equivalence. Equivalent viscous damping can match loss at one point. Diagnostic: Compare phase, waveform, and scaling away from calibration.
T5: Contact damping versus joint motion. A nominal joint may microslip before gross sliding. Diagnostic: Establish the actual interface regime.
T6: Autonomy versus generic damping. Damping removes motion energy, while Coulomb damping fixes opposing constant dry friction. Diagnostic: Replace the force with velocity-proportional loss and test the signatures.

Structural–Framed Character¶

The opposing-force, work-loss, and piecewise-reversal structure is invariant; coefficients and the validity range are empirically framed. The five framing criteria point in a consistent direction. Evaluative weight is limited to whether the defining conditions are met, not whether the outcome is desirable. Human practice matters to the extent that experts choose conventions, instruments, or reporting thresholds, but those choices do not make every verdict arbitrary. Institutional history explains the name and standard use; it does not replace the recognition rule. The operative vocabulary travels within the home field and closely adjacent subfields, while transfer farther away requires translation to the parent prime. Thus recognition remains disciplined even where interpretation is defeasible.

Structural Core vs. Domain Accent¶

What is skeletal. Coulomb Damping instantiates Damping because dry-friction work irreversibly removes mechanical energy from motion. This is the part that can be expressed without the candidate's specialist nouns.

What is domain-bound. The domain accent consists of dry contact, friction coefficient, normal force, oscillators, reversal events, energy envelopes, and stick-slip behavior. Remove those elements and the result is no longer Coulomb damping; it is only the parent relation or a loose analogy.

Why this does not clear the prime bar. The name does not recur with unchanged diagnostics across three independent domains. What transfers is already represented by prime:damping. The candidate remains autonomous because its in-domain recognition rule, failure modes, and consequences are stable, but its vocabulary and interventions do not float free of the home substrate.

Coulomb Damping instantiates Damping because dry-friction work irreversibly removes mechanical energy from motion.

The prospective workspace queue contains one strict upward edge to prime:damping. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Coulomb damping Domain-specific

Parents (1) — more general patterns this builds on

Coulomb damping is a kind of Damping Prime

Coulomb Damping instantiates Damping because dry-friction work irreversibly removes mechanical energy from motion.

Hierarchy paths (2) — routes to 2 parentless roots

Coulomb damping → Damping → Dissipation → Irreversibility → Reversibility and Irreversibility

Show alternative path (1)

Neighborhood in Abstraction Space¶

Coulomb damping sits in a sparse region of the domain-specific corpus (99^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Traction (mechanics) — 0.78
Stoneley wave — 0.75
Fault — 0.74
Law of the wall — 0.74
Verlet Integration — 0.73

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Viscous damping. Uses velocity-proportional force and exponential free decay.
Structural damping. Represents material or joint hysteresis through loss factors.
Coulomb friction. The broader friction law need not occur in an oscillatory damping role.
Stick-slip. Alternation between static and kinetic regimes rather than the whole damping model.
Equivalent damping ratio. An amplitude-dependent surrogate, not literal force-law identity.
Rolling resistance. A different contact-loss mechanism.

References¶

[1] Inman, D. J. (2022). Engineering Vibration, 5^th ed. Pearson. ISBN 978-0-13-680953-1. registry ↩

[2] Rao, S. S. (2017). Mechanical Vibrations, 6^th ed. Pearson. ISBN 978-0-13-436130-7. registry ↩

[3] Den Hartog, J. P. (1985). Mechanical Vibrations, Dover reprint. ISBN 978-0-486-64785-2. registry ↩