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Damped oscillations

Damping removes modeled mechanical energy by applying a force opposite velocity. The state still evolves through a force and integrator loop; decay is an outcome, not a visual envelope pasted over an undamped trajectory. The model must say where lost mechanical energy goes—an implicit bath, internal heating ledger, or a richer coupled system.

Think like a programmer

Add damping as a force function, not a visual amplitude multiplier. Keep position, velocity, mass, stiffness, damping coefficient, time step, integrator, and energy ledger separate. Calculate spring and damping force from the same state snapshot, then record velocity, mechanical energy, damping power, and cumulative transferred energy. Check a numerical trace against the sign of the power law, not against a pretty decaying curve.

Model checklist

Inputs
Mass m, stiffness k, viscous damping c, initial position/velocity, external-drive policy, time step/integrator, output times, and reservoir/loss convention.
State
Timestamped position, velocity, spring/damping forces, mechanical energy, damping power, and cumulative mechanical-to-reservoir transfer.
Rule
Evaluate Fspring = −kx and Fdamp = −cv from current state; sum forces, integrate acceleration, then update the energy/transfer ledger with declared timestamps.
Output
Trajectory, mechanical energy, damping-power trace, transfer residual, regime classification, and step-refinement comparison.
Check
With c = 0 and no drive, ideal mechanical energy is bounded under a suitable refined integrator; with c > 0, vFdamp = −cv² is nonpositive; zero velocity gives zero instantaneous viscous force; increasing c under fixed setup increases early loss rate; cumulative loss explains mechanical-energy decrease within numerical tolerance.
\[m\ddot{x}+c\dot{x}+kx=0\]

Oscillator state sampler

Change stiffness, damping, and displacement. The force is restoring; damping affects velocity once the system is moving.

Stiffness 20 N/m; damping 1; displacement 0.40 m; acceleration -8.00 m/s²; natural angular frequency 4.47 rad/s; energy 1.60 J.

The force sign is a compact energy test. Mechanical power from damping is Pdamp = Fdamp v = −cv², so it cannot add mechanical energy when c ≥ 0. At a release point with zero initial velocity, viscous damping is initially zero even though the spring force is not; the particle accelerates first, then damping grows with speed. A model that multiplies position by a shrinking factor cannot reproduce this timing or a power ledger.

\[m\ddot x+c\dot x+kx=0,\qquad P_{damp}=(-c\dot x)\dot x=-c\dot x^2\le0\]

The damping ratio ζ = c/(2√(mk)) organizes the ideal linear regimes: underdamped systems oscillate with decaying envelope, critical damping returns without oscillation as quickly as this model allows, and overdamped systems return more slowly without crossing equilibrium. It is a derived diagnostic, not a substitute for checking a numerical solution. Compare output at fixed physical times as the step shrinks; explicit updates can add energy or overdamp a system for numerical reasons.

Try this experiment

Prediction: Greater damping removes energy faster, while a zero-velocity initial state has zero instantaneous viscous loss.

Release the same displaced mass with several c values. At t = 0, record spring force, damping force, and damping power. Then compare mechanical-energy change plus cumulative loss at shared output times for Δt and Δt/2. Identify the regime from ζ and distinguish it from what a coarse plotted trace merely appears to do.

Where this model breaks

Linear viscous damping is only one loss model. Air drag can scale with speed squared, dry friction behaves differently near zero speed, and real damping can have memory, frequency dependence, temperature coupling, or nonlinear material loss. Treating lost mechanical energy as heat requires a reservoir model if temperature or entropy is an output.

Summary

Express damping as a velocity-dependent force and confirm its signed power and transfer ledger numerically instead of faking an envelope. State the loss model, regime, integrator, and refinement evidence.

Glossary

Self-check

  1. Which state value drives viscous damping?
  2. Why is damping power never positive for c ≥ 0?
  3. Why is zero initial velocity compatible with a later decaying amplitude?
  4. What ledger term explains the mechanical-energy decrease?

Sources

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]
Inputs
Quantities you set or measure, with units and useful bounds.
State
Values the program must retain to reproduce the next result.
Rule
The relationship or update that turns inputs and state into a result.
Check
A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Damped Oscillations, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Oscillations, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Damped Oscillations as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

  • Input: a measured value or chosen parameter supplied to a model.
  • State: the smallest set of values needed to continue or reproduce a model.
  • Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

  1. Which values are inputs, and which values must remain state?
  2. What observable result would tell you the model is behaving as expected?
  3. Which assumption would you test first before applying the model to a real system?

Model review: turn Damped Oscillations into a test

Add a velocity-opposing force to an oscillator and test its predicted mechanical-energy loss.

  1. Name the inputs and units that the oscillations model needs.
  2. Separate the state you must keep from values you can calculate when needed.
  3. Write one rule that maps the current state and inputs to an observable result.
  4. Choose a limiting case, unit check, invariant, or known result before trusting an output.
  5. State one assumption you would change before using this simplified model for a real decision.
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