Duffing Oscillator Laboratory

Why can the same forcing frequency produce different responses? A nonlinear spring can support more than one stable response to the same drive. Which one you get depends on how the motion started, and in a frequency sweep on whether you came from below or from above. Sweep up and then down, and the response jumps at different places, leaving a band where either response can appear.

A nonlinear spring can sustain two amplitudes under the same drive. Explore hysteresis, phase portraits and motion that repeats every second drive cycle.

Displacement $0$

Velocity $0$

Last eight-cycle fundamental $0$

Retained strobe repetition $0$

Physical time $0$

Mechanical energy $0$

Signed input work $0$

Viscous dissipation $0$

Energy balance error $0$

Half-step displacement change · first four cycles $0$

Integration step $0$

Spring force $0$

Damping force $0$

Additive force $0$

Record boundary $0$

Last eight-cycle nonharmonic RMS $0$

Last eight-cycle mean $0$

Centered single-harmonic approximation $0$

Retained strobes $0$

Absolute displacement limit $0$

Same drive, different motion

Force arrowsArrows share a fixed force scale in each trial.Cubic stiffness

\alpha

160.00\,\mathrm{N\,m^{-3}}

Drive amplitude

F_0

0.60\,\mathrm{N}

Drive / reference frequency

\omega/\omega_0

1.30\,

Linear stiffness sign

\operatorname{sgn}k

1.00\,

Discard first strobe cycles

N_d

88.00\,

Sweep lower bound

r_{\min}

0.80\,

Sweep upper bound

r_{\max}

1.80\,

Damping ratio

\zeta

0.03\,

Release displacement

x_0

0.00\,\mathrm{m}

Release velocity

v_0

0.00\,\mathrm{m\,s^{-1}}

Mass

m

1.00\,\mathrm{kg}

Simulated cycles

N

120.00\,

Maximum step

\Delta t_{\max}

10.00\,\mathrm{ms}

Position is measured rightward in metres from the spring-law origin, time in seconds, angular frequency in radians per second, force in newtons, mass in kilograms, and cubic stiffness in newtons per cubic metre. Linear stiffness has magnitude sixteen newtons per metre and a selectable sign. The reference frequency uses its magnitude. The full cubic force law, viscous damping and cosine forcing are integrated; collisions and wire deformation are excluded. Positive linear and negative cubic stiffness define only a local well; the record stops before its saddle or the two-metre displacement limit. A complete finite simulation is loaded on entry; playback and the last-eight-cycle harmonic measurement have separate time meanings. The harmonic residual includes higher harmonics and transients, not measurement uncertainty. Centered single-harmonic roots are approximate and have no stability classification. When shown, the amber carriage is a separate full simulation with identical parameters and forcing phase, released at the largest approximate harmonic amplitude with zero velocity. Its sustained response is measured from that simulation. Retained strobes are sampled at whole drive periods after the selected discard. The last twenty-four strobes are tested for repetition after one to eight periods at a relative phase-state tolerance of one part in ten thousand, with at least sixteen samples; this finite test does not prove an attractor or chaos. Each sweep point runs forty-eight drive cycles, discards forty, measures eight and passes its endpoint to the next point at the same forcing phase. Finite directional differences are not an exact bifurcation diagram. Work and viscous heat are integrated separately. Both carriages share one fixed length scale. The coil shape, actuator, chosen rest length and carriage sizes are schematic. Rails balance gravity. Net-force and velocity arrows use separate fixed scales, with tails lifted above the mass for visibility. Dragging prepares a new release and retains the selected initial velocity; sliders provide equivalent keyboard input.

A spring that remembers the sweep direction

BackgroundA cubic stiffness term makes the frequency depend on the amplitude. With damping and forcing, the same settings can settle into different bounded responses, depending on the history.

Why it mattersBrown University’s Duffing tutorial and York University’s computational physics notes cover cubic restoring forces, phase portraits and forcing. This Lab adds an explicit harmonic projection of the simulated motion, escape from a local well, and a finite frequency sweep in which each run starts from the end state of the previous one.

Focus question: When sweeping up and sweeping down give different results, is that a real stable branch or an effect of the finite recording?
One-sentence intuition: Measure the fundamental, the mean and the nonharmonic residual separately, and compare sweeps that carry the state forward at the same drive phase.

Core mathematical model

Spring and damping forces, and the wells without forcing

F_s=-kx-\alpha x^3,\quad F_d=-c\dot x,\quad x_*=\pm\sqrt{-k/\alpha}\quad(k<0,\ \alpha>0)

These are the unforced potential minima. A driven mass can move between the wells; the separate potential plot shows their shape.

Equation of motion with a cubic spring

m\ddot x+c\dot x+kx+\alpha x^3=F_0\cos\omega t

The sign of the linear stiffness and the cubic stiffness together decide whether the spring hardens, softens locally or forms a double well.

Potential and energy

U(x)=\tfrac12kx^2+\tfrac14\alpha x^4,\quad E=\tfrac12m\dot x^2+U(x)

The model uses the full potential, not its quadratic approximation near a minimum.

Single-harmonic amplitude relation

A^2\left[\left(k-m\omega^2+\tfrac34\alpha A^2\right)^2+(c\omega)^2\right]=F_0^2

This approximation assumes motion centered on zero and ignores the mean and higher harmonics. It does not say which solutions are stable.

Saddle of the locally softening spring

x_b=\sqrt{-k/\alpha}\quad(k>0,\ \alpha<0)

Outside this local well the potential falls without limit, so recording stops at the saddle.

Work and heat integrated separately

\dot W=F_0\cos(\omega t)\dot x,\quad \dot Q=c\dot x^2,\quad E-E_0=W-Q

The drive work, which can be positive or negative, and the heat, which is nonnegative, are integrated along with the motion.

Common difficulties

An approximate solution is not necessarily an attractor

Typical misconceptionEvery solution of the harmonic-balance equation is a stable response you will measure.

Better mental modelThe approximation drops higher harmonics. A simulated release lands in one particular basin and still carries transients.

The memory effect depends on the sweep settings

Typical misconceptionA gap between sweep directions proves an adiabatic bifurcation.

Better mental modelEach point uses only 48 cycles, so the frequency spacing and the settling time can change the result.

Irregular points need stronger evidence

Typical misconceptionA scattered Poincaré record proves chaos.

Better mental modelA finite record that may still contain transients cannot, by sampling alone, establish a positive long-term exponent.

Run the experiment

01
Check the linear limit
Choose Linear resonance and compare the loaded harmonic amplitude with the linear formula. Open Potential, work and dissipation to inspect work and heat.
What to observe: With the cubic term at zero, the response settles onto the linear result, which is known independently.
02
Measure the effect of sweep direction
Choose Two sustained amplitudes. Compare the independent carriages and the automatically calculated upward and downward sweeps.
What to observe: Each run starts from the end state of the previous one, at the same drive phase, so the history carries over.
03
Compare an open well with a closed one
Compare Softening spring with Stay in one well, then drag the carriage to change its release.
What to observe: The softening spring has an edge the cart can escape over, while the quartic double well keeps it confined.
04
Record a finite section
Choose Alternate every cycle. The retained strobes form two groups; compare this finite repetition with Across both wells and its residuals.
What to observe: The mean, the fundamental and the retained fixed-phase points each describe a different part of the signal.