M069 · Vibration design

Duffing Oscillator Laboratory

A cubic spring and force actuator create hardening, local softening and double-well experiments. Drag the cart to choose a release basin, acquire phase portraits and fixed-phase samples, compare the measured fundamental with a single-harmonic approximation, and run both directions of a state-carrying frequency sweep.

Acquired time $0\,\mathrm{s}$

Primary displacement $0\,\mathrm{m}$

Primary velocity $0\,\mathrm{m\,s^{-1}}$

Mechanical energy $0\,\mathrm{J}$

Integrated signed actuator work $0\,\mathrm{J}$

Independent viscous dissipation $0\,\mathrm{J}$

Independent energy ledger defect $0\,\mathrm{J}$

Position change with half step · first four cycles $0\,\mathrm{m}$

Actual integration step $0\,\mathrm{ms}$

Model domain / record status $\text{Ready}$

Retained eight-cycle fundamental amplitude $0$

Nonharmonic residual RMS $0$

Acquired mean displacement $0$

Approximate centered harmonic amplitudes $0$

Acquired retained strobes $0$

Continuation frequency records $0$

Continuation status $0$

Absolute displacement domain limit $0$

DESIGN / ACQUIRE / AUDIT

Why can the same forcing frequency produce different responses?

Moving mass

m

1.00\,\mathrm{kg}

Linear stiffness sign

\operatorname{sgn}k

1.00\,

Cubic stiffness

\alpha

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

Reference damping ratio

\zeta

0.03\,

Drive force amplitude

F_0

0.60\,\mathrm{N}

Drive to reference rate

\omega/\omega_0

1.30\,

Release displacement

x_0

0.00\,\mathrm{m}

Initial velocity

v_0

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

Recorded drive cycles

N

80.00\,

Discard initial strobe cycles

N_d

48.00\,

Sweep lower frequency ratio

r_{min}

0.80\,

Sweep upper frequency ratio

r_{max}

1.80\,

Maximum integration step

\Delta t_{\max}

10.00\,\mathrm{ms}

Inspect acquired record

t/t_{\mathrm{end}}

0.00\,\%

Drag the mass or use the keyboard ranges. Acquire the full record before comparing late-cycle measurements. Changing the experiment cancels incompatible scans.

The linear stiffness magnitude is fixed at sixteen newtons per metre, with a selectable sign; the reference rate uses its magnitude. The cubic term is the sole nonlinearity. A negative cubic spring with positive linear stiffness is a local well with an unbounded exterior; recording stops at its saddle, or at the two-metre display domain, without clipping. Eight-cycle projections remove the mean; residuals include higher harmonics and transients and are not uncertainty estimates. Retained strobes alone do not prove chaos. Continuation uses 48 cycles per frequency, discards the first 40 and carries the endpoint into the next integer-cycle run. Its finite direction dependence does not certify all attractors or an adiabatic bifurcation. Single-harmonic roots are approximate and have no assigned stability.

Start with the essentials

Focus question: Is a sweep-direction difference a stable branch or a finite acquisition effect?
One-sentence intuition: Measure the fundamental, its mean and its nonharmonic residual separately; compare state-carrying sweeps at identical drive phase.

Core mathematical model

Cubic spring dynamics

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

The linear sign and cubic stiffness distinguish hardening, local softening and double wells.

Potential and energy

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

The potential is not replaced by its local quadratic approximation.

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 centered approximation neglects the mean and higher harmonics; root stability is not assigned.

Local softening saddle

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

The polynomial potential is unbounded outside this local well; recording stops at the saddle.

Independent power ledger

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

The signed drive work and positive heat are integrated alongside motion.

Common difficulties

A root is not an attractor certificate

Typical misconceptionEvery harmonic-balance root is a stable measured response.

Better mental modelThe approximation drops harmonics; numerical releases select basins and retain transients.

History has a finite protocol

Typical misconceptionA direction gap proves an adiabatic bifurcation.

Better mental modelEach point uses only 48 cycles; sweep spacing and settling can affect the result.

Irregular points need stronger evidence

Typical misconceptionA scattered Poincaré record proves chaos.

Better mental modelFinite transient records and sampling alone do not establish a positive asymptotic exponent.

Run the experiment

01
Check the linear limit
Acquire the linear benchmark and compare harmonic amplitude.
What to observe: The nonlinear model converges to the independently known linear response.
02
Measure direction dependence
Choose hardening and run the full up/down sweep.
What to observe: The endpoint carries history into the next run at a matching force phase.
03
Inspect the potential boundary
Compare local softening with a double-well release.
What to observe: An escape boundary differs from a confining quartic well.
04
Acquire a finite section
Use the irregular preset, acquire the full record and inspect residuals.
What to observe: Mean, fundamental and retained fixed-phase points describe different parts of the signal.