Parameters
Natural frequency ω₀ [rad/s]1.00
Damping ratio ζ0.050
Nonlinear coefficient ε+0.30
Excitation amplitude F0.20
Freq ratio range Ω/ω₀0.1–2.5
Duffing equation:
$\ddot{x} + 2\zeta\omega_0\dot{x} + \omega_0^2 x + \varepsilon x^3 = F\cos(\Omega t)$
$\ddot{x} + 2\zeta\omega_0\dot{x} + \omega_0^2 x + \varepsilon x^3 = F\cos(\Omega t)$
—
Peak amplitude X_max
—
Peak freq ratio Ω/ω₀
—
Jump freq ratio
—
Spring type
▲ Nonlinear Frequency Response Curve (FRC) and Backbone Curve
▲ Nonlinear Potential V(x) = ½ω₀²x² + ¼εx⁴
Theory — Harmonic Balance Method
First-order amplitude equation ($x \approx A\cos\Omega t$):
$$\left[(\omega_0^2 + \tfrac{3}{4}\varepsilon A^2 - \Omega^2)^2 + (2\zeta\omega_0\Omega)^2\right]A^2 = F^2$$Backbone curve (undamped free vibration):
$$\Omega_{bb} = \omega_0\sqrt{1 + \frac{3\varepsilon A^2}{4\omega_0^2}}$$Jump phenomenon occurs at fold bifurcation points of the three-valued response region.
CAE Note: Geometric nonlinearity in FEM (large displacement) naturally produces Duffing-type stiffening. ANSYS uses Full Newton-Raphson + Newmark-β; LS-DYNA uses explicit integration. Nonlinear springs: COMBIN39 (ANSYS), MAT_SPRING_NONLINEAR_ELASTIC (LS-DYNA), SPRING1 with nonlinear force-displacement (ABAQUS).