In a linear oscillator, the natural frequency does not depend on amplitude. If the stiffness parameter is \(\alpha\), then the small-amplitude angular frequency is
\[
\omega_0=\sqrt{\alpha}.
\]
That is why ideal simple harmonic motion has the same period no matter how large or small the oscillation is.
In a nonlinear oscillator, this is no longer true. A common model is the Duffing oscillator, which contains a cubic restoring term. For weak nonlinearity, perturbation theory shows that the oscillation frequency shifts with amplitude. A useful approximation is
\[
\omega_{\mathrm{eff}}=\omega_0\sqrt{1+\frac{3}{4}\frac{\beta A^2}{\alpha}},
\]
where \(A\) is the oscillation amplitude, \(\alpha\) is the linear stiffness parameter, and \(\beta\) measures the cubic nonlinearity.
This formula shows immediately that the frequency depends on amplitude through the factor \(A^2\). If \(\beta>0\), then the term inside the square root increases as amplitude increases, so the effective frequency becomes larger. This is called a hardening spring. If \(\beta<0\), then the effective frequency decreases as amplitude increases. This is called a softening spring.
For the sample case
\[
\alpha=1,\qquad \beta=0.2,\qquad A=1,
\]
the linear natural frequency is
\[
\omega_0=\sqrt{1}=1.
\]
The correction factor is
\[
1+\frac{3}{4}\frac{\beta A^2}{\alpha}
=
1+\frac{3}{4}\frac{0.2\cdot 1^2}{1}
=
1+0.15
=
1.15.
\]
Therefore
\[
\omega_{\mathrm{eff}}=\sqrt{1.15}\approx 1.072.
\]
So the nonlinear resonance frequency is slightly larger than the linear one. That is a typical hardening-spring result.
The difference
\[
\Delta\omega=\omega_{\mathrm{eff}}-\omega_0
\]
is called the frequency shift. Another useful quantity is the percent shift
\[
100\frac{\Delta\omega}{\omega_0}.
\]
These values help quantify how strongly the resonance curve bends away from the linear prediction.
The graph of effective frequency against amplitude is called the backbone curve. It is a central concept in nonlinear vibration theory because it shows the amplitude-frequency trend followed by resonance peaks. In hardening systems the backbone bends upward. In softening systems it bends downward.
This calculator uses the weakly nonlinear approximation, so it is most reliable when the cubic term is not too strong and the correction remains moderate. At more advanced level, one can derive the same result using multiple-scales analysis, averaging, or harmonic balance, and one can also study hysteresis, jump phenomena, and forced response curves.
Even so, this simple formula already captures the main physical message: in nonlinear oscillators, the resonance frequency is no longer fixed, but shifts with oscillation amplitude.
