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For conservative forcing in averaged oscillator equations, $f=f(y)$, the average determining $R_T$ vanishes. If $F^{\prime }=f$, thenwhose period average is zero. Thus $R=R_0$ to this order, while a position-dependent force generally shifts the oscillation phase and frequency. The exact equation also conserves the energy $\frac{1}{2}{\dot{y}}^2+\frac{1}{2}{y}^2-\epsilon F(y)$, consistent with the absence of amplitude drift for bounded orbits.

For the cubic position force, $⟨f\cos \varphi ⟩=R^3⟨{\cos }^4\varphi ⟩=3R^3/8$. ThereforeThe leading oscillation is $R_0\cos [(1-3\epsilon R_0^2/8)t+{\theta }_0]$. For positive $\epsilon$ it is a softening cubic oscillator, obtained by weakly perturbing a harmonic oscillator. The weak-force condition is $\epsilon R_0^2\ll 1$; the constant $R$ is the leading harmonic amplitude, not a claim that the exact waveform remains sinusoidal.

Introduce the slow time $T=\epsilon t$ and treat $t,T$ independently in the method of multiple scales. WriteAt the next order,For fixed $R$ define the period average byHere $f$ inside the averages is evaluated at the leading position and velocity $R\cos \varphi ,-R\sin \varphi$. Orthogonality to both fundamental harmonics is the solvability condition in the method of multiple scales: it removes the resonant forcing that would otherwise generate a secular term. Since the squared sine and cosine averages are $1/2$, the amplitude-phase equations areEquivalently $\dot{R}=-\epsilon ⟨f\sin (t+\theta )⟩$ and $\dot{\theta }=-\epsilon ⟨f\cos (t+\theta )⟩/R$ at first order. These equations describe a bounded, weakly perturbed oscillation over $t=O({\epsilon }^{-1})$ while the amplitude remains in a range where the expansion is ordered. At zero amplitude the oscillation phase coordinate is singular; Cartesian harmonic coefficients or an equilibrium analysis should replace it.

Take the oscillation to be $z(t)=L\cos \omega t$; a different initial phase does not affect a period average. In the mass-density normalization of the quadrupole formula, the second mass moment tensor of the point mass has only $I_{33}=m{z}^2$ nonzero. Its trace-free mass quadrupole moment is thereforeThe constant part does not radiate. Since $d^3({\cos }^2\omega t)/d{t}^3=4{\omega }^3\sin 2\omega t$, the tensor contraction isUsing $⟨{\sin }^2(2\omega t)⟩=1/2$ in the quadrupole formula gives the averaged positive radiated powerThe mass quadrupole moment oscillates at twice the source's angular frequency, and the source energy loss has the opposite sign. The expression uses the leading slow-motion weak-field approximation, $\omega L\ll c$. With SI stress-energy tensor components, the mass-density integrand is $T_{00}/c^2$; the displayed $c^{-5}$ quadrupole formula uses that mass normalization. This is the point-mass contribution requested by the model: an apparatus maintaining an accelerating mass would also contribute its own mass quadrupole moment.