= Solution
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 $I_{33}=mz^2$ nonzero. Its trace-free <mass quadrupole moment> is therefore
$$
Q_{ij}=mL^2\cos^2\omega t\;\operatorname{diag}\left(-\frac13,-\frac13,\frac23\right)_{ij}.
$$
The constant part does not radiate. Since $d^3(\cos^2\omega t)/dt^3=4\omega^3\sin2\omega t$, the <tensor contraction> is
$$
\dddot Q_{ij}\dddot Q_{ij}
=16m^2L^4\omega^6\sin^2(2\omega t)\left(\frac19+\frac19+\frac49\right)
=\frac{32}{3}m^2L^4\omega^6\sin^2(2\omega t).
$$
Using $\langle\sin^2(2\omega t)\rangle=1/2$ in the <quadrupole formula> gives \b[the averaged positive radiated power]
$$
\boxed{\left\langle\frac{dE_{\rm rad}}{dt}\right\rangle=\frac{16Gm^2L^4\omega^6}{15c^5}.}
$$
The <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>.
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