We choice $alpha=bar{alpha}+deltaalpha$ to linearize, and then we get
$$begin{aligned}
-iomegadeltaalpha[omega]&=(iDelta-frac{kappa}{2})deltaalpha[omega]+iGbar{alpha}x[omega]\
-m_{eff}omega^2x[omega]&=-m_{eff}Omega_m^2x[omega]+iomega m_{eff}Gamma_m x[omega]\
&+hbar G(bar{alpha}^{dagger}deltaalpha[omega]+bar{alpha}deltaalpha[-omega]^{dagger})
end{aligned}$$

From equation one, we get
$$begin{aligned}
deltaalpha[omega]&=iGbar{alpha}chi_{opt}[omega]x[omega]\
deltaalpha[-omega]^{dagger}&=-iGbar{alpha}^{dagger}chi_{opt}[-omega]^{dagger}x[-omega]^{dagger}
end{aligned}$$

Insert them back to equation two, we get the modified mechanical susceptibility
$$begin{aligned}
chi^{-1}_{m,eff}(omega)&=m_{eff}((Omega_m^2-omega^2)-iomegaGamma_m)\
&+hbar G^2|bar{alpha}|^2(frac{1}{(Delta+omega)+ikappa/2}+frac{1}{(Delta-omega)-ikappa/2})\
&=chi_m^{-1}(omega)+Sigma(omega)
end{aligned}$$
By using
$$hbar G^2|bar{alpha}|^2=hbar g_0^2frac{2m_{eff}Omega_m}{hbar}n_{cav}=2m_{eff}Omega_mg^2$$
we get
$$Sigma(omega)=2m_{eff}Omega_mg^2(frac{1}{(Delta+omega)+ikappa/2}+frac{1}{(Delta-omega)-ikappa/2})$$
From the equation above we can get the effective mechanical damping rate
$$begin{aligned}
Gamma_{eff}&=Gamma_m+Gamma_{opt}\
&=Gamma_mleft(1+frac{Omega_m}{omega}left(frac{1}{(frac{Delta+omega}{g})^2frac{Gamma_m}{kappa}+frac{1}{C}}-frac{1}{(frac{Delta-omega}{g})^2frac{Gamma_m}{kappa}+frac{1}{C}}right)right)
end{aligned}$$
Now in weak laser drive, then $gllkappa$, we choice $omega=Omega_m$, then we get
$$Gamma_{eff}=Gamma_mleft(1+left(frac{1}{(frac{Delta+Omega_m}{g})^2frac{Gamma_m}{kappa}+frac{1}{C}}-frac{1}{(frac{Delta-Omega_m}{g})^2frac{Gamma_m}{kappa}+frac{1}{C}}right)right)$$
In the blue detuning(red detuning), $Delta=pmOmega_m$, then
$$Gamma_{eff}|_{Delta=pmOmega_m}=Gamma_mleft(1mp Cleft(1-frac{1}{frac{16Omega_m^2}{kappa^2}+1}right)right)$$
In the resolved-sideband regime, $kappallOmega_m$, we can get two conclusions below

1. In red detuning, $Gamma_{eff}=Gamma_m(1+C)$, the mechanic mode line width will increase
2. In blue detuning, $Gamma_{eff}=Gamma_m(1-C)$, the mechanic mode line width will decrease. Especially, when C is greater than 1, then we will get an instability.