<p>The purpose of this research is the prediction of the natural frequencies of nanospheres radial vibrations. An innovative analytical method, based strain theory, is introduced to explore the influence of small-scale effects on the radial vibrations of nanospheres. To address the scale effects, three second-gradient models were devised, incorporating variations with negative and positive signs, as well as an inertial gradient. The derived natural frequency equations extend the classical continuum model initially proposed by Lamb. This study highlights the impact of the gradient model on the vibrational behavior of nanospheres. The key numerical findings indicate that the second-gradient model with a negative sign is physically unrealistic, whereas the model with a positive sign, though more realistic, exhibits instability. To address this instability, a second gradient model with inertia gradient is proposed. Moreover, the derived frequency equations are essential for analyzing the impact of scale effects on the natural frequencies of nanospheres radial. Ultimately, the findings play a vital role in interpreting experimental Raman spectra.</p>