Real-time evaluation of soil compaction quality is crucial for intelligent compaction. To evaluate compaction quality scientifically

the dry density under impact compaction was previously derived by coupling the compaction envelope equation with the equation of motion. The same modeling approach can be applied to vibratory compaction. However

the operating principles of vibratory compaction and impact compaction are different. These differences yield distinct compaction envelope equations for soils under the two conditions. Therefore

this study focuses on deriving the compaction envelope equations for vibratory compaction. By observing the soil’s compressive deformation during vibration compaction and integrating this with theoretical analysis

we establish the compaction envelope equation in a double-logarithmic coordinate system. From the envelope asymptote

we obtain the relationship between peak impact stress and the corresponding pore ratio. Based on the equation of motion for the vibrating steel wheel

the relationship equation between the peak vibration acceleration and the peak impact stress is established. By coupling the above two equations

a real-time equation to predict the dry density of fill soil under vibratory compaction is deduced. Considering the existence of both vertical and lateral deformation on the soil surface during compaction

the resulting real-time dry density equation is approximate. The calculation deviation caused by the difference between the actual constraint conditions and the ideal conditions can be reflected by adjusting the parameters. Finally

the newly proposed real-time dry density formula was applied to the vibratory compaction test to predict the dry density of the soil. The results demonstrate that the newly proposed method for evaluating the quality of fill compaction can accurately predict the dry density of soil under vibratory compaction.