The fractional model considering geometric factor of viscoelastic damping systems is proposed by adopting fractional viscoelastic oscillator. To obtain dynamic responses of the fractional model, a numerical method is derived based on matrix function theory and Grumwald–Letnikov discrete form of fractional derivative. As a special engineering application example, the vibration response of the viscoelastic suspension installed in heavy crawler-type vehicles is studied through the proposed model. Furthermore, the parameter influence on the vibration control capability of the viscoelastic suspension is researched. The results indicate that the fractional viscoelastic oscillator is a favorable choice to characterize the dynamic behavior of viscoelastic damping structures. Additionally, the parameters in fractional viscoelastic oscillator namely geometric factor and fractional order exert considerable impact on the dynamic response of viscoelastic damping structures.