The phenomenon of vibrational resonance holds significant application value in weak signal detection and energy harvesting. However, existing studies predominantly focus on symmetric bistable systems or linear dissipation scenarios. To address this gap, this paper investigates vibrational resonance (VR) in an asymmetric bistable system with nonlinear dissipation,driven by biharmonic forces at two different frequencies. The fast and slow variable separation method is applied to derive the response amplitude of the system at low frequencies. Moreover,based on the analytical expression of the response amplitude,the effects of the dual-frequency periodic signal,the nonlinear dissipation and the asymmetric parameter on the VR are investigated. The results indicate that,first,the larger the nonlinear dissipation factor,the weaker the VR of the system occurs. Second,for the systems with nonlinear dissipative term,tuning the asymmetric parameter can change the shape of the VR,that is,the symmetric bistable systems emerge the double resonance,and the asymmetric bistable systems have the single resonance. Third,the asymmetric parameter does not influence the location of the VR,but it can weaken the response amplitude. Finally,the theoretical predictions are in good agreement with the numerical simulation results,verifying the validity of the theoretical analysis.