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• The 13 DOFs vehicle dynamic model with semi-active HDS is built. • The experiment results show that the acceleration of the seat rail in all directions are reduced. • The active side acceleration of torque strut calculated is basically consistent with the measured value. • The 13 vehicle dynamic model and the evaluation indexes are feasible in this study. • The semi-active HDS introduced in this paper can improve the vehicle NVH to a great extent. On the basis of previous research, this paper takes a more comprehensive study on the vibration of a vehicle with automatic transmission when engine start-stop and automatic start-stop. A 13 degrees of freedom (DOFs) vehicle dynamic model is established, and the dynamic response evaluation method is put forward. Three methods to reduce the vibration of powertrain and vehicle are analyzed, and the results show that increasing the damping of the mounting system can effectively reduce the vibration in these three methods. A novel semi-active hydraulic damping strut (HDS) as the fourth mount is designed and developed, the installation position and its outer channel size are optimized and calculated respectively based on its damping force calculation method and the design method of the powertrain mounting system (PMS). The calculation results of evaluation indexes with engine start-stop are analyzed in-depth. The optimized parameters of the semi-active HDS are taken into the 13 DOFs dynamic model to analyze the dynamic response of the engine automatic start-stop process, and the values of the evaluation indexes are decreased. To validate the effectiveness of the design method of the semi-active HDS, the experiment of the vehicle with and without the semi-active HDS are carried out, and the test results show that the seat rail acceleration are reduced with semi-active HDS introduced. The active side acceleration of the torque strut calculated by the 13 DOFs model is consistent well with the measured value with and without the semi-active HDS, which proves that the model is reasonable. [ABSTRACT FROM AUTHOR]