This paper studies the factors that affect the critical speed of a shock absorber to improve its performance in avoiding the serpentine movement of the vehicle. Particularly, the effects of the series stiffness and structural damping under different values of equivalent conicity are investigated. First, the Maxwell simplified model of the absorber is introduced. A rigid bogie with a shock absorber is created to simulate the operation of the vehicle. Then, the analytical expression of the linear critical speed of a rigid bogie is deduced based on the theory of the hunting motion stability. The calculation results show that there is a unique structural damping threshold for each series stiffness, and they are positively correlated. To obtain a desired critical speed, a small series stiffness and large structural damping is better for a small equivalent conicity, and vice versa.