This paper deals with the transient vibration of a rotating shaft with nonlinear spring characteristics during constant acceleration passing through a major critical speed. In a theoretical analysis, we solve the equation of motion numerically and discuss this phenomenon paying attention to nonlinear components represented by the polar coordinates. As a result, it is clarified that, when the angular acceleration λ increases, the maximum amplitude of the shaft decreases gradually for a small value of λ and extremely at a certain value of λ=λ0. This characteristic is different from that of a linear system, where the maximum amplitude decreases markedly in the range of small angular acceleration. We ascertained these phenomena in experiments. Generally speaking, it is more difficult to pass through a critical speed in a nonlinear system than in a linear system.