Most centrifugal pendulum vibration absorber (CPVA) research focuses on the horizontal or vertical plane, ignoring the influence of gravity. However, with the wide application of CPVAs in the automobile industry, some gravity-related problems have been encountered in practice. In this study, employing the second kind of Lagrange equation, the differential equation of motion of a CPVA is established, and the first-order approximate analytical solution is solved using the method of multiple scales. The mathematical relations among the excitation torque amplitude and phase, gravity influence, absorber trajectory shape, absorber position, viscous damping coefficient, and mistuning level parameters are provided for study. Specifically, the second-order responses of four absorbers and two absorbers in a gravity field are studied, and the influence of the change in the torque excitation phase on the response of the absorber is thoroughly analyzed. The grouping phenomenon in four absorber applications is found during the simulation, and a mathematical explanation is provided. The influence of the absorber viscous damping coefficient on the amplitude and the rotor angular velocity is discussed. To maximize the performance of the CPVA, a damping quality factor Q is defined to compare different end stop positions, and a design principle for the end stop is finally determined. From a mathematical point of view, this explains why a CPVA affects the angular acceleration of the rotor, and the influence of the design parameters on the synchronization and stability of the CPVA is discussed. Using the CPVA differential equation of motion considering gravity, a MATLAB Simulink model is created to simulate the influence of the torque excitation phase on CPVA performance. Especially, one set of CPVA is tested on dynamometer and vehicle, and the test results were compared with simulation results.