Transient dynamics of centrifugal pendulum vibration absorbers, which are used for reducing torsional vibrations in rotating machines, are investigated using analysis and simulations of a dynamical model. These absorbers are being implemented in automotive engines to smooth vibrations and aid with fuel saving technologies, such as cylinder deactivation and torque converter lockup. In order for the absorbers to operate effectively with minimal mass, they must be designed to accommodate large amplitude, nonlinear responses, and in automotive engines they will experience a variety of transient environments. Here we consider the most severe transient environment, that of sudden activation near resonance, which leads to beating behavior of a nonlinear oscillator coupled to a driven rotor. An approximate method for predicting the percent overshoot of the beating transient response is derived, based on perturbation analysis of the system equations of motion. The main result is expressed in terms of the system and excitation parameters, and is found to accurately predict results from direct simulations of the model equations of motion. It is shown that absorbers with near-tautochronic paths behave much like linear absorbers, and when lightly damped and start from small initial conditions, they have an overshoot close to 100%. For absorbers with softening paths, such as the commonly used circular path absorbers, the overshoot can reach up to 173%, depending on system and input parameters, far exceeding predictions from linear analysis. These results provide a useful tool for design of absorbers to meet transient response specifications. In the following companion paper an experimental investigation is used to verify the analytical predictions.