Valveless pumping assists in fluid transport in various biomedical and engineering systems. Here we focus on one factor that has often been overlooked in previous studies of valveless pumping, namely the impact that a compression actuator exerts upon the pliant part of the system when they collide. In particular, a fluid-filled closed-loop system is considered, which consists of two distensible reservoirs connected by two rigid tubes, with one of the reservoirs compressed by an actuator at a prescribed frequency. A lumped-parameter model with constant coefficients accounting for mass and momentum balance in the system is constructed. Based upon such a model, a mean flow in the fluid loop can only be produced by system asymmetry and the nonlinear effects associated with actuator impact. Through asymptotic and numerical solutions of the model, a systematic parameter study is carried out, thereby revealing the rich and complex system dynamics that strongly depends upon the driving frequency of the actuator and other geometrical and material properties of the system. In particular, a number of critical frequencies that characterize the interactions between the actuator and the system are calculated asymptotically. Guided by such critical frequencies, the numerical results are categorized into different types of dynamical responses, and the parameter regions for their existence are systematically determined. Moreover, the transition of different system responses are observed through critical phase (which corresponding to the moment when different system responses occur) tracking.