Abstract: The vibro-impact dynamics due to loose constraints have become one of the key scientific problems in the dynamical system of pipes conveying fluid. The impact force modeled by smoothed nonlinear springs varies continuously with time and displacement of the vibrating pipe, which cannot exactly capture the non-smooth characteristics of the saltation of state vectors for the pipes before and after an impact. In this paper, a non-smooth mathematical model of simply supported pipes conveying pulsating fluid, subjected to a rigid constraint somewhere along the pipe length is established, with consideration of the effect of the values of clearance and coefficient of restitution of the constraint. Especially, the periodic and aperiodic oscillations are investigated under various pulsating frequencies of the internal fluid. The transition matrices of the displacement and velocity of all nodes along the pipe before and after impact were derived based on a Galerkin's approach. The nonlinear equations of motion are solved via a fourth-order Runge-Kutta method, by applying the impact boundary conditions. Results show that the pipe is capable of displaying interesting vibro-impact behaviors in the presence of the rigid clearance constraint with the variation of pulsating frequency of the flowing fluid. With a clearance close to the maximum displacement of the pipe without constraint, periodic vibro-impact behaviors are observed with multiple impacts. The vibration velocities before the pipe impacts on the edge of the rigid clearance constraint decrease to zero gradually with the displacement invariant, which is called a dynamical stick-slip motion, also known as a typical non-smooth phenomenon. By decreasing the value of coefficient of restitution, the responses of the pipe may change from periodic vibrations to chaotic ones. This work provides an attractive strategy for further understanding of the nonlinear impact dynamics of pipes conveying fluid subjected to rigid clearance constraint based on non-smooth theories.