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This paper is concerned with the distributed flocking control problem for second-order dynamic multiple polygonal agents with limited communication area. When dealing with such a problem, in order to avoid collisions between agents with non-smooth geometry, this paper redefines the relative distance between two specific agents and then establishes an anti-collision condition that ensures continuity of control inputs. In addition, different from other studies, this paper proposes a method of taking advantage of finite cut-off potential functions to ensure that no collisions occur within a group of agents despite the use of bounded control inputs. Consequently, this paper opens up the possibility that the proposed flocking control algorithm can be used for input-constrained polygon agents with different numbers of vertices and edge lengths. In this light, the validity of the proposed flocking control algorithm is clearly verified through three illustrative examples.