This paper establishes some necessary and sufficient conditions for the average-consensus problem of second-order multi-agent sampled control system. It is assumed that the sampled interval of each agent is independent of the others, I.e., the sampling interval is time-varying. By algebraic graph theory and matrix theory, a necessary and sufficient condition is derived to ensure the average-consensus. It is found that the eigenvalues of the corresponding Laplacian matrix play a key role in reaching consensus. Based on this result, an approach of how to choose the scaling parameter and time-varying sampling interval is given to guarantee the consensus. Finally, an example is provided to illustrate the effectiveness of the theoretical results.