Dynamic community discovery is an important research area in Social Network Analysis （SNA）. As nodes joining or leaving social networks， the relationships between nodes establish or terminate， which affects community structure changes. The discovery algorithms of static communities in social networks lack of the essential historical information of community nodes， resulting in the insufficient network structure analysis as well as clustering information and the high computational cost. Aiming at these problems， based on the division of the community network evolution events， according to the analysis of the major community events， a Spectral Clustering based Dynamic Community Discovery Algorithm （SC-DCDA） was proposed. Firstly， according to the experimental observation， the dimensionality of high-dimensional data was reduced by using the method of spectral mapping. At the same time， the improved Fuzzy C-Means clustering （FCM） algorithm was adopted to determine the correlation between the nodes in the dynamic social network and the communities to be discovered. Secondly， the community structures were analyzed according to the evolutionary similarity matrix. Finally， the real network datasets and community discovery algorithm indicators， such as modularity score and Silhouette coefficient， were used to evaluate the effects of the proposed algorithm. Experimental results show that the computational cost of SC-DCDA is reduced by 8.37% compared with traditional spectral clustering， the average modularity score of the algorithm on all datasets is 0.49， and the qualitative analysis results of other algorithm metrics are also good， indicating that the proposed algorithm performs well in information interaction， clustering effect， and accuracy.