The concept on generalized consensus problem of second-order multi-agent systems is first proposed. Distributed linear generalized consensus protocols with and without communication delay are desiged. Based on algebraic graph theory and stability theory，the necessary and sufficient conditions for second-order multi-agent system to achieve linear generalized consensus are established. It is shown that both the coupled gain parameters and the ei-genvalues of the Laplacian matrix play key roles for reaching generalized consensus in directed network. Finally，numerical simulation are given to verify and illustrate the theoretical analysis．