In the consensus problem, the communication network describing the interactions between agents may change due to link failures and environmental conditions, which plays an important role in the stability and overall performance of the system. Also, faster convergence is always of interest in some practical engineering applications. For this purpose, the design of a fixed-time controller is of interest, because for this type of controller, the convergence time of the system states does not depend on the initial conditions of the agents. This paper investigates the problem of leaderless consensus control for second-order linear multi-agent systems (MASs) under a switching topology with link failures. To address this problem, we proposed an improved fixed-time distributed controller for second-order consensus in MASs with velocity and position coupling that guarantees convergence within a specified time interval. Compared to previous studies, the effects of simultaneous switching topology with link failures on second-order linear MASs are considered, so that the communication graphs are considered piecewise fixed and undirected jointly-connected ones. The stability of the proposed controller is proven based on graph theory, Lyapunov theorem and bi-limit homogeneity method. Finally, the effectiveness of the proposed method is demonstrated through numerical simulations, considering different periodicity for changes between topologies.