Explicit symplectic integrators play a vital role in computational structural dynamics due to their unique ability to preserve phase-space invariants and eliminate artificial numerical dissipation. Although the classical Central Difference Method (CDM) is symplectic, its practical application is often complicated by the need for an auxiliary starting procedure for the initial step. This paper develops a new class of self-starting explicit time integration schemes for linear structural dynamics and wave propagation problems. The proposed schemes are constructed such that the one-step discrete mapping can be written as: ${z_{n + 1}} = {\mathbf{A}}{z_n}$ , where \[{z_n} = {[{{\mathbf{u}}_n},{\mathbf{M}}{{\mathbf{\dot u}}_n}]^T}\] represents the state vector of displacements and velocities. For a general multi-degree-of-freedom system, the transition matrix A is derived as:${\mathbf{A}} = \left[ {\begin{array}{*{20}{c}} {{\mathbf{I}} - \alpha {h^2}{{\mathbf{M}}^{ - 1}}{\mathbf{K}}}&{h{{\mathbf{M}}^{ - 1}}({\mathbf{I}} - \alpha \beta {h^2}{\mathbf{K}}{{\mathbf{M}}^{ - 1}})} \\ { - {\mathbf{K}}h}&{{\mathbf{I}} - \beta {h^2}{\mathbf{K}}{{\mathbf{M}}^{ - 1}}} \end{array}} \right]$ where h denotes the time step, M is the lumped mass, and K is the stiffness matrix, alpha and beta are positive algorithmic parameters satisfying alpha + beta = 1. We rigorously prove that this mapping satisfies the discrete symplectic condition ${{\mathbf{A}}^T}{\mathbf{JA}} = {\mathbf{J}},$ exactly, ensuring the preservation of the system's geometric structure without requiring any multi-step information or auxiliary starting algorithms. Theoretical analysis demonstrates that the proposed schemes maintain the same computational efficiency and stability limits as the CDM. Numerical simulations involving long-term vibrations and wave propagation demonstrate that the proposed integrators exhibit superior numerical behavior over the CDM by eliminating starting errors while maintaining identical stability and efficiency. Consequently, this class of schemes serves as a more robust and streamlined alternative to the CDM for high-fidelity dynamic analysis in large-scale engineering applications.