No literature has reported an explicit integration algorithm to achieve controllable numerical dissipation and identical third-order accuracy simultaneously. This paper develops two novel explicit integration algorithms to achieve this goal well without increasing computational complexity. In detail, two novel methods are identical third-order accuracy, so avoiding the order reduction for solving damped and forced vibrations, and both of them provide a full range of dissipation control via adjusting their unique algorithmic parameter. Apart from these two highlights, two novel methods embed another two enjoyably advantages. One is to present a significantly larger stability limit than the published third-order explicit schemes. Two novel methods provide the maximum stability limit, reaching 2 3 in the non-dissipative case, getting close to four. The other is to maintain a relatively little computational cost. Two novel methods require explicit solutions only twice within each time step. Therefore, two novel methods are significantly superior to other composite sub-step explicit schemes with respect to the accuracy, stability, dissipation control, and computational cost. Numerical examples are also performed to confirm the numerical performance and superiority of two novel explicit methods. • Two novel fully explicit methods are proposed based on the composite two-sub-step technique. • Two novel methods are identically third-order accurate for solving general structures. • Two novel methods can control numerical dissipation imposed at the bifurcation point. • Two novel methods provide a larger stability limit and require a relatively little computational cost. [ABSTRACT FROM AUTHOR]