Abstract: This paper uses the simplest linear finite elements of the Galerkin type and gives a compact and efficient recurrence solution formula for equations of motion. Further, based on the EEP (Element Energy Projection) super-convergence technique, two critical techniques, i.e. adaptive time-step size and recovery of nodal displacement accuracy, have been developed, enabling a linear finite element solution with errors uniformly distributed and satisfying the pre-specified error tolerance at any moment in the whole time domain. Numerical examples of both single and multiple degreed systems are given to verify the validity of the proposed method.