This investigation is concerned with the development, implementation and application of variational inequalities (VI) to treat the general elastodynamic contact problem. Two VI algorithms are used to treat the general frictional contact formulation and to identify the candidate contact surface and contact stresses. The first algorithm, which decomposes the proposed problem into two sub-problems, is based upon the iterative use of mathematical programming and Lagrange multipliers. The second algorithm treats the VI formulation in a single step, using nondifferentiable optimization algorithm. In order to ensure that the dynamic contact problem is modeled accurately, a generalized-\(\alpha\) method is adopted for time marching. The selected time integration parameters significantly reduce the spurious high-frequency modes, which are present in the traditional Newmark method in impact and dynamic frictional problems. In order to demonstrate the versatility and accuracy of the proposed finite element algorithms, a number of numerical examples are examined.