Periodic solutions of autonomous and conservative second-order dynamical systems of finite dimension $n$ undergoing a single unilateral contact condition are investigated in continuous time. This unilateral constraint is complemented with an purely elastic impact law, which conserves total energy. The dynamics is assumed linear when there is no contact. The number of impacts per period, $k$, arises as a natural parameter of the proposed formulation. Interestingly, the existence of the targeted periodic solutions is essentially governed by a system of only $k-1$ nonlinear equations with $k$ unknowns, regardless of the number of degrees-of-freedom. This serves to prove that the phase-space is primarily populated by one-dimensional continua of periodic solutions, generating invariant manifolds which can be understood as a non-smooth mode of vibration in the context of vibration analysis. Additionally, these equations provide an efficient and systematic way of calculating non-smooth modes of vibration. They also demonstrate the existence of interesting properties: symmetries of trajectories and emergence of unique features, such as unreported constant-frequency manifolds. All results are illustrated through a simple in-line spring-mass system whose last mass undergoes the impact law. Note The presentation file includes several animations which can only be viewed if the file is opened with Adobe Acrobat Reader 9 or more.