The paper deals with the dynamics of a one-dimensional mechanical system involving a clutch or a brake. At first, the author shows that this mechanical problem can be described by the differential inclusion \((*)\) \(A(q)\ddot q+ F(t, q, \dot q)\in \Gamma(- q)\), where \(\Gamma(u)= \{- a\}\) if \(u< 0\), \(\Gamma(u)= \{b\}\) if \(u> 0\), and \(\Gamma(0)= [- a, b]\) (\(a\) and \(b\) are friction coefficients). If \(A\) and \(F\) are continuous functions and if \(\ddot q\) is right-continuous, then \((*)\) holds everywhere if and only if the following differential equation holds everywhere: \((**)\) \(A(q)\ddot q+ F(t, q, \dot q)= \text{proj}_{\Gamma(- \dot q)} F(t, q, \dot q)\), where \(\text{proj}_c x\) denotes the projection of \(x\) on a convex set \(\mathcal C\).
In a weak formulation, the above problem admits a variational formulation. The author subsequently proves, by using two discretization procedures, the existence of a solution of \((*)\) and \((**)\) in the weak sense. The regularity of such solution is studied, and the uniqueness is proved by making some assumptions on the structure of \(A\) and \(F\).