Assembled structures are characterized by contact interfaces that introduce a local non-linearity and affect the dynamics of the assembly in terms of resonance frequencies and vibration levels. To assess the forced response levels of the assemblies during the design, nonlinear dynamic analyses are performed and, in order to reduce the computation time, spatial and temporal reductions of the governing equations must be used. A classical way to achieve temporal reduction is to implement the harmonic balance method to turn the time-domain differential governing equations into frequency-domain algebraic equations. Due to the local nature of contact interfaces, which usually involve a subset of degrees of freedom (dofs) of the structure, a common strategy to achieve spatial reduction is to use component mode synthesis (CMS), by retaining the contact dofs as master dofs. In this paper, a recent CMS approach, named dual Craig–Bampton method (Rixen in J Comput Appl Math, 2004. doi: 10.1016/j.cam.2003.12.014 ), is applied to the nonlinear forced response of structures with contact interfaces. The spectral orthogonality of the two subsets of mode shapes used as a projection basis is exploited to write a set of algebraic equations of the contact dofs in the frequency domain, with no need to compute the reduced matrices of the system. Different formulations of the governing equations are proposed for different configurations (i.e., outer contacts, inner contacts and structures with floating components), and two academic numerical test cases are used to demonstrate the method.