Most systems consist on dynamical substructures connected at a number of constraining points. Moreover, constraints often display intermittent contact phenomena, such as arising from clearance supports. A significant difficulty when computing time-domain responses is the manner to enforce such coupling constraints. Here, we explore the Udwadia-Kalaba (U-K) formulation, which has been very seldom used in this context. By extending the basic U-K analytical framework, we address continuous flexible subsystems modelled by their unconstrained modes and coupled through the highly nonlinear intermittent point-constraints. For continuous flexible systems, a modal U-K formulation is implemented such that the constraint is applied when contact is detected at the clearance location. A crucial aspect is that constraint violations must be prevented, not only at the acceleration level, but also at the velocity and displacement levels, in order to avoid computational drift. This is achieved through a constraint violation correction method. For single gap-constraints, a convenient formulation is obtained, in which the constraint matrix is pre-computed prior to the simulation time-loop and applied whenever an intermittent contact is detected, leading to an efficient computation of vibro-impact responses. For systems with several intermittent constraints, an essential difficulty within the context of the proposed formulation is that every possible combination of contact/non-contact conditions is expressed by a different constraint matrix. A pragmatic solution is to keep track of the current system contact configuration and rebuild the constraint matrix whenever a change in the constraint state is detected. We formulate and illustrate such computational strategy, as applied to random-excited multi-supported beams with a significant number of clearance supports. Results are compared with dynamical computations performed using a classic penalty technique for enforcing the nonlinear support constraints, emphasizing the viability of the proposed technique for performing predictive analysis of flexible structures with multiple clearance supports.