The vibration isolation effect of linear and nonlinear single degree of freedom oscillators attached to a structure is studied. This system of oscillators, directly attached to the main structure, induces a decrease to the amplitude of the structural response, over a wide excitation frequency band. This is achieved by distributing the natural frequencies of the attachments over an a priori specified frequency interval, which is related to the excitation power spectrum density. The use of nonlinear Duffing type oscillators increases the vibration isolation effect of the attachments. The study of the vibration isolation system behavior is conducted using the Monte Carlo technique. An implicit Newmark integration scheme is used for the linear attachment case. In the case of nonlinear attached oscillators, the system of equations of motion is integrated by an iterative decoupled Newmark integration technique. This scheme improves significantly the efficiency of the Monte Carlo method.