<p>This study deals with a longitudinally excited chain of external linear oscillators, enhanced with internal linear oscillators. The undamped case is considered first as a benchmark, and then a viscously damped chain is examined. Qualitatively different vibration attenuation regions are identified, including the cases of no attenuation region, a finite and semi-infinite attenuation region, which represents a new result for this type of chain of interest for vibration control. The influence of the number of unit cells, the mass and the damping ratio on their appearance is examined thoroughly and illustrated in the form of novel and original 2D and 3D behavioural maps, which can be used as a design criterion for metastructures, but also for other systems modelled as this type of chains. The nonlinear boundary between the finite and semi-infinite vibration region involving the non-dimensional damping coefficient and the number of units is determined for the first time. Besides, the nonlinear change of the critical non-dimensional damping ratio, which assures the appearance of the semi-infinite vibration attenuation region in the system response, with the non-dimensional total mass, is also innovatively obtained. The theoretical identification of three distinct regions of vibration attenuation is experimentally validated through an original damping control method utilizing an air track and an air pump with a variable power output capability.</p>