On some properties of mass points systems linked by springs and dampers

Here we consider physical systems consisting of a finite chain of point masses consecutively linked by springs and dampers. An external control force acts at one of the end points which is aligned with the chain whereas the system is observable from the other end point. We show that, whatever is the number of point masses, if the sequence of elastic coefficients is proportional to the sequence of damping coefficients, then the mechanical system is completely controllable, completely observable, completely reachable and completely reconstructible. For introduction we note. Before in our research a finite chain of mass points consecutively linked by springs was considered; the mechanical system is externally controlled by a control force aligned with the chain, applied at one of the end points, and is observable from the other end point. Then shows that such a system is completely controllable, completely observable, completely reachable and completely reconstructible in the sense of control theory for all choices of physical parameters. Here we consider a spring-mass system similar to the one described above but which in addition includes viscous damping generated by the presence of dampers connecting consecutive points of the chain. Here we show that such a system is completely controllable, completely observable, completely reachable and completely reconstructible provided the sequence of damping coefficients is proportional to the sequence of elastic coefficients. This is true for any choice of the number of points, of their masses, and of elastic coefficients of the springs.