On some properties of mass points systems

linked by springs and dampers

Adriano Montanaro

Department of Mathematical Methods and Models for Scientific Applications

University of Padua

via Trieste, 63, 35121, Padova, Italy

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.

6. On applications of the above results

In conclusion we indicate some applications of obtained results. The system of original equations studied here is a mathematical model suitable to describe various rectilinear physical systems as chains of three-dimensional bodies undergoing translational motions.

Using the results of obtained in this research theorems people involved in the design of such physical systems with љN > 2љ can choose the material constants in order to render the system completely controllable, observable, reachable and reconstructible;љ instead in the case ofљ N = 2љ any such a system has such properties for any choice of the material constants. The key advantage of the theorems proved here for engineering applications is that when control is a property useful for the system under design, one can design the system by choosing the material parameters in such a way as to have controllability.

A further theoretical task will be to construct or to find the controlling input functions for the system: the present paper clarifies when such a problem has a solution.

An example of a physical object, which can be modelled by considered system, is given by a chain of љNљ pistons consecutively connected by springs and constrained inside a cylindrical cavity containing fluid. Assuming that an external input force u(t) can act normally on the first piston, and taking as output the position of the latter piston, the system will be controllable and observable by the history of the latter provided that the elastic springs are chosen proportional to the constants of viscous damping between consecutive pistons.љ Hence, (i) the system can reach any prescribed state by a suitable input force љu(t) љmoreover, љ(ii) љevery state of the system (that is, position and velocity of each piston) can be determined by observing the excursions of the latter piston.

In particular, this system can also be used to model the behavior of car-wheel suspensions. In more detail, when a car travels along a bumpy road the wheel tyre copies roughness of the road surface. The wheel is thus driven up or down in the vertical direction along the z axis. In this case, the rectilinear system under consideration consists of four bodies: a spring, a shock absorber, a wheel and a quarter of a car body. The system excitation by the road surface and gravitational attraction of the wheel and the quarter-car body form the system surroundings. The model is excited by the source of vertical velocity resulting from the tyre copying the road bumps.

The mathematical model of considered object is a particular system. Hence, using our results we obtain that it is observable and reconstructible for arbitrary choices of all the material parameters.

If one wants to know how e.g. the wheel center W behaves in response to the external force road input, without a priori knowing the latter, he may observe the output past excursions of the body-suspension interaction point: they determine the actual state of the system, thus the actual position ofљ W, љtoo. Thus he can go back to the external force road input which generates the vertical motion of the suspension.

In conclusion we note that it is very important to extend the investigated here statement with consideration of nonlinear properties of the suspension bracket of car.