# Concentrated Parameters Model

The bending and torsion behaviour of the continuous strip is an integral part of the whole system. In order to have a unique kinematic model of the deformation system, which will allow us to analyse and control the behaviour of the strip, we have developed a concentrated-parameters model of this component. In this model, continuum is discretized and modelled as a string of rigid sectors. Each sector is connected to the previous and the next by means of two types of hinges, as shown in Fig.4.3. In order to make the image clearer, the two systems have been represented separated but in the model they work simultaneously. In particular, Fig. 4.3a shows a schematisation of the flexural model and Fig.4.3b presents the torsional one.

To represent the bending behaviour of the strip each rigid sector is connected with the next one by means of a hinge, which allows the sector to rotate around the ideal axis that lies on the transversal plane of the strip. In the same way, in order to allow the sectors to represent the torsional rotations, each sector is connected with the next one with a hinge allowing the sector to rotate around the ideal axis, which lies on the centre line of the strip. To represent the stiffness of the continuum material springs are combined to the hinges.

**Fig. 4.3 ****Representation of the flexural and torsional models**

The value of the stiffness of the springs is analytically calculated by concentrating the continuous stiffness of each sector in a point. Similarly, the mass of the continuous material is concentrated in the centre of gravity of the sector. The centres of each sector represent the nodes of the geodesic trajectory to represent. Thanks to the developed model, each sector is connected to the previous and the next ones by the constraints and stiffness relationships imposed.

To check the accuracy of the model we have used also a finite elements analysis. As shown in Fig. 4.4, we have considered a hypothetic load, which is applied to the finite elements model of the strip. This analysis allows us to obtain displacements, which would be obtained as a result of the hypothetical load. we have imposed these displacements to the concentrated parameters model, which allow us to obtain the load needed to obtain the imposed displacements. This load has resulted to be congruent with the hypothetic load imposed to the finite elements model. Therefore, we can consider the model as accurate without forgetting that both methods discretise

**Fig. 4.4 **Validation of concentrated parameters model the continuous and that they are numerical methods, which introduce an intrinsic error, even if in the form infinitesimal.

The developed model allows us simulating the behaviour of the strip. When a sector is moved in the space, the whole chain will move according to the behaviour of the continuous material. As a consequence, it is possible to control the behaviour of the model, and, hence, the geodesic trajectory, by controlling the position and the rotations of only few sectors. From now on, these sectors will be referred to as control sectors.