Walter D’Ambrogio and Annalisa Fregolent from which, being[LE] T[BE] T =0 since BT E is the nullspace of LT E (see Eq. 17): [LE] T [Z][HA] D =[LE] T (37) By comparing Eq. (37) and Eq. (33), it is found that: [HA] D =[LC][HA] P (38) The former equation shows that, if NC =NE, primal and dual formulations provide the same result. In fact, by considering the primal formulation, the relation between the unique set of DoFs {q} and the redundant set of DoFs {u} is given by {u}=[LC]{q}. On the contrary, if NC >NE, Eq. (33) can not be obtained from Eq. (13) since: [LE] T [Z][LC] [LE] T [Z][LC] + =[I] due to the definition of pseudoinverse. In fact in case of overdetermined system, the pseudoinverse of a matrix[A] (number of rows >number of columns) is: [A]+= [A] T [A] −1 [A] T from which: [A]+[A]= [A] T [A] −1 [A] T [A]=[I] [A][A]+=[A] [A] T [A] −1 [A] T =[I] Therefore, if NC >NE, the primal and the dual formulation provide different results. 4 Application A relatively simple application is considered on a torsional system that represents a model of a two-speed transmission. The complete system consists in three shafts: an input shaft, a layshaft and an output shaft (see Fig. 3). The layshaft is coupled both to the input shaft and to the output shaft by helical gears. Power flows through the gear that is locked to the output shaft by the shift collar (e.g. through gears 5 and 7 in Fig. 3). Input and output shafts are assumed to be fixed at the outer ends. Such a boundary condition is a good approximation whenever the mass moments of inertia upstream and downstream the transmission are very large compared to those within the transmission. 60
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