PSI - Issue 37

Rogério Lopes et al. / Procedia Structural Integrity 37 (2022) 73–80 R. F. Lopes et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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b) Lateral (drift) oscillation Respecting the lateral, drift , oscillation, it may involve lateral translation and/or roll rotation, about the longitudinal vehicle’s axis . The presented model is a kind of 2-DOF, in which the motions and masses are distributed at the axes and roof levels, respectively. It can be a suitable model to explain the lateral and roll vibrational modes. Suspension stiffness is now exclusively due to drift or shear mode with tire deformation. In this figure, the vertical springs represent the radial stiffness of the vehicle tires when contacting the track, while the horizontal springs are associated to the lateral (or shear) stiffness of the tires, as result of the dynamic effect of the centripetal acceleration (when the vehicle takes a curve), or as result of lateral forces due for example, to side wind gusts. The setup of the stiffness matrix for the force/displacement relation due to transverse displacements only, the process described next is known as the “Direct Stiffness Method”, where the procedure is summarized as follows: when the chassis midpoint labelled as “2” is horizontally blocked, as shown in the Fig. 3-b) and a unitary transverse displacement is prescribed at the vehicle roof midpoint, the necessary force to achieve this deformation body frame at the level of DOF “1” is 11 . Due to the prescribed displacement here mentioned, an internal reaction of the structure at the level of DOF “2” is identified as 12 . Conversely, in Fig. 3-c), when node “1” is blocked and a unitary transverse displacement is prescribed the level of node “2” (chassis midpoint), we obtain at node “1” the internal reaction 21 = 12 (for the principle of reciprocity of forces in any structure) and force 22 at node 2. It is important to remind that, in this last case, the reaction force by prescription of a unitary displacement at “2” is due to a joint reaction by radial deforma tion of the tires (generating a moment counteracting the transverse force at node “1”) but also to the shear reaction of the tires by transverse deformation at the soil level, as graphically suggested in figure above by the spring model deformation. The stiffness parameters are presented in the following equations, Eq. (5) to Eq. (9).

a) c) Fig. 3. a) General overview of the bus, b) effect of horizontal factor ate the roof level and c) effect of the horizotnal factor at the lower level. 11 = 2 2 ( ⁄2) 2 2 = ( ⁄ ) 2 (5) 22 = 4 + ( ⁄ ) 2 (6) 12 = 21 = 11 (7) ‖[ 11 − 11 2 12 21 22 − 22 2 ]‖ = 0 (8) 1 = 0.530 ; 5 = 1.800 (9) c) Vertical oscillation In this vibrational mode, it can be assumed that there are two distinct natural modes: the vertical translation and the transverse rotation. In these natural frequencies, only in serial by the suspension/tire stiffness participate in the vibration modes. It must be noted that there exist two sets of tire/spring-damper per axis. The 1 mode is a rotational motion similar to a “dive - rise” angular motion, represented in Fig. 4 by the dash-dot-line. With the regard to the 2 mode, this is a vertical translation induced by four equivalent springs and all the bus mass, indicated in Fig. 4 by the dash line. b)