21 Experimental Modal Analysis and Modelling of an Agricultural Tire 215 Fig. 21.2 Normal contact stress as function of the history of load cycles 00.511.522.533.54 −20 0 20 40 60 80 100 120 140 160 relative displacement [mm] τ [kPa] experimental σ = 50 kPa numerical σ = 50 kPa experimental σ = 150 kPa numerical σ = 150 kPa experimental σ = 200 kPa numerical σ = 200 kPa Fig. 21.3 Relation between tangential contact stress and relative displacement between tread and soil 21.2.3 Soil Model The soil model is made up of a layer of springs whose compression, caused by the sinkage of the tire, gives rise to a normal contact stress σ: the stress is assumed to be hydrostatic, i.e. even though sinkage is computed along a vertical axis, the resulting pressure acts along three directions [9]. The relation between sinkage and normal stress depends on the time history of the deformation of the terrain. Figure 21.2 describes the adopted model: the load phase follows a linear relation whose slope KI is determined according to the experimental tests carried out with a standard penetrometer. The model takes into account the increase of the soil stiffness associated with its compaction: assuming that a soil has been compressed at z , the unload phase from z toz and further load phases back to z are described through a different linear relation whose slope KII is assumed to be five times higher than KI. As far as the tangential contact stress is concerned, a direct shear test has been carried out on a soil sample, to measure the shear stress as a function of the relative displacement between soil layers and the normal stress [10]; thus an analytical relation has been stated to fit the experimental data. As it is shown in Fig. 21.3, the tangential contact stress τincreases along with the normal contact stress σand the relative displacement of the soil, until a value of 4 mm of displacement is reached, at this point the soil is supposed to fail. 21.2.4 Generalized Forces Recalling Eq. (21.1), this paragraph is devoted to the determination of the terms Qhub and Qj which account for the effect of external forces (weight and contact forces) acting on the system d.o.f.s. Figure 21.4 depicts the absolute position of point Cik, geometrical center of thek−tharea (Aik) of thei−th element of the grid (Fig. 21.1); each element is in fact characterized by four surfaces (top, base, front, rear) so that k varies between 1 and 4. Figure 21.4 reports also the position of the tire hub (point O) and the distance between Cik and O, namedRik. If point Cik happens to be below the undeformed soil level, it will be characterized by a sinkage sik.
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