Direct decoupling of substructures using primal and dual formulation Fig. 3 Sketch of the test system: a two speed transmission in first gear Rotational inertias and torsional stiffnesses, as well as torsional dampings, are shown in Table 1 together with the number of teeth z of each gear. Table 1 Physical parameter values and number of teeth Item J [kgm2] c [Nms/rad ] k [Nm/rad] z 1 3· 10−3 8.25· 10−4 82.5 – 2 1.98· 10−4 8.25· 10−4 82.5 30 3 7.62· 10−4 1.65· 10−3 165 42 4 8.13· 10−5 1.65· 10−3 165 24 5 2.57· 10−5 6.875· 10−3 687.5 18 6 1.3· 10−3 3.435· 10−3 343.5 48 7 2.08· 10−3 3.435· 10−3 343.5 54 8 1· 10−2 – – – The rotations of meshing gear pairs are related by their gear ratio, e.g. the rotations θ2, θ6 and θ7 are related to rotations θ3, θ4 and θ5, respectively. The whole set of rotations θ1··· θ8 can be expressed through a reduced set of five independent rotations, as shown by Eq. (39): ⎧⎪ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 ⎫⎪ ⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎭ = ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ 1 0 0 0 0 0 z3/z2 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 z4/z6 0 0 0 0 0 z5/z7 0 0 0 0 0 1 ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦ ⎧⎪ ⎪⎨ ⎪⎪⎩ θ1 θ3 θ4 θ5 θ8 ⎫⎪ ⎪⎬ ⎪⎪⎭ (39) Therefore, the coupled system can be modelled as a 5 DoFs lumped parameter system. 61
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