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The results of theoretical numerical-analytical investigation of some kinematic characteristics of the nonlinear second harmonics for monochromatic axisymmetric normal elastic torsional waves along the axial direction in transversely isotropic cylinder of circular cross section with a rigid fixed of lateral surface. Frequency-parametric analysis of amplitude and shape of wave motion into the second harmonic for normal waves of the investigated type was conducted for the cylinder made from a gadolinium using the proposed method. The regularities inherent in wave processes of this type are described.