By Leping Yang, Qingbin Zhang, Ming Zhen, Haitao Liu
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4 Reflection and Transmission of Stress Wave Between Cable Sections To research the stress wave propagating in a cable, the cable is normally regarded as a homogeneously continuous elastic or viscoelastic material with isotropy. If the research goals are multiple cable units with different parameters, then when the 26 2 Cable Dynamics Elements stress wave propagates to a section, because of the obvious differences in the materials at the section, only some of the energy crosses the section and propagates forward, while the rest is reflected at the section.
1) a ¼ 1 The generalized impedance of the connected cables matches each other, and ðq1 c1 A1 Þ ¼ ðq2 c2 A2 Þ ð2:41Þ According to Eqs. 40), the reflected wave satisﬁes the following equation, vR ¼ rR ¼ uR ¼ 0 ð2:42Þ The transmitted wave satisﬁes the following equations, 8 < vT ¼ vI ðr A Þ ¼ ðrI A1 Þ : T 2 uT ¼ uI ð2:43Þ As Eqs. 43) show, the stress wave will not be reflected at the conjunction for cables with completely different materials and structures with their generalized impedances matching each other.
For two cables, according to governing equation (Eq. 31) of the propagating stress wave, the corresponding characteristic line equation is as follows: dx ¼ Æci dt ð2:48Þ Substituting Eq. 48) into the governing equation, the compatible equation is as follows: dri Ç qi ci dvi ¼ 0 ð2:49Þ Integrating both sides, we obtain the following equation: ri Ç qi ci vi ¼ const ð2:50Þ The stress and velocity maintain a constant relationship along the characteristic line. Based on this, the stress, strain, and velocity in the time domain and spatial domain can be solved along the characteristic line with a given boundary condition and initial condition.