Primary parametric resonance of current-carrying conductor

Abstract

In order to study on nonlinear vibration of current carrying conductor, a differential equation is established by means of dynamics theory. Based on the method of multiple scales, the first approximate solution and corresponding solution of the primary parametric resonance is obtained.Numerical analysis results show that the amplitude and the topological structure changed with the increase of direct current. The characteristics and laws of the primary parametric resonance excited by the other system parameters such as current, detuning, tension and damping are also analyzed. It has also been found that the natural frequency of the system increases with the increasing tension but decreases with the increasing length.

Chaotic dynamics and subharmonic bifurcations of current‐carrying conductors subjected to harmonic excitation

“…Consider two straight parallel conductors 𝐴𝐵, 𝐶𝐷 with a distance of 2𝑎 as shown in Figure 1, the direct current is 𝐼, a tension wire with both ends fixed, its equilibrium is along the 𝑜𝑥 axis, the length is 𝑙, the alternating current is 𝑖, the harmonic excitation is 𝑃 cos(Ω𝑡), where 𝑃 and 𝜔 are the excitation amplitude and frequency, respectively, and 𝑢(𝑥, 𝑡) is the transverse displacement. Assume the conductor density and tension do not change with space-time, the equation of motion is [4,5]…”

“…In references [4,5], with the method of multiple scales, the first approximate solution and corresponding solution of the primary parametric resonance and 1/3 subharmonic resonance for Equation (4) were obtained, respectively. The characteristics and laws of the primary parametric resonance excited by the system parameters such as current, detuning, tension, and damping were analyzed there.…”

“…Assume the conductor density and tension do not change with space‐time, the equation of motion is [4, 5] $$\begin{equation} \frac{T}{\sqrt {1+\left(\frac{\partial u}{\partial x}\right)^2}}\frac{\partial ^2u}{\partial x^2}+\frac{\mu Iiu}{\pi (a^2-u^2)}[1+\left(\frac{\partial u}{\partial x}\right)^2]=\rho _{0}\frac{\partial ^2u}{\partial x^2}+c\frac{\partial u}{\partial t}-P\cos (\Omega t), \end{equation}$$where ρ 0 is the density of conductor, T is the tension, μ is the magnetic permeability in air, and c is the damping coefficient.…”

“…With the method of multiple scales, Xi and Yang studied the primary resonance and primary parametric resonance of current‐carrying conductor subjected to harmonic excitation and ampere force [3]. Based on the method of multiple scales, the first approximate solution and corresponding solution of the primary parametric resonance for current‐carrying conductor subjected to harmonic excitation and ampere force was investigated by Yang and Xi [4]. They also studied 1/3 subharmonic resonance of current‐carrying conductor subjected to harmonic excitation [5].…”