1/3 Subharmonic Resonance of Current-Carrying Conductor Subjected to Harmonic Excitation

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.…”

“…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]. With the Lindstedt-Poincaré method and Newton-harmonic balance method, Sun et al [6] investigated analytical approximate solutions to oscillation of a current-carrying wire in a magnetic field.…”

“…Using generalized weighted residual method, Xiang [2] investigated the lateral vibration of two long straight conducting wires carrying electric current. 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].…”