多阶梯梁系统的3：1内共振

3:1 Internal Resonance in Multiple Stepped Beam Systems

研究了具有三次非线性项的多阶梯梁的振动.讨论了该系统3∶1内共振情况.运用多重尺度法,即一种摄动技术,得到该问题的一般近似解,并得到两种模型的振幅和相位调制方程.这些方程组用来确定稳态解及其稳定性.假设外加的强迫频率接近于较低的频率.在研究的数值部分,讨论固有频率中的3∶1情况.对两端固支和一端固支另一端简支,观测到的频率位于第一和第二固有频率之间;对两端简支,观测到的频率位于第二和第三固有频率之间.最后,利用数值算法求解3∶1内共振.第一模型为两端固支和一端固支另一端简支梁的外激励模型;第二模型为两端简支梁的外激励模型.然后,当外激励第一模型时,研究第一、二模型的振幅.当外激励第二模型时,研究第二、三模型的振幅.对振动的内共振模型,画出强迫响应、阻尼响应和频率响应曲线.同时进行这些曲线的稳定性分析.

Vibrations of multiple stepped beams with cubic nonlinearities were considered. 3：1 internal resonance case was investigated for the system. A general approximate solution of the problem was found using the method of multiple scales, a perturbation technique. The modulation equations of the amplitudes and the phases were derived for two modes. These equations were utilized to determine steady state solutions and their stabilities. It was assumed that external forcing frequency is near to the lower frequency. For numeric part of the study, 3： 1 ratio in natural frequencies was investigated. These values were observed to be between first and second natural frequencies in cases of clamped-clamped, clamped-pinned support and between second and third natural frequencies in case of pinned-pinned support. Finally, a numeric algorithm was used to solve 3：1 internal resonance. The first mode is externally excited for clamped-clamped, clamped-pinned support and the second mode is externally excited for pinned-pinned support. Then, amplitudes of first and second modes were investigated when the first mode is externally excited. Amplitudes of second and third modes were investigated when the second mode is externally excited. Force-response, damping-response and frequency-response curves were plotted for internal resonance modes of vibrations. Stability analysis was carried out for these plots.