本研究所建構的模式是一個3D非線性樑，其一端為fixed end，另一端為free end，此研究可以運用模擬一般之鐵軌、橋樑、海底電纜結構等，而且就一個三維 (3D) 的情況而言，相對的研究就比較複雜。首先，吾人利用Nayfeh and Pai [12]的非線性fixed-free 3D beam為基本架構，以Newton’s Laws將此beam的y方向及z方向的非線性彈性介質以三次方彈簧的方式導入運動方程式 (E.O.M.)之中，再以多尺度法解析此非線性系統。吾人發現系統中之第一模態(Mode Shape)及第二模態存在一對三(1：3)之內共振情形(Internal Resonance)。而為達避開I.R.及達到減振的效益，裝置一單一自由度的調質(量)減振器 (Tuned mass damper (TMD))，嘗試改變TMD之質量、位置以及其彈簧係數、阻尼係數，利用各模態(Mode)之系統最大振幅的3D圖及3D Maximum Amplitude Contour Plot(3D MACP)觀察其TMD的最佳組合，達到本系統之最佳減振效果。 Vibration has long played a crucial role in engineering due to its effects on structural stability, metal fatigue, and structural damage to materials. Beams are used in a wide range of engineering problems and the internal resonance of nonlinear beam vibrations is a popular research topic. Internal resonance is unique to nonlinear systems in which integer relationships exist among the natural frequencies with various modes. This study examined the vibrations of a 3D fixed-free nonlinear beam placed on a nonlinear elastic foundation. We found that specific combinations of elastic modulus in the elastic foundation resulted in 1:3 internal resonances in the 1st and 2nd modes of the beam. This prompted us to add a Tuned mass damper (TMD) on the elastic beam in order to prevent internal resonance and suppress vibrations. We analyzed this nonlinear system using the method of multiple scales (MOMS). Fixed points plots were also used to facilitate the observation of internal resonance. This made it possible for us to study the influence of nonlinear geometry and nonlinear inertia associated with the vibration of the elastic beam. In this work, we examined the combination of optimal mass ratio and elastic modulus as well as the location of the TMD in order to prevent internal resonance and achieve optimal damping effects. Finally, the numerical results were compared with the fixed points frequency plots to confirm the findings from this study.
