This study used a fluid-conveying nonlinear beam and a nonlinear spring to simulate the vibration of a fluid-conveying tube placed on an elastic foundation. The centripetal force and tangential force of fluid acting on the tube wall were considered. In this paper, Hamilton’s principle is used to derive the equation for the nonlinear flow-structure coupled motion, where the method of multiple scales is used to derive the frequency response of each mode under the fixed point (steady state), and the amplitude of each mode is used to examine internal resonance. This study added (tuned mass dampers) TMDs of different masses, spring constants and damping coefficients at different locations in the system to observe the effect of the shock absorber in avoiding the internal resonance of the flow-structure coupled system and reducing the vibration of the system. Poincaré map, maximum amplitude contour plots, and basin of attraction are used to analyze and compare the system to verify the correctness of our theory. The stability of the system is analyzed by changing the flow velocity of the fluid. The results show that under a certain combination of elastic foundation spring constants and flow speeds, the 1:3 internal resonance between the first and third modes of the main system will occur. In addition, the stability range of any case will increase significantly after TMD is added, indicating that TMD plays an important role.