Nonlinear Forced Vibrations of a Fluid-Conveying Pipe with a Lumped-Mass Intermediate Support and Asymmetric Elastic Boundaries

Although general elastic boundaries, additional springs and lumped masses, nonlinear end supports, and internal nonlinear supports have been investigated in previous studies, the mode-dependent forced response arising from the interaction between unequal nonlinear end restraints and a lumped-mass–damper nonlinear intermediate support remains insufficiently quantified. This study develops an Euler–Bernoulli model of a fluid-conveying pipe incorporating unequal linear and cubic translational end springs, linear rotational end springs, and an intermediate support with independently specified mass, viscous damping, linear stiffness, and cubic stiffness. The governing equation is derived using Hamilton’s principle and reduced by a four-mode Galerkin approximation based on elastic-boundary-adapted modal functions constructed using a modified Fourier–Ritz procedure. Frequency-by-frequency time integration using an adaptive Dormand–Prince scheme is employed to examine the effects of flow velocity, intermediate-support mass, location, linear and cubic stiffnesses, and boundary cubic stiffnesses. Numerical convergence and agreement with an independently formulated Lagrange-equation solution are demonstrated for the principal frequency-response characteristics. The results show that flow velocity reduces and shifts the first resonance toward lower frequencies, whereas it amplifies nonlinear response features near the second resonance. The support mass produces a non-monotonic change in the second-resonance amplitude, while increasing the support linear stiffness can strongly suppress the first resonance. Increasing either the intermediate or boundary cubic stiffness produces hardening-type bending of the attracting response and suppresses the second-resonance peak, while strong nonlinearity gives rise to multiple long-time response extrema and non-period-one phase-space trajectories within the attracting responses reached by direct integration. These results clarify the mode-dependent roles of local inertia and nonlinear constraints and provide guidance for tuning supports in fluid-conveying piping systems.