<p>The dynamics of a flexible pipe conveying fluid has attracted much attention for a long time, and many researchers have studied the nonlinear phenomena that occur when the flexible pipe undergoes self-excited vibration caused by non-conservative forces when the flow velocity exceeds a critical speed. Because of the non-self-adjointness and non-orthogonality of the system, a corresponding adjoint function orthogonal to the eigenmodes is necessary for nonlinear analysis. In this study, we consider a model where the upper end of a cantilevered elastic water pipe is connected to a moving mass attached to a spring and damper. We theoretically and experimentally show that veering in nonlinear self-excited oscillation depends on the boundary stiffness. Because of such a boundary condition, conventional methods cannot obtain the adjoint function. Thus, we propose a method to find the associated adjoint functions to perform nonlinear analysis. Using the adjoint functions, we derive amplitude equations considering nonlinear inertia and restoring forces to clarify the bifurcations due to the change in flow velocity. Depending on the upper-end stiffness and damping, two modes are simultaneously destabilized through nearly-perfect 1:1 internal resonance and veering occurs, or only a single mode is destabilized and there is no veering. Experiments using an apparatus to freely set the stiffness and the damping through position and velocity feedback demonstrate the theoretically predicted nonlinear characteristics of a self-excited pipe conveying fluid in a veering situation.</p>