The Euler-Bernoulli equation with distributional coefficients and forces

Abstract

In this work we investigate a very weak solution to the initial-boundary value problem of an Euler-Bernoulli beam model. We allow for bending stiffness, axial-and transversal forces as well as for initial conditions to be irregular functions or distributions. We prove the wellposedness of this problem in the very weak sense. More precisely, we define the very weak solution to the problem and show its existence and uniqueness. For regular enough coefficients we show consistency with the weak solution. Numerical analysis shows that the very weak solution coincides with the weak solution, when the latter exists, but also offers more insights into the behaviour of the very weak solution, when the weak solution doesn't exist.

An integrated method for the leakage fault mode diagnosis and life prediction of the reactor coolant pump

“…According to Bernoulli’s equation [ 42 ], for an ideal fluid to stably flow, at any point in the same flow tube, the sum of the kinetic energy and potential energy per unit volume of the fluid and the pressure at that point is constant [ 43 ], as shown in Fig 4 . where p is the pressure at a certain point in the fluid, v is the flow velocity of the fluid at that point, ρ is the fluid density, g is the acceleration of gravity, h is the height of the point, and C is a constant number.…”