A Hybrid Ale Formulation for the Investigation of the Stability of Pipes Conveying Fluid and Axially Moving Beams

The present work addresses pipes conveying fluid and axially moving beams undergoing large deformations. A novel two dimensional beam finite element is presented, based on the Absolute Nodal Coordinate Formulation (ANCF) with an extra Eulerian coordinate to describe axial motion. The resulting formulation is well known as Arbitrary Lagrangian Eulerian (ALE) method, which is often used to model axially moving beams and pipes conveying fluid. The proposed approach, which is derived from an extended version of Lagrange's equations of motion, allows for the investigation of the stability of pipes conveying fluid and axially moving beams for a certain axial velocity and stationary state of large deformation. Additionally, a multibody modeling approach allows us to extend the beam formulation for co-moving discrete masses, which represent concentrated masses attached to the beam, e.g., gondolas in ropeway systems, or transported masses in conveyor belts. Within numerical investigations, we show that axially moving beams and a larger number of discrete masses behave similarly as the case of (continuously) distributed mass.

Investigation of the Stability of Axially Moving Beams With Discrete Masses

The present paper addresses axially moving beams with co-moving concentrated masses while undergoing large deformations. For the numerical modeling, a novel beam finite element is introduced, which is based on the absolute nodal coordinate formulation extended with an additional Eulerian coordinate to represent the axial motion. The resulting formulation is well known as Arbitrary Lagrangian Eulerian (ALE) method, which is often used for axially moving beams and pipes conveying fluids. As compared to previous formulations, the present formulation allows us to introduce the Eulerian part by an independent coordinate, which fully incorporates the dynamics of the axial motion, while the shape functions remain independent of the beam coordinates and are thus constant. The proposed approach, which is derived from an extended version of Lagrange’s equations of motion, allows for the investigation of the stability of axially moving beams for a certain axial velocity and stationary state of large deformation. A multibody modeling approach allows us to extend the beam formulation for co-moving discrete masses, which represent concentrated masses attached to the beam, e.g., gondolas in ropeway systems, or transported masses in conveyor belts. Within numerical investigations we show that a larger number of discrete masses behaves similarly as the case of (continuously) distributed mass along the beam.

On Travelling Wave Modes of Axially Moving String and Beam

The traditional vibrational standing-wave modes of beams and strings show static overall contour with finite number of fixed nodes. The travelling wave modes are investigated in this study of axially moving string and beam although the solutions have been obtained in the literature. The travelling wave modes show time-varying contour instead of static contour. In the model of an axially moving string, only backward travelling wave modes are found and verified by experiments. Although there are n − 1 fixed nodes in the nth order mode, similar to the vibration of traditional static strings, the presence of travelling wave phenomenon is still spotted between any two adjacent nodes. In contrast to the stationary nodes of string modes, the occurrence of galloping nodes of axially moving beams is discovered: the nodes oscillate periodically during modal motions. Both forward and backward travelling wave phenomena are detected for the axially moving beam case. It is found that the ranges of forward travelling wave modes increase with the axially moving speed. It is also concluded that backward travelling wave modes can transform to the forward travelling wave modes as the transport speed surpasses the buckling critical speed.