PhD Defence Karlijn van Voorthuizen | A generalised superelement formulation for fluid-conveying flexible multibody systems

A generalised superelement formulation for fluid-conveying flexible multibody systems

Karlijn Voorthuizen is a PhD student in the Department of Applied Mechanics & Data Analysis. (Co)Promotors are prof.dr.ing. B. Rosic, dr. M.I. Abdul Rasheed and dr.ir. J.P. Schilder from the Faculty of Engineering Technology.

As mechanical systems grow more and more complex, the need for advanced and efficient modelling techniques to support their design, optimization and control, among other engineering challenges, increases. In particular, many dynamic systems consist of multiple components which experience rigid body motions relative to one another. Modelling of these multibody systems requires formulations capable of handling the geometric non-linearity resulting from these large rigid body motions. Moreover, to accurately capture the dynamic behaviour, it is often essential to account for the elastic deformation of the components in a computationally efficient way.

If the elastic deformations remain small compared to the overall size of the body, an assumption that is valid in many practical applications, the floating frame of reference (FFR) formulation is a commonly used and computationally attractive modelling approach. This formulation separates the motion of each component into a rigid body motion and elastic deformations by introducing a floating frame that remains close to the component. As the elastic deformation is defined locally with respect to the floating frame, it can, under the assumption of small deformations, be expressed as a linear superposition of mode shapes following well-established model order reduction techniques. This enables efficient and accurate simulation of flexible multibody systems.

A subclass of flexible multibody system includes fluid-conveying pipes or hoses alongside structural components. For example, in firefighting drones, automated fuelling arms and concrete printing systems. These pipes can exhibit highly dynamic behaviour, making it essential for realistic and predictive modelling of such systems to capture the fluid-induced inertia effects and their interaction with the structural components. However, no existing fluid-conveying pipe elements are suitable for use in the FFR formulation. Therefore, in this work, a modelling approach is developed for fluid-conveying pipes within the FFR formulation. This enables the unified simulation of systems containing both structural components and fluid-conveying pipes within a single consistent formulation.

Implementing fluid-conveying pipe elements in the FFR formulation requires careful treatment of the inertia forces. For structural elements, implementation of the volume integrals that compose the inertia forces typically involves complicated summation operations of inertia shape integrals. An alternative method exists that defines the volume integrals using the consistent finite element (FE) mass matrix without the need for complicated summation operations. However, this method is limited to elements with purely translational degrees of freedom (DoF). Since fluid-conveying pipes are beam-like elements that include rotational DoF, this method is not suitable. To address this, a novel approach is developed that approximates the inertia forces using the consistent FE mass matrix in a way that is straightforward to implement and applicable to elements with rotational DoF.

Furthermore, in systems containing fluid-conveying pipes as well as those composed solely of structural elements, the DoF of the FFR formulation do not typically correspond to the global coordinates of points of interest (boundary nodes). Examples of such points are constraints, actuators, sensors and end-effectors. For many systems, expressing the formulation in terms of the global boundary node coordinates rather than global floating-frame coordinates and local elastic deformations offers definite advantages. This superelement approach reduces the overall model size, simplifies the incorporation of constraints yielding a system of ordinary differential equations (ODEs) rather than differential-algebraic equations (DAEs), and provides direct access to time histories at key locations. Various methods exist to transform the FFR formulation to a superelement formulation. Because these methods are typically presented as separate and individual approaches, their common basis is often unclear. A main difference between those methods lies in the definition of the position and orientation of the floating frame, as well as the implemented reduction technique, both being closely related. Therefore, this thesis presents a generalised framework that unifies these methods, offering greater flexibility and understanding in their application.

The developed methods provide a practical modelling framework for flexible multibody systems with embedded fluid-conveying components. Under the assumptions of small elastic deformations and incompressible, inviscid plug flow for the fluid, the proposed methods enable efficient computation of inertia forces using the consistent FE mass matrix exclusively. For fluid-conveying pipe elements, this also requires the derivatives of the interpolation functions along the centreline of the element. Additionally, the multibody system may be modelled using superelements that support a range of reference conditions and reduction bases.