Rigid Finite Element Method in Analysis of Dynamics of Offshore Structures

Following the depletion of land natural resources (oil, gas, minerals etc.), methods of obtaining them from beds of seas and oceans are gaining more and more importance. However, exploiting undersea resources poses a number of challenging technical problems pertaining to their extraction, transport and processing in a specific, inhospitable environment. Important groups of machines used in offshore engineering are cranes applied to reloading and assembly works and specialized devices for laying pipes which transport oil and gas. A characteristic feature of their working conditions is sea waves causing significant movement of the base on which offshore structures are installed. This is a phenomenon which must be taken into consideration in design of such machines.

Extraction of undersea natural resources, particularly oil and gas, has expedited the progress in offshore technology for a few decades, including the construction of platforms as well as the development of new extraction techniques and methods of laying underwater pipelines. Various types of cranes are an important aid in the construction of extraction infrastructure as well as its operation and servicing. The current chapter describes some most important elements of the infrastructure necessary for extracting oil and gas and methods of installation of offshore pipelines. Specific conditions pertaining to offshore cranes’ operation and their basic typology are presented.

A characteristic phenomenon in offshore engineering is the impact of water on individual devices and elements of dynamic systems. This impact is a very complex phenomenon composed, among other things, of wave action, sea currents, hydrostatic and hydrodynamic forces. These processes are difficult to describe, and there exists a handful of approaches to modelling them, which differ in the level of idealization [Newman J. N., 1977], [Mei C. C., 1989], [Faltinsen O. M., 1990]. A possibly simple description of the motion of an offshore structure’s base is often desirable, e.g. in the problems of control [Fossen T. I., 1994].

Basic models of bodies used in dynamic analysis of mechanical systems, including multibody systems, are a material point and a rigid body. They have, respectively, three and six degrees of freedom. To describe their positions either three or six independent coordinates must therefore be given. Usually, the position is given in a rectangular clockwise Cartesian system. It is then convenient to express the position of a point as a vector, also called a radius vector. To describe a body’s position, an additional coordinate system is attached to it in a fixed way. The position of this coordinate system, thus also of the body, is defined by giving the position vector of a selected point of the body (usually coinciding with the origin of the coordinate system attached to the body) and additionally a 3×3 matrix called a rotation matrix. In classical mechanics, displacement of a body from one position to another is treated as a superposition of two motions: translation and rotation. As a consequence, if a position vector of a point in the movable coordinate system attached to the body is given, and a position vector of this point in the reference system is to be determined, two mathematical operations are necessary: multiplication of the rotation matrices and addition of two vectors. By introducing the method of homogeneous transformations, the notation can be simplified. Such transformations are described by 4×4 matrices and take into account both a translation of a coordinate system and its rotation. The convenience of such interpretation makes it highly popular in robotics [Craig J. J., 1988], [Morecki A., et al., 2002], [Spong M. W., et al., 2006], [Jezierski E., 2006], which is a domain where multibody systems commonly occur.

The current chapter offers a basic introduction to describing positions and orientations of coordinate systems, transformations of vectors and joint coordinates. Application of homogeneous transformations and joint coordinates to describe the geometry of multibody systems is also discussed.

In the current chapter the main steps of determining the components of the equation of motion for open kinematic chains consisting of rigid links are presented [Wittbrodt E., et al., 2006]. The method is based on the Lagrange equations of the second order, homogeneous transformations and joint coordinates.

Individual links of a kinematic chain are often interconnected by elastic or damping (or both) elements. Among these are mainly: springs, dampers, absorbers, actuators. Components expressing the potential energy accumulated in such elements and its dissipation need to be introduced to the system’s equations of motion. The present chapter discusses a method of modelling spring-damping elements treated as massless objects. Constraint equations occurring when kinematic subchains are joined in certain systems are also presented.

In numerous technical applications the supporting structure of a device is assumed to be subjected to stresses within the limits of proportionality, i.e. where the Hooke’s law is applicable. It is also the case with offshore cranes. In the installation process of underwater pipelines with the reel method, however, the pipes are commonly deformed plastically when they are wound onto the reel. Furthermore, material exposed to prolonged deformation may show a tendency to creep. Hence, the present chapter which briefly introduces these models of construction materials: elasto-plastic and visco-elastic.

Actual kinematic chains commonly contain links whose flexibility greatly exceeds that of other links. It may then be necessary to take that flexibility into account. Booms of cranes and certain links of manipulators count among those. A large number of approaches in analysis of multibody systems can be found in literature with with one and more flexible links [Zienkiewicz O. C., 1972], [Wittbrodt E., 1983], [Wojciech S., 1984], [Huston R. L., Wanga Y., 1994], [Arteaga M. A., 1998], [Zienkiewicz O. C., Taylor R. L., 2000], [Berzeri M., et al., 2001], [Adamiec-Wójcik I., 2003], [Wittbrodt E., et al., 2006]. Chapter 9 introduces models of offshore cranes (a column one and an A-frame) which enable taking into account the flexibility of the supporting structure.

Each offshore structure is unique in the sense that it is built only after a customer with a specific need actually places an order. Design companies and manufacturers of engineering systems of this type are often small and medium enterprises, which cannot afford purchasing costly computer software packages for numerical computation involved in dynamics of mechanical systems. Therefore, they often employ custom, in-house dynamic models of the structures designed. In the present chapter, dynamic models of the following are presented: a gantry suited for relocating sets of BOP valves on an extraction platform, a column crane and a device for laying pipes on the seabed. The formulation of models thereof leverages the methods described in earlier chapters.

Dynamic analyses of mechanical systems are often considered together with problems related to their control. With traditional ways of operating machines it the operator who decides what the working motions are. In contemporary machines, it is becoming commonplace to support the process of control. Control systems based on microprocessor technology (programmable drivers, onboard computers) are supposed to facilitate human work or even replace it. They enable realization of various strategies unachievable with manual control. Automated control is used also in offshore structures, including cranes. The criteria of control strategies may be different, for example:

■ minimal duration of motion,

■ minimal consumption of energy,

■ accuracy of load positioning, including minimization of oscillations after the motion has ended,

■ minimization of dynamic loads,

■ stabilization of the load’s position,

■ minimization of the influence of sea waves on the device’s dynamics.

The present chapter describes the basics of the method of selecting the drive functions based on dynamic optimization. Control of the drum of a winch of an A-frame type crane allowing it to compensate for vertical movements of the base due to sea waves is presented. For an offshore jib crane, an auxiliary system is proposed enabling the load to be positioned in three directions. In the last part, a concept of active compensation of waves for a drum’s drive of a device for laying pipelines is discussed.

In the process of new designs of mechanical structures or systems and control strategy development, a great role for numerical modeling and simulation can evidently be identified. The most comprehensive verification provided by an experiment is naturally the best solution. It is however tedious and costly and in many cases difficult to perform. Offshore structures are often produced as single specimen for a specific order. Carrying out detailed empirical research would raise the final price of a device considerably. Therefore many design companies, including ones in the business of offshore engineering, are interested in access to appropriate calculation software. Such programmes have different purposes. Some of them are suited for strength analysis, others to simulate operation of a device or its control system. In addition to accurate calculations of precise values which are necessary when designing a given machine, companies also need quick and rough simulations, e.g. when preparing an offer (during initial negotiations with a counterparty). Calculations performed at the design stage are not significantly constrained by allowed duration of the simulation. On the other hand, control systems of devices must perform real-time calculations which requires using sufficiently numerically efficient models and methods. In many cases, to obtain satisfactory correspondence to reality flexibility of links must be taken into consideration by their discretisation. In some problems, nonlinear properties of the material or other specific conditions may be important. At present, different discretisation methods are used in calculations of machines’ dynamics. The most widely known is the finite element method. The authors of this book have been involved for many years in the development of the rigid finite element method. Based on their experience, it is their position that this method allows developing models of structures adequately reflecting the actual features of the dynamics involved while keeping the number of generalized coordinates small. It is also fairly simple to implement on a computer. It furthermore enables quick and convenient changes of the number of rigid finite elements in the discretized links. This allows both the calculations in real time (for small numbers of RFEs) necessary for control and more time consuming ones which better reflect the flexibility of the system (assuming more RFEs) to be carried out.