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Dynamic behaviour and stability of marine propulsion systems M Figari* and M Altosole Naval Architecture and Marine Engineering (DINAV), Universita` di Genova, Genova, Italy The manuscript was received on 10 March 2006 and was accepted after revision for publication on 22 June 2007. DOI: 10.1243/14750902JEME58
Abstract: The paper describes an approach used to study the dynamic behaviour of marine propulsion systems. The method consists of three main steps: analytical modelling of the ship propulsion system, stability analysis of the system, and dynamic behaviour of the propulsion plant. The model is based on non-linear first-order differential equations. The concepts of ‘geometric non-linear dynamics’ are used to highlight some important properties of the model. One of the main advantages of the method is that it enables some important dynamic properties of the propulsion system to be highlighted without solving the differential equations of motion. In particular circumstances an analytical solution of the proposed model is possible; the solution includes the steady state behaviour of the system, which is useful for the engine-propulsion matching. Keywords: marine propulsion, engine matching, dynamic analysis, stability analysis
1 INTRODUCTION Design and optimization of the propulsion system are crucial tasks of ship design; in fact, the behaviour of the propulsion system is a key aspect of the global behaviour of a ship, particularly if the ship is a naval vessel. Marine propulsion systems are required to behave efficiently and safely, not only in steady state conditions, but also in transient conditions, where the dynamic behaviours of the single component and of the whole system play a crucial role. Different operational speeds, acceleration, deceleration, crash stop, heavy turning, and faults are some examples of transient situations that a propulsion system has to sustain without reducing the ship safety and reliability. Generally, the ship has to demonstrate the ability to perform some of the above-mentioned manoeuvres by sea trials. One of the most useful ways for the prediction of the dynamic behaviour of marine propulsion systems * Corresponding author: Naval Architecture and Marine Engineering, Universita` di Genova, Via Montallegro 1, Genova, 16145, Italy. email: fi
[email protected]
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is numerical simulation. It gives the possibility to foresee, at the design stage, the behaviour of the ship propulsion plant during manoeuvres; it gives the designer the possibility to optimize the choice of the system parameters (choice of a suitable pitch–speed combinator law, engine governor calibration, scantling of the shaft line, etc.) and allows good development of the propulsion control systems in order to prevent engine and mechanical overloads or faults. Numerical simulation of propulsion systems requires a detailed knowledge of the system and great efforts in human and computational resources. Because of the abovementioned factors, at the moment simulation is not a usual task of ship propulsion plant design. References [1] to [13] give an idea of the simulation approach adopted by the present authors and co-workers and some realized applications. In the present paper a different approach to studying the dynamic behaviour of the marine propulsion systems is adopted. In particular a possible application to marine propulsion systems of the so-called ‘geometrical approach’ to non-linear dynamics is described. The concepts of attractor, phase portrait, stability, and bifurcation may help the propulsion plant designer to understand the phenomena under study and the effects of different design choices. The
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focus of the paper is on the preliminary design phase when the proposed method can be used to assess the engine–propulsor matching and the propulsion system transient performances. In particular, the equilibrium of the non-linear model gives an indication of the engine matching and its optimization with respect to energy requirements. The stability analysis suggests information about the overall dynamic properties of the system, without solving the equations. The solutions of the non-linear model for different initial conditions give an indication concerning the dynamic effects for the sizing of the propulsion drive train. This study is a step in an ongoing investigation about the design of a proper control system for marine propulsion plants. The main purposes of the investigation are to evaluate the effectiveness of methods able to characterize the stability of the propulsion systems, to study the effect of system modelling on the results, and to determine the role of the control system in marine propulsion dynamics. Considering the complexity of the whole problem, in the first phase of the work, attention has been focused on the analytical structure of the developed non-linear equations, without taking into account the influence of the control system. This is the principal topic of the present paper; however, the investigation of the whole interaction among prime movers, propulsors, hull, and control system has already been started and first results will be readily available.
2 BASIC CONCEPTS OF NON-LINEAR DYNAMICS The geometrical theory of non-linear dynamics has spawned a multitude of specialized concept and terminologies not familiar to most engineers. This paragraph provides a quick overview of the central concepts and ideas of the non-linear dynamics with the aim of highlighting a few items of particular importance to the present application. Interested readers should refer to the ample and available specific literature [14, 15].
t as an extra phase coordinate governed by the dummy equation t=1. The evolution of the system is given by the time histories x(t) in the n-dimensional phase space Rn, spanned by the n components of vector x. The geometry of phase space contains important information about the system behaviour. The vector field in the phase space consists, for each point {x }, of a vector of components { f } evaluated in j i a point {x } at instant t. For autonomous systems the j vector field does not change in direction or magnitude with time. The evolution of the dynamic system is completely characterized by a stationary vector field. An equilibrium or fixed point of equation (1) is characterized by x˙(t)= f (x )=0 0
Fixed points can be stable or unstable; the local stability analysis near equilibrium point starts with the linearized equations describing small variations around the point. In order to analyse the dynamic behaviour of the system, it is useful to identify the trajectories of the solutions depending on the starting conditions; these trajectories fill the phase space to form a phase portrait. 2.2 Two-dimensional linear systems Consider the linear autonomous two-dimensional system x˙ =ax+by y˙ =cx+dy (3) where a, b, c, and d are constant parameters. It can be written in the matrix form x˙ =Ax
2.1 Dynamic systems
x˙(t)= f (x)
(1)
Non-autonomous equations, in which time t appears explicitly, can be rendered autonomous by identifying
(4)
where A=
The general type of a continuous dynamic system pertinent to ship applications is described by an autonomous set of n first-order ordinary differential equations (ODEs)
(2)
A B a b c
d
,
x=
AB x y
Assume that each solution is of the type x(t)=eltz
(5)
where z≠0 is the eigenvector of A, and l is the corresponding eigenvalue. Substituting equation (5) into equation (4) gives the eigenvalue equation Az=lz
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The eigenvalues of A are the solutions of the characteristic equation det
A
a−l
b
c
d−l
B
=0 [ l2−tl+D=0
(D>0, t>+√4D) and the region D
