Generation of the equations of motion of MBS

For a multibody system, the equations of motion are a set of ordinary differential equations (ODE) relating the accelerations to the time, the positions, the velocities, and the parameters of the system. There are various methods to derive the equations of motion of multibody systems. In order to be able to understand the differences between these methods, we will first highlight the roots of multibody system dynamics.

Roots of multibody system dynamics

As mentioned, the dynamics of multibody systems is based on classical mechanics. The most simple element of a multibody system is a free particle (or point mass) which can be treated by Newton's laws. The rigid body as principle element of a MBS was introduced in 1776 by Euler in his contribution entitled "Nova methodus motum corporum rigidarum determinandi" [59]. For the modeling of constraints and joints, Euler already used the free body principle resulting in reaction forces. The equations obtained are known in the multibody dynamics as Newton-Euler equations.

A system of constraint rigid bodies was considered in 1743 by d'Alembert in his "Traite de Dynamique"[52] where he distinguished between applied and reaction forces. D'Alembert called the reaction forces "lost forces" having the principle of virtual power in mind. A mathematical consistent formulation of d'Alembert's principle is due to Lagrange [149] combining d'Alemberts fundamental idea with the principle of virtual work. As a result a minimal set of ordinary differential equations of second order is found.

A systematic analysis of constraint mechanical systems was established in 1788 by Lagrange [149], too. The variational principle applied to the total kinetic and potential energy of the system considering its kinematical constraints and the corresponding generalized coordinates result in the Lagrangian equations of the first and second kind. Lagrange's equations of the first kind represent a set of differential-algebraic equations (DAE), while the second kind leads to a minimal set of ordinary differential equations (ODE).

An extension of d'Alembert's principle valid for holonomic systems only was presented in 1909 by Jourdain [125]. For non-holonomic systems the variations with respect to the translational and rotational velocities resulting in generalized velocities are required. Then, a minimal set of ordinary differential equations of first order is obtained. The approach for generalized velocities, identified as partial velocities, was also introduced by Kane and Levinson in 1985 [130]. The resulting Kane's equations represent a compact description of multibody systems. Interestingly, when Kane's equations were introduced in 1961, there was little - if any - interest in multibody dynamics. More details on history of classical mechanics including rigid body dynamics can be found in Pasler [213] and Szabo [288].

Summarizing, either one of the following methods can be used to derive the equations of motion of multibody systems:

• Hamilton (or state space form of Lagrange);

The question is: "Which method is most suitable for multibody system dynamics analysis?" [292], or in other words: "Which method formulates the equations of motion most efficiently? Kane's method (sometimes called "Langrange's form of d'Alembert's principle", "Jourdain's principle", or the "Principle of virtual power"), is preferable to the use of each of the other mentioned methods, particularly for the automated numerical analysis of large multibody systems [127], because it leads directly to the simplest possible equations of motion (i.e. Euler's dynamical equations when applied to a single rigid body) [131, 243]. The Euler dynamical equations are exceedingly compact, uncoupled in the highest derivatives, and free of trigonometric functions. Furthermore, since Kane's method focuses its attention on motions rather than on configurations, it offers the designer maximum physical insight [130].

Automatic versus manual generation

In general, the equations of motion can be generated either by hand, or automatically. Generating the equations of motion for complex multibody systems with a large number of degrees of freedom (DOF) is very time-consuming with pencil and paper, even if the best suited method is used. For this reason, various computer programs for automatic equation generation have been developed.

The first programs were on an numerical basis (i.e. the equations produced are given as implicit formulas, or in other words: numerical formalisms combine the generation and solution of the equations of motion). Examples are ADAMS [4, 56], SIMPACK, and SPACAR [121]. This approach has obviously a number of drawbacks: it requires starting the computations all over again for each new set of input data, and it does not provide all the insight provided by analytical equations. Moreover, for controller design numerically generated equations are of no help. Therefore, most recent program are analytical based (i.e. the equations produced are given as explicit formulas, or in other words: symbolic formalisms generate the equations of motion independent of the integration routine used). Examples are AUTOLEV [129, 234, 250], MESA VERDE, NEWEUL [253], and SD/FAST [109].

The symbolic representation has the advantage that the equations of motion are to be generated only once, and the expressions have only to be evaluated during timeintegration. The cost for the formulation of one symbolic set of equations is higher than once running through a numerical formalism. However, symbolic equations are an exact basis for the model under consideration, and combined with a package for numerical analysis, and simulation (e.g. MATLAB®/SlMULINK®) they result in ten or even more times faster computations than a purely numerical approach would require [62, 140, 201, 243]. See Schiehlen [252] for a more detailed comparison of important multibody system dynamics software.

We have decided to use SD/FAST® for the generation of the equations of motion of flexible wind turbines within DAWIDUM's mechanical module. The main reasons are twofold. First, SD/FAST®® uses Kane's method for the derivation of the equations of motion. Second, it is able to generate SIMULINK® MEX-files.

Main features of SD/FAST

SD/FAST1®, a product of Symbolic Dynamics, Inc., is a general-purpose multibody program whose function it is to create special-purpose simulation code employing explicit equations of motion for particular multibody configurations of interest. Computer symbol manipulation is used to simplify the general form of the equations of motion as appropriate to the system at hand (i.e. repeated terms are removed in order to arrive at the computationally simplest equations).

Any mechanical system that can be described as a collection of hinge-connected rigid bodies can be modeled in SD/FAST®®. The system topology can either be open-loop, tree, or closed-loop (see Fig. 3.17 for a graphical illustration). The systems can be "free-flying" (e.g. a spacecraft) or "grounded" (e.g. a wind turbine). The system complexity is limited to 300 rigid bodies and 1000 degrees of freedom.

Hydroelectric Storage Closed Loop
Figure 3.17: Topological structures of multibody systems: (a) open-loop (or chain), (b) tree, and (c) closed-loop.

The connection between the bodies, or between a body and the ground is established by means of joints (e.g. translational joints, rotational joints, or spherical joints). These connections typically impose constraints on the relative motion be tween the bodies. SD/FAST® has eleven pre-defined joint types, ranging from a weld joint (a zero DOF joint) to a free joint (a completely free, 6-DOF joint with three translational plus three rotational DOF's). It should be noted that all joints can be used as a tree or a loop joint. For a more detailed survey the reader is referred to the SD/FAST User's Manual [109].

SD/FAST®® generates the equations of motion describing the dynamic behavior of the system in either C or FORTRAN as requested, based on a user-written input file (sdinputs()), a system description file (System Descripion), and a user-written output file (sdoutputs()). In the system description file the basic geometry, mass properties, and gravity are specified. In the sdinputs() file, the inputs to the SD/FAST®® block are defined, while in the sdoutputs() file the outputs from the SD/FAST®® block are defined. Using the SD/FAST®® Interface for SIMULINK® it is possible to built out of these three files a single MEX-file, which appears as a SIMULINK®® block [255, 295]. This implies that it is possible to simulate virtually any mechanical system within the SIMULINK®® environment. This relationship is illustrated in Fig 3.18. Examples of these files can be found in Molenaar [189].

SD/FAST Block (Masked S-function MEX-file)

SD/FAST Block (Masked S-function MEX-file)

Figure 3.18: Relationship between the SD/FAST block interface files.
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  • medhanie
    What is the difference between D'Alembert's principle of motion and euler motion?
    6 years ago

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