Dynamics of Multibody Systems

The Seventeenth Century saw Galileo, Stevin, Huygens and Newton first shape the science we call rational mechanics. Statics had developed, at least in part, by the end of the Sixteenth Century. Thereafter, the early goals were to understand and describe in mathematical terms the translational motion of material bodies, usually idealized as particles.

In classical mechanics the motion of bodies takes place in Euclidean space. To describe geometrical and mechanical relations a frame of axes spanning the space is required. It is common practice to use the Cartesian frame as shown in Fig.1. It is characterized by the location of its origin O and by the orientation of three perpendicular axes, each of which has a scale of distance established by making a correspondence between the real numbers and the Euclidean distance of each point on the axis from the origin O. Orientation and scale may be visualized by three directed line segments e_α, α = 1, 2, 3¹ extending from the origin O to the points on the axes whose distance from O is 1. The set {e_α} is sometimes called an axis triad, and is denoted by symbol e (see also Eq.2.1.4-1 later). The complete frame, consisting of an origin and a triad, is designated by the notation {O, e} in which its location and orientation are explicit. When referring to both location and orientation, we use the term position with this generalized meaning. As regards the numbering of the triad axes, in this book we use only right-handed or dextral frames. A dextral frame is one in which a rotation about the 3-axis in a positive sense given by the right-hand rule moves the 1-axis toward the 2-axis, as shown by the bold curved arrow in Fig.1.

The position of a body with respect to any reference frame {O ¹,e¹} is known if the locations of all the points of the body are known. Any frame having a known, possibly variable, position in space can be used as such a reference. If the body is rigid, the description of its position is very simple. A frame {O ², e²} can be embedded rigidly in the body. In this frame all points of the body have constant and known coordinates whatever the position of the body. Hence the location of all the points of a rigid body with respect to the reference frame can be determined when the position of the body-fixed frame {O ², e²} is known with respect to the reference frame {O ¹, e¹}. This relative position can be considered to be composed of the location of O ² with respect to O ¹ and of the orientation of e² with respect to e¹. Motions of the rigid body are described by parameters characterizing the relative location and orientation of the two frames as a function of time. Motion during which the relative orientation of the triads e² and e¹ does not change is called translation. When O ² remains in a fixed location with respect to O ¹ the motion is called rotation. The most general motion of a rigid body can be composed of these two simple motions.

The velocity of a body with respect to a reference frame is known if a set of variables has been specified from which the velocity of any point of a body can be derived. These variables are called velocity variables. Possibilities for describing the velocities of points of a rigid body can be based on the several methods of representing position discussed in Ch.3. Here we examine some of the basic concepts for representing velocity of translation.

The position state and velocity state of a rigid body are known once position and velocity of every point of the body can be determined from sets of variables which have been called position variables and velocity variables. These variables may be independent or interrelated, but at least six independent variables are required to represent the corresponding states of an unconstrained rigid body.

Consider a rigid or gyrostatic body. Following the concepts of classical mechanics (for historical remarks, see [1]) we postulate that there exists an inertial frame {O ^I , e ^I } such that the mathematical equations based on the laws of Newton and Euler hold, namely: Here _I P is the linear momentum of the body with respect to the inertial frame and _I H is the corresponding angular momentum. Dots indicate time derivatives with respect to the inertial frame. Quantities F and _I L are resultants of force and torque on the body, the reference point for the torque being point O ^I . Equations 1 are the basic dynamical equations of motion, founded on independent laws of nature [1]. All subsequent manipulation simply recasts Eqs.1 in alternative forms. Initially we assume that the motions are unconstrained.

Consider a system of N rigid or gyrostatic bodies enumerated i = 1, 2, … N in an arbitrary way. The i ^th body is shown in Fig.1. Its mass is m ⁱ its center of mass is designated CM ⁱ and its inertia dyadic with respect to the center of mass is l ⁱ . Internal angular momentum is h ⁱ , nonzero for a gyrostat and zero for a rigid body. All vector bases are orthonormal and dextral. Base vectors e ⁱ = [e _α ⁱ ] are rigidly embedded in the body at CM ⁱ , establishing a body-fixed frame {CM ⁱ , e ⁱ }. An inertial frame is {O ^I , e ^I }.

Vector-dyadic forms of the kinematical and dynamical equations of a multibody system are given as Eqs.8.1.1–2 to 4. Representing vectors r ⁱ , ω ⁱ , v ⁱ and matrices A ⁱ by any suitable set of coordinates and velocities those vector-dyadic equations go over into corresponding matrix equations whose general forms are The specific forms of matrix X̂ _I in the kinematical equations, Eqs.2, and of matrices J, Q, and Λ in the dynamical equations, Eqs.3, depend on the choice of variables x _I and x _II .

To provide the modeling options for both kinematic and dynamic reference frames, the location of any body must be described with respect to points O ^o or CM ⁺ respectively.

Heretofore it has been assumed tacitly that the applied interactions depend only on the relative motion across the joints. Two important exceptions to this are interconnections with dry friction and with feedback control. In the former case the known interactions depend on the unknown constraint forces and torques. In the latter case they depend on additional states of the multibody system and the control system.

In some applications one is interested just in small motions of a multibody system about some known nominal motion. The small motions can be described by the linearized system equations. It is well known, of course, that there are special nonlinear phenomena such as self-excitation, limit cycles, entrainment of frequency and subharmonic oscillations which can appear even for small motions, but cannot be gotten as responses of linear systems. Nevertheless, the linearized equations often are sufficient to represent motion in the vicinity of a nominal motion. The basic equations are first collected here, and variables are then separated into those describing the known nominal motion and others describing the unknown small deviations from it. Ultimately, a linearized state space representation of the motion is developed.

Simulation of multibody system motion needs to be done for various problems of system dynamics investigation, as described in Ch.1. In particular, it is useful for system analysis and optimization, for identification of system parameters and for verification of assumptions on the modeling. Important approaches to system design involve the addition of feedback control based on real time plant models. In such cases and in situations where simulation is used to evaluate a cost function to be minimized by optimization, the computer time required for simulation must be minimized. Though less important, time-efficient simulation is desirable in all other cases as well. But it is not the only requirement when developing a general-purpose multibody computer program. Others might be to minimize storage requirements (especially when implementing on a small computer), portability of code or the amount of time and labor needed to develop the computer code.