Jens Wittenburg

Dynamics of multibody systems — A brief review

Abstract The subject of multibody dynamics is the simulation of large motions of complex systems of bodies interconnected by kinematical joints and by force elements such as springs, dampers and actuators. Typical technical multibody systems are vehicles, spacecraft, robots, mechanisms of all kinds, biomechanical systems and others. For simulations exact nonlinear equations of motion are required. The paper describes how equations can be formulated in an efficient way. One of the main ideas is the definition of a parameter describing the system topology. The resulting formalism can be programmed in such a way that a computer will automatically generate a minimal set of differential equations of motion for any multibody system. The program requires a standard set of input data describing the system topology as well as individual bodies, joints and force elements.

Angular Velocity. Angular Acceleration

This is the first chapter devoted not to position theory, but to continuous motion. The essential kinematical quantities are velocity, acceleration, angular velocity and angular acceleration.

Planar Four-Bar Mechanism

The solid lines in Fig. 17.1 are the links of a planar four-bar mechanism or briefly planar four-bar. The link lengths l (base or fixed link), r 1 (input link), r 2 (output link) and a (coupler) are free parameters. They determine, whether individual links can rotate relative to others full cycle (i.e., unlimited) or through an angle smaller than 2π.

Kinematic Differential Equations

The angular velocity ω of a body-fixed basis \( {\b{e}^2} \) relative to a reference basis \( {\b{e}^1} \) cannot, in general, be expressed as time derivative of some other vector. This is possible only in the special case when the direction of ω is constant in \( {\b{e}^2} \) and, thereby, also in \( {\b{e}^1} \).

Spatial Simple Closed Chains

Subject of this chapter is the kinematics of 1-d.o.f. spatial simple closed chains with axial joints. The simple closed chain is explained in Fig. 4.1b. Axial joints are the cylindrical joint (C), the revolute joint (R) and the prismatic joint (P). Overconstrained mechanisms are excluded from consideration. Then it is known from Theorem 4.1 that a 1-d.o.f. spatial simple closed chain has seven joint variables. The variable of either one revolute joint or one prismatic joint is declared independent. The problem to be solved is to determine the six dependent variables in terms of the independent variable and of constant mechanism parameters. The solution to this problem is of great engineering importance because 1-d.o.f. spatial simple closed chains are basic elements of machine mechanisms.

Finite Screw Displacement

Subject of this chapter are relationships between two positions of a rigid body without a fixed point. These two positions, referred to as initial and final position, respectively, are assumed to be arbitrary subject only to the restriction that the final position cannot be produced from the initial position by pure translation. Motions leading from the initial to the final position are not investigated.

Rotation about a Fixed Point. Reflection in a Plane

Subject of this chapter are relationships between two positions of a rigid body with a fixed point. Note: Motions leading from one position to the other are not investigated. Consequently, terms such as velocity or angular velocity do not occur.

Displacements in a Plane

Subject of this chapter are relationships between positions of a plane Σ in a reference plane Σ 0. Motions leading from one position to another are not considered. Displacements in a plane are translation, rotation, reflection and resultants of these three. All of them are special cases of the spatial displacements investigated in Chaps. 1 and 3. Consequently, theorems governing the latter remain valid, normally in simplified form. In addition, special theorems exist which are valid for planar displacements only.

Direct Kinematics of Tree-Structured Systems

In Fig. 11.1 a general spatial tree-structured system is shown. Its bodies i = 0, …, n and its joints i = 1, …, n are regularly labeled. By this is meant that joint i (i = 1, …, n) connects body n to a body b(i) < i. Example: In Fig. 11.1 b(5) = 3 and b(1) = 0. Regular labeling is always possible and, in general, in more than one way. The simplest tree-structure is a serial chain (a system without side branches). In a serial chain joint i (i = 1, …, n) connects body i to body b(i) = i – 1.

Theory of Gearing

Gears are wheels with teeth shaped so as to transmit rotational motion from one wheel to another. One of the wheels may be a rack (a wheel of infinite radius). In this case, rotational motion of a wheel is transformed into translatory motion of the rack or vice versa. The present chapter is restricted to gears transforming uniform motion of one wheel into uniform motion of the other wheel or rack. This restriction eliminates from investigation specialties such as elliptical wheels, Geneva wheels etc.