Multibody system

The systematic treatment of the dynamic behavior of interconnected bodies has led to a large number of important multibody formalisms in the field of mechanics. The simplest bodies or elements of a multibody system were treated by Newton (free particle) and Euler (rigid body). Euler introduced reaction forces between bodies. Later, a series of formalisms were derived, only to mention Lagrange’s formalisms based on minimal coordinates and a second formulation that introduces constraints.

Basically, the motion of bodies is described by their kinematic behavior. The dynamic behavior results from the equilibrium of applied forces and the rate of change of momentum. Nowadays, the term multibody system is related to a large number of engineering fields of research, especially in robotics and vehicle dynamics. As an important feature, multibody system formalisms usually offer an algorithmic, computer-aided way to model, analyze, simulate and optimize the arbitrary motion of possibly thousands of interconnected bodies.

While single bodies or parts of a mechanical system are studied in detail with finite element methods, the behavior of the whole multibody system is usually studied with multibody system methods within the following areas:

• Aerospace engineering (helicopter, landing gears, behavior of machines under different gravity conditions)
• Biomechanics
• Combustion engine, gears and transmissions, chain drive, belt drive
• Dynamic simulation
• Hoist, conveyor, paper mill
• Military applications
• Particle simulation (granular media, sand, molecules)
• Physics engine
• Robotics
• Vehicle simulation (vehicle dynamics, rapid prototyping of vehicles, improvement of stability, comfort optimization, improvement of efficiency, ...)

The following example shows a typical multibody system. It is usually denoted as slider-crank mechanism. The mechanism is used to transform rotational motion into translational motion by means of a rotating driving beam, a connection rod and a sliding body. In the present example, a flexible body is used for the connection rod. The sliding mass is not allowed to rotate and three revolute joints are used to connect the bodies. While each body has six degrees of freedom in space, the kinematical conditions lead to one degree of freedom for the whole system.

The motion of the mechanism can be viewed in the following gif animation

A body is usually considered to be a rigid or flexible part of a mechanical system (not to be confused with the human body). An example of a body is the arm of a robot, a wheel or axle in a car or the human forearm. A link is the connection of two or more bodies, or a body with the ground. The link is defined by certain (kinematical) constraints that restrict the relative motion of the bodies. Typical constraints are:

• cardan joint or Universal Joint ; 4 kinematical constraints
• prismatic joint; relative displacement along one axis is allowed, constrains relative rotation; implies 5 kinematical constraints
• revolute joint; only one relative rotation is allowed; implies 5 kinematical constraints; see the example above
• spherical joint; constrains relative displacements in one point, relative rotation is allowed; implies 3 kinematical constraints

There are two important terms in multibody systems: degree of freedom and constraint condition.

The equations of motion are used to describe the dynamic behavior of a multibody system. Each multibody system formulation may lead to a different mathematical appearance of the equations of motion while the physics behind is the same. The motion of the constrained bodies is described by means of equations that result basically from Newton’s second law. The equations are written for general motion of the single bodies with the addition of constraint conditions. Usually the equations of motions are derived from the Newton-Euler equations or Lagrange’s equations.

The motion of rigid bodies is described by means of

${\displaystyle \mathbf {M(q)} {\ddot {\mathbf {q} }}-\mathbf {Q} _{v}+\mathbf {C_{q}} ^{T}\mathbf {\lambda } =\mathbf {F} ,}$ (1)

${\displaystyle \mathbf {C} (\mathbf {q} ,{\dot {\mathbf {q} }})=0}$ (2)

These types of equations of motion are based on so-called redundant coordinates, because the equations use more coordinates than degrees of freedom of the underlying system. The generalized coordinates are denoted by ${\displaystyle \mathbf {q} }$, the mass matrix is represented by ${\displaystyle \mathbf {M} (\mathbf {q} )}$ which may depend on the generalized coordinates. ${\displaystyle \mathbf {C} }$ represents the constraint conditions and the matrix ${\displaystyle \mathbf {C_{q}} }$ (sometimes termed the Jacobian) is the derivative of the constraint conditions with respect to the coordinates. This matrix is used to apply constraint forces ${\displaystyle \mathbf {\lambda } }$ to the according equations of the bodies. The components of the vector ${\displaystyle \mathbf {\lambda } }$ are also denoted as Lagrange multipliers. In a rigid body, possible coordinates could be split into two parts,

${\displaystyle \mathbf {q} =\left[\mathbf {u} \quad \mathbf {\Psi } \right]^{T}}$

where ${\displaystyle \mathbf {u} }$ represents translations and ${\displaystyle \mathbf {\Psi } }$ describes the rotations.

There are several cases in which it is necessary to consider the flexibility of the bodies. For example in cases where flexibility plays a fundamental role in kinematics as well as in compliant mechanisms.

Flexibility could be take in account in different way. There are three main approaches:

• Discrete flexible multibody, the flexible body is divided into a set of rigid bodies connected by elastic stiffnesses representative of the body's elasticity
• Modal condensation, in which elasticity is described through a finite number of modes of vibration of the body by exploiting the degrees of freedom linked to the amplitude of the mode
• Full flex, all the flexibility of the body is taken into account by discretize body in sub elements with singles displacement linked from elastic material properties

• Dynamic simulation
• Multibody simulation (solution techniques)
• Physics engine

• J. Wittenburg, Dynamics of Systems of Rigid Bodies, Teubner, Stuttgart (1977).
• J. Wittenburg, Dynamics of Multibody Systems, Berlin, Springer (2008).
• K. Magnus, Dynamics of multibody systems, Springer Verlag, Berlin (1978).
• P.E. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall (1988).
• E.J. Haug, Computer-Aided Kinematics and Dynamics of Mechanical Systems, Allyn and Bacon, Boston (1989).
• H. Bremer and F. Pfeiffer, Elastische Mehrkörpersysteme, B. G. Teubner, Stuttgart, Germany (1992).
• J. García de Jalón, E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems - The Real-Time Challenge, Springer-Verlag, New York (1994).
• A.A. Shabana, Dynamics of multibody systems, Second Edition, John Wiley & Sons (1998).
• M. Géradin, A. Cardona, Flexible multibody dynamics – A finite element approach, Wiley, New York (2001).
• E. Eich-Soellner, C. Führer, Numerical Methods in Multibody Dynamics, Teubner, Stuttgart, 1998 (reprint Lund, 2008).
• T. Wasfy and A. Noor, "Computational strategies for flexible multibody systems," ASME. Appl. Mech. Rev. 2003;56(6):553-613. doi:10.1115/1.1590354.

• http://real.uwaterloo.ca/~mbody/ Collected links of John McPhee