Page 112 - Mechatronic Systems Modelling and Simulation with HDLs

6.2 MULTIBODY MECHANICS 101

Figure 6.1 Multibody system with four bodies, springs, dampers, suspensions, joints, and
inertial and body-related frames of reference

In the consideration of the structure of a multibody system, an abstracted descrip-
tion such as that given in Figure 6.1 is generally sufficient. Decisive factors are
the topography of the system and the parameters of the individual elements, such
as mass, centre of gravity, moments of inertia with respect to the main axes or the
point of application of forces.
For the consideration of point-shaped masses we start from Newton’s second
law, which identifies the product of mass m and acceleration in the x, y, and z
direction a x ,a y ,a z of a particle with the forces F x ,F y ,F z acting upon it:

F x = ma x , F y = ma y , F z = ma z (6.1)
Let us now consider a system of N particles. These may be subject to additional
limitations to their movement, so-called constraints. This state of affairs can be
taken into account by the introduction of the so-called reaction forces, which ensure
that the constraints are adhered to. The total force acting upon a body is divided
r
e
into two components, the force applied from outside F and the reaction force F .
i i
In total this yields the following equation system:
e
m i a ix = F + F r
ix ix
e
m i a iy = F + F r (i = 1, 2,... , N) (6.2)
iy iy
e
m i a iz = F + F r
iz iz
This can be formulated as a vector equation as follows:
e
m i a i = F + F r i (6.3)
i