System degree of freedom

In the case of a totally unconstrained 'free' system of p rigid bodies, the degrees of freedom (DOF) of the system are 6p. It stems from the fact that 6p independent coordinates are necessary to describe the kinematical configuration (the position and orientation of the system's bodies) uniquely [#!schi1990!#]. In Table 4.1 typical constraint elements and their characteristics are depicted.

Table 4.1 Types and valency of bearings

If q holonomic constraints are added to the system, its degree of freedom is reduced.

• If all q constraints are independent, the degrees of freedom of the system are f = 6p - q.

• If only r of the q constraints are independent, the degree of freedom of the system is f = 6p - r. The number r of independent constraints is equal to the rank of matrix Q in equation (4.82).

If the system possesses f DOF, there are f independent coordinates necessary to describe the configuration of the system uniquely. These coordinates are called generalised coordinates and can be chosen in different ways appropriate to the particular problem. The choice of a set of generalised coordinates may strongly influence the process of mathematical modelling as well as the process of solving the equations (see Chapter 4.3). Degree of freedom of double pendulum is determined in Figure 4.4.

Figure 4.4: Degree of freedom of mechanical system: double pendulum

Figure 4.5: Forces in mechanical system: a double pendulum