Page 117 - Mechatronic Systems Modelling and Simulation with HDLs

106 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

Where the 6k × 6k matrix M takes the form

M = diag(m 1 E,..., m k E, I ,... , I k ) (6.20)
1
and forms a block diagonal matrix of masses and inertia tensors. The k denotes the
number of bodies. The 6k × nmatrix J is the global Jacobi matrix and consists of
a stack of k translational and k rotational 3 × n Jacobi matrices of the individual
bodies, where n is the number of generalised coordinates:
T
T
T
T
J = [J |··· |J |J |··· |J ] T (6.21)
T1 Tk R1 Rk
Again k is the 6k × 1 vector of gyroscopic and centrifugal forces as well as Cori-
olis forces. Finally, the applied forces and moments and the reaction forces and
e r
moments are located in the 6k × 1 vectors p and p :
eT T
e
eT
eT
eT
p = [F |··· |F |M |··· |M ] (6.22)
1 k 1 k
r
rT
rT
rT
rT T
p = [F |··· |F |M |··· |M ] (6.23)
1 k 1 k
Finally, we multiply the equation system (6.19) from the left with the transposed,
T
global Jacobi matrix J , so that it is formulated completely in generalised coordi-
nates. This yields equilibrium of forces in matrix form:
M¨q + k = Q (6.24)
T T
The product J MJ yields the mass matrix M. Similarly, J k yields k, the vector
T
e
of the generalised gyroscopic forces, and J F yields the vector of the generalised
forces Q.Here M is dependent upon the generalised coordinates q and t, and k
and Q are dependent upon q, ˙q and t. Last but not least, we should note that
T
the reaction forces are dispensed with as a result of the multiplication by J .We
therefore have a system of ordinary differential equations to solve because the
algebraic equations of the constraints have disappeared with the reaction forces.
Lagrange approach
The focus of the Newton–Euler approach described in the previous section was
the drawing up of Newton and Euler equations for each body and the conversion
of the resulting overall system into generalised coordinates so that the constraint
equations are dispensed with. The Lagrange approach takes a different and partic-
ularly elegant route. It starts from the premise that the generalised inertial forces
and the generalised applied forces cancel each other out. For the formulation of the
I
generalised inertial forces Q we require the total kinetic energy T of the system,
i
which of course must also be formulated in the form of generalised coordinates:
∂T d ∂T
I
Q = − (6.25)
i
∂q i dt ∂ ˙q i