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126 Mutibody Systems Approach to Vehicle Dynamics
The predicted values in [N th ] lie on the polynomial described by [N t ] but
are subject to further change as the state equations have not yet been corrected
using the process shown in Figure 3.39. During the corrector phase
the starting values of x and (dx/dt) are taken from [N th ]. As the value of x
changes during the Newton–Raphson iteration process the components of
the Nordsieck vector are updated by [N] where
[N] [c][x] (3.85)
and
[c] T [c 0 c 1 c 2 c 3 … c k ] (3.86)
The matrix [c] contains constants with values dependent on the polynomial
order k (Orlandea, 1973).
3.4 Systems of units
As with all engineering analysis it is important that consistent units are
used throughout any calculation or simulation exercise. For static analysis
the choice of a system of units can be quite forgiving as long as consistency
is observed. For dynamic analysis more care is required. At a basic level if
we consider the application of Newton’s second law to a body n with mass
m n and acceleration {A Gn } 1 at the mass centre G n we have with the vector
convention used here:
Σ{F n } 1 m n {A Gn } 1 (3.87)
It is important to note that equation (3.87) is only valid for certain systems
of units. For a metric system of SI units (force N, mass kg, acceleration
m/s 2 ) equation (3.87) can be used as it stands. The system of units
used mainly in this text, and popular in Europe, is to use mm as the unit of
choice for length. Clearly equation (3.87) would produce incorrect forces
unless the right-hand side of (3.87) was divided by a constant, in this case
1000, to ensure consistency.
Early users of programs such as MSC.ADAMS needed to define such a
constant when defining gravitational forces, even for dynamic models
operating in a zero gravity environment. As such users were reminded of
some of the basic fundamentals of dynamics when using such software. More
modern programs require the users to define only a set of units and apply the
correction internally. The following table is provided to show the required
corrections to various systems of units. Although for most users of MBS
this will be for reference only, advanced users developing subroutines that
link with MBS may still be required to implement a constant to ensure consistency.
An early term used to define the constant was the ‘gravitational
constant’ although a more recent and applicable definition is the ‘units
consistency factor’ or UCF where for any system of units the following
equation is valid for the given units consistency factor in Table 3.6:
Σ F
n
mn
AGn
UCF
{ } 1
{ } 1
(3.88)
In the following table the units consistency factor is for the systems of units
common to both metric and imperial systems.