2
2.1 Introduction
The application of a modern multibody systems computer program requires a good understanding of the underlying theory involved in the formulation and solution of the equations of motion. Due to the three-dimensional nature of the problem the theory is best described using vector algebra. In this chapter the starting point will be the basic definition of a vector and an explanation of the notation that will be used throughout this text. The vector theory will be developed to demonstrate, using examples based on suspen- sion systems, the calculation of new geometry and changes in body orienta- tion, such as the steer change in a road wheel during vertical motion relative to the vehicle body. This will be extended to show how velocities and accel- erations may be determined throughout a linked three-dimensional system of rigid bodies. The definition of forces and moments will lead through to the definition of the full dynamic formulations typically used in a multibody systems analysis code.
2.2 Theory of vectors
2.2.1 Position and relative position vectors
Consider the initial definition of the position vector that defines the loca- tion of point P in Figure 2.1.
In this case the vector that defines the position of P relative to the reference frame O1may be completely described in terms of its components with magnitude Px, Pyand Pz. The directions of the components are defined by
P {RP}1/1
Z1
Px X1
Y1
Pz
O1
Py Fig. 2.1 Position vector
attaching the appropriate sign to their magnitudes:
(2.1)
The use of brackets { } here is a shorthand representation of a column matrix and hence a vector. Note that it does not follow any quantity that can be expressed as the terms in a column matrix is also a vector.
In writing the vector {RP}1/1the upper suffix indicates that the vector is measured relative to the axes of reference frame O1. In order to measure a vector it is necessary to determine its magnitude and direction relative to the given axes, in this case O1. It is then necessary to resolve it into com- ponents parallel to the axes of some reference frame that may be different from that used for measurement as shown in Figure 2.2.
In this case we would write {RP}1/2where the lower suffix appended to {RP}1/2indicates the frame O2in which the components are resolved. We can also say that in this case the vector is referred to O2. Note that in most cases the two reference frames are the same and we would abbreviate {RP}1/1 to {RP}1.
It is now possible in Figure 2.3 to introduce the concept of a relative pos- ition vector {RPQ}1. The vector {RPQ}1is the vector from Q to P. It can also be described as the vector that describes the position of P relative to Q.
{R }/
Px Py Pz
P 1 1
24 Multibody Systems Approach to Vehicle Dynamics
P
Py2 Px2 {RP}1/2
Z1
X1
Y1 O1
Pz2
X2
Y2
Z2 O2
Fig. 2.2 Resolution of position vector components
P Z1
X1
Y1 O1
Q
{RQ}1
{RPQ}1
{RP}1
Fig. 2.3 Relative position vector
These vectors obey the triangle law for the addition and subtraction of vec- tors, which means that
(2.2) It also follows that we can write
(2.3) Application of Pythagoras’ theorem will yield the magnitude |RP| of the vector {RP}1as follows:
(2.4) Similarly the magnitude |RPQ| of the relative position vector {RPQ}1can be obtained using
(2.5) Consider now the angles X, Yand Zwhich the vector {RP}1makes with each of the X, Yand Zaxes of frame O1as shown in Figure 2.4. This gives the direction cosines lx, lyand lzof vector {RP}1where
These direction cosines are components of the vector {lP}1where
(2.7) It can be seen that {lP}1has unit magnitude and is therefore a unit vector.
{ { }
| |
l R
p RP
P
} 1
lx x Px
R
ly y Py
R
lz z Pz
R
P
P
P
cos cos cos
| |
| |
| |
RPQ (Px Qx)2 (Py Qy)2 (Pz Qz)2
RP Px2 Py2 Px2
{ }
{ }
R R R
R R R
QP Q P
Q P QP
1 1
1
{ } { }
or { } { }
1
1 1
{ }
{ }
R R R
R R R
PQ P Q
P Q PQ
1 1
{ } { }
or { } { }
1 1
1 1
Kinematics and dynamics of rigid bodies 25
P {RP}1
Z1
Px X1
Y1
Pz
O1
Py θz
θy θx
Fig. 2.4 Direction cosines
(2.6)
2.2.2 The dot (scalar) product
The dot, or scalar, product {A}1• {B}1of the vectors {A}1and {B}1yields a scalar Cwith magnitude equal to the product of the magnitude of each vec- tor and the cosine of the angle between them.
Thus:
{A}1• {B}1|C||A| |B| cos (2.8)
The calculation of {A}1• {B}1requires the solution of
{A}1• {B}1{A}1T{B}1AxBxAyByAzBz (2.9)
(2.10)
The T superscript in {A}T1indicates that the vector is transposed.
Clearly {A}1• {B}1{B}1• {A}1and the dot product is a commutative operation. The physical significance of the dot product will become appar- ent later but at this stage it can be seen that the angle between two vectors {A}1and {B}1can be obtained from
(2.11) A particular case which is useful in the formulation of constraints repre- senting joints and the like is the situation when {A}1and {B}1 are perpen- dicular making cos 0.
As can be seen in Figure 2.6 the equation that enforces the perpendicular- ity of the two spindles in the universal joint can be obtained from
{A}1• {B}10 (2.12)
2.2.3 The cross (vector) product
The cross, or vector, product of two vectors, {A}1and {B}1, is another vec- tor {C}1given by
{C}1{A}1{B}1 (2.13)
The vector {C}1is perpendicular to the plane containing {A}1and {B}1as shown in Figure 2.7.
cos { } A1 B A B
• { }
| | | |
1
where { }B and { } [ ]
Bx By Bz
AT Ax Ay Az
1 1
26 Multibody Systems Approach to Vehicle Dynamics
{B}1
{A}1 θ
Fig. 2.5 Vector dot product