11.1 Fundamental Terms
11.1.5 Kinematics
11.1.5.1 Coordinate Systems and Coordinates Definition 11.6:
A coordinate system is a set of three vectors which are orthogonal to each other. These build the basis of the vectors which are represented in the imaginable space R3. This vector
space base will be added to one origin in an Euclidian space. We refer to the new defined coordinate system as
{
; , ,}
= x y z
K O e e e (11.4)
Definition 11.7:
A coordinate system K is called inertial system if the basis vectors are time-constant. K is body fixed if it is fixed to one body point (e.g. the midpoint of the mass) and if the coordinates of the body points have the same coordinates with respect to this coordinate system.
Annotation 11.1:
The position and the orientation of a rigid body can be described by six coordinates, e.g. the specification of the translator coordinates x y, and z of the mass midpoint and by three Bryant angles α β γ, , towards the inertial system.
The rotation of the body fixed coordinate system with respect to the inertial system is only then defined if an arbitrary vector a≅a a axK Ky Kz T ≅a a axI yI zI T can be defined in both the coordinate systems.
This connection is described by a matrix product:
I = K
a Sa .
Note that aK and aI refer to the same vector. The elements of the transformation matrix (rotary matrix) S must be independently calculated from the considered case.
Example 11.2:
cos sin
The respective rotations around the x− and y− axis respectively will be described by the following rotary matrices:
The determined transformation matrices between Cartesian coordinate systems have the following characteristics
1( )γ T( ) (γ ( γ))
− = = −
S S S ,
i.e. the rotary matrix is an orthogonal matrix.
Generally, we can reach any position of a rigid body by means of a maximum three successive rotations (i.e. by a specification of three rotary axes and three rotary angles).
Depending on the order of the rotations, the rotary angles are referred as Briant angle (succession x y z, , sometimes also vice versa) and as Euler angle (succession z y z, , ). We obtain the resultant matrix by multiplying the singular matrices, e.g.:
( , , ) ( ) ( ) ( )
cos cos cos sin sin
sin sin cos cos sin sin sin sin cos cos sin cos cos sin cos sin sin cos sin sin sin cos cos cos
Kardan α β γ x α y β z γ
Fig. 11.6: Coordinate transformation by rotation around the z-axis.
If, in a coordinate system, the position and the orientation of multiple bodies K K1, 2,,Kp are described by the corresponding position vector to the mass midpoint ri and the rotary matrix Si with three angles α β γi, i, i (e.g. Briant and Euler angles), we combine these coordinate into a position vector
1, 1, ,1 1, 1, ,1 , p, p, p, p, p, p T x y z α β γ x y z α β γ
=
z (11.5)
Because of the given bearing elements, relations between the body coordinates to each other are defined by q algebraic equations:
( ) 0, 1, , , 6 ;
i i q q p
Φ z = = < (11.6)
The position of the MBS can be uniquely described by f =6p−q independent coordinates
1
T
y yf
=
y . In mechanics, these coordinates yi are referred to as generalized coordinates (cp. Section 11.1.4). This also means that in a general approach, it does not necessarily deal with coordinates in the imagined space.
Generalized coordinates must be
• independent,
• uniquely define the position of a system,
• reconcilable with constraints.
This coordinates are also referred to as minimal coordinates or position variables.
The introduction of minimal coordinate makes it necessary that the position vectors of the system bodies be expressed independent to y. The explicit representation of the position vector z out of (11.3) wrt y is a prerequisite.
In order to completely define the MKS, we must add the velocities y = y1,,yf T. Thus, one mechanical degree of freedom leads to two state quantities. The state vectors of a MKS results into:
The vector x
( )
t uniquely described the position and the velocity of the multi-body system at each time point t. The space describe by x is referred to as state space (cp. Section 2.2).11.1.5.2 Translation
The position of a rigid body i is described by a position vector to its mass midpoint ( , ).
i = i t
r r y (11.8)
We obtain the velocity and the acceleration by successive differentiation in respect to time
, 1, ,
where JTi is the 3× f functional matrix or the Jacobi matrix of translation. Under rheonom conditions, the local 3 1× velocity vector vi additionally appears. The vector ai is
describes the centrifugal, Coriolis and gyroscopic parts of the translation acceleration of the system.
Similar to the translation of mass midpoint, the rigid body can be restricted by constraints. In this case, we obtain:
, 1, , the centrifugal, Coriolis and gyroscopic parts of the rotations acceleration. In rheonom constraints, the local 3 1× velocity vector ωi furthermore appears.
11.1.5.3 Kinematical Differentials
The calculation of the Jacobi Matrices by means of formal differentiation according to equation (11.9) and (11.13) can be very complex. Even the calculation by symbolic formula manipulators (e.g. MATHEMATICA, MAPLE) can be problematic because the provisional results can be so voluminous that they practically can not be utilised later. That is why we
consider an alternative solution approach in which no analytical derivation need to be accomplished.
This shall be exemplified in the following on a translation.
First Derivation
Time derivatives ri of absolute coordinates of all bodies for arbitrary values of generalized velocities y, for given positions, can be specified by means of elementary-kinematic expressions. This is made possible by the incorporation of global kinematics into general mechanism.
Especially pseudo-velocities ri(j) for special dimensionless pseudo-velocities of generalized coordinates else only zeroes. Since the real time derivations ri are linear combinations of the generalized velocities y, these are again independent from each other and it yields:
(j) (j)
i i
j
=
∑
r r y . (11.15)
The comparison of equation (11.15) with equation (11.9) finally provides the simple rule:
i
(j)
column i
jth Jr =r (11.16)
Second derivation
At given position and velocity of the system the acceleration of all bodies can also be determined for arbitrary values of the generalized acceleration y by means of simple kinematic applications. Especially pseudo-accelerations ri for different generalized accelerations, i.e. where y=0 can be determined. Out of equation (11.10) we directly obtain
i= i
ar r. (11.17)
Equation (11.16) and equation (11.17) imply all the needed relations between the differentials of the generalized coordinated and the absolute coordinates of the bodies. They are only determinable by elementary-kinematic expressions (particularly laws of Relative Kinematics). Thus, they are referred to as kinematic differentials.
Kinematic Differentials
The time derivatives of the summarized absolute coordinates can now be split into their translator parts ri, ri. The corresponding statements are as follows:
(j) (j)
Equation (11.18) points out a further advantage: The kinematic approach allows a representation of the relations by means of “physical” vectors which are independent of the choice of the coordinate systems used. In contrast to that, in the analytical approach it is only possible to establish these relationships between differentials, by differentiating the specific components when they are all represented in a common coordinate system.
The transition to a component representation can be arbitrarily “retarded”, i.e. the choice of the coordinate system can be adjusted to the analyzed term, which also leads to a reduction in calculation efforts. This in a very compact form is made possible by the formulation of the equations of motion for general multi-body systems.
Example 11.3: cp. Section 11.1.5, “Equations of Motion of a Double Pendulum”.