The framework for simulation of dynamics of mechanical aggregates

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The framework for simulation of dynamics of
mechanical aggregates.
Petr R. Ivankov, Nikolay P. Ivankov
February 1, 2008
Abstract
A framework for simulation of dynamics of mechanical aggregates has
been developed. This framework enables us to build model of aggregate
from models of its parts. Framework is a part of universal framework for
science and engineering.
1 Introduction
A set of engineering problems are concerned with aggregates simulation. As a
rule models of every part of aggregate are simple. However model of whole ag-
gregate may be very complicated. That is the purpose to which the framework
has been developed. You can download source code and evaluate examples from
http://www.genetibase.com/universal-engineering-framework-9.php. It
is worth to note that typical engineering problems have many aspects that lay
outside pure mechanics. Therefore this framework is a part of the universal engi-
neering framework http://www.genetibase.com/universal-engineering-framework-1.php,
that enables us to simulate complicated engineering phenomena.
2 Math Background.
Described framework has a simple background. There exists a set of connected
parts of aggregates. If we have two parts connected at the point A then linear
a and an angular ǫ accelerations of one part at point A coincides with corre-
sponding accelerations of the. If one part acts on another one with force F and
momentum M then the second part acts to first one with force −F and mo-
mentum −M . This is set of conditions is sufficient for construction of aggregate
equations. Let us consider that aggregate contains n parts numbered by 1, ..., n
and S is a set of such pairs (i, j) that i-th part is connected to j- th one. We
shall consider only the case when the graph corresponding to S [3] is a forest
[4]. According to Lagrangian mechanics [1] vector q of Generalized coordinates
[2] of i -th part satifies to the following ordinary differential equation:
1

q¨i = A(qi, q˙i) +Qi. (1)
Where A(qi, q˙i) is a part’s specific term and Qi is a generalized force. We
can decompose generalized force in the following way:
Qi = Qi0 +
∑
(i,j)∈S
Qij . (2)
Where Qi0 is an external force and Qij is generalized force caused by action of
part j to part i. Let us denote vector W as:
Wij =
(
Fij
Mij
)
. (3)
Where Fij and Mij are ordinary force and momentum of action of j - th part
to i - th one. Then we have
Qij = Bij(qi, q˙i)Wij . (4)
Where Bij is a connection specific matrix. Let us denote vector wij as
wij =
(
aij
ǫij
)
. (5)
Where aij and ǫij are linear and angular acceleration at the place of connec-
tion between i - th and j - th part. Then
wij = Cij(qi, q˙i) +Dij(qi, q˙i)q¨i. (6)
Using previous equations we can obtain a system of linear equations whose
variables are vectors q¨i i ∈ {1, ..., n} and Wij (i, j) ∈ S. Resolving of these
equations give as q¨i and full system of differential equation of aggregate. Note
that if we use numerical methods to solve such equations we should the latter
should be normalized. It means that we should make such corrections to gen-
eralized coordinates that coordinates and velocities in i - th and j - th part’s
connection point should be equal for all (i, j) ∈ S.
3 Program Implementation.
All equations of previous section are elementary exercises for first year engineer-
ing student. The main advantage of this work is its program implementation.
This advantage may be exhibited by the following example. Let us consider a
spacecraft with two nonrigid photovoltaics and three flywheels (See Figure 1).
This example contains spacecraft with 5 connections C 1, ..., C 5. Flywheels
F 1, F 2, F 3 and photovoltaics P 1, P 2, P 3 are connected to the spacecraft.
We can obtain full mechanical model of the spacecraft at once using designer
(See Figure 2).
2

Figure 1: A spacecraft with two photovoltaics and three flywheels
The idea of software is very simple. All mechanical objects should implement
interface [5] IAggregableMechanicalObject. You can download source code of this
interface from http://www.genetibase.com/universal-engineering-framework-9.php.
Then designer enables us to construct full models of aggregates from objects
those implement this interface.
4 Implemented mechanical objects.
Now three types of mechanical objects are implemented. You can develop your
own type of object. To do this you should implement IAggregableMechanicalOb-
ject interface. Let us consider types of implemented objects.
3

Figure 2: A full model of specectaft
4.1 Rigid body.
Rigid body is a simplest object of 6D dynamics. It is characterized by mass and
moment of inertia. Key properties of our rigid body implementation are places
of connections. Connections are defined by those coordinates and orientation.
You can define those coordinates X , Y , Z and components Q0, Q1, Q2, Q3 of
quaternions of orientation.
4.2 Flywheel.
Flywheel has general properties of rigid body and additional ones. Additional
properties of flywheel include a moment of inertia of a wheel, initial angular
velocity of a wheel and a moment that acts to wheel.
4.3 Elastic vibrations body.
Elastic vibrations body is a mechanical system of infinite degree of freedom.
Usually math model of this object contains finite degree of freedom with finite set
of valuable harmonic oscillations. Every harmonic oscillation may be described
by following second order ordinary differential equation:
Aq¨ + ǫq˙ + cq = Q. (7)
where q is a generalized coordinate and Q is a corresponding generalized force.
4

5 Advanced example. Controlled spaceraft.
Let us consider the following example. We have a spacecraft (See Figure 3).
Figure 3: A controlled spacecraft.
Currents of its equipment interact with Earth’s magnetic field. Spacecraft
has a photovoltaic that is an elastic vibrations body. The is a flywheel that
is used for angular stabilization of the spacecraft. Using the designer we may
simulate this situation in the way represented at Figure 4. Let us briefly explain
this situation. First of all we setup Coordinates of Spacecraft. We need
them for definition of gravitational acceleration and magnetic field. Then we
construct Magnetic Field by formulas. It is convenient to represent magnetic
field with a vector. Therefore we’ve used Field Vector. Its usage enables us
to define mechanical moment as a vector product of magnetic induction and
magnetic moment of spacecraft:
M = d×B. (8)
where d is magnetic dipole momentum of spacecraft and B is vector of magnetic
induction of th Earth. Vector product operation is supported by formula editor
embedded into framework. The Gravity component is used for simulation of
gravitational accelerations. Using above components we have defined inertial
accelerations and mechanical moments. Now let us constuct control system. Its
first element is a sensor. We will use a sensor of local vertical. To do this we de-
fine Spacecraft frame and Earth’s center frame. Relative position (6D)
5