The framework for simulation of dynamics of mechanical aggregates
Genetibase
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 aggregate 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 engineering framework
http://www.genetibase.com/universal-engineering-framework-1.php,
that enables us to simulate complicated engineering phenomena.
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 ε accelerations of one part at point A coincides with corresponding 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 momentum -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 is a forest. According to Lagrangian mechanics vector q of Generalized coordinates of i-th part satisfies to the following ordinary differential equation:
Where 
is a part's specific term and
is a generalized force. We can decompose
generalized force in the following way:
Where 
is an external force and
is generalized force caused by action of
part j to part i. Let us denote vector W as:
Where 
and
are ordinary force and momentum of action of
j- th part to i- th one. Then we have
Where 
is a connection specific matrix. Let us
denote vector 
as
Where 
and
are linear and angular acceleration at
the place of connection between i- th and j- th part. Then
Using previous equations we can obtain a system of linear equations whose variables are vectors
, i in {1, ..., n} and
. Resolving of these
equations give as 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 generalized coordinates that coordinates and
velocities in i- th and j- th part's connection point should be equal for all (i, j) in S.
Program Implementation
All equations of previous section are elementary exercises for first year engineering 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 C1, ..., C5. Flywheels F1, F2, F3 and photovoltaics P1, P2, P3 are connected to the spacecraft. We can obtain full mechanical model of the spacecraft at once using designer (See Figure 2).
Figure 1: A spacecraft with two photovoltaics and three flywheels
The idea of software is very simple. All mechanical objects should implement interface 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.
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 IAggregableMechanicalObject interface. Let us consider types of implemented objects.
Figure 2: A full model of specectaft
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
of quaternions of orientation.
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.
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:
where q is a generalized coordinate and Q is a corresponding generalized force.
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. There 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:
where d is magnetic dipole momentum of spacecraft and B is vector of magnetic induction of the 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 define Spacecraft frame and Earth's center frame. Relative position (6D)
Figure 4: A controlled spacecraft design.
enables us to simulate the sensor of local vertical. The Control law uses it. And at last moment of Flywheel is calculated by the Control law. You can download this example from http://www.genetibase.com/universal-engineering-framework-9.php and evaluate it.
Conclusion
This framewok is very useful for simulation of complicated phenomena that include mechanical ones. Authors of this article think that present-day engineering software should (potentialy) include all kinds of physical phenomenons. Authors would like to aknowldge the Genetibase company that kindly sited source code and examples on their site.
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