**Full paper in compressed Postscript *.ps.gz
**

**In the report long-term experience of the authors on development of algorithms
and automation of a research of complicated mechanical and controlled systems
is considered. We have created specialized systems of computer algebra
and software packages "Dynamics", "Mechanic", "Normalization" possibility
and the algorithms of which are described in [1-5 and other].
The base of the algorithms, which are realized in these packages,
was formed by classical methods of analytical mechanics and stability theory.
**

**The indicated software allows after geometric description of a system of rigid
and deformable bodies to make their finite-dimensional models as the Lagrange
equations of the second kind (in generalized coordinate) and Euler - Lagrange
(in quasi-coordinates) ones and also will obtain the linear equations in a
neighbourhood of steady-state motion.
The application packages differ from another software by that
they contain large units, permitting to carry out in a symbolic form
qualitative researches of the differential equations : to build the first
integrals, linear and quadratic in generalized velocities (quasi-velocities) ;
for linear (ignorable) and quadratic integrals to determine invariant manifolds
of steady motions and to investigate their stability ; to fulfill
normalization in Poincaré sense in a neighbourhood of singular points
of the equations.
**

**While solving of these problems the computer algebra system (CAS) fulfills :
operations of matrix - vectorial algebra, partial differentiation,
expansion of functions in to Taylor series,
selection of main diagonal minors, calculation of determinant,
evaluation of conditions of a property of having fixed sign of the quadratic
forms, replacement and substitution, removal of brackets, collecting terms,
trigonometric transformations, symbolic-numerical interface and other.
The algorithms of analytical transformations are simplified considering the
specificity of the expressions of data domain.
**

**The developed software allows to automatize, and consequently,
essentially to speed up processes of modelling and dynamic analysis
of complicated systems, to avoid errors at all stages of researches.
**

**At the moment we develop the software "Analysis" for IBM PC/AT on basis
of system "Mathematica" [6,7]. In the report we consider some problems
and difficulties, connected to application of modern systems of computer
algebra for solving mentioned above and other problems of common mechanics
and stability theory.
**

**For the package "Analysis" new algorithms for a research in symbolic-numerical
form of stability and stabilization of mechanical systems under an
operation of gyroscopic, dissipative, conservative and nonconservative
positionary forces are developed [8].
Experience, accumulated at creation and maintenance of systems for the analysis
of mechanical systems, was transferred by authors to electrical and
electromechanical systems.
The following operations are fulfilled for these systems :
**

**
- the graph of a circuit is analyzed ;
- Routh or Hamilton function is created ;
- the equations of motion are written in the Lagrange, Hamilton or Routh form ;
- the first integrals are created ;
- linear in velocities integrals are reduced to ignorable ones.
**

**Figure 1.
**

**Our experience with symbolic computation packages allows us to conclude
that CAS are perspective tool for researches in the field of theoretical
mechanics. Here we will not concentrate on specific software implementations.
Instead we will give a brief description of the functionality of mentioned
software packages ( Fig. 1 ).
**

**1.1. Model of the mechanical system.**
The mechanical system is a system of bodies **connected by one-two-three-degree joints, i.e. for every ****there exists the point **** и
****, or joints allowing
translational displacements **** relative to **** ( Fig. 2 ).
**

**
**

**Figure 2.
**

**
The body **** is a carrier, **** is carried.
Let us introduce the systems of coordinates :
a system **** is connected with **** (with hardening body
which will be received if all deformations would be equal to zero);
a system **** is connected with **** in a
small vicinity of point **** and such that if **** is an absolute
rigid body then **** and **** are parallel.
The vocation of **** in the inertial space ****(or in some system of coordinates **** with given motion)
assigns by coordinates of point **** and by matrix of rotation
**** of axes ****.
The location of body **** assigns by the matrix of rotation
**** with reference to
**** or by coordinates of pole
****. It is assumed that for non-rigid body **** in every point
**** of body
****the deviation reflected due to deformation is represented by a vector
**

**1.1.1. The kinetic energy of a system.**
The kinetic energy of a system is calculated as a sum of kinetic energies
** for every body ****, connected with ****:
**

**
**

**
**

**
**

**
is the relative velocity of point ,
if moves translationally (or freely) relative to ;
**

**
**

**
**

**
**

**1.1.2. The force function of a system.**
Let the system is under Newton's gravity ( approximate ) to the motionless
centre **. The force function of a system equals a sum of
force functions **** of each body ****:
**

**The force function of a body **** in field of constant gravity
is calculated under the formula :
**

**If body **** is non-rigid, then the force function of deformation
is assumed as the quadratic form of the generalized coordinates of deformation :
**

**1.2. Description of an electric circuit.**
Electromechanical analogies [10] allow to use the same
apparatus - methods of analytical mechanics - for describing
and investigation of both mechanical systems and electric
circuits.

**Consider a linear electric circuit, in which
resistors (R), inductors (L), capacitors (C),
power sources of current (I) and voltage (E) are
interconnected arbitrarily. For the purpose of describing the
circuit let us choose a set of independent variables
characterizing its state at any time moment.
Such variables may be represented by either currents in
the loops or voltages in the nodes or else
currents and nodal voltages. Selection of a set of variables
defines the structure of the equations.
**

**Consider an algorithm of constructing the Lagrangian for
the electric circuit. A set of currents will be used as
the state variables. The computations are conducted in
the two stages. On the 1st stage, the set of state variables
of the electrical circuit is determined by the method of
loop currents; on the 2d - the Lagrangian is computed.
**

**The general idea of the method of loop currents consists
in decomposing the electrical circuit into independent geometric
loops and assigning the current in each of the loops.
This problem may be solved via finding a set of
fundamental cycles in a graph.
**

**Let us use the graph of the electrical circuit ( Fig. 3 ) for
finding the fundamental cycles.
**

**Let **** be a list of constructed fundamental
cycles, S contains a list of contiguous vertices
for each of the graph's vertex, D a list "free" graph's edges:
**

**Choose the 1st element **** from D and remove
it from the list. Using S find the vertex **

**If on the **** th step it appears that
there is not any unused vertex contiguous with ****, we
return to the vertex **** and continue the process.
When **** the construction of the cycle is over. The
latter is always possible since the graph corresponds to the
closed electrical circuit. The constructed cycle is stored in
the list ****.
All the elements included in the new cycle are removed
from D . If **

**It may happen that the number of loops constructed by
such algorithm is less than that required by the method of
loop currents. In this case the list **** is complemented
up to the necessary number under the condition that the new constructed
loop does not coincide with any of existing loops.
**

**When denoting by **** the current in the
***i*-** th loop and considering it positive in the edge
****, when **** and otherwise negative,
we shall find the magnitude of the current in each branch as
the sum or the difference of the corresponding loop currents.
**

**According to [10] the kinetic energy ****, the force
function ****, the Rayleigh function
**** for the linear electrical circuit has the form :
**

**For constructing the function characterizing the state of
the nonlinear electric circuit we use the algorithm [11,12].
**

**After an evaluation of kinetic energy and force functions we discover
characteristic function of the system and create the equations of motion.
**

**2.1. Lagrange's equations.**
Let characteristic function be a Lagrangian **, which is
shown in the form:
**

**2.2. Euler-Lagrange's equations.**
Let the configuration of a system be set by independent parameters
**and quasi-velocities - linear forms on generalized velocities
**** (in common case inhomogeneous):
**

**Jt is supposed, that
**** is nondegenerate.
The linear forms of differentials of generalized coordinates
**

**
The kinetic energy of system **** in quasi-velocities in common case
looks like :
**

**2.3. Hamilton's equations.**
Let characteristic function be Hamilton's function

**In electric circuits it is often the case that
**

**3.1. Investigations in the first approximation.**
The differential equations obtained after their expanding in
series in the neighbourhood of the given or found
steady-state solution may be represented in the form

**Now construct the characteristic equation (CE):
**

**3.2. Construction of first integrals for differential equations of motion.**
Algorithm to construct first integrals for the systems described is based on
invariant relations (differential consequences) of equations of motion [5].
For equations (2.2.3) we have invariant correlation :

*Statement 1.* If lagrangian (2.1.1) and kinematic correlations (2.2.1)
do not depend explicitly on time and nonpotential forces
** are absent, there exists generalized Jacobi integral:
**

*Statement 2.* If for
**
and
**

*Statement 3.* If in stationary mechanical system following correlation

*Statement 4*. If body system is described by Lagrange equations in
generalized coordinates
** and velocities ****, and lagrangian does not contain coordinates
****, there exist ignorable integrals
**

**Let for carrying body of the system generalized coordinates
**** describe rotational motion,
**** - motion of point ****, and
coordinates **** describe relative motions of
carried bodies. We choose quasi-velocities for carrying body equal to
projections of absolute angular velocity ****and linear velocity **** of the point **** to axes associated
with body. Let quasi-velocities of carried bodies be equal to projections of
relative velocities or to generalized velocities. In these conditions for
Euler-Lagrange equations following correlations take place :
**

*Statement 5.* If
**, then
**

*Statement 6*. If
**, then
**

*Statement 7*. If
**where **** is a constant matrix ****, then
**

*Statement 8*. If ** ,
then ****
**

**
Statement 9. If
, then
**

**For equations of the form (3.1.1) algorithms of obtaining differential
consequences and invariant correlations are described in [6, 8].
For linear systems (3.1.1) obtaining quadratic first integrals can be
reduced to systems of linear algebraic equations.
**

**3.3. Investigation of stability by Lyapunov's second method.**
Invariant relations, first integrals obtained from it and
other invariants may be used for finding steady motions
or invariant manifolds of steady motions as well as for
constructing Lyapunov functions in investigation of stability
of the latter [14]. To this end, an algorithm of
Routh-Lyapunov's theorem is used. In accordance with the
latter theorem we come to the problem of conditional extremum.

**Let for the system of ordinary differential equations
**

**In the regular case, for a system of interconnected bodies
the steady motions are found as solutions of equations of
stationarity of the bundle **** of first integrals of
motion differential equations.
Sufficient conditions of stability for these motions may be
found as the conditions of sign-definiteness of the second variation
**** for constant values of the rest of the first
integrals. This problem has a good algorithm of solving in
the case when side by side with the energy integrals there are
linear first integrals with respect to velocities or
quasi-velocities (e.g. ignorable ones) [2, 4].
In the case, when there are (i) first integrals of another
structure or (ii) singularity of the system (2) (this is
characteristic of electromechanical systems), difficulties of
the investigation increase. When there are no first integrals in the
system, for the purpose of constructing the Lyapunov functions
in the process of investigation of stability by Lyapunov's
method it is possible to use differential corollaries of
equations of disturbed motion [6].
**

**3.4. Reduction to the normal form.**
The preliminary reduction of the system to one of the
normal forms is an efficient aid in solving problems of
qualitative investigation of ordinary differential equations.
Poincare's form [15, 16] is one such normal forms for
differential equations of the general form (3.3.1).

**Normalization of systems of analytical ordinary differential
equations
**

**As a result, the system of ordinary differential equations
takes the form
**

**In practical investigations the precision of computing of the
eigenvalues was chosen ****Zero and multiple roots were determined with the same precision.
Eigenvectors and adjunct vectors were found with the precision
of ****
**

**
Linear normalization has been conducted only in the numerical form.
**

**Nonlinear normalization has been conducted by Poincare's method [15,16],
which retains only the resonance terms of the form
**** where
**** integer numeric vector of nonnegative indices
of powers of the variables.
Nonlinear transformation is realized in the form of the formal series:
**

**Nonlinear normalization is executed in both numerical and symbolic forms.
**

**With the help of packages in a symbolic form the authors investigated rather
large number of concrete mechanical systems. In particular:
**

**
- nonlinear equations of motion of a series of robotic systems are obtained
(for example, robot - manipulator of 6 links, walking platform on four
legs with two degrees of freedom each) ;
**

**
- problems of dynamics and stability of steady motions in space
systems were considered (for example, satellite with a gravitational
system of the stabilization on circular orbit are considered at various
control laws by a gravitational stabilizer, gyroscopic frame in Newtonean
central field of forces );
**

**
- linear equations of motion for a mechanical system with 32 degrees of
freedom consisting of 20 absolutely of rigid bodies are constructed.
**

**Application of the software system "Normalization" has allowed
us:
**

**
- to investigate stability of a nonlinear autonomous
mechanical system with two pairs of purely imaginary multiple roots
(with two groups of solutions);
**

**
- to solve the problem of stabilization of one mechanical
Hamilton-type system with the resonance of 1:3 and with
nonlinear control.
**

**Let us illustrate our reasoning with a few applied examples
of usage of the above software.
**

**Example 4.1.**
Let us consider gyroscopic frame, placed in Newtonean central
field of forces.
The mechanical system consists of a carrying frame
(a body with a fixed point) and two identical connected two-degree gyroscopes,
symmetrically installed in a frame (Fig. 4).
Masses and the moments of inertia of housing of gyroscopes are neglected.

**Figure 4.
**

**Before exposition of input information about the system we note that
the matrix of rotation **** and relative angular velocities
of bodies **** are not required
to be introduce. They are calculated automatically on specific "sequence of
rotations". Hereby we specify :
**

**
a) Number of axis of rotation (one of numbers 0, 1, 2, 3),
i.e. 1 - the rotation is carried out around the axis ****2 - around the axis **** 3 - around the axis ****0 - there is no rotation;
**

**
b) angle of rotation.
**

__Input data :__

**
Number of bodies in the system : ****.
**

**
Body 1 is a frame. **** is mass ;
**

**"sequence of rotations" :
****.
**

**
Body 2 is the housing of the first gyroscope, it is connected to the
frame. ****;
**

**"sequence of rotations" :
****.
**

**
Body 3 is the rotor of the first gyroscope, it is connected to the second body.
**** is mass of a rotor;
**

**"sequence of rotations" :
****.
**

**
Body 4 is the housing of the second gyroscope, it is connected to a
frame. ****;
**

**"sequence of rotations" :
****.
**

**
Body 5 is the rotor of the second gyroscope, it is connected to the fourth body.
**** is mass of a rotor;
**

**"sequence of rotations" :
****.
**

**
Program output :
**

**
Matrices of rotation :
**

**
Relative angular velocities :
**

**
The kinetic energy of the system :
**

**
где ****
**

**
**

**
are the projections of
absolute angular velocity of the frame on the axes
. They are selected as quasi-velocities.
**

**
Approximate force function **** (1.1.13) in Newtonean field
of gravitation to the fixed center ****
**

**
The force function of elastic forces of the spring device :
**

**Further four first integral of motion equations (integral of an energy
and three ignorable on coordinates ****) are found,
equations of steady motions
and sufficient conditions of stability of steady motions are obtained
by computer [17].
All the expressions obtained by computer have the symbolic form.
**

**Example 4.2.**
The modelling of the electrical circuit consisting of two parts
connected inductively (Fig. 5).

**Figure 5.
**

__Input data.__

**The number of parts of the electrical circuit connected inductively: 2
**

**The first part of the electrical circuit:
**

**
**

**
The second part of the electrical circuit:
**

**
**

**
The order of path-tracing the graph:
**

**"+" - to trace the graph's vertices in the direction of
growth of their numbers ("-" - inverse order).
**

__Program output.__

**The Lagrange's function:
**

**The Rayleigh's function:
**

**The differential equations of the electrical circuit in Lagrange's form
of 2-d kind:
**

**Example 4.3.** In the process of investigation of the phase
space of a mechanical or electric system there often appears
the need to solve algebraic equations (or those reducible to
them). In the contemporary universal computer algebra systems
there exists a tool for solving such problems, which allows to
simplify such systems. Gröbner's bases or their analogies may be
considered as such tools.

**In applied problems, direct usage of Gröbner's bases does
not always entail in satisfactory result. For example, for the
systems containing parameters the investigation requires
greater efforts and does not guarantee a success even when a
computer is used. In such situations the following technique of
analysis of algebraic systems with parameters may be useful.
**

**Let a family of solutions be known for an algebraic
system. The existence of such a family in concrete cases may
be grounded by the mechanical or physical character of the
problem. We take an interest in such solutions of the
system which (i) contain some elements of this family or (ii) adjunct
to the family. Such problems arise in different directions of
bifurcation theory. Now, let there be given a system of
algebraic equations with parameters
**

**Assume that the rank of the Jacoby matrix for the system
(2) with respect to
**** is **** and
consider the system of equations
**

**Analysis of the system (3) allows us to find, generally
speaking, those relationships between
****, for which the Jacoby
matrix's rank ****on the chosen family ****becomes smaller.
**

**These bifurcation values of the parameters may be
substituted into the initial system (1), what allows
to simplify it and find new solutions or their families.
**

**The proposed procedure may successfully be employed, for
example, in analysis of equations of stationarity of
mechanical systems' first integrals.
**

**Consider a concrete example of application of above
approach in analysis of bifurcations in the neighbourhood of
the family of permanent rotations of S.V.Kovalevskaya's top
about the vertical axis **** [14].
**

**To this end, let us compose a complete linear bundle of
the problem's first integrals:
**

**Now write the steady-state conditions **** with
respect to the phase variables:
**

**It can be easily determined that the system of algebraic
eqs. (4), (5) has the elements of the family of the body's
permanent rotations about its vertical axis **** in the
capacity of its solutions:
**

**Having substituted the solutions (6) into eqs. (5),
we obtain the condition for the parameters
**

**The Jacobian for the system (5) with respect to
**** for the
solution (6) writes:
**

**Hence, for obtaining bifurcation values of the parameters
**** it is necessary
to find solutions of the algebraic equations (7), which are
complemented with the condition ****.
Furthermore, as obvious from (8), we managed to factorize the
expression of the determinant.
**

**Complete analysis of the system (7), (8) is rather
cumbersome. So, we restrict our investigation, while reducing
it to only the first obvious subcase. Consider the following
system of algebraic equations with unknown variables
**** :
**

**The standard procedure puts the following collection of
equations in correspondence to the system in the capacity of
its Gröbner basis for the corresponding ordering of
unknown variables (computations have been conducted with the CAS
on IBM PC/486):
**

**Analysis of these equations allows one to find the
bifurcation values of parameters, which correspond to the
branches from the family of permanent rotations (6) of
invariant manifolds of the problem's steady-state motions.
**

**With the aid of the CAS it is possible to find all the
solutions of the above nonlinear algebraic system. Without
suggesting the result in complete form, let us outline the
plan of further actions. From the last equation of (9) we
have the following solutions for ****:
**

**The penultimate equation of (9) for ****gives the following value ****:
**

**Having removed ****, we obtain
**

**
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IMACS ACA'98 Electronic Proceedings