A reduction of the equations of the linear stationary vibrating systems without limitations of dissipation using the method of modal truncation/Neturinciu disipacijos apribojimu tiesiniu stacionariuju virpamuju sistemu lygciu prastinimas modalines atkirtos budu.
Augustaitis, V.K. ; Gican, V. ; Sesok, N. 等
1. Introduction
For computer-aided analytic research of complex vibrating systems
with a large finite number of degrees of freedom, including linear
stationary systems, various methods, for example, the method of finite
elements and so on, are applied [1-5].
In the Paper, the mentioned linear vibrating systems are discussed
upon. They are described by linear differential equations with constant
coefficients. For digital integration of such equations, Runge-Kutta and
other methods are applied. In a majority of cases, such systems have a
very wide spectrum of natural frequencies; however, an investigator
takes an interest in the much narrower range of the lowest natural
frequencies within the said spectrum only. The high natural frequencies
considerably increase the time of digital integration, so it is
important to have a system of differential equations for describing the
object under investigation where such frequencies are absent in the
roots of its characteristic equations. Such equations can be obtained by
reducing the number of degrees of freedom in the dynamical model of
object under investigation. However, in many case, this task is
difficult or even impossible. For example, such a problem appears when
the method of finite elements is applied.
Other methods of elimination of high natural frequencies are
obtained on the relevant reduction of the equations describing the
vibrating system [6-14]. Among those methods, the method of modal
truncation where the initial system of equations is divided to a number
of independent equations is widely used; each of such independent
equations describes modal vibrations of one natural frequency and then
the equations corresponding to high natural frequencies are rejected
(see, for example [10-14]). It is considered that the errors appearing
in this case would be permissible, if the frequencies of the rejected
modal vibrations are at least 1.5-2 times higher than the natural or
resonance frequencies the investigator takes an interest in [15]. In
some methods (see, for example, [16]), the errors of calculation
appearing because of the rejected modal vibrations are counterbalanced
by special components in the equations.
Although the known methods of modal truncation are effective, they
are applicable only to vibrating systems with a proportional damping,
when in the matrix equation describing the system, the elements of the
damping matrix are proportional to the mass and stiffness matrixes or
any of them. However, in many cases, damping cannot be considered
proportional. This statement is applicable, for example, to complex
mechatronical, mechanical and other systems where modal truncation is
important for their investigation.
The purpose of the paper: to propose a method for reduction of the
equations used for describing linear stationary vibrating systems with
any damping in state variables and reducing the time of digital
integration. The method is based on formation of the equations of the
system in state variables [17-19] using the normal Bulgakov's
coordinates [20, 21] and application of modal truncation when the
equations corresponding to high natural frequencies are eliminated (see
below). The value of the error resulted by the truncation is assessed by
comparing the frequency response of the vibrating system before and
after the reduction in the frequency range under the interest of the
investigator.
2. Initial version of the equations used for describing the system
in state variables
Let's consider that such a nonreduced system of differential
equations describing vibrations of the system (object) under
investigation is formed
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where
[{q} = {[q.sub.1],[q.sub.2], ..., [q.sub.n]}.sup.T] (2)
the vector of the generalized coordinates qi (i = 1, 2, n) defining
motions of the system; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII]; t is time or, more rarely, another argument; [A], [B], [C] are
square matrixes of the n-th degree with constant (stationary) elements;
the--matrix [A] may include r [less than or equal to] n zero lines and
columns with the same serial numbers, i.e. in the system (1), r is
differential equations of the first degree that include only those
generalized coordinates [q.sub.j] (j = 1, 2, r) and their fluxions
[[??].sub.j] not presented in the equation of the second fluxions
[[??].sub.j] (Eq. (1)) can exist; {h(t)}={[h.sub.1] (t), [h.sub.2]
(t),..., hn(t)}T is the vector of the n-th degree of external
generalized forces involved in excitation of the system.
For examination of the vibrating system in state variables, the Eq.
(1) are reduced to the normal (Cauchy) form, i.e. when the generalized
coordinates [q.sub.i] are replaced for other variables, the system of
equations is reduced to a system of differential equations of the first
degree solved in respect of the first derivates of the said variables.
Such variables are considered phase or state variables of the system
described by the Eq. (1) defining all states of the system in the time t
[greater than or equal to] [t.sub.0] ([t.sub.0] is the initial time of
observance of the system). For investigation of the said equations,
special softwares are used [19, 21].
The most frequently applied method of introducing variables
defining the states of a system under investigation is based on
replacing the derivates [[??].sub.k] of the generalized coordinates
[q.sub.k] (with their derivates of the second degree [[??].sub.k]
included in the Eq. (1)) for new variables. It is may be carried out as
described in [21]. New variables are introduced
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
Their vector
[{x} = {[x.sub.1], [x.sub.2], ..., [x.sub.r], [x.sub.1+1],...,
[x.sub.n], [x.sub.n+1], ..., [x.sub.s]}.sup.t] (4)
To simplify the following description, it is considered that r
equations having no second-degree derivates of coordinates are in the
beginning of the system (1), so the matrixes [A] and [B] can be divided
into the following submatrixes.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [[O].sub.1,1], [[O].sub.1,2], [[O].sub.2,1] are zero matrixes
of r x r , r x (n - r), (n - r)x r degree; [[A].sub.2,2] is nonsingular
square matrix of the (n - r)degree; [[B].sub.1] , [[B].sub.2] are
matrixes of the n x r ir n x (n - r)degree.
Taking into account the Eqs. (3) and (5), we obtain the following
instead of the system of Eq. (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
here the derivates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] are specified only formally. They are multiplied by the zero
submatrixes [[O].sub.1,1] and [[O].sub.2,1] of the matrix[A], so really
they are not presented in the equations (6). These zero submatrixes are
replaced for the submatrix [[B].sub.1], and the derivates [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] are replaced for the derivates
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then instead of the
Eq. (6), we find
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
From it, we find
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
here the square matrix of the nth degree
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[[G].sub.1], [[G].sub.2] are the submatrixes where the lines are
formed of r first lines and the remained (n - r) lines of the matrix [G]
. It is accepted that the inverse matrix it nonsingular and the Eq. (9)
is valid. In such a case, the matrix [[A].sub.2,2] as well as the matrix
[[B].sub.1,1] formed of r first lines of the matrix [[B].sub.1] should
be nonsingular.
In addition, the system of Eq. (8) does not include the equations
defining the values of the fluxions [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII]. They are obtained from the part [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] of the Eq. (3) taking into account
that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
After uniting the Eqs. (8) and (10), we obtain the following normal
equations describing the system under examination in its state variables
instead of the Eq. (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
where {x} is the vector defined by the Eq. (4); its components are
the coordinates describing the location of the system in its state
variables
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
The square matrix of 2 n - r = s degree consists of: the zero
submatrix [[O].sub.2] of (n - r) x n degree and the unit submatrix
[[E].sub.1] of (n - r) degree; the submatrixes [[G].sub.1] [C] and
[[G].sub.1] [[B].sub.1] r x n of r x(n - r) degree; the submatrix
[[G].sub.2] [C] of (n - r)x n degree and the square submatrix
[[G].sub.2] [[B].sub.2] of (n -r) degree
{k(t)}= [[G].sub.*] {h(t)} (13)
The vector of 2n - r = s degree; [[G].sub.*], is the matrix of (2 n
- r)x n = s x n degree obtained from the matrix [G] by inserting (n -r)
zero lines between its submatrixes [[G].sub.1] and [[G].sub.2].
On investigation of the vibrating system in its state variables,
its output coordinates (output signals) ye (e = 1,2, ..., m) (that
describe the state of the object as well) bound with its variables {x}
by linear algebraic equations are added to its equation of state [18,
19]
{y}=[D ]{x}+[H ]{h(t)} (14)
where {y} is m dimensional vector (m = 1,2, ...); [D], [H ] are
matrixes of m x s and m x n degree with constant or varying in course of
time elements; the values and character of variation of such elements
depend on a system under investigation.
3. The proposed version of the equations of state variables and
their reduction
The obtained mathematical model Eqs. (11) and (14) of the system
under investigation in its state variables is based on the application
of variables for defining the state of the system provided in the Eq.
(4).
As it was mentioned above, on investigation of the vibrating
systems described herein (Eq. (1)), it is not reads Eq. (11) because of
a considerable time of integration, when the spectrum of natural
frequencies of the system includes the range of high natural frequencies
(out of the interests of the investigator), not only the range of low
natural frequencies. It would be purposeful to reduce the Eq. (11) by
eliminating components with high natural frequencies from their
solutions and simultaneously maintaining an adequacy of the obtained
results with a permissible error.
For such reduction of the Eq. (11) and saving the computer time, it
is proposed in this paper to use the normal Bulgakov's coordinates
for variables of the system's state of variables and the method of
modal truncation on the base of [20, 21]. For this purpose, the
below-described procedure is used.
The method is applicable, if the matrix [R] has multiple natural
values; in such a case, the element divisors related to them should be
linear (this condition is equivalent to the condition on absence of any
secular terms in the solutions of homogenous Eqs. (1) and (11), when {h}
[equivalent to] 0 .
In addition, the vector [??] should have no zero components, i.e.
the systems of Eqs. (1) and (11) should include no algebraic equations.
It is considered that the said conditions were satisfied (if secular
members appear in solutions of the equations, they usually may be
avoided by a slight correction of the dynamic model of the system under
examination without losing its adequacy).
It is considered that the matrix included in the Eq. (11) has s
natural values [[lambda].sub.j] (j = 1,2, ... s), including s' real
natural values [[lambda].sub.j] = [[CHI].sub.[sigma]] ([sigma] = 1,2,
..., s') and s" couples of complex conjugates roots
[[lambda].sub.s'+h], = [[epsilon].sub.h] + i[[omega].sub.h],
[[lambda].sub.s'+s"+h] = [[epsilon].sub.h] - i
[[omega].sub.h]; where [[epsilon].sub.h], [[omega].sub.h] are real and
imaginary parts of roots, h = 1, 2, ..., s", i = [square root of
-1].
The natural values [[lambda].sub.j] of the matrix [r] are also the
roots of the characteristic equations of the systems of Eqs. (1) and
(11), the values [[omega].sub.h] is the natural frequencies of the
system under investigation; [[epsilon].sub.h] is defines damping of free
vibrations with frequency [[omega].sub.h] and [[chi].sub.[sigma]]defines
aperiodical nonvibrating processes. It can be seen that s = s' +
2s" = 2n - r .
Each real natural value [[lambda].sub.[sigma]] of the matrix [R]
corresponds to its own vector [{V}.sub.[sigma]] =
[{[upsilon]}.sub.[sigma]] with s components, each complex couple of
roots [[lambda].sub.s'+h] and [[lambda].sub.s'+s"+h]--to
own vectors [{V}.sub.s'+h] = [{[upsilon]}.sub.s'+h] +
i[{[upsilon]}.sub.s'+s"+h] = with s components and,
correspondingly, [{V}.sub.s'+s"h] =
[{[upsilon]}.sub.s'+]--. -i{[upsilon]}.sub.s'+s"+h] The
natural values [[lambda].sub.s'+s"+h] and the own vectors
[{V}.sub.s+s"+h] corresponding to them hereinafter will not be
used.
Each natural vector [{V}.sub.s'+h] +h is simultaneously the
mode of natural vibrations corresponding to the natural frequency
[[omega].sub.h] of the homogenous part of the system under investigation
Eq. (1) according to the generalized coordinates [q.sub.1],
[q.sub.1],..., [q.sub.n] and the mode of speeds of natural vibrations
according to the derivates of the generalized coordinates [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII]. The complex form of these vectors
shows that they were formed taking into account the damping in the
system; however, no special requirements are set for the character of
the latter [20, 21]. Each natural vector [{V}.sub.[sigma]] is
simultaneously the mode of aperiodic motions corresponding to the real
root [[lambda].sub.[sigma] of the homogenous part of the system under
investigation (1) according to the generalized coordinates [q.sub.1],
[q.sub.2], ..., [q.sub.n] and their derivates [MATHEMATICAL EXPRESSION
NOT REPRODUCIBLE IN ASCII].
When the natural vectors are known, a square modal matrix of the s
-th degree [[upsilon]] consisting of s columns of the vectors
[{[upsilon]}.sub.[sigma]] and 2s" columns of the real
[{[upsilon]}.sub.s'+h] and imaginary
[{[upsilon]}.sub.s'+s"-+h] parts of the vectors
[{V}.sub.s'+h] is formed [15] .When the said columns are laid out
in the way where s' first columns of the matrix [[upsilon]] are the
vectors {u}a and then couples of the vectors [{[upsilon]}.sub.s'+h]
and [{[upsilon]}.sub.s'+s"+h] follow, the following structure
of this matrix is obtained [20, 21]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
Considering that the matrix [ R] satisfies the above-mentioned
conditions, a replacement of the variables of the system (11) is carried
out [20, 21]
{x} = [[upsilon]]{[xi]} (16)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
is the vector of s th degree in the normal Bulgakov's
coordinates (NBC).
After inserting the value Eq. (16) of the vector {x} into the Eq.
(11), we find [15]
{[??]}=[Q]{[xi]} + [z]{h (t)} (18)
where, if the order of components of the vector {[xi]} provided in
the Eq. (17) is preserved
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
[[O].sub.3] , [[O].sub.4] are zero submatrixes of the s' x
2s" and 2s" x s' degree;
[[chi]] = diag {[[chi].sub.1], ..., [[chi].sub.s'],} (20)
[[OMEGA]] is the block square submatrix of the 2s" degree;
square submatrixes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
are situated along its diagonal and all remained elements of the
said submatrix are zero elements;
[z] = [[[upsilon]].sup.-1] [[G].sub.*] (22)
matrix of the (2 n - r)x n = s x n degree.
The scalar expression of the Eq. (18) will be as follows. To each
real root [[chi].sub.[sigma]], an equation not bound with other
equations
[[??].sub.[sigma]] - [[chi].sub.[sigma][[zeta].sub.[sigma]] =
[[PHI].sub.[sigma]](t), ([sigma] = 1,2 , ..., s') (23)
will correspond, and to each complex joint root [[epsilon].sub.h]
[+ or -][[omega].sub.h], a couple of equations not bound with other
equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
will correspond, where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
So, the initial system of equations was divided to independent
equations with easily findable solutions. The variables {x} as well as
the coordinates {q}, if the normal Bulgakov's coordinates {[xi]}
are known, should be found from the Eq. (16). It should be noted that
the division of the equations of the system under investigation into
independent equations was carried out without applying any limitations
to the structure and elements of the matrix [B] included in an equation
of the system (1).
The other version of equations of the system under investigation in
the state variables consists of the system of Eq. (18) with the joined
system of algebraic linear Eq. (14) where the value of the vector {x} is
taken from the Eq. (16)
{y}=[D][ [upsilon]]{[xi]}+[H]{h((t)} (26)
This version differs from the above-mentioned one (see the Eqs.
(11) and (14)) in a considerably simpler structure of the differential
equations; in addition, it provides an opportunity of a major reduction
of these equations using the method of modal truncation.
It may be made sure that upon investigating of a dynamical system
in its state variables and solving the equations that describe it by
numerical integration, the process of the integration will be longer
when the natural frequencies of the system are higher and processes of
higher frequencies are explored (the step of integration becomes
smaller). If the range of natural frequencies of the system under
investigation is very large and the investigator takes an interest in
the range of low frequencies only, the process of integration can be
shortened considerably by eliminating the systems of Eqs. (24) that
correspond to higher natural frequencies [[omega].sub.h] from the
process of calculation, i.e. by applying the method of modal truncation.
In addition, often all Eq. (23) or a part of them can be neglected. In
such a case, instead on the matrix (19) and the system of Eq. (18), we
obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)
{[??]} = [[Q].sub.*] [{[xi]}.sub.*] + [[z].sub.*] {h(t)} (28)
where [{[xi]}.sub.*] is the vector obtained from the vector {[xi]}
after elimination of the neglected NBCs; [[[chi]].sub.*], ;
[[[OMEGA].sub.*], are square submatrixes of the [alpha] and 2[beta]
degree obtained from the submatrixes (19) and (20) when a part of the
Eqs. (23) and [alpha] part of the couples of Eqs. (24) are neglected;
[alpha] < s', [beta] < [beta] < s"--the number of the
remained Eqs. (23) and, correspondingly, the remained couples of Eqs.
(24); [[O].sub.5], [[O].sub.6] are zero submatrixes; [[Q].sub.*] is the
square matrix of ([alpha] + 2[beta]) degree analogous to the matrix [Q],
but without the excluded elements of the Eqs. (23) and (24); [[z].sub.*]
is the matrix of ([alpha] + 2[beta])x n degree obtained from the matrix
[z] after elimination of the lines with the serial numbers corresponding
to the eliminated coordinates [[xi].sub.[sigma]], [[xi].sub.s'+h],
[[xi].sub.s'+s"+h].
[FIGURE 1 OMITTED]
The approximate values of the variables [x.sub.p], i.e.
[x'.sup.p] (p = 1, 2, ..., s), are found from the equation that is
analogous to the Eq. (16)
{x'} = [[[upsilon]].sub.*] [{[xi]}.sub.*] (29)
where {x'} is the vector of the approximate variables
[x.sub.p] of the state variables, [[[upsilon]].sub.*] is the reduced
modal matrix of s x ([alpha] + 2[beta]) degree found from the matrix
[[upsilon]] where only the columns corresponding to the assessed natural
values [[lambda].sub.j] of the matrix [R] are left.
The v--dimensional vector {y'} of approximate coordinates of
the output of the system is found from the equation that is analogous to
the Eq. (14)
{y'} = [[D].sub.*] {x'} + [H]{h(t)} (30)
After insertion of the value of the vector {x'} from Eq. (29)
into the equation, we find
{y'}=[[D].sub.*] [[D].sub.*][[[upsilon]].sub.*][{[xi]}.sub.*]
+ [H] {h(t)} (31)
So, the reduced version of the equations of the system under
investigation with NBCs in the state variables consists of the systems
of Eqs. (28) and (31).
The issue of the numbers [alpha] and 2[beta] remained in NBC
[[xi].sub.j] is important. In a majority of cases, a reduction of the
number [alpha] of the Eqs. (23) can be avoided and only the number
[beta] of the Eqs. (24) can be reduced; however, a specific decision
should be passed for each individual system under investigation. For
defining the number of the remained NBC, it is proposed to compare the
amplitude-frequency responses (AFRs) and the phase responses (PRs)
obtained upon certain selection of the components of the vector {h(t)}
for the non-reduced system (1) or (11), (18) with the respective
responses of the reduced system (28) calculated in the range of
frequencies the investigator is interested in. Nonzero components of the
vector {h(t)} can be following
[h.sub.a] (t) = [A.sub.a] sin vt + [B.sub.a] cos vt (a = 1, 2, ..,,
n) (32)
here the values of the constant the coefficients [A.sub.a],
[B.sub.a] and the range [DELTA]v of changing of the excitation
frequencies v are chosen by the investigator. Upon comparison of AFRs
and PRs of the whole and reduced systems, the value of inadequacy is a
base for making a conclusion on the level of the errors of the reduction
and the required number of remained NBC. One of the possible algorithms
for AFRs and PRs calculation that does not require much time for the
calculation is provided in [15]. The algorithm is based on the
application of solutions of the equations (18) in the analytic form and
the equation (16) in the case of harmonic excitation of the system. The
scheme of formation of the reduced equations in Bulgakov's
coordinates is shown in Fig. 1.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
4. Example
In our paper [22], a system of equations of the type Eq. (18) for
investigating transversal vibrations of the unit of plate cylinders and
blanket cylinders in a section of offset printing press was described
and used. Using the results of the said work, we'll show an
efficiency of the proposed reduction. The above-mentioned unit (Fig. 2)
consists of two plate cylinders 1, 2 with printing forms attached to
them (Fig. 2, not shown herein) and two blanket cylinders 3, 4 covered
by a special elastic textile (blanket) 5. The blankets are fixed to the
said cylinders by special oblong locks along the generatrices of the
cylinders. The length of all cylinders is 1040 mm, the diameter
[empty]200 mm. The printing process runs as follows: while the cylinders
connected with gearwheels 7 and pressed against each other along their
generatrices rotate and deform the blanket, the prints from the
ink-moistened printing forms are transferred to the blankets 5 and from
them--to the both sides of the paper tape 8 moving between the blanket
cylinders 3, 4. The position of the blanket cylinders 3, 4 pressed
against each other through the blanket 5 is regulated in such a way that
ensures getting of the locks 6 into collision while rotation of the
cylinders. It causes blows and transversal vibrations of the cylinders
pressed against each other (similar vibrations appear on getting of the
locks into collision with surfaces of plate cylinders; however, the
intensity of such vibrations is lower). For description of the
vibrations, the equations of the type (1), then also the equations of
the types (11), (18) are used; in [17], these equations are further used
nonreduced. For formation of the equations, all cylinders are divided
into finite elements. In addition, elasticity and damping of the
blankets 5 as well as deformation of the bearing units of cylinders are
taken into account. Thus, a system with 168 degrees of freedom is
formed. Its natural frequencies are situated in the 171.5 - 530183 Hz
range. Blows are simulated by rectangular 100 N force impulses with
duration of 0.003 s (the period 0,1 s); the said impulses impact the
pressed against each other plate cylinders in the radial direction in
units of finite elements entering into contact in the zones of the plate
cylinders (each of two plate cylinders is impacted by 10 impulses). The
transversal vibrations generated by the said impulses in the middle of
the plate cylinder 2 are calculated.
It is accepted that a person engaged in the calculation takes an
interest in vibrations with the frequencies up to 700 Hz, so only 12
couples of equations in normal Bulgakov's coordinates are left in
the reduced system.
They correspond to all natural frequencies of the system up to 750
Hz.
Amplitude-frequency and phase responses of the complete and the
reduced system are calculated for the range 0-1000 Hz, where the force
impulses are replaced by harmonic excitation. The calculations were
carried out using the mathematical simulation set MATLAB. When the
equation (11) is applied for the calculations, the time of numerical
integration equals to 14 min 14 s (the step 5e-7), and when the reduced
equation of the type (30) is applied, it equals to 8 s (the step 1e-5),
i.e. the time reduces 106 times. The conversion of the system (11) into
equations of the type (18) took 1.5 s, and the calculation of the
frequency responses took 2 s.
The obtained results are shown in the Fig. 3. It can be seen that
the error of the reduction in the range 0750 Hz is very small.
5. Conclusions
1. For shortening the time of numerical integration of the
differential equations usable for describing linear stationary vibrating
systems with a wide spectrum of their natural frequencies, it is
purposeful to convert the said equations in to the normal
Bulgakov's coordinates and then to reduce them eliminating the
equations to high natural frequencies by the method of modal truncation.
2. For the proposed method of calculation, no limitations for
dissipation of the vibrating system are required.
3. The value of the error appeared during the reduction is assessed
by a comparison of the amplitude-frequency and phase responses of the
system under investigation obtained from the initial and reduced
equations with harmonic excitation.
Received February 11, 2011
Accepted August 30, 2011
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V. K. Augustaitis, Vilnius Gediminas Technical University,
Basanaviciaus 28, 03224 Vilnius, Lithuania, E-mail: pgkatedra@vgtu.lt
V. Gican, Vilnius Gediminas Technical University, Basanaviciaus 28,
03224 Vilnius, Lithuania, E-mail: pgkatedra@vgtu.lt
N. Sesok, Vilnius Gediminas Technical University, Basanaviciaus 28,
03224 Vilnius, Lithuania, E-mail: pgses@vgtu.lt
I. Iljin, Vilnius Gediminas Technical University, Basanaviciaus 28,
03224 Vilnius, Lithuania, E-mail: pgilj@vgtu.lt