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механика. транспорт. машиностроение. технологии
но ближе к рабочему диапазону, чем это показывают расчеты на монолитном аналоге ротора. По амплитудной составляющей результаты также показывают существенную разницу результатов расчетов монолитной и сборной моделей.
В целом, на основании представленных результатов, а также сравнения их с экспериментальными данными, можно утверждать, что теоретические расчеты динамических параметров сборного ротора без учета его контактной жесткости могут давать погрешность порядка 50%.
Математическое моделирование сборных конструкций с применением решения контактной задачи механики деформируемого тела является одной из ключевых позиций в современной концепции разработки изделий транспортных систем, когда использование их виртуальных прототипов отводит натурному эксперименту роль проверки на завершающих этапах проектирования, что в конечном итоге повышает в них параметры надежности и долговечности.
БИБЛИОГРАФИЯ
1. Григорьев Н. В. Нелинейные колебания элементов машин и сооружений. М. : Машгиз, 1961. 256 с.
Eliseev S. V., Upyr' R. Yu.
2. Ден-Гартог Дж. Механические колебания. М. : Физматгиз, 1960. 580 с.
3. Зенкевич О. С. Метод конечных элементов в технике. М. : Мир, 1975. 542 с.
4. Левина З. М., Решетов Д. Н. Контактная жесткость машин. М. : Машиностроение. 1971. 264 с.
5. Пыхалов А. А., Высотский А. В. Расчет сборных роторов турбомашин с применением него-лономных контактных связей и метода конечных элементов // Компрессорная техника и пневматика. 2003. № 8. С. 25-33.
6. Пыхалов А. А., Милов А. Е. Математическое моделирование динамического поведения сборных роторов турбомашин // Компрессорная техника и пневматика. 2006. № 3. С. 16-23.
7. Пыхалов А. А., Милов А. Е. Контактная задача статического и динамического анализа сборных роторов турбомашин : моногр. Иркутск : Ир-ГТУ, 2007. 192 с.
8. Рыжов Э. В. Контактная жесткость деталей машин. М. : Машиностроение, 1966. 196 с.
9. Хронин Д. В. Теория и расчет колебаний в двигателях летательных аппаратов. М. : Машиностроение, 1970. 412 с.
УДК 531.3
ELASTIC ELEMENTS OF VIBROPROTEC-TION SYSTEM WITH NEGATIV RIGIDITY. POSSIBILITIES OF PHYSICAL REALIZATION
While working out vibroproof systems on the basis of structural approaches [1, 2, 3], it is supposed that, to the initial calculation model in the form of some mechanical oscillatory system, the dynamically equivalent system of automatic control (SAC) is being constructed by certain rules. The further researches of initial system are related to the use of the transfer functions reflecting both features of initial calculation models, and features of external impacts.
Even considering the generality of two approaches (in the theory of automatic control and the structural theory of vibroproof systems) that lies in the fact that in their basis there lie representations about the use of a set of typical elementary links, there are
also distinctions which, in particular, are connected with formation of an additional feedback chain. Such a link within the limits of structural theory of VPS is formed as the additional chain, introduced in parallel to an elastic link of the base model. In this case, the physical nature of an additional chain as some generalized spring is predetermined, and also does the specificity of connection of links in the chain by rules of consecutive and parallel connection of springs [4, 5] that does not contradict the switching rules in the automatic control theory.. Thus, it is necessary to specify such a circumstance that the set of typical elementary links in both cases is different. The majority of typical links in SAC can be received by connections
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of elementary VPS links, whereas in VPS elementary links represent a set of elements which are not compound [3].
The development of theory and practice of vi-broprotection is due to the complication of calculation models and complication of additional feedback links [6-11], that is reflected not only in structure of transfer functions which often take the form of fractional-rational expressions of a high order, but also necessity of considering the features brought by signs of factors. In the automatic control theory this question is connected with the fact that typical SAC links are divided into stable and unstable ones, [14] whereas the same question in structural theory VPS is treated a bit differently.
In the VPS theory, springs with negative rigidity have been considered in a number of works [1, 2], attempts of their constructive realization were made [3, 9] and that finally leads to formation of conceptions that elements of the expanded typical set can have negative parameters, as well as the SAC links .
1. Let's consider a vibroproof system (VPS), consisting of an object of protection (Fig. 1) with a mass m, a base elastic element with a rigidity k{)
and of an elastic element represented as a two-element mechanism rotating around a vertical axis and having two additional masses m . Such a mechanism is used
in devices of speed regulation in turbines, internal combustion engines, etc. We will designate the rigidity of a rotating two-element mechanism through k; elastic properties of such an additional link are formed by centrifugal forces of inertia during the rotation of elements with masses m at a constant rotation speed
negative rigidity k as a rotating two-element mechan-
ism.
Vibrations in the system in Fig. 1 occur in case of a basis displacement z = z0 Sin Ot, that is, the basis vibrates. The two-element mechanism with two spheres m rotates with constant angular speed O0 .
An object with a mass m0 moves in relation to the fixed coordinates and is characterized by a coordinate y . Spheres with a mass m participate in the complex motion defined by participation in rotation around the vertical axis AB with angular speed O0 = const; the vector of linear speed of spheres is
perpendicular to the plane of the drawing and can be designated as
V = (AEl ■ sina = oJ ■ sina. (1)
In this regard, let's suppose that a = CX0+ Aa, where a0 is an angle characterizing stationary position, relative to which small fluctuations Aa occur. In its turn, Aa is connected with a relative displacement of object m relative to the basis (y — z). Let's introduce into consideration the auxiliary model (Fig. 2) in which the relative positioning of points AB and E , participating in motion, is shown.
Believing Aa small, it can be written down
that
y — z = 2l ■Aa ■ sina0, (2)
O0 . In Fig. 1 bars of levers are considered identical whence. ..
and equal to l . We will suppose that external impact has a kinematic character, that is, the basis vibrates according to the known law (the motion is considered harmonious).
эЬ
m.
P
Aa =
y-
2l ■ sin^
(3)
E,
Fig. 1. The calculation model of a mechanical oscillatory system with an elastic element k and a spring with
Fig. 2. The model for defining the displacement and speeds of motion ml.
механика. транспорт. машиностроение. технологии
The similar result can be obtained in a different
way:
y - z = 2l ■ cosa0 - 2l ■ cos(a0 + Aa), (4) whence we can find that (4) coincides with (3).
The speed of points E and E in absolute
motion can be found using the velocity addition theorem. One of the types of flat motion of spheres is a motion in plane, consisting of a translational motion with a speed V = y — z and a relative motion as
rotation of point E (and E) relative to point B -
V . Then
отн
V
2 Vnep + Vomn . (5)
Speed of points E and E during the rotation around the AB axis can be represented as
V =®0
, . l( y - z)cosan
l srna0 --0
l sinaA
(6)
Speed m during the absolute motion is
V* = V + V .
As to V , the speed is equal to
V„
A à-l,
v 2
(7)
(8) (9)
(10)
It's necessary to know Vagc to define the kinetic energy of spheres mx. The kinetic energy of the system is written as
V22={y-zY + {Aâ-l)' 2(y- z}Aà -icos(a0 + Aa).
In its turn,
V 2а& = V2 + V22.
1 2 1 2
Т = 2т°У +
абс
or
T 1 1
T = -mQy + 2-—ml :
(j-z)2 +(Aci; •¡Y -2(j-z)Aa: -/cosa0 +
rnn
, . (y - z)cosan
l sin^ --0
2sina„
Let's develop expression (12)
T 1 • 2 I • 2 I -2 T ' • , /2 (У Z)
1 =—m0y +OTjZ -2OTj3>Z + mj л 2 . 2
2
4/ sin я.
(13)
-2ml cosa0
+ mlw0 ■ I sin cc0 +
- {У-*) ■ (y~é)
V—-— z--
2/sinee0 2/sinee0
2mpllsina0 (y - z)cosa0 m^02 (y - z)2 cos2 a(
4 sin a
Let's find necessary relations:
dT . . m,y m,z
— = m0y + Im^y - Im^z + ——----—---
dy 2sin a0 2sin a0 (14)
ff7jcos«02j + ml cos a0z + ml cos a0z
or
sina„
ÔT_ _ ..
dy
sina„
sma„
m0 + 2щ
>m
2 sin a.
-- 2 щ —
cos«n
sin«„
(15)
—z
dT dy
2щ
m
2 sin a,
cosa0 2_. -2m^—~-
sin«„
2 m y cos2 a : -щco0 cos a — „-- +
"0
2
2 sin a
0
+
mx a>l z cos2 a0
ô • 2
2 sin a
(16)
(11)
(12)
The potential energy of system is defined by the deformation of springs and position of gravity forces. The potential energy of elastic elements looks like
1 2
n = ~K (y - z) ; n = -2mg. (17)
Let's suppose that the system vibrates in relation to its balance position, therefore the component of the generalized force, due to the impact of gravity forces, can be left out in a first approximation, though inertia forces of spheres during rotation change the balance position.
If needed, potential energy of gravity forces can be found:
П = m0g (y - z ) + 2mlg (y - z -1 cos a0 ) +
( У - z )l
2l sin a
2mg (y - z) sin a.
(18)
The system of the differential equations of motion in coordinate system y, in the end, takes the following form:
x
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У
да,, + 2т, + -
m,
2 sin «
2m cosa0
sina„
m с2 cos2«
+ K У--2-0 У =
2 sin «
= z
2m, +■
m
2 sin2 a,,
2m cos«
sin«
(19)
с
+ z
V
, m С cos2 « ^ k0
2 sin «
i —
(20)
"-o y
+m (2 cosa — (mg+2mg)— —2mgi cos a — mg sina0.
Having accepted constant members
a=m(o( cos a — (mg+2mg) ■ —2mgi cos <p0—mg sin a
in the right part of expression (19), it is possible to
pass to a condition A = 0, or to introduce the system of coordinates considering the displacement of the static balance position, and then the system of the differential equations (18) can be transformed in the following way:
У
дап + 2/7?, +
m,
2 sin «
2m cos« sin«
+У
m с2 cos2 «^
Kr,
2 sin «
= z
2 да, + -
+z
m с2 cos2 «^
K0
m 2sin2 «
У
2m cos«
(21)
sin«
V
2 sin «
У
Transfer function in such a system can be written down as
w = у =
(22)
m
2 sin2 «
cosa0 sina„
m®0 cos « 2 sin2 a,.
m+2m +
m
2 sin «
- 2m -
+ к -
m®0 cos «
2 sin2 «
_ mj с2 cos2 « ki=
2 sin «
(23)
o
Presence of a spring with negative rigidity allows creating modes with quasizero rigidity, with m ,
O0 and a0 chosen as adjusting parameters irrespective of frequency of external impact O. The resulted rigidity of vibroproof systems on the whole, is defined by expression
K = K + k = K
пр 0 i 0
mC cos2 «
2 sin2«
(24)
In Fig. 3 the dépendance of the resulted rigidity of vibroproof system on angular speed Knp (®0) is shown.
Let's note that in such a vibroproof system the realization of a mode of dynamic clearing is possible
k0 -
с =-
mxa>l cos2 « 2sin2«
(
m,
4 sin2«
2 sin «
1 2 cos«
sin a,
A + ki , (25)
m • y
0 У
where y =
4 sin2 « +1
2cos«0
is a dimen-
From the analysis of a transfer function (23) follows the fact that the mechanism consisting of the rotating masses and introduced in parallel to the elastic element of the base model k is a spring with a
negative rigidity k
2sin2a0 sina
sionless factor.
In case of a consistent development of starting positions in the structural theory of vibroproof systems one should agree (and it has been shown in works [3], etc.) that an elementary VPS link with the corresponding transfer function, simple enough, when physically realized, can be much more complex.
In this case the spring with negative rigidity is realized through the mechanism of a rotation regulator. However, such a fact should not be of surprise, as elementary VPS link with transfer function
W (p) = ap, (that is, the differentiating link of the
first order, or viscous friction damper) represents the hydraulic mechanism consisting of a piston, cylinder and one more element: a operating fluid, that is, an element of the new type with special distributed (instead of concentrated) properties.
Coming back to the analysis of properties of a vibroproof system with an elastic element having negative rigidity, we note that presence of weight in elements introduces additional properties to the vibroproof system.
z
2
0
During the kinematic perturbation in a system with one degree of freedom there is a mode of dynamic damping, and an approach to the quasizero rigidity mode is also possible.
At the same time, it is necessary to point out that while inferring the equations of motion a number of simplifications has been made. Actually, the arising processes are of more difficult nature. Therefore, technical realization of the suggested approaches can demand certain efforts. However, the positive moment in the suggested approaches is a potential possibility of other representations about properties of known solutions, considering that under certain realization conditions, elementary typical elements can have negative values of transfer functions, but perform necessary tasks in the structure of a vibroproof system..
BIBLIOGRAPHY
1. Eliseev S. V. Structural theory of vibroproof systems. Novosibirsk : Nauka, the Siberian branch, 1978.224 p.
2. Eliseev S. V., Volkov L. N., Kukharenko V. P. Dynamics of mechanical systems with additional links. Novosibirsk : Nauka, the Siberian branch, 1990.214 p.
3. Dynamic synthesis in the generalized problems of vibroprotection and vibroinsulation of technical objects / Eliseev S. V., Reznik J. N., Khomenko A. P., Zasyadko A. A. Irkutsk : Publishing House of Irkutsk State University, 2008. 523 p.
4. Eliseev S. V., Upyr R. Yu. Mechatronic Approaches in Problems of Vibroprotection of Cars and Equipment // Modern Technologies. System Analysis. Modelling. 2008. № 4 (20). P. 8-16.
5. Ermoshenko Yu. V. Management of Vibrating Condition in Problems of Vibroprotection and Vi-broinsulation // Dissertation of Candidate of Sciences (Engineering). Irkutsk : Irkutsk State Transport University, 2003. 196 p.
6. Lukyanov A. V. Methods and Means of Control Considering the Technical Systems of Variable Structure of Objects // Author's abstract of doctoral dissertation. Irkutsk : Irkutsk State Transport University, 2002. 320 p.
7. Kuznetsov N. K. Methods of Dynamic Errors Decrease in Operated Cars with Elastic Links Based on the Concept of Additional Links // Author's abstract of doctoral dissertation. Irkutsk : Irkutsk State Transport University, 2006. 32 p.
8. Drach M. A. Dynamic Synthesis and Modelling in Problems of Evaluation and Changing the Vibrating Condition in Torsional Oscillatory Systems // Dissertation of Candidate of Sciences (Engineering). Irkutsk : Irkutsk State Transport University, 2006.177 p.
9. Dimov A. V. Modelling and Dynamic Processes in the Generalized Problems of Vibroprotection and Vibroinsulation of Technical Processes // Dissertation of Candidate of Sciences (Engineering). Irkutsk : Irkutsk State Transport University, 2005. 210 p.
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10. Banina N.V. Structural methods of dynamic syn- 11. Upyr R.Yu. Dynamics of mechanical oscillatory
thesis of the oscillatory mechanical systems considering features of physical realization of the feedback links// Dissertation of Candidate of Sciences (Engineering). - Irkutsk: Irkutsk State Transport University, 2006. - 192 p.
systems considering spatial forms of connection of elementary links.// Dissertation of Candidate of Sciences (Engineering). - Irkutsk: Irkutsk State Transport University, 2009. - 189 p.
Gao Jian-ping, Pan Yueyue
Y^K 624.042.6
ENERGY-BASED PARAMETER OPTIMIZATION OF ADDING-STOREY STRUCTURE USING ADDING-STOREY AS A TMD SYSTEM
1. INTRODUCTION
Before 1980s, most of the residential and civil buildings in China were multi-storey buildings. However, with the development of urban construction, urban land available becomes less and less. Many practical cases prove that, adding stories to old building is a kind of building upgrade technique with obvious comprehensive benefit, which accords also with the national conditions of China, i.e. less land and more people. Compared with traditional method of solving seismic problem, it is a new development direction to apply modern structure control technique to adding-storey construction of old building, and the combination of siesmic resistance with siesmic reduction is an inevitable trend.
Adding-storey seismic reduction technology is a passive control method similar to TMD(Tuned Mass Damper) proposed by Zhou(1997) in view of seismic retrofit of old buildings. The difference from TMD system is that, spring and damper are repalced with seismic isolation bearing, and additional mass blocks are repalced with new adding-storey structure, as shown in Fig.1(a).This technology belongs to high position inter-storey seismic isolation essentially but with the characteristics and functions of TMD system somewhat. Xie and Zhou (1998) had studied this structure system, including modeling, testing and
practical application. Similarly, Niu and Shi(2002), Liu et al(2008), Roberto Villaverde(2002) proposed that roof slabs or thermal insulating roof be used as the mass block of TMD, see Fig.1 (b).
It should be pointed out that, the seismic reduction effect of this technology depends on the selection of its device parameters. Therefore, Qian(1998), Li (1999), Luo(2000), and Zheng(2007) had proposed different optimal design method based on different objective functions and optimization criteria, respectively.
In this paper, the lateral rigidity and damping ratio of isolation device are taken as optimization variables, and the proportion of strain energy expectation value of isolation device to the total strain energy expectation value of the structure as the objective function, which attains its maximum. Thus, the proportion of strain energy dissipated by seismic isolation device to the strain energy input into structure is always the maximum, while the strain energy absorbed and dissipated by main structure is relatively less. The goal of protecting the main structure as much as possible is achieved, consequently, strengthening original structure can be reduced to the maximum extent or even avoided.
2. OPTIMIZATION PRINCIPLE
2.1 Expectation Value of Total Strain Energy
