CC BY
19
УДК 531.8:075.8
Candidate of technical sciences A. Kolyada, T. Sokol, D. Prokopenko, L. Ishkova, V. Bezhenova
National technical university, Zaporizhzhia
VIBRATIONS DECREASE IN MACHINES WITH CONTINUOUS IMPACT ACTION BY INTRODUCTION OF NON-LINEAR RESILIENT ELEMENTS
The influence of non-linearity in resilient characteristics on oscillation processes in machines drives working in continuous shock action for the purpose to improve its reliability and service life was considered. The condition of vibrations diminishing in a drive is shown.
Key words: machine drive, non-linear oscillation, service life.
The drive composed of electric motor, fly-wheel, reduction gear and two couplings with non-linear characteristics is considered. The drive is designed to work together with presses and casting-breakers in cyclic duty. The presence of periodic disturbances of resistance forces is the design feature of this kind of the drive. Loads in the drive are of shock character. The moment of useful resistance forces brought to incoming shaft of a working machine exceeds several times the rated drive (nominal) forces moment brought to the same shaft (for press - 2-4 times, for casting - breakers - 10-15 times, fig. 1, a). Disturbances spectrum is of a continuous nature in a wide range of frequencies (fig. 1, b).
As a consequence with continuous machine operation the possibility of resonance arises. Damping of such vibrations is usually achieved through dissipation of eneigy
that results in its losses as well as in high wear out of resilient elements and decrease in reliability and service life of a drive. Introduction of non-linearity results in connection of natural vibration frequency and amplitude and this may influence on origination of resonance phenomena.
General view of a drive and its kinematical scheme together with working machine are shown in fig. 2. Here, Jx is moment of inertia of a fly-wheel, J2 is brought to output shaft the inertia moment of revolving parts of reduction gear, J3 is moment of inertia of moving parts of working machine brought to the incoming shaft; cv c2 are rigidity coefficients of resilient couplings; kv k2 are energy dissipation coefficients in the couplings resilient elements; Uu is a gear ratio of a reduction gear.
50 100 150 200
Fig. 1. Force characteristics of a drive loading: cyclic loading (a) and loading spectrum (b)
Fig. 2. General view of a drive and working machine
©A. Kolyada, T. Sokol, D. Prokopenko, L. Ishkova, V. Bezhenova, 2012
ISSN 1607-6885 Нові матеріали і технології e металургії та машинобудуванні №1, 2012
137
For non-linear elements the power relationship of the torsion moment and angular deformation is taken. Generalized coordinates are: is the angle of rotation of
motor shaft, are angles °f deformation of
couplings. Afly-wheel with high inertia moment on a drive shaft enables to overcome peak loadings with a low value of vibration factor. In this case it is possible to accept t/cpj jdt = roj = const and to describe the drive vibrations with two equations
v|>! + a ■ + b ■ + c ■ \j/i + d ■ \j/2 + e = 0;
^2 + f-Wi +g'H'2 + h-\\i1 + k-<\i2 + l(([>3) = 0;
q>3 =(9i-H'iy£/i2-H'2> C1)
where e, /(cp3), cp3 are disturbances from the engine, working machine and angle of incoming shaft rotation of working machine, and factors of equations are connected with system parameters like a following ratios:
2 2 u12 1 c2 i u\2 7 ^2
a — Ci------------; b —--------------’u\2? c — k\------------------\ d —----------------u\2?
J'
M,
2 1 1
Cl mU\2 ,1 1 .
Jx J-> g~
1 1
h =--------k\-u\2, k = k2'(--------1----); / = —
M
resist
J
J J 'i
(2)
J 3
The solution of this problem on the whole is of great complexity. That is why features of machines class being examined are used. They include stationary areas of loading. First area where power disturbances of the engine act upon M = const and the second one where disturbances
1 m
of an engine M = const and working machine M = const
c? m o re&st
act upon. Initial disturbances of eveiy area are caused by the transient features between the areas. Presence of these disturbances only within the limits of eveiy area allows to consider this problem as a piece-stationaiy one. On the other hand, the existent of continuous nature disturbances and relation of self-disturbances to frequency and amplitude allow to influence upon the resonance.
The solution of the first problem includes the search of stationary points in eveiy area and estimation of their stability. The computation of stationary points of coordinates is realized with equations of motion (1) on
eveiy area with 14^ = 0, 14/2 = Vi = V2 = 0
provision of conditions, e 0, / = 0 (on the first area),
and e ^ 0, 1^0 (on the second area).
Stability of stationary points has been estimated by the Rauth-Hurwitz criterion. Then the Hurwitz determinants for linearized system being considered by the method of a harmonic linearization have the form:
63 = c + k\ = 0>75 ■ a ■ yQ + 0.75 ■ g - y\ — d -h + c - k\ &1 = 0,75 ■ y\(c ■ g- —b -h)— 0,75 ■ y$ -(d ■ f — a ■ £);
b0 = 0,5625 ■ yl ■ yi ■ (a ■ g - b ■ /); R = 61 'h^3
b 1 +63 -b0
(3)
where _y0> yj are the stationary points.
For stability of stationary points it is necessary that that Hurwitz determinants b3, b7 bvb0 and/? be more than zero.
With chosen parameters of working machine and a drive, that provide the stability of stationary points,
J1 = °°, J2 = 0,25 Kgm2,/3 = 0,1 Kgm2,q = 4-104Nmirad, Ui 2 = 15,7, ki - 1 Nms! rad,
c2 =3,14-106jV/«/W
k2 = 20 Nms I rad, MM = 40 Nm, Mconp =3140 Nm 3
through numerical methods the solution for disturbed motion was obtained. Results of the solution, the angles of deformation in couplings and loads in a working machine are shown in fig. 3.
Computations have proved that including of non-linearity into the system and provision of stability for stationaiy points enables to obtain little vibrations in drives of machines being considered. Fig. 4 shows the alteration in vibrations at transition to operating loading regime (A^) and an idle operation duty (A^) with respect to degree (n) of polynomial of non-linearity of resilient elements. So, when changing the linear system for cubic one the nonlinear vibration amplitude decreases for an order. This is related as to the shifting of vibration frequency at the increase of amplitude in areas of high frequencies where the eneigy of disturbance spectrum is significantly lower, as well as with progressive dissipation of vibration eneigy in area of high frequencies, and with positive effect of transient processes on the rate of amplitude growth at origination of the resonance.
Fig. 3. Dynamics of a system with non-linear elements: angle of deformation in couplings (a) and change of forces in working
machine (7) ) with present of vibrations
о.і
о.
Аі ♦ ♦♦
8
List of literature
1. Теория автоматического управления. Книга 1 / [под ред. 2. Коляда А. Ф. Формирование случайных процессов в Me-
B. В. Солодовникова]. - М. : Машиностроение, 1967. - ханических системах : монография / Коляда А. Ф.,
C. 455-537. Шевченко В. Г. - М. : Просвещение, 2008. - С. 14-21.
Одержано 05.07.2011
Коляда О.Ф., Сокол Т.О., Прокопенко Д.С., Ішкова Є.О., Бєженова B.C. Зменшення коливань у машинах безперервної ударної дії введенням нелінійних пружних елементів
У статті розглядається вшив пружних нелінійних характеристик на коливальні процеси в приводах машин, що працюють в ударному режимі, зметою підвищення надійності і довговічності. Показані умови зменшення коливань у приводі.
Ключові слова: привод машини, нелінійні коливання, термін життя.
Коляда А.Ф., Сокол Т.А., Прокопенко Д.С., Пшкова Е.А., Беженова B.C. Уменьшение колебаний в машинах непрерывного ударного действия введением нелинейных упругих элементов
В статье рассматривается влияние нелинейностей упругих характеристик на колебательные процессы в приводах машин, работающих в ударном режиме, с целью повышения его надежности и долговечности. Показаны условия уменьшения колебаний в приводе.
Ключевые слова: привод машин, нелинейные колебания, срок службы.
УДК 631.316.022
Канд. техн. наук 3. М. Шанина, канд. техн. наук Л. В. Гальченко,
канд. техн. наук Л. М. Мартовицкий
Национальный технический университет, г. Запорожье
МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ПОВЕРХНОСТИ ЗУБЧАТОГО РАБОЧЕГО ОРГАНА ДЛЯ ОБРАБОТКИ ПОЧВЫ
Предложена математическая модель, которая при определенных условиях позволит разработать зубчатый орган с такой формой рабочей поверхности, который будет удовлетворять агротехническим, технологическим и экономическим показателям при обработке почвы. Получены уравнения, которые описывают поверхность рабочего органа на участках впадин и выступов.
Ключевые слова: зубчатый рабочий орган, почва, выступы и впадины зубьев, логарифмическая спираль, парабола четвертого порядка, математическая модель, матрично-векторное решение.
Плодородие черноземных почв как основного эко- качества обработки почвы является ее крошение. Уро-
номическош ресурса Украины зависит от качества их вень крошения почвы зависит от геометрии рабочею
обработки. Известно, что определяющим критерием органа и от кинематики его движения в процессе ра-
© 3. М. Шанина, Л. В. Гальченко, Л. М. Мартовицкий, 2012
1607-6885 Нові матеріали і технології в металургії та машинобудуванні №1, 2012 1 39
Fig. 4. Relation of amplitude vibrations to polynomial degree
