The analytical solution and the dynamic characteristics of the system model velocity control vibrating roller Текст научной статьи по специальности «Строительство и архитектура»

Journal of Siberian Federal University. Engineering & Technologies 4 (2014 7) 480-488

УДК 625.084/085:625.855.3

The Analytical Solution and the Dynamic Characteristics of the System Model Velocity Control Vibrating Roller

Andrey P. Prokopiev*, Vladimir I. Ivanchura and Ruric Т. Emelianov

Siberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia

Received 27.12.2013, received in revised form 14.01.2014, accepted 02.02.2014

Operation is devoted a problem of identification of dynamic system of the volume hydraulic actuator of the running gear of a vibrating roller. The mathematical model of control process by speed of driving of a vibrating roller taking into account dynamics of hydrostatic transmission and a moment of resistance to roller driving is observed. Assay values of dynamic responses, a frequency analysis of a model of a guidance system are resulted by speed of driving of a vibrating roller.

Keywords: a vibrating roller, hydrostatic transmission, a mathematical model, a state space, dynamic responses, a frequency analysis.

Introduction

Most common in modern technologies of road construction in the final compaction of Hot Mix Asphalt (HMA) were vibrating road rollers. The need to improve the compaction asphalt concrete pavement process is the development of automated control vibrating roller on the basis of modern science and technology [1].

The purpose of research is to develop a model of the process speed control vibrating roller and analysis of the dynamic responses in preparation for the task of system synthesis control modes of compaction.

The original mathematical description of the system and the formulation of the problem

Roller is a machine, which consists of: engine, the front and back frames, the cabin, the mechanism of asphalt concrete edging. The working body of the roller is smooth metal drum with vibrations [2].

The hydraulic actuator of a driving of the vibration road roller with two power-driven rollers, Fig. 1, includes the variable capacity pump of the course drive and in parallel joint two hydromotors. The hydraulic actuator of the roller as control system, it is possible to present in the form of two

© Siberian Federal University. All rights reserved

* Corresponding author E-mail address: prok1@yandex.ru

16

Fig. 1. Hydraulic circuit design VHT of the drive of the mechanism of movement [2]: 1, 2, 3, 5, 6 - back-pressure valves; 4, 13 - relief valves; 7, 1 1 - the spool-type allocator; 8, 10 -a cooler; 9 - the pump; 12 - an actuator; 14 -the feed pump; 15 -a release valve; 16 - the variable capacity pump; 17 - noncontrollable hy-romotors

subsystems: the hydraulic and the hydromechanical. The state variable characterising a hydraulic subsystem, magnitude of hydraulic pressure P(t) of a stream of the operating fluid, created by the pump concerning pressure in a drain forecastle is. The state variable characterising a hydromechanical subsystem, magnitude <m(t) of speed of twirl of the hydromotor, under the pressure influence of a fluid stream is.

The volume hydrostatic transmission (VHT) the drive of a driving of a running roller (Fig. 1) switches on the variable capacity pump 16 and two noncontrollable hydromotors 17 connected in parallel to a hydraulic line.

As a result of transformations the mathematical model in terms of state-space presenting working process of system of automatic control by speed of driving of a vibrational running roller [3] is gained.

Analytical transformation of transfer functions

For the purpose of reduction of gained models transfer functions [3] to the shape of the most matching to conditions of computer (imitative) simulation of processes in the volume hydraulic actuator of the road roller following analytical transformations are executed.

Transfer functions [3] be presented two types of transfer functions Wi(s) or W2(s):

Wi(s)=-2-Kl 2 2; (i)

s2 + 2-a • s + a2 + p2

W2(s) = 2 K2-(S + Y> 2 . (2)

s2 + 2 • a • s + a2 + p2

Here Ki, K2 - factors; a, p, y - the real positive numbers. Numbers a and p define the real and imaginary parts of poles, and y - value of null of a matching transfer function.

- 481 -

Impulse response k^t) and step response h^t) performances for Wi(s) are defined by an inverse transformation method of Laplace (transition from images to originals):

) = K--e~a'' -sin(ß-1);

w) = K

ß-(ß2 +a2)

— ^ (ß2 + a2)

sin| ß-1 + arctgI — ||-ß

Impulse response k2(t) and step response h2(t) performances for W2(s) are defined by an inverse transformation method of Laplace:

K2• e~a t ((Y - a)• sin(ß • t) + ß • cos(ß • t));

K2

ß •( + a2)

Y • ß + e

(a2+ß2 )sin(ß • t) - y • (a • sin(ß • t )+ß • cos(ß • t))

From expressions for time responses follows, that -value a defines fading, and value p - an angular frequency of dynamic processes.

Through factors of the state- space equation [3] express as follows:

. "On + a22 ).

ß = 1 -(ü2 +alC'atl-an- a22f

"fpu- ~a22 . Torn - -all .

Kpu =hl> Kpm -al2'b22,

Krnu = afl '^f 1; K®m = b22-

Definition of fransUer functions of installation of control on the basis of transfer functions W1(s) or W2(s) gained by an analytical method:

Wpu (s) =

Kpuis + Ypu)

pu)

s2 + 2-a-s + a2 + ß2

Wpm (s) =

K

pm

22 + 2-a-2 + a2 +ß2

Wma>u(-) = -

Wa>m(s) = "

s2 +2-=■ s + 022 +ß2 '

Kgm ' Us + Trnm ) s2 + 2-a-s + a2+ßn

(3)

(4)

(5)

(6)

Here Wpu(s) - a transfer function defining change of pressure concerning change of control action; Wpm(s) - a transfer function defining change of pressure concerning change of dithering impact; Wm„(s) - a transfer function defining chcnge of an angular velocity of twirl of rollers concerning

change of control action; Wmm(s) - a transfer function defining change of an angular velocity of twirl of rollers concerning change of dithering impact; Kpe, Kpm, Km„, Kmm - factors; a, p, yp„, ymm - the real positive numbers. Numbers a and p define the real and imaginary parts of poles, and yp„, ymm - value of null of a matching transfer function.

Impulse response kpu(f) and sfep response hpi( performances -or W^is):

KP

kpu (0 = -jP" • i • ( - a) • sin (ß • t) + ß • cos (p • -) ) ;

hpu(t)--

K

pu

P>2 + a2)

Y pu'ß + e"

(a2+p2) -smfr-Y^^oosmPß^+ß^ cos(ß^ t))

Impulse response kpm(t) and step response hpm(t) performances for Wpm(s):

kpm (t ) "

К

pm -at

e~aK -sin(ß •t);

hpm (t1 "

к

pm

e~a■ * ■ П( +a2) • sin||ß(t + arctgj^-j j-ß

p(a2)

Impulse response kœu(=) and step re spooose hœu(i) performances for Wœu(s):

k

Ku (t) = ■ e~a t ■ sm(ß • t);

Km (t)'-

ß>2+ a2 )

■^(ß2 + a2) ■smfß■ t + arctg^£jj-P

Impulse response kmm(t) and step response hmm(t) performances for Wmm(s):

K«

Ks (?) = ^ • • ((Y«f - «) • sm (P • ß) + h • cos)|3-r)) :

Km (t) =

K«

P-( + a2 )

Ta,m-ß + e

(<x2+p2)- sm( ß-i )-y(°m-(ocsin(P^ 0+P-cos (ß • t ))

Computing Experiment

For check of gained models (3), (4), (5), (6) time dynamic characteristics of matching transfer functions are define d.

Type of road roller DU-96 [2] Open Joint Stock Company «Raskat» (Rybinsk, URL: http://www. raskat.yarosla~vl.ru): vibrational with two power-driven rollers. Running roller mass: mkl - operation, Mp-2 - constructive: mkl = 7200 kg; mk2 = 6600 kg. Diameter Db smooth вальца: Db = 1,07 m. Width вальца Lb (width of an obturated strip): Lb = 1,5 m. Linear pressure smooth вальца accordingly Plb - fast-head, P2b - back:

P1b = 23000 Nm-1; P2b = 224000 Mmr1.

Facto rst and their values for the state-space equation of a hydraulic subsystem (the hydrau lic pump, the drive of rollers and a transmitting hydraulic line): Ke! - factor of pressure of a transmitting hydraulic line, Kq = 34011m3P£r1 ; Wfoss - the factor considering pressure losse s in a hydraulic line, Kloss = 9,843f9-11m3Pa-1s-0 ca - a propeller angular velocity, œe = 29)3,2 s-1; qp - the maximum swept volume of the pump, qi^, = 35,iH0-6m3; qm - the maximum swept volume of a hydraulic engine of the drive roller, qm = 28740-<sm3; Jb- a running roller moment of inertia, Jb = 2058 kg m2.

Factors of mathematical model [2] vinrational tunning rollers ,3,yc96 counted on the basis of specifications:

for the first state-space equalion

lee = -Kloss ■ Kefl = - 3,281 ; au = -2qnt -Kel s e = -1,9133-10 7 Pa ;

b11 = ep • ■ Kel^ = 3,49904• 108 kgm~ls-3,

for the second state-space equation

"21 = 2qm ■ Jb_1 = 2,7891240-7 m■ kg-1; a22 = -1 f-1;

b22 = -Jb-= =-r,r(i049 kg~1m-2; ¿ai?imax = 14-103 Nm.

Following value s of factois and real numbers are gained

i

(afa + a7-) „ r . 2n"I7

a =--2-' P = L-( a12 ' a21 - all ' a22 + a ) J 2; Y pu =- a22;

Kpu 8 ¿11 ' Kpm 8 a12 • b22 ' Yram = -a11; Krau = a21 ' ' Kram 8 b22 '

Computing experiment with following initial data is put:

time range (s) process t = 0.. .6/a, a time step 1/100 s;

a = 2,14 s^1 ; P = 2,009 s-1 ; ypu =1,0 s-1 ; ymm = 3,281 s-1 ;

Kpu t 3,499 • 108 Kpm = 9,297 • 103 m-3.;

m • s

Kmu = 9-7,592 s-3; Kmn! = -4,85 9-10-4 mkg- 1.

Calculations are executed with application of mathematical program MathCAD.

Graphs of the time responses are presented in Fig. 2, 3, 4, 5.

Calculation oa time responses of change of pressure es operating and disturbing affe cting

kp(f) = kpu(f) + kpm(f), hp(f) = hpu(f) + (f).

The time responses of changes of pressure to controlling and to disturbing affecting are presented in Fig. 2.

kp(t), Pa ■ s hp(t), Pa

4x10'

kp (t ) 3x10

hp(t) 2x10

1x10

- 1x10

t, s

Fig. 2. The time responses ofchanges of pressure to controlling and to disturbing affecting

s"2

hm(t), s

t, S

Fig. 3. The time responses of change of an angular velocity to controlling and disturbing affecting

0

0

1

2

3

Calculation of time responses of change of an angular velocity to controlling and disturbing affecting at Ml™© = 14-103 N ■ m

K (t) (t) + ka>m (t)' r«max (t) = K (t) = ha>u (t) + ha>m (t)'M «max (t).

The time responses of change of an angular velo city to controlling and disturbing affecting are presented in Fig. 3.

Let's define system feequency responses on the channel o1 speed concerning control action. We will make change ofevariable 5 = /' ■ ce. Asaeesullof Irangformations, we will gain:

97+59 , » -97,59

Wwu (s) = -; Wmu (ro) =

7 5 9

s2 +4,28-s + 8,62 W-4,28-i-co-8,62

We build a logarithmic amplitude-frequency response (LAFR) and a logarithmic phase frequency response (LPFR): frepuency varies ovee the range co = 0,1- 100

180

= 20far ((u H) > 9cm (CO) = aig Wu (m))-gr ■ - 485 -

log o r

Fig. 4. Logarithmic frequency responses of system for the channel of control action - speed

Logarithmic frequency responses of dynamic system for the; channel of control action - velocity are displaf ed on Fig. 4.

Crossover Orequency: log0:0,^ =0,992!; ceMC.o. =lf °'992 = 9,f0 rad/s.

Vibration frequency oi1 asphalt running rollers usually makes from l = 30 Hz tof = "70 Hz [4]. At a speed conteol o;f driving erf a runningroller vibratkon is a handicap. Values LAFR for this frequency range hove made: ii0ti(27uJf1r =-f 1,2 dB; Liu(9nf2) = -6f,f dB.

At frequencies ovef tht i^^ng^e; from 30 Hz to 70 Hz influence of control action on speed is relaxed in hundreds and thousand timms in eelation to influence i- crossover frequency.

L et's define system frequency responses on the channel of speed concerning disturbing affecting, having made of a vf riable 5 = i • cy

-0,00048591 ( 5 + 3,281) „. 0,00048591-0• co + S,SSS59427S7S Wfm(5) =-2---¿, Wcm(f) =-2-.

52 + 4,28-5 + 8,62 -f2 + 4,28 • z -co + 8,62

We construct a logarithmic amplitude and phase frequency responses: frequency ranges c = 0,1 - 100 s-1

180

Llom(o) = 201eg(K<xraM l40»0»0), cpic„H(0^) = arg(wai1orwt(o))---- .

Logarithmic frequency responses of system on the channec disturbing affecting - velocity are presented in Fig. 5

Crossover frequency:locordncp_ = 0,89); cddc./ = 10 0'89 = 77, 8 rad/s.

Valuer LAFR foy this frequenry range Iraivei made: n^^nof) = -28,85 0B ha d (2n /2)=-3 6,22 0f.

- 4816 -

Llmm(ro), dB 9iram(ra), degr.

50

0

Lhm (ra )

* Um (<° ) - 50 - 180 0

- 100

- 150

- 200

- 1

0

1

log huc. f.

2

log (® )

Fig. 5. Logarithmic; frequency responses of system on the channel disturbing affecting - velocity

3

At vibration frequencies over the range from 30 Hz to 70 Hz moment influence vibration for the velocity is relaxed in tens and hundreds times in relation to influence of this moment in crossover frequency.

Conclusions

The problem of construction of mathematical model of contasl process by velocify of movement of an asphalt roller, taking into account dynamic responses of system of the volume hydraulic drive of transmission and a roller movement resistance on a rigid pavement is solved. Analytical methods of working out of mathematical model of a control system are applied. Results of research of the time responses and frequency responses on channels are gained: control action - velocity of motion; disturbing influence - velocity of motion.

At frequencies ovea the range faom 30 Hz io 70 Hz that is characteristic for systems vibration rumaing rolleas, isfluence of control actios an speed it relaxed in hundtefs and tfousand times in relation to influence in crossover frequency.

At vibration frequencies over the range from 30 Hz to 70 Hz moment influence vibration for the speed is relaxed in tens and hundreds times in relation to influence of this moment in crossover frequency.

The gained transfer functions can be used for research of dynamic responses of systems of the drive of a motion of the road roller and automatic-control system model building by working process of the vibration roller.

References

[1] ИванчураВ.И., ПрокопьевА.П., ЕмельяновР.Т., ПетровА.Д. // Строительные и дорожные машины. 2012. № 9. С. 39 - 45.

[2] Каток вибрационный двухосный двухвальцовый ДУ-96. Каток вибрационный комбинированный двухосный ДУ-97. Руководство по эксплуатации. ДУ-96.000.000 РЭ2. Рыбинск: ОАО «Раскат», 2012. 86 с.

[3] Иванчура В.И., Прокопьев А.П. // Журнал Сибирского федерального университета. Техника и технологии. 2013. Т. 6. № 2. С. 192 - 202.

[4] Казмиренко В.Ф. Электрогидравлические мехатронные модули движения. М.: Радио и связь, 2001. 431 с.

Аналитическое решение и динамические характеристики модели системы управления скоростью движения вибрационного катка

А.П. Прокопьев, В.И. Иванчура, Р.Т. Емельянов

Сибирский федеральный университет, Россия, 660041, Красноярск, пр. Свободный, 79

Работа посвящена задаче идентификации динамической системы объёмного гидравлического привода ходовой части вибрационного катка. Рассмотрена математическая модель процесса управления скоростью движения вибрационного катка с учетом динамики гидрообъёмной трансмиссии и момента сопротивления движению катка. Приведены результаты анализа динамических характеристик, частотный анализ модели системы управления скоростью движения вибрационного катка.

Ключевые слова: вибрационный каток, гидрообъемная трансмиссия, математическая модель, пространство состояний, динамические характеристики, частотный анализ.