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Controlled Passage through Resonance for Two-Rotor Vibration Unit: Influence of Drive Dynamics1
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Controlled Passage through Resonance for Controlled Passage through Resonance for Controlled Passage through Resonance for 11 Controlled Passage through Resonance for Controlled Passage through Resonance for Two-Rotor Vibration Unit: Influence of Drive Dynamics 1 Two-Rotor Vibration Unit: Influence of Drive Dynamics Controlled Passage through Resonance for Two-Rotor Two-Rotor Vibration Vibration Unit: Unit: Influence Influence of of Drive Drive Dynamics Dynamics111 Two-Rotor Vibration Unit: Influence of Drive Dynamics Two-Rotor Vibration Unit: Influence ofOlga Drive Dynamics Dmitry P. Dmitry V. V. Gorlatov*. Gorlatov*. Dmitry Dmitry A.Tomchin**. A.Tomchin**. Olga P. Tomchina*** Tomchina***
Dmitry Olga Dmitry V. V. Gorlatov*. Gorlatov*. Dmitry Dmitry A.Tomchin**. A.Tomchin**. Olga P. P. Tomchina*** Tomchina*** Dmitry V. Gorlatov*. Dmitry A.Tomchin**. Olga P. Tomchina*** Dmitry V. Gorlatov*. Dmitry A.Tomchin**. Olga P. Tomchina*** * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), Russian Federation, St. Petersburg, (e-mail: [email protected]) * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), * Saint Russian Petersburg State University of Architecture and Civil Engineering (SPSUACE), Federation, St. Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), ** Institute of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), Russian Federation, St. Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) ** Institute of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), ** of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), Federation, St. Petersburg, (e-mail: [email protected]) Federation, St. Petersburg, (e-mail: [email protected]) ** Institute Institute Russian of Russian Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), ** Institute of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), Russian Federation, St. Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) ** *** Institute of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), *** Saint Saint Petersburg Petersburg State University University of Architecture Architecture and Civil Engineering Engineering (SPSUACE), (SPSUACE), Russian Federation, Federation, St. Petersburg, Petersburg, (e-mail: [email protected]) Russian St. (e-mail: [email protected]) State of and Civil *** Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), Russian Federation, St. Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) *** Saint Saint Petersburg Petersburg State University University of Architecture Architecture [email protected]) Civil Engineering Engineering (SPSUACE), (SPSUACE), *** State of and Civil Russian Federation, St. Petersburg, (e-mail: Russian Federation, St. Petersburg, (e-mail: *** Saint Petersburg State University of Architecture [email protected]) Civil Engineering (SPSUACE), Russian St. (e-mail: [email protected]) Russian Federation, Federation, St. Petersburg, Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) Abstract: The problem of controlled passage through resonance zone for mechanical systems with Abstract: The problem of controlled passage through resonance zone for mechanical systems with Abstract: The problem of controlled passage through resonance zone for mechanical systems with several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating Abstract: The problem of controlled passage through resonance zone for mechanical systems with Abstract: The problem of controlled passage through resonance zone for mechanical systems with several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating Abstract: The problem of controlled passage through resonance zone for mechanical systems with efficiency of the control algorithm based on speed-gradient method are presented. The novelty of the several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating efficiency of the control algorithm based on speed-gradient method are presented. The novelty of the efficiency ofevaluation theofcontrol control algorithm basedsystem on simulation speed-gradient method are presented. The novelty of the several degrees freedom is analyzed. The results for two-rotor vibration unit illustrating results is in of the closed loop performance when the electric drive dynamics are taken efficiency of the algorithm based on speed-gradient method are presented. The novelty of the efficiency of the control algorithm based on speed-gradient method are presented. The novelty of the results is in evaluation of the closed loop system performance when the electric drive dynamics are taken results is in evaluation of the closed loop system performance when the electric drive dynamics are taken efficiency the control algorithm based onvibration speed-gradient method presented. novelty the into account. An interesting fact is that for unit model into account the electric drive results is of closed loop system performance when the electric drive dynamics are taken results is in inofevaluation evaluation of the the closed system performance whentaking theare electric drive The dynamics areof taken into An interesting fact is that for vibration unit model taking into account the electric drive into account. account. An interesting fact is loop thatresonance for vibration unit model taking intoless account the the electric drive results is in evaluation of the closed loop system performance when the electric drive dynamics are taken dynamics the time of passage through zone sometimes may be than for simplified into account. An interesting fact is that for vibration unit model taking into account the electric drive into account. interesting fact is thatresonance for vibration model taking intoless account the the electric drive dynamics the time of passage through zone sometimes may than simplified dynamics the An timethe ofdrive passage through resonance zoneunit sometimes may be be less than for for the simplified into account. An interesting fact is thatresonance for vibration unit model taking intoless account the the electric drive model neglecting dynamics. dynamics the time of passage through resonance zone sometimes may be less than for the simplified dynamics the time of passage through zone sometimes may be than for simplified model neglecting the drive dynamics. model neglecting the drive dynamics. dynamics the time of passage through resonance zone sometimes may be less than for the simplified model neglecting neglecting the the drive drive dynamics. dynamics. model Keywords: vibration unit, dynamics. resonant frequencies, frequencies, drive drive dynamics, dynamics, resonance resonance zone, zone, control control algorithms, algorithms, model the drive Keywords: vibration unit, resonant © 2015,neglecting IFAC (International Federationfrequencies, of Automaticdrive Control) Hosting resonance by Elsevier zone, Ltd. Allcontrol rights reserved. Keywords: vibration unit, resonant dynamics, algorithms, unbalanced rotor s. Keywords: vibration unit, resonant resonant frequencies, drive drive dynamics, dynamics, resonance zone, zone, control control algorithms, algorithms, Keywords: vibration unit, frequencies, resonance unbalanced rotor s. unbalanced rotor s. vibration unit, resonant frequencies, drive dynamics, resonance zone, control algorithms, Keywords: unbalanced s. unbalanced rotor rotor s. unbalanced rotors. evaluated, and the necessary number of dampers and their evaluated, and the necessary number of dampers and their 1. INTRODUCTION evaluated, and the necessary number of dampers and their 1. INTRODUCTION optimal location were determined in (Viderman and Porat, evaluated, and the necessary number of dampers and their 1. INTRODUCTION evaluated, and the necessary number of dampers and their optimal location were determined in (Viderman and Porat, 1. INTRODUCTION optimal location were determined in (Viderman and Porat, 1. INTRODUCTION evaluated, and the necessary number of dampers and their 1987). In (Kinsey and Mingori, 1992) a nonlinear controller optimal location were determined in (Viderman and Porat, An important problem for systems based on vibration optimal location were determined in (Viderman and Porat, 1987). In (Kinsey and Mingori, 1992) a nonlinear controller 1. INTRODUCTION An important problem for systems based on vibration 1987). (Kinsey and Mingori, 1992) aa nonlinear controller An important problem for systems based on vibration optimal were determined indespin (Viderman Porat, reducing resonance during of aa and dual-spin 1987). In Inlocation (Kinsey andeffects Mingori, 1992) nonlinear controller technologies is the passing of vibroactuators through An important problem for systems based on vibration 1987). In (Kinsey and Mingori, 1992) a nonlinear controller reducing resonance effects during despin of dual-spin An important problem for systems based on vibration technologies is the passing of vibroactuators through reducing resonance effects during despin of aasuppression dual-spin technologies is the passing of vibroactuators through 1987). In (Kinsey and Mingori, 1992) a nonlinear controller spacecraft was designed. A method of vibration reducing resonance effects during despin of dual-spin An important for systems based on vibration resonance for systems in post-resonance modes. technologies is the passing of vibroactuators through reducing resonance effects during of despin of asuppression dual-spin spacecraft was designed. A method vibration technologies isproblem the operating passing ofthe vibroactuators through resonance for systems operating in the post-resonance modes. spacecraft was designed. A method of vibration resonance for systems operating in the post-resonance modes. reducing resonance effects during resonances of by asuppression dual-spin for rotating shafts passing switching spacecraft was designed. Athrough method ofdespin vibration suppression technologies is the passing of vibroactuators through Such a problem needs special attention in the case when the resonance for systems operating in the post-resonance modes. spacecraft was designed. A method of vibration suppression for rotating shafts passing through resonances by switching resonance for systems operating in the post-resonance modes. Such aa problem needs special attention in the case when the for rotating shafts passing through resonances by switching Such problem needs special attention in the case when the spacecraft was designed. A method of vibration suppression shaft stiffness was proposed in (Wauer and Suherman, 1998). for rotating shafts passing through resonances by switching resonance for systems operating in the post-resonance modes. power of aa motor not sufficient passage through Such aa problem problem needsis special attention for in the the case when when the for rotating shafts passing through resonances by switching shaft stiffness was proposed in (Wauer and Suherman, 1998). Such needs special attention in case the power of motor is not sufficient for passage through shaft stiffness was proposed in (Wauer and Suherman, 1998). power of a motor is not sufficient for passage through for rotating shafts passing through resonances by switching In (Balthazar et al., 2001) the dynamics of passage through shaft stiffness was proposed in (Wauer and Suherman, 1998). Such a problem needs special attention in the case when the resonance zone due to tois Sommerfeld effect (Blekhman, 2000). shaft power of of zone a motor motor is Sommerfeld not sufficient sufficient for(Blekhman, passage through through stiffness was proposed in (Wauer and Suherman, 1998). In (Balthazar et al., 2001) the dynamics of passage through power a not for passage resonance due effect 2000). In (Balthazar et al., 2001) the dynamics of passage through resonance zone due to Sommerfeld effect (Blekhman, 2000). shaft stiffness was proposed in (Wauer and Suherman, 1998). resonance of a vibrating system with two degrees of freedom In (Balthazar et al., 2001) the dynamics of passage through power of a motor is not sufficient for passage through Dynamics of near resonance behavior are nonlinear and very resonance zone zone dueresonance to Sommerfeld Sommerfeld effect (Blekhman, 2000). In (Balthazar al., 2001) the dynamics of passage through resonance of aetvibrating system with two degrees of freedom resonance due to effect (Blekhman, 2000). Dynamics of near behavior are nonlinear and very resonance of system with two degrees of freedom Dynamics of behavior are nonlinear and very In (Balthazar al., 2001) theearly dynamics passage through was examined. However the did have resonance of aaaetvibrating vibrating system withalgorithms two of degrees ofnot freedom resonance dueresonance to Sommerfeld effect (Blekhman, 2000). complicated. Publications related to this problem can be Dynamics zone of near near resonance behavior are nonlinear and very resonance of vibrating system with two degrees of freedom was examined. However the early algorithms did not have Dynamics of near resonance behavior are nonlinear and very complicated. Publications related to this problem can be was examined. However the early algorithms did not have complicated. Publications related to this problem can be resonance of a vibrating system with two degrees of freedom enough robustness with respect respect to algorithms uncertainties and were was examined. examined. However the early earlyto algorithms did and not were have Dynamics of near resonance behavior are nonlinear and very found in the literature during about 50 years (Kononenko, complicated. Publications related to this problem can be was However the did not have enough robustness with uncertainties complicated. Publications related to this problem can be found in the literature during about 50 years (Kononenko, enough robustness with respect to uncertainties and were found in the literature during about 50 years (Kononenko, was examined. However the early algorithms did not have hard to design. enough robustness with with respect respect to to uncertainties uncertainties and and were were complicated. Publications related to this problem can be 1964; Quinn et al., 1995; Cvetićanin, 2010). found in the literature during about 50 years (Kononenko, enough robustness hard to design. found in the et literature about 2010). 50 years (Kononenko, hard 1964; Quinn al., 1995;during Cvetićanin, 1964; al., Cvetićanin, robustness with respect to uncertainties and were hard to to design. design. found in the et about 2010). 50 years (Kononenko, enough 1964; Quinn Quinn etliterature al., 1995; 1995;during Cvetićanin, 2010). hard to design. 1964; Quinn et al., 1995; Cvetićanin, 2010). For practical implementation of control system it is important Perhaps the first approach to the problem of controlled hard to design. For practical implementation of control system it is important 1964; Quinn etfirst al., 1995; Cvetićanin, 2010). Perhaps the approach to the problem of controlled For practical implementation of control system it is important Perhaps the first approach to the problem of controlled to develop reasonably simple passing through resonance zone For practical implementation of control system it is important passage through resonance zone was the so-called "double Perhaps the first approach to the problem of controlled For practical implementation of control system it is important to develop reasonably simple passing through resonance zone Perhaps the first approach to the problem of controlled passage through resonance zone was the so-called "double to develop reasonably simple passing through resonance zone passage through resonance zone was the so-called "double For practical implementation of control system it is important control algorithms, which have such robustness property: to develop reasonably simple passing through resonance zone Perhaps the first approach to the problem of controlled start" method (Gortinskii et al., 1969). This and other passage through resonance zone was the so-called "double to develop reasonably simple passing through resonance zone control algorithms, which have such robustness property: passage through resonance zone was the so-called "double start" method (Gortinskii et al., 1969). This and other control algorithms, which have such robustness property: start" method (Gortinskii et al., 1969). This and other to develop reasonably simple passing through resonance zone keeping high quality of the controlled system (vibration unit) control algorithms, which have such robustness property: passage through resonance zone was the so-called "double nonfeedback methods are characterized characterized byThis difficulties in control 1969).by This and other other start" method methodmethods (Gortinskii et al., al., 1969). algorithms, have suchsystem robustness property: keeping high quality which of the controlled (vibration unit) start" (Gortinskii et and nonfeedback are difficulties in keeping high of the system (vibration unit) nonfeedback methods are characterized by difficulties in control algorithms, which have such robustness property: under of and external conditions. keepingvariation high quality quality ofparameters the controlled controlled system (vibration unit) start" method (Gortinskii et al., This and other calculation of switching instants of aa1969). motor and sensitivity to nonfeedback methods are characterized by difficulties in keeping high quality of the controlled system (vibration unit) under variation of parameters and external conditions. nonfeedback methods are characterized by difficulties in calculation of switching instants of motor and sensitivity to under variation of parameters and external conditions. calculation of switching instants of a motor and sensitivity to keeping high quality of the controlled system (vibration unit) Perhaps the first such a simple controller was proposed in under variation of parameters and external conditions. nonfeedback methods are characterized by difficulties in inaccuracies of model and to interferences. In (Leonov, 2008) calculation of switching instants of a motor and sensitivity to variation of parameters and external conditions. Perhaps the first such aa simple controller was proposed in calculation ofof switching instants of a motorIn and sensitivity to under inaccuracies model and to interferences. (Leonov, 2008) Perhaps the controller was in inaccuracies of model and to interferences. In (Leonov, 2008) under of based parameters andspeed-gradient external conditions. (Tomchina, 1997)such based on the the speed-gradient method Perhapsvariation the first first such a simple simple controller was proposed proposed in calculation of switching instants of a motor and sensitivity to it is proved that the Sommerfeld effect does not occur for any inaccuracies of model and to interferences. In (Leonov, 2008) Perhaps the first such a simple controller was proposed in (Tomchina, 1997) on method inaccuracies of model and to interferences. In (Leonov, 2008) it is proved that the Sommerfeld effect does not occur for any (Tomchina, 1997) based on the speed-gradient method it is proved that the Sommerfeld effect does not occur for any Perhaps the first such a simple controller was proposed in previously used for control of nonlinear oscillatory systems (Tomchina,used 1997) based on on the speed-gradient speed-gradient method inaccuracies of model and to interferences. In (Leonov, 2008) passage through resonance for a synchronous electric motor it is proved that the Sommerfeld effect does not occur for any (Tomchina, 1997) based the method previously for control of nonlinear oscillatory systems it is proved that the Sommerfeld does notelectric occur for any previously passage through resonance for aaeffect synchronous motor for control of nonlinear oscillatory systems passage through resonance for synchronous electric motor (Tomchina, 1997) based on the speed-gradient method et al., 1996). number of speed-gradient previously used used for control ofA nonlinear oscillatory systems it is proved that the Sommerfeld does notelastic occurbase. for any (Andrievskii with an asynchronous start-up, on an passage through resonance formounted aeffect synchronous electric motor previously used for control of nonlinear oscillatory systems (Andrievskii et al., 1996). A number of speed-gradient passage through resonance for a synchronous electric motor with an asynchronous start-up, mounted on an elastic base. (Andrievskii et al., 1996). A number of speed-gradient with an asynchronous start-up, mounted on an elastic base. previously used for control of nonlinear oscillatory systems algorithms for passage through resonance in 2-DOF (Andrievskii et al., 1996). A number of speed-gradient passage through resonance for mounted a synchronous electric motor (Andrievskii with an an asynchronous asynchronous start-up, mounted on an an elastic elastic base. etpassage al., 1996). Aresonance number in of 2-DOF speed-gradient algorithms for through systems with start-up, on base. algorithms for through in 2-DOF systems A prospective approach to the problem is on feedback (Andrievskii al., 1996). Aresonance number ofThe speed-gradient were proposed in (Fradkov et al., 2011). approach of algorithms foretpassage passage through resonance in 2-DOF systems with an asynchronous start-up, mounted onbased an elastic base. A prospective approach to the problem is based on feedback algorithms for passage through resonance in 2-DOF systems were proposed in (Fradkov et al., 2011). The approach of A prospective approach to the problem is based on feedback were proposed in (Fradkov et al., 2011). The approach of control. Feedback control algorithms for passing through algorithms for passage through resonance in 2-DOF systems A prospective approach to the problem is based on feedback (Tomchina, 1997; Fradkov et al., 2011) was applied to twowere proposed in (Fradkov et al., 2011). The approach of A prospective approach to the problem is based on feedback control. Feedback control algorithms for passing through were proposed in (Fradkov et al., 2011). The approach of (Tomchina, 1997; Fradkov et al., 2011) was applied to twocontrol. Feedback control algorithms for passing through (Tomchina, 1997; Fradkov et al., 2011) was applied to twoA prospective approach to the problem is based on feedback resonance zone of ofcontrol mechanical systems were studied in were proposed in (Fradkov et al., 2011). The approach of control. Feedback Feedback control algorithms for were passing through rotor vibration set-up in (Fradkov et al., 2014). (Tomchina, 1997; Fradkov et al., 2011) was applied to twocontrol. algorithms for passing through resonance zone mechanical systems studied in (Tomchina, 1997; Fradkov et al., et 2011) was applied to tworotor vibration set-up in (Fradkov al., 2014). resonance zone of mechanical systems studied in rotor set-up in al., control. algorithms for were passing through (Malinin and 1983; Viderman and Porat, 1997; Fradkov et al., et was applied to tworesonanceFeedback zonePervozvansky, ofcontrol mechanical systems were studied in (Tomchina, rotor vibration vibration set-up in (Fradkov (Fradkov et2011) al., 2014). 2014). resonance zone of mechanical systems were studied in (Malinin and Pervozvansky, 1983; Viderman and Porat, rotor vibration set-up in (Fradkov et al., 2014). (Malinin and Pervozvansky, 1983; Viderman and Porat, In this paper the system designed in (Fradkov et al., is resonance zone of mechanical systems were studied in 1987; Kinsey and Mingori, 1992). In (Malinin and rotor vibration set-up in (Fradkov et al., 2014). (Malinin and Pervozvansky, 1983; Viderman and Porat, In this paper the system designed in (Fradkov et al., 2014) 2014) is (Malinin and Pervozvansky, 1983; Viderman and Porat, 1987; Kinsey and Mingori, 1992). In (Malinin and In the designed (Fradkov et 2014) 1987; Kinsey and Mingori, 1992). In (Malinin and analyzed taking into account account thein electric drive dynamics In this this paper paper the system system designed in electric (Fradkovdrive et al., al.,dynamics 2014) is is (Malinin and Pervozvansky, 1983; Viderman and Porat, Pervozvansky, 1983) an optimal control algorithm for 1987; Kinsey and Mingori, 1992). In (Malinin and In this paper the system designed in (Fradkov et al., 2014) is analyzed taking into the 1987; Kinsey and Mingori, 1992). In (Malinin and Pervozvansky, 1983) an optimal control algorithm for analyzed taking into account the electric drive Pervozvansky, 1983) an optimal control algorithm for In this paper the system designed (Fradkov et al.,dynamics 2014) is particularly the DC EMF. (Leonov, 2008), analyzed taking into motor account thein Unlike electric drive dynamics 1987; Kinsey and Mingori, 1992). In (Malinin and passage of an unbalanced rotor through critical speed was Pervozvansky, 1983) an optimal control algorithm for analyzed taking into account the electric drive dynamics particularly the DC motor EMF. Unlike (Leonov, 2008), Pervozvansky, 1983) an rotor optimal control algorithm for particularly passage of an unbalanced through critical speed was the DC motor EMF. Unlike 2008), passage an unbalanced rotor through critical speed taking intoabsence account theSommerfeld electric (Leonov, drive dynamics which proved of effect for aa particularly the the DC motor EMF. Unlike (Leonov, 2008), Pervozvansky, 1983) an optimal control algorithm for analyzed proposed. same problem several control methods are passage of ofFor an the unbalanced rotor through critical speed was was particularly the DC motor EMF. Unlike (Leonov, 2008), which proved the absence of Sommerfeld effect for passage of an unbalanced rotor through critical speed was proposed. For the same problem several control methods are which proved the absence of Sommerfeld effect for a proposed. For the same problem several control methods are particularly the DC motor EMF. Unlike (Leonov, 2008), which proved the absence of Sommerfeld effect for passage an the unbalanced rotor several throughcontrol criticalmethods speed was proposed.ofFor For the same problem problem several control methods are which proved the absence of Sommerfeld effect for aa proposed. same are proposed. For the same problem several control methods are which proved the absence of Sommerfeld effect for a
1 1 The work was supported by Russian Scientific Foundation (project 14-29-00142) 1 The work was supported by Russian Scientific Foundation (project 14-29-00142) The work was supported by Russian Scientific Foundation (project 14-29-00142) 1 1 The work was supported by Russian Scientific Foundation (project 14-29-00142) The work©was supported by Russian Scientific (project 14-29-00142) 2405-8963 2015, IFAC (International FederationFoundation of Automatic Control) Hosting by 1
Thereview work©was supported by Russian ScientificFederation Foundationof(project 14-29-00142) Copyright IFAC 2015 317 Peer responsibility of International Automatic Control. Copyright ©under IFAC 2015 317 Copyright © IFAC 2015 317 10.1016/j.ifacol.2015.09.204 Copyright © © IFAC IFAC 2015 2015 317 Copyright 317 Copyright © IFAC 2015 317
Elsevier Ltd. All rights reserved.
MICNON 2015 314 Dmitry V. Gorlatov et al. / IFAC-PapersOnLine 48-11 (2015) 313–318 June 24-26, 2015. Saint Petersburg, Russia
where H1 is the energy of rotating subsystem, H2 is the energy of a carrier subsystem, H12 is the energy of interaction.
synchronous motor, this effect is topical for a DC motor. In Section 2 the problem statement for control of passage through resonance zone for mechanical systems with several degrees of freedom adopted from (Fradkov et al., 2011) is presented. The control algorithm based on the speed-gradient method for two-rotor vibration units is exposed in Section 3. The simulation results illustrating efficiency and robustness of the proposed algorithm in presence of drive dynamics are described in Section 4.
The idea of control algorithms described below, is in that slow motion (t) is being isolated and then "swinging" starts to obtain rise of energy of a rotating subsystem. To isolate slow motions, low-pass filter is being inserted into energy control algorithms. Let slow component appears in oscillations of angular velocity of a rotor. Then we start with the control algorithm proposed in (Tomchina, 1997) is:
2. PROBLEM STATEMENT AND IDEA OF SOLUTION To describe the dynamics of a mechanical system with n degrees of freedom the standard Euler-Lagrange equations are used:
), T , (8) u sign (( H H )
where is the variable of the filter, performing slow motions, T is the time constant of the filter. At low damping, slow motions also fade out slowly, that gives control algorithm an opportunity to create suitable conditions to pass through resonance zone. Thus the effect of "feedback resonance" (Fradkov, 1999) is created. After passing the resonance zone it is suggested to turn off the "swinging" and then to switch the algorithm to controlling with constant torque. For a proper work of a filter, it should suppress fast oscillations with frequency and pass slow oscillations with B frequency, where B is the Blekhman frequency (Blekhman et al., 2008). I.e. time constant of a filter T should be chosen from the inequality
A(q )q C (q, q ) G (q) Bu, (1)
where u=u(t) is n-dimensional input vector; q=q(t) is n-vector of generalized coordinates, A(q) is nn inertia matrix; C (q, q ) is the n-vector of Coriolis and centrifugal forces; G(q) is the n-vector of gravity forces; B is the mn control matrix. However, for the purpose of control algorithm design it is often convenient to use equations in Hamiltonian form: H p q
T
H Bu, q p
T
, (2)
T 2 B . (9)
where p=p(t) is the generalized momenta vector, H=H(p, q) denotes the Hamiltonian function total energy of the free (uncontrolled) system: H ( p, q )
Algorithms of passing through resonance zone for the tworotor vibration units are described in (Fradkov et al., 2014). Below we analyze the closed loop system with the algorithm of (Fradkov et al., 2014) taking into account the electric drive dynamics.
1 T 1 p A(q ) p Π (q ), (3) 2
where (q) is the potential energy. First of all, let us define the auxiliary control goal as the approaching of energy of the free system to a surface of the given energy level H ( p (t ), q (t )) H
3. MODEL OF TWO-ROTOR VIBRATION UNIT TAKING INTO ACCOUNT THE DRIVE DYNAMICS Consider the problem of two-rotor vibration unit start-up (spin-up) (Blekhman et al., 1999). The unit SV-2 consists of two rotors installed on the supporting body (Fig. 1).
when t . (4)
Introducing the objective function Q ( p, q )
1 H ( p, q) H 2 , (5) 2
the goal (4) can be reformulated as: Q(q(t ), p (t )) 0 when t . (6)
Control algorithm for passing through resonant frequencies for an unbalanced rotor is based on the speed-gradient method (Balthazar et al., 2001), which allows to synthesize control algorithms for significantly nonlinear plants. The model of the system is split into two subsystems: carrier and rotating bodies. Then the total energy can be represented in the following form:
H ( p (t ), q (t )) H 1 ( p(t ), q(t )) H 2 ( p(t ), q(t )) H 12 ( p(t ), q(t )),
Fig. 1. Schematics of two-rotor vibration unit: 1 motors; 2 – motor supports; 3 – frame of the unit; 4 – unbalanced rotors; 5 – vibrating platform; 6 – carrying springs; 7 – rotor bearings; 8 – cardan shafts.
(7)
318
MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Dmitry V. Gorlatov et al. / IFAC-PapersOnLine 48-11 (2015) 313–318
motor; kr is the friction coefficient in the bearings; M r1 k r 1 is resistance torque of a rotor, caused by resilient friction, Mu1 is the unbalanced rotor’s own torque, M u1 (t ) mg cos( 1 ); Mm1 is the control torque arriving ~ at the input of the "unbalanced rotor"; M si is the moment, caused by the influence of the supporting body:
The kinematic scheme of vibration unit is presented in Fig. 2.
y M
φ1 a
yc
M
xc
315
~ M si xc m sin( i ) yc m cos( i )
φ2
( J i (1) i rm cos i ) (1) i 2 rm sin i ,
a
where Ji, i = 1, 2 are the inertia moments of the rotors.
c02; β
c01
Since the laboratory setup SV-2 used the DC motors, the electric drive structure is selected as the traditional singlecircuit system with current loop and proportional-integral (PI) current controller WCR(р) = b(τр+1) / τр is configured to optimum modulo; b, are dynamic gain and time constant of the regulator.
c02; β
c01
Fig. 2. Two-rotor vibration unit with DC motors. , 1, 2 are angle of the supporting body and rotation angles of the rotors, respectively, measured from the horizontal position; xc, yc are the horizontal and vertical displacement of the supporting body from the equilibrium position; mi = m, i = 1, 2 and mn are the masses of the rotors and supporting body; ρi = ρ, i = 1, 2 are the rotor eccentricities; c01, c02 are the horizontal and vertical spring stiffness; β is the damping coefficient. It is assumed that rotor shafts are orthogonal to the motion of the support. φψ
To convert the scheme into the state space equations, assume that the whole system dynamics may be considered in the vertical plane. Then the equations of dynamics have the following form (Fradkov et al., 2014): m(sin( 1 ) sin( 2 )) m0 xc 1 m sin( 1 ) 2 m sin( 2 ) 2 m(cos( 1 ) cos( 2 )) 12 m cos( 1 ) 22 m cos( 2 )
Unbalanced rotor structural diagram taking into account the drive dynamics is shown in Fig. 3. M1
2 1 m cos( 1 ) 2 2 m cos( 2 ) 2c 01 x c x c 0;
PI-CR TC DC-Motor ETC U1 1 Ra p 1 UCR kTC b kU TTC p 1 Ta p 1 p CS Ia UCS kCS Electric Drive TCS p 1 Em kF 1 1 p
1 1 Jp
mgρ cos(·)
m(cos( 1 ) cos( 2 )) m0 yc 1 m cos( 1 ) 2 m cos( 2 ) 2 m(sin( 1 ) sin( 2 )) 12 m sin( 1 ) 22 m sin( 2 )
km kr
Mr1
Mu1
2 1 m sin( 1 ) 2 2 m sin( 2 ) m0 g 2c 02 y c y c 0;
Mm1
(10)
xc m(sin( 1 ) sin( 2 )) yc m(cos( 1 ) cos( 2 ))
Rotor
Supporting Body Ms
J J 1 J 2 2mr (cos 1 cos 2 ) 1 ( J 1 mr cos 1 ) 2 ( J 2 mr cos 2 )
Fig. 3. Vibration unit block diagram with drive dynamics.
12 mr sin 1 22 mr sin 2 2mr 1 sin 1 2mr 2 sin 2 mg (cos( 1 ) cos( 2 ))
Fig. 3 include scheme for first drive only, the second drive scheme is similar. The following notation is used here: CR is the current regulator; TC is the power (thyristor type) converter; CS is the current sensor; Ia is the armature current; ЕTC and Еm are converter and motor EMFs; kTC and kCS are converter and current feedback gains; kF is the motor torque (EMF) coefficient; TTC and TCS are converter and current sensor time constants; Ta is the armature time constant; Ra is the armature curcuit resistance; UCR and UCS are current controller and current sensor output voltages; M1 is the first drive control torque; U1 is the voltage corresponding to the calculated torque, km = kF; 1 is the angular velocity of a
c 03 0; xc m sin( 1 ) yc m cos( 1 ) ( J 1 mr cos 1 ) 1 J 1 2 mr sin 1 mg cos( 1 ) k r 1 M m1 ; xc m sin( 2 ) yc m cos( 2 ) ( J 2 mr cos 2 ) 2 J 2 2 mr sin 2 mg cos( 2 ) k r 2 M m 2 ; 319
MICNON 2015 316 Dmitry V. Gorlatov et al. / IFAC-PapersOnLine 48-11 (2015) 313–318 June 24-26, 2015. Saint Petersburg, Russia
where g is the gravity acceleration; m0 is the total mass of the unit, m0 = 2m + mn. When taking into account the dynamics of the drive in the simulation process the control torques Mmi, i = 1, 2 are formed in accordance with the structural diagram of the "electric drive" (Fig. 3). The "unbalanced rotor" structural diagram prepared in accordance with the fourth equation of system (10).
satisfy the relation (9). The larger values of T lead to decrease in average power of the control signal and to slowing down the algorithm. M1, obtained in accordance with the equations (12). 5. SIMULATION RESULTS Efficiency of the proposed control algorithm was studied in the MATLAB environment. Fig.4 Fig.7 correspond to the model (10) with the drive dynamics neglected. The simulation results for the system (10) are shown in Fig. 4 with basic system parameters: Ji = 0.014 kgm2, m = 1.5 kg, mn = 9 kg, ρ = 0.04 m, kr = 0.01 J/s, = 5 kg/s, c02 = 5300 N/m, c01=1300 N/m and constant torque M0 = 0.82 Nm (internal curves, "capture") and M0 = 0.83 Nm (external curves, passage).
In accordance with the structure in Fig. 3 each motor torque Mmi determined by the value Mi, calculated in the algorithm and obeys a system of differential and algebraic equations (index "i" is omitted for simplicity):
1 1 Ia I a ( ETC E m ) , Ta Ra 1 ETC k TCU CR , E TC TTC
d 1 , s -1
1 U CS U CS k CS I a , (11) TCS
b U 1 U CS ,
0
5
10
t,s 20
15
The graphs of the angular velocity 1 and filter variable 1(t) for the first rotor are shown in Fig.5 and Fig.6. The algorithm (12) is used for control for two values M0, namely M0 = 0.8 Nm (Fig.5, T = 0.8 s), M0 = 0.5 Nm (Fig. 6, T = 0.1.2 s). Both values are less than those corresponding to capture if the controlling torque is constant. Fig.7 demonstrates the graphs of 1 and 2 for operating value of the torque in the algorithm (12) M0 = 0.4 Nm. Smaller values of M0 do not provide passage through resonance. In addition the graphs of the supporting body coordinates xc, yc, are shown in Fig. 5. As seen from the pictures application of the algorithm (12) allows the designer to significantly expand the range of operating frequencies beyond resonance and provides stable amplitude of the platform vibrations after passing the resonance zone.
4. CONTROL ALGORITHM At the low levels of constant control action M i (t ) (1) i M 0 , i = 1, 2 in the near-resonance zone the rotor angle is "captured", while increase of the control torque leads to passage through resonance zone towards the desired angular velocity. Synthesis of the control algorithm Mi = U(z), i = 1, 2 is needed for acceleration of the unbalanced rotors, before the system passes through the resonance zone, where T z x c , x c , y c , y c , , , 1 , 1 , 2 , 2
is the state vector of the system. It is assumed that the level of control action is limited and does not allow system to pass through the resonance zone using constant action.
x(t)
x, m
0
-0.05 0
1,
(12)
5
, rad
where Н = Т+, i(t) are the variables of the filters, T > 0, T = const and H* are the parameters of the algorithm, (t ) max sgn ( H () H ) , where sgn[z] = 1 with z > 0,
t,s 5 10 15 20 d1/dt(t), (t)
60 d /dt(t) 1 40 20 (t) 0 t,s t , s -20 0 5 10 15 20 10 15 20 -1
0
-0.01 0
y(t)
0
t , s -0.1 10 15 20 0 (t)
0.01
i 1,2 ,
0.1 y, m
0.05
In this paper we analyze the following modification of the control algorithm (8) initially proposed by (Fradkov et al., 2014) for passing through resonance zone: 0 & H H * i i 0 ,
M0 = 0.82 Nm
Fig. 4. Simulation results for various constant torques.
E m kF , M m k m I a .
1i M 0 , if i ui 1 M 0 , if 0, else, T i i i ,
50 0
U CS bU 1 U CS U CS 1 ,
M0 = 0.83 Nm
d1, , s
U CS 1
d 1 /dt(t)
100
5
Fig. 5. Simulation results for systems (10), (12) with M0 = 0.5 Nm.
0 t
sgn[z] = 0 with z ≤ 0. Time constant of the filters T should 320
MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Dmitry V. Gorlatov et al. / IFAC-PapersOnLine 48-11 (2015) 313–318
d 1 , , s -1
100
d 1 /dt(t)
0
(since Mm = kmIa). An appropriate choice of the drive parameter kCS and PI-current controller gains, defining M0 and I a may realize the same range of the drive torques as in the previous case. In Fig. 8 the graphs of 1 , 1(t), the graph of the current Ia1 and the graph of EMF ЕTC1 are shown. As seen from the graphs taking into account the drive dynamics does not change the lower bound of achievable stable operating velocities. Moreover the filter time constants T also do not change their value.
d 1 /dt(t), (t)
50
(t) 0
5
10
15
20
t,s
dφ1 / dt, dφ2 / dt, s
-1
Fig. 6. Simulation results for systems (10), (12) with M0 = 0.8 Nm.
50 40 30 20 10 0 -10 -20 -30 -40 -50
dφi / dt
0
2
4
6
Table 1. Quantitative indicators of the vibration unit dynamics in the vertical plane without the motor dynamics
8
10
t1, s t2, s Ia , A T, s 0.8 0.8 3.5 8 0.7 0.9 2.29 8.5 0.6 1 4.47 9 0.65 1.1 5.71 9.5 0.5 1.2 9.71 14 0.4 1.5 11.73 15.5 t1 – the time of passage through resonance; t2 – transient time.
t,s
The control algorithm requires specifying the desired value of the total energy H* based on the parameter values of the vibration unit. It does not depend on the steady-state velocity Em kF () that in turn depends only on the value of the controlling torque Mi = M0i. To evaluate H* it is suggested to specify some frequency * such that * > r where r = 28 s1 is resonance frequency depending on the parameters of the vibration unit.
Fig. 7. Simulation results for systems (10), (12) with M0 = 0.4 Nm. Simulation results for the model (10), (11) taking into account the drive dynamics are shown in Fig.8 and in Table 1.
d 1 /dt(t), 1 (t) 60 d 1 /dt(t) 40 1 (t) 20 0 -20 0 5 10 15 Ia1 (t) 8 6 4 2 0 0 5 10 15 ET C1 (t) 80 60 40 20 0 0 5 10 15
6. CONCLUSION Usage of the proposed control algorithm allows system designer to extend the area of reachable working frequencies in the near resonance zone. It is demonstrated for the unit under consideration by simulation that without control (for constant torque) the area of reachable working frequencies is determined by the relation Mi > 0.82 Nm. On the other hand using the proposed control algorithm one can reach working frequencies satisfying Mi > 0.4 Nm. Therefore the lower bound of the range of reachable frequencies for controlled system is 40 s1 that is 1.5 times greater than resonance frequency r = 28 s1, while for nonfeedback system with constant torque the lower bound of the range of reachable frequencies for controlled system is 82 s1, i. e. 3 times greater than resonance frequency. An interesting fact is that for vibration unit model taking into account the electric drive dynamics the time of passage through resonance zone sometimes may be less than for the simplified model neglecting the drive dynamics.
t,s 20
Ia1 , A
d 1 , , s -1
317
ET C1 , V
t,s 20
t,s 20
The obtained results are important for implementation of the mechatronic vibration units.
Fig. 8. Simulation results for systems (10), (11), (12) taking into account drive dynamics, with M0 = 0.5 Nm. In this case the value of the steady state rotor speed is measured with the average steady-state armature current I a 321
practical
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averaging. Proc. Int. Conf. "Control of Oscillations and Chaos", IEEE, v.1, pp.138-141. St. Petersburg. Viderman, Z, Porat, I. (1987). An Optimal-Control Method For Passage of a Flexible Rotor Through Resonances. Journal of Dynamic Systems Measurement And Control Transactions of the ASME, V. 109, Is. 3, pp.216-223. Wauer, J., Suherman, S. (1998). Vibration suppression of rotating shafts passing through resonances by switching shaft stiffness. Journal of Vibration And AcousticsTransactions of the ASME, V. 120, Is. 1, pp. 170-180.
REFERENCES Andrievskii, B.R., Guzenko, P.Yu., Fradkov, A.L. (1996). Control of nonlinear vibrations of mechanical systems via the speed gradient method. Autom. Rem. Control, V.57, No 4, pp. 456-467. Balthazar, J.M., Cheshankov, B.I., Ruschev, D.T., Barbanti, L., Weber, H.I. (2001). Remarks on passage through resonance of a vibrating system with two degrees of freedom, excited by a non-ideal energy source, Journal of Sound and Vibration, 239 (5), pp. 1075-1085. Blekhman, I.I., Bortsov, Yu.A., Burmistrov, A.A., Fradkov A.L., Gavrilov, S.V., Kononov, O.A., Lavrov, B.P., Shestakov, V.M., Sokolov, P.V., Tomchina, O.P. (1999). Computer-controlled vibrational set-up for education and research. In: Proc. of 14th IFAC World Congress, Vol. M, pp. 193-197. Blekhman, I.I. (2000). Vibrational mechanics. World Scientific, Singapore. Blekhman, I.I., Indeitsev, D.A., Fradkov, A.L. (2008). Slow Motions in Systems with Inertial Excitation of Vibrations. Journal of Machinery Manufacture and Reliability, No 1, pp. 21-27. Cvetićanin, L. (2010). Dynamics of the non-ideal mechanical systems. A review Journal of the Serbian Society for Computational Mechanics, V. 4, No. 2, pp. 75-86. Fradkov, A.L. (1999). Exploring nonlinearity by feedback. Physica D, 128, pp. 159-168. Fradkov, A.L, Tomchina, O.P., Tomchin, D.A. (2011). Controlled passage through resonance in mechanical systems. Journal of Sound and Vibration, V. 330, Is. 6, pp. 1065-1073. Fradkov, A.L., Tomchin, D.A., Tomchina, O. P. (2014). Controlled Passage Through Resonance for Two-Rotor Vibration Unit. Mechanics and Model-Based Control of Advanced Engineering Systems, pp. 95-102. SpringerVerlag, Wien. Gortinskii, V.V., Savin, A.D., Demskii, A.B., Boriskin, M.A., Alabin, E.A. (1969). A technique for reducing resonance amplitudes during start of vibration machines. Patent of USSR No 255760, 28.10.1969, Bull. No 33 (in Russian). Kinsey, R.J., Mingori, D.L., Rand, R.H. (1992). Nonlinear controller to reduce resonance effects during despin of a dual-spin spacecraft through precession phase lock. Proc. 31th IEEE Conf.Dec.Contr, pp. 3025-3030. Kononenko, V.O. (1964). Oscillating Systems with Bounded Excitation. Nauka, Moscow (In Russian). Leonov, G.A. (2008). The passage through resonance of synchronous electric motors mounted on an elastic base. Journal of Applied Mathematics and Mechanics, V. 72, Is. 6, pp. 631-637. Malinin, L.N., Pervozvansky, A.A. (1983). Optimization of passage an unbalanced rotor through critical speed. Mashinovedenie, No. 4, pp. 36-41 (in Russian). Quinn, D., Rand, R., Bridge, J. (1995). The Dynamics of Resonant Capture. Nonlinear Dynamics, V. 8, Is.: 1, pp. 1-20. Tomchina, O.P. (1997). Passing through resonances in vibratory actuators by speed-gradient control and 322
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IFAC-PapersOnLine 48-11 (2015) 313–318
Controlled Passage through Resonance for Controlled Passage through Resonance for Controlled Passage through Resonance for 11 Controlled Passage through Resonance for Controlled Passage through Resonance for Two-Rotor Vibration Unit: Influence of Drive Dynamics 1 Two-Rotor Vibration Unit: Influence of Drive Dynamics Controlled Passage through Resonance for Two-Rotor Two-Rotor Vibration Vibration Unit: Unit: Influence Influence of of Drive Drive Dynamics Dynamics111 Two-Rotor Vibration Unit: Influence of Drive Dynamics Two-Rotor Vibration Unit: Influence ofOlga Drive Dynamics Dmitry P. Dmitry V. V. Gorlatov*. Gorlatov*. Dmitry Dmitry A.Tomchin**. A.Tomchin**. Olga P. Tomchina*** Tomchina***
Dmitry Olga Dmitry V. V. Gorlatov*. Gorlatov*. Dmitry Dmitry A.Tomchin**. A.Tomchin**. Olga P. P. Tomchina*** Tomchina*** Dmitry V. Gorlatov*. Dmitry A.Tomchin**. Olga P. Tomchina*** Dmitry V. Gorlatov*. Dmitry A.Tomchin**. Olga P. Tomchina*** * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), Russian Federation, St. Petersburg, (e-mail: [email protected]) * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), * Saint Russian Petersburg State University of Architecture and Civil Engineering (SPSUACE), Federation, St. Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) * Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), ** Institute of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), Russian Federation, St. Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) ** Institute of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), ** of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), Federation, St. Petersburg, (e-mail: [email protected]) Federation, St. Petersburg, (e-mail: [email protected]) ** Institute Institute Russian of Russian Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), ** Institute of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), Russian Federation, St. Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) ** *** Institute of Problems in Mechanical Engineering Russian Academy of Sciences (IPME RAS), *** Saint Saint Petersburg Petersburg State University University of Architecture Architecture and Civil Engineering Engineering (SPSUACE), (SPSUACE), Russian Federation, Federation, St. Petersburg, Petersburg, (e-mail: [email protected]) Russian St. (e-mail: [email protected]) State of and Civil *** Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), Russian Federation, St. Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) *** Saint Saint Petersburg Petersburg State University University of Architecture Architecture [email protected]) Civil Engineering Engineering (SPSUACE), (SPSUACE), *** State of and Civil Russian Federation, St. Petersburg, (e-mail: Russian Federation, St. Petersburg, (e-mail: *** Saint Petersburg State University of Architecture [email protected]) Civil Engineering (SPSUACE), Russian St. (e-mail: [email protected]) Russian Federation, Federation, St. Petersburg, Petersburg, (e-mail: [email protected]) Russian Federation, St. Petersburg, (e-mail: [email protected]) Abstract: The problem of controlled passage through resonance zone for mechanical systems with Abstract: The problem of controlled passage through resonance zone for mechanical systems with Abstract: The problem of controlled passage through resonance zone for mechanical systems with several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating Abstract: The problem of controlled passage through resonance zone for mechanical systems with Abstract: The problem of controlled passage through resonance zone for mechanical systems with several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating Abstract: The problem of controlled passage through resonance zone for mechanical systems with efficiency of the control algorithm based on speed-gradient method are presented. The novelty of the several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating several degrees of freedom is analyzed. The simulation results for two-rotor vibration unit illustrating efficiency of the control algorithm based on speed-gradient method are presented. The novelty of the efficiency ofevaluation theofcontrol control algorithm basedsystem on simulation speed-gradient method are presented. The novelty of the several degrees freedom is analyzed. The results for two-rotor vibration unit illustrating results is in of the closed loop performance when the electric drive dynamics are taken efficiency of the algorithm based on speed-gradient method are presented. The novelty of the efficiency of the control algorithm based on speed-gradient method are presented. The novelty of the results is in evaluation of the closed loop system performance when the electric drive dynamics are taken results is in evaluation of the closed loop system performance when the electric drive dynamics are taken efficiency the control algorithm based onvibration speed-gradient method presented. novelty the into account. An interesting fact is that for unit model into account the electric drive results is of closed loop system performance when the electric drive dynamics are taken results is in inofevaluation evaluation of the the closed system performance whentaking theare electric drive The dynamics areof taken into An interesting fact is that for vibration unit model taking into account the electric drive into account. account. An interesting fact is loop thatresonance for vibration unit model taking intoless account the the electric drive results is in evaluation of the closed loop system performance when the electric drive dynamics are taken dynamics the time of passage through zone sometimes may be than for simplified into account. An interesting fact is that for vibration unit model taking into account the electric drive into account. interesting fact is thatresonance for vibration model taking intoless account the the electric drive dynamics the time of passage through zone sometimes may than simplified dynamics the An timethe ofdrive passage through resonance zoneunit sometimes may be be less than for for the simplified into account. An interesting fact is thatresonance for vibration unit model taking intoless account the the electric drive model neglecting dynamics. dynamics the time of passage through resonance zone sometimes may be less than for the simplified dynamics the time of passage through zone sometimes may be than for simplified model neglecting the drive dynamics. model neglecting the drive dynamics. dynamics the time of passage through resonance zone sometimes may be less than for the simplified model neglecting neglecting the the drive drive dynamics. dynamics. model Keywords: vibration unit, dynamics. resonant frequencies, frequencies, drive drive dynamics, dynamics, resonance resonance zone, zone, control control algorithms, algorithms, model the drive Keywords: vibration unit, resonant © 2015,neglecting IFAC (International Federationfrequencies, of Automaticdrive Control) Hosting resonance by Elsevier zone, Ltd. Allcontrol rights reserved. Keywords: vibration unit, resonant dynamics, algorithms, unbalanced rotor s. Keywords: vibration unit, resonant resonant frequencies, drive drive dynamics, dynamics, resonance zone, zone, control control algorithms, algorithms, Keywords: vibration unit, frequencies, resonance unbalanced rotor s. unbalanced rotor s. vibration unit, resonant frequencies, drive dynamics, resonance zone, control algorithms, Keywords: unbalanced s. unbalanced rotor rotor s. unbalanced rotors. evaluated, and the necessary number of dampers and their evaluated, and the necessary number of dampers and their 1. INTRODUCTION evaluated, and the necessary number of dampers and their 1. INTRODUCTION optimal location were determined in (Viderman and Porat, evaluated, and the necessary number of dampers and their 1. INTRODUCTION evaluated, and the necessary number of dampers and their optimal location were determined in (Viderman and Porat, 1. INTRODUCTION optimal location were determined in (Viderman and Porat, 1. INTRODUCTION evaluated, and the necessary number of dampers and their 1987). In (Kinsey and Mingori, 1992) a nonlinear controller optimal location were determined in (Viderman and Porat, An important problem for systems based on vibration optimal location were determined in (Viderman and Porat, 1987). In (Kinsey and Mingori, 1992) a nonlinear controller 1. INTRODUCTION An important problem for systems based on vibration 1987). (Kinsey and Mingori, 1992) aa nonlinear controller An important problem for systems based on vibration optimal were determined indespin (Viderman Porat, reducing resonance during of aa and dual-spin 1987). In Inlocation (Kinsey andeffects Mingori, 1992) nonlinear controller technologies is the passing of vibroactuators through An important problem for systems based on vibration 1987). In (Kinsey and Mingori, 1992) a nonlinear controller reducing resonance effects during despin of dual-spin An important problem for systems based on vibration technologies is the passing of vibroactuators through reducing resonance effects during despin of aasuppression dual-spin technologies is the passing of vibroactuators through 1987). In (Kinsey and Mingori, 1992) a nonlinear controller spacecraft was designed. A method of vibration reducing resonance effects during despin of dual-spin An important for systems based on vibration resonance for systems in post-resonance modes. technologies is the passing of vibroactuators through reducing resonance effects during of despin of asuppression dual-spin spacecraft was designed. A method vibration technologies isproblem the operating passing ofthe vibroactuators through resonance for systems operating in the post-resonance modes. spacecraft was designed. A method of vibration resonance for systems operating in the post-resonance modes. reducing resonance effects during resonances of by asuppression dual-spin for rotating shafts passing switching spacecraft was designed. Athrough method ofdespin vibration suppression technologies is the passing of vibroactuators through Such a problem needs special attention in the case when the resonance for systems operating in the post-resonance modes. spacecraft was designed. A method of vibration suppression for rotating shafts passing through resonances by switching resonance for systems operating in the post-resonance modes. Such aa problem needs special attention in the case when the for rotating shafts passing through resonances by switching Such problem needs special attention in the case when the spacecraft was designed. A method of vibration suppression shaft stiffness was proposed in (Wauer and Suherman, 1998). for rotating shafts passing through resonances by switching resonance for systems operating in the post-resonance modes. power of aa motor not sufficient passage through Such aa problem problem needsis special attention for in the the case when when the for rotating shafts passing through resonances by switching shaft stiffness was proposed in (Wauer and Suherman, 1998). Such needs special attention in case the power of motor is not sufficient for passage through shaft stiffness was proposed in (Wauer and Suherman, 1998). power of a motor is not sufficient for passage through for rotating shafts passing through resonances by switching In (Balthazar et al., 2001) the dynamics of passage through shaft stiffness was proposed in (Wauer and Suherman, 1998). Such a problem needs special attention in the case when the resonance zone due to tois Sommerfeld effect (Blekhman, 2000). shaft power of of zone a motor motor is Sommerfeld not sufficient sufficient for(Blekhman, passage through through stiffness was proposed in (Wauer and Suherman, 1998). In (Balthazar et al., 2001) the dynamics of passage through power a not for passage resonance due effect 2000). In (Balthazar et al., 2001) the dynamics of passage through resonance zone due to Sommerfeld effect (Blekhman, 2000). shaft stiffness was proposed in (Wauer and Suherman, 1998). resonance of a vibrating system with two degrees of freedom In (Balthazar et al., 2001) the dynamics of passage through power of a motor is not sufficient for passage through Dynamics of near resonance behavior are nonlinear and very resonance zone zone dueresonance to Sommerfeld Sommerfeld effect (Blekhman, 2000). In (Balthazar al., 2001) the dynamics of passage through resonance of aetvibrating system with two degrees of freedom resonance due to effect (Blekhman, 2000). Dynamics of near behavior are nonlinear and very resonance of system with two degrees of freedom Dynamics of behavior are nonlinear and very In (Balthazar al., 2001) theearly dynamics passage through was examined. However the did have resonance of aaaetvibrating vibrating system withalgorithms two of degrees ofnot freedom resonance dueresonance to Sommerfeld effect (Blekhman, 2000). complicated. Publications related to this problem can be Dynamics zone of near near resonance behavior are nonlinear and very resonance of vibrating system with two degrees of freedom was examined. However the early algorithms did not have Dynamics of near resonance behavior are nonlinear and very complicated. Publications related to this problem can be was examined. However the early algorithms did not have complicated. Publications related to this problem can be resonance of a vibrating system with two degrees of freedom enough robustness with respect respect to algorithms uncertainties and were was examined. examined. However the early earlyto algorithms did and not were have Dynamics of near resonance behavior are nonlinear and very found in the literature during about 50 years (Kononenko, complicated. Publications related to this problem can be was However the did not have enough robustness with uncertainties complicated. Publications related to this problem can be found in the literature during about 50 years (Kononenko, enough robustness with respect to uncertainties and were found in the literature during about 50 years (Kononenko, was examined. However the early algorithms did not have hard to design. enough robustness with with respect respect to to uncertainties uncertainties and and were were complicated. Publications related to this problem can be 1964; Quinn et al., 1995; Cvetićanin, 2010). found in the literature during about 50 years (Kononenko, enough robustness hard to design. found in the et literature about 2010). 50 years (Kononenko, hard 1964; Quinn al., 1995;during Cvetićanin, 1964; al., Cvetićanin, robustness with respect to uncertainties and were hard to to design. design. found in the et about 2010). 50 years (Kononenko, enough 1964; Quinn Quinn etliterature al., 1995; 1995;during Cvetićanin, 2010). hard to design. 1964; Quinn et al., 1995; Cvetićanin, 2010). For practical implementation of control system it is important Perhaps the first approach to the problem of controlled hard to design. For practical implementation of control system it is important 1964; Quinn etfirst al., 1995; Cvetićanin, 2010). Perhaps the approach to the problem of controlled For practical implementation of control system it is important Perhaps the first approach to the problem of controlled to develop reasonably simple passing through resonance zone For practical implementation of control system it is important passage through resonance zone was the so-called "double Perhaps the first approach to the problem of controlled For practical implementation of control system it is important to develop reasonably simple passing through resonance zone Perhaps the first approach to the problem of controlled passage through resonance zone was the so-called "double to develop reasonably simple passing through resonance zone passage through resonance zone was the so-called "double For practical implementation of control system it is important control algorithms, which have such robustness property: to develop reasonably simple passing through resonance zone Perhaps the first approach to the problem of controlled start" method (Gortinskii et al., 1969). This and other passage through resonance zone was the so-called "double to develop reasonably simple passing through resonance zone control algorithms, which have such robustness property: passage through resonance zone was the so-called "double start" method (Gortinskii et al., 1969). This and other control algorithms, which have such robustness property: start" method (Gortinskii et al., 1969). This and other to develop reasonably simple passing through resonance zone keeping high quality of the controlled system (vibration unit) control algorithms, which have such robustness property: passage through resonance zone was the so-called "double nonfeedback methods are characterized characterized byThis difficulties in control 1969).by This and other other start" method methodmethods (Gortinskii et al., al., 1969). algorithms, have suchsystem robustness property: keeping high quality which of the controlled (vibration unit) start" (Gortinskii et and nonfeedback are difficulties in keeping high of the system (vibration unit) nonfeedback methods are characterized by difficulties in control algorithms, which have such robustness property: under of and external conditions. keepingvariation high quality quality ofparameters the controlled controlled system (vibration unit) start" method (Gortinskii et al., This and other calculation of switching instants of aa1969). motor and sensitivity to nonfeedback methods are characterized by difficulties in keeping high quality of the controlled system (vibration unit) under variation of parameters and external conditions. nonfeedback methods are characterized by difficulties in calculation of switching instants of motor and sensitivity to under variation of parameters and external conditions. calculation of switching instants of a motor and sensitivity to keeping high quality of the controlled system (vibration unit) Perhaps the first such a simple controller was proposed in under variation of parameters and external conditions. nonfeedback methods are characterized by difficulties in inaccuracies of model and to interferences. In (Leonov, 2008) calculation of switching instants of a motor and sensitivity to variation of parameters and external conditions. Perhaps the first such aa simple controller was proposed in calculation ofof switching instants of a motorIn and sensitivity to under inaccuracies model and to interferences. (Leonov, 2008) Perhaps the controller was in inaccuracies of model and to interferences. In (Leonov, 2008) under of based parameters andspeed-gradient external conditions. (Tomchina, 1997)such based on the the speed-gradient method Perhapsvariation the first first such a simple simple controller was proposed proposed in calculation of switching instants of a motor and sensitivity to it is proved that the Sommerfeld effect does not occur for any inaccuracies of model and to interferences. In (Leonov, 2008) Perhaps the first such a simple controller was proposed in (Tomchina, 1997) on method inaccuracies of model and to interferences. In (Leonov, 2008) it is proved that the Sommerfeld effect does not occur for any (Tomchina, 1997) based on the speed-gradient method it is proved that the Sommerfeld effect does not occur for any Perhaps the first such a simple controller was proposed in previously used for control of nonlinear oscillatory systems (Tomchina,used 1997) based on on the speed-gradient speed-gradient method inaccuracies of model and to interferences. In (Leonov, 2008) passage through resonance for a synchronous electric motor it is proved that the Sommerfeld effect does not occur for any (Tomchina, 1997) based the method previously for control of nonlinear oscillatory systems it is proved that the Sommerfeld does notelectric occur for any previously passage through resonance for aaeffect synchronous motor for control of nonlinear oscillatory systems passage through resonance for synchronous electric motor (Tomchina, 1997) based on the speed-gradient method et al., 1996). number of speed-gradient previously used used for control ofA nonlinear oscillatory systems it is proved that the Sommerfeld does notelastic occurbase. for any (Andrievskii with an asynchronous start-up, on an passage through resonance formounted aeffect synchronous electric motor previously used for control of nonlinear oscillatory systems (Andrievskii et al., 1996). A number of speed-gradient passage through resonance for a synchronous electric motor with an asynchronous start-up, mounted on an elastic base. (Andrievskii et al., 1996). A number of speed-gradient with an asynchronous start-up, mounted on an elastic base. previously used for control of nonlinear oscillatory systems algorithms for passage through resonance in 2-DOF (Andrievskii et al., 1996). A number of speed-gradient passage through resonance for mounted a synchronous electric motor (Andrievskii with an an asynchronous asynchronous start-up, mounted on an an elastic elastic base. etpassage al., 1996). Aresonance number in of 2-DOF speed-gradient algorithms for through systems with start-up, on base. algorithms for through in 2-DOF systems A prospective approach to the problem is on feedback (Andrievskii al., 1996). Aresonance number ofThe speed-gradient were proposed in (Fradkov et al., 2011). approach of algorithms foretpassage passage through resonance in 2-DOF systems with an asynchronous start-up, mounted onbased an elastic base. A prospective approach to the problem is based on feedback algorithms for passage through resonance in 2-DOF systems were proposed in (Fradkov et al., 2011). The approach of A prospective approach to the problem is based on feedback were proposed in (Fradkov et al., 2011). The approach of control. Feedback control algorithms for passing through algorithms for passage through resonance in 2-DOF systems A prospective approach to the problem is based on feedback (Tomchina, 1997; Fradkov et al., 2011) was applied to twowere proposed in (Fradkov et al., 2011). The approach of A prospective approach to the problem is based on feedback control. Feedback control algorithms for passing through were proposed in (Fradkov et al., 2011). The approach of (Tomchina, 1997; Fradkov et al., 2011) was applied to twocontrol. Feedback control algorithms for passing through (Tomchina, 1997; Fradkov et al., 2011) was applied to twoA prospective approach to the problem is based on feedback resonance zone of ofcontrol mechanical systems were studied in were proposed in (Fradkov et al., 2011). The approach of control. Feedback Feedback control algorithms for were passing through rotor vibration set-up in (Fradkov et al., 2014). (Tomchina, 1997; Fradkov et al., 2011) was applied to twocontrol. algorithms for passing through resonance zone mechanical systems studied in (Tomchina, 1997; Fradkov et al., et 2011) was applied to tworotor vibration set-up in (Fradkov al., 2014). resonance zone of mechanical systems studied in rotor set-up in al., control. algorithms for were passing through (Malinin and 1983; Viderman and Porat, 1997; Fradkov et al., et was applied to tworesonanceFeedback zonePervozvansky, ofcontrol mechanical systems were studied in (Tomchina, rotor vibration vibration set-up in (Fradkov (Fradkov et2011) al., 2014). 2014). resonance zone of mechanical systems were studied in (Malinin and Pervozvansky, 1983; Viderman and Porat, rotor vibration set-up in (Fradkov et al., 2014). (Malinin and Pervozvansky, 1983; Viderman and Porat, In this paper the system designed in (Fradkov et al., is resonance zone of mechanical systems were studied in 1987; Kinsey and Mingori, 1992). In (Malinin and rotor vibration set-up in (Fradkov et al., 2014). (Malinin and Pervozvansky, 1983; Viderman and Porat, In this paper the system designed in (Fradkov et al., 2014) 2014) is (Malinin and Pervozvansky, 1983; Viderman and Porat, 1987; Kinsey and Mingori, 1992). In (Malinin and In the designed (Fradkov et 2014) 1987; Kinsey and Mingori, 1992). In (Malinin and analyzed taking into account account thein electric drive dynamics In this this paper paper the system system designed in electric (Fradkovdrive et al., al.,dynamics 2014) is is (Malinin and Pervozvansky, 1983; Viderman and Porat, Pervozvansky, 1983) an optimal control algorithm for 1987; Kinsey and Mingori, 1992). In (Malinin and In this paper the system designed in (Fradkov et al., 2014) is analyzed taking into the 1987; Kinsey and Mingori, 1992). In (Malinin and Pervozvansky, 1983) an optimal control algorithm for analyzed taking into account the electric drive Pervozvansky, 1983) an optimal control algorithm for In this paper the system designed (Fradkov et al.,dynamics 2014) is particularly the DC EMF. (Leonov, 2008), analyzed taking into motor account thein Unlike electric drive dynamics 1987; Kinsey and Mingori, 1992). In (Malinin and passage of an unbalanced rotor through critical speed was Pervozvansky, 1983) an optimal control algorithm for analyzed taking into account the electric drive dynamics particularly the DC motor EMF. Unlike (Leonov, 2008), Pervozvansky, 1983) an rotor optimal control algorithm for particularly passage of an unbalanced through critical speed was the DC motor EMF. Unlike 2008), passage an unbalanced rotor through critical speed taking intoabsence account theSommerfeld electric (Leonov, drive dynamics which proved of effect for aa particularly the the DC motor EMF. Unlike (Leonov, 2008), Pervozvansky, 1983) an optimal control algorithm for analyzed proposed. same problem several control methods are passage of ofFor an the unbalanced rotor through critical speed was was particularly the DC motor EMF. Unlike (Leonov, 2008), which proved the absence of Sommerfeld effect for passage of an unbalanced rotor through critical speed was proposed. For the same problem several control methods are which proved the absence of Sommerfeld effect for a proposed. For the same problem several control methods are particularly the DC motor EMF. Unlike (Leonov, 2008), which proved the absence of Sommerfeld effect for passage an the unbalanced rotor several throughcontrol criticalmethods speed was proposed.ofFor For the same problem problem several control methods are which proved the absence of Sommerfeld effect for aa proposed. same are proposed. For the same problem several control methods are which proved the absence of Sommerfeld effect for a
1 1 The work was supported by Russian Scientific Foundation (project 14-29-00142) 1 The work was supported by Russian Scientific Foundation (project 14-29-00142) The work was supported by Russian Scientific Foundation (project 14-29-00142) 1 1 The work was supported by Russian Scientific Foundation (project 14-29-00142) The work©was supported by Russian Scientific (project 14-29-00142) 2405-8963 2015, IFAC (International FederationFoundation of Automatic Control) Hosting by 1
Thereview work©was supported by Russian ScientificFederation Foundationof(project 14-29-00142) Copyright IFAC 2015 317 Peer responsibility of International Automatic Control. Copyright ©under IFAC 2015 317 Copyright © IFAC 2015 317 10.1016/j.ifacol.2015.09.204 Copyright © © IFAC IFAC 2015 2015 317 Copyright 317 Copyright © IFAC 2015 317
Elsevier Ltd. All rights reserved.
MICNON 2015 314 Dmitry V. Gorlatov et al. / IFAC-PapersOnLine 48-11 (2015) 313–318 June 24-26, 2015. Saint Petersburg, Russia
where H1 is the energy of rotating subsystem, H2 is the energy of a carrier subsystem, H12 is the energy of interaction.
synchronous motor, this effect is topical for a DC motor. In Section 2 the problem statement for control of passage through resonance zone for mechanical systems with several degrees of freedom adopted from (Fradkov et al., 2011) is presented. The control algorithm based on the speed-gradient method for two-rotor vibration units is exposed in Section 3. The simulation results illustrating efficiency and robustness of the proposed algorithm in presence of drive dynamics are described in Section 4.
The idea of control algorithms described below, is in that slow motion (t) is being isolated and then "swinging" starts to obtain rise of energy of a rotating subsystem. To isolate slow motions, low-pass filter is being inserted into energy control algorithms. Let slow component appears in oscillations of angular velocity of a rotor. Then we start with the control algorithm proposed in (Tomchina, 1997) is:
2. PROBLEM STATEMENT AND IDEA OF SOLUTION To describe the dynamics of a mechanical system with n degrees of freedom the standard Euler-Lagrange equations are used:
), T , (8) u sign (( H H )
where is the variable of the filter, performing slow motions, T is the time constant of the filter. At low damping, slow motions also fade out slowly, that gives control algorithm an opportunity to create suitable conditions to pass through resonance zone. Thus the effect of "feedback resonance" (Fradkov, 1999) is created. After passing the resonance zone it is suggested to turn off the "swinging" and then to switch the algorithm to controlling with constant torque. For a proper work of a filter, it should suppress fast oscillations with frequency and pass slow oscillations with B frequency, where B is the Blekhman frequency (Blekhman et al., 2008). I.e. time constant of a filter T should be chosen from the inequality
A(q )q C (q, q ) G (q) Bu, (1)
where u=u(t) is n-dimensional input vector; q=q(t) is n-vector of generalized coordinates, A(q) is nn inertia matrix; C (q, q ) is the n-vector of Coriolis and centrifugal forces; G(q) is the n-vector of gravity forces; B is the mn control matrix. However, for the purpose of control algorithm design it is often convenient to use equations in Hamiltonian form: H p q
T
H Bu, q p
T
, (2)
T 2 B . (9)
where p=p(t) is the generalized momenta vector, H=H(p, q) denotes the Hamiltonian function total energy of the free (uncontrolled) system: H ( p, q )
Algorithms of passing through resonance zone for the tworotor vibration units are described in (Fradkov et al., 2014). Below we analyze the closed loop system with the algorithm of (Fradkov et al., 2014) taking into account the electric drive dynamics.
1 T 1 p A(q ) p Π (q ), (3) 2
where (q) is the potential energy. First of all, let us define the auxiliary control goal as the approaching of energy of the free system to a surface of the given energy level H ( p (t ), q (t )) H
3. MODEL OF TWO-ROTOR VIBRATION UNIT TAKING INTO ACCOUNT THE DRIVE DYNAMICS Consider the problem of two-rotor vibration unit start-up (spin-up) (Blekhman et al., 1999). The unit SV-2 consists of two rotors installed on the supporting body (Fig. 1).
when t . (4)
Introducing the objective function Q ( p, q )
1 H ( p, q) H 2 , (5) 2
the goal (4) can be reformulated as: Q(q(t ), p (t )) 0 when t . (6)
Control algorithm for passing through resonant frequencies for an unbalanced rotor is based on the speed-gradient method (Balthazar et al., 2001), which allows to synthesize control algorithms for significantly nonlinear plants. The model of the system is split into two subsystems: carrier and rotating bodies. Then the total energy can be represented in the following form:
H ( p (t ), q (t )) H 1 ( p(t ), q(t )) H 2 ( p(t ), q(t )) H 12 ( p(t ), q(t )),
Fig. 1. Schematics of two-rotor vibration unit: 1 motors; 2 – motor supports; 3 – frame of the unit; 4 – unbalanced rotors; 5 – vibrating platform; 6 – carrying springs; 7 – rotor bearings; 8 – cardan shafts.
(7)
318
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motor; kr is the friction coefficient in the bearings; M r1 k r 1 is resistance torque of a rotor, caused by resilient friction, Mu1 is the unbalanced rotor’s own torque, M u1 (t ) mg cos( 1 ); Mm1 is the control torque arriving ~ at the input of the "unbalanced rotor"; M si is the moment, caused by the influence of the supporting body:
The kinematic scheme of vibration unit is presented in Fig. 2.
y M
φ1 a
yc
M
xc
315
~ M si xc m sin( i ) yc m cos( i )
φ2
( J i (1) i rm cos i ) (1) i 2 rm sin i ,
a
where Ji, i = 1, 2 are the inertia moments of the rotors.
c02; β
c01
Since the laboratory setup SV-2 used the DC motors, the electric drive structure is selected as the traditional singlecircuit system with current loop and proportional-integral (PI) current controller WCR(р) = b(τр+1) / τр is configured to optimum modulo; b, are dynamic gain and time constant of the regulator.
c02; β
c01
Fig. 2. Two-rotor vibration unit with DC motors. , 1, 2 are angle of the supporting body and rotation angles of the rotors, respectively, measured from the horizontal position; xc, yc are the horizontal and vertical displacement of the supporting body from the equilibrium position; mi = m, i = 1, 2 and mn are the masses of the rotors and supporting body; ρi = ρ, i = 1, 2 are the rotor eccentricities; c01, c02 are the horizontal and vertical spring stiffness; β is the damping coefficient. It is assumed that rotor shafts are orthogonal to the motion of the support. φψ
To convert the scheme into the state space equations, assume that the whole system dynamics may be considered in the vertical plane. Then the equations of dynamics have the following form (Fradkov et al., 2014): m(sin( 1 ) sin( 2 )) m0 xc 1 m sin( 1 ) 2 m sin( 2 ) 2 m(cos( 1 ) cos( 2 )) 12 m cos( 1 ) 22 m cos( 2 )
Unbalanced rotor structural diagram taking into account the drive dynamics is shown in Fig. 3. M1
2 1 m cos( 1 ) 2 2 m cos( 2 ) 2c 01 x c x c 0;
PI-CR TC DC-Motor ETC U1 1 Ra p 1 UCR kTC b kU TTC p 1 Ta p 1 p CS Ia UCS kCS Electric Drive TCS p 1 Em kF 1 1 p
1 1 Jp
mgρ cos(·)
m(cos( 1 ) cos( 2 )) m0 yc 1 m cos( 1 ) 2 m cos( 2 ) 2 m(sin( 1 ) sin( 2 )) 12 m sin( 1 ) 22 m sin( 2 )
km kr
Mr1
Mu1
2 1 m sin( 1 ) 2 2 m sin( 2 ) m0 g 2c 02 y c y c 0;
Mm1
(10)
xc m(sin( 1 ) sin( 2 )) yc m(cos( 1 ) cos( 2 ))
Rotor
Supporting Body Ms
J J 1 J 2 2mr (cos 1 cos 2 ) 1 ( J 1 mr cos 1 ) 2 ( J 2 mr cos 2 )
Fig. 3. Vibration unit block diagram with drive dynamics.
12 mr sin 1 22 mr sin 2 2mr 1 sin 1 2mr 2 sin 2 mg (cos( 1 ) cos( 2 ))
Fig. 3 include scheme for first drive only, the second drive scheme is similar. The following notation is used here: CR is the current regulator; TC is the power (thyristor type) converter; CS is the current sensor; Ia is the armature current; ЕTC and Еm are converter and motor EMFs; kTC and kCS are converter and current feedback gains; kF is the motor torque (EMF) coefficient; TTC and TCS are converter and current sensor time constants; Ta is the armature time constant; Ra is the armature curcuit resistance; UCR and UCS are current controller and current sensor output voltages; M1 is the first drive control torque; U1 is the voltage corresponding to the calculated torque, km = kF; 1 is the angular velocity of a
c 03 0; xc m sin( 1 ) yc m cos( 1 ) ( J 1 mr cos 1 ) 1 J 1 2 mr sin 1 mg cos( 1 ) k r 1 M m1 ; xc m sin( 2 ) yc m cos( 2 ) ( J 2 mr cos 2 ) 2 J 2 2 mr sin 2 mg cos( 2 ) k r 2 M m 2 ; 319
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where g is the gravity acceleration; m0 is the total mass of the unit, m0 = 2m + mn. When taking into account the dynamics of the drive in the simulation process the control torques Mmi, i = 1, 2 are formed in accordance with the structural diagram of the "electric drive" (Fig. 3). The "unbalanced rotor" structural diagram prepared in accordance with the fourth equation of system (10).
satisfy the relation (9). The larger values of T lead to decrease in average power of the control signal and to slowing down the algorithm. M1, obtained in accordance with the equations (12). 5. SIMULATION RESULTS Efficiency of the proposed control algorithm was studied in the MATLAB environment. Fig.4 Fig.7 correspond to the model (10) with the drive dynamics neglected. The simulation results for the system (10) are shown in Fig. 4 with basic system parameters: Ji = 0.014 kgm2, m = 1.5 kg, mn = 9 kg, ρ = 0.04 m, kr = 0.01 J/s, = 5 kg/s, c02 = 5300 N/m, c01=1300 N/m and constant torque M0 = 0.82 Nm (internal curves, "capture") and M0 = 0.83 Nm (external curves, passage).
In accordance with the structure in Fig. 3 each motor torque Mmi determined by the value Mi, calculated in the algorithm and obeys a system of differential and algebraic equations (index "i" is omitted for simplicity):
1 1 Ia I a ( ETC E m ) , Ta Ra 1 ETC k TCU CR , E TC TTC
d 1 , s -1
1 U CS U CS k CS I a , (11) TCS
b U 1 U CS ,
0
5
10
t,s 20
15
The graphs of the angular velocity 1 and filter variable 1(t) for the first rotor are shown in Fig.5 and Fig.6. The algorithm (12) is used for control for two values M0, namely M0 = 0.8 Nm (Fig.5, T = 0.8 s), M0 = 0.5 Nm (Fig. 6, T = 0.1.2 s). Both values are less than those corresponding to capture if the controlling torque is constant. Fig.7 demonstrates the graphs of 1 and 2 for operating value of the torque in the algorithm (12) M0 = 0.4 Nm. Smaller values of M0 do not provide passage through resonance. In addition the graphs of the supporting body coordinates xc, yc, are shown in Fig. 5. As seen from the pictures application of the algorithm (12) allows the designer to significantly expand the range of operating frequencies beyond resonance and provides stable amplitude of the platform vibrations after passing the resonance zone.
4. CONTROL ALGORITHM At the low levels of constant control action M i (t ) (1) i M 0 , i = 1, 2 in the near-resonance zone the rotor angle is "captured", while increase of the control torque leads to passage through resonance zone towards the desired angular velocity. Synthesis of the control algorithm Mi = U(z), i = 1, 2 is needed for acceleration of the unbalanced rotors, before the system passes through the resonance zone, where T z x c , x c , y c , y c , , , 1 , 1 , 2 , 2
is the state vector of the system. It is assumed that the level of control action is limited and does not allow system to pass through the resonance zone using constant action.
x(t)
x, m
0
-0.05 0
1,
(12)
5
, rad
where Н = Т+, i(t) are the variables of the filters, T > 0, T = const and H* are the parameters of the algorithm, (t ) max sgn ( H () H ) , where sgn[z] = 1 with z > 0,
t,s 5 10 15 20 d1/dt(t), (t)
60 d /dt(t) 1 40 20 (t) 0 t,s t , s -20 0 5 10 15 20 10 15 20 -1
0
-0.01 0
y(t)
0
t , s -0.1 10 15 20 0 (t)
0.01
i 1,2 ,
0.1 y, m
0.05
In this paper we analyze the following modification of the control algorithm (8) initially proposed by (Fradkov et al., 2014) for passing through resonance zone: 0 & H H * i i 0 ,
M0 = 0.82 Nm
Fig. 4. Simulation results for various constant torques.
E m kF , M m k m I a .
1i M 0 , if i ui 1 M 0 , if 0, else, T i i i ,
50 0
U CS bU 1 U CS U CS 1 ,
M0 = 0.83 Nm
d1, , s
U CS 1
d 1 /dt(t)
100
5
Fig. 5. Simulation results for systems (10), (12) with M0 = 0.5 Nm.
0 t
sgn[z] = 0 with z ≤ 0. Time constant of the filters T should 320
MICNON 2015 June 24-26, 2015. Saint Petersburg, Russia Dmitry V. Gorlatov et al. / IFAC-PapersOnLine 48-11 (2015) 313–318
d 1 , , s -1
100
d 1 /dt(t)
0
(since Mm = kmIa). An appropriate choice of the drive parameter kCS and PI-current controller gains, defining M0 and I a may realize the same range of the drive torques as in the previous case. In Fig. 8 the graphs of 1 , 1(t), the graph of the current Ia1 and the graph of EMF ЕTC1 are shown. As seen from the graphs taking into account the drive dynamics does not change the lower bound of achievable stable operating velocities. Moreover the filter time constants T also do not change their value.
d 1 /dt(t), (t)
50
(t) 0
5
10
15
20
t,s
dφ1 / dt, dφ2 / dt, s
-1
Fig. 6. Simulation results for systems (10), (12) with M0 = 0.8 Nm.
50 40 30 20 10 0 -10 -20 -30 -40 -50
dφi / dt
0
2
4
6
Table 1. Quantitative indicators of the vibration unit dynamics in the vertical plane without the motor dynamics
8
10
t1, s t2, s Ia , A T, s 0.8 0.8 3.5 8 0.7 0.9 2.29 8.5 0.6 1 4.47 9 0.65 1.1 5.71 9.5 0.5 1.2 9.71 14 0.4 1.5 11.73 15.5 t1 – the time of passage through resonance; t2 – transient time.
t,s
The control algorithm requires specifying the desired value of the total energy H* based on the parameter values of the vibration unit. It does not depend on the steady-state velocity Em kF () that in turn depends only on the value of the controlling torque Mi = M0i. To evaluate H* it is suggested to specify some frequency * such that * > r where r = 28 s1 is resonance frequency depending on the parameters of the vibration unit.
Fig. 7. Simulation results for systems (10), (12) with M0 = 0.4 Nm. Simulation results for the model (10), (11) taking into account the drive dynamics are shown in Fig.8 and in Table 1.
d 1 /dt(t), 1 (t) 60 d 1 /dt(t) 40 1 (t) 20 0 -20 0 5 10 15 Ia1 (t) 8 6 4 2 0 0 5 10 15 ET C1 (t) 80 60 40 20 0 0 5 10 15
6. CONCLUSION Usage of the proposed control algorithm allows system designer to extend the area of reachable working frequencies in the near resonance zone. It is demonstrated for the unit under consideration by simulation that without control (for constant torque) the area of reachable working frequencies is determined by the relation Mi > 0.82 Nm. On the other hand using the proposed control algorithm one can reach working frequencies satisfying Mi > 0.4 Nm. Therefore the lower bound of the range of reachable frequencies for controlled system is 40 s1 that is 1.5 times greater than resonance frequency r = 28 s1, while for nonfeedback system with constant torque the lower bound of the range of reachable frequencies for controlled system is 82 s1, i. e. 3 times greater than resonance frequency. An interesting fact is that for vibration unit model taking into account the electric drive dynamics the time of passage through resonance zone sometimes may be less than for the simplified model neglecting the drive dynamics.
t,s 20
Ia1 , A
d 1 , , s -1
317
ET C1 , V
t,s 20
t,s 20
The obtained results are important for implementation of the mechatronic vibration units.
Fig. 8. Simulation results for systems (10), (11), (12) taking into account drive dynamics, with M0 = 0.5 Nm. In this case the value of the steady state rotor speed is measured with the average steady-state armature current I a 321
practical
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averaging. Proc. Int. Conf. "Control of Oscillations and Chaos", IEEE, v.1, pp.138-141. St. Petersburg. Viderman, Z, Porat, I. (1987). An Optimal-Control Method For Passage of a Flexible Rotor Through Resonances. Journal of Dynamic Systems Measurement And Control Transactions of the ASME, V. 109, Is. 3, pp.216-223. Wauer, J., Suherman, S. (1998). Vibration suppression of rotating shafts passing through resonances by switching shaft stiffness. Journal of Vibration And AcousticsTransactions of the ASME, V. 120, Is. 1, pp. 170-180.
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