The subject of research is the longitudinal vibrations of stab knife refiners. A model of the fibrous layer between the rotor and stator is developed. It is shown that it is advisable to use the Maxwell-Thomson model for liquid friction of the rotor and stator, and the Hooke elastic model for boundary friction. The formula for determining the dynamic stiffness of the fibrous layer is obtained. Dynamic and mathematical models of the refiner in the longitudinal direction are developed. The mathematical model is a system of linear differential equations with periodically varying coefficients (Mathieu-Hill equations). A method for calculating the amplitude of oscillations of the refiner rotor and stator is developed. According to this technique, Andritz 54-60-1c stab refiner was designed. Oscillation parameters of the refiner rotor and stator are determined theoretically and experimentally. The stator oscillation amplitude is 1.6-2.3 times smaller than the rotor oscillation one. In case of boundary friction, the amplitude of oscillations of the stator and rotor increases by 2-3 times in comparison with liquid friction. The oscillation amplitude of the rotor and stator in the longitudinal direction is comparable with the gap between these elements. Therefore, it is recommended while designing refiners, to develop methods and means of vibration protection, and during operation, to prevent boundary friction between headsets. The developed calculation procedure can be used in other industries, for example, mining and metallurgy. © Published under licence by IOP Publishing Ltd

Vikharev, S. N.

Zasypkina, S. A.

English

Electronic archive of the Ural State Forest Engineering University

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSStudy of longitudinal vibrations of stab knife refinersTo cite this article: S Vikharev and S A Zasypkina 2019 J. Phys.: Conf. Ser. 1399 044011 View the article online for updates and enhancements.This content was downloaded from IP address 79.110.248.251 on 27/01/2020 at 06:33Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing LtdAPITECH-2019Journal of Physics: Conference Series 1399 (2019) 044011IOP Publishingdoi:10.1088/1742-6596/1399/4/0440111      Study of longitudinal vibrations of stab knife refiners S Vikharev and S A Zasypkina Faculty of technical mechanics and the equipment of pulp and paper manufactures, Ural State Forest Engineering University, 36 Siberian tract, Ekaterinburg, 620100, Russia  E-mail: cbp200558@mail.ru Abstract. The subject of research is the longitudinal vibrations of stab knife refiners. A model of the fibrous layer between the rotor and stator is developed. It is shown that it is advisable to use the Maxwell-Thomson model for liquid friction of the rotor and stator, and the Hooke elastic model for boundary friction. The formula for determining the dynamic stiffness of the fibrous layer is obtained. Dynamic and mathematical models of the refiner in the longitudinal direction are developed. The mathematical model is a system of linear differential equations with periodically varying coefficients (Mathieu-Hill equations). A method for calculating the amplitude of oscillations of the refiner rotor and stator is developed. According to this technique, Andritz 54-60-1c stab refiner was designed. Oscillation parameters of the refiner rotor and stator are determined theoretically and experimentally. The stator oscillation amplitude is 1.6–2.3 times smaller than the rotor oscillation one. In case of boundary friction, the amplitude of oscillations of the stator and rotor increases by 2–3 times in comparison with liquid friction. The oscillation amplitude of the rotor and stator in the longitudinal direction is comparable with the gap between these elements. Therefore, it is recommended while designing refiners, to develop methods and means of vibration protection, and during operation, to prevent boundary friction between headsets. The developed calculation procedure can be used in other industries, for example, mining and metallurgy. 1.  Introduction Knife refiner is the main technological equipment for grinding fibrous materials in the pulp and paper industry. The most unreliable element of these machines is the headset [1]. The gap between the grinding sets of the rotor and stator is a fraction of a millimeter and depends on the mill operating mode, type and concentration of pulp [1-4]. During grinding, axial forces act on the rotor and stator, which consist of constant, periodic and random components [5,6].  The headset can work with fluid or boundary friction. With boundary friction, the technical resource of the headset decreases [7.8].  For an effective grinding process in knife refiners, it is necessary to ensure the stability of the gap between the rotor and stator headsets [9,10]. The purpose of the work is to study the oscillations of the refiner in the longitudinal direction. APITECH-2019Journal of Physics: Conference Series 1399 (2019) 044011IOP Publishingdoi:10.1088/1742-6596/1399/4/0440112      2.  Methods and materials 2.1. Model of the fibrous layer between the rotor and stator A distinction is made between grinding with liquid friction of the rotor and stator, when the rubbing surfaces are completely separated by the fibrous layer, and boundary friction, when the rubbing surfaces are not completely separated by this layer, there is a metal contact [8, 11, 12]. For liquid friction of the rotor and stator, it is advisable to use the Maxwell – Thomson model [1, 3, 13], and for boundary friction, the elastic Hooke model (figure 1). The elastic element С2 with the Еr module determines the instantaneous deformation of the system. A viscoelastic element comprising С1 spring with Еk module and a damper with viscosity b(t) connected in parallel with it characterizes a delayed elastic deformation. С3 spring "turns on" only with boundary friction of the rotor and stator. It should be noted that the model parameters depend on factors influencing the course of grinding [14]. In the general case, the relationship between stress and deformation of the fibrous layer can be described in the form of a differential equation [15]   nllmkkkktqltP010.                                                     (1) It is known that when the knives of the rotor headset pass over the stator knives, the concentration of the fibrous layer changes [1,3]. The modulus of elasticity of a mass of various concentrations remains almost constant, but its viscosity changes to a large extent [2]. Analysing the foregoing in relation to the model of the fibrous layer, it is concluded that when grinding the semi-finished products, the damping coefficient of the model changes. Moreover, the change in damping will occur with the same frequency with which the rotor and stator knives intersect - with headset frequencies 𝜔𝑓 [16]. Damping in the pulp occurs due to the friction of the surfaces of the fibers against each other, resistance to the movement of fibers in water, the passage of water through the pores of the fibers, the presence of adhesion forces between the fibers. The presence of damping forces causes the manifestation of nonlinear properties in the system. This complicates the study of such systems. In practice, various damping approximation methods are used. This approach allows nonlinear systems to be reduced to simpler linear ones, in particular, using the energy balance method [17, 18]. With boundary friction, the contact area of the headset changes, that is, the stiffness of the metal contact of the rotor and stator changes [13,19,20]. If the force сила PA(t) acts in section A of the fiber layer (figure 1), then the same force will act in sections B and C, since the model does not contain inertial elements )()()( tPtPtP СBA  . Let уa(t), ув(t), ус (t) – be А, В and C cross sections, then 𝑃𝐴(𝑡) =  𝐶1(𝑦𝑐 − 𝑦𝑎) +  𝑏(𝑡)(𝑦?̇? − 𝑦?̇?) = 𝑃𝐶(𝑡) = 𝐶2(𝑦𝑏 − 𝑦𝑐) = 𝑃𝐵(𝑡)                    (2)             We believe that harmonic oscillations with 𝜔𝑓 frequency occur in the system. Having substituted ,cossin;cossin2211tBtAytBtAyffвffa and having eliminated ус, we will find solution (1) in the following form APITECH-2019Journal of Physics: Conference Series 1399 (2019) 044011IOP Publishingdoi:10.1088/1742-6596/1399/4/0440113      ,cossin)(;cossin)(tPtPtPtPtPtPfBfBBfAfAA whence we determine the dynamic stiffness C(t) of the fibrous layer       ;)()()()()()(222222221122221122212ffffftbCtbCCCtbCtbCCCtbCCtC                  (3) 2.2. The model of a disk refiner in the longitudinal direction The model of a disk refiner in the longitudinal direction is a rotating shaft with a rotor disk, which interacts with the stator disk through a fibre interlayer, which is modelled by liquid friction - the Maxwell-Thomson model, and with boundary friction - by Hooke's body (figure 2) [21]. We will make the following designations: ω is the rotational speed of the rotor disk; тр is the mass of the rotor disk with the reduced mass of the shaft; тс is the mass of the stator disk. Stator and rotor disks are interconnected by means of elastic С4, С5 and inelastic b1, b2 resistances. The refiner body is taken absolutely rigid, and the parameters of С4, С5, b1 и b2 elements are constant in time.    Figure 1. Model of the fibrous layer between the rotor and stator. Figure 2. Model of the longitudinal refiner. Using the D’Alembert’s principle, we obtain differential equations describing the oscillations of the refiner. With fluid friction ,0))((;0))((1212422221115111yytCyCybymyytCyCybymрc                                        (4) where у1, у2 – are mс, mр mass movements. With boundary friction ,0))((;0))((1232422221315111yytCyCybymyytCyCybymрс                                       (5) where C1(t), C3(t) - stiffness of the fibrous interlayer during fluid and boundary friction.  We will denote: APITECH-2019Journal of Physics: Conference Series 1399 (2019) 044011IOP Publishingdoi:10.1088/1742-6596/1399/4/0440114      25ссmC ;   24 ррmC ;    11 2сmb;    22 2рmb,                                     (6) where λс, λр - partial frequencies of free oscillations of the stator and rotor in the longitudinal direction; 1  2  - partial damping coefficients of the stator and rotor. Putting (6) into (4) and (5) we will get with liquid friction ,0)()(2;0)()(21212222221112111yymtCyyyyymtCyyyррcс                                          (7) with boundary friction .0)()(2;0)()(21232222221312111yymtCyyyyymtCyyyррсс                                         (8) C1(t) and C3 (t) parameters can be arranged in a Fourier series 𝐶1(𝑡) = 𝐶0 + ∑ 𝐶𝑖𝑛𝑖=1 cos (𝑖𝜔𝑓𝑖 + 𝛽𝑖);  𝐶3(𝑡) = С΄0 + ∑  С΄0𝑛𝑖=1 cos (𝑖𝜔𝑓𝑖 + 𝛽΄𝑖) ,                                          (9)                                                               where Со , С΄о - permanent stiffeners; Сi  , С΄i – amplitude of i-th  harmonic component;  βi ,  β΄i - shear angle of phases of i-th  harmonic component. Equations (7, 8) are a system of linear differential equations with C1(t) and С3(t) periodically varying coefficients (Mathieu-Hill equations). To consider the stability of their solutions, we will substitute expressions (9) into (7), and without taking into account the dispersion of vibrational energy, we will obtain 𝑦1̈ + 𝜆𝑐2(1 + 𝛼1𝑐𝑜𝑠𝜔𝑓𝑖𝑡)𝑦1 = 0 𝑦2̈ + 𝜆𝑝2 (1 + 𝛼2𝑐𝑜𝑠𝜔𝑓𝑖𝑡)𝑦2 = 0                                               (10) where 21211ссmyy ;    22121ррmyy . We will introduce τ = W0(t) dimensionless time, where dt = dτ / W0 , then ,0)cos(;0)cos(222222111212yWdydyWdyd                                                         (11) where 2111ynW  ;   2221ynW   ;    2111yn  ;     2222yn   .   APITECH-2019Journal of Physics: Conference Series 1399 (2019) 044011IOP Publishingdoi:10.1088/1742-6596/1399/4/0440115      Equations (11) are Mathieu ones, whose stability regions are determined by the Ains-Strett diagram. For α1, α2 << l, according to the diagram, instability will occur in narrow frequency ranges at ny1 = 0.5, 2/3, 1.0 и ny2 = 0.5, 2/3, 1.0. The solution of the equations in the field of stable states is a periodic function in the form of a Fourier series 𝑦1 = ∑ [𝐴1𝑖𝑛𝑖=1 𝑐𝑜𝑠(𝑖𝜔𝑓𝑖𝑡) +  𝐵1𝑖𝑠𝑖𝑛(𝑖𝜔𝑓𝑖𝑡)]  𝑦2 = ∑ [𝐴2𝑖𝑛𝑖=1 𝑐𝑜𝑠(𝑖𝜔𝑓𝑖𝑡) +  𝐵2𝑖𝑠𝑖𝑛(𝑖𝜔𝑓𝑖𝑡)]                                    (12) Substituting (12) into (10), we will obtain a system of algebraic equations from which А1i,2i, B1i,2i, coefficients are found, and from them the amplitudes and angles of phase shift are determined     niniiianiniiiaBASBAS1 1222221 121211,;                                                                   (13) iiiBAtg11 ;    iiiBAtg22.                                                           (14)                                                          For engineering calculations, it suffices to restrict ourselves to the first two numbers of the series i=1,2. 3.  Results and discussion A study of oscillations of 54-60-1c disk refiner of the Andritz company was carried out according to the developed technique experimentally. This refiner grinds waste sorting of thermomechanical pulp at JSC Solikamskbumprom. The study of oscillations was performed using “Maple 15” mathematical software complex. The main research results are presented in figures 3 and 4.                                                 a)                                                                               b) Figure 3. The amplitude of the longitudinal oscillations of the rotor at i=1: а) depending on В1coefficient;  b) depending on А1.  APITECH-2019Journal of Physics: Conference Series 1399 (2019) 044011IOP Publishingdoi:10.1088/1742-6596/1399/4/0440116                                                    а)                                                                                  b)    Figure 4. The amplitude of the longitudinal oscillations of the rotor at i=2: а) depending on В1coefficient;  b) depending on А1 The amplitude of oscillations of the rotor and stator of 54-60-1c refiner of the Andritz company when grinding waste sorting thermomechanical pulp in the longitudinal direction is presented in table 1. Table 1. The amplitude of oscillation of the elements of 54-60-1c refiner of the Andritz company in the longitudinal direction. Refiner element Element friction Amplitude of vibrations, microns Theory Experiment Error, % Rotor Liquid 12 16 25 Boundary 42 48 13 Stator Liquid 8 10 20 Boundary 16 21 24 Analyzing the data obtained, we can conclude that the amplitude of the stator oscillations is 1.6 - 2.3 times less than the amplitude of the rotor. This is due to the greater rigidity of the stator mounting to the refiner body. The rotor has less rigidity due to the presence of bearings. The amplitude of oscillations of the rotor and stator depends on the type of friction between the headsets. In case of boundary friction, the amplitude of oscillations of the stator and rotor increases by 2–3 times in comparison with liquid friction. The error in determining the amplitude of oscillations of the rotor and stator does not exceed 25%. The error is caused by the error in determining the parameters of the elements of the refiner model and the error in the experiment. The oscillation amplitude of the rotor and stator in the longitudinal direction is comparable with the gap between these elements. The gap between the rotor and the stator during operation of the refiner is regulated by the additive mechanism and can be hundredths of a millimeter. Therefore, it is recommended that when designing refiners, the development of methods and means of vibration protection, and during operation, to prevent boundary friction between headsets, i.e. when the amplitude of oscillations of the rotor and stator exceeds the gap between them. 4.  Conclusion    A method for calculating the longitudinal vibrations of the mill has been developed and tested. The stator oscillation amplitude is 1.6–2.3 times smaller than the rotor oscillation amplitude. This is due to the greater rigidity of the stator mounting to the refiner body. APITECH-2019Journal of Physics: Conference Series 1399 (2019) 044011IOP Publishingdoi:10.1088/1742-6596/1399/4/0440117      The amplitude of oscillations of the rotor and stator depends on the type of friction between the headsets. In case of boundary friction, the amplitude of oscillations of the stator and rotor increases by 2–3 times in comparison with liquid friction. The oscillation amplitude of the rotor and stator in the longitudinal direction is comparable with the gap between these elements. Therefore, it is recommended that when designing refiners, to develop methods and means of vibration protection, and during operation, to prevent boundary friction between headsets. The developed calculation procedure can be used in other industries, for example, mining and metallurgy. References [1] Legotskij S S and Goncharov V N 1990 Refining equipment and pulp preparation (Moscow:  Forest industry)  [2] Ivanov S N 2006 Paper technology (Moscow: Forest industry) [3] Byvshev A V and Savitskiy E E 1991 Mechanical dispersion of fibrous materials (Krasnoyarsk: Krasnoyarsk university publ) [4] Gorski D, Hill J, Engstrand P and Johansson L 2010 Reduction of energy consumption in TMP refining through mechanical pre-treatment of wood chips Nord. Pulp Pap. Res. J. 25(2) 156 61 [5] Olender D and Wild P 2007 Forces on Bars in High-Consistency Mill-Scale Refiners. Trends in Primary and Rejects Stage Refiners J. Pulp Paper Sci. 33 (3) 163 [6] Eriksen O 2013 Mechanism in refining zone for development of physical properties of TMP                fibers in low-consistency refiner (Norwegian University of Science and Technology) 64 p [7] Vuorio P and Bergquist P 2000 New refiner segments technology to optimize fiber quality and energy consumption of refiner mechanical pulp Metso Paper Service 18-25 [8] Muhic D, Huhtanen J P and Sundstrom L 2011 Energy efficiency in double disc refining Influence of intensity by segment design Nord. Pulp Pap. Res. J. 26(3) 224-31 [9] Per-Erik Ohls and Syrjanen A 1976 The importance of disk parallelism in single rotating disk refiners Tappi 59 100-3 [10] Strand B and Mokvist A 1987 Control and optimization of conical disk refiner International   Mechanical Pulping Conference pp 11-8 [11] Gorski D, Hill J, Engstrand P and Johansson L 2010 Reduction of energy consumption in TMP refining through mechanical pre-treatment of wood chips Nord. Pulp Pap. Res. J. 25(2) 156-61 [12] Fernando D, Muhic D, Engstrand P and Daniel G 2011 Fundamental understanding of pulp                 property development under different thermomechanical pulp refining conditions as       observed by a new method and SEM observation of the ultra structure of fiber surfaces Holzforschung 65(6) 777-86 [13] Vikharev S N 2019 IOP Conf. Ser.: Earth Environ. Sci. 226 012048 [14] Vikharev S N 2014 Vibration protection of knife refining machines (Ekaterinburg: UGLTU) [15] Aleksandrov V M and Chebakov M I 2007 Introduction in mechanics of contact interactions                (Rostov on Don publishing house of Open Company TSVVR) p114 [16] Vikharev S N 2018 IOP Conf. Ser.: Mater. Sci. Eng. 450 032020 [17] Karlstrom A and Eriksson K 2014 Fiber energy efficiency Part II: Forces acting on the refiner bars Nord. Pulp Paper Res. J. 29(2) 332 [18] Johnson K 1989 Mechanic of contact interaction (Moscow: World) 509 [19] Vikharev S N 2019 IOP Conf. Ser.: Earth Environ. Sci. 226 012010 [20] Vikharev S N 2019 IOP Conf. Ser.:Mater. Sci. Eng. 537 032008 APITECH-2019Journal of Physics: Conference Series 1399 (2019) 044011IOP Publishingdoi:10.1088/1742-6596/1399/4/0440118      [21] Zasypkina S A 2017 Development of methods for vibration calculation of disk refiners (Yekaterinburg)          

Study of longitudinal vibrations of stab knife refiners

https://elar.usfeu.ru/bitstream/123456789/9259/1/2-s2.0-85077439510.pdf

Study of longitudinal vibrations of stab knife refiners

Abstract

Similar works

Full text

Available Versions

Electronic archive of the Ural State Forest Engineering University