The subject of the study is the longitudinal fluctuations of the rotors of disk knife refiners. The dynamic and mathematical models of rotors of disk mills are developed. As a result of the research, a methodology for calculating the longitudinal fluctuations of rotors is proposed and tested. The error in determining the lower frequencies of free fluctuations of the mill rotors does not exceed 9%. The lowest frequencies of free longitudinal fluctuations of the rotors of existing mills are in the range of 85-165 Hz, and the second and subsequent frequencies of free fluctuations are in the frequency range above 1 kHz. When designing and operating disk knife refiners, it is necessary that the minimum inter-knife gap between the rotor and stator does not exceed the amplitude of fluctuations of the mill disk. 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.

IOP Conference Series Materials Science and Engineering

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Electronic archive of the Ural State Forest Engineering University

IOP Conference Series: Materials Science and EngineeringPAPER • OPEN ACCESSLongitudial fluctuations of rotors of disk knife refinersTo cite this article: S Vikharev 2020 IOP Conf. Ser.: Mater. Sci. Eng. 734 012044 View the article online for updates and enhancements.This content was downloaded from IP address 79.110.248.251 on 04/03/2020 at 05:39Content 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 LtdMIST: Aerospace 2019IOP Conf. Series: Materials Science and Engineering 734 (2020) 012044IOP Publishingdoi:10.1088/1757-899X/734/1/0120441      Longitudial fluctuations of rotors of disk knife refiners S Vikharev  Faculty of technical mechanics and the equipment of pulp and paper manufactures, Ural State Forest Engineering University, Siberian tract, 37, Ekaterinburg, 620100, Russia  E-mail: cbp200558@mail.ru Abstract. The subject of the study is the longitudinal fluctuations of the rotors of disk knife refiners. The dynamic and mathematical models of rotors of disk mills are developed. As a result of the research, a methodology for calculating the longitudinal fluctuations of rotors is proposed and tested. The error in determining the lower frequencies of free fluctuations of the mill rotors does not exceed 9%. The lowest frequencies of free longitudinal fluctuations of the rotors of existing mills are in the range of 85-165 Hz, and the second and subsequent frequencies of free fluctuations are in the frequency range above 1 kHz. When designing and operating disk knife refiners, it is necessary that the minimum inter-knife gap between the rotor and stator does not exceed the amplitude of fluctuations of the mill disk. The developed calculation procedure can be used in other industries, for example, mining and metallurgy. 1.  Introduction Knife refiners are the main technological equipment for refining fibrous materials in the pulp and paper industry [1-4]. The inter-knife gap in these machines is a fraction of a millimeter [5.6]. When designing and operating knife refiners, it is necessary that the minimum inter-knife gap between the rotor and stator does not exceed the amplitude of disk fluctuations in the longitudinal direction [7]. Therefore, it is necessary to study the longitudinal fluctuations of the mill rotors and to develop a method of fluctuation calculation. The purpose of the work is to study the longitudinal fluctuation of the mill rotor. 2.  Methods and materials 2.1. Dynamic rotor model The following assumptions are made: the study is conducted in a linear formulation; the dispersion of fluctuations in the rotor design is not taken into account; the mill rotor shaft is modeled as a uniform rod; disk deformations, the effect of radial bearings and couplings are not taken into account. It is assumed that the centers of mass of the cross sections of the rod lie on the same axis, and the sections themselves remain flat, their transverse deformations are not taken into account; the effects of bending and torsional fluctuations are not taken into account. The main factor in the excitation of fluctuations is considered to be the forceful effect of the milled semi-finished product. The dynamic model is represented as a homogeneous rod with a cross-sectional area S, elastic modulus E, mass mв and equivalent length l (figure 1). The mass of the disk md is taken as a concentrated MIST: Aerospace 2019IOP Conf. Series: Materials Science and Engineering 734 (2020) 012044IOP Publishingdoi:10.1088/1757-899X/734/1/0120442      one. A dissipative force, proportional to the vibration speed of the disk, acts on the disk. The rotor rotates at a constant frequency ω. The total stiffness of the thrust bearing and its mounting are denoted by Cy. The mechanical and hydrodynamic effect of the milled material on the disk is modeled in the form of axial force F(t), consisting of constant, periodic and random components [8-10]. 2.2. The mathematical model of the rotor Let us select the element dy of the rod (figure 2) and consider its equilibrium in the direction of the Y axis. If the rod has constant characteristics ES const , S const   along the length, then the homogeneous equation describing its free fluctuations takes the following form: 2 22 2 210u uy a t   ,                                                                       (1) where a is velocity of longitudinal waves in the rod.    Figure 1. Dynamic rotor model. Figure 2. The elementary portion of the rod.  We divide the variables by introducing a time factor harmoniously changing with time ( , ) ( )cosu y t y t  .                                      (2) With the joint solution of (1) and (2), we will obtain 220a   .                             (3) We will represent the solution (3) in the form 𝑣(𝑦) =  𝐴 cos𝜔𝑎𝑦 + Bsin𝜔𝑎y.                                (4) Boundary conditions for the study of the model: if y=0 MIST: Aerospace 2019IOP Conf. Series: Materials Science and Engineering 734 (2020) 012044IOP Publishingdoi:10.1088/1757-899X/734/1/0120443       220gu uES m F ty t    .      (5) if  y = l 0yuES C uy .                       (6)  3.  Results and discussion Substitution of (2) and (4) into the boundary conditions (5) and (6) gives a system of linear homogeneous equations. Given only the first harmonic of the axial force   1 cosF t F t  and equating the main determinant of the system of equations to zero, we find the frequency equation, the real roots of which at 0  are the frequencies of the longitudinal fluctuations of the rotor. The lowest is the frequency of free longitudinal fluctuations   222' 2001 2 22' 20( )( )y вoвд в goвy вд в gC Сm m mC Сm m m                           , where  1/2'0 /y g вC m m      is the lowest partial frequency of free fluctuations of an absolutely rigid rotor in the longitudinal direction,  1/20 /в в gС m   is lower partial frequency of free fluctuations of the disk at the sealed end of the rod in the longitudinal direction, вESCl  is longitudinal stiffness.  The lower frequencies of the free longitudinal fluctuations of the mill rotors are presented in table 1. Table 1. Frequencies of free longitudinal fluctuations of mill rotors. Mill brand The lowest frequency of free fluctuations, Hz Calculation  Experiment  Error, %   MD-31 98  92  7   MD-5Sh1А 86  80   8  MD-4Sh6 94  86   9  MD-23 127  136   7   The lowest frequencies of free longitudinal fluctuations of the rotors of existing mills are in the range of 85-165 Hz, and the second and subsequent frequencies of free fluctuations are in the frequency range above 1 kHz. When developing methods and means of vibration protection of rotors in the longitudinal direction, it is necessary to take into account only their lowest frequency of free fluctuations. The error in determining this frequency does not exceed 9%. MIST: Aerospace 2019IOP Conf. Series: Materials Science and Engineering 734 (2020) 012044IOP Publishingdoi:10.1088/1757-899X/734/1/0120444      The dispersion of rotor vibrations occurs in the bearings, shaft material, and the medium in which the rotor rotates. It is impossible to analytically account for all types of dispersion. The introduction of a member that generalizes all types of dispersion of vibrations is known in [11–13]. The parameters of inelastic resistances in this case are determined experimentally. The amplitude of the displacements of the rotor disk in the longitudinal direction (y = 0) in the low- and mid-frequency region, taking into account the dissipative forces in the system 𝑆𝑎 = 𝑆𝑐𝑡 [(1 +𝐶𝑦𝐶𝑒)] / 〈{1 −𝜔2𝜔𝑒2 −𝜔2(𝜔𝑒´ )2 }2+  𝜔2/(𝜔0´ )2𝜒𝑝𝑛2 〉1/2, where 𝑺𝒄𝒕 = 𝑭𝟏/𝑪𝒚 - static displacement of an absolutely rigid rotor in the longitudinal direction, 𝝌𝒑𝒏   - dynamic stress coefficient at resonance in the longitudinal direction. The amplitude-frequency characteristic of the rotor in the longitudinal direction with reference to the MD-31 mill is presented in figure 3.            Figure 3. The amplitude-frequency characteristic of the rotor in the longitudinal direction in relation to the mill MD-31 if F1=103 Н:  о - excluding inelastic resistances; x  - taking into account non-elastic resistances.   The longitudinal fluctuations of the mill rotor are excited not only at the set frequencies [14], but, as shown by experimental studies, at the reverse frequencies and their harmonics. The source of these excitations is the imperfection of the structural elements of mills, in particular the end runout of the disk [15]. When designing and operating knife refiners, it is necessary that the condition A> Sa is fulfilled, where A is the minimum inter-knife gap between the rotor and the mill stator. If this condition is not met, boundary friction occurs between the rotor and stator, which leads to a decrease in the reliability of the knife headset. 4.  Conclusion    A method for calculating the longitudinal fluctuations of the mill rotor has been developed and tested. The error in determining the frequencies of free bending fluctuations of the mill rotors does not exceed 9%. The lowest frequencies of free fluctuations of the rotors of existing mills are in the range of 85-165 Hz, and the second and subsequent frequencies of free fluctuations are in the frequency range above 1 kHz. MIST: Aerospace 2019IOP Conf. Series: Materials Science and Engineering 734 (2020) 012044IOP Publishingdoi:10.1088/1757-899X/734/1/0120445      When designing and operating knife refiners, it is necessary that the minimum inter-knife gap between the rotor and stator does not exceed the amplitude of fluctuations of the mill disk. The developed calculation procedure can be used in other industries, for example, mining and metallurgy. References [1] Ivanov S N 2006 Paper technology (Moscow: Forest industry) [2] Byvshev A V and Savitskiy E E 1991 Mechanical dispersion of fibrous materials (Krasnoyarsk:   Krasnoyarsk university publ) [3] Legotsky S S and Goncharov V I 1990 Refining equipment and pulp preparation (Moscow: Forest Industry)  [4] Korda I and Libnar Z 1967 Paper mass grinding (Moscow: Forest production) [5] 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 [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)  [7] Vikharev S N 2019 IOP Conf. Ser.: Mater. Sci. Eng. 537 032015 [8] Vikharev S N 2019 IOP Conf. Ser.: Mater. Sci. Eng. 537 032008 [9] 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 [10] Berg J K, Sandberg C, Engberg B A 2015 Low consistency refining of mechanical pulp in the light of forces on fibers Nord. Pulp Pap. Res. J. 30 (2) 225 [11] Chelomey V N 1980 Vibration and Engineering: A Handbook (Мoscow: Machinebuilding) [12] Levitsky N I 1988 Oscillations in the mechanisms (Moscow: Nauka)  [13] Frolov K V 1995 Vibration and Engineering: A Handbook (Мoscow: Machinebuilding)  [14] Vikharev S N 2019 IOP Conf. Ser.: Earth and Environ. Sci. 316 012079 [15] Vikharev S N 2018 IOP Conf. Ser.: Mater. Sci. Eng. 450 032020 

Longitudial fluctuations of rotors of disk knife refiners

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Longitudial fluctuations of rotors of disk knife refiners

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