The dissertation aims to study to reduce torsional vibration for the shaft The machine is equipped with DVA (dynamic vibration absorber) oscillator, mass disk type - spring - viscous. To know more details of the content please refer to the thesis summary.
MINISTRY OF VIETNAM ACADEMY EDUCATION AND OF SCIENCE AND TRAINING TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Vu Xuan Truong STUDY ON REDUCTION OF TORSIONAL VIBRATION OF SHAFT USING DYNAMIC VIBRATION ABSORBER Major: Engineering Mechanics Code: 62 52 01 01 SUMMARY OF DOCTORAL THESIS IN MECHANICAL ENGINEERING AND ENGINEERING MECHANICS Hanoi - 2018 The doctoral thesis was completed at Institute of Mechanics, Graduate University of Science and Technology, Vietnam Academy of Science and Technology Supervisors: Assoc.Prof.Dr Khong Doan Dien Dr Nguyen Duy Chinh Reviewer 1: Prof.Dr Hoang Xuan Luong Reviewer 2: Assoc.Prof.Dr Nguyen Phong Dien Reviewer 3: Dr La Duc Viet This doctoral thesis will be defended at Graduate University of Science and Technology, Vietnam Academy of Science and Technology on hour , date month 2018 This doctoral thesis can be found at: - Library of the Graduate university of Science and Technology - National Library of Vietnam INTRODUCTION Necessary of doctoral thesis With the development of human history, technology is gradually developing breakthroughs One of the most important periods that opened up the early beginnings of the modern era was the industrial revolution During this time, the machinery industry has been formed, contributed an important role in supporting production activities Machinery allows production of a variety of items It is not only speed performance but also high efficiency beyond the human ability In addition, machines perform better in long-term jobs and achieve higher consistency The quality of work can be changed when human is influenced by emotional factors, health, etc In addition, the machines help to carry out various dangerous tasks on behalf of humans The machines are widely used in various fields such as: manufacturing, construction, agriculture, industry, mining, Today, many machines are even designed to operate without controlling human With the support of machines, the world becomes more modernizing and growing Particularly in the context, the industrial revolution is developing on the world and affecting to the global economy The research, manufacture and enhance longevity and ability to work of machinery that has contributed significantly to the industrial revolution Shaft is one of the most important parts in machine The Shaft is used to transmit torque and rotation from a part to another part of the machine through other machine parts assembly on the shaft such as the gear, belt, key, shaft couplings The characteristic movement of shaft is rotary motion During operation, the shaft is subjected to torque induced by the engine or system to the transmission shaft [21], [22], [25], [26], [28], [35] In particular, the shafts and other components are generally made of elastic materials So under the influence of torque, the axis will be subjected to twisting This deformation is changed over time and repeated at each rotation cycle of the shaft that is called shaft oscillation This oscillation is particularly harmful, undesirable It is cause of fatigue damage and effects to the longevity and workability of the shaft and machine [21], [22], [25], [26], [ 28], [35] Specifically, this deformation is cause of the vibration, machine noise, and fatigue damage of shaft It effects not only to the shaft itself, but also damages other important machines mounted on the shaft Thereby it induces damaging of the machine The study to reduction of shaft vibration is an important and timely task [21], [22], [25], [26], [28], [35] By wishing to inherit and develop previous research results and applying the research results in practice to improve lifespan, ability to work, accuracy of the shaft in general and machine in particular Author chose topic: "Study on reduction of torsional vibration of shaft using dynamic vibration absorber" to study in my thesis Research propose of the thesis As discussed above, torsional vibrations are particularly harmful to the durability, longevity and performance of the shaft in general and the machine in particular During the working process, it is cause of vibration and noise It is not only affects the life span and working ability of the shaft but also directly affects the quality of the machining parts In particular, is has no research used the calculation method to optimize the parameters of oscillator for the main system under torsion oscillator Therefore, the purpose of the thesis is research to reduce the torsion oscillator of the shaft by the dynamic vibration absorber (DVA) Object and research scope Object of research The object of the thesis is the optimal parameters of the passive DVA to reduce the torsional vibration of shaft when it is subjected to different types of agitation: harmonic, impact and random excitation Research scope In the scope of this thesis, author find out the optimal parameters of the DVA to reduce the torsional vibration for the SDOF (single degree of freedom) shaft-DVA system and to develop the fixed point method for N-th degree of the MDOF (multi degrees of freedom) shaft model The thesis only focuses on reducing the torsional vibration, without bending and axial vibration,… Calculations include these vibration are mentioned in part “Further research directions” Methodology of research Based on the actual shaft, author has transformed a real shaft model into a theoretical model that attached the DVA absorber From the calculation model of shaft attached the DVA, author used Lagrange equation to set the vibration differential equations of the system From the differential equations obtained, author researches, analyses, calculate to reduce the torsional vibration for the shaft and find out analytic solutions of system by the methods: fixed point method (FPM), minimization of quadratic torque (MQT), maximization of equivalent viscous resistance method (MEVR) and minimization of kinetic energy method (MKE) To perform the calculations and evaluation the effect of oscillation reduction in thesis, author built the computer programs on Maple software to simulate oscillations of the system so that the reader has a visual view on the efficiency of the DVA This software is used by scientists around the world for it can be obtained the reliable results Structure of the thesis The thesis consists of the beginning, four chapters and concluding section, next study with 139 pages, 12 tables and 45 figures and graphs Chapter presents an overview about researches on the reduction of torsional vibration and the calculation methods to determine the optimal parameters of the DVA Chapter establishes the computational model and determines the differential equations system of motion that describes the vibration of the mechanic system Chapter solves problem to calculate the reduction of torsional vibration for the shaft and determines the optimal parameters of the DVA by some different methods Chapter analyzes, evaluates the efficiency of oscillation reduction according to the optimal results defining in Chapter Besides, the chapter simulates the numerical results of the research to reduce the oscillation of machine And the chapter also develops research results for shaft model that has MDOF The main and new contributions and further research direction of the thesis are summarized in the concluding section CHAPTER OVERVIEW ON REDUCTION OF TORSIONAL VIBRATION AND METHODS FOR DETERMINING THE OPTICAL PARAMETERS 1.1 Overview on reduction of torsional vibration researches 1.2 Overview of DVA and vibration reduction methods 1.2.1 General introduction 1.2.2 The basic principles of DVA 1.2.3 Calculate the DVA for the undamped structure 1.2.4 Calculate the DVA for the damped structure 1.2.5 Optimal parameter calculations for MDOF structure 1.2.6 Some criterias for determining DVA 1.3 Conclusion for chapter Chapter gave an overview of domestic and international studies on the reduction of torsional vibration for the shaft; overview of the DVA The chapter presents the basic principles of passive DVA, provides methods for calculating passive DVA applying on damped and undamped main system; Overview on studies for determining optimal parameters in case the main system that has many degrees of freedom At the end of the chapter, author figure out some criterias for identifying passive DVA These are the basis for author’s study to determine the optimal parameters of the passive DVA that can be reduced the torsional vibration of the shaft when it is subjected to different excitations such as: Air, collision, accidental incitement in the following chapters CHAPTER TORSIONAL VIBRATION DIFFERENTIAL EQUATIONS FOR SHAFT ATTACHED DVA 2.1 Analyzing the model of torsional vibration of the shaft-DVA structure in the thesis From the researches in Chapter 1, author finds out that there are many studies on the reduction of torsional vibration with or without CPVA (centrifugal pendulum vibration absorber), CDR (centrifugal delay resonant) and DVA (dynamic vibration absorbers) But these studies just focus on the stability and motion control of oscillating absorber systems, and it has no research that uses the optimum arithmetic calculations to calculate the optimal parameters of DVA for main system under torsional vibration There are some studies to reduce the torsional vibration of shaft by setting a DVA in different forms In these studies, authors also focused on determining optimal parameters for the DVA design However, the methods used in these studies are always numerical methods such as the Taguchi method, the Gauss-Newtonian nonlinear regression method So optimization results only can be applied to a detail shaft and it can be not applied to any shaft with variable parameters [7], [9], [10], [13], [14] Therefore, in this thesis, author proposed to set a DVA in disk type –spring- damper on the shaft to reduce the torsional vibration of the shaft as shown in Figure 2.1 In fact, the DVA disc-type spring-breaker is a particular type of DVA, which applies the results of the CPVA oscillator [40], [43], [44] and the DVA should be designed symmetrically through the center of the shaft This model overcomes the limitations of [7], [9], [10], [13], [14] and inherits the advantages of the absorption design in [21], [26], [54] with the DVA design that has center is centered on axis of the shaft, so that no eccentricity occurs and the structure achieves the greatest stability Particularly, in this thesis, author concentrates on studying, calculating and determining the optimal parameters of the DVA in analytical form with the aim of reducing the angular displacement of the main system (torsional angle of shaft) by using the fixed point method [29], [59], [60], minimization of quadratic torque approaches [60], [67], maximization of equivalent viscous approaches [39], [60] ] and the minimizing energy method [6], [63], [64] of the system to determine the optimal parameters of the DVA, such as the damping and the tuning ratio From this, author calculate the results obtained for evaluating the effect of shaft oscillation reduction with different types of excitation, according to different criteria Figure 2.1 shows a schematic diagram of the rotary-disk-mounted DVA The modeled shaft consists of a spring with a torsional rigidity of ks (Nm), and a disc with mass momentum inertia is Jr [35], [59] (shaft and rotor rigid with shaft via hub) The machine shaft rotates with angular velocity Ω0 (s-1) The shaft affected by the damped coefficient cs ks ka 0 Jr cs ca Ja Figure 2.1 The shaft-DVA structure e1 r e2 ka M(t) r a ca Figure 2.2 Model of dynamic vibration absorber (DVA) In order to reduce the torsional vibration for the shaft, author set up a mass-spring-dics (DVA) oscillator on the shaft through the hub of the DVA The connection between the shaft and DVA is a spline shaft So the rotor of the DVA will rotate with the shaft The structure diagram of the DVA is discussed in the thesis, as shown in Figure 2.2 The DVA consists of a rotor (fixed with shaft through a hub) and a passive disk The rotor and the passive disk are connected together through spring-damper Inertial radii and inertia momentum of rotor and disc are ρr, Jr, ρa, Ja, respectively The stiffness of each spring is ka (N/m), the viscosity of each damper is ca (Ns/m) The rotational angle of the rotor is φr (rad), the relative rotation between the passive disk and the rotor is rad (rad) The torsion angle θ(t) between the two shaft ends is defined as θ(t) = φr-Ω0t The motor shaft is affected by the excitation torque M(t) due to the system mounted behind the impact axis [35] 2.2 Establish vibration differential equations for shaft-DVA strucrure By using the Lagrange equation for the torsional shaft model with the DVA, author obtained the differential equation system describing the torsional vibration of the shaft as follows: (2.29) (mr r2 ma a2 ) ma a2a cs ks M(t) 2 (2.30) m m nk e nc e a a a a a a a a a Demonstreting equations (2.29) and (2.30) in matrix form, it can be obtained as: Cq Kq F (2.31) Mq where the general coordinate vector, mass matrix, damping matrices, stiffness matrix and vector of excitation forces are expressed as follows: m m ma a2 c T q a M r r 2a a C s 2 2 ma a nca e2 ma a T M (t ) k K s 0 2 F nkae1 mr r In case undamped primary system, the differential equation is rewritten as follows: (2.37) (mr r2 ma a2 ) ma a2a ks M(t) (2.38) ma a2 ma a2a nkae12a ncae22a 2.3 Simulate torsional vibration of the shaft-DVA system In this section, author performs the simulation of the torsional vibration of the non-retractable shaft with the DVA absorber with any parameter (without DVA the optimum parameters) To simulate numerical shaft model, author used simulation data in the publication [35] of Prof Hosek (Figure 1.2) Figure 2.3 Torsional vibration response with harmonic excitation at resonance frequency Figure 2.4 Torsional vibration response with impact excitation Figure 2.5 Torsional vibration response with random excitation In this chapter, the simulation purpose for the torsional vibration of the shaft is: If the selected design parameters are unreasonable, it may have effect to reduce oscillation but with low efficiency (Figures 2.4 and 2.5) It is not only unable vibration reduction effects but also increases the amplitude of this harmful vibration It can see that determining the optimal parameters of the DVA absorber to improve the efficiency of reducing the torsional vibration for the shaft is a very meaningful and practical application technique 2.4 Conclusions for chapter Chapter establishes a mechanical model and a mathematical model to determine the torsional vibration of the shaft using a nonblocking, disk-retaining spring-loaded DVA To establish the differential equation system for shaft model, author uses the type II Lagrange equation The differential equation system is linear From the torsional vibration rule of the shaft, it contains the design parameters of the DVA Which is the scientist basis to study, analyze, calculate optimal parameters of DVA with different optimum standards At the end of the chapter, author simulates the torsional vibration response of the shaft in case of with and without the DVA using any selected parameters It finds out that the installation of the DVA into the shaft has the effect of changing the amplitude of the shaft oscillation Hoauthorver, does not imply that the amplitude of the oscillation is reduced by the vibration amplitude of the motor shaft The shaft is not reduced but also increased According the results, author find that the study of determining the optimum parameters of the DVA is very necessary and meaningful The calculation of optimal parameter is presented in Chapter CHAPTER RESEARCH, ANALYSIS, CALCULATION AND DETERMINATION OPTIMAL PARAMETERS OF DVA For the purpose of research, author calculates the optimal oscillator to reduce the displacement of the main system The optimal parameters of the DVA include the spring coefficient and the viscous resistance Identification of these parameters allows to choice the spring and viscous oil for the DVA design with the best vibration reduction performance while still ensuring the technical and economical requirements when designing DVA 3.1 Determination of optimal parameters in case the shaft is subject to harmonic excitation Under the harmonic excitation, the fixed point method (FPM-Fixed Points Method) is used to determine optimal parameters In this part, author finds the optimal parameters of the DVAs for the purpose of reducing the displacement of the main system (torsional vibration of shaft) From differential equation system (2.37) and (2.38), the nature frequency of the DVA is presented as: a ka ma (3.1) and the nature frequency of shaft: ks (3.2) Jr Introducing dimensionless parameters: μ = ma /mr, η = ρa / ρr, λ = e1 / ρr, α = ωa /Ωs, β = ω /Ωs, ξ = ca /(ma ωa) Thus the differential equations (2.25) and (2.26) become: s b) Damped primary system (cs = 22.5 kgm2/s) Figure 4.4 Torsional vibration response of shaft with and without DVA Table 4.4 Effectively of reduction vibration with harmonic excitation Effectively of reduction vibration (%) DVA design cs = kgm2/s cs = kgm2/s cs = 22.5 kgm2/s DVA-FPM 99.987 94.939 68.178 4.1.2 In case the shaft excited by impact excitation During the work of the machine many, the gears some time occur on the collision during the match, or overload occurs locally to the system Therefore, the efficiency of the reduction of the optimal DVA design needs to consider the obtained results in Chapter when the system encounters a collision When the main system is impacted by the DVA-MKE design impact collision (which is determined by the energy minimization method), it is most appropriate The impact-resistant system is equivalent to the system with an initial zero velocity In this section, author performs the simulation with the initial state: y 0 T The simulation results and calculations are as follows: Figure 4.7 Torsional vibration response of DVA-MKE design with impact excitation 16 Damped primary system cs = kgms-2 Figure 4.9 Torsional vibration response of DVA-MKE design with impact excitation and damped primary system Table 4.5 Effectively of reduction vibration with impact excitation Effectively of reduction vibration (%) DVA design cs = kgms-2 cs = kgms-2 cs = 22.5 kgms-2 DVA-MKE 99.473 95.461 81.674 4.1.3 In case the shaft is excited by random excitation In the case of the main system, the DVA-MQT (defined by MQT) or DVA-MEVR design (determined by MEVR) are used The simulation results and calculations are as follows: Figure 4.11 Torsional vibration response of DVA-MQT design with random excitation 17 Damped primary system cs = 22.5 kgms-2 Figure 4.12 Torsional vibration response of DVA-MQT design with random excitation Table 4.6 Effectively of reduction vibration with random excitation Effectively of reduction vibration (%) DVA design cs = kgms-2 cs = kgms-2 cs = 22.5 kgms-2 DVA-MQT 97.058 95.464 95.758 DVA-MEVR 96.988 95.909 96.013 4.2 Development of research results for the case of MDOF structure 4.2.1 Research model and motion equation system J rN 0 k sN J r1 J r ( N 1) ks ( N 1) Jr2 ks k s1 ka ca Ja Figure 4.17 Model of the MDOF shaft – DVA By considering the shaft model of machine (MDOF), each of the freedom degrees is modeled as a torsion spring with a stiffness of ksi and the mass moment inertia is Jri, as shown in Figure 4.17 In order to reduce the torsional vibration for the shaft, a DVA-type oscillator for oscillation is inserted into the shaft section through the hub of the DVA The connection between the shaft and the DVA is the spline shaft, so that the rotor of the DVA will rotate in the same direction as the rotary shaft The configuration of the DVA is shown in Figure 2.2 (chapter of this thesis) The system has (N+1) degrees of freedom In which, the 18 principal N-wave and free-band oscillator DVA have a degree of freedom The independent co-ordinate is the angle of rotation of the i-th degree (i = 1, 2, , N) and the relative rotation angle of the DVA disk over the first degree of freedom + The system of differential equations of motion of the system Appling the Lagrange equation to the system with N+1 freedom degrees: J a (1 a ) nca e22a nka e12a J J ( ) k M (t ) r1 a a s1 J r 22 k s1 1 ks 3 J rjj ks ( j 1) j j 1 ksj j 1 j J k r ( N 1) N 1 s ( N 2) N 1 N 2 (4.19) ks ( N 1) N N 1 J rNN ks ( N 1) N N 1 ksN N 4.2.2 Determination of the optimum parameters of the oscillator reduces the oscillation of the spindle for multiple degrees of freedom From the system of differential equations (4.19), the nature frequency of the DVA is: (4.20) a2 ka / ma Set: (4.21) 2si ksi / J ri and introducing dimensionless parameters: μ = ma /mr, η = ρa / ρr, λ = e1 / ρr, α = ωa /Ωs, β = ω /Ωs, ξ = ca /(ma ωa) In the case, the shaft model is subjected to a harmonic torque (Figure 4.17), it can be expressed as complex: ˆ I t (4.24) M (t ) Me Determine the amplitude function of amplitude A of the Nth degree Replace (4.20), (4.21), (4.24) and the dimensionless factors above into the differential equations system (4.19) and solve for this system of differential equations with N = 1,2,3 , and draw the general expression of the complex amplitude of the degree of freedom N as follows: M A1 I A2 (4.55) ˆN AN 2 A1 AN 1 A3 I AN 2 A2 AN 1 A4 ks 19 These calculations are done on the Maple Math software Performing complex transformations has: 2 M A12 A22 (4.56) ˆN A A A A 2 A A A A 2 ks N 1 N 2 N 1 N 2 The amplitude-frequency amplifier of the N-th degree of the form is obtained as: 2 A12 A22 (4.57) A AN A A A A 2 A A A A 2 N 1 N 2 N 1 N 2 Where A1, A2, A3, A4 are coefficients and they are defined from the corresponding a freedom degree shaft These coefficients are defined in Equations (3.20), (3.21), (3.22) and (3.23) Set yN = AN-1 then xN = AN-2 kN N 1 k ( N k )! k yN 1 22 ; xN yN 1 N 2k !k ! k 0 If N is even: kN = (N-2)/2; if N odd: kN = (N-1)/2 Determine the optimum α tuning ratio With the amplitude A defined in formula (4.29), it depends on the eight dimensionless parameters of n, μ, η, λ, α, β and ξ It is possible to define these parameters so that the amplitude-frequency amplification function is minimized Determination of fixed points The generalized expression (4.57) defines the amplitude function of the Nth degree of freedom in the N-factorial shaft model Fig 4.18 and Fig 4.19 respectively describe the change in frequency amplitude of the Nth degree of freedom with respect to the frequency of β with second degree freedom (N=2) and the principal third degrees freedom (N=3) are determined from the formulas (4.60) and (4.61) From Fig 4.18 and Fig 4.19, all curves with values of viscous resistances ξ pass all through fixed points This fixed points are 2N When simulating the amplitude graph of the Nth degree of freedom in the frequency domain with the main system there are different degrees of freedom and for all different values of the ratio of resistances ξ the author finds that the curves are Described by (4.60) and (4.61) always go through fixed points and in the general case, the elevations of these points are different 20 Figure 4.18 Change of amplitude amplitude curve when changing damping ratio with N = 2, = 0.02, = 1, = 0.5, = 0.8,n = and = 0.2 Figure 4.19 Change of amplitude amplitude curve when changing damping ratio with N = 3, = 0.02, = 1, = 0.5, = 0.8,n = and = 0.2 From Figure 4.18, it shows that the main system there are degrees of resonance (with ξ = 0) and two resonance peaks (with ξ = ∞) The main system having degrees of freedom, the resonant number with ξ = is vertices and vertices with ξ = ∞ (Figure 4.19) In general, if the main system has an N degree of freedom, it will has N+1 resonance with non-blocking states (ξ = 0) corresponds to the resonant N of the free-radical N of the main system and an additional peak resonance of DVA (subsystem); In the critical state ξ = ∞, N resonances will exist There are fixed points between these resonators Similarly to the case of the main system having a degree of freedom (Section 3.1 of this thesis), the βj degree of these fixed points is determined by solving the equation: A 0 The derivative of function A in (4.57) is given by the variable ξ A2 A A AN 1 A4 (4.65) 22 N 2 A1 AN 2 A1 AN 1 A3 2 21 Equation (4.65) is used to determine the magnitudes of fixed points in the general case Solving the equation (4.65), the values of βj for the twodegree shaft are obtained Similarly, for solving the equation (4.67), the values of βj for the three-degree shaft are obtained To determine the optimal parameter α, the value of the amplitude function of amplitude A at two fixed points (corresponding to β1 and β2) must be equal There are some values of β, such as β = 0.26, β = 0.67 and β = 1.62, where resonance occurs (Figure 4.18) The resonance region is controlled and determined in the optimization design are one of the closest points to β = Therefore, the β1 and β2 ratios are chosen so that the β-ratio is controlled between them In this way, the two fixed points are chosen as S and T The solution of the equation AS = AT obtains the optimal parameter α Table 4.7 lists the results of the tuning ratio for N = 1, N = 2, and N = Table 4.7 Optimal tuning ratio of the DVA DOF αopt N=1 n 2n 3 N=2 2 n 2n 6 10 N=3… 2 n Determining the optimum damping ratio To determine optimal damping ratio, solve the equation A From the equation (4.57), the minimal expression of the resistance ratio is obtained as follows: A A A A A A2 A1 AN 2 A3 AN 1 A3 N 1 A1 N 2 AN 1 AN 2 A1 A A A A A A2 A2 AN 2 A4 AN 1 A2 N 2 A4 N 1 AN 2 AN 1 A2 (4.71) 22 According to Brock [14], the results of the calculated viscosity resistances ξ obtained and listed in table 4.8 Table 4.8 Optimal damping ratio of the DVA DOF closed-form of the damping ratio ξ N=1 3 2n(1 ) AB C N=2 A (4 3 6 2 5 2) B (2 2 5 2) 2 ; C n (2 3 )(1 )3 2 N=3… ABC DE A 2 ; B 2 48 7 3 6 2 2 C 5 3 14 2 10 2; D n E 2 48 13 3 26 2 18 4.2.3 Numerical simulation of the results of the study for the MDOF Figure 4.23 The amplification function with = opt and = with N=2 Figure 4.20 shows the variation of the amplitude curve in terms of the damping Figure 4.20 shows that there are some resonances, such as frequencies = 0.392, 0.873 and 1.648 It is clear that the value of the optimal viscosity resistance ratio ξopt = 0.23 at the peak amplitudes are the loauthorst of the A curves relative to the different values of ξ 23 a) Without DVA b) with optimal DVA design Figure 4.24 Torsional vibration response of system with N=2 at = 0.88 a) without DVA b) with optimal DVA design Figure 4.25 Torsional vibration response of system with N=2 at = 0.46 24 a) without DVA b) with optimal DVA design Figure 4.26 Torsional vibration response of system with N=2 at = 1.58 Figure 4.27 The amplification function with = opt, = and N=3 25 a) without DVA b) with optimal DVA design Figure 4.29 Torsional vibration response of system with N=3 at = 0.77 From Figures 4.28 and 4.29, in the case of optimal DVA, the system oscillation is significantly reduced compared to the case when DVA is not installed The vibration performance of the optimal DVA in this case is not only shown by the short-term harmonic oscillation stabilization, but also shown by a markedly reduced vibration amplitude and small value Whereas, the optimal DVA is not attached, the oscillating torque is unstable, with a maximum amplitude of 1.6 rad (first degree freedom) at resonant frequency = 0.36 4.3 Conclusions for chapter - Author has studied, analyzed and evaluated the effect of reducing the oscillation of the shaft with and without DVA and if the DVA is mounted with the optimal analysis solution found of the DVA by four methods as follows: 26 Excitation DVA design harmonic impact (DVA-FPM) (DVA-MKE) (DVA-MQT) (DVAMEVR) random Effectively of reduction vibration (%) cs = cs = 22.5 cs = kgm2/s kgm2/s kgm2/s 99.987 94.939 68.178 99.473 95.46 81.674 97.058 95.464 95.758 96.988 95.909 96.013 This result confirms the optimal parameters of DVA found in the dissertation effectively reduce the fluctuations authorll in the case of main plagued and unobstructed Author applied research results, calculated optimal parameters of DVA and simulated research results From the results simulation results of the study, in the case of the conditioned, shock and random stimulus systems, it is found that the effect of vibration reduction of DVA designs is very good In case of harmonic stimulation, even if the system is working in resonant area, the efficiency decreases when the main system is not obstructed and has good resistance This meets the technical requirements set out The research results have been developed for the main case with many degrees of freedom Author has established a differential equation system that describes the oscillation of the system and finds the optimum parameter of the DVA decreasing the oscillation of the MDOF system by a fixed points method CONCLUSIONS AND RECOMMENDATIONS The main results of thesis The thesis focuses on calculating the optimal parameters of the vibration absorbing DVA for the torsional model with one and many degrees of freedom The optimal parameters of the DVA include the ratio α (tuning ratio) and the damping ξ Author has established a mechanical model and a mathematical model to determine the torsional vibration of the shaft using the DVA To find the equation of oscillation of the system, author uses Lagrange equations The system of differential equations obtained is linear From differential equation system describing the torsional vibrations of the shaft, it is found that the quantities of the oscillator are oscillating, which is the basis for the scientists to study, analyze and 27 calculate the parameters The absorbers of the oscillator vary according to different optimal standards Author has determined the optimal parameters of the DVA set in the form of expressions, reduce the torsional vibration for the shaft has a degree of freedom from the effects of different excitation In case of the shaft is excited by harmonic moment opt n opt ; 2 3 n (1 ) In case of the shaft is excited by random excitation - The optimal parameters are solved by the minimum of quadratic moment n (2 ) ; n (1 ) 2 (4 ) n (1 )(2 ) - and solved by maximum of equivalent viscous resistant opt ; opt n n (1 ) opt opt In case of the shaft is excited by impact excitation opt n (2 ) n (1 ) ; opt (4 2 ) (1 ) 2 n (2 ) Author has studied, analyzed and evaluated the effect of reducing the oscillation of the shaft in the case of without and with DVA and the absorber is mounted oscillating with the optimal analysis solution found DVA From the simulation of the oscillation amplitude over time, in the case of the harmonic-induced system, collision and random agitation, it is found that the amplitude of the oscillation of the shaft when designing the DVA according to the parameters The optimum look in the thesis is very good In case of harmonic stimulation, even if the system is working in resonant area, the effect of vibration reduction is very good This meets the technical requirements set out The results of the study have been developed in the case of the main system with many degrees of freedom (MDOF) Author has established a differential equation system describing the oscillation of the system, setting up the general expression of the amplitude function of the degree of freedom of N-th in the MDOF shaft model and finding optimal 28 parameters of the DVA is the reduction of the torsional vibration for the MDOF system in close-form solution - Optimal parameters of the DVA with 2DOFs shaft model: 2n 3 opt opt 2 n (4 3 6 2 5 2)(2 2 5 2) 2 2 n (2 3 )(1 )3 - Optimal parameters of the DVA with 3DOFs shaft model: 2n 6 10 opt 2 n opt 2 ABC DE A 2 ; B 2 48 7 3 6 2 2 C 5 3 14 2 10 2; D n ; E 2 48 13 3 26 2 18 The results of this thesis have been compiled and simulated by Maple software Which are used by scientists in the world for reliable results Further research directions Continuing research to find the optimal parameters of the oscillator to reduce the oscillation for the shaft when the main obstacle Appling the research results into the practical reality need to study experiments Develop the research results of the thesis when it comes to the bending and axial vibration, NEW POINTS OF DOCTORAL THESIS An analytical approach has been proposed to solve the optimal design problem of DVA, which is used to enhance the vibration performance of a primary shaft under time-varying torsional moment The closed-form formulae of DVA’s optimal parameters have been obtained analytically 29 The effectiveness of the proposed optimal parameters has been evaluated The computation results show the robustness of the proposed method with different types torsional excitations even in resonant condition The obtained optimal formulae can be used for the system having a primary structure with single degree of freedoms (SDOF) and multi-degree of freedoms (MDOF) The major academic contribution of the thesis is the development of analytical method to calculate the optimal parameters and to simulate the vibration efficiency of the MDOF primary system LIST OF PUBLIC WORKS OF AUTHOR Vu Xuan Truong, Nguyen Duy Chinh, Khong Doan Dien, Tong Van Canh (2017), Closed-form solutions to the optimization of dynamic vibration absorber attached to multi degree-of-freedom damped linear systems under torsional excitation using the fixed-point theory, Journal of Mutibody Dynamics (ISI, IF 1.242), First Published August 4, 2017, DOI: 10.1177/1464419317725216 Vu Xuan Truong, Khong Doan Dien, Nguyen Duy Chinh, Nguyen Duc Toan (2017), Optimal Parameters of Linear Dynamic Vibration Absorber for reduction of torsional vibration, Journal of Science and Technology (Technical Universities), Vol 119B, pp.37-42 Nguyen Duy Chinh, Vu Xuan Truong, Khong Doan Dien (2017), Study on reduction for torsional vibration of shaft using minimization of kinetic energy, Journal of Structure Engineering and Contruction Technology, ISSN 1859-3194, Vol 25, pp 5-12 Khong Doan Dien, Vu Xuan Truong, Nguyen Duy Chinh (2017), The fixed-points theory for shaft model by passive mass-spring-disc dynamic vibration absorber, Proceedings of The 2nd National Conference on Mechanical Engineering and Automation, ISBN 978-604-95-0221-7, pp 82-86 Vu Xuan Truong, Khong Doan Dien, Nguyen Duy Chinh (2017), Calculate and simulate the effective of torsional vibration reduction for the shaft using dynamic vibration absorber, UTEHY Journal of Science & Technology, ISSN 2354-0575, Vol 15, tr 9-15 Khong Doan Dien, Nguyen Duy Chinh, Vu Xuan Truong (2014), Research to reduce vibration for shaft of machines using tuned mass damper, Proceedings of The Regional Conference on Mechanical and Manufacturing Engineering, ISBN 978-604-911-942-2, pp 132136 30 ... direction of the thesis are summarized in the concluding section CHAPTER OVERVIEW ON REDUCTION OF TORSIONAL VIBRATION AND METHODS FOR DETERMINING THE OPTICAL PARAMETERS 1.1 Overview on reduction of torsional. .. Torsional vibration response with impact excitation Figure 2.5 Torsional vibration response with random excitation In this chapter, the simulation purpose for the torsional vibration of the shaft. .. kgm2/s) Figure 4.4 Torsional vibration response of shaft with and without DVA Table 4.4 Effectively of reduction vibration with harmonic excitation Effectively of reduction vibration (%) DVA design