Summary Of Doctoral Thesis In Mechanical Engineering And Engineering Mechanics: Study on reduction of torsional vibration of shaft using dynamic vibration absorber

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.

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY

The doctoral thesis was completed at Institute of Mechanics, Graduate University of Science and Technology, Vietnam Academy

of Science and Technology

Supervisors: 1 Assoc.Prof.Dr Khong Doan Dien

2 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

1 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

2 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)

3 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”

4 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

5 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 1 presents an overview about researches on the reduction

of torsional vibration and the calculation methods to determine the optimal parameters of the DVA Chapter 2 establishes the computational model and determines the differential equations system of motion that describes the vibration of the mechanic system Chapter 3 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 4 analyzes, evaluates the efficiency of oscillation reduction according to the optimal results defining in Chapter 3 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 1 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 1

Chapter 1 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 2 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 k s

(Nm), and a disc with mass momentum inertia is J r [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

a k

a c

    

( )

M t

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 ρ r ,

J r , ρ a , J a , respectively The stiffness of each spring is k a (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:

q

M   (2.31) where the general coordinate vector, mass matrix, damping matrices,

stiffness matrix and vector of excitation forces are expressed as follows:

s a

00

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 2

Chapter 2 establishes a mechanical model and a mathematical model to determine the torsional vibration of the shaft using a non-blocking, 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 3

CHAPTER 3 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 a

a

k m

  (3.1)

and the nature frequency of shaft:

s s

r

k J

2 2

2

2)1

(

r r s a

2 2 2

Present equations (3.3), (3.4) in the matrix form as:

F Kq q C

q

M   (3.5) where

I s a

t I s a

t

I

s

t I s s

t I s a t I s t

I s

s s

s s

s s

s s

e e

e n I e

n

e k

M e

e e

ˆ ˆ

ˆ ˆ ˆ

ˆ )

1

(

2 2 2 2

2 2 2

2 2

2

2

2 2

2 2 2 2

2 2

A  B    n  (3.20)

2 2

2 2 2 2 1





A A

A A A





 (3.27)

Figure 3.1 depicts the graph of the amplitude-frequency amplifier

A according to the damping ξ It finds out that two critical cases ξ = 0

(unbound) and ξ = ∞ (critical constraint) lead to edge of the amplifier graph

going to infinity This shows that there is an optimum value of the

resistance ratio ξ between these two values In addition, the non-blocking

nature of the main system leads to the existence of two fixed points S and T

independent of the resistance ratio ξ of the vibration absorber DVA The

first step of the fixed point method is to find two fixed points S, T

Assuming that the two points S, T have the magnitudes of β1, β2

Figure 3.1 The graph of amplification amplitude-frequency

functional with α=0.9, μ=0.04, η=1, γ=0.5, λ=0.8 and n=4

The conditions to A do not depend on ξ then A 0

n n

n

n n

n n

n

2 2

3 2

2 3 2 2 2 2 2 4 2 2 2 2

Solving the equation (3.41) author get the optimal values of β as follows:

2

)1(2

)1(

2

2

1 2 2 2 2 4

2 2 2 4 2 2 2

)1(

2

2 1 2 2 2 2 4

2 2 2 4 2 2 2

Substituting β = β1,opt and β = β2,opt in to the expression of the

amplification function (3.27), the value of the amplification function at two

fixed points S and T can be obtained According to Den Hartog [29], the

amplitude of A does not change much in the interval between the two

vertices First, two points S and T need to give equal height, so that the

optimal parameter α of the DVA is obtained:

Next, the damping ratio ξ is looking to maximize the

amplitude-frequency curve at the fixed points S and T

To satisfy this condition, it has equation: 0

Replace the optimal values α = αopt and β1,2 = β1,2,opt from equations

(3.42), (3.47) into equation (3.51) and apply Brock's theory [24] Optimal

damping ratio are obtained:

3.2 Determination of optimal parameters in case the shaft is subject to

random excitation

In the case of calculating the optimum parameter of the DVA when

the shaft is subjected to random excitation, author uses the minimization of

quadratic torque (MQT) and maximization of equivalent vicous resistance

(MEVR)

 Minimization of quadratic torque (MQT)

The MQT applied to the impactor of the random excitation moment

with white noise M (t) has the spectral density Sf

From the oscillator equation in matrix form (3.5), the equation of

state is constructed:

( )t  ( )t  f M t( )

y By H (3.56)

where: y(t) is the state vector corresponding to the response of the system

and is defined as follows:

The components of the matrix P are determined by solving the equation

(3.62) Optimal parameters are sought to optimize the quadratic momentum

of the P11 primary response [67] Minimum conditions are:

 Determination of optimal parameters according to the MEVR

The differential equation system (2.37), (2.38) are transformed into the

Considering the case where the main system is affected by white noise, the

equivalent resistance is:

3.3 Determination of optimal parameters in case of impact excitation

This section presents the optimal parameter calculation for DVA by

using the energy minimization method with the objective function of the

energy integral causing the minimum system oscillation The main purpose

is to reduce vibration component of the main system

From the oscillation differential equation (3.5), it is presented in equation:



y By + F (3.89b)

where: y a  aT

Matrix B is defined in (3.61) In this section, author applies the

energy minimization method using the kinetic energy function of materials

[63], [64] The problem is to design the parameters of the DVA to calculate

the energy function

to reach the minimum value, where y0 is the initial condition vector, the

matrix P is the solution of the Lyapunov algebraic equation

To minimize the objective function L, the minimum condition of the

function of two variables is applied:

3.4 The design steps of the DVA for the torsional vibration reduction

3.5 Conclusions for chapter 3

Chapter 3 has researched a theory and calculated the optimal parameters of

the DVA The obtained results are as follows:

Case 1 The shaft of machine is subjected to harmonic agitation The

optimal parameters are determined by fixed point method