DESIGN PARAMETERS OF A DYNAMIC VIBRATION ABSORBER WITH TWO SPRINGS IN PARALLEL

International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163 Issue 12, Volume 4 (December 2017) www.ijirae.com

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DESIGN PARAMETERS OF A DYNAMIC VIBRATION

ABSORBER WITH TWO SPRINGS IN PARALLEL

Giman KIM* Department of Mechanical System Engineering, Kumoh National Institute of Technology, Korea

[email protected] Manuscript History Number: IJIRAE/RS/Vol.04/Issue12/DCAE10088 DOI: 10.26562/IJIRAE.2017.DCAE10088 Received: 22, November 2017 Final Correction: 07, December 2017 Final Accepted: 17, December 2017 Published: December 2017 Citation: KIM Giman. (2017). DESIGN PARAMETERS OF A DYNAMIC VIBRATION ABSORBER WITH TWO SPRINGS IN PARALLEL . IJIRAE::International Journal of Innovative Research in Advanced Engineering, Volume IV, 38-44. doi: 10.26562/IJIRAE.2017.DCAE10088 Editor: Dr.A.Arul L.S, Chief Editor, IJIRAE, AM Publications, India Copyright: ©2017 This is an open access article distributed under the terms of the Creative Commons Attribution License, Which Permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

Abstract: The optimal design parameters of a Dynamic vibration absorber (DVA) in a vibrating bar were discussed in this paper. It has been controlled passively by a Dynamic vibration absorber which consists of a mass and two springs in parallel. A bar, which is fixed at the left end and free at the right end is subjected to a harmonically excited force being positioned at the free end. To define the motion of a bar with a DVA, the equation of motion for a bar was derived by using a method of variation of parameters. To define the optimal design conditions of a DVA, the reduction of a vibrational intensity which is known as the time averaged power flow was evaluated and discussed. The possibility of reduction of the vibrational intensity was found to depend on the mass of a DVA, the positions, and the stiffness of two combined springs.

Keywords— Passive control; a method of variation of parameters; the force transmissibility; a Dynamic Vibration Absorber (DVA); spring stiffness;

I. INTRODUCTION In the industrial fields, lots of machines and structures had been troubled by vibration and unbalance. Numerous attempts have been made to control the vibrations of the machines in operation in the past decades. Hence the fast and the accurate motions of the machine are known to be the optimal design parameters. In the high speed operation, the vibration which may occurs at each parts of machine should be controlled actively or passively for the stability of the machine structure. The vibration problems of machines and structures had been studied by lots of researchers. The vibration energy flow and the dynamic response of a beam, plate, shell or some compound system have been analyzed. By using a model of an elastic beam, the vibrational intensity and control skills had been presented [1]-[3]. Furthermore, the complex frame which is the assembly of several beams and plates had been used to analyze and control the vibration characteristics [4]-[5]. The Active control method had been proven to become the convenient control method for the known forcing frequency zone [6]-[8]. The passive control of the vibrations of the machine tools had been introduced [9]-[12].

A working arm or part of machinery or robots in the industrial fields is known to experience the longitudinal vibration being induced by the constant reciprocal motions.

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The Dynamic Vibration Absorber (DVA) [13] is known to be effective over narrow band of frequencies and is designed to make the natural frequencies of the main system be away from the forcing frequency. Hence in this study a theoretical model of a vibrating bar which is subjected to an external force at the free end is employed to be controlled passively by a Dynamic Vibration Absorber which consists of a mass and two springs (spring 1 and spring 2) in parallel. The edges condition of a bar is fixed at one end and free at the other end. Based on the wave equation, the vibration of a bar will be discussed. To define the optimal conditions for the passive control of the bar, the cost function which is the ratio of the vibrational intensity of a bar without DVA to that of a bar with DVA will be discussed. Fig. 1 shows the theoretical model of a uniform bar which is subjected to a harmonically excited force F(t) at the free end and has a Dynamic Vibration Absorber (DVA) which is connected by two springs at two positions of x1 and x2 being distanced from the left end of the bar, respectively.

Fig. 1: Theoretical model of a vibrating bar with fixed and free ends F(t) (the external force), Dynamic Vibration Absorber (DVA)[mass (ma), spring stiffness (k1, k2), locations (x1, x2)], BAR properties [length (L), density (),

constant CROSS Section area (A), Young’s modulus (e)]

II. THE GOVERNING EQUATIONS

A uniform bar of a cross section area A, mass density and length L, which is fixed at the left end and free at the right end is subjected to a harmonically excited force F(t) at the free end and has a Dynamic Vibration Absorber (DVA) which is connected by two springs at two positions of x1 (spring 1) and x2 (spring 2) being distanced from the left end of the bar, respectively. The governing equations of a vibrating bar and a Dynamic Vibration Absorber (DVA) can be written as:

222111 ),()(),()(2),(2

2),(2

E xxtxutykxxtxutykt

txuA

x

txuA

(1)

22211121 ),(),()(2)(2

xxtxukxxtxuktykkt

tyma

(2)

where u(x,t) and y(t) represent the displacement of a bar and a DVA, respectively. In the above equations, the displacements of a bar and a DVA are given as the time harmonic function. Hence the displacements (u(x,t) and y(t)) are expressed into the terms of the time harmonic motion as,

Y and )(u),( titi ey(t)extxu (3)

where u(x) and Y mean the deflection of a bar and a DVA, respectively and is the circular frequency. By inserting Eq. (3) into Eq. (1) and Eq. (2), then suppressing the time term, Eq. (2) can be rearranged as,

uu2

2211 )()( 21

ak-m

xxxkxxxkY

(4)

Where k=k1+k2. Then by using Eq. (4), Eq. (1) can be rewritten as

2221112

22

2

2

)(u )(u)(u )(uxxxkxxxk

a

a

mkEAmx

xx

(5)

where the wave number = 2/E. The boundary conditions of a bar with fixed and free ends are given as ti

oLx

xef

xEu

uA and 0 0

(6)





where fo is the magnitude of the external force. The total solution of Eq. (6) can be expressed into the sum of a homogeneous solution (uh) and a particular solution (up) as

)(u)(u)(u xxx ph (7)

The homogeneous solution is determined by letting the right side of Eq. (5) be zero and then becomes as xCxCxh cossin)(u 21 (8)

where C1 and C2 are the unknown parameters. The particular solution can be solved by means of a method of variation of parameters and then be assumed as

xxVxxVxp cos)(sin)()(u 21

(9)

Where the coefficients V1 and V2 are can be determined by means of the method of variation of parameters and defined as follows

dfxVanddfxVxx

sin1)(cos1)( 21 (10)

., 2221112

2

),( ),( xxtxukxxtxuka

a

mkEAkmffunctionforcingthewhere

The final form of the particular solution can be obtained by using Eqs. (9) and (10). The complete solution of Eq. (7) can be written as

2221212111 cossin)(u CCxCCCxCx

(11)

where

2

2

a

a

mkEAm

,

2222111111 sinsinsinsin xxHxxxkxxHxxxkC , 221222112 sinsinsin xxHxxxxxkkC ,

2222111121 sincossincos xxHxxxkxxHxxxkC 221222122 sinsincos xxHxxxxxkkC , and

H(x) represents the Heaviside unit step. By using the boundary conditions given in Eq. (6), C2 becomes zero and the final solution of the dynamic response of a bar becomes as,

1211sin)( CCxDEN

fxu o (12)

where }]cossinsin

cossincossin{[cos

212121

222111

xLxxxkkxLxkxLxkLEADEN

Here Eq. (12) introduces the deflection of a bar with a DVA, which is caused by the harmonically excited force applied to the free end.

III. NUMERICAL RESULTS AND DISCUSSION All values obtained in this study have been expressed into the dimensionless forms. In Table 1, the dimensionless properties and the given values are introduced as

TABLE I - DIMENSIONLESS PROPERTIES AND THE GIVEN VALUES

Expression form Value A mass ratio m = ma/ L 0.1 ~ 0.3 External force ratio Fo=fo/EA 1 A structural damping 0.001 Two spring positions X1= x1/L , X2= x2/L X2 - X1= 0.1 Stiffness ratios Ks=k/(EA/L), K1=k1/k, K2=k2/k K1+K2 = 1





Here Ks means the ratio of the total stiffness (k1+k2) of a DVA to the elasticity of a bar. The summation (K1+K2) of two stiffness ratios and the external force ratio keep constant as a unit and also the span (X2-X1) of two spring ratios keeps constant as 0.1. The ratio of a mass of a DVA to that of a bar is assumed to be equal or less than 0.3. A. Resonance Frequency Coefficients( ) In Eq. (12), the denominator part vanishes for certain specific values of the wave number . The roots of denominator which are expressed in r by 2= 2/E are called the resonance frequencies for the bar with a DVA. So the resonance frequency equation becomes as,

0}cossinsin

cossincossin{cos)(

212121

222111

xLxxxkkxLxkxLxkLF

(13)

Here Eq. (13) can be rewritten in terms of the dimensionless form as,

0}1cossinsin

1cossin1cossin{cos)(

212121

222111

XXXXKKXXKXXKF

(14)

Where = L and sKm

m/1 2

.

Equation (14) is a transcendental equation in and its roots must be obtained numerically. For two stiffness ratios (Ks = 1 and 5) of mass ratio (m=0.1), the variations of the first resonance frequency coefficient ( ) versus stiffness ratio (K1) of a spring 1 for four position ratios (X1 = 0.1, 0.3, 0.5, 0.8) of spring 1 are plotted in Fig. 2 (A) and (B). In Figs. 2 (A) and (B), the first resonance frequency coefficient ( ) is found to be increasing slightly along the increasing stiffness ratio (K1) but getting lower as X1 goes to the higher position ratio. So it is noted that as the location of a DVA is close to the free end, the values of the first resonance frequency coefficient ( ) are decreasing gradually. The similar result was reported in the previous study [13]. In Fig. 3, the above mentioned notations might be proved. For two mass ratios (m = 0.1 and 0.3) and two stiffness ratios (Ks = 1 and 5) of position ratio (X1 = 0.5), the variations of the first resonance frequency coefficient ( ) versus stiffness ratio (K1) of a spring 1 for four cases are plotted in Fig. 3 (A). In case of the different stiffness with the same mass, the higher stiffness gives the higher values of . But in case of the different mass with the same stiffness, the higher mass gives the lower values of . Fig. 3 (B) shows the variations of the first resonance frequency coefficient ( ) versus position ratio (X1) for three cases of stiffness ratio (K1 = 0.2, 0.5, 0.8) with stiffness ratio (Ks = 5) and mass ratio (m=0.1). It is confirmed in Fig. 3 that the values of is decreasing along the further distance from the fixed end of a bar and the higher stiffness gives the higher values of .

0.0 0.2 0.4 0.6 0.8 1.0

1.2

1.3

1.4

1.5

1.6

1.7

1.8m=0.1, Ks=1

Freq

uenc

y co

effic

ient

(

1)

Stiffness ratio ( K1 )

X1=0.1 X1=0.3 X1=0.5 X1=0.8

0.0 0.2 0.4 0.6 0.8 1.01.2

1.3

1.4

1.5

1.6

1.7

1.8m=0.1, Ks=5

Fre

quency

coeff

icie

nt(

1)


X1=0.1 X1=0.3 X1=0.5 X1=0.8

(A) (B)

Fig. 2 Resonance frequency coefficient ( 1) vs. Stiffness ratio (K1) : (A) Ks = 1 and (B) Ks = 5





0.0 0.2 0.4 0.6 0.8 1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7X1=0.5

Fre

quency

coeff

icie

nt(

1)


Ks=5, m=0.1 Ks=1, m=0.1 Ks=5, m=0.3 Ks=1, m=0.3

0.0 0.2 0.4 0.6 0.8 1.01.40

1.42

1.44

1.46

1.48

1.50

1.52

1.54

1.56

1.58

1.60

Fre

quency c

oeff

icie

nt(

1)

m=0.1, Ks=5

Position of X1

K1=0.8 K1=0.5 K1=0.2

(A) (B)

Fig. 3 Resonance frequency coefficient ( 1) distribution (B) vs. Position of (X1) : m = 0.1 and Ks=5 (A) vs. Stiffness ratio (K1): X1 = 0.1

Reduction of Vibrational Intensity The Vibrational intensity (VI) is known as the time averaged power flow. So the vibrational intensity of a vibrating bar [15] is expressed as follows;

2/)()(Re *xu

xxuEAiVI (15)

Where ‘*’ is the conjugate of a complex number. In Eq. (15), the structural damping (= 0.001) is introduced to avoid being infinite at the resonance. The reduction level of the vibrational intensity (VI) is evaluated in comparison with VI of a bar without DVA and VI of a bar with DVA. The ratio of VI of a bar without DVA to VI of a bar with DVA is expressed in terms of decibel [dB] as follows;

DVA

noDVA

VIVI

LogVI][

][10 10[dB] of Reduction (16)

For the arbitrary value of the stiffness ratio (Ks = 5) ratio, Fig. 4 shows the reductions of the vibrational intensity (VI) of two mass ratios (m=0.1 and 0.3) versus the forcing frequency coefficient ( f). As shown in Fig. 4, the levels of the vibrational intensity (VI) are observed to be good or bad irregularly along the forcing frequency coefficient ( f). It means that a DVA with the arbitrary value of stiffness would be harmful to a main system which is subjected to any forcing frequency. In Fig. 5, the reductions of the vibrational intensity (VI) of two mass ratios (m=0.1 and 0.3) are plotted along the stiffness ratio (Ks) for the forcing frequency coefficient ( f=5). As shown in Fig. 5, the levels of the vibrational intensity (VI) are observed to be bad almost along the stiffness ratio (Ks) except zone around Ks=2.5 for m=0.1 and Ks=7.5 for m=0.3, respectively.

Fig. 4 Reduction of VI vs. forcing frequency coefficient ( f) Fig. 5 Reduction of VI vs. Stiffness ratio ( s) ; X1=0.3, K1=0.5, Ks=5, Solid (m=0.1), Dot (m=0.3) ; X1=0.3, K1=0.5, f =5, Solid (m=0.1), Dot (m=0.3)

2 4 6 8 10

-40

-20

20

40

60

80

100

dB

2 4 6 8 10 12 14

-50

-25

25

50

75

100

dB





According to the previous study [13]-[14], the optimal design condition of a DVA was confirmed such that the natural frequency of DVA should be set equal to the forcing frequency of a main system. So the optimal value of the stiffness ratio can be given as;

2

s foptimalmK (17)

For the given forcing frequency coefficient, the optimal stiffness ratio of a DVA can be determined by using Eq. (17). In Fig. 5, the most proper stiffness ratio for the good reduction zone of the forcing frequency coefficient ( f = 5) is found to become Ks=2.5 for m=0.1 and Ks=7.5 for m=0.3, which are equal to the optimal values of the stiffness ratio of Eq. (17). In Figs. 6 (A) and (B), the reductions of VI for four position ratios of spring 1 where X1 = 0.1, 0.3, 0.5, 0.8 are plotted versus stiffness ratio (K1) of a spring 1 for two mass ratios (m = 0.1 and 0.3) with f = 5 , respectively. As shown in Fig. 6, the reduction loci show the bilaterally symmetric curve for four position ratios. It is observed that the highest reduction occurs at the position ratio 0.3 and the lowest does at the position ratio 0.5 for both mass ratios. It is stated [13] that the position ratio might be close to nodes of the normal mode corresponding to the second natural frequency of the bar ( 2 = 4.71). By comparison with (A) and (B) in Fig. 6, the reductions of mass ratio 0.3 are to be greater than those of mass ratio 0.1. In Figs. 7 (A), the detailed variations of the reduction of VI along the position ratio (X1) are plotted for the cases where K1 =0.2, K1 =0.5 and K1 =0.8. As observed in Figs. 6 (A) and (B), the ratio where K1=0.6 provides the lower reduction than others and the symmetric ratios where K1 =0.2 and K1 =0.8 are observed to have the same reductions. In Fig. 7 (B), the reduction of VI for the cases where f = 1, f = 5 and f = 10 are plotted versus position ratio (X1). As goes to higher frequency, the fluctuation of the variation curve is getting sever.

0.0 0.2 0.4 0.6 0.8 1.0110

115

120

125

130

135

140

145

150

Red

uctio

n of

VI (

dB

)

Stiffness ratio ( K1)

m=0.1, f=5

X1=0.8 X1=0.5 X1=0.3 X1=0.1

0.0 0.2 0.4 0.6 0.8 1.0130

135

140

145

150

155

160

165

170

Red

uctio

n of

VI (

dB

)


m=0.3, f=5

X1=0.8 X1=0.5 X1=0.3 X1=0.1

(A) (B)

Fig. 6 Reduction of VI vs. Stiffness ratio (K1): f=5 (A) m=0.1, Ks = 2.5 and (B) m=0.3, Ks = 7.5

0.0 0.2 0.4 0.6 0.8 1.090

100

110

120

130

140

150m=0.1, f=5

Red

uctio

n of

VI (

dB

)

Position of X1

K1=0.2 K1=0.5 K1=0.8

0.0 0.2 0.4 0.6 0.8 1.060

70

80

90

100

110

120

130

140

150m=0.1, K1=0.5

Red

uctio

n of

VI (

dB

)

Position of X1

f=10 f=5 f=1

(A) (B)

Fig. 7 Reduction of VI vs. Position of (X1): (A) m = 0.1, f=5, Ks = 2.5 and (B) m = 0.1, K1=0.5





IV. CONCLUSIONS In this paper, the passive control of a vibrating bar with a Dynamic Vibration Absorber is studied to determine the optimal design parameters of a DVA. On the bases of the analyses in this study, the conclusions are obtained as follows, A Dynamic Vibration Absorber (DVA) is observed to have the satisfactory results for the reduction of the

vibrational intensity which is the time averaged power flow generated by the external force and motion. The location of a DVA which is defined as positions of two springs is confirmed to become the significant factor

for the control strategy. Compared to a single spring used in a DVA, the combination of two springs in parallel provides the variety of

control conditions. It is proved that the optimal stiffness of DVA which is determined by Eq. (17) is still conservative as the design

parameter of a DVA. ACKNOWLEDGMENT

This paper was supported by Research Fund, Kumoh National Institute of Technology.

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