Journal of Sound and Vibration (1988) 124(2), 381-384

L E T F E R S T O T H E E D I T O R

VOIGT DYNAMIC VIBRATION ABSORBER

1. i NTRODUCTION

Many studies involving dynamic vibration absorbers are available in the literature. Den Hartog [1] has given the classical analysis of undamped and damped vibration absorbers. Snowdon [2] has shown that a Voigt vibration absorber has increased effectiveness compared to the conventional absorber. Srinivasan [3] has given an analysis of different combination of absorbers. In the present study the response of a Voigt dynamic absorber is obtained. It is shown that in the limiting cases of the damping ratio, r/-o 0 and 7/~ oo, the results reduce to those of standard two degree of freedom systems.

2. ANALYSIS

Consider the system shown in Figure 1. The main system consists of the mass M and spring of stiffness K. The parameters of the Voigt dynamic vibration absorber are the spring stiffnesses k and kl, the dashpot coefficient c and the mass m (a list ofnomenclature is given in the Appendix).

The equations of motion of the system are

M~ + Kx + k ( x - x2) + k , ( x - x ,) = P cos cat,

m d ~ 2 + k ( x 2 - x ) + c ( : G - ~ t ) =0, c ( ~ t - Y c 2 ) + k l ( x , - x ) = O , (1)

Assuming solutions as

x = Z e ~'~ x , = Z~ e i~'', x2 = Z~ ei~ ', (2)

one obtains -Mca2Z + ( K + k + k i ) Z - k lZl - kZ2 = P,

- n t c a 2 Z 2 - k Z - i c c a Z l + ( k + i c c a ) Z 2 = O , i c c a ( Z t - Z 2 ) + k t Z t - k l Z = O . (3)

Equations (3) can be written as {z} [r(ca)] Z, = , (4)

Z~

7 =~'-- M Ipcos~t

Figure 1. Voigt dynamic vibration absorber with undamped main mass.

381 0022-460X/88/140381 +04 $03.00/0 O 1988 Academic Press Limited

382

where

L E T T E R S T O T H E E D I T O R

" - M a , 2 + K + k + k l - k l

[ T(to)] = - k - i ca ,

- k~ icto + k~

Premult iplying equat ion (4) by [T( to) ] -~ gives

Iil Z, = [T( to) ] -1 .

Z2

_k j - m t o 2 + k +icto .

- i c e ,

(5)

After calculat ing [ T( to)] -~, one can insert it into equat ion (5) and solve for the response Z, to get

Z = E + iD, (6)

where

E = [ P / ( A 2 + B2)][A(moj2k l - k k l ) + B(cmoJ s - ctoki - cwk)] ,

D = [ P / ( A 2 + B2)][A(cmto 3 - c w k - c w k ~ ) - B (mto2k l - kkl)] ,

A = k~k[ t t , v 2 + ( v 2 - 1 ) (/.t 2 - p/x, v2)] = k 2 k A ',

B = k ~ k [ 2 r l v ( p q v : - I z , - 1 ) ( t t 2 - p . , p v 2) + 2 r l ( k / k , ) v 3 ( l x ~ + 1)]

= k 2 k B '

p2 = k / m , #2 = K / k l , v 2 = w / p , c = 2mprl,

tzl = k / k ~ , X~, = P / K , p = M / m . (7)

F rom the first and second o f equat ions (7) one obtains

E #2 r A , t v 2 1 ) + 2 B , r l v ( i . t l v 2 _ l . t l _ i ) ] , - X~, - An"-fiB'2 i. , -

D - D Ix2 Xs, - A '2 + B'2 [2A'r/v(p.~ v 2 -/.t~ - 1 ) - B ' ( v 2 - 1)]. (8)

F rom equat ions (6) and (8) one obtains

[Z /X~ , I = H ( w ) = x/(/~2 + E3 2) and r = tan - I ( /9 / /~) , (9)

which give the ampl i tude and phase angle of the response Z. Case (i). When r/-~ 0, the isolator spr ing is ineffective and the system reduces to the

one shown in Figure 2. Putt ing r / = 0 in equat ion (7) results in B - - 0 . From equat ion (8)

"///.,6

K~

r d - k!

Y/ / / / .

r'~ LPcos~Jt

Figure 2. Reduction of Voigt absorber (r/= 0).

LETrERS TO TIlE EDITOR

/ 5 = 0 and /~ =/z2(v 2 - 1)/A', so that

H(~o) =/_t2(v z - 1)/[tztv2+(v 2 - 1)(/~2- p/z, v2)],

r = t a n - t (0) = 0,

383

(10)

which is the response for the reduced two degree of freedom system shown in Figure 2. Case (ii). When 7/--> eo the system reduces as shown in Figure 3. Dividing by r/ both

numerator and denominator in the first of equations (8) gives

#2(/,tl P 2 - - ~ l - - 1)

E , - ( m 1' ~ - ~ , - 1 ) ( ~ - m p ~ ~) + ~ , , ' ~ ( m + I ) '

and a l so /5 = O, so that

(11)

H(to) =/~1 and ~b = 0. (12)

It can be seen that equations (11) and (12) give the standard response of the two degree of freedom system shown in Figure 3.

r/././A "/I/A

Ki

Figure 3. Reduction of Voigt absorber (77 ~ oo).

3. CONCLUSIONS

The analysis of a" Voigt dynamic vibration absorber has been carried out and the response obtained. In the two limiting cases when the damping parameter r/--> 0 and 77 --> co, it has been shown that the result reduces to the standard response of a two degree of freedom system.

Department of Mechanical Engineering Indian Institute of Technology, Powai, Bombay, 400076 India.

(Received 23 November 1987)

U. V. TAMBE G. CHANDRASEKHARAPPA

H. R. SRIRANGARAJAN

REFERENCES

1. J. P. DEN HARTOG 1956 Mechanical Vibrations. New York: McGraw-Hil l . 2. J. C. SNOWDON 1974 Journal of EngiJeering for Industry 96, 940-945. Dynamic vibration

absorbers that have increased effectiveness. 3. A. V. SRINIVASAN 1969 Journal of Engineering for Industry 91, 282-287. Analysis of parallel

damped dynamic vibration absorbers.

384 LETTERS TO THE EDITOR

APPENDIX: NOMENCLATURE

M ~ m K , k , k , X, X l , X 2

5i t P C

z , z , , z2 i p2

to

i, 2

I.t, tz2 rl x~,

P 6 HCto)

main and absorber mass spring stifinesses displacements d 2 x / d t 2

time force amplitude . coefficient of damping complex quantities =4-zi = k / m forcing frequency =to /p = k / k ~ = K / k t = c /2mp = P / K = M / m , mass ratio phase angle of Z amplitude of Z