The optimal design of a dynamic absorber for an arbitrary planar structure

Applied Acoustics 23 (1988) 85-98

The Optimal I ~ . g n of a Dy!mic" Absorber for an Arbitrary P k m r Structure Y. Z. Wang and K. S. Wang Department of MechanicalEngineering,National Central University, Chungli (Taiwan) (Received 15 January 1987; revised version received30 April 1987; accepted 11 May 1987)

SUMMARY The optimal absorber design for a randomly excited planar structure is presented in this paper. The Lagrange equations are derived with generalized coordinates. The transfer functions in closed form are obtained thereafter. Based on the band-limited whitenoise excitation, the optimum tuning and damping ratios of the absorber are determined by minimizing the variance of the response of the structure. For simplicity, square-shaped, plane-frame structures are studied in numerical examples. The effects on the design parameters of the mass ratio of the absorber to the structure, the position of the absorber, and the position of the applied force are studied extensively.

NOTATION

F, Fo H(o,)

external harmonic excitation and its amplitude, respectively transfer function L/h length to width ratio of the frame elements M^,M, mass of the absorber and frame structure, respectively SF(O~),S~(O~) power spectral densities of excitation and response, respectively displacement and velocity of absorber u,0 amplitude of U uo displacement of frame at point (x~, y j) damping factor of absorber cA 85 AppliedAcoustics 0003-682X/88/$03.50 © ElsevierAppliedSciencePublishersLtd, England• 1988. Printed in Great Britain

86 KA

q, Era]

(Ua)r

(UF)r (Uj), (xA,YA) (XF,YF) (xj, yj)

~A, mA [~, (.0 ~'~, 03i (-0u ~A, ~S (00, ~0

Y. Z. Wang, K. S. Wang

spring stiffness of absorber rth natural coordinate of frame in vibration mass matrix of frame column matrix of normalized amplitude ratio corresponding to the rth mode normalized amplitude ratio of point (x^,yA) normalized amplitude ratio of point (xF,YF) normalized amplitude ratio of point (xj, yj) coordinates where absorber is attached to frame coordinates where external loading acts on frame coordinates of the point at which we desired to minimize the response dimensional and dimensionless undamped natural frequencies of absorber, respectively dimensional and dimensionless frequencies of excitations, respectively ith dimensional and dimensionless natural frequencies of the frame, respectively the upper-bound frequency of the band-limited spectral density damping ratio and material damping ratio of absorber and frame, respectively optimum values of c02 and ~2, respectively mass ratio of absorber to frame variance of frame response at point W(x~,yj) INTRODUCTION

The application of a dynamic absorber to machinery or a structure has been studied by many investigators over the past 50 years. For a single degree-offreedom system either without or with primary damping, the optimal absorbers were designed by Den Hartog i and Ioi and Ik~a, 2 respectively. The use of the dynamic absorber in reducing the vibration ofstructures was studied by Warburton and Ayorinde 3 and Van de Vegte. 4 The design of an optimal random vibration absorber for a single degree-of.freedom system was introduced by Jacquot: The optimum parameters of an absorber are obtained by minimizing the variance of the response over the frequency range. Furthermore, the use of the dynamic absorber in a randomly excited structure was also studied by Jacquot, 6 Ayotinde and Warburton 7 and CampbeU. s In the latter c a s e s 7.s the structure considered was a single degree-of-freedom system. The optimum design of a dynamic absorber for a

The optimal design of a dynamic absorber

87

randomly excited machine mounted on a plate-like structure foundation has been studied by the authors. 9 An effective method is suggested for obtaining the optimum parameters of the absorber, as the transfer function of the multi-degree-of-freedomsystem is obtained in closed form. In this paper, the optimum design of a dynamic absorber for a randomly excited Plane structure is studied. The structure is excited at a point 'F' by a random force and the absorber is attached at another point 'A'. The design of the optimum absorber is based on the procedure of minimizing the variance at some point 'J'. In certain of the numerical examples in this study, these points are considered to be coincident. When the natural frequencies and mode shapes of the structure are given (no matter whether numerically decided or experimentally determined), the governing equations of the whole system, i.e. the structure with dynamic absorber, can be derived via the Lagrangian approach. In this work, the plane-frame structures are analyzed numerically to obtain the natural frequencies. The influences of primary damping and the position of the absorber on the design parameters (e.g. tuning ratio and damping ratio of absorber) are also studied. Furthermore, the present approach can be extended to the case of 3-D machinery structure. Two additional assumptions are made: (1) that the material damping ratio ~s of the structure is small; and (2) that only the transverse motion of the structure is considered.

THEORY FOR HARMONIC RESPONSE Consider a plane structure with mass M,, as shown in Fig. 1, which is exci.ted by a harmonic force F0 e~'~tat the point (xF,y~). An absorber with mass MA is attached to the structure at point (x,,y,). The motion of the absorber is constrained in the direction perpendicular to the plane ofthe structure. For a small material damping ratio ~,, the modes ofthe structure may be evaluated

W.

{xj.y|)

U

T (z~.ya)

(xF'YF) W

F~. 1. Random force applied to planar structure with dynamic absorber.

88

Y. Z. Wang, K. S. Wang

as if/;, were absent. In the present work, the natural frequencies and mode shapes of the planar structure are obtained numerically. The structure of mass Ms is divided into n dements, and the corresponding transverse displacement W of these elements can be written as n

{w} =

{u},q,

(1)

where {W}=

W2

and

{u},=

r = 1 , 2 .... .n

tu.J, where {u}, is the amplitude ratio of elements corresponding to the rth mode, and q, is the coordinate of the rth mode. For this case themodes should be orthogonal to each other, and {u}, is defined as

{u}T,[m]{u}, = Ms

r = 1, 2,..., n

(2)

where [m] is the mass matrix of the planar structure, and {u}T is called the normalized amplitude ratio. Using eqn (2), the total kinetic energy E of the system including the absorber is E = ½~', M.g,2 + ½M^0 2

(3)

r~l

where/.J is the velocity of the absorber. The total strain energy V of this system is 3 n

n

V = ½ M . ~ (~)2 q2 + ½XA[ U-- ~ (U,),.q,;2 r--1

(4)

r~l

where t'~ is the rth natural frequency of the structure foundation, s:Ais the stiffness of the spring, and (u,), is the normalized amplitude ratio at the point (xA,y^). When the effect of material damping is considered, the stiffness x of the structure foundation becomes complex and may be written as x (1 + i~,), where i = ~ and ~, is the materialdamping ratio of the structure. Hence eqn (4) becomes n

n

V= ½M,Z (1 + i ~ . X D ~ ) Z q Z , + ½ x , [ U - ~ ( u . ) , q , ] r----1

r=l

2

(5)

The optimal design of a dynamic absorber

89

and the dissipation function F is defined as n

2

r=l

Applying the L a g r ~ g e equation with respect to the coordinate q,, we have n

M.~,+ M.(I + i~,D~)2q,+ tc^(u.),[X (u.),q,- U] rffil n

+C,,(u,),[~(u,),6-O]=Fo(u,,),e'* (7)

r=l

where (uF), is the normalized amplitude ratio at the point (XF, YF)" For the absorber coordinate U, we have n

n

r:l

r:l

Introducing the following parameters: (q,, U) = (q,o, Go) e'nt ~2 = Ic~/M^ 2~^~^ = CA~M^ (co^, co, co*) = (t~A, t~, ~ ) / f ~ * St = M ~ / M , (q,o, 00) = (q,o, Uo)/h T= ~*t Fo = Fo/M, hn .2 the dimensionless forms of governing eqns (7) and (8) become (for convenience the overbars are dropped) [co*2(l + i~,) - co2]qvo +/~(Ua) v (9) r----I

and

(CO,~-COz+ i2~AO~,OUo = (co,~+ i2co^co~A~ (U,):/,o]

(10)

Substituting eqn (10) into (9), we have q,o = [#(u,),CO2Uo "4"F0(UF),][CO*z(I4-i~J --coZl- I

(11)

90

Y. Z. Wang, K. S. Wang

when q,o is substituted into eqn (10), Uo is obtained as n

U o = {(o~2 + i2co^co(^)Z F

(UF)'(U')~

+ i2C~ACO~A)

,=1 n

_

~ c0.2~( u +---~)-v),(u,), ~c~2(co2 + i,~c -- ° ^ c°rsA,, [L/, c02]}-

'

(12)

r=l

Then q,o becomes n

q,o = /~2(u,),(°~ + i2co^co~^)

co'2(1 + i~,) - to 2. Fo ,----1

x t[c0"2(1 + i~) - c02](c0~ - co2 + i2C0ACO¢^)-- #C02(CO~+ i2CO^C0¢A)

[2 n

×

6~'2(1 + i~,) - co2

]}_1

+ {(UF),Fo/[O~*2(1 + i~,) -- co2]}

(13)

,---1

The displacement of point j is n

W~(xj,y~) -- >', (uAq,

(14)

r--I

where (u~), is the amplitude ratio of point j of the rth mode. The transfer function of the input F and response W~ is defined as H(o,) = W / F

(15)

From eqns (13), (14) and (15),//(to) is obtained as //(co) =

~ / O J 2 ( f D i "{- i2coAtOG,)[l][2] + [3] {(co2 - co2 + iEto^to(^) - gto2(to 2 + i2C0ACO(A)[4]}

where

(16)

n

[1] = L

(u.),(u A

6~72('~'~s) -- (D2

(17a)

(u.),(u~), to 2

(17b)

(Uj),(UF), co.25~co2

(17C)

e=l n

[2] =

~ ----7-2-7~ , co,, 2 (1 + 1~,) -

r=l /I

[31= e--1

The opt~al design of a dynamic absorber

91

and n

[-4] ----

(u.)y tot*2(1+ i~.) -- to2

(17d)

r:=l

OPTIMUM DESIGN PROCEDURE FOR RANDOM RESPONSE For the stationary input-output problem the power spectral density between the response and excitation is Sw(to)-- Sv(to)]H(to)[ 2

(18)

If the forcing function has zero mean, then the mean squared response is just the variance of the response. This can be obtained by integrating the power spectral density function, i.e.

= f = &4to)ln(to)12 dto

(19)

C 0 m i d e t i a g the random excitation force as a band-limited white noise with spectral density SF(to) as Sdto) = So = 0

for 0 < to _< tou otherwise

(20)

the mean squared response of eqn (20) becomes trw__2 __

S0lH(to)[ 2 dto

(21)

where Hl(to)l 2 can be obtained from eqn (16) as iH(to)12 = (QR 1)2 + (QI 1)2 (QR 2) 2 + ( Q / 2 ) 2

(22)

where

QR 1 = Q R A + Q R B QR A =/~to2{(to2[1R][2R] - [1/][2/]) - 2toAtO~A([1/][2R] + [1R][2/])} QR • -- {(ta2 - to2)[3R] -/aa2(to2[4R] [3R] - 2toAOKAE4/'J[3R])} -- {2to^to~Al'3/] --/~to2(toi[4/][3/] + 2tOAtO(A[4R][3/])} QII=QIA+QIB

"

QIA = ~to2{to,~([1/][2R] + [1R][2/]) + 2to^to~^([IR][2R] - [1/][2/])} Q I B = {(to2 _ 0 2 ) [ - 3 / ] _~/aoatto2[4R][3/] _ 2toAto~A[4/][3/])} + {2toAto[A[3R] --/ao2(toi[4/][3R] + 2toAto~A[4R][-3R])}

QR 2 = {(to2 _ to2) _ #to2(to2[4R] _ 2to^to(A'[4/])}

Y. Z. Wang, K. S. Wang

92

and

QI2= {2WAta~'~,--/~w2(coIE4/] + 2coaco(,[4R])} In the above expressions, [XR] = real part of [X] and [X/-J = imaginary part of IX'J, with l-l], [2], [3] and [4"] as shown in eqn (17). The optimum tuning ratio coo and damping ratio ~o of the dynamic absorber are determined by minimizing the variance, i.e. 2o'2/2coA = 0

(23a)

2a~/2~A = 0

(23b)

and

Since coA and ~A are independent of co, the partial differentiations can be carried out before the integral operation of o-~. After n e c e ~ rearrangement, the final forms of eqns (23a) and (23b) become

j.o= Q Q 2 [ Q R 1

c~QR1 . . . . aQ117. ~ QQ2

I [ Q R 2 0 Q R 2 ....

oQI2~

=0

(24a)

and

QQ2IQR OQR1

fo"

OQI11

IIQR_OQR2

___c~QI2"]

QQ2 =0

(24b)

where

QQ 1 = (QR 1)2 + (Q/l) 2 and

QQ 2 = (QR 2) 2 + (QI2) 2 N U M E R I C A L RESULTS AND DISCUSSIONS Square-shaped plane-frame structures are studied in this work. The crosssection of the frame elements are square with a l ~ - t o - w i d t h (slenderness) ratio L/h--30. The method of steepest descent is used for finding the optimum parameters coo and ~o. For convenience, ~= is chosen to be 10. It is slightly larger than the fifth natural frequency of the frame structure. The

The optimal design of a dym~mic absorber

//~

~J (a)

93

h " (b)

A2

(c) (d) Fig. 7. (a) Plane-frame structure A--A, F and Wj at the same point; (b) plane-frame structureA--A, Fand Wj at differentpoints;(c)plane-frame structureB--A, Fand Wj at the same point; (d) plane-frame structure B--A, F and Wj at differentpoints.

numerical studies show that the contribution of higher modes in the response is less than 1% for white noise excitation. The frame structures studied in this section are shown in Fig. 2(a)-(d). The capital letters F, A and J represent the positions of the random force, absorber, and the place at which the variance is designed to be minimized, respectively. The optimum parameters O~o,~o, and the relevant response variance ~ of the example in Fig. 2(a) are shown in Fig. 3. In this case, the points F, A and J are taken to be at the geometric center of the central frame element. Figure 3(a) shows the variation of~oo with increasing/~ for several ~,. It is found that ~oo decreases with respect to #, in general, but the change is greater for small ~s. The optimum damping ratio ~o increases with increasing ~t (see Fig. 3(b)). The variations of ~o are little sensitive to the choice of ~,, especially for /t > 0.1. Figure 3(c) shows that the differences of variance among these curves become small as/t becomes large.

94

Y. Z, Wang, K. $. Wang

O.6 ---.-

0.4 0

Ci ,G01 r.I :0.03 ih :0.0S

I 0.1

I G2

/,

I

(~3

(a) 300

06

----.-

"~

Co

Cl : 0 0 ! Cl sO.03 ¢1 : O0 S

---

-'I 0.1

1:1 -'001 C) *GO3 ~:) :O-OS

I 02

(b)

I 0.3

0.4

0

I 0.1

I 0.2

I 0.3

0.~

(c)

Fig. 3. The relations between the optimum parameters and/~ for various {~ (L/h = 30, structure A).

The optimum parameters o~o, ~o and its relevant response variance of the example shown in Fig. 2(b) are plotted in Fig. 4, wherein the absorberis attached at either the point A, C or B, as indicated in the figures. Due to the symmetry of the frame structure, the data for A and C coincide. Figure 4{a) shows that o~o increasesl(decreases)with respect to/~ for case B (A or C). The changes of ~o with increasing/z are shown in Fig. 4(b). Figure 4(c) gives the

The optima/design of a dynamic absorber

95

!.2

f w.

oe

B

03 ---

A and C

i 0;I

l 0.2

i 0,3

0.4

(a) !00C

1.0 ~ B --A ond C

~ 9

---

A and C

01

aoc

60C

4~ 04

/

20C

02

I

I

I

0.1

O2

O3

(b) Fig. 4.

I

04

0

0.1

i

0~2

O)

G4

v (c)

The relations between the optimum parameters and # for various response points (~, = 0-01, L/h = 30).

variance between these cases, and the results are quite different. They indicate that the choice of the absorber attached at point Bis the best. Figure 5 shows the optimum parameters coo, ~o, and the corresponding variance CT 2 of the frame shown in Fig. 2(c). The trends ofthese figures are similar to those given in Fig. 3. Due to the larger stiffness of the frame of Fig. 2(c) as compared with that of Fig. 2(a), the variance is much .smaller for this case.

96

¥. Z. Wang,K. $. Wang I.©

OI

OI ---'-

OJ. 0

Cl *001 Cl :003 Cl ,O.OS

l

I

I

0.1

02

0.3

O.t

# (a)

O4 ~1 '0.01 - * - C1 ' 0.0) --'--C 1,0.OS

C,

10¢

02

r//

,:, .ool

~'

---

SC

I:1 ,0.03

- - ' - - ~1 IO.OS

I

I

I

0.1

02

0.3

0~,

0

I

I

I

0.I

G2

0.3

(c) S. The relations between the optimum parameters and ~ for various ~. (Lib structure B).

0.4

(b)

= 30,

Figure 6 shows the optimum parameters of theframe shown in Fig. 2(d). In this case the absorber is attached at either the points A~,I A 2 or A3, respectively. Point A2 is chosen to be at the middle between A~ and A3. The optimal tuning eoo and damping ratio ~o are plotted in Figs 6(a) and (b), respectively. The relations between variance ~rZrand mass ratio/~ are shown in Fig. 6(c). It shows that the nearer the absorber position to the geometric

The optimal design of a dynamic absorber

97

1.0

O)o 0.8

" " " ~•' ~ '~" ~'~" ~~*~.. ~'~'~...~

O6

AI ---A2 ~--A3 ~

~0

!

|

!

0.1

0.2

(~3

04

(a) m

--^1

02

0

-''

A2

--.-

A3

~

60

.~'~ 40

•

%%

20

!

I

0.1

~2

! 0.3

04

0

!

I

!

0.1

0.2

&3

~4

Y (c)

r (b) Fig, 6.

AI

__. A 2 --.-- A 3

The relations between the optimum parameters and # for various mounting positions of the absorber (~s ffi 0-01, L/h = 30).

center of frame, the smaller the variance is. Since the velocity response of the frame is larger at the geometric center, the absorber dissipates the energy of the system more efficiently when it is attached at this point. CONCLUSIONS The optimum design of a dynamic absorber for minimizing the random response variance at a prescribed position on a planar structure has been

98

Y. Z. Wang, K. S. Wang

studied in this paper. The effects on the design parameters of mass ratio/~, material damping ~, and the position of absorber are discussed. Several conclusions are made as follows: (1) To have a small variance, it is better to keep the mass ratio/~ _>0.1 for a stiff frame and/~_> 0.3 for a soft frame (see Figs 3(c) and 6(c)). (2) For the sake of a small variance, it is better to have the absorber attached to the frame at the softest part; in our examples, the geometric center is the best choice of position of absorber. (3) The increase in frame stiffness leads to a decrease in the response of the variance.

REFERENCES 1. J. P. Den Hartog, Mechanical vibration, 4th edn, McGraw-Hill, New York, 1967. 2. T. Ioi and K. Ikeda, On the dynamic damped absorber of the vibration system, Bull. Japan. Soc. Mech. Engrs., 21 (1978), 64. 3. G. B. Warburton and E. O. Ayorinde, Optimal absorber parameters for simple system, Earthq. Engng Struct. Dynam., 8 (1987), p. 197. 4. J. Van de Vegte, Optimal dynamic absorbers for the plate vibration control, J. Dynam. Systems, Meas. Control, 97 (1975), p. 432. 5. R. G. Jacquot and D. L. Hoppe, Optima! random vibration absorber, J. Engng Mech. Div., Am. Soc. Mech. Engrs, 99 (1973), p. 612. 6. R.G. Jacquot, Optimal dynamic vibration absorbers for general beam systems, J. Sound Fib., 60 (1978), p. 535. 7. E. O. Ayorinde and G. B. Warburton, Minimizing structural vibrations with absorbers, Earthq. Engng Struct. Dynam., $ (1980), p. 219. 8. P. H. Wirsching and G. W. Campbell, Minimal structural response under random excitation using the vibration absorber, Earthq. Engng Struct. Dynam., 2 (1974), p. 303. 9. K. S. Wang, Y. Z. Wang and R. T. Wang, Optimum design of dynamic absorber for random-excited machine mounted on a platelike structure foundation, Int. J. Mech. Sci., 29 (1985), p. 335.