Steady-State Response of a Multidegree System With an Impact Damper

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Steady-State Response of a Multidegree System With an Impact Damper

Sami Masri

To cite this version:

Sami Masri. Steady-State Response of a Multidegree System With an Impact Damper. Journal

of Applied Mechanics, American Society of Mechanical Engineers, 1973, �10.1115/1.3422910�. �hal-

01511958�

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S. F. MASRI

Associate Professor, Department of Civil Engineering, University of Southern California, Los Angeles, Calif.

Steady-State Response of a lultidegree System With an Impact Damper

An exact solution is presented for the steady-state motion of a sinusoidally excited n- degree-of-freedom system that is provided with an impact damper. Both the excitation and the damper may be independently applied to any point in the system. Experi- mental studies with an analog computer and with a mechanical model corroborate the theoretical results. Results of the analysis are applied to the lumped parameter repre- sentation of a modern 10 story building, and the effects of various system parameters, including mode shape, excitation frequency, damper location, and force location are determined. It is found that the impact damper is an efficient device for reducing the vibrations of mullidegree-of-freedom systems, particularly in structures such as tall

R.

IECENT studies [1, 2 ]1 have shown t h a t chain-type impact dampers offer a simple and reliable method for attenuat- ing wind-induced vibrations of tall flexible structures such as launch vehicles, antennas, smoke stacks, etc. However, although impact dampers have been investigated both analytically and experimentally, no analytical results are available in the literature regarding the response of multidegree-of-freedom systems utilizing impact dampers. I t will be shown in this paper t h a t the use of impact vibration dampers is feasible to suppress earthquake-induced or wind-induced oscillations of tall buildings.

This paper presents the exact solution for the steady-state motion of a series-type multidegree-of-freedom system equipped with an impact damper and subjected to a sinusoidal force input.

T h e lumped parameter model of the n-degree-of-freedom system being considered is shown in Fig. 1. T h e sinusoidal forcing function is applied to the fcth mass and the impact damper is attached to the jth mass. I t is assumed that, during a period of the forcing function, two symmetric impacts occur at equal time intervals and at opposite ends of the damper container. This assumed

1 Numbers in brackets designate References at end of paper.

Contributed by the Applied Mechanics Division and presented at the Applied Mechanics Summer Conference, University of Cali- fornia, La Jolla, Calif., June 26-28, 1972, of T H E AMERICAN SOCIETY OP MECHANICAL ENGINEERS.

Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N. Y. 10017, and will be accepted until April 20, 1973. Discussion received after this date will be returned. Manu- script received by ASME Applied Mechanics Division, September,

1971; final revision, January, 1972. Paper No. 72-APM-39.

motion is consistent with t h a t which has been found to pre- dominate in the experimental studies of impact dampers as ob- served by most investigators in this field.

Steady-State Solution

T h e equation of motion of the system in the absence of the damper is

[m]x + [c]x + Wx = F(0 (1) where [m], [c], [k] are the mass, damping, and stiffness matrices,

respectively, and F(<) = col {0, 0, . . . , 0, Fh(t), 0, . . ., 0}.

Assume t h a t the damping mechanism is the proportional-type satisfying the condition

[c] = a[m] + 0[k] (2)

where a and /3 are constants.

Consider the steady-state motion of the system shown in Pig. 1 with the origin of the time axis shifted to coincide with the time of occurrence of an impact at t = to. T h e result of this shift is to modify the excitation force, giving in the new time scale

Fh(t) = F0 sin (Qt + ao) (3) where a0 = ilto is a phase angle to be determined from the steady- state motion.

Using the normal mode approach, equation (1) can be trans- formed into the form

l^M<\q + [^xC \ ] q + [ ^ \ ] q = Q « M (4) where the n-components of q are the normal coordinates, [xiW\], [XC \ ] , and [X.K\] are diagonal matrices corresponding to the

Journal of Applied Mechanics

M A R C H 1 9 7 3 / 127

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Fig. 1 Model of system

generalized mass, damping, stiffness matrices, respectively, and where Qes ( 0 = [<p]TF(t), a nd [p] is the modal matrix.

T h e i t h equation of system (4) is

M&i + C4i + K,Qi = Qex,» = <PkiFo sin (tit + a0) (5) and its solution is

?,« = exp(- — tit)\- UiSiny-Slt

+ 77; cos —'- tit ) qt>i H I sin —- tit ) q^ai Ai ( . Vi o , . i)i n \ •

~ — I f • s l n — tit + Vi c o s — tit I sin Ti Vi \ ri '•» /

— — rv I sin —- tit) cos Tt-} + A< sin (tit +

1i V r( I J

i = l , 2 , . . . , n (6) where

- V f ;

Vi = Vi - $-,-2

fi = VuFo

2£V,-

^ == t a n "¹ — ~^{L ±}- 1 - r,-»

?oi = ?i(0)

f« =

^C, ti

At = fi/K<

V(i

(1 - rW + (2f<r⁴)»

g«i = g,(0+)

and the + subscript indicates conditions immediately after the

specified time. Letting the initial displacement and velocity at t = 0+ be

x(0) = xo = [<p]qo, x(0+) = x„ = [<p]qa (7) then

x(0 = [Bn(.t))ka + [B,s(*)]xo + [B2 3«]Si

+ [S24(01S2 + M S „ ( 0 (8) k(t) = [B31(*)]X. + [B32(«)]xo+ [B„a(0]Si

+ [iJ34(*)]S2 + [^]S4(«) (9) where the undefined matrices and vectors are functions of the system parameters.

Letting y{t) be the relative displacement of the damper mass md with respect to mjt then

y{t) = z(t) - XjOL). (10) Since the time origin coincides with the time of occurrence of an impact,

2/(0)^=2/0 = z(0) - xj(0) = ± - (11) During an impact (idealized to be a discontinuous process), the conditions of the system remain unchanged except for the velocities of md and nij, whose instantaneous changes are calcu- lated using the m o m e n t u m equation and the coefficient of restitution. Noting t h a t in steady-state motion

2ti ,

z(0)+ = (xoy + ?/o)

7T (12)

then the velocity vectors before and after an impact are related by

itb = [s.Bes]x0 (13)

-Nomenclature"

rrit = i t h mass in multidegree-of-free- dom system

hi — ith spring constant in multi- degree-of-freedom system a, fi = constants used to specify pro-

portional-type damping Xj(t) = absolute displacement of ith

mass

Xj(t) = absolute velocity of ith mass z(t) = absolute displacement of damper

md

y(t) = relative displacement of damper with respect to mass m,j md = damper mass

Ft, = amplitude of sinusoidol exciting force Fo sin tit

ti = frequency of sinusoidol excita- tion Fo sin tit

#o = phase angle, initially unknown CO; = i t h natural frequency of multi-

degree-of-freedom system f t = ratio of critical damping corre-

sponding to i t h mode

fiT = mass ratio of damper to total n mass of structure, md/ 2_i mi

» = 1

H = mass ratio of damper to mass rtij to which it is attached, mjnij

e = coefficient of restitution

d = clearance in which damper is free to oscillate

j = mass number to which damper is attached

k = mass number to which force is applied

n = total number of masses in n- degree-of-freedom system Xst; = static deflection of i t h mass in an

n-degree-of-freedom system when force is applied to mass mk

Xpi = peak displacement amplitude of mass m,- in the absence of damper

£mox; = maximum displacement amplitude of mass mv with damper operating

128 / MARCH 1 9 7 3

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n = 4 9

(a)

(W

n=7

m o o o o o o

\\V\\\\N

n=IO

• <>

o o

<>

o o o

<>

(cl i i1-' *}Sv\\\\\\\

-m«-8.6—

10

9 8 7 6 5 4 3 Z 1

(a =

lb -s

— x i o in

3.74 7.55 10.75 14.20 16.90 20.40 22.80 27.00 38.50 47.25

lb-sec*x,5»

in

™i 6.73 5.05 5.05 5.05 5.05 5.0 5 5.05 5.05 6.30 5.90 0 , £=0.00188)

— H 3 I—

(d) O ^^^.

Fig. 2 Example structure: (a) four-degree-of-freedom system; (b) seven-degree-of-freedom system; (c) 10-degree-of-freedom system;

(d) first three modes of 10-degree-of-freedom system

where [^.BON] is a constant diagonal matrix whose elements are equal to unity, except for the jth element which is equal to (1 - e - 2/*)/(l - e - 2ne).

I n steady-state motion with two symmetric impacts/cycle of excitation on opposite ends of damper container,

x(«)|«=»- = - x ( 0 ) = - x o (14) x(«)|w=^_ = - x ( 0 ) _ = - x , = - [ \ B6s ] x „ (15) Using (14) and (15) in conjunction with (8) and (9)

xo = S7 sin a0 + Sg cos a0 (16)

ka = S9 sin a0 + Sw cos ao (17)

Equations (16) and (17), together with (13), result in

a^0± = t a n "¹ [(hh, ± kJi^/ihzh, =F hju)] (18) where the h's are functions of S7 — Sio. With a0 determined from

(18), the rest of the unknowns can be found b y back substitution.

Applications

Consider the 10-degree-of-freedom lumped parameter system shown in Fig. 2(c) whose natural frequencies, mode shapes, and damping characteristics approximate those obtained experimentally from full-scale dynamic tests of a modern 10-story building [3].

T o investigate the effects of various system parameters, the

models shown in Figs. 2(a) and (6) were also studied. T h e hypothetical systems in Figs. 2(a) and (6) consist of t h e first 4 and first 7 stories, respectively, of t h e system in Fig. 2(c). Fig.

2(d) shows the first three mode shapes and the corresponding frequencies for t h e lumped parameter representation of the 10- story building in Fig. 2(c).

Fig. 3 shows a typical solution curve for t h e 7-story building in Fig. 2(6). T h e left-hand-side (LHS) ordinate in Fig. 3 is the ratio of t h e maximum displacement of t h e i t h floor, for t h e structure using t h e damper divided by the peak displace- ment, xvi, of t h e same floor with no damper being used. This ratio is, therefore, a measure of t h e effectiveness of the damper (without a damper the solution curve is a straight line a t ima*,/

xv% — ! ) • T h e abscissa in Fig. 3 is t h e clearance ratio d/xst„, where x„t„ is the static deflection of t h e top floor with the excitation force and t h e damper both applied a t t h e top.

F o r a given set of parameters there are two possible solutions to equation (18) labeled a0 + and a0~, which correspond to two distinct steady-state solutions. Experiments with an electronic analog computer were conducted to check the theoretical results and to determine which branch of the solution, if any, was asymptotically stable. As indicated by the experimental results shown in Fig. 3, t h e solution curve associated with a0+ (which, in general, leads to lower response amplitude than solution curve a⁰~) was found to be stable throughout most of its range.

T h e validity of the analytical results were further compared to, and were found to be in excellent agreement with, published

Journal of Applied Mechanics

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j=k=n=7 a=o /? = 0.00I9 - ^= l

0.5 (i-l-,7)

(a)

X

n

(b)

XP>

Q5 -

0.5

(e)

at,

j = k = n=|0 a = ° /? =00019 fiT = Q05 e = Q25 Q2I5 Q32 0.43

^T = O . O I 3 5

e=0.075

-|LLT=0.0067 - T h e o r y

Analog computer

20 4 0 6 0 8 0

*.t„

e = 0 2 5

- H T - 0 . 0 1 3 5

• / i r - 0 0 0 6 7

2 8 Xst„

14

*max;

XPi

0.5 •

n - 1 0 - ^

^-SDOF(n-l) £a=Q02l

•

• J U i

d / X s t n

r l T - Q O I 3 5 -

1.55 2 0 5 d x xP n

0.8 d / X s +"

Fig. 4 Effect* of roods shape: / = It = n = 1 0 , HT = 0 . 0 5 , • Fig. 3 Comparison of analytical and experimental results; / = k = (a) il/on = 1 ; (b) fi/wj = 1 ; and (e) 0 / « 3 = 1

n = 7 ; n/toi = 1—(a) e = 0 . 0 7 5 ; (b) e = 0 . 2 5 ; and (e) e = 0 . 7 5

3.73

Xmaxn

Xst„

1.86

0.39 Xmoxn

Xstn

0.19

1.2

= 0.25—

analytical and experimental results pertaining to single-degree- of-freedom systems provided with impact dampers [4].

Effects of System Parameters. A detailed investigation of t h e independent p a r a m e t e r s on t h e response of t h e system was under- taken, and t h e results can be summarized as follows:

1 Mass Ratio Mr. T h e system under discussion is effective even with mass ratios on t h e order of 1 percent. N o t e from Fig.

3 t h a t even with a mass ratio nT = 0.005 t h e response amplitude can be reduced b y as much as 80 percent.

For " t a l l " structures operating in t h e vicinity of their first mode, similar to t h e one under discussion, t h e optimum clearance ratio (in regard to vibration attenuation) satisfies t h e condition Hjd/xttn » 0.6 and results in a reduced amplitude which satisfies the relationship /xT^mai,/a;p, » 0.001, (i = 1, 2, . . . , re) for rela- tively high values of e.

2 Natural Mode Shapes. T h e response of t h e 10-story building under discussion vibrating in each of its first three modes is given in Pig. 4. I t can be seen that, with a given damper mass, t h e maximum percentage reduction in t h e response of t h e structure is achieved in t h e first mode. T h e maximum reduction de- creases with mode number from 90 percent in t h e first, to 85 percent in t h e second, and 60 percent in t h e third. One of t h e reasons for this reduction in t h e efficiency of t h e damper is t h a t f „ the ratio of critical damping in t h e ith mode, increases with mode number from ft » 0.01 to f 2 » 0.02 and f 8 « 0.03. Figs.

4(a), (6), and (c) also show t h e corresponding response of an equivalent single-degree-of-freedom (SDOF) system whose ratio of critical damping is t h e same as t h e 10-story building in a given mode. I t is clear from Fig. 4 t h a t factors in addition to damping are involved in modifying t h e performance of the damper. Since the "moment a r m " of t h e damper with respect to t h e base of t h e building is not affected b y the mode shape, i t appears t h a t the main factor t h a t influences t h e efficiency of t h e damper is t h e magnitude of the displacement of m, to which t h e damper is attached.

3 Damper Location /'. Fig. 5 illustrates some of t h e possible

(a) innnt

k=5

J - i

k = I O -

J-i > J - i

I k - i — .

(t>) /nfo? M?

> J - 5 k - 5 -

k-io-

• J=5

mw

. J - 5

10 ^{. J - I O} • J = I 0 k - 1 0 - • J - 1 0

k - 5 -

, I k- l - |

Fig. 5 Different combinations of force and damper location for 10-story building; (a) / = 1 ; (b) / = 5; a n d (e) / = 1 0

damper locations in t h e 10-story (re = 10) building t h a t is vibrating in its first mode.

With the force imposed on mass TO6 (i.e., k = 5), then, for t h e parameters in Fig. 6, t h e maximum reduction in amplitude will be = 2 percent if t h e damper is attached to mi, Fig. 6(a), about 50 percent if t h e damper is a t m⁵, Fig- 6(6), and more t h a n 90

130 / MARCH 1 9 7 3

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a= 0 p=0.0019 £ • - Xrnaxl

0.5

(b)

1

1 , . .

j = 5

i

— -

k^xPj Xstn

I 0.81 5 6.4 10 17

Q5 1.0 d / XD

Fig. 6 Effects of force and damper location: n = 10, Qfui = 1, HT = 0.05, e = 0.25; (a) / = 1; (b) / = 5; and (e) / = 10

Xmax

*Pi 0.5 (i-1,2,-,7)

(b) 0 I

5 10 0.18 0.36

^-SOOF(n-l)

n = 7 -^/

15 0.54

n=7

20 x ^ 3 0 - ^ &

£,= 0.015

___-

"

2 8 XmaXn

Xst„

14

5 0.11

10 0.22

15 2 0 0.33 0.44

30

,-SD0F(n=i)

^ / ^ ^ - n = | 0

n=|0 £,= 0.0089

"

4S

«rrv

Xstn

2 3

LO 15 2 0

xfr„

0 3 0 Fig. 7 "Telescoping" effect: 12/wj = l , j u r = 0.05, e = 0.25, /' = fc = n; (a) n = 4; (b) n = 7; and (c) n = 10

percent if the damper is placed a t the top floor, Fig. 6(c). This progressive increase in efficiency with the damper "elevation"

above the ground floor is to be expected since the moment arm increases with increasing j to a maximum a t the top floor (i.e., j = n).

For modes other t h a n the first, increase in moment a r m must be modified to account for the relative locations of the damper and the nodal points.

4 Force Location k. Although the application of an exciting force of a given amplitude a t different points along the structure will affect the magnitude of the response of the structure, it was found t h a t the response curves for an impact damped structure are virtually independent of the point of application of the exciting force, provided t h a t the results are expressed in terms of d/xpj and Zmaxj/xp,-.

5 Telescopic Effect. Fig. 7 shows the relation between the amplification ratio xmoxt'/xst,-, (i = 1, . . . , n) and the clearance ratio d/x^st„ for the three buildings shown in Fig. 2, where each one is vibrating in its first mode and with both force and damper acting on the top floor so t h a t j = k = n.

I t is clear t h a t if the damper mass ratio fiT is kept constant, the effectiveness of the damper is proportional to the height of the building. This is due to t h e fact t h a t as the height of the building is increased, the moment arm of the impact damper is simultaneously increased to amplify the dampening effects.

T h e deviation of t h e results from t h e corresponding ones for a n equivalent S D O F system increases as the number of stories increase.

6 Damping in the Primary System (a,/3). An increase in the vis- cous damping of t h e primary system is detrimental to the efficiency of the impact damper.

7 Excitation Frequency fi. If the structure is excited at a fixed frequency, the design parameter can be optimized according to the aforementioned criteria. However, if the response is to be controlled effectively over a relatively wide frequency band, an increase in /J.T and damping or a decrease in e will improve the performance of the damper.

8 Base Excitation. T h e present theory can also analyze cases

where the excitation is supplied through support displacement rather t h a n force excitation. The base-excited two-degree-of- freedom system in Fig. 8(a) is equivalent to the force-excited three-degree-of-freedom system in Fig. 8(6) where mi replaces the support and the excitation amplitude is adjusted so t h a t

So/Wh) = 1.

T h e theoretical predictions and the corresponding experimental measurements obtained by using an actual mechanical model are shown in Fig. 8(c). As mentioned earlier, of the two possible steady-state solutions, the one associated with ott,+ is the stable one. This is true in spite of the fact that, for the case in t h e frequency range O/coi < 0.95, the solution « o+ leads to larger amplitudes than a0~. N o t e also the existence of a peak in the response curve at 0/cOi » 1/(1 + Mr)-

Practical Design Considerations. Some of the design problems inherent in the application of practical impact dampers are (a) large accelerations developed during impact; (6) the accompany- ing noise; (c) design of the concentrated impacting mass (in massive structures, such as buildings, even a damper mass ratio of ~ 1 percent is still a large weight to have suspended in the structure). Analytical and experimental studies have shown t h a t the foregoing problems can be alleviated to some extent by the use of multiple-unit impact dampers operating in parallel, in- stead of one single "particle."

Summary and Conclusions

An exact theory is presented for determining the steady-state motion of a damped multidegree-of-freedom lumped parameter system t h a t is provided with an impact damper attached to some arbitrary point in the structure and is subsequently excited by a sinusoidal force, also applied to an arbitrary point.

T h e results were applied to several multidegree-of-freedom systems, including a model of a modern 10-story building. A detailed investigation of the effects of various parameters of the system on the response was conducted. I t was found t h a t re- gardless of where t h e excitation is applied (including the base) the efficiency of t h e damper increases as its moment a r m in-

Journal of Applied Mechanics

M A R C H 1 9 7 3 / 131

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J=3 k=l -n=3 a=0 /3 =l.79x|0

E/k,

100

^ s i n J l t (a)

k2/m2=k3--'m3

-

« k,^m.

r

al. / A

jh\

'^^ a+ '

' ,

0

^ N o

•

S-S solution Experiments

damper M-r=o.05 e = 0.25 d/S.=23

(c)

• I » 1 — • -

0.90 Q95 1.0 1.05 LIO A

Fig. 8 Bate excitation: (a) base-excited two-degree-of-freedom model; (b) equivalent force excited model; and (c) frequency response of a base-excited two-degree-of-free- dom system

creases. Due t o t h e "telescoping effect," it is feasible to use impact dampers, with mass ratios on t h e order of 1 percent of the mass of a lightly damped structure, to considerably attenuate the response of the structure over a relatively broad suppression band around any frequency of interest.

Predictions of t h e theory were corroborated by experimental studies with an electronic analog computer and with a mechanical model.

Acknowledgment

This work was supported by t h e National Science Foundation Grant N o . GK-11553. T h e assistance of Dr. D . S. Margolias in the preparation of this manuscript is appreciated.

References

1 Reed, W. H., I l l , and Duncan, R. L., "Dampers to Suppress Wind-Induced Oscillations of Tall Flexible Structures," Developments in Mechanics, Vol. 4, Cermak, J. E., and Goodman, J. R., eds., Johnson Publishing Co., 1968, pp. 881-897.

2 Reed, W. H., I l l , "Hanging-Chain Impact Dampers: A Simple Method for Damping Tall Flexible Structures," Wind Effects on Buildings and Structures, Vol. II, University of Toronto Press, 1968, pp. 283-321.

3 Kuroiwa, J. H., Vibration Test of a Multistory Building, Earth- quake Engineering Research Laboratory, California Institute of Technology, Pasadena, Calif., 1967.

4 Masri, S. F., and Caughey, T. K., "On the Stability of the Im- pact Damper," JOURNAL OF APPLIED MECHANICS, Vol. 33, TRANS.

ASME, Vol. 88, Series E, 1966, pp. 586-592.

132 / MARCH 1 9 7 3