The multi-unit impact damper: Equivalent continuous force approach

Journal of Sound and Vibration (1974) 34(2), 199-209

THE MULTI-UNIT IMPACT DAMPER: EQUIVALENT CONTINUOUS FORCE APPROACH Cz. CEMPEL Technical Uni~'ersity of Poznaft, 61-138 PoznatJ, Poland (Received 10 August 1973, and h~ revised form 23 November 1973) The problem of the multi-unit impact damper is analyzed by approximating the impulsive impact interactions with an equivalent continuous force. This approach simplifies the analysis of the problem in a basic way. Moreover the theoretical results obtained show sufficient agreement with the experimental results obtained for the harmonic excitation case. 1. INTRODUCTION In many cases concerned with vibration neutralizing such as the minimization of vibrations in lathe tools, airplane ailerons, helicopter tension rods, etc. [1, 2], and especially in low amplitude vibrations [3], it fs best to apply impact dampers. However, at high vibration velocities the system generates high level ifnpulsive forces. Those forces may have undesirable influence on the construction under consideration. Moreover such continuous impacts a r e sources of high level noise. In reference [4] the possibility is considered of substituting, for a one-unit impact damper, a multi-unit one having identical percussion parameters and satisfying the condition/7 = ~ ill, where p, is the ratio of the single free mass to the total mass of the damped object (a full list of notation is given in the Appendix). For the multi-unit system 0t = 2) it was proved that the effectiveness of vibration neutralizing is the same as for the case when n = 1 and moreover that the impulsive forces and the noise level are much lower. Additional analysis for n = 3 proved [4] that the individual free masses strike at the primary mass in equal time intervals, 3. It should be obvious that general analysis of such a system is very complicated and for the case ofn free masses it is quite impossible. The question arises whether for the n free masses it is possible to substitute an equivalent continuous interaction force for the sequence of impulsive forces. Such an approach should yield an important simplification of the system motion analysis without obscuring the accuracy of the results achieved. 2. THE SYSTEM MOTION EQUATION Consider a system with one degree of freedom, as a model of a primary damped system (Figure 1). Let the mass M of this container-shaped primary system include n (>>1) equal masses m, = m. These free masses may move in proper channels passing, in between impacts, through the free path s = 2d, = 2d. Assume, moreover, that the impact phenomenon can be characterized by the coefficient of restitution R, with R t = ,R, and i = 1. . . . . n. The effect of such assumptions is to subject the primary mass to a periodic exciting force F(t) = F(t + T), which forces T-periodic motion of the whole system. From reference [4] and the author's experiments there is evidence that within the above assumptions and the stable system motion, the impacts will occur one after another, so that in the period T one will observe n impacts on each side of the container (M). Assuming an 199

200

CZ. CEMPEL

I

,

g',- 4

, F(t).F(t+T)

--

--

, I<

--

L

_

M

I I '~.F-~, ~

2~.

:,1

[

>x

Figure 1. Model of the damped system.

equidistant lay-out of impacts occurring in the period T, one can calculate the time space, ~, between two successive impacts: T = T/2n. Consider now the system motion equation. Taking into consideration the impuls!ve kind of motion, with the help of the Dirac function 6(x) [5, 6] one may write

MS~ + c.~ + k x + /_.,

t- 1

mjk +

(1 + R) Mm~ M +

(I + R)Mmk M+

mt

(~ -.~,)I~ -),l [~(x - y t

- dr) + cS(x - y t + dt)] = F(t),

(P~ - :~)l Pk - Xl [,~(x -yk - d~) + ~(x -y~ + d~)] = 0,

mk

Ix-y~l =dl=d;

k, i = 1. . . . . n.

(2.1)

On the further assumption that the system motion is T-periodic, the same equation can be written for the time range 0 ~< t ~< T. At this point note that, within the limits of the assumptions, the arguments of each Dirae function, 6(z j), will become zero once in the period, and impacts at the walls of the container will occur every z = T[2n. Allowing for this and transforming the distribution arguments, 6(z j), from the displacement domain to the time domain [7] will result in .-i (1 + R ) Mmi

M.# + cYc + k x + ~ , t-o rnkyk +

(1 + R) Mmk _M+ mk

(Pk - .~)[6(t - it) + 6(t - ir - 7"]2)] Ulk = 0,

Ix(iO O <<.t <<.T,

(/c - - y , ) [ 6 ( t - - it) + 6 ( t -- i r - - :/'/2)] = F ( t ) ,

M + m~

yk(iO ut~l = d,

l,k=i ul~=

0,

k4i;

k, i = l , . . . , n .

(2.2)

As one knows x = T[2n, n >>1, x / T ~ I ;so one can assume that the quantities x, ~?, 5?, )k, Yk and F(t) remain almost unchanged in the time interval ir ~< t ~< (i + 1)3. Using such assumptions to integrate both differential equations (2.2), and additionally assuming that the ith

201

T H E blULTI-UNIT I M P A C T D A M P E R

impact occurs for t = ir + e, e ,~ 3, one has

zM~(it) + zc~(ir) + ~kx(iz) +

(1 + R) Mnh

,~F(rc),

[ $ ( i r ) - J',(iO] =

M + nh

xm~ Yk(iO +

(1 + R)Mm~ [.fk(i0 - 2 ( i 0 ] Ulk = 0, M + mk

ir<<.t<<.(i+l)t,

i;k=l

(2.3)

. . . . . n.

Allowing for the equality o f all free masses nh = m and introducing notation such as c/M = 2~coo, k / M = co~, m / M = It, one obtains e ( i ~ ) + 2~O~o

~(it) + o~gx(it)

+ R)It [~(i0 ~(1 + ~)

(1

I+R

- ),(it)]

F(i~) m

= - -

i; k = 1. . . . ,n.

x~k(ir) + ..-"7-- [.fk(iz) -- x(i~)] u,~ = 0, 1+/:

(2.4)

The second o f equations (2.4) can be transformed into an equivalent one, by summing it up for every mk mass, k = 1. . . . , n. Performing this sum gives ~. zfi~(i~) +

I+R

[ ) , ( i 0 - 2(iz)] = 0.

(2.5)

k-n

The expression ~ - 1 ZYk(iO is the unit o f mass impulse that each mass mk receives in an interval o f time 1"/2 = hr. Because o f this, one can substitute for equation (2.5) an equivalent one. This equation will be true in the mean value sense for each mass, according to the relation T ~. z.~kCiO = ~" Y,(iO-

(2.6)

k=l

By virtue o f this, the primary equation will assume the following f o r m : T

I+R fi,(it) + ..- 7 l+p

(2.7)

[ ) , ( i 0 - 2 ( i 0 ] = 0.

It is easy to prove that this equation is the equation o f balance o f linear m o m e n t u m in a half-period for each mass ink. It can be deduced from the above considerations that thanks to equations (2.5)-(2.7) the problem o f n equal masses' motion can be reduced to the problem o f one representative moving mass. In connection with this it can be written in equation (2.7) that y~(ir) = y(i~). Linking the equation o f primary mass m o t i o n (2.4) with the equation o f the representative mass (2.7) and, moreover, allowing for the fact that 9 = T[2n, one has

I + R 2nla

5/(i0 + 2~O~o-x(i0 + to2 x(iO + -1 -+ it 21+R .i~(i~) + - - - -

TI+p

Ix(it) -)
r

F(iO [:~(i~) - j'(ir)] = 9 . ~

,

[J~'(it) -- 2(i~)] = 0, = d,

i = 1 . . . . . n.

(2.8)

202

cz. CEMPEL

The above set of equations is true for one half-period motion, i = 1. . . . . n. However, because of the assumption of symmetry of the motion, one may extrapolate the validity of equation (2.8) for 0 ~< t ~< T, and further, because of the periodicity of motion, for 0 ~< t < oo. Thus, according to the previous assumptions of stability and symmetry of motion as well as a large number o f impacts, n >> 1, instead of the input system equations (2.1) one may write the following equations, equivalent in the mean value sense: I + R 21m

5/+ 2~too 2 + o902x + . - -

--

1 + It

y q

T

2 I+R - [ ) - Yc] = O, T 1+//

[x(t)-)(t)l

=,4,

[2 -- y] =-

F(t)

,

M

F(t) = F(t + T),

n>> 1,

0,< t < oo.

(2.9)

It is evident that this equivalence o f t h e equations will be more accurate the more frequent and the smaller the impact impulses between the masses M and ink. Within limits, when n -§ co, /m = constant, both sets of equations should yield the same qualitative and quantitative solutions. 3. THE ANALYSIS OF THE EQUATIONS OF MOTION Consider now the motion of a system with harmonic forcing. In complex form, one has

F(t)=Focostot=

Re [FoCm],

i = ~ec-1,"

27r

x ; y = R e [.~; )],

T=

(3.1)

m.

to

The equation of motion then can be written as + 2~too,~ + tog ~ 4 1 + R ~ . t o ( ; _ ~) = Fo e,O, '

l+/t l+Rto

; + - 1-+ / 1

rr

M

(~-~)=o, (3.2)

The solution of the system (3.2) can be assumed to have the following form:

= A (ito) e '~',

(3.3)

5' = B(ito) e ''~'.

U p o n substituting these forms into equations (3.2) and after proper transformations one obtains il+R)Fo I

A(~) = (l-t52+2i~6)

1

- -

~l+p

k

l l_+_R_l + i - -

~z 1 + / ~ ]

1 +/t

rc

il+RFo ~l+lt

B(i3) = (I -- 62 + 2i~6) 1

k

il+R /

l+R~,,3~

--_ +i l +it ] 1 +l t

where 6 = co]too is-a dimensionless forcing frequency.

n

,

(3.4)

203

THE MULTI-UNIT IMPACT DAMPER

For the amplitudes of the quantities of interest (Ampl[x(t)] = ]A(i6)I, Ampl[y(t)] = in terms of the static deflection of the primary system, Xo = Folk, one has

IB(i6)l) and

IA(i6)I -- Xo 1-6

2+

-r~

+

l'~pJ

[

2~6 .

.

. . rc 1 + p

+ - -

-

1 +#

ll+R n l+p IB(i6)l = xo

(3.5) It is now of interest to turn attention to the behaviour of the amplitudes (3.5) on the frequency scale. A simple limit crossing yields

IA(i~)I ~

xo, Xol+R r~ l + p

IB(i~5)l ~o' l+

l+p}

IA (i~)l /

(3.6)

I B(i6)I J ~

o.

It can be easily observed that the behaviour of both amplitudes remain almost the same within limits and is independent of the number of free masses, n. The application o f this system to the primary mass vibration damper can now be considered. It is known that the problem of vibration damping arises when the primary system finds itself in a resonance state, 6 ~ 1. Allowing for this in the relationships (3.5) one obtains

1+ ,4

1-~-~ /

IA(il)I = Xo

1

I+R

X0 ~

/r 1 + / /

B = ]B(il) I = zc l + / t

+

2~+ l+p

(3.7)

204

cz. CEMPEL

The first of relations (3.7) determines the vibration amplitude of the primary system with the application of the multi-unit impact damper. The second one, in turn, determines the vibration amplitude of the substitute representative mass. If the amplitude of resonant vibrations of the primary system is denoted by H, then it can be easily proved that H = Xo/2~. Comparing this quantity with the amplitudes in equation (3.7) gives

A

l;p]

m_-

_

H

I+R

< 1,

z

+

l+n

1 I+R

pn

n l+p

I I+R B

n 1+/~ 1 I-t-R_

+

< 1,

(3.8)

1 + 1 1 + R pn] 2

~ 1 +p] where, as can easily be seen, B < A. All the speculations so far, as well as the results obtained, are true only on the assumption that the motion of the system is stable and depends on successive, equidistant, free masses' impacts. This kind of motion is possible under the condition of the proper choice of the clearance, d, equal for all masses ink. Thus, now, the relationship between properties of the system (R, p, ~), and the optimal value of clearance, do, should be found. The conditions of impacts on the primary mass which are expressed in equations (2.1) and (2.9) must be fulfilled for every discrete moment of time, t = it, i = I . . . . , n. Thus, with n large enough, one can say that in the equivalent continuous analysis this condition is to be fulfilled for each t e (0, co). In connection with this one can write (3.9)

d = I x ( t ) - y ( t ) l = I~(t)-.f'(t)].

Allowing in the above for the relationships (3.3) and (3.4) and connecting the result achieved with the relation (3.8) one will finally get l[

I+R

A

2

1 <1, 1 1 +R

n 1+

+

d D = --=

1+

/m] ~ /

24/

1

l+l J +

< 1.

(3.10)

l + p 2~J

The first of relations (3.10) determines the degree of reduction of the resonant amplitude, K, and, as can be easily observed, is valid for all it; also, for ~ ~ 0 it establishes, as an important inequality for practical purposes, that K < 1. The second relation determines, in turn, the dimensionless clearance of the free masses, which yields the Kvalue predicted by the preceding

THE MULTI-UNIT IMPACT DAMPER

205

formula. O f course, from the physical and constructional point of view, one always has p.,~ 1. Because of this, the values o f K a n d D will only be in a small degree dependent on the value of the restitution coefficient, R, and will mainly depend on the terms involving lan[24 in the quotients. Thus, the degree of reduction of the amplitude, K, and the dimensionless clearance, D, depend mainly on the quality factor of the primary system (I/24) and on the joint dimensionless free masses of the multi-unit impact damper. F o r the purposes of qualitative analysis one can thus write the proportion relations as 24

2~ ,.,

Im

(3A1)

- - .

Im

'

It can be deduced from the above that the application of the multi-unit impact neutralizer is feasible for a primary system with a high quality factor and with a proper value of the free masses, i.e., 24,.~ 1, n,u >> 2~, (3.12) because only then, with l m < 1, can one obtain K = Km~, '~ 1. 4. A NUMERICAL EXAMPLE AND RESULTS OF THE EXPERIMENT In order to verify the results achieved for K a n d D a numerical example valid for harmonic forcing was calculated. The following data were accepted: a number of free masses n = 18; a dimensionless free mass/t = 0.01, or 0.001, which gives a joint moving mass I m = 0.18, or 0.018, and a viscotic damping rate in the primary system ~ = 0.01, or 0.002, which yields a resonant amplitude H = Xo[2~ = 50Xo, or 250Xo. For this data the behaviour of K and D as a function of the restitution coefficient R was calculated for the range 0 ~
!

I

1

I

I

I

I

I

I

I0

05

3

z______.

2

I 0

I

I

I

f O5

I

!

!

I -I 1.0

R

Figure 2. Calculated dependence of the resonant amplitude reduction, K, on the coefficientofrestitution, R. n= 18. Curve 1: ,u = 0-001, ~ =0.002; curve 2: it =0"01, ~=0-01; curve 3: ~ =0.001, ~= 0-002; curve 4: p = 0-001, ~ = 0-01.

206

CZ. CEMPEL I

i

I

I

I

i

I

I

I

I

I'0

4 C~ 0 5

I I

I

I

I

I 0-5

I

I

1

I

--I I-0

R

Figure 3. Calculated dependence of the dimensionless clearance, D, on the restitution coefficient, R. n = 18. Curve I : / t =0.01, ~=0-002; curve 2: It =0.01, ~=0-01; curve 3:/t =0.001, ~=0-002; curve 4: It 0.00l, ~ = 0.01. =

contained in expressions (3.11) and (3.12) are confirmed: that is, the reduction ofthe amplitude of the vibrations of the primary system becomes important only when nil ~ 2~ (see curve l, Figure 2). Of great importance also is the fact of weak dependence of K and D on the value of the restitution coefficient R, especially in the case of a high-value quality factor (2~ <~ 1). This result differs greatly from the results of a series of theoretical papers on this subject (see, for example, references [6], [8], [9] and [10]), which imply that the degree of reduction of amplitude, K, is strongly dependent on the value of the restitution coefficient, R. This result, on the other hand, agrees with the experiments conducted by Skachkov [11 ], in which impact masses made of rubber and steel, respectively, gave almost the same degree of reduction of vibrations of the primary system. In order to test the described results and conclusions, an experiment was conducted. A multi-unit impact damper with n = 18 was attached at the end of a cantilever beam. A view of the main element of the measuring system together with the exciter and vibration transducer is shown in Plate 1. The main stress in the experiments was laid on proving the K and D dependency on R. In order to do this, three measurement series were conducted. The first test was performed in an original multi-unit impact damper with free masses made of brass and the motion limiting stops made of steel. For this case one may assume R _~ 0-7. In the second series of tests, stiff cardboard, 1 m m thick, was pasted over all the motion limiting stops. The restitution coefficient was estimated at R "" 0-5. In the last series of tests the motion limiting stops were covered with stiffwool fabric 2 m m thick. In this case R _ 0.3 was accepted as the restitution coefficient. The first steps in research were to determine the damping rate, ~, by using the decay rate method, and the dimensionless mass,/t, by Rayleigh's method. Each time the error ofmeasuremerits was estimated. Then, after measuring the resonant amplitude, H(6 = 1), the clearance, d, was increased gradually until the minimum stable vibration amplitude, )i'm, was obtained. For this case, the value of d = do was used. The results achieved in this manner, Km = X, d H and Do = do/H, are presented in graphic form in Figures 4 and 5, where K and D curves for these it and ~ values are marked.

ty

Plate 1. View of the experimental apparatus. Plate 2. Oscillograph of acceleration at a motion limiting stop.

(facing p. 206)

207

THE MULTI-UNIT IMPACT DAMPER I

i

i

i

i

I

I

I

05I.~~i ~

i

O9

I

I 0"2

I

I 0-4

I

I O6

I

I O 8

I

I I-O

R Figure 4. Measuredva~ues~fthedependence~ftheres~nantamp~itudereducti~n~K~n~hec~e~cient~f :stitution, R. n = 18. Curve A : It = 0.0068, ~ = 0-0116; curve B: `U= 0.0067, ~ = 0-0061. $ , error o f K m e a s u r e lent; : : : : : , e r r o r of,u, ~.

0.5~ i

i i

i i

i

I

I I

I

I. 03

~

02

B OI

0

0-2

0.4

0.6

0.8

I-0

/?

9 Figure 5. Measured values o f the dependence o f the dimensionless clearance, D, on the restitution oefficient, R. n = 18. Curve A: `U= 0"0068, ~ = 0-0116; curve B:/1 = 0'0067, ~ = 0.0061. ~ , error o f D measureaent; : : : : : , e r r o r of`u, ~.

208

CZ. CEMPEL

As can be seen on the graphs almost all results of the measurements are contained within the error limits ofthe K a n d D curves. Moreover, in agreement with the theoretical predictions, the dependency of these quantities, especially of D on the value of the coefficient R, is minimal. Certain divergences between the value o f K a n d D obtained from measurements and calculations, respectively (see Figures 4 and 5), probably can be explained by the influence of the free masses' friction and by unavoidable errors in the degree to which the motion limiting stops are parallel (see Plate 1). The time dependence of the accelerations measured on one of the motion limiting stops was the last result of the experiment and an important argument for the theory presented in this paper. It can be seen, from the oscillographic record of the acceleration (Plate 2), that the acceleration jumps due to the several impacts of the free masses are spaced almost evenly and that the envelope of the impulse peaks has a sinusoidal character. 5. CONCLUSIONS The equivalent continuous-force approach to the problem of multi-unit impact damper and the results obtained in this paper lead to the following conclusions. (a) The introduction ofthe so-called representative mass diminishes the number of equations from n + 1 to two. This, in turn, largely simplifies their analysis. These equations, written primarily in the distribution form are, in this equivalent continuous-force approach, ordinary differential equations. (b) The results obtained for the degree of amplitude reduction, K, and for the dimensionless clearance, D, in the case of the harmonic forcing, are almost independent of the coefficient R. This result agrees with that of reference [11] as well as with the experiments presented in this paper. This fact is very important for practical purposes. (c) The K and D formulas derived show sufficient agreement with the results of the experiment, so that they can be applied in designing multi-unit impact dampers. ACKNOWLEDGMENTS The author wishes to express his thanks to Dr U. Kosiel and Mr Z. Golec for their assistance in making the numerical calculations and during the experiment. REFERENCES 1. A. E. KOBRINSKU1964 Mechanisms with Elastic Links. Moscow: Nauka (in Russian). 2. I. V. ANANIEV, N. M. KOLBINand N. P. SIEREBRINSKU1972 The Dynamics of,4irplane Constructions. Moscow: Machinostroienie (in Russian). 3. V. M. KOBIELEV,JU. F. KOBIELEVand V. F. RIEWA 1972 Machhze Mechanics 33-34, 103-110. Impact dampers (in Russian). 4. S. F. MASRI 1969 Journal of the Acoustical Society of America 45, 1111-1117. Analytical and experimental studies of multi-unit impact damper. 5. Cz. CEMPEL1969 Engbzeering Reports 17, 639-649. The equations of motion of a system with impacts (in Polish). 6. Cz. CEMPEL1970 Reports of the Technical Unicersity of Poznasi 44. The periodical vibrations with impacts in mechanical discrete systems (in Polish). 7. J. MIKOSINSKIand R. SmORSK11964Elementary Distribution Theory. Warsaw: PWN (in Polish). 8. 1957Journalof.4ppliedMechanics24, 322-324. On the theory ofacceleration damper (discussion). 9. M. I. F~oIN 1969 Izv. Vuzov. RADIOFIZIKA II, 4, 607-625. Towards a theory of non-linear dampers (in Russian). 10. Cz. CEMPEL1971 Engineering Reports 19, 301-307. The percussive vibration of two independent systems (in Polish). 11. A. F. GUROV 1966 Calculations of Strengths and Vibrations in Rocket Engines. Moscow: Machinostroienie (in Russian).

THE MULTI-UNIT IMPACT DAMPER

APPENDIX LIST O F SYMBOLS

/tl, ,u dimensionless particular and total free mass, respectively n number of free masses l" time interval between successive impacts M mass of the primary system free mass number i ml d,D clearance and dimensionless clearance, respectively R restitution coefficient F(t) exciting force T period of motion of the system C damping coefficient k stiffness of the spring x(t), yl(t) displacement co-ordinates of the primary system and ith free mass, respectively ,~(z~) Dirac delta function

(3 damping ratio of the primary system natural frequency of the primary system A(ico) complex amplitude of the primary system I~(ico) complex amplitude of the representative mass dimensionless forcing frequency K effectiveness of the primary system damping 0..) o

209