Journal of Sound and Vibration (1988) 121(I), 178-184

I M P A C T D A M P I N G OF R A N D O M VIBRATIONS

1. INTRODUCTION

Impact damping is a passive vibration control technique to attenuate the vibrations of lightly damped, usually resonant structures having little weight. It has been used fairly successfully to reduce the excessive vibrations of turbine blades, radar antennae, machine- tools, electrical relays and tall light poles [1-4]. An impact damper is normally a solid slug which is placed in a slightly larger cavity attached to the primary system as shown in Figure l(a). The damper is inactive until the vibrations of the primary system surpass the initial clearance, d*, between the two systems. Then a collision can occur from which the slug reverses direction while the mass of the primary system decelerates if the two are moving in opposite directions immediately before the contact. Ifthis motion opposition is sustained before each repeated collision, then the impact process acts like an intermittent brake or damper on the primary system. Hence, the primary system's maximum deflections are reduced by the momentum exchanges and, to some extent, by the energy dissipated at the collisions.

(a) Impact damper

. External ~ i M f~c. s 3

\ # 2 Primary system

!

O. 0

(b)

+

+

§

+

§ .+

+ §

§

, § I , I , 1 2 3

Deformation (ram)

'ci ,2 T',.2

,L I7 Figure 1. Showing (a) the vibroimpact system; (b) the characteristic static force-deformation curve of the

bean bag damper; and (c) the experimental set-up: I, hollow cantilever beams; 2, slotted primary mass; 3, impact damper; 4, light cotton thread; 5, white noise generator in Hewlett Packard 3582A Spectrum Analyzer; 6, Ling amplifier, Type PA 300; 7, Krohn Hite filter, Model 3750; 8, Ling electromagnetic shaker, Model 400, Series 192; 9, soft spring; 10, Bruel & Kjaer accelerometer, Type 4333; 11, Briiel & Kjaer conditioning amplifier, Type 2626; 12, Wayne Kerr capacitance transducer, Type M E 1; 13, rigid support for the capacitance transducer; 14, Wayne Kerr amplifier, Type TE 100 MK 11; 15, IBM/PC with WAVEPAK.

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0022-460X/88/070178 +07 $03.00/0 �9 1988 Academic Press Limited

L E T T E R S T O T H E E D I T O R 179

The requirement of motion opposition for successful impact damping may be satisfied straightforwardly, at least for steady state conditions, when the time-varying external force is a sinusoid with a known phase. It is achieved by merely adjusting the clearance of a given damper. Such a simple procedure is unlikely to be effective, however, when the frequency (and phase) of the external force is no longer unique. Indeed, it has been already demonstrated theoretically [5] that a solid slug does not effectively attenuate random forces. Such forces, however, cannot be ignored easily because they characterize a group of potentially significant practical applications involving earthquakes, wind loads and the like. Therefore, it is of more than passing interest to determine if the bean bag described in reference [6] retains its superiori ty over the solid slug damper when the external force is changed from the previously considered sinusoidal to a random fluctu- ation.

2. EXPERIMENTS

The bean bag damper is a spherical plastic bag full of 2 mm diameter lead shot. Its name is derived from the similarity with a pouffe. The fit of the plastic cover is critical and should not be quite so tight that all the relative movement of the shot is prevented. Then the resulting flexibility is categorized most easily by measuring the bean bag's deformations in the direction of progressively larger static forces which pass through its center. This quite convenient categorization is presented in Figure l(b) for the bean bag used here.

An impact damper was suspended from a sturdy frame as indicated in Figure l(c) so that it could move horizontally in a 17-78 mm long and much wider slot contained in the primary mass. The bean bag had to be initially compressed very slightly indeed in order to fit within the length of this slot. Repeated collisions, however, reduced the voids between the shot within the bag so that a small (and initially changing) clearance was created [6]. The ensuing effects on the primary system will be compared later with those produced by a mild steel slug having the same mass but a constant 0.27 mm clearance. This slug was constructed like a doughnut wrapped around a vertical cylinder to ensure a horizontal translational motion [7].

The cantilevered structure described in reference [8] and labelled as items 1 and 2 in Figure l(c) was used to simulate the single degree-of-freedom (SDOF) primary system. Its damping corresponded to a small 0.5% critical (viscous) damping ratio and the equivalent mass was 35.7 times that of each individual impact damper. The primary system was excited directly by an electromagnetic shaker driven by amplified white noise with a spectral content purely between 0 and 40 Hz. (Details of the experimental equipment are given in Figure l(c).) The 0-40 Hz range encompassed the 19.85 Hz fundamental natural frequency of the primary system when this system acted alone. On the other hand, it was much lower than the second natural frequency of 420 Hz to ensure an essentially SDOF behaviour. The resulting compromise, however, should still provide a stringent test of the performance of both impact dampers because a fair proportion of the excitation energy was below the primary system's fundamental frequency [9].

An external force may induce one of three types of structural failure. Fatigue failure becomes increasingly likely as the average dynamic stress grows [10]. Failure can also occur, however, when too much time is spent above a critical stress level. Its possibility is suggested by the frequency of crossing that stress level. Finally, a structure's displace- ment could be so severe that exceeding a particular value just once may cause an instantaneous failure. This likelihood is indicated by the probability of first exceeding that particular displacement level [11].

180 LETTERS TO THE EDITOR

The displacement and acceleration of the primary mass are much easier to measure than its stress. However, the displacement will be indicative of the stress when the primary system behaves linearly between collisions. The acceleration history, on the other hand, will reflect the forces generated by the collisions. For the sake of brevity, statistics presented here will be based exclusively upon the displacement of the primary mass. Typical acceleration histories will be given, however, to indicate their fundamental features. The displacement and acceleration were found separately by using a Wayne Kerr non-contact probe and a Briiel and Kjaer, Type 4333 accelerometer. Voltages output from these transducers were amplified and analyzed by employing an IBM PC-XT outfitted with the commercial measurement package called WAVEPAK [12].

WAVEPAK was used to obtain the R.M.S. displacement of the primary mass from the square root of the area under the displacement's frequency spectrum. The frequencies of crossing different displacement levels were determined by supplementing the digitization hardware of WAVEPAK with custom-written software. In this software, a search was initiated to find first the maximum and minimum from the record of a signal's fluctuation. The resulting difference between the maximum and minimum values was divided by 200 to give a level increment, dx. Then the search was reinitialized to count the number of times the signal went beyond integer multiples of dx. Counts were conducted over 75 separate displacement records for a particular excitation and an arithmetic mean was calculated for each level, x = n(dx), n = • •

The determination of the probability of first exceeding various displacement levels was performed from the instant of first applying the excitation rather than waiting several minutes for the damped primary system to settle as before. First, 50 individual histories of the primary system's displacement were recorded when the primary system was acting alone. Each record started with the primary system at rest and ended at 300 times its fundamental natural period, 2~r/to,. A search was undertaken next to find the maximum displacement in all of the 50 recordings, Xma~- The recording procedure was then repeated exactly after first inserting the bean bag and allowing it to become motionless. These last recordings were used to determine the probabilities, with the bean bag in action, of first exceeding different fractions, p, of the Xmax- The probability for a given fraction was found at a particular instant by calculating the number of recordings in which that fraction was exceeded during the prior period and dividing this value by the total number of recordings, 50.

A comparison of the performance of the bean bag and the slug will be presented next for the three previously described behaviour characteristics.

3. RESULTS

The ratio of the R.M.S. displacement of the primary mass measured before (tTXo) and after (trx) the separate insertion of each impact damper at a given excitation level is presented in Figure 2. Different levels of random white noise excitation were used to produce the ~rx o and corresponding tTx/~rXo shown. The more conventionally employed, non-dimensional ~rxo/d* is noted for reference at the top of Figure 2. This non-dimension- alization advantageously makes the curve for the solid slug independent of the clearance, d*, used experimentally [5]. However, the bean bag, unlike the slug, has a changeable d* which increases non-linearly with the growing excitations producing the larger trXo. Consequently, the bottom O-xo scale is more appropriate for the bean bag.

An impacting body behaves as an attenuator when the ratio O'x/trxo is less than one. Hence, Figure 2 shows that the bean bag is clearly a better attenuator then the solid slug except for the very smallest O'xo below 30 x 10 -6 m. However, the erratic performance of

LETTERS TO THE EDITOR 181

Cr~o /d* 0"00 0"25 0"50 0 '75 1"00 1"25

I I I I

1 .0 -o o

0"8 - o ' e r - - - 1 - -

o u ~: 0 .6 o o

0-4 A B C D

0.2

0-O J I J l ~ I 0 50 100 150 200 250 300 350

O-Xo xlOe (m)

Figure 2. Showing the variation of Ox/Oxo with O-x0 for both the slug (1) and bean bag (O) dampers.

the bean bag in this region should not be a problem because the trxo is so small that control is not warranted anyway. Elsewhere, the O'x/O'x o produced by the bean bag changes slowly from a low of around 0.4 to about 0.6. These ratios translate to reductions of 60% and 40%, respectively. Such reductions are encouraging but probably insufficiently high to justify using the bean bag, in practice, as the sole control device.

Results for the frequency of crossing various displacement levels are presented in Figure 3 when the primary system acts alone and when it interacts with the slug or bean bag. The vertical axis is given in terms of the frequency of crossing, tOCF, divided by to,. The horizontal, on the other hand, shows the non-dimensionalised displacement levels, X~ CrXo, at which the crossing frequencies of the displacement of the primary mass are measured. Only the positive X/o'xo are shown in Figure 3 but, in fact, all the curves are symmetrical about the vertical axis. The third axis essentially indicates the four levels of white noise excitation used. These levels gave the crx o or, in the case of the slug, the Crxo/d*, corresponding to points A, B, C and D in Figure 2. A comparison of the three plots at point B and then at point C and D indicates that the alternating dash-dot curve for the

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2 5 4 5

X/crx 0 Figure 3. The frequencies of crossing various displacement levels for the primary system acting alone ( );

interacting with the slug (- - -) and with the bean bag damper ( . . . . . ).

182 LETTERS TO T H E E D I T O R

bean bag invariably decays much more rapidly than the other two curves. There is very little difference, on the other hand, between the other two curves at any of these points. Clearly, the bean bag reduces virtually all the maximum displacements, including the maximum of these maxima by about 40%, whereas the slug is quite ineffective. Thus, the bean bag advantageously attenuates not only the previously noted R.M.S. but also the peak displacements of the primary mass. Both Figures 2 and 3, however, suggest that the bean bag is not useful at the practically insignificant point A where trxo is low regardless.

At point A, all three curves intercept the vertical axis at a noticeably higher value than elsewhere. This phenomenon happens because, at times, the primary mass "dwells" around zero displacement and produces a correspondingly low O'x0 when the excitation level is small. The dwell is observed as frequent zero crossings by the high sampling rate of the analog-to-digital converter in the WAVEPAK package. The intercept on the vertical axis is even higher at point A in the case of the solid slug because of sharp impulses generated at the collisions of the two practially rigid bodies. This behaviour is best illustrated by the typical acceleration histories given in Figure 4.

2 5[(o)

J I

o o

i- o| - / 2 .5 (c)

- 2 - 5 I t I I I

L

0 50 100 150 200 250 3 0 0

oa n t / 2 "tr

Figure 4. The acce le ra t ion of the p r imary mass for (a) the p r imary sys tem a lone; the p r imary with (b) the s lug and (c) with the bean bag damper , o-x~, = 102 x 10 -6 m.

Figure 4 presents the acceleration histories of the primary mass at the excitation level corresponding to point B in Figures 2 and 3. The primary system acts alone in Figure 4(a), whereas it interacts with the slug and bean bag in Figure 4(b) and 4(c), respectively. The almost instantaneous, noticeably higher spikes produced by the colliding slug are readily apparent in Figure 4(b). These large collision accelerations, of course, indicate the presence of comparable contact forces and detrimentally high noise levels. Thus, the slug clearly amplifies the basic acceleration history of the primary mass given in Figure 4(a) even though it might somewhat reduce the corresponding RMS displacement. A similar comparison of Figure 4(a) and 4(c), on the other hand, indicates that the bean

L E T T E R S TO T H E E D I T O R 183

bag attenuates the acceleration as well as the displacement of the primary mass. Con- sequently, its operation is quiet [9].

Figure 5 presents the percentage probability, within a period to,t/2~r starting from a totally at rest initial state, of first exceeding, when using the bean bag, a given fraction p of the Xmax of the primary sytem without the bean bag. The level of excitation corresponds to point B in Figure 2. It can be seen that the use of the bean bag leads to only a 10% probability of exceeding 0.5 Xm,x during the first 6007r/to~ seconds of the white noise excitation. However, a displacement greater than 0.2 Xmax is attained after about 200 7r/to, seconds. On the whole, it can be concluded that the bean bag should produce reasonable attenuations for transient excitations like earthquakes.

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~ 6o /0.3 .~ '~-*--* ' -*"

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Figure 5. The p robab i l i ty of the p r ima ry mass first exceed ing given f rac t ions of Xmax in con junc t ion wi th the bean bag.

4. D I S C U S S I O N A N D C O N C L U S I O N S

The bean bag was both noticeably quieter and superior to the comparable solid slug for the impact damping of random vibrations. The gains stem largely from the bean bag's flexibility which ensures appreciably smaller contact forces having longer durations [6]. Therefore, chances are better for a collision to be happening with the bean bag when the primary system attains its greatest speed. Consequently, the exchange of momentum will then be greatest. In addition, the bean bag dissipates somewhat more energy at a collision than the steel slug because it has a more plastic-like behaviour.

Although the previously noted gains are substantial, they are probably still insufficient to justify using the bean bag except as a supplement to other vibration controls. The following points are then worth remembering:

(1) Momentum transfer will be enhanced if the bean bag has the largest feasible mass and is located where the greatest speed of the original system occurs.

(2) The original system should have well separated natural frequencies because short duration collisions not only dissipate but also spread energy over a wide frequency band. If this requirement is not fulfilled, additional energy could well be input to nearby modes which could detrimentally enlarge their response contributions.

(3) Unlike a solid slug, the bean bag is relatively insensitive to slightly different clearances if a just less than completely tight bag is employed.

(4) Bean bags have been used continuously in excess of a year without breaking. However, very high temperatures will undoubtedly adversely affect the plastic cover. Effects of the environment as well as size enlargements have still to be investigated.

184 LETTERS TO THE EDITOR

A C K N O W L E D G M E N T S

F inanc ia l suppo r t f rom the Na tu ra l Sciences and Eng inee r ing Research Counc i l o f C a n a d a is a c k n o w l e d g e d grateful ly . The expe r imen ta l p r o c e d u r e and c o m p u t e r p rog rams for the f r equency o f c ross ing analys is were d e v e l o p e d by Y. Desa i and P. Tra inor .

Department of Mechanical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3 T 2N2

(Received 4 November 1987)

S. E. SEMERCIGIL N. POPPLEWELL

R. TYC

R E F E R E N C E S

1. J. B. HUNT 1979 Dynamic Vibration Absorbers. London: Mechanical Engineering Publications. 2. D. S. STEINBERG 1977 Machine Design, March 24, 71-73. Snubbers calm PCB vibration. 3. Y. KATO et al. 1976 Bulletin of the Japanese Society of Mechanical Engineering 19, 103-109.

Study on impact damper having a spring supported additional mass. 4. W. H. REED 1967 Proceedings of International Research Seminar; Wind Effects of Buildings and

Structures, Ottawa 2, 283-321 (University of Toronto Press). Hanging chain impact dampers; a simple method of damping tall flexible structures.

5. S. F. MASRI and A. M. IBRAHIM 1973 Journal of The Acoustical Society of America 53, 200-211. Response of the impact damper to stationary random excitation.

6. N. POPPLEWELL and S. E. SEMERCIGIL 1988 (to be submitted) Journal of Sound and Vibration. Performance of the bean bag impact damper for sinusoidal external forces.

7. C. N. BAPAT 1982 Ph.D. Thesis, University of Manitoba. A study of vibroimpact systems. 8. C. N. BAPAT and S. SANKAR 1985 Journal of Sound and Vibration 99, 85-94. Single unit impact

damper in free and forced vibration. 9. N. POPPLEWELL and S. E. SEMERCIGIL 1986 Proceedings of the International Conference on

Vibration Problems in Engineering, June 1986, Xian, China. Examples of impact damping. 10. D.E. NEWLAND 1984 An Introduction to Random Vibrations and Spectral Analysis (2nd edition).

New York: Longman. 11. S. AOKI and K. SUZUKI 1985 Bulletin of the Japanese Society of Mechanical Engineering 28,

no. 240, 1226-1232. Reduction of first excursion probability of mechanical systems under earthquake excitation by inelastic restoring force-deformation relation.

12. Wavepak User's Manual 1986. Tennesse: Computational Systems Inc.