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Journal of Sound and Vibration

Journal of Sound and Vibration 332 (2013) 6254–6264

0022-46http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/jsvi

Free vibration analysis of linear particle chain impact damper

Mohamed Gharib n, Saud GhaniMechanical and Industrial Engineering Department, Qatar University, PO Box 2713, Doha, Qatar

a r t i c l e i n f o

Article history:Received 3 April 2013Received in revised form10 July 2013Accepted 15 July 2013

Handling Editor: H. Ouyang

accelerations, and noise levels. To that end, the authors of this paper have developed a

Available online 9 August 2013

0X/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jsv.2013.07.013

esponding author. Tel.: +974 3307 2364; faxail addresses: gharib@qu.edu.qa, mgharibsm

a b s t r a c t

Impact dampers have gained much research interest over the past decades that resultedin several analytical and experimental studies being conducted in that area. The mainemphasis of such research was on developing and enhancing these popular passivecontrol devices with an objective of decreasing the three parameters of contact forces,

novel impact damper, called the Linear Particle Chain (LPC) impact damper, which mainlyconsists of a linear chain of spherical balls of varying sizes. The LPC impact damper wasdesigned utilizing the kinetic energy of the primary system through placing, in the chainarrangement, a small-sized ball between each two large-sized balls. The concept of theLPC impact damper revolves around causing the small-sized ball to collide multiple timeswith the larger ones upon exciting the primary system. This action is believed to lead tothe dissipation of part of the kinetic energy at each collision with the large balls.

This paper focuses on the outcome of studying the free vibration of a single degreefreedom system that is equipped with the LPC impact damper. The proposed LPC impactdamper is validated by means of comparing the responses of a single unit conventionalimpact damper with those resulting from the LPC impact damper. The results indicatedthat the latter is considerably more efficient than the former impact damper. In order tofurther investigate the LPC impact damper effective number of balls and efficient geometrywhen used in a specific available space in the primary system, a parametric study wasconducted and its result is also explained herein.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Active and passive control techniques are the main approaches used to suppress vibrations in mechanical and structuralsystems. In practice, passive control is favorite due to its simplicity and low energy consumption. The impact damper isa common passive control devices which consists of a freely moving mass constrained by two stops inside a rectangularcontainer mounted on the primary system (see Fig. 1a) [1–3]. When the moving mass impacts the stops, exchange ofmomentum and energy dissipation occur. The impact damper has found wide use in industry because it is simple, low-cost,robust, effective in vibration suppression over a wide range of accelerations and frequencies and suitable to operate in harshenvironments. It is used to dissipate the vibration energy and control the dynamic response in mechanical and structuralsystems [4].

Vibrations can cause inaccurate process performance in mechanical instrumentation and can cause fatigue damage instructural systems. Typical applications of impact dampers are vibration attenuation of manufacturing machines, structures,

All rights reserved.

: +974 4403 4301.u@gmail.com (M. Gharib), s.ghani@qu.edu.qa (S. Ghani).

Fig. 1. Common types of impact dampers; (a) single unit impact damper; (b) multiunit impact damper; (c) bean bag impact damper; (d) particle/granularimpact damper; (e) resilient impact damper; and (f) buffered impact damper.

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–6264 6255

cutting tools, tubing, plates, turbine blades, heavy armored personnel carrier, television antennas, digger machines, andshafts. In addition, impact dampers play an important role in damping the vibrations due to wind and wave loads on highrise building, towers, chimneys, offshore structures and bridges. Moreover, they are used to attenuate earthquake excitationsthat can cause damage in different types of structures [5–13].

On the other hand, some problems restrict the use of impact dampers. One problem is the high noise level caused byrepeated impacts. If the primary system is a building or a vehicle, the noise will cause discomfort to the residents. Additionaldifficulties stem from the high contact forces caused by the collisions. Additionally, the large contact forces may shorten theoperational life of the dampers. Furthermore, system parameters and the type of excitation affects the performance of animpact damper.

Researchers attempted to develop the impact damper for many decades. Their objective was to reduce the highaccelerations, absorb the contact forces, and attenuate the noise levels. Fig. 1 depicts the most common types that were presentedin the literature, which are:

(i)

Single unit impact damper [14–16]. (ii) Multiunit impact damper [17,18]. (iii) Bean bag impact damper [19,20]. (iv) Particle/granular impact damper [21,23,22]. (v) Resilient impact damper [24]. (vi) Buffered impact damper [25–27].

Multiunit impact damper consists of multiple masses instead of a single mass. This produces a smaller contact force for

each mass while maintaining the same effect of the single unit impact damper. The analytical and experimental workshowed that the multiunit impact damper is more functional than the conventional single unit impact damper in reducingnoise and vibration [17]. The bean bag impact damper is considered as another form of multiunit impact damper. It consistsof a flexible bag packed with small spherical particles (e.g. lead shots). The resilience of the damper can be varied byadjusting the tightness of the flexible bag. It is found that the bean bag impact damper is better than the conventionalimpact damper in vibration suppression, contact forces reductions, and noise attenuation [19]. The particle/granular impactdamper consists of a cavity(s) filled with ceramic/metal particles or powders with small granule sizes. Better dampingperformances are achieved when using metal particles with high density (lead or tungsten steel) [23]. Other investigationsrecommended using multiple particle impact dampers that involve friction, impact and shear mechanisms to achieveoptimal damping effect [22]. The resilient impact damper is similar to the conventional impact dampers. The only differenceis that the deformation of the impact damper with the stops during the collision is taken into account [24]. The bufferedimpact damper is an extension of the resilient impact damper by adding a flexible buffer layer to the stops to absorb theenergy of the moving mass. The experimental work shows that the buffer zone reduces the impact forces, avoids highacceleration and reduces the contact forces by absorbing more of the impact energy and increasing the contact time [25].

Over the last century, six types of impact dampers were introduced and developed to be used in industrial applications.We hereby propose a novel seventh type of impact dampers, called Linear Particle Chain (LPC) impact damper. The newimpact damper consists of a linear chain of spherical balls with different sizes (see Fig. 2). The motivation behind this workstems from the work of Gharib et al. [28]. The authors proposed a novel energy absorption model based on the repeatedmultiple impacts that occur in an impacted linear arrangement of particles with various sizes. The LPC impact damperrevolves around causing the small-sized balls to collide multiple times with the larger ones upon exciting the primarysystem. This action is believed to lead to the dissipation of part of the kinetic energy at each collision with the large balls.The results of a preliminary numerical study into the efficiency of the LPC impact damper for improving the dynamicresponse of a single degree of freedom (SDOF) structure are presented.

Fig. 2. LPC impact damper.

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–62646256

2. The impact law

Impact is the collision between two bodies that occurs over a short time interval, during which the bodies exert largeforces on each other [29,30]. Impact is characterized by very short time interval, presence of large reaction forces, rapiddissipation of energy and rapid increase and decrease in acceleration and deceleration. During impact, two phases occur, thefirst one is the compression phase when the two bodies come into contact and press against each other. The second phase isthe restitution phase, during which the bodies move away from one another while remaining in contact. The latter phaseends when the two bodies are separated.

The energy loss due to impact can be expressed in terms of a parameter called the Coefficient of Restitution (COR),e, where typically 0≤e≤1. The limiting values correspond to the perfectly elastic (e¼1) and the perfectly plastic (e¼0)collisions. The COR depends on pre-impact velocities, geometry and material properties of the colliding bodies, contact time,and friction. Three common models for the COR were introduced: (i) the kinematic, Newton's model [31]; (ii) the impulsive,Poisson's model [32]; the energetic model [33]. In each model, the equation defining the COR is known as the “Impact Law”.

kinematic e¼�Δvf =Δvi (1)

kinetic e¼ τr=τc (2)

energetic e2 ¼�Wr=Wc (3)

where Δvf and Δvi are the final and initial differences in normal velocities, respectively, τr and τc are the normal impulsesduring the restitution and compression phases, respectively and Wr and Wc are the work done during the restitution andcompression phases, respectively. The three models of the impact law produce the same solution in case of frictionlessimpact.

In general, two approaches are common in solving the impact problems. The first approach is the impulse-momentumbased method. The underlying assumption of this method is that the impact takes place in a very short time. As a result, theposition and configuration of the colliding bodies do not change during the collision period. The process is divided intotwo intervals, before impact and after impact. Experimentally measured collision parameter, COR, is needed to solve theproblem. The post collision velocities are obtained by solving the conservation of momentum and the impact law equationssimultaneously.

The second approach is the compliance based methods, also known as continuous or contact force based methods. Thecolliding bodies are assumed to be rigid with localized compliance and damping elements (such as springs and dashpots)inserted at the contact points of the colliding bodies. The problem is solved by obtaining the equations of motion of thesystem and computing the position and velocity profiles through time integration [34–36].

In this paper, the impulse momentum approach and the kinematic COR are used to solve impact problem between thecolliding masses. The friction is neglected and the balls are treated as particles. Hence, the rotation effects of the balls are notconsidered.

3. Multiple impact based energy dissipation scheme

The motivation of developing an impact damper using a chain of spherical balls initially stems from the results that wereobtained in the work of Ceanga and Hurmuzlu [37]. The authors introduced a new parameter called the Impulse CorrelationRatio (ICR) to solve the multiple impacts for a linear chain of spherical balls initially in contact. In their work, the authorsdiscovered that trapping a small ball between two larger balls would lead to a significant increase in the number ofintermittent collisions among the three balls. Based on that conclusion, Gharib et al. [28] proposed a scheme for energyabsorption using a linear chain of spherical balls. This scheme relies on dissipating the kinetic energy by using theappropriate arrangement of large and small balls. In this section, a brief overview of the energy absorption scheme proposedby Gharib et al. [28] will be presented.

The linear chain energy absorption scheme is based on the multiple collisions in a chain of spherical balls where a smallintermittent ball is placed between each two large balls. It is deduced that each impact causes an energy loss. Consequently,more impacts lead to more energy dissipation as a result of the collision process. Hence, one can exploit this feature touse this arrangement for energy absorption. The multiple impact problem in this scheme is solved using the impulsemomentum based method.

In the analysis of the linear chain energy absorption scheme, a horizontally staggered linear chain of N spherical balls(initially at rest) is considered. The chain of balls is composed of an assortment of large and small balls and is impacted with

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–6264 6257

an incident ball (see Fig. 3). To simplify the analysis, identical small and large balls are assumed with masses of mS and mL,respectively. The incident ball is assumed to have mass of mf (see Fig. 3). The efficiency of the shock absorption is quantifiedby defining a Kinetic Energy Ratio (KER) as

KER¼ Total KE of all balls after impactThe KE of the incident ball before impact

(4)

This ratio represents the ratio of the total kinetic energy of the ball that strikes the chain and the remaining kinetic energy inthe system after collision. Hence, smaller values of KER represent more efficient energy absorption.

An experimental set-up was used to verify the theoretical outcomes from the new energy absorption scheme. Theclassical collision experiment known as “Newton's Cradle” is constructed with alternating large and small balls (see Fig. 4).

Five chains for the experiments were used; chains that have three to seven large balls. For each chain, a small ball isplaced between contiguous pairs of large balls. The experimental data were captured by using a high speed digital videocamera. The relative positions of the balls before and after the impact were determined with respect to fixed markers, usedas references. The acquired video images were transferred to a computer, where MATLAB digital image processing toolboxprogram was used to digitize the markers positions. The digitized positions were used to compute the dropping height ofthe first ball and the maximum post-impact heights attained by each ball. Finally, the pre-impact velocity of the first ball and

Incident Ball Small Ball

Initial Velocity

Large Ball

Fig. 3. The model for the impact based energy absorption scheme [28].

Fig. 4. The experiment setup for the impact based energy absorption scheme [28].

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

543 76

Kin

etic

ene

rgy

ratio

(KER

)

Number of large balls

Theory: Without Small Balls;

Experiment: Without Small Balls

Theory: With Small Balls; e = 0.94

Experiment: With Small Balls

e = 0.94

Fig. 5. The results for the impact based energy absorption scheme [28].

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–62646258

post-impact velocities of all balls were computed from the calculated heights. The experiments were conducted usingChrome Steel balls with two sizes, large (L) and small (S) balls. Steel material is selected because it could be used in mostof the applications due to its favorable mechanical properties. The COR is experimentally determined to be e¼0.94. Then,five sets of experiments are conducted: four for each of 3L�2S, 4L�3S, 5L�4S, 6L�5S, 7L�6S, balls, respectively. The dropheight of the first ball produced approximately 1.4 m/s pre-impact velocity v�f . Fig. 5 depicts the experimental andtheoretical results.

The graph depicts the results of five sets of experiments conducted with specific sequences of balls. The results clearlydemonstrate that the new energy absorption scheme is efficient in dissipating energy due to collision. In addition, the figuredepicts that, from the point of view of energy dissipation, the benefit of increasing the number of large balls diminishes assize of the chain increases. The adverse effect of increasing the number of balls is creating a bulkier shock absorber. It isneeded to increase the amount of energy absorption while keeping the mass and the size of the absorber small. This maylead to the idea of using more than one chain with fewer numbers of balls or possibly extending the study for 2D and 3Dconfigurations.

4. Structure damper model

To analyze the dynamic characteristics of the impact damper vibration system, a model with SDOF is considered (seeFig. 6). The SDOF system is assumed to vibrate freely due to an initial condition. The impact event is divided into two phases;pre-impact and post-impact. A velocities jump occur during this event which make the motion discontinuous for the ballsand the system. In case if no collision detected, the system continues vibrating and each ball moves freely at constant speed.The balls are treated as particles. Hence, the rotation effects of the balls are not considered. Neglecting friction, the systemEquations of Motion (EOM) is written as

ms €xsðtÞ þ cs _xsðtÞ þ ks xsðtÞ ¼ FeðtÞ (5)

mi €xiðtÞ ¼ 0; i¼ 1;2;…;n (6)

where, xsðtÞ, ms, ks and cs are the position, mass, lateral stiffness and damping of the structure, respectively, FeðtÞ is theexternal excitation force on the structure, and xi(t) and miði¼ 1;2;…;nÞ are the positions and masses of balls, respectively.The subscript “s” refers to the structure quantity.

If a collision is detected, the impact event is treated as a discontinuous process governed by the conservation of linearmomentum equation for the two masses, given by

miv�i þmjv�j ¼mivþi þmjvþj (7)

The kinematic COR equation is given by

eðv�i �v�j Þ ¼ �ðvþi �vþj Þ (8)

where v� and vþ are the velocity magnitudes just before and immediately after the impact event, respectively. SolvingEqs. (7) and (8) for vþi and vþj , we get

vþi ¼ 1mi þmj

mi�emj� �

v�i þ 1þ eð Þmjv�j

h i(9)

vþj ¼ 1mi þmj

mj�emi� �

v�j þ 1�eð Þmiv�ih i

(10)

CsKs/2 Ks/2

ms

XsFe

Cs

Ks

ms

Fe

X2 XnX1Xs

Fig. 6. Schematics of the SDOF structure with the LPC impact damper; (a) SDOF system with LPC impact damper; and (b) equivalent mass-spring-dampersystem.

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–6264 6259

where

fi; jg ¼ fs;1g ∀ ms�m1 impactsfi; jg∈f1;2;…;ng; i≠j ∀mi�mj impacts

fi; jg ¼ fn; sg ∀ mn�ms impacts

By knowing the pre-impact velocities and the COR, the post-impact velocities of the colliding masses can be obtained.The spherical balls are placed in a rectangular container fixed at the top of the structure. The characteristic dimensions

are related by the following relationship (see Fig. 7):

Le ¼ nΔþ 2 ∑n

i ¼ 1Ri þ Lc (11)

where Le is the effective length of the impact damper, Lc is the clearance, and Δ is the gap between each two balls. A uniformgap is assumed between all the colliding bodies (including the right wall) as an initial condition. Practically, the gaps andclearance cannot (very difficult to) be customized before the excitation. However, the values can be determined by capturingan image for the initial positions of the balls. Then, the image can be digitized and scaled to calculate the gaps values andhence the clearance.

5. Numerical results

Numerical analysis is performed using conventional and LPC impact dampers to control the response of a SDOF framestructure subject to free vibration. The structure is subjected to an initial lateral displacement, xso. In the analysis, it isassumed that each ball has the same material but different masses. Also, the end stops are assumed to have the same materialas the balls. The impact problem for the arrangement is solved using the impulse momentum method. The kinematic COR isassumed to be constant for the same material. Hence, the effect of pre-impact velocities, contact time and friction during theimpact events is neglecting [28]. The COR is chosen to approximate the values corresponding to the steel balls that would besuggested for practical applications. The steel material is selected because it could be used in most of the applications due to itsfavorable mechanical properties. The linear chain consists of two types of balls, large balls each with mass mLarge and smallballs each with mass mSmall. The mass ratio of the LPC impact damper balls, μb, is defined as

μb ¼mSmall=mLarge (12)

A reference radius, Rr , refers to the radius of the large ball in the arrangement. The damper clearance, Lc, and the gap betweenballs, Δ, will be functions of the reference radius. The numerical values of the structure are experimentally estimated in thework of [26]. Table 1 summarizes the numerical values used in the numerical analysis.

The solution is obtained by solving the EOM Eqs. (5) and (6) numerically using the explicit Runge–Kutta method with adifference order of 5. The solution code is developed using the Mathematica software. The EOM is solved from an initial timeto the time at which an impact occurs. If an impact event detected, the solution stops, the impact law is used, new initial

X1 X2 XnXs

Δ Δ Lc

Le

Fig. 7. Geometry of the LPC impact damper.

Table 1Numerical values.

Property Value Unit Property Value Unit

ms 1.35 kg mLarge 0.11 kgks 865 N/m mSmall mLarge � μb kgcs 1.092 N s/m ρ 8000 kg/m3

xso 18.5 mm e 0.9

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–62646260

conditions are imposed using Eqs. (9) and (10), then the solution of the EOM continues with the new initial conditions. Thevibration response of numerical simulation of the system with and without the conventional impact damper is shown inFig. 8. The figure shows the displacement and acceleration responses of the SDOF structure.

Three arrangements of balls are simulated to investigate the efficiency of the LPC impact damper:

(a)

Fig.and

Fig.and

Figand

n¼3 with arrangement LSL;

(b) n¼5 with arrangement LSLSL; and (c) n¼3 with arrangement LLL.

The vibration response of the numerical simulation of the three arrangements is shown in Figs. 9–11, respectively. Thefigures depict the role of placement of small balls in energy dissipation of a chain composed of large balls.

Time (sec)Time (sec)

Dis

plac

emen

t (m

m)

Acc

eler

atio

n (m

/s2 )

without impact damperwith impact damper

without impact damperwith impact damper

1 2 3 4 50

5

10

15

20

25

30

1 2 3 4 50

5

-5

10

-10

15

9. Time response for the SDOF system with LPC impact damper; (a) displacement; (b) acceleration; n¼3 (LSL); μb ¼ 10; Lc ¼ 2Rr; Δ¼ 0:25Rr;e¼0.9.

Time (sec)Time (sec)

Dis

plac

emen

t (m

m)

Acc

eler

atio

n (m

/s2 )

without impact damperwith impact damper

without impact damperwith impact damper

1 2 3 4 50

5

10

15

20

25

30

1 2 3 4 5

-10

-5

0

5

10

15

10. Time response for the SDOF system with LPC impact damper; (a) displacement; (b) acceleration; n¼5 (LSLSL); μb ¼ 10; Lc ¼ 2Rr; Δ¼ 0:25Rr;e¼0.9.

1 2 3 4 50

5

10

15

20

25

30

1 2 3 4 5

-10

-5

0

5

10

15

Time (sec)Time (sec)

Dis

plac

emen

t (m

m)

Acc

eler

atio

n (m

/s2 )

without impact damperwith impact damper

without impact damperwith impact damper

. 8. Time response for the SDOF structure with conventional impact damper; (a) displacement; (b) acceleration; n¼1 (L); Lc ¼ 2Rr; Δ¼ 0:25Rr;e¼0.9.

1 2 3 4 50

10

20

30

40

50

60

1 2 3 4 50

20

40

60

80

100

1 2 3 4 50

20

40

60

80

100

120

1 2 3 4 50

20

40

60

80

100

120

140

Time (sec)

Dis

plac

emen

t (m

m)

Time (sec)

Dis

plac

emen

t (m

m)

Time (sec)

Dis

plac

emen

t (m

m)

Time (sec)

Dis

plac

emen

t (m

m)

Fig. 12. Impacts of the Damper masses with the primary system; (a) n¼1 ðLÞ ; (b) n¼3 ðLSLÞ; (c) n¼5 ðLSLSLÞ; and (d) n¼3 ðLLLÞ. In each graph, the upperand lower curves represent the left and right internal edges of the primary system, respectively. The included curves from the top to the bottom representthe order of the balls from left to right, respectively.

Time (sec)Time (sec)

Dis

plac

emen

t (m

m)

Acc

eler

atio

n (m

/s2 )

without impact damperwith impact damper

without impact damperwith impact damper

1 2 3 4 50

10

20

30

40

1 2 3 4 5

-10

0

10

Fig. 11. Time response for the SDOF system with LPC impact damper; (a) displacement; (b) acceleration; n¼3 (LLL); μb ¼ 1:0; Lc ¼ 2Rr; Δ¼ 0:25Rr;and e¼0.9.

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–6264 6261

Fig. 9 shows that the LPC impact damper that consists of 3 balls with LSL arrangement significantly improve the dampingcharacteristics compared to the conventional impact damper. Compared to the conventional impact damper, when usingthe LPC impact damper, the peak displacements and peak acceleration are reduced with about 17 percent and 33 percent,respectively. Also, the steady state is reached much faster.

Fig. 10 depicts that increasing the number of balls to five with LSLSL arrangement leads to a significant improvement inthe steady-state system response compared to the LSL arrangement. However, the adverse effect of increasing the number ofballs is creating a longer and heavier impact damper. The small balls mass can be neglected compared to the large ballsmasses. The effect of placing a small ball between each two large balls can be concluded from the results shown in Figs. 10and 11. It is clear that the vibration of the small ball which is trapped between the large balls leads to relatively larger energyabsorption by the large balls. The effect of trapping a small ball between two large balls is also depicted in Fig. 12; where ahigher number of collision is observed compared to the case with no small balls in the arrangement.

Time (sec)Time (sec)

Dis

plac

emen

t (m

m)

Acc

eler

atio

n (m

/s2 )

1 2 3 4 5

-5

0

5

10

15

20

25

30

1 2 3 4 5

-10

-5

0

5

10

Fig. 14. The effect of the gap length; (a) displacement; (b) acceleration; n¼3 (LSL); μb ¼ 10; Lc ¼ 1:0Rr; and e¼0.9.

Time (sec)Time (sec)

Dis

plac

emen

t (m

m)

Acc

eler

atio

n (m

/s2 )

1 2 3 4 5-5

-50

5

10

15

20

25

30

1 2 3 4 50

5

10

15

Fig. 15. The effect of the mass ratio; (a) displacement; (b) acceleration; n¼3 (LSL); Δ¼ 0:25Rr; Lc ¼ 1:0Rr; and e¼0.9.

Time (sec)Time (sec)

Dis

plac

emen

t (m

m)

Acc

eler

atio

n (m

/s2 )

1 2 3 4 50

10

20

30

1 2 3 4 5

-10

-5

0

5

10

Fig. 13. The effect of the clearance; (a) displacement; (b) acceleration; n¼3 (LSL); μb ¼ 10; Δ¼ 1:0Rr; and e¼0.9.

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–62646262

6. Parametric study

The clearance in an impact damper is highly affecting the response of the primary system. Fig. 13 captures the effect ofincreasing/decreasing the clearance of the LPC impact damper. It is clear that decreasing the clearance values lead to betterdamping of the system response. However, the shorter impact damper may not be sufficient for the desired momentumexchange between the primary system and the damper.

The effect of the gap length between the balls in the LPC impact damper is investigated. Fig. 14 shows the systemresponse for different values of the gap lengths.

The figure depicts that the system's response is not highly sensitive to the gap length. The overshoot slightly increases asthe length of the gap increases. However, the steady-state response can be significantly improved for an optimum relativeintermediate value of the gap length.

The mass ratio in the linear chain is the key parameter for the proposed impact damper. Fig. 15 captures the effect ofvarying the mass ratio. The figure depicts that the smaller the mass ratio, the better response characteristics of the system.However, from the practical point of view, using a relatively very small mass ratio may lead to permeant deformation in thesmall balls. Consequently, the small balls will not vibrate effectively between the large balls.

Time (sec)Time (sec)

Dis

plac

emen

t (m

m)

Acc

eler

atio

n (m

/s2 )

e = 0.85e = 0.90e = 0.95

e = 0.85e = 0.90e = 0.95

1 2 3 4 50

5

10

15

20

25

30

1 2 3 4 5

-10

-5

0

5

10

Fig. 16. The effect of the COR; (a) displacement; (b) acceleration; n¼3 (LSL); μb ¼ 10; Δ¼ 0:25Rr; and Lc ¼ 1:0Rr .

M. Gharib, S. Ghani / Journal of Sound and Vibration 332 (2013) 6254–6264 6263

The COR of a spherical ball may change due to deformation and may vary based on the material properties of the balls.Fig. 16 shows the system response for different values of the COR. The figure depicts that, lower COR leads to better responsecharacteristics. However, lower COR corresponds to softer materials. Hence, an optimization between the COR and thedesired response characteristics should be considered during the material selection of the spherical balls.

Noise is one of the main disadvantage of the conventional impact damper. The noise level is proportional to the massesand velocities magnitudes of the colliding bodies. In the case of multiunit impact damper, replacing a larger mass with asmaller masses led to reduction in the noise level [17]. In the case of LPC impact damper, it is expected that trapping smallball between the large balls, will reduce the noise level generated from the impacts.

7. Conclusion

In this paper, a new passive vibration device called the Linear Particle Chain (LPC) impact damper is introduced. Theproposed impact damper provides new tools to design effective structure damping systems and can be used in a wide rangeof applications in various industries. The LPC impact damper consists of linear chain of spherical balls with an alternatingarrangement of large and small balls. The new impact damper is based on a new energy absorption scheme utilizing themultiple impacts between spherical balls. The energy absorption scheme is based on placing a small ball between each twolarger balls in the chain. Hence, the small ball will have very high number of collisions with the larger balls when theprimary system is excited. This behavior leads to dissipation of the kinetic energy due to the numerous collision of the smallball on the large balls.

The behaviors of a vibration SDOF structure with the LPC impact damper are investigated. The LPC impact damper isvalidated by comparing its responses with the conventional single unit impact damper. The numerical results showed thatthe LPC impact damper produces significant reduction of the vibration compared to the conventional impact damper. Aparametric study is conducted to specify the effect of the clearance in the damper, the gaps between balls, the mass ratioof the balls, and the coefficient of restitution. It is found that the system response enhanced by decreasing the damperclearance and increasing the damper mass ratio and using spherical balls with low coefficient of restitution. However, adesign compromise is needed in selecting these parameters based on the damper application.

References

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