Control of flexural waves on a beam using distributed vibration neutralisers

Journal of Sound and Vibration 330 (2011) 2758–2771

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Control of flexural waves on a beam using distributed vibration neutralisers Yan Gao a,n, Michael J. Brennan a,1, Fusheng Sui b a b

Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK Institute of Acoustics, Chinese Academy of Sciences, Beijing 100080, PR China

a r t i c l e in f o

abstract

Article history: Received 26 August 2010 Received in revised form 21 December 2010 Accepted 5 January 2011 Handling Editor: D.J. Wagg

This paper is concerned with the use of distributed vibration neutralisers to control the transmission of flexural waves on a beam. Of particular interest is an array of beam-like neutralisers and a continuous plate-like neutraliser. General expressions for wave transmission and reflection metrics either side of the distributed neutralisers are derived. Based on transmission efficiency, the characteristics of multiple neutralisers are investigated in terms of the minimum transmission efficiency, the normalised bandwidth and the shape factor, allowing optimisation of their performance. Analytical results show that the band-stop property of the neutraliser array depends on various factors, including the neutraliser damping, mass, separation distance in the array and the moment arm of each neutraliser. Moreover, it is found that the particular attachment configuration of an uncoupled force–moment-type neutraliser can be used to improve their overall performance. It is also shown that in the limit of many neutralisers in the array, the performance tends to that of a continuous neutraliser. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Flexural wave motion in beam-like components can lead to undesirable noise/vibration transmission and radiation, which is present in many practical engineering structures such as bridges, cranes and buildings. The suppression of flexural waves can be achieved by passive means such as using a vibration neutraliser. Following the terminology discussed in [1], the vibration neutraliser rather than vibration absorber is used in this paper to signify that the device is being used to control vibration at a troublesome excitation frequency rather than a resonance frequency. The fundamental theory of the vibration neutraliser was first presented by Ormondroyd and Den Hartog [2]. Ideally, a neutraliser consists of a mass–spring–damper system. In practise, however, it can be made of any resonant system, for examples a centrifugal pendulum [3], a cruciform beam [4,5] and a plate-like system [6,7]. It is also possible to use piezoelectric patches along with passive, resonant electronic circuits to form vibration neutralisers [8]. Control of flexural waves on a beam using a single vibration neutraliser has been discussed in previous work by Brennan [5] and Clark [9]. It has been found that the maximum reduction in the amplitude of a flexural wave occurs at a frequency just above the natural frequency of a force-type neutraliser and just below the natural frequency of a momenttype neutraliser. This frequency is called the tuned frequency. If an undamped neutraliser is tuned to meet the condition that its resonance frequency coincides with the forcing frequency, then it would ‘‘pin’’ the beam at this frequency and

n

Corresponding author. Tel.: + 44 1372 722574; fax: + 44 845 127 5121. E-mail addresses: [email protected] (Y. Gao), [email protected] (M.J. Brennan), [email protected] (F. Sui). 1 Now at the Department of Mechanical Engineering, UNESP, Ilha Solteira, SP15385-000, Brazil.

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.01.002

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consequently only half of the incident energy in a propagating flexural wave would be reflected (corresponding to a 3 dB reduction). Such vibration neutralisers are tunable narrow-band control devices. Hence they generally are only effective at their tuned frequencies. In practice, there is often drift in the excitation frequency that induces vibration. The application of such devices attached at a single point on a beam becomes less attractive, if it is required to suppress wave propagation over a reasonably wide frequency range. However, it was observed in [9] that a coupled force–moment-type neutraliser could improve the bandwidth. Following up on this work, a particular configuration of a beam-like neutraliser was designed to exert an uncoupled force and moment to improve the performance [10]. Compared to the other three configurations (the force-type, the moment-type and the coupled force–moment-type neutralisers), the uncoupled force– moment-type neutraliser has proven to be more effective since it has two tuned frequencies and hence a wider bandwidth. Alternatively, an interesting wideband neutraliser was designed by Brennan [1] by grouping several neutralisers in an array at a point that were all tuned to slightly different natural frequencies. For the same overall mass, as more neutralisers were added to the array, the maximum impedance was reduced but the bandwidth was increased by a greater factor. Therefore, the main advantage of this device over a single neutraliser was the possibility of a greater maximum impedance-bandwidth product. It should be noted that a vibration neutraliser discussed above would have to be attached at a single point on a beam to suppress flexural waves. However, less attention has been paid to the application of distributed neutralisers over a structure. Kashina and Tyutekin [11] proposed a set of undamped resonators located over a certain length of a beam or plate to reduce the longitudinal or flexural waves. The optimum mass and number of oscillators for attenuation in a specific frequency bandwidth were also derived. Previous work by Smith et al. [12] has shown the potential of a continuous layer of neutralisers added to a beam or plate to suppress flexural waves. Numerical results have shown that adding damping to the device can reduce wave attenuation at its peak value but spread the effect over a wider bandwidth. Additional benefits were demonstrated experimentally when the neutraliser mass was distributed between two different tuned frequencies. Although certainly useful in practical situations, this device has not quite lived up to its promises since its introduction in the mid-1980s. More recently a damped mass–spring vibration neutraliser system attached continuously along the beam length has been investigated by Thompson [13] in order to attenuate structural waves in beams. Although a tuned system, it was designed to be effective over a wide frequency range by using a high damping loss factor and multiple tuned frequencies. The development of such a system was motivated by the desire to reduce structural waves along a railway track over a broad frequency band. This paper expands on the previous work of a single vibration neutraliser described in [10] and investigates the application of distributed vibration neutralisers to control the transmission of flexural waves on a uniform beam. An infinite Euler–Bernoulli beam is chosen to avoid the unwanted resonant behaviour of the beam. Whilst the vibration devices studied in [11,12] are both referred to as distributed neutralisers, their performance for flexural wave attenuation follows very different principles. For convenience, the term ‘multiple neutralisers’ is used for an array of neutralisers attached to the beam at slightly different locations; the term ‘continuous neutraliser’ is used for a continuous mass– spring–damper system covering a certain length of the beam. This paper is set out as follows. Section 2 describes an analytical model of a beam-like vibration neutraliser with specific configurations attached on an infinite beam for flexural wave suppression. Based on this model, the analysis is extended in Section 3 to wave suppression on a beam using multiple neutralisers. Using a recursive algorithm, general expressions for wave transmission and reflection metrics are derived. Rather than focus on wave attenuation at discrete tuned frequencies, the multiple neutralisers provide a relatively wide stop band due to the interfering effect of the waves among each neutraliser in the array. In Section 4, the performance of the multiple vibration neutralisers is characterised by the minimum transmission efficiency, the normalised bandwidth and the shape factor. In Section 5, a continuous neutraliser is considered under the assumption that it only applies a force to the beam and expressions are also given to calculate wave transmission and reflection metrics. Its performance in relation to flexural wave suppression is further compared to the array of neutralisers. Finally, some conclusions are drawn in Section 6. 2. A model of a beam-like neutraliser Before studying distributed neutralisers attached to a beam, it is helpful to review the control of flexural wave motion on a beam using a single beam-like neutraliser. Salleh and Brennan [10] described four ways in which a beam-like neutraliser can be attached to the beam. They are such that (a) it applies a force only to the beam; (b) it applies a moment only to the beam; (c) it applies both a force and a moment to the beam, but these are not independent, i.e., they are coupled; and (d) it applies a force and a moment which are independent, i.e., they are uncoupled. They showed that the uncoupled force–moment-type neutraliser has some advantages over the other three configurations, since it can significantly attenuate both translational and rotational motions and has two tuned frequencies. Due to the simple implementation of the force-type and uncoupled force–moment-type neutralisers, these two neutraliser configurations are of particular interest in this paper. They are shown in Fig. 1. The neutraliser consists of a beam with a mass fitted to each end as shown in Fig. 1(a). It is attached in the middle to the infinite beam. If the device lies across the beam, i.e., y =901 as shown in Fig. 1(b), then it will act as a force-type neutraliser. If it is rotated so that it is in-line with the beam (y = 01) as shown in Fig. 1(c), then it will act as an uncoupled force–moment-type neutraliser. Therefore, these two neutralisers can

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 m/2

m/2

Attachment point Upstream a

Force-type neutraliser

+

a-

x=0

Downstream

m Kt

b+

∞

∞ Infinite beam Uncoupled forcemoment-type neutraliser x = 0 a+

m/2

-

a

a

a

m/2 Kt

b+

∞

∞

Fig. 1. Diagram showing neutraliser configurations for controlling flexural waves on an infinite beam: (a) attachment configuration of a beam-like neutraliser; (b) force-type neutraliser; (c) uncoupled force–moment-type neutraliser. The total mass and the translational dynamic stiffness of the neutraliser are denoted by m and Kt, respectively; a denotes the length of the moment arm, which is assumed to be rigid and massless; a þ denotes a set þ of positive-going waves incident upon the neutraliser; b and a denote the sets of waves transmitted downstream and reflected upstream, respectively.

be implemented by simply adjusting the orientation of a single beam-like device. In general, the displacement at any point along the beam consists of four components including positive- and negative-going propagating waves and two near-field components, which can be regarded as positive- and negative-going attenuating waves. When a vibration neutraliser is attached to an infinite beam as shown in Fig. 1, if the beam is excited upstream of the neutraliser and a vector of positivegoing waves a þ including a propagating wave Ai and a near-field wave Ani is incident upon the neutraliser, waves are þ transmitted downstream as vector b and reflected upstream as vector a . These are 2  1 vectors of positive- and negative-going waves with the time dependence ejot suppressed for clarity. The relationships between these waves are given by b

þ

¼ Ta þ ;

a ¼ Ra þ ,

(1a,b)

where T and R are the transmission and reflection matrices, respectively. Applying the continuity conditions at the attachment point, i.e., the beam displacement wð0Þ and slope w’ð0Þ for the positive and negative regions are the same, gives " # " # " # 1 1 1 1  1 1 þ (2) aþ þ a ¼ b , j 1 j 1 j 1 pffiffiffiffiffiffiffi where j ¼ 1. The relationships between the corresponding force F and moment M exerted by the neutraliser, and the displacement and slope of the beam at the point with the neutraliser fitted are given by [10] 8 9 F >  ( wð0Þ ) = < EIk3f > 1 s wuð0Þ ¼  e , (3) t M > s s2 kf : EIk2 > ; f

where E and I are the Young’s modulus and the second moment of area of the beam, respectively; kf ¼ ðrA=EIÞ0:25 o0:5 ¼ 2p=l is the flexural wavenumber of the beam where l, r and A are the flexural wavelength, the density and the crosssectional area of the beam, respectively; the non-dimensional translational dynamic stiffness of the neutraliser et is given by [5]

et ¼

Kt cð1þ jZÞ ¼ 2 , EIk3f O 1jZ

(4)

where Kt is the translational dynamic stiffness of the neutraliser; O ¼ o=on is the ratio of the excitation frequency to the natural frequency of the neutraliser; Z denotes the loss factor of the hysteretically damped spring of the neutraliser; c is the ratio of the neutraliser mass to the beam mass of approximately one-sixth of a wavelength of the beam and is given by

c¼

o2 m EIk3f

¼

m

rAl=2p

(5)

and s is the ratio of the neutraliser moment arm length a to approximately one-sixth of a wavelength of the beam and is given by s ¼ a=ðl=2pÞ. It should be noted that both the mass ratio c and the moment arm ratio s are frequency dependent and proportional to o0:5 . Damping of the beam can easily be introduced if necessary by means of a complex Young’s

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Modulus E, and hence a complex flexural wavenumber kf . In this paper the natural energy loss in the beam due to internal damping, however, is not taken into account so that the effect of neutralisers on wave suppression can be seen more easily. Damping is included in the neutralisers however, as this is an important parameter and needs to be considered along with other neutraliser parameters. Applying the equilibrium conditions of the shear force and bending moment at the support, obtained by Eq. (3), gives ( #)       " 1 s 1 1 j 1 þ j 1  j 1 þ a þ a ¼ þ et (6) b : j 1 s s2 1 1 1 1 1 1 Combining Eqs. (1a,b), (2) and (6) gives the transmission and reflection matrices as ( #)1   " 1 s 1 1 j 1 e t T ¼ Iþ 4 ; j 1 s s2 1 1 " #    1 s 1 1 j 1 R ¼ e4t T, 2 j 1 s s 1 1

(7a,b)

    1 0 1 s reduces to for the force-type where I is the identity matrix; the matrix of the moment arm ratio 2 0 0 s s   1 0 for the uncoupled force–moment-type neutraliser. T and R are 2  2 ‘scattering’ matrices, which neutraliser and 0 s2 generally contain four complex elements. The first columns of T and R are the transmission and reflection coefficients, which indicate the contributions made by the incident propagating wave Ai , whereas the second columns contain the coefficients due to the incident near-field wave Ani . In general, the near-field effects are localised around discontinuities, and no energy is transmitted by a near field unless it interacts with another near-field wave. Because it decays rapidly with increasing distance from the point at which it is generated, its effects can be neglected at distances greater than about 0.75l from a discontinuity or source. Thus, vibration neutralisers are mostly designed to suppress propagating flexural waves. Nevertheless, the effect of near-field waves is taken into account in Eqs. (7a,b), since the energy conversion at the discontinuities along the beam is significantly influenced by the near-field. There is further discussion on this in Section 4, for a beam with multiple neutralisers attached. The energy carried by a propagating wave is proportional to the square of the wave amplitude. Therefore, the effectiveness of a neutraliser in attenuating the incident flexural wave can be evaluated by the transmission efficiency t, which is defined as the square of the modulus ratio of the transmitted propagating wave to the incident wave, and is given by  2 At  t ¼ jT11 j2 ¼   (8) Ai Similarly, the square of the modulus ratio of the reflected propagating wave to the incident wave gives the reflection efficiency r by  2 Ar  (9) r ¼ jR11 j2 ¼   Ai For an undamped neutraliser (Kt is real), substituting Eqs. (7a,b) into (8) and (9), the trasmission and reflection effficiencies satisfy the relationship t +r = 1. This is because energy incident at the attachment point must balance that leaving it for an undamped neutraliser. Fig. 2 is a plot of the transmission efficiency t as a function of frequency ratio O for an uncoupled force–moment-type neutraliser with parameters cn ¼ 0:5, Z ¼ 0:001, and s2n ¼ 0, 0.5, 1.0, and 1.5. Throughout this paper, the subscript n denotes that the parameter is evaluated at the undamped natural frequency of translational motion of the neutraliser. In particular, for a force-type neutraliser to be effective (s2n ¼ 0), it has to behave in a stiffness-like manner at its tuned frequency [5]. Eq. (4) shows that the dynamic stiffness of the force-type neutraliser behaves as a mass-like element below its natural frequency and stiffness-like element above its natural frequency. Thus, the excitation frequency at which wave suppression can be achieved is always above the natural frequency of the neutraliser. As can be seen from the figure, the minimum transmission efficiency occurs above O = 1. For the uncoupled force–moment-type neutraliser of s2n ¼ 0:5, however, two tuned frequencies can be seen in the figure. It also shows that the first tuned frequency coincides with the natural frequency of the neutraliser, i.e., O = 1. When an undamped neutraliser is attached to the beam, previous work [5,9] has shown that half of the incident energy in a propagating flexural wave is reflected at the neutraliser natural frequency for both the force-type and moment-type neutralisers. Hence for the uncoupled force–moment-type neutraliser, an independent force and a moment are generated, which can be used to reflect the incident wave completely. In practice, complete wave suppression cannot be achieved since a neutraliser always contains some damping. Clearly, the corresponding minimum transmission efficiency at O =1 is largely controlled by damping in the neutraliser. As discussed above, the tuned frequency of a force-type neutraliser occurs above its natural frequency. When an additional moment is applied to the beam independent of the force, the tuned frequency of the force-type neutraliser is shifted towards its

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0

Transmission efficiency t (dB)

-10 -20 -30 -40 -50 -60 -70 -80 -90 0.9

0.95

1

1.05

1.1

1.15

1.2

Frequency ratio Ω Fig. 2. Transmission efficiency t of an uncoupled force–moment-type neutraliser as a function of frequency ratio O. cn ¼ 0:5 and Z ¼ 0:001: ———, s2n ¼ 0; , s2n ¼ 0:5; , s2n ¼ 1; , s2n ¼ 1:5.

–––

······

–·–·

natural frequency to give the second tuned frequency of the uncoupled force–moment-type neutraliser. As shown in Fig. 2, a lower level of transmission efficiency of the uncoupled force–moment-type neutraliser implies that the performance of the beam-like neutraliser for wave suppression can be enhanced by using the additional rotational discontinuity. Moreover, it can be seen from the figure that when s2n ¼ 1, a significant amount of wave attenuation occurs at the natural frequency of the neutraliser. In this case, the second tuned frequency coincides with the first tuned frequency. For the uncoupled force–moment-type neutraliser of s2n ¼ 1:5, Fig. 2 shows that the minimum transmission efficiency can be found below its natural frequency. Thus, the moment arm ratio sn is a critical parameter in the design of the uncoupled force– moment-type neutraliser. When s2n o 1, the performance of the neutraliser is predominantly governed by the translational constraint with the second tuned frequency above its natural frequency. In contrast, when s2n 4 1, the rotational constraint plays a dominant role for wave suppression so that the second tuned frequency occurs below its natural frequency. In practical situations, however, a large value of s2n 41 is difficult to achieve. Therefore the first case when s2n o1 is of particular interest in this paper. Compared to the performance of the force-type neutraliser, the main advantage of the uncoupled force–moment-type neutraliser is that the incident flexural waves can be effectively attenuated over a wider frequency range between the two tuned frequencies, as reported by Salleh and Brennan [10].

3. Wave suppression using multiple neutralisers In general, propagating waves incident on discontinuities can experience reflection (some energy in the waves is reflected), absorption (some energy is ‘absorbed’, i.e., energy losses at the discontinuities) and transmission (waves are transmitted past these features). Consider the array of N neutralisers attached to an infinite beam shown in Fig. 3. The subscript i denotes the ith neutraliser in the array. The separation distance between the ith and (i+ 1)th neutralisers is li. A diagonal field transfer matrix " # ejkf li 0 Fi ¼ 0 ekf li is introduced between two adjacent neutralisers, the diagonal elements of which present wave propagating and decaying along the distance li, respectively. To investigate the performance of this array on flextural wave suppresion it is helpful to initially introduce the concepts of net transmission and reflection matrices when the array contains two neutralisers. Consider now a set of positive-going waves a1þ incident upon the 1st neutraliser. The wave amplitudes at the 1st neutraliser are related by 

þ a 1 ¼ R1 a1 þ T1 b1 ;

þ



b1 ¼ T1 a1þ þ R1 b1 :

(10a,b)

þ b1

The transmitted waves past the 1st neutraliser, which are incident upon the 2nd neutraliser, are further reflected and transmitted along the beam. Similarly the reflected and transmitted wave amplitudes at the 2nd neutraliser are given by 

þ a 2 ¼ R2 a2 þ T2 b2 ;

þ



b2 ¼ T2 a2þ þ R2 b2 :

(11a,b)

Y. Gao et al. / Journal of Sound and Vibration 330 (2011) 2758–2771

x=0

Upstream + a1

2763

Downstream

Kt1

+

Kt2

+ ai

b1

−

+ bi

−

−

a1

Kti

+ aN −

−

ai

b1

Kt(N-1)

aN

bi

KtN

+ bN −

bN

∞

∞ Fig. 3. An infinite beam with an array of N neutralisers.

The wave amplitudes upstream a2þ and a 2 at the 2nd neutraliser are related to those downstream at the 1st neutraliser þ  b1 and b1 by the propagation relations þ



a2þ ¼ F1 b1 ;

b1 ¼ F1 a 2: 12

 b2

(12a,b) 11

Assuming ¼ 0, the net transmission and reflection matrices T and R are related to the vectors of wave amplitudes þ downstream of the 2nd neutraliser b2 and those upstream at the 1st neutraliser a 1 , and the vector of incident wave þ amplitudes a1 by b2 ¼ T12 a1þ ;

11 þ a 1 ¼ R a1 :

þ

(13a,b)

Combining Eqs. (10a,b)–(13a,b), the net transmission and reflection matrices are given by T12 ¼ T2 ðIF1 R1 F1 R2 Þ1 F1 T1 ;

R11 ¼ R1 þ T1 ðIF1 R2 F1 R1 Þ1 F1 R2 F1 T1 :

(14a,b)

Next, based on the results of the transmission and reflection matrices determined by Eqs. (7a,b) and the net transmission and reflection matrices given by Eqs. (14a,b), the behaviour of the beam with N neutralisers in the array,  in response to an incident flexural wave, is determined in a similar way. Referring to Fig. 3, setting bN ¼ 0, the vectors of þ wave amplitudes downstream of the array bN and upstream of the array a are related to the vector of incident waves a1þ 1 by the net transmission matrix T1N and the net reflection matrix R11 by bN ¼ T1N a1þ ;

11 þ a 1 ¼ R a1 :

þ

Similar to the analysis of two discontinuities, the net transmission and reflection matrices T by using the net transmission and reflection matrices T2N and R22 , and are given by T1N ¼ T2N ðIF1 R1 F1 R22 Þ1 F1 T1 ; where the net transmission and reflection matrices T reflection matrices T3N and R33 by

2N

T2N ¼ T3N ðIF2 R2 F2 R33 Þ1 F2 T2 ;

(15a,b) 1N

and R

R11 ¼ R1 þT1 ðIF1 R22 F1 R1 Þ1 F1 R22 F1 T1 ; and R

22

11

can be expressed (16a,b)

are further expressed by using the net transmission and

R22 ¼ R2 þT2 ðIF2 R33 F2 R2 Þ1 F2 R33 F2 T2 ; ^

(17a,b)

where the net transmission and reflection matrices TiN and Rii can be further expressed by using the net transmission and reflection matrices Tði þ 1ÞN and Rði þ 1Þði þ 1Þ , and are given by TiN ¼ Tði þ 1ÞN ðIFi Ri Fi Rði þ 1Þði þ 1Þ Þ1 Fi Ti ; ii

R ¼ Ri þ Ti ðIFi Rði þ 1Þði þ 1Þ Fi Ri Þ1 Fi Rði þ 1Þði þ 1Þ Fi Ti ,

(18a,b)

^ where the net transmission and reflection matrices TðN1ÞN and RðN1ÞðN1Þ can be finally expressed by using the transmission matrix TN (corresponding to the net transmission matrix TNN ) and the reflection matrix RN (corresponding to the net reflection matrix RNN ) by TðN1ÞN ¼ TN ðIFN1 RN1 FN1 RN Þ1 FN TN ; R

ðN1ÞðN1Þ

¼ RN1 þTN1 ðIFN1 RN FN1 RN1 Þ1 FN1 RN FN1 TN1 :

(19a,b)

The reflection and transmission matrices R1 ,R2 ,. . .,RN , and T1 ,T2 ,. . .,TN , can be obtained by Eqs. (7a,b) for discontinuities i= 1–N. The corresponding transmission and reflection efficiencies t and r are given by the square of the modulus ratios of the transmitted and reflected propagating waves to the incident wave, which are the first terms of the net matrices T1N and R11 , respectively. 4. Characteristics of a neutraliser ARRAY Consider the array of N neutralisers with a fixed distance of l (li =l) between two adjacdent neutralisers as shown in Fig. 3. Each neutraliser has the same configuration, and hence the same nautral frequency. The total mass m of the mulitple neutralisers is kept constant, i.e., mi ¼ m=N. The transmission efficiency of multiple force-type neutralisers (s2n ¼ 0) with parameters cn ¼ 0:5, Z ¼ 0:001, and l ¼ 0:16ln is plotted in Fig. 4 for an array with up to ten neutralisers

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0

Transmission efficiency t (dB)

-20

-40

-60

-80

-100

-120 0.9

0.95

1

1.05 1.1 Frequency ratio Ω

1.15

1.2

Fig. 4. Transmission efficiency t of a force-type neutraliser array as a function of frequency ratio O. cn ¼ 0:5, Z ¼ 0:001 and l ¼ 0:16ln : ——, N=1; , N=5; , N=10.

······

–·–·

– – –, N=2;

0 -3

Transmission efficiency t (dB)

-10 -20 -30 -40 -50 -60 -70 0.95

1

1.05

1.1

Frequency ratio Ω Fig. 5. Diagram showing the calculation of the shape factor: filter.

area of the neutraliser array in its bandwidth;

area of the corresponding ideal

attached. When two neutralisers are in the array (N =2), two tuned frequencies can be seen in the figure. The first tuned frequency occurs just above the neutraliser natural frequency, and the second one just below the tuned frequency of a single force-type neutraliser (N = 1). In contrast to a single neutraliser, the two closely spaced neutralisers can provide a wider frequency bandwidth over which the transmission efficiency is lower. An important quantity to evaluate the performance of multiple neutralisers for wave suppression is the bandwidth, which is defined here as the frequency range for which the transmission efficiency is less than 3 dB, as suggested in [10]. When N = 5 and 10, more ripples appear in the tuned frequency bandwidth. The local minimum transmission efficiency occurs at successively increasing frequencies due to wave dispersion kf po0:5 . Moreover, it is observed that the minimum transmission efficiency becomes less by splitting a single device into a greater number of neutralisers in an array at the expense of the tuned frequency bandwidth. Thus there is a trade-off between the minimum transmission efficiency and the tuned frequency bandwidth. A further quantity worthy of consideration is the ‘shape factor’. Clearly, the multiple neutralisers have the property of a band-stop filter. The ideal characteristic of such a filter is that it passes most frequencies unchanged but attenuates those in its stop band to a very low level. The shape factor can be determined by dividing the area of the bandwidth of the array under the curve by that of the ideal filter as illustrated in Fig. 5. No simple formulae exist to characterise the dynamic behaviour of the neutraliser array. In terms of the minimum transmission efficiency, the normalised bandwidth and the shape factor, several factors that can affect the performance of the array for N =1–10 are studied in the following subsections.

Y. Gao et al. / Journal of Sound and Vibration 330 (2011) 2758–2771

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4.1. Effect of the neutraliser damping For multiple neutralisers with cn ¼ 0:5 and l ¼ 0:16ln , the characteristics for different loss factors Z =0.001, 0.01 and 0.1 are plotted in Fig. 6. It can be seen from Fig. 6(a,b) that both the minimum transmission efficiency and the normalised bandwidth increase by adding damping to the device. For small value of loss factor Z = 0.001 and 0.01, the minimum transmission efficiency decreases with an increasing number of neutralisers in an array whilst the penalty of the reduction on the normalised bandwidth is more obvious. When Z = 0.1, the change of the minimum transmission efficiency is insignificant whilst the normalised bandwidth decreases slightly with the number of neutralisers. Fig. 6(c) shows that the shape factor is improved by adding damping to the vibration neutralisers (except when Z = 0.01 and N 47). When Z = 0.1, the shape of the transmission efficiency for the array is worse than for a single neutraliser, but when Z = 0.001 and 0.01, the shape factor is generally increased (except when Z =0.01 and N 45). 4.2. Effect of the neutraliser mass When Z = 0.001 and l ¼ 0:16ln , the effect of the neutraliser mass on the characteristics of multiple neutralisers are shown in Fig. 7 for mass ratios cn = 0.1, 0.5 and 1.0. It can be seen in the figures that the minimum transmission efficiency decreases as the neutraliser mass is increased whilst the normalised bandwidth of the overall neutraliser array increases although the natural frequency of an individual neutraliser decreases. Fig. 7(a, b) also show clearly the trade-off between the minimum transmission efficiency and the tuned frequency bandwidth with an increasing number of neutralisers in the array. As shown in Fig. 7(c), the shape factor is worse for larger mass ratio (except when cn ¼ 0:1 and N 47). Multiple

0.2

-20 Normalised bandwidth

Minimum transmission efficiency (dB)

0

-40 -60 -80

0.15

0.1

0.05

-100 0

-120 1

2

3

4 5 6 7 Number of neutralisers

8

9

10

1

2

3

4 5 6 7 Number of neutralisers

8

9

10

0.8

Shape factor

0.7 0.6 0.5 0.4 0.3 0.2 1

2

3

4 5 6 7 Number of neutralisers

8

9

10

Fig. 6. Comparison of some characteristics of a neutraliser array for different loss factors. cn ¼ 0:5, l ¼ 0:16ln , and ———, Z = 0.001; , Z = 0.1: (a) minimum transmission efficiency; (b) normalised bandwidth; and (c) shape factor.

······

– – –, Z = 0.01;

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0.25

-40

0.2 Normalised bandwidth

Minimum transmission efficiency (dB)

-20

-60 -80 -100

0.15

0.1

0.05

-120 -140

0 1

2

3

4 5 6 7 Number of neutralisers

8

9

10

1

2

3

4 5 6 7 Number of neutralisers

8

9

10

0.6

Shape factor

0.5

0.4

0.3

0.2 1

2

3

4

5

6

7

8

9

10

Number of neutralisers Fig. 7. Comparison of some characteristics of a neutraliser array for different mass ratios. Z ¼ 0:001, l ¼ 0:16ln , and ———, cn = 0.1; , cn =1.0: (a) minimum transmission efficiency; (b) normalised bandwidth; and (c) shape factor.

······

– – –, c

n = 0.5;

neutralisers can improve the shape of the transmission efficiency compared to a single vibration neutraliser and the values of the shape factor are generally increased with the number of neutralisers (except when cn ¼ 0:1 and N 45). 4.3. Effect of the separation It has been shown in Section 3 that the band-stop property of multiple neutralisers is caused by the coherent interactions of wave motion due to the discontinuities. Thus the separation l between two neutralisers in the array is an important parameter in controlling the overall effectiveness for wave suppression. Fig. 8 plots some characteristics of multiple neutralisers with cn ¼ 0:5, Z =0.001 and different separation ratios l ¼ 0:016ln , 0:08ln , and 0:16ln . For small separation ratio l ¼ 0:016ln , the change of the characteristics with the number of neutralisers is indistinguishable. As can be seen from the figures, when the separation ratio becomes larger, the minimum transmission efficiency and the shape factor are improved at the expense of the normalised bandwidth. The effect of the separation on the characteristics can be accentuated by increasing the number of neutralisers in the array. However, when the neutralisers are well separated, e.g., the separation is of the order of the wavelength ln , the interference of the waves is not obvious and therefore out of scope of our investigation in this paper. 4.4. Effect of the moment arm As discussed in Section 2, an uncoupled force–moment-type neutraliser outperforms a force-type neutraliser in two respects, that being the minimum transmission efficiency and the occurrence of two tuned frequencies. This implies that

Y. Gao et al. / Journal of Sound and Vibration 330 (2011) 2758–2771

0.14 0.12

-40 Normalised bandwidth

Minimum transmission efficiency (dB)

-20

2767

-60

-80

-100

0.1 0.08 0.06 0.04

-120

0.02 1

2

3

4 5 6 7 Number of neutralisers

8

9

10

1

2

3

4 5 6 7 Number of neutralisers

8

9

10

Shape factor

0.5

0.4

0.3

0.2 1

2

3

4 5 6 7 Number of neutralisers

8

9

10

Fig. 8. Comparison of some characteristics of a neutraliser array for different separation ratios. cn ¼ 0:5, Z ¼ 0:001, and ———, l ¼ 0:016ln ; l ¼ 0:08ln ; , l ¼ 0:16ln : (a) minimum transmission efficiency; (b) normalised bandwidth; and (c) shape factor.

······

– – –,

the device is more effective for wave suppression over a tuned frequency bandwidth. Furthermore, it has been found that the performance of such a device is largely affected by the moment arm. It must be noted that for a single neutraliser a reasonable length of moment arm in practice is when s2n o 1 as suggested in Section 2. However, this is further restricted to satisfy s2n rðpl=ln Þ2 when these neutralisers are used in an array. Their characteristics are plotted in Fig. 9 for cn ¼ 0:5, Z = 0.001, l ¼ 0:16ln and different moment arms s2n ¼ 0, 0.1 and 0.25. As can be seen from Fig. 9(a, b), the minimum transmission efficiency decreases largely as the moment arm ratio is increased with no penalty in the reduction of the normalised bandwidth. Compared to the force-type neutralisers, Fig. 9(c) shows that the shape of the transmission efficiency becomes worse when the uncoupled force–moment-type neutralisers are used. It can generally be improved by splitting the single device into more neutralisers configured to form a multiple neutraliser array. A vibration neutraliser generally has small mass and damping ratios. Consequently, it is more beneficial to have an array of closely spaced uncoupled force–moment-type neutralisers for flexural wave suppression in a beam. 5. Wave suppression using a continuous neutraliser Flexural wave suppression on an infinite beam using a neutraliser array has been investigated in Sections 3 and 4. If a single neutraliser is divided into more neutralisers in the array so that the relative distance between two adjacent neutralisers li -0 (without overlap of the attachment configurations in the array), then the multiple neutralisers become a continuous plate-like neutraliser, as shown in Fig. 10. Simliar to the anlysis previously, the total mass of a continuous neutraliser remains unchanged. The bending stiffness of the device is assumed to be much more flexible than the beam itself and so is ignored. Thus the continuous neutraliser studied in this section exerts a force only upon the beam. Referring

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Y. Gao et al. / Journal of Sound and Vibration 330 (2011) 2758–2771

0.14

-60

0.12

-80

Normalised bandwidth

Minimum transmission efficiency (dB)

-40

-100 -120 -140 -160

0.1 0.08 0.06 0.04

-180 -200

0.02 1

2

3

4 5 6 7 Number of neutralisers

8

9

10

1

2

3

4 5 6 7 Number of neutralisers

8

9

10

Shape factor

0.5

0.4

0.3

0.2 1

2

3

4

5

6

7

8

9

10

Number of neutralisers Fig. 9. Comparison of some characteristics of a neutraliser array for different moment arm ratios. cn ¼ 0:5, Z = 0.001, l ¼ 0:16ln and ———, s2n ¼ 0; , s2n ¼ 0:1; , s2n ¼ 0:25: (a) minimum transmission efficiency; (b) normalised bandwidth; and (c) shape factor.

–––

······

Upstream

m′

+

a

a-

x=L

x=0 Continuous neutraliser

Downstream +

b K t′

∞

∞ Fig. 10. An infinite beam attached with a continuous neutraliser.

to Fig. 10, the continuous neutraliser is assumed to have mass per unit length m0 and translational dynamic stiffness per unit length Kt u . In the region 0 rxrL where the neutraliser is attached, the distributed force FuðxÞ exerted by the device on the beam is given by FuðxÞ ¼ KtuwðxÞ:

(20)

Regardless of the effects of shear deformation and rotary inertia, for time harmonic motion of the form ejot , the forced vibration of the beam satisfies EI

d4 wðxÞ rAo2 wðxÞ ¼ FuðxÞ: dx4

(21)

Y. Gao et al. / Journal of Sound and Vibration 330 (2011) 2758–2771

2769

Subsituting Eqs. (20) into (21) gives EI

d4 wðxÞ ðrAo2 KtuÞwðxÞ ¼ 0: dx4

(22)

Eq. (22) shows that the continuous mass–spring–damper system changes the flexural wavenumber kf of the original untreated beam in the region 0 rxrL to be " k ¼ kf 1

mð1þ jZÞ O2 1jZ

#0:25 ,

(23)

where m is the mass density ratio of the neutraliser mass m per unit length to the beam mass per unit length, which is independent of frequency and is related to c by

m¼

c

(24)

:

kf L

Although damping is not considered in the infinite beam (kf is real), the attachment of such a device results in a change of both the propagating wavespeed and wave attenuation since the new wavenumber due to a continuous neutraliser has both real and imaginary parts. Thus, for a sufficient length L, the use of such a distributed neutraliser will be effective for the control of propagating flexural waves in the beam at any tuned frequency. A particular application of a continuous damped mass–spring system has been studies for the reduction of noise from a railway track [13]. The focus in this paper, however, is on the effectiveness of a continuous neutraliser of length L for flexural wave attenuation. Similar to the study of wave motion in Section 2, the flexural wave amplitudes at the discontinuity x =0 can be obtained from four equations expressing continuity and equilibrium, and are given by "

#

1

1

j

1

"

a0þ

þ

1

1

j

1

"

# a 0

¼

#

1

1

jw

w

" þ b0

þ

1

1

jw

w

# 

b0

(25)

and 

"  jw3 a ¼ 0 w2 1

  j a0þ þ 1 1

j 1

w3

1

1

#

w2

" þ

b0 þ

#

jw3

w3  b , w2 0

w2

(26)

where the wavenumber ratio w on the beam with and without the continuous neutraliser attached can be obtained from Eq. (23) by

w¼ þ

" #0:25 k mð1 þjZÞ ¼ 1 2 : kf O 1jZ

(27)



The wave amplitudes b0 and b0 at x= 0 are related to those downstream aLþ and a L at the discontinuity x =L by the propagation relations þ

aLþ ¼ FL b0 ;



b0 ¼ FL a L :

(28a,b)

Here the diagonal field transfer matrix is " FL ¼

ejkL

0

0

ekL

# :

Applying the continuity and equilibrium conditions at the discontinuity x ¼ L gives "

#

1

1

jw

w

"

aLþ þ

1

1

jw

w

#

"

a L ¼

1

1

j

1

#

" þ

bL þ

1

1

j

1

# 

bL

(29)

and "

jw3 w2

w3

w2

#

" aLþ þ

jw3 w2

#

w3   j 1  þ  j 1   bL þ b : a ¼ w2 L 1 1 1 1 L

(30)

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Y. Gao et al. / Journal of Sound and Vibration 330 (2011) 2758–2771

Eqs. (25)–(30) can be combined to give the following relations: "

a0þ

#

a 0

2

1

6 j 6 ¼6 4 j 1 2 1 6 6 jw 6 3 6 jw 4 w2

1

1

1

j

1 1

j 1

31 2 1 6 jw 17 7 6 7 6 3 15 6 4 jw w2 1

1

1

w

jw

w3

jw3 w2

w2

1

1

w

jw

w3

jw3

1

1

2

31 2

w7 7 7 w3 7 5 w2

w2

w

1 6 j 6 6 4 j 1

1

3 "

w7 7 F1 0 7 L 37 w 5 0 FL w2 1

#

3

1

1

1

j

1

j

" þ# 17 7 bL 7  : 1 5 bL

1

1

1

(31)

 bL

¼ 0, the transmission and reflection matrices can be determined from Eq. (31), which are related to the wave Setting þ þ amplitudes downstream bL and upstream a 0 , and the incident waves a0 by þ

bL ¼ Ta0þ ;

þ a 0 ¼ Ra0 :

(32a,b)

0

Transmission efficiency t (dB)

-10 -20 -30 -40 -50 -60 -70 -80 0.9

0.95

1

1.05

1.1

1.15

1.2

Frequency ratio Ω Fig. 11. Transmission efficiency t of a continuous neutraliser as a function of frequency ratio O. cn ¼ 0:5 and Z = 0.001: ———, m = 0.1 (L ¼ 0:8ln ); , m = 0.25 (L ¼ 0:32ln ); , m = 0.5 (L ¼ 0:16ln ); , m =1 (L ¼ 0:08ln ).

–––

–·–·

······

0

Transmission efficiency t (dB)

-10 -20 -30 -40 -50 -60 -70 -80 0.9

0.95

1

1.05

1.1

1.15

1.2

Frequency ratio Ω Fig. 12. Comparison of the transmission efficiency t of distributed neutralisers as a function of frequency ratio O. cn ¼ 0:5 and Z = 0.001: continuous neutraliser ———, m = 0.1 (L ¼ 0:8ln ); neutraliser array , N = 10 (l ¼ 0:09ln ); , N = 5 (l ¼ 0:2ln ); , N = 2 (l ¼ 0:8ln ); , N = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

–––

······

–·–·

––––

Y. Gao et al. / Journal of Sound and Vibration 330 (2011) 2758–2771

2771

For the continuous neutraliser with parameter cn ¼ 0:5 and Z ¼ 0:001, when the total mass of the neutraliser is kept constant, different mass density ratios m = 0.1, 0.25, 0.5 and 1 are obtained by adjusting length L ¼ 0:8ln ,0:32ln ,0:16ln , and 0:08ln . The corresponding transmission efficiency is plotted in Fig. 11. Clearly, when m = 0.1, an effective tuned frequency bandwidth and a lower level of transmission efficiency are achieved by using the continuous neutraliser. As the neutraliser increases in size (increase m), Fig. 11 shows that the band-stop properties of the device becomes undistinguishable. In particular when m b1, only one tuned frequency can be found in the figure. In this case, the overall continuous neutraliser acts as a single neutraliser. Alternatively, the multiple neutralisers consisting of an array of neutralisers can be attached to the beam over a certain length L. Different numbers of neutralisers can be chosen to make up the multiple neutralisers with the same combined mass in the array. Their performances for wave attenuation are shown in Fig. 12 in comparison with the continuous neutraliser which has a mass density ratio of m = 0.1 (L ¼ 0:8ln ). As discussed in Section 2, when a single neutraliser is applied (N = 1), a sharp peak transmission efficiency occurs at the tuned frequency above the neutraliser natural frequency. When N = 2, a broader peak with smaller minimum transmission efficiency is found in Fig. 12. Although the performance for wave attenuation is improved using two neutralisers, the bandwidth is still very narrow. In fact, the tuned frequencies of the device coincide with those of each neutraliser (which are the same and are thus overlaid) because the effects of the coherent interference of the waves become negligible due to the large separation distance compared to the wavelength (l ¼ 0:8ln ). As more neutralisers are used in the array, the band-stop property of the neutralise array becomes more distinct. Good agreement of the transmission efficiency for the neutraliser array and the continuous neutraliser can be achieved using ten neutralisers with the separation distance of 0:09ln between two adjacent neutralisers. 6. Conclusions The use of distributed vibration neutralisers to control flexural waves on a beam has been investigated theoretically in this paper. General expressions for wave transmission and reflection metrics have been derived for both multiple neutralisers in an array and a continuous neutraliser attached to an infinite beam. In contrast with a single vibration neutraliser, the effectiveness of the distributed devices are characterised by the minimum transmission efficiency, the normalised bandwidth and the shape factor. For the neutraliser array to be effective over a wide frequency band, the separation distance in the array must be relatively small compared to the wavelength of the beam at the excitation frequency, due to the beneficial effects of the coherent interference of the waves on the band-stop properties. It has also been found that the performance of such a device can be enhanced by the introduction of a moment arm, which leads to an uncoupled force–moment-type device that can be realised using a particular configuration of a beam-like neutraliser. For a continuous neutraliser attached to the beam, a new wavenumber has been determined for the attachment region. The continuous neutraliser can effectively act as a band-stop filter when it has a small mass density ratio compared to the beam. Analytical results have shown that the transmission efficiency given by the continuous neutraliser is in good agreement with that achieved from the neutraliser array for a sufficiently large number of neutralisers. References [1] M.J. Brennan, Characteristics of a wideband vibration neutralizer, Noise Control Engineering Journal 45 (1997) 201–207. [2] J. Ormondroyd, J.P.D. Hartog, Theory of the dynamic vibration absorber, Transactions of the ASME 50 (1928) 9–22. [3] M. Sharif-Bakhtiar, S.W. Shaw, The dynamic response of a centrifugal pendulum vibration absorber with motion-limiting stops, Journal of Sound and Vibration 126 (1988) 221–235. [4] J.C. Snowdon, A.A. Wolfe, R.L. Kerlin, The cruciform dynamic absorber, Journal of Acoustical Society of America 75 (1984) 1792–1799. [5] M.J. Brennan, Control of flexural waves on a beam using a tunable vibration neutraliser, Journal of Sound and Vibration 222 (1998) 389–407. [6] J.C. Snowdon, Platelike dynamic absorbers, Transactions of ASME Series b, Journal of Engineering for Industry 97 (1975) 88–93. [7] J.C. Clemens, Plate-like dynamic absorbers-comparison of measurement and theory, Journal of Acoustical Society of America 75 (1984) 638. [8] C. Maurinia, F. dell’Isola, D.D. Vescovo, Comparison of piezoelectronic networks acting as distributed vibration absorbers, Mechanical Systems and Signal Processing 18 (2004) 1243–1271. [9] P. Clark, Devices for the Reduction of Pipeline Vibration, PhD Thesis, Institute of Sound and Vibration Research, University of Southampton, UK, 1995. [10] H. Salleh, M.J. Brennan, Control of flexural waves on a beam using a vibration neutraliser: effects of different attachment configurations, Journal of Sound and Vibration 303 (2007) 501–514. [11] V.I. Kashina, V.V. Tyutekin, Waveguide vibration reduction of longitudinal and flexural modes by means of a multi-element structure of resonators, Soviet Physics Acoustics 36 (1990) 383–385. [12] T.L. Smith, K. Rao, I. Dyer, Attenuation of plate flexural waves by a layer of dynamic absorbers, Noise Control Engineering Journal 26 (1986) 56–60. [13] D.J. Thompson, A continuous damped vibration absorber to reduce broad-band wave propagation in beams, Journal of sound and vibration 311 (2008) 824–842.