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The development of a tunable vibration absorbing isolator
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International Journal of Mechanical Sciences 47 (2005) 983–997 www.elsevier.com/locate/ijmecsci
The development of a tunable vibration absorbing isolator N.F. du Plooya,, P.S. Heynsa, M.J. Brennanb a
Dynamic Systems Group, Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria 0002, South Africa b Institute of Sound and Vibration Research, University of Southampton, Southampton, UK Received 3 September 2002; received in revised form 24 March 2005; accepted 6 April 2005
Abstract This paper describes a tunable vibration absorbing isolator. The device is based on the liquid inertia vibration eliminator, which is modified so that the frequency at which maximum isolation occurs can be changed in real time. This is achieved by using adjustable pneumatic springs. A theoretical model describing the device is derived using Lagrange’s equations. The model is used to design a practical device and experimental results confirm the validity of the model. With the experimental device a roll-off of about 90 dB/decade over a 12 Hz frequency band was achieved. r 2005 Elsevier Ltd. All rights reserved. Keywords: Tunable vibration absorbing isolator
1. Introduction The ideal vibration isolator has high static stiffness to support the isolated structure and low dynamic stiffness to ensure low transmission of dynamic forces [1]. Isolation is achieved at frequencies greater than the square root of two times the mounted resonance frequency. Above this frequency, and provided that damping in the isolator is light, the transmission of vibration reduces at 40 dB/decade. To improve on this situation, Flannely incorporated a vibration absorber in the isolator [2] and made a vibration absorbing isolator (VAI), which he called a dynamic anti-resonant vibration isolator (DAVI). The absorber modifies the dynamic behaviour Corresponding author. Tel.: +27 823314091; fax: +27 126633052.
E-mail address: [email protected] (N.F. du Plooy). 0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.04.009
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Nomenclature kj stiffness j damping j cj Z loss factor o circular frequency isolation circular frequency oi p ffiffiffiffiffiffiffi i 1 mass j mj diameter j dj area j Aj u; x; xB ; y displacement X , Y displacement amplitude force j fj force transmitted FT P pressure E Young’s modulus experimental transmissibility aX analytical transmissibility aA t thickness n ratio of specific heats h air spring height T kinetic energy V potential energy B bandwidth R robustness transmissibility Tr n Poisson’s ratio t thickness h air spring height n ratio of specific heats of the conventional isolator, providing a frequency band over which the dynamic stiffness is less than the static stiffness, and hence reduces vibration transmission in this frequency band. The device contains another innovation, in that it uses a pendulum to amplify the acceleration of the vibration absorber mass. A number of other devices utilising hydraulic mass amplification followed, for example those described by Braun [3] and Halwes [4]. Halwes introduced the liquid inertia vibration eliminator (LIVE), a device that employed hydraulic fluid as the vibration absorber mass, resulting in a compact arrangement. Halwes showed that the size of the device is inversely proportional to the square of the density of the fluid used, which means that the use of heavy liquids is attractive. All the devices discussed above were designed to be most effective at a single frequency, however, applications often exist where the excitation frequency is a function of environmental or operating
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conditions, for instance load. The main drawback of VAIs is their limited effective frequency range, defined as the range of frequencies where the transmissibility of the device is below a certain value, which might not cover the range of excitation frequencies encountered. Increasing this frequency range can be achieved in a number of ways, for instance by reducing the damping in the device. There are, however, physical limits on the amount of passive damping reduction that can be achieved practically, which has led to the development of methods to actively reduce damping as described by Kidner and Brennan [5]. It has also been found that the effective frequency range is affected by the ratio of natural frequency of the mounting system to the isolation frequency of the device [6]. A small ratio is desirable which means a low natural frequency is required since the isolation frequency is application driven and therefore fixed. It is also possible to increase the band by using a non-linear softening spring, but this may be difficult to realise in practice. Finally, the effective suppression band can be increased by tuning the VAI so that the isolation frequency is coincident with the excitation frequency. This method differs from those discussed above since the effective frequency range is not increased, but rather the isolation frequency is adjusted (with a fixed bandwidth) to create a wider suppression band. The tuning method assumes that excitation frequency can be determined with sufficient accuracy and that a system parameter can be adjusted so that the isolation and excitation frequencies coincide. This device is called a tunable vibration absorbing isolator (TVAI), and the aim of this paper is to describe the development of such a device. Following this introduction a mathematical model of the TVAI is presented in Section 2. In Section 3 experimental work on a prototype device is described, and this is followed with some concluding remarks in Section 4.
2. The tunable vibration absorbing isolator 2.1. Description of the device The TVAI is discussed with reference to the LIVE system, which is shown in Fig. 1. The displacement transmissibility for this device is given by [6]: Ab Ab 2 kð1 þ iZÞ þ ioc þ o mB 1 X Aa Aa " (1) ¼ 2 # , Y A b 2 kð1 þ iZÞ þ ioc o m þ mB 1 Aa where the absorber mass mB is the mass of the fluid in the port, m is the mass of the central part of the device, Aa ¼ pd 2a =4; Ab ¼ pd 2b =4; c is the viscous damping in the device, o is circular frequency, k is the stiffness of resilient connection between the moving parts of the device and Z is the loss factor of the resilient connection. The transmissibility pffiffiffi of the device is less than that of a conventional isolator of equivalent static deflection up to 2oi , where oi is the isolation frequency given by [6] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k u . (2) oi ¼ u t Ab Ab 1 mB Aa Aa
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mB h
x l
xB Port
m
k
da
c y
Fig. 1. Schematic of the LIVE.
This is the frequency when the transmissibility is a minimum, which occurs when the real part of the numerator in Eq. (1) equals zero. It can be seen that the stiffness, absorber mass (mB ) and port and reservoir geometry can be used to tune the absorber. It can also be seen that the isolation frequency is very sensitive to port diameter, but this parameter is difficult to adjust in practice because of the incompressibility of the absorber fluid. Nevertheless, a device using this tuning mechanism has been patented by Smith and Stamps [7]. They also describe a device that changes the port length, thereby changing the absorber mass. Several examples of variable stiffness devices exist in the literature. Examples are shape memory alloy helical springs [8] and electromagnetic springs [9]. There is a theoretical advantage in using spring stiffness since it will not affect the ratio of natural to isolation frequencies, which will affect the transmissibility at the isolation frequency. In this paper, however, the reservoir stiffness is adjusted. The main advantage of this tuning method is that it results in a robust and simple device. As with the LIVE system, the TVAI consists of two fluid reservoirs located on both sides of a port as shown in Fig. 2a. Relative motion between the port and the reservoirs forces the fluid through the port. However the TVAI differs from the LIVE system in that flexible rubber membranes terminate both reservoir cavities. The purpose of the membrane is to allow tuning of the device to the excitation frequency, as the stiffness of the membrane controls the amount of fluid that is pumped through the port, which in turn varies the isolation frequency. The stiffness of the membrane is a function of the air pressure in the air spring cavity and the stiffness of the rubber membrane. To allow a large tuning band and low damping the stiffness of the rubber must be minimised. A small choke is fitted between the air spring and the air supply to allow the exchange of air at low frequencies and to minimise flow at high frequencies. This allows tuning of the device since the static pressure across the choke equalises while the pressure difference caused by the movement of the membrane is too fast to allow substantial airflow. This maximises the stiffness of the air spring and minimises the damping. The trade-off is the tuning rate that can be achieved with a small choke diameter. Applications requiring fast tuning will call for a more
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Isolated mass (my)
Air spring cavity Pressure Control valve
Port Primary spring
Choke Reservoir (a)
Membrane
(b)
Fig. 2. TVAI.
complicated valve arrangement as described by Longbottom et al. [10]. For the device described in this paper the average pressure was adjusted manually. The TVAI can conceptually be split in two parts by the primary spring. The isolated mass can be connected to either the port or the reservoir side of the device. In the device discussed in this paper the isolated mass is connected to the reservoir side while the port is excited as shown in the photograph of the prototype depicted in Fig. 2b. Without the fluid present the isolated mass is simply suspended by the primary spring. The inertia of the mass in the port modifies the dynamic stiffness to be less than the static stiffness over a given frequency range. One advantage of this device is that the static deflection is constant regardless of the pressure present in the air spring cavity. 2.2. Mathematical model A model of the TVAI is shown in Fig. 3. Since the two membranes are coupled through the incompressible fluid, their stiffness can be modelled as having one effective value ðku Þ. The mass of the fluid in the two reservoirs is added to the mass of the casing my . Several theoretical models with various levels of complexity were developed during the course of this study. It was found that some assumptions were essential to achieve a good fit between measured data and predictions. The most important assumption regards the shape of the membrane separating the air spring from the reservoir, which is loaded on both sides by a constant pressure. Relative motion occurs because the membrane bulges due to the pressure differential acting on it. Assuming a fixed boundary condition for the membrane edge the volume of fluid displaced by the membrane can be written as a function of its absolute centre (u) and edge (y) displacements as well as the reservoir area [11]. V ¼ 13 Ab ju yj.
(3)
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½ku
½cu db do xB(t) x(t)
mx
l
my
k
da
½ku
c
u(t)
½cu y(t)
Fig. 3. Model of the TVAI.
It was found that the deformation of the primary spring must be modelled accurately. The equations of motion for the LIVE system were based on the assumption that the volume displaced by the port is simply the difference between the port and reservoir area multiplied by the relative displacement, but for the TVAI this did not result in a good model. The conical geometry of the primary spring must be taken into account as can be seen by comparing Figs. 1 and 3. The equation describing the continuity of motion for the fluid in the device can be written as:
1 Ab d b Ao d o Aa ðx yÞ þ ðAb Aa Þy þ Aa xB , Ab y þ Ab ðu yÞ ¼ 3 3ðd b d o Þ
Ab d b Ao d o 1 Ab Ab d b Ao d o 1 Ab u xB ¼ yþ 1 xþ 3 Aa 3Aa ðd b d o Þ 3 Aa 3Aa ðd b d o Þ ¼ C 1 y C 2 x þ C 3 u,
ð4Þ
where 1 Ab C1 ¼ 3 Aa
" # do 2 do 1 Ab þ þ1 ; 3 Aa db db
1 Ab C2 ¼ 3 Aa
" # do 2 do þ þ 1 1; db db
C3 ¼
1 Ab . 3 Aa
The left-hand side is the volume swept by the displacement of the reservoir (y) plus the volume displaced by the membrane (relative to the reservoir). The first two terms on the right-hand side of the equation is the volume displaced by the relative motion between the port and reservoir. The third term describes the volume that is forced through the port. The ratio d o =d b describes the primary spring geometry and Ab =Aa the hydraulic advantage. The intake geometry of the port also has some effect, but is neglected for this study because the effect is small [6]. Assuming the mass of the membrane is small, the kinetic energy of the system can be written as: T ¼ 12 mx x_ 2 þ 12 my y_ 2 þ 12 mB x_ 2B ,
(5)
_ and the where x_ B is given by x_ 2B ¼ C 21 y_ 2 þ C 22 x_ 2 þ C 23 u_ 2 2C 1 C 2 x_ y_ þ 2C 1 C 3 y_ u_ 2C 2 C 3 x_ u, absorber mass is mB ¼ rAa l, where r is the density of the fluid and l is the length of the central
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port. The potential energy of the system is V ¼ 12 kðx yÞ2 þ ku ðu yÞ2 .
989
(6)
The matrix equation of motion can be found using the Lagrange approach, which, including damping forces gives 2 32 3 2 32 3 2 32 3 2 3 c þ cu c cu k þ ku k ku M1 M2 M3 y€ y_ 0 y 6 7 6 6 M M M 76 € 7 6 c 7 6 7 6 7 c 0 54 x_ 5 þ 4 k k 0 54 x 5 ¼ 4 0 7 5 6 54 x 5 þ 4 5, 4 4 c k M7 M8 M9 0 c 0 k u€ u_ 0 u u u u u (7) where 2
M1
M2
6M 4 4 M7
M5 M8
3
2
my þ mB C 21 6 M6 7 5 ¼ 4 mB C 1 C 2 M9 mB C 1 C 3 M3
mB C 1 C 2 mx þ mB C 22 mB C 2 C 3
3 mB C 1 C 3 mB C 2 C 3 7 5. mB C 23
Assuming simple harmonic motion, the transmissibility of the device can be determined from Eq. (7) to give ku þ iocu þ o2 M 3 o2 M 8 þ ku þ iocu o2 M 9 k þ ioc þ o2 M 2 Y . (8) ¼ X ðku þ iocu o2 M 9 Þ½k þ ku þ ioðc þ cu Þ o2 M 1 ðku þ iocu þ o2 M 3 Þ2 If the device is blocked such that Y ¼ 0, the transfer dynamic stiffness can be derived to give F T o2 M 8 ku þ iocu þ o2 M 3 ¼ þ k þ ioc þ o2 M 2 , (9) ku þ iocu o2 M 9 X where F T the amplitude of the transmitted force. Both Eqs. (8) and (9) can be used to study the TVAI. To conduct the study the parameters given in Table 1 were used. These parameters result in a device that could be usefully employed. For instance it was assumed that the absorber mass would be smaller than the system mass and that the natural frequency of the device would be less than the isolation frequency. To adjust the isolation frequency the membrane stiffness can be changed, and the effect of changing this on the modulus of the transmissibility is shown in Fig. 4a. The dashed line connects the points of minimum transmissibility for each isolation frequency. The point of minimum transmissibility does not necessarily coincide with the isolation frequency because the minimum transmissibility is also a function of reservoir stiffness. The frequency at which isolation occurs can only be adjusted over a finite range. The lowest frequency will occur when the reservoir stiffness is at a minimum. In this case the system will behave like a single degree of freedom with mass my isolated by the primary spring only. As the stiffness increases so does the isolation frequency. For large values of reservoir stiffness no noticeable additional shift is achieved.
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Table 1 Parameter values for sensitivity discussion Parameter
Value
Primary system mass Absorber mass Primary spring stiffness Membrane stiffness Primary spring damping Reservoir damping Area ratio Cone diameter ratio
my mB k ku c cu Ab =Aa d o =d b
10 0:001my 1 0:01 100 0.1 0.01 10 0.5
The envelope of the modulus of the transmissibility values can be plotted as shown in Fig. 4b. Three curves are shown, where ku ¼ 0; ku ¼ 1 and for values in between. For the case with no membrane stiffness the curve is the same as that of a single degree of freedom system as could be anticipated when inspecting Fig. 3. For the case of infinite stiffness the transmissibility is the same as for the LIVE type absorber shown in Fig. 1. If an increase in excitation frequency occurs, the device can be tuned so that the isolation frequency is coincident with the excitation frequency and the transmissibility will be reduced until the frequency reaches point A on Fig. 4b. At frequencies higher than this the device must be tuned to the lowest achievable frequency (i.e. ku small) to achieve the minimum transmissibility. The highfrequency roll-off (i.e. at frequencies greater than the cross-over point A) is however only 20 dB/decade compared to the 40 dB/decade for a conventional isolator with low damping indicating poor high-frequency performance of the device and emphasizing the limited useful bandwidth. The effect of damping in the device is to reduce the transmissibility at the natural frequency and the isolation at the isolation frequency. The damping in the primary spring has a large influence on the transmissibility at the first natural frequency and the isolation frequency, while the reservoir damping influences mainly the second natural frequency. The advantage of the TVAI when compared to a conventional isolator with the same static deflection can be seen in Fig. 5 which shows the improvement of the device, defined as =T ðisolatorÞ . Also shown is the improvement over the case where a fixed frequency VAI is T ðTVAIÞ r r used when tuned to the highest frequency. The improvement for this theoretical case is seen to be substantial over a limited frequency range and warrants further investigation and experimental verification. pffiffiffi The bandwidth is defined as frequency range over which the transmissibility is less than 2 times the minimum transmissibility, i.e. : B¼
o2 o1 . oi
(10)
The bandwidth is calculated numerically as no explicit equation exists for the frequencies in Eq. (10). In practice it is important to maximise both bandwidth and attenuation, which is defined
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10 Increasing membrane stiffness
Transmissibility [dB]
0
-10
-20
-30
-40 Minimum envelope -50 0
0.1
0.2
(a)
0.3 Frequency [Hz]
0.4
0.5
0.6
20
10
Transmissibility [dB]
0 ku → 0
-10
-20 ku → ∞ A
-30 0 < ku < ∞
-40
-50 0.05 (b)
0.1
0.15
0.2
0.25 0.3 Frequency [Hz]
0.35
0.4
0.45
0.5
Fig. 4. Set of transmissibility curves obtained by varying the membrane stiffness. The values used are listed in Table 1.
as robustness given by [12]: R ¼ B ½1 jT r ðoi Þj . The bandwidth and robustness are plotted for different isolation frequencies in Fig. 6.
(11)
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Over a tuned absorber
Improvement
0.7 0.6 0.5 0.4 0.3
Over a conventional isolator
0.2 0.1 0 0.1
0.15
0.2
0.25
0.3 0.35 0.4 Frequency [Hz]
0.45
0.5
0.55
0.6
Fig. 5. Improvement in isolation achieved by tuning the device compared to a conventional isolator and an absorber tuned to the highest isolation frequency. Improvement is defined as the transmissibility ratio (i.e. T ðTVAIÞ =T ðisolatorÞ ). r r
0.2 Bandwidth Robustness
0.18
0.16
0.14
0.12
0.1
0.08 0.1
0.12
0.14
0.16
0.18 0.2 0.22 Frequency [Hz]
0.24
0.26
Fig. 6. General trends of bandwidth and robustness.
0.28
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The stiffness of the membrane is a function of the rubber and the properties of the air spring. Since the objective is simply to minimise the contribution of the membrane to the reservoir stiffness only a maximum value is estimated. The membrane stiffness can be calculated using the pressure deflection equation in [11]. km ¼
64pEt3 3d 2b ð1 n2 Þ
(12)
with t the thickness, d b is the diameter, E is the Young’s modulus and n is Poisson’s ratio of the membrane, respectively. The air spring stiffness for a closed double acting air spring is a function of displacement: " gþ1 gþ1 # 1 1 g for x5h þ ka ðxÞ ¼ gP0 Ab h hx hþx 2gP0 Ab ð13Þ h with g the ratio of specific heats for air, P0 the initial air pressure and h the height of the undeflected air spring. Evaluation of Eqs. (12) and (13) shows that the membrane stiffness is more than an order of magnitude smaller than the air spring stiffness (as designed) for a 2 mm thick membrane with modulus of elasticity of less than 20 MPa, which is considered to be the maximum value. The air spring stiffness and damping are also a function of the choke size. The choke size chosen for the experimental work was 0.15 mm, which should minimise its effect on the loss in stiffness at high pressure. The main source of damping in the spring is the loss through the orifice, and the heat transfer losses will be minimal at frequencies above 3 Hz [13]. ka ¼
3. Experimental work A prototype TVAI was designed, built and tested to prove the principle of operation and to validate the mathematical model. An isolation frequency below 40 Hz was chosen so that the TVAI was relatively small and lightweight, and the resulting device is shown in Fig. 2b. The experimental set-up consisted of the TVAI fitted to a Zonic hydraulic actuator and loaded with weights in 10 kg increments as shown in Fig. 2b. The TVAI was excited at single frequencies over the frequency range from 10 to 70 Hz with a constant acceleration input using a Spectral Dynamics SigLab 20-42 FFT analyser. Acceleration of the actuator and the isolated mass was measured using 100 mV/g PCB ICP accelerometers. A set of measured transmissibilities for the TVAI with a 47.5 kg isolated mass is shown in Fig. 7. The points of minimum transmissibility are connected with a linear curve fit, which has a slope of 95 dB/decade. The slope of the curves for the various values of isolated mass used in testing are summarised in Table 2. Thus, over a limited band of frequencies it is possible to achieve a roll-off that is considerably more than that of a conventional isolator. The parameters in the model described in Section 2 were adjusted so that the experimental data fitted the model. A standard MATLAB optimisation algorithm was used with objective function
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15 10
Transmissibility [dB]
5 0
Increasing membrane stiffness 89 dB/decade
-5 -10 -15 -20 -25 -30 10
20
30 Frequency [Hz]
40
50
60
70
Fig. 7. The roll-off line fitted through the minimum transmissibility points on the set of transmissibility curves obtained by varying the membrane stiffness. A conventional isolator is shown for comparison.
Table 2 Roll-off slopes for tested isolated masses Mass (kg)
Roll-off slope (dB/decade)
17.5 27.5 37.5 47.5
94.7 95.9 93.0 89.3
being the least squares error between the measured data and the theoretical model. Only the absolute values of the transmissibilities were used. Comparisons between the optimised model and the experimental data are shown in Fig. 8. The signature assurance criterion (SAC) given by 2 H H SAC ¼ jaH X aA j =ðaX aX ÞðaA aA Þ and the average amplitude error value given by !, M N N X X X ERROR ¼ 100 aAj aX i aX i j¼1
i¼1
i¼1
were used to assess the quality of the curve fits [14]. aX and aA are the experimental and predicted transmissibilities respectively, and H denotes the Hermitian transpose. The SAC and ERROR values are shown in Fig. 9. These values indicate very good correlation between the model and the measured data.
ARTICLE IN PRESS N.F. du Plooy et al. / International Journal of Mechanical Sciences 47 (2005) 983–997 20 Curve fit Measured data
15 10 my = 10 kg
Transmissibility [dB]
5 0 -5 -10 -15
my = 40 kg
-20 -25 -30 10
20
30
40 Frequency [Hz]
50
60
70
Fig. 8. Transmissibility curve fits at a pressure of 584 kPa (40 kg) and 592 kPa (10 kg).
1
SAC
0.95 0.9 0.85 0.8 0.75 0
100
200
300
400
500
600
300 Pressure [kPa]
400
500
600
4 my = 17.5 my = 27.5 my = 37.5 my = 47.5
ERROR [%]
2 0 -2 -4 -6 -8 0
100
200
Fig. 9. SAC and ERROR values.
995
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Primary system stiffness (k) [N/m]
5
x 105
550 Primary system damping (c) [N.s/m]
996
4.8 4.6 4.4 4.2 4 3.8
400
my = 17.5 my = 27.5 my = 37.5 my = 47.5
350 300 250
200
400
600
x 104 Membrane damping coefficient (cu) [N.s/m]
Membrane stiffness (ku) [N/m]
450
200 0
6
500
5 4 3 2 1 0
200 400 Pressure [kPa]
600
0
200
400
600
0
200 400 Pressure [kPa]
600
30 25 20 15 10 5
Fig. 10. Physical properties obtained from fitting the analytical models to experimental data.
The model parameters are shown in Fig. 10. The primary system spring stiffness is a function of material property and geometry. It was observed that the geometry of the spring changes substantially due to static deflection, which explains the different values obtained. The maximum difference was 20%. The difference in values with pressure can also be explained by a change in geometry due to the pressure differential between the reservoir and ambient. The variation in primary system damping was less and this is probably because it is not a function of geometry. For the four isolated masses used the reservoir stiffness was found to be similar in each case and consistently increased with pressure. The relationship between pressure and isolation frequency is roughly given by f i ¼ 0:280p2s þ 3:787ps þ 22:473, where ps is the supply pressure in bar. The isolation frequency could be varied by 12 Hz by changing the supply pressure from 0 to 600 kPa.
4. Conclusions This paper has described an adaptive isolator that incorporates a vibration absorber. The device is called a tunable vibration absorbing isolator (TVAI) and is based on the liquid vibration eliminator. It is made adaptive by incorporating adjustable pneumatic springs. A mathematical model of the TVAI has been developed. Once the parameters of this model were optimised, there was excellent agreement between the model and experimental data as confirmed by visual inspection of the transmissibility curves as well as error estimates. It has been shown that the
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TVAI can achieve significantly lower transmissibility than a conventional isolator over a limited frequency band. This was confirmed by experimental data which showed that the prototype device achieved a roll-off of about 90 dB/decade over a 12 Hz frequency band. Acknowledgements The authors gratefully acknowledge financial support of this project by the Safety in Mines Research Advisory Committee of South Africa (SIMRAC) and Boart Longyear.
References [1] Sciulli D. Dynamics and control for vibration isolation. PhD dissertation, Virginia Polytechnic Institute and State University; 1997. [2] Flannely WG. The dynamic anti-resonant vibration isolator, 22nd Annual AHS national forum, Washington; 1966. p. 152–8. [3] Braun D. Development of anti-resonance force isolators for helicopter vibration reduction. Sixth European rotorcraft and powered lift aircraft forum, Bristol, England; 16–19 September 1980. [4] Halwes DR. Total main rotor isolation system analysis. Bell helicopter textron, NASA contractor report no. 165667. Langley Research Center, Hampton, Virginia; June 1981. [5] Kidner M, Brennan MJ. Improving the performance of a vibration neutraliser by actively removing damping. Journal of Sound and Vibration 1999;221(4):587–606. [6] Du Plooy NF. The development of a vibration absorber for vibrating screens. M.Eng. dissertation, University of Pretoria; 1999. [7] Smith MR, Stamps FN. Vibration isolation system. US patent no. 5,788,029; 1988. [8] Williams KA, Chiu T-C, Bernhard RJ. Controlled continuous tuning of an adaptively tunable vibration absorber incorporating shape memory alloys. Proceedings of SPIE, Mathematics and control in smart structures, vol. 3984; March 2000. [9] Mizuno T, Araki K. Control system design of a dynamic vibration absorber with an electromagnetic servomechanism. Mechanical Systems and Signal Processing 1993;7(4):293–306. [10] Longbottom CJ, Day MJ, Rider E. A self tuning vibration absorber. Certificate GB 2189573 B, The Patents Office, UK; 1990. [11] Roark RJ, Warren CY. Formulas of stress and strain. New York: McGraw-Hill; 1989 [chapter 10]. [12] Brennan MJ. Vibration control using a tunable vibration neutralizer. Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science 1997;211:91–108. [13] Harris CM, editor. Shock and vibration handbook, 3rd ed. New York: McGraw-Hill; 1988 [chapter 33]. [14] Dascotte E, Strobbe J. Updating finite element models using FRF correlation functions. Proceedings of the 17th international modal analysis conference (IMAC), Kissimmee, Florida; February 1999.
International Journal of Mechanical Sciences 47 (2005) 983–997 www.elsevier.com/locate/ijmecsci
The development of a tunable vibration absorbing isolator N.F. du Plooya,, P.S. Heynsa, M.J. Brennanb a
Dynamic Systems Group, Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria 0002, South Africa b Institute of Sound and Vibration Research, University of Southampton, Southampton, UK Received 3 September 2002; received in revised form 24 March 2005; accepted 6 April 2005
Abstract This paper describes a tunable vibration absorbing isolator. The device is based on the liquid inertia vibration eliminator, which is modified so that the frequency at which maximum isolation occurs can be changed in real time. This is achieved by using adjustable pneumatic springs. A theoretical model describing the device is derived using Lagrange’s equations. The model is used to design a practical device and experimental results confirm the validity of the model. With the experimental device a roll-off of about 90 dB/decade over a 12 Hz frequency band was achieved. r 2005 Elsevier Ltd. All rights reserved. Keywords: Tunable vibration absorbing isolator
1. Introduction The ideal vibration isolator has high static stiffness to support the isolated structure and low dynamic stiffness to ensure low transmission of dynamic forces [1]. Isolation is achieved at frequencies greater than the square root of two times the mounted resonance frequency. Above this frequency, and provided that damping in the isolator is light, the transmission of vibration reduces at 40 dB/decade. To improve on this situation, Flannely incorporated a vibration absorber in the isolator [2] and made a vibration absorbing isolator (VAI), which he called a dynamic anti-resonant vibration isolator (DAVI). The absorber modifies the dynamic behaviour Corresponding author. Tel.: +27 823314091; fax: +27 126633052.
E-mail address: [email protected] (N.F. du Plooy). 0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.04.009
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Nomenclature kj stiffness j damping j cj Z loss factor o circular frequency isolation circular frequency oi p ffiffiffiffiffiffiffi i 1 mass j mj diameter j dj area j Aj u; x; xB ; y displacement X , Y displacement amplitude force j fj force transmitted FT P pressure E Young’s modulus experimental transmissibility aX analytical transmissibility aA t thickness n ratio of specific heats h air spring height T kinetic energy V potential energy B bandwidth R robustness transmissibility Tr n Poisson’s ratio t thickness h air spring height n ratio of specific heats of the conventional isolator, providing a frequency band over which the dynamic stiffness is less than the static stiffness, and hence reduces vibration transmission in this frequency band. The device contains another innovation, in that it uses a pendulum to amplify the acceleration of the vibration absorber mass. A number of other devices utilising hydraulic mass amplification followed, for example those described by Braun [3] and Halwes [4]. Halwes introduced the liquid inertia vibration eliminator (LIVE), a device that employed hydraulic fluid as the vibration absorber mass, resulting in a compact arrangement. Halwes showed that the size of the device is inversely proportional to the square of the density of the fluid used, which means that the use of heavy liquids is attractive. All the devices discussed above were designed to be most effective at a single frequency, however, applications often exist where the excitation frequency is a function of environmental or operating
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conditions, for instance load. The main drawback of VAIs is their limited effective frequency range, defined as the range of frequencies where the transmissibility of the device is below a certain value, which might not cover the range of excitation frequencies encountered. Increasing this frequency range can be achieved in a number of ways, for instance by reducing the damping in the device. There are, however, physical limits on the amount of passive damping reduction that can be achieved practically, which has led to the development of methods to actively reduce damping as described by Kidner and Brennan [5]. It has also been found that the effective frequency range is affected by the ratio of natural frequency of the mounting system to the isolation frequency of the device [6]. A small ratio is desirable which means a low natural frequency is required since the isolation frequency is application driven and therefore fixed. It is also possible to increase the band by using a non-linear softening spring, but this may be difficult to realise in practice. Finally, the effective suppression band can be increased by tuning the VAI so that the isolation frequency is coincident with the excitation frequency. This method differs from those discussed above since the effective frequency range is not increased, but rather the isolation frequency is adjusted (with a fixed bandwidth) to create a wider suppression band. The tuning method assumes that excitation frequency can be determined with sufficient accuracy and that a system parameter can be adjusted so that the isolation and excitation frequencies coincide. This device is called a tunable vibration absorbing isolator (TVAI), and the aim of this paper is to describe the development of such a device. Following this introduction a mathematical model of the TVAI is presented in Section 2. In Section 3 experimental work on a prototype device is described, and this is followed with some concluding remarks in Section 4.
2. The tunable vibration absorbing isolator 2.1. Description of the device The TVAI is discussed with reference to the LIVE system, which is shown in Fig. 1. The displacement transmissibility for this device is given by [6]: Ab Ab 2 kð1 þ iZÞ þ ioc þ o mB 1 X Aa Aa " (1) ¼ 2 # , Y A b 2 kð1 þ iZÞ þ ioc o m þ mB 1 Aa where the absorber mass mB is the mass of the fluid in the port, m is the mass of the central part of the device, Aa ¼ pd 2a =4; Ab ¼ pd 2b =4; c is the viscous damping in the device, o is circular frequency, k is the stiffness of resilient connection between the moving parts of the device and Z is the loss factor of the resilient connection. The transmissibility pffiffiffi of the device is less than that of a conventional isolator of equivalent static deflection up to 2oi , where oi is the isolation frequency given by [6] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k u . (2) oi ¼ u t Ab Ab 1 mB Aa Aa
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mB h
x l
xB Port
m
k
da
c y
Fig. 1. Schematic of the LIVE.
This is the frequency when the transmissibility is a minimum, which occurs when the real part of the numerator in Eq. (1) equals zero. It can be seen that the stiffness, absorber mass (mB ) and port and reservoir geometry can be used to tune the absorber. It can also be seen that the isolation frequency is very sensitive to port diameter, but this parameter is difficult to adjust in practice because of the incompressibility of the absorber fluid. Nevertheless, a device using this tuning mechanism has been patented by Smith and Stamps [7]. They also describe a device that changes the port length, thereby changing the absorber mass. Several examples of variable stiffness devices exist in the literature. Examples are shape memory alloy helical springs [8] and electromagnetic springs [9]. There is a theoretical advantage in using spring stiffness since it will not affect the ratio of natural to isolation frequencies, which will affect the transmissibility at the isolation frequency. In this paper, however, the reservoir stiffness is adjusted. The main advantage of this tuning method is that it results in a robust and simple device. As with the LIVE system, the TVAI consists of two fluid reservoirs located on both sides of a port as shown in Fig. 2a. Relative motion between the port and the reservoirs forces the fluid through the port. However the TVAI differs from the LIVE system in that flexible rubber membranes terminate both reservoir cavities. The purpose of the membrane is to allow tuning of the device to the excitation frequency, as the stiffness of the membrane controls the amount of fluid that is pumped through the port, which in turn varies the isolation frequency. The stiffness of the membrane is a function of the air pressure in the air spring cavity and the stiffness of the rubber membrane. To allow a large tuning band and low damping the stiffness of the rubber must be minimised. A small choke is fitted between the air spring and the air supply to allow the exchange of air at low frequencies and to minimise flow at high frequencies. This allows tuning of the device since the static pressure across the choke equalises while the pressure difference caused by the movement of the membrane is too fast to allow substantial airflow. This maximises the stiffness of the air spring and minimises the damping. The trade-off is the tuning rate that can be achieved with a small choke diameter. Applications requiring fast tuning will call for a more
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Isolated mass (my)
Air spring cavity Pressure Control valve
Port Primary spring
Choke Reservoir (a)
Membrane
(b)
Fig. 2. TVAI.
complicated valve arrangement as described by Longbottom et al. [10]. For the device described in this paper the average pressure was adjusted manually. The TVAI can conceptually be split in two parts by the primary spring. The isolated mass can be connected to either the port or the reservoir side of the device. In the device discussed in this paper the isolated mass is connected to the reservoir side while the port is excited as shown in the photograph of the prototype depicted in Fig. 2b. Without the fluid present the isolated mass is simply suspended by the primary spring. The inertia of the mass in the port modifies the dynamic stiffness to be less than the static stiffness over a given frequency range. One advantage of this device is that the static deflection is constant regardless of the pressure present in the air spring cavity. 2.2. Mathematical model A model of the TVAI is shown in Fig. 3. Since the two membranes are coupled through the incompressible fluid, their stiffness can be modelled as having one effective value ðku Þ. The mass of the fluid in the two reservoirs is added to the mass of the casing my . Several theoretical models with various levels of complexity were developed during the course of this study. It was found that some assumptions were essential to achieve a good fit between measured data and predictions. The most important assumption regards the shape of the membrane separating the air spring from the reservoir, which is loaded on both sides by a constant pressure. Relative motion occurs because the membrane bulges due to the pressure differential acting on it. Assuming a fixed boundary condition for the membrane edge the volume of fluid displaced by the membrane can be written as a function of its absolute centre (u) and edge (y) displacements as well as the reservoir area [11]. V ¼ 13 Ab ju yj.
(3)
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½ku
½cu db do xB(t) x(t)
mx
l
my
k
da
½ku
c
u(t)
½cu y(t)
Fig. 3. Model of the TVAI.
It was found that the deformation of the primary spring must be modelled accurately. The equations of motion for the LIVE system were based on the assumption that the volume displaced by the port is simply the difference between the port and reservoir area multiplied by the relative displacement, but for the TVAI this did not result in a good model. The conical geometry of the primary spring must be taken into account as can be seen by comparing Figs. 1 and 3. The equation describing the continuity of motion for the fluid in the device can be written as:
1 Ab d b Ao d o Aa ðx yÞ þ ðAb Aa Þy þ Aa xB , Ab y þ Ab ðu yÞ ¼ 3 3ðd b d o Þ
Ab d b Ao d o 1 Ab Ab d b Ao d o 1 Ab u xB ¼ yþ 1 xþ 3 Aa 3Aa ðd b d o Þ 3 Aa 3Aa ðd b d o Þ ¼ C 1 y C 2 x þ C 3 u,
ð4Þ
where 1 Ab C1 ¼ 3 Aa
" # do 2 do 1 Ab þ þ1 ; 3 Aa db db
1 Ab C2 ¼ 3 Aa
" # do 2 do þ þ 1 1; db db
C3 ¼
1 Ab . 3 Aa
The left-hand side is the volume swept by the displacement of the reservoir (y) plus the volume displaced by the membrane (relative to the reservoir). The first two terms on the right-hand side of the equation is the volume displaced by the relative motion between the port and reservoir. The third term describes the volume that is forced through the port. The ratio d o =d b describes the primary spring geometry and Ab =Aa the hydraulic advantage. The intake geometry of the port also has some effect, but is neglected for this study because the effect is small [6]. Assuming the mass of the membrane is small, the kinetic energy of the system can be written as: T ¼ 12 mx x_ 2 þ 12 my y_ 2 þ 12 mB x_ 2B ,
(5)
_ and the where x_ B is given by x_ 2B ¼ C 21 y_ 2 þ C 22 x_ 2 þ C 23 u_ 2 2C 1 C 2 x_ y_ þ 2C 1 C 3 y_ u_ 2C 2 C 3 x_ u, absorber mass is mB ¼ rAa l, where r is the density of the fluid and l is the length of the central
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port. The potential energy of the system is V ¼ 12 kðx yÞ2 þ ku ðu yÞ2 .
989
(6)
The matrix equation of motion can be found using the Lagrange approach, which, including damping forces gives 2 32 3 2 32 3 2 32 3 2 3 c þ cu c cu k þ ku k ku M1 M2 M3 y€ y_ 0 y 6 7 6 6 M M M 76 € 7 6 c 7 6 7 6 7 c 0 54 x_ 5 þ 4 k k 0 54 x 5 ¼ 4 0 7 5 6 54 x 5 þ 4 5, 4 4 c k M7 M8 M9 0 c 0 k u€ u_ 0 u u u u u (7) where 2
M1
M2
6M 4 4 M7
M5 M8
3
2
my þ mB C 21 6 M6 7 5 ¼ 4 mB C 1 C 2 M9 mB C 1 C 3 M3
mB C 1 C 2 mx þ mB C 22 mB C 2 C 3
3 mB C 1 C 3 mB C 2 C 3 7 5. mB C 23
Assuming simple harmonic motion, the transmissibility of the device can be determined from Eq. (7) to give ku þ iocu þ o2 M 3 o2 M 8 þ ku þ iocu o2 M 9 k þ ioc þ o2 M 2 Y . (8) ¼ X ðku þ iocu o2 M 9 Þ½k þ ku þ ioðc þ cu Þ o2 M 1 ðku þ iocu þ o2 M 3 Þ2 If the device is blocked such that Y ¼ 0, the transfer dynamic stiffness can be derived to give F T o2 M 8 ku þ iocu þ o2 M 3 ¼ þ k þ ioc þ o2 M 2 , (9) ku þ iocu o2 M 9 X where F T the amplitude of the transmitted force. Both Eqs. (8) and (9) can be used to study the TVAI. To conduct the study the parameters given in Table 1 were used. These parameters result in a device that could be usefully employed. For instance it was assumed that the absorber mass would be smaller than the system mass and that the natural frequency of the device would be less than the isolation frequency. To adjust the isolation frequency the membrane stiffness can be changed, and the effect of changing this on the modulus of the transmissibility is shown in Fig. 4a. The dashed line connects the points of minimum transmissibility for each isolation frequency. The point of minimum transmissibility does not necessarily coincide with the isolation frequency because the minimum transmissibility is also a function of reservoir stiffness. The frequency at which isolation occurs can only be adjusted over a finite range. The lowest frequency will occur when the reservoir stiffness is at a minimum. In this case the system will behave like a single degree of freedom with mass my isolated by the primary spring only. As the stiffness increases so does the isolation frequency. For large values of reservoir stiffness no noticeable additional shift is achieved.
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Table 1 Parameter values for sensitivity discussion Parameter
Value
Primary system mass Absorber mass Primary spring stiffness Membrane stiffness Primary spring damping Reservoir damping Area ratio Cone diameter ratio
my mB k ku c cu Ab =Aa d o =d b
10 0:001my 1 0:01 100 0.1 0.01 10 0.5
The envelope of the modulus of the transmissibility values can be plotted as shown in Fig. 4b. Three curves are shown, where ku ¼ 0; ku ¼ 1 and for values in between. For the case with no membrane stiffness the curve is the same as that of a single degree of freedom system as could be anticipated when inspecting Fig. 3. For the case of infinite stiffness the transmissibility is the same as for the LIVE type absorber shown in Fig. 1. If an increase in excitation frequency occurs, the device can be tuned so that the isolation frequency is coincident with the excitation frequency and the transmissibility will be reduced until the frequency reaches point A on Fig. 4b. At frequencies higher than this the device must be tuned to the lowest achievable frequency (i.e. ku small) to achieve the minimum transmissibility. The highfrequency roll-off (i.e. at frequencies greater than the cross-over point A) is however only 20 dB/decade compared to the 40 dB/decade for a conventional isolator with low damping indicating poor high-frequency performance of the device and emphasizing the limited useful bandwidth. The effect of damping in the device is to reduce the transmissibility at the natural frequency and the isolation at the isolation frequency. The damping in the primary spring has a large influence on the transmissibility at the first natural frequency and the isolation frequency, while the reservoir damping influences mainly the second natural frequency. The advantage of the TVAI when compared to a conventional isolator with the same static deflection can be seen in Fig. 5 which shows the improvement of the device, defined as =T ðisolatorÞ . Also shown is the improvement over the case where a fixed frequency VAI is T ðTVAIÞ r r used when tuned to the highest frequency. The improvement for this theoretical case is seen to be substantial over a limited frequency range and warrants further investigation and experimental verification. pffiffiffi The bandwidth is defined as frequency range over which the transmissibility is less than 2 times the minimum transmissibility, i.e. : B¼
o2 o1 . oi
(10)
The bandwidth is calculated numerically as no explicit equation exists for the frequencies in Eq. (10). In practice it is important to maximise both bandwidth and attenuation, which is defined
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10 Increasing membrane stiffness
Transmissibility [dB]
0
-10
-20
-30
-40 Minimum envelope -50 0
0.1
0.2
(a)
0.3 Frequency [Hz]
0.4
0.5
0.6
20
10
Transmissibility [dB]
0 ku → 0
-10
-20 ku → ∞ A
-30 0 < ku < ∞
-40
-50 0.05 (b)
0.1
0.15
0.2
0.25 0.3 Frequency [Hz]
0.35
0.4
0.45
0.5
Fig. 4. Set of transmissibility curves obtained by varying the membrane stiffness. The values used are listed in Table 1.
as robustness given by [12]: R ¼ B ½1 jT r ðoi Þj . The bandwidth and robustness are plotted for different isolation frequencies in Fig. 6.
(11)
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Over a tuned absorber
Improvement
0.7 0.6 0.5 0.4 0.3
Over a conventional isolator
0.2 0.1 0 0.1
0.15
0.2
0.25
0.3 0.35 0.4 Frequency [Hz]
0.45
0.5
0.55
0.6
Fig. 5. Improvement in isolation achieved by tuning the device compared to a conventional isolator and an absorber tuned to the highest isolation frequency. Improvement is defined as the transmissibility ratio (i.e. T ðTVAIÞ =T ðisolatorÞ ). r r
0.2 Bandwidth Robustness
0.18
0.16
0.14
0.12
0.1
0.08 0.1
0.12
0.14
0.16
0.18 0.2 0.22 Frequency [Hz]
0.24
0.26
Fig. 6. General trends of bandwidth and robustness.
0.28
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The stiffness of the membrane is a function of the rubber and the properties of the air spring. Since the objective is simply to minimise the contribution of the membrane to the reservoir stiffness only a maximum value is estimated. The membrane stiffness can be calculated using the pressure deflection equation in [11]. km ¼
64pEt3 3d 2b ð1 n2 Þ
(12)
with t the thickness, d b is the diameter, E is the Young’s modulus and n is Poisson’s ratio of the membrane, respectively. The air spring stiffness for a closed double acting air spring is a function of displacement: " gþ1 gþ1 # 1 1 g for x5h þ ka ðxÞ ¼ gP0 Ab h hx hþx 2gP0 Ab ð13Þ h with g the ratio of specific heats for air, P0 the initial air pressure and h the height of the undeflected air spring. Evaluation of Eqs. (12) and (13) shows that the membrane stiffness is more than an order of magnitude smaller than the air spring stiffness (as designed) for a 2 mm thick membrane with modulus of elasticity of less than 20 MPa, which is considered to be the maximum value. The air spring stiffness and damping are also a function of the choke size. The choke size chosen for the experimental work was 0.15 mm, which should minimise its effect on the loss in stiffness at high pressure. The main source of damping in the spring is the loss through the orifice, and the heat transfer losses will be minimal at frequencies above 3 Hz [13]. ka ¼
3. Experimental work A prototype TVAI was designed, built and tested to prove the principle of operation and to validate the mathematical model. An isolation frequency below 40 Hz was chosen so that the TVAI was relatively small and lightweight, and the resulting device is shown in Fig. 2b. The experimental set-up consisted of the TVAI fitted to a Zonic hydraulic actuator and loaded with weights in 10 kg increments as shown in Fig. 2b. The TVAI was excited at single frequencies over the frequency range from 10 to 70 Hz with a constant acceleration input using a Spectral Dynamics SigLab 20-42 FFT analyser. Acceleration of the actuator and the isolated mass was measured using 100 mV/g PCB ICP accelerometers. A set of measured transmissibilities for the TVAI with a 47.5 kg isolated mass is shown in Fig. 7. The points of minimum transmissibility are connected with a linear curve fit, which has a slope of 95 dB/decade. The slope of the curves for the various values of isolated mass used in testing are summarised in Table 2. Thus, over a limited band of frequencies it is possible to achieve a roll-off that is considerably more than that of a conventional isolator. The parameters in the model described in Section 2 were adjusted so that the experimental data fitted the model. A standard MATLAB optimisation algorithm was used with objective function
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15 10
Transmissibility [dB]
5 0
Increasing membrane stiffness 89 dB/decade
-5 -10 -15 -20 -25 -30 10
20
30 Frequency [Hz]
40
50
60
70
Fig. 7. The roll-off line fitted through the minimum transmissibility points on the set of transmissibility curves obtained by varying the membrane stiffness. A conventional isolator is shown for comparison.
Table 2 Roll-off slopes for tested isolated masses Mass (kg)
Roll-off slope (dB/decade)
17.5 27.5 37.5 47.5
94.7 95.9 93.0 89.3
being the least squares error between the measured data and the theoretical model. Only the absolute values of the transmissibilities were used. Comparisons between the optimised model and the experimental data are shown in Fig. 8. The signature assurance criterion (SAC) given by 2 H H SAC ¼ jaH X aA j =ðaX aX ÞðaA aA Þ and the average amplitude error value given by !, M N N X X X ERROR ¼ 100 aAj aX i aX i j¼1
i¼1
i¼1
were used to assess the quality of the curve fits [14]. aX and aA are the experimental and predicted transmissibilities respectively, and H denotes the Hermitian transpose. The SAC and ERROR values are shown in Fig. 9. These values indicate very good correlation between the model and the measured data.
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15 10 my = 10 kg
Transmissibility [dB]
5 0 -5 -10 -15
my = 40 kg
-20 -25 -30 10
20
30
40 Frequency [Hz]
50
60
70
Fig. 8. Transmissibility curve fits at a pressure of 584 kPa (40 kg) and 592 kPa (10 kg).
1
SAC
0.95 0.9 0.85 0.8 0.75 0
100
200
300
400
500
600
300 Pressure [kPa]
400
500
600
4 my = 17.5 my = 27.5 my = 37.5 my = 47.5
ERROR [%]
2 0 -2 -4 -6 -8 0
100
200
Fig. 9. SAC and ERROR values.
995
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Primary system stiffness (k) [N/m]
5
x 105
550 Primary system damping (c) [N.s/m]
996
4.8 4.6 4.4 4.2 4 3.8
400
my = 17.5 my = 27.5 my = 37.5 my = 47.5
350 300 250
200
400
600
x 104 Membrane damping coefficient (cu) [N.s/m]
Membrane stiffness (ku) [N/m]
450
200 0
6
500
5 4 3 2 1 0
200 400 Pressure [kPa]
600
0
200
400
600
0
200 400 Pressure [kPa]
600
30 25 20 15 10 5
Fig. 10. Physical properties obtained from fitting the analytical models to experimental data.
The model parameters are shown in Fig. 10. The primary system spring stiffness is a function of material property and geometry. It was observed that the geometry of the spring changes substantially due to static deflection, which explains the different values obtained. The maximum difference was 20%. The difference in values with pressure can also be explained by a change in geometry due to the pressure differential between the reservoir and ambient. The variation in primary system damping was less and this is probably because it is not a function of geometry. For the four isolated masses used the reservoir stiffness was found to be similar in each case and consistently increased with pressure. The relationship between pressure and isolation frequency is roughly given by f i ¼ 0:280p2s þ 3:787ps þ 22:473, where ps is the supply pressure in bar. The isolation frequency could be varied by 12 Hz by changing the supply pressure from 0 to 600 kPa.
4. Conclusions This paper has described an adaptive isolator that incorporates a vibration absorber. The device is called a tunable vibration absorbing isolator (TVAI) and is based on the liquid vibration eliminator. It is made adaptive by incorporating adjustable pneumatic springs. A mathematical model of the TVAI has been developed. Once the parameters of this model were optimised, there was excellent agreement between the model and experimental data as confirmed by visual inspection of the transmissibility curves as well as error estimates. It has been shown that the
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TVAI can achieve significantly lower transmissibility than a conventional isolator over a limited frequency band. This was confirmed by experimental data which showed that the prototype device achieved a roll-off of about 90 dB/decade over a 12 Hz frequency band. Acknowledgements The authors gratefully acknowledge financial support of this project by the Safety in Mines Research Advisory Committee of South Africa (SIMRAC) and Boart Longyear.
References [1] Sciulli D. Dynamics and control for vibration isolation. PhD dissertation, Virginia Polytechnic Institute and State University; 1997. [2] Flannely WG. The dynamic anti-resonant vibration isolator, 22nd Annual AHS national forum, Washington; 1966. p. 152–8. [3] Braun D. Development of anti-resonance force isolators for helicopter vibration reduction. Sixth European rotorcraft and powered lift aircraft forum, Bristol, England; 16–19 September 1980. [4] Halwes DR. Total main rotor isolation system analysis. Bell helicopter textron, NASA contractor report no. 165667. Langley Research Center, Hampton, Virginia; June 1981. [5] Kidner M, Brennan MJ. Improving the performance of a vibration neutraliser by actively removing damping. Journal of Sound and Vibration 1999;221(4):587–606. [6] Du Plooy NF. The development of a vibration absorber for vibrating screens. M.Eng. dissertation, University of Pretoria; 1999. [7] Smith MR, Stamps FN. Vibration isolation system. US patent no. 5,788,029; 1988. [8] Williams KA, Chiu T-C, Bernhard RJ. Controlled continuous tuning of an adaptively tunable vibration absorber incorporating shape memory alloys. Proceedings of SPIE, Mathematics and control in smart structures, vol. 3984; March 2000. [9] Mizuno T, Araki K. Control system design of a dynamic vibration absorber with an electromagnetic servomechanism. Mechanical Systems and Signal Processing 1993;7(4):293–306. [10] Longbottom CJ, Day MJ, Rider E. A self tuning vibration absorber. Certificate GB 2189573 B, The Patents Office, UK; 1990. [11] Roark RJ, Warren CY. Formulas of stress and strain. New York: McGraw-Hill; 1989 [chapter 10]. [12] Brennan MJ. Vibration control using a tunable vibration neutralizer. Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science 1997;211:91–108. [13] Harris CM, editor. Shock and vibration handbook, 3rd ed. New York: McGraw-Hill; 1988 [chapter 33]. [14] Dascotte E, Strobbe J. Updating finite element models using FRF correlation functions. Proceedings of the 17th international modal analysis conference (IMAC), Kissimmee, Florida; February 1999.