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Shock isolation systems using non linear stiffness and damping D.F. Ledezma-Ramirez 1 , M. Guzman-Nieto 1 , P.E. Tapia-Gonzalez 1 , N.S. Ferguson 2 1 Universidad Autonoma de Nuevo Leon, Facultad de Ingenieria Mecanica y Electrica, Centro de Investiga- cion e Innovacion en Ingenieria Aeronautica, Nuevo Leon, Mexico. e-mail: [email protected] 2 Institute of Sound and Vibration Research, University of Southampton, UK, SO17 1BJ. Abstract An overview of the use of non linear shock isolators is briefly presented, discussing possibilities for the development of a more efficient shock isolation system. The models discussed are a low dynamic stiffness concept with cubic hardening nonlinearity, and a dry friction isolation system. The stiffness and damping of these models are quantified experimentally, then the shock response of both systems is evaluated experimen- tally, discussing their advantages over a classic linear system. The combination of these two properties in a further mathematical model is suggested for the modelling of dry friction isolators available off the shelf. 1 Introduction Transient vibration is defined as a temporarily sustained vibration of a mechanical system. It may consist of forced or free vibrations, or both [1]. Transient loading, also known as impact or mechanical shock, is a nonperiodic excitation, which is often characterised by a sudden and severe application. In real life, mechanical shock is very common. Examples of shock could be a forging hammer, an automobile passing across a road bump, the free drop of an item from a height, etc. The description of the input requires knowledge of the variation of the input displacement versus time, from which the velocity and acceleration profiles can be derived. In general terms, a specific input can be qualitatively described as being of short or long duration, and in engineering terminology, such descriptions will be more explicitly defined later. For sensitive or supported equipment the response might cause damage through exceeding the allowable lev- els of stress or strain resulting from the transmitted displacement, velocity or acceleration. Alternatively, the equipment might be positioned in a finite space and a large relative displacement could cause the equipment to impact another structure. This shock isolator has the objective of reducing or modifying the vibratory forces transmitted to the receiver, and it normally takes the form of a resilient element. These anti-vibratory mounts are readily available in many different forms, such as helical spring and shock absorber combinations, rubber pads, leaf springs, etc. When the physical properties of the isolator, i.e. stiffness and damping are fixed for a particular application, it is said that the isolators are passive [2]. This form of vibration isolation is generally a low cost and reliable solution, but is normally designed for a particular problem and there might not be good performance for different situations for example under very unpredictable excitation. In general, passive isolation is the most commonly used solution for shock excitation problem. Although many of the mathematical models consider linear passive elements used to describe the properties of the isolator, most of the times real isolators experience a non linear behaviour, which can also been taken into the mathematical models. However, the use of linear passive elements is limited. For instance there is the compromise, between isolation performance and space limitations, when low isolation stiffness is adopted to obtain a lower mounted system natural 4111

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Page 1: Shock isolation systems using non linear stiffness and dampingpast.isma-isaac.be/downloads/isma2014/papers/isma2014... · 2014-10-27 · of mechanical springs and electromagnetic

Shock isolation systems using non linear stiffness anddamping

D.F. Ledezma-Ramirez 1, M. Guzman-Nieto 1, P.E. Tapia-Gonzalez 1, N.S. Ferguson 2

1 Universidad Autonoma de Nuevo Leon, Facultad de Ingenieria Mecanica y Electrica, Centro de Investiga-cion e Innovacion en Ingenieria Aeronautica, Nuevo Leon, Mexico.e-mail: [email protected]

2 Institute of Sound and Vibration Research, University of Southampton, UK, SO17 1BJ.

AbstractAn overview of the use of non linear shock isolators is briefly presented, discussing possibilities for thedevelopment of a more efficient shock isolation system. The models discussed are a low dynamic stiffnessconcept with cubic hardening nonlinearity, and a dry friction isolation system. The stiffness and damping ofthese models are quantified experimentally, then the shock response of both systems is evaluated experimen-tally, discussing their advantages over a classic linear system. The combination of these two properties in afurther mathematical model is suggested for the modelling of dry friction isolators available off the shelf.

1 Introduction

Transient vibration is defined as a temporarily sustained vibration of a mechanical system. It may consistof forced or free vibrations, or both [1]. Transient loading, also known as impact or mechanical shock,is a nonperiodic excitation, which is often characterised by a sudden and severe application. In real life,mechanical shock is very common. Examples of shock could be a forging hammer, an automobile passingacross a road bump, the free drop of an item from a height, etc. The description of the input requiresknowledge of the variation of the input displacement versus time, from which the velocity and accelerationprofiles can be derived. In general terms, a specific input can be qualitatively described as being of short orlong duration, and in engineering terminology, such descriptions will be more explicitly defined later.

For sensitive or supported equipment the response might cause damage through exceeding the allowable lev-els of stress or strain resulting from the transmitted displacement, velocity or acceleration. Alternatively, theequipment might be positioned in a finite space and a large relative displacement could cause the equipmentto impact another structure.

This shock isolator has the objective of reducing or modifying the vibratory forces transmitted to the receiver,and it normally takes the form of a resilient element. These anti-vibratory mounts are readily available inmany different forms, such as helical spring and shock absorber combinations, rubber pads, leaf springs, etc.When the physical properties of the isolator, i.e. stiffness and damping are fixed for a particular application,it is said that the isolators are passive [2]. This form of vibration isolation is generally a low cost andreliable solution, but is normally designed for a particular problem and there might not be good performancefor different situations for example under very unpredictable excitation. In general, passive isolation isthe most commonly used solution for shock excitation problem. Although many of the mathematical modelsconsider linear passive elements used to describe the properties of the isolator, most of the times real isolatorsexperience a non linear behaviour, which can also been taken into the mathematical models. However, the useof linear passive elements is limited. For instance there is the compromise, between isolation performanceand space limitations, when low isolation stiffness is adopted to obtain a lower mounted system natural

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frequency. Moreover, the isolation system could behave non-linearly due to the large deformations resultingfrom shocks.

In this article, two approaches for shock isolation are discussed. The first one considers a nonlinear stiffnessdevice following the idea of a low dynamic high static stiffness isolator. This strategy is sometimes calledquasi zero stiffness, and usually involves the combination of a linear spring with a negative stiffness in orderto get a very low stiffness value around the equilibrium point. Previous work regarding the use of suchisolators is briefly discussed, and the experimental device is presented following recent theoretical findings.The device comprises electromagnetic forces in order to achieve the desired nonlinear stiffness, and it ischaracterised in terms of the frequency response and shock response for different situations. Then, the useof dry friction isolators is considered, based on wire rope springs, which are widely available as commercialoff the shelf isolators. However, the properties and shock response of these devices has not been properlystudied, and this paper is intended to give some insight into the damping quantification and shock responseof wire rope isolators.

2 Literature review

Snowdon was one of the first researchers to incorporate non linear elements into the theory of shock isolation,by considering tangent stiffness in a SDOF system subjected to rounded step and pulse functions [3, 4],finding that softening elastic elements could lead to improved isolation. The use of nonlinear stiffness hasbeen investigated recently as alternative means for passive vibration isolation. Particularly, the idea of lowdynamic high static stiffness isolators is of interest, since it can provide excellent isolation at a particularequilibrium point where the dynamic stiffness of the system is very low [5]. Carella et al have studied thestatic and dynamic behaviour of these mechanisms and developed experimental devices to validate the theory[6, 7] and Kovacic has further investigated the effect of nonlinearities in the elastic elements [8]. Combinationof mechanical springs and electromagnetic forces, have also been used to apply the concept of quasi zerostiffness to tunable vibration absorbers [9]. However, in the previous studies no shock response has beenconsidered. Nevertheless, this idea presents great potential for shock isolation systems since one wants toideally have a low frequency mount capable of high energy storage, but this is not always possible due torestrictions on space and supporting weight. A recent published work by Xingtian [10] shows theoreticallythat this strategy is beneficial for shock isolation in some situations, but might increase the amount of relativemotion. It remains a further task to validate experimentally and investigate the practical implications of thesesystems in shock isolation. They found that a system with very low tangent stiffness has better shock isolationperformance regarding absolute acceleration response, but the absolute displacement response is increasedunless the shock duration is short in relation to the natural period of the system.

The use of shock isolators based on dry friction has been also explored by Mercer [11] who developed anoptimum shock isolator with great advantages over a common resilient mount. The device is passive innature, but the friction force can be varied. For low frequency inputs, the device depends solely upon viscousdamping given by a dashpot. However, in the case of high frequency and large amplitude inputs, the pistonacts like a rigid link, and friction damping is predominant.

3 Shock response of passive systems

Consider a single degree-of-freedom system subjected to a transient excitation in the form of a pulse function.The response of the system to a shock input is presented as a function of the duration of the pulse comparedwith the natural period of the system and the resulting plot is called the Shock Response Spectra (SRS)[1]. Typical response parameters are the relative and absolute displacement, related to the available space orclearance, and the maximum acceleration, which is an indicator of the forces transmitted.

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νmaxξ p

ξ p

ν(t )

ξ(t)

τT

Figure 1: Example of a shock response spectra (SRS) for an undamped SDOF system, excited at the baseby a versed sine pulse. Three typical regions are shown, namely the isolation, amplification and quasi-staticregions. The variables υ and ξ represent the time histories of the response and the input pulse respectively,whilst ξp represents the maximum pulse amplitude.

A typical SRS plot is presented in Fig. 1 for a base excited system where the input is a versed sine displace-ment pulse x(t). The response of the system is given as y(t) and it is normalized considering the maximumamplitude of the shock pulse xp as a reference. The horizontal axis is the period ratio, where τ is defined asthe shock duration and T is the natural period of the system. When the pulse is of short duration comparedto the natural period for instance, when τ

T is less than 0.5, the shock is said to be impulsive and the responseis smaller in magnitude than the excitation. For longer duration inputs, approximately similar to the naturalperiod of the system the maximum response is larger than the amplitude of the input. Finally, for pulsesof much longer duration compared to the natural period, the shock is applied relatively very slowly and itbecomes a quasi-static response. In order to achieve shock isolation it is required to have flexible supportsresulting in large relative displacements and static deformations resulting in a low natural frequency for thesupported mass.

4 Non-linear stiffness experimental model

The experimental device considers electromagnetic forces and mechanical stiffness to achieve a system witha very low natural frequency. The following sections describe the model, its properties and the shock re-sponse. The model presented here is very similar to a previously proposed system used for a differentisolation strategy [12, 13]. For this work, the model was built again, maintaining the same dimensions, butreducing the weight of the supporting frame.

The experimental device is shown in Fig. 2. The device includes two permanent, disk shaped neodymiummagnets fitted inside an aluminium ring and suspended between two electromagnets using four tensionednylon wires attached to the main frame. The nylon wires give positive stiffness in parallel with the stiffnessprovided by the magnets. The total mass of the permanent magnets and the aluminium ring was 0.0753 kg,which acted as the isolated mass (payload) in the experimental system. By configuring the DC voltage sup-plied to the electromagnets with the latter having opposite poles the magnetic force is essentially a negativestiffness force. The range of operation for the electromagnets is 0 to 24 DC volts, but a maximum of 20 V

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was used in the experiments to avoid overheating. The degree of the effective negative stiffness force canbe controlled by moving the electromagnets closer to the suspended magnets. During the experiments thedistance used was 10mm in order to prevent accidental impacts on the magnets.

Figure 2: Experimental rig used for the tests. The system comprises a suspended mass made up of analuminum support disk and two permanent magnets, and two electromagnets in line with the suspended massare mounted at a separation distance of 10mm. The setup is shown mounted on a shaker in the horizontalposition.

4.1 Properties of the system

The system was subjected to a random excitation test in order to estimate the equivalent natural frequencyfor a linear approximation and assess the effective stiffness change for different voltage configurations. Thefrequency response function (FRF) was obtained using a similar configuration to the setup used in the fre-quency sweept. However, in this test the amplitude was kept small and a random excitation was used inorder to approximate the system to a linear behaviour. In this case, the frequency span of interest was 0 to80 Hz, covering the first natural frequency of the system under different configurations. For the system withno electromagnets attached the effective natural frequency measured was 17 Hz. The system with electro-magnets attached and turned off registered a natural frequency of 16.9 Hz. The case with electromagnetsturned on and having an attraction force with respect to the suspended magnets reported a natural frequencyis 10.7 Hz for a supply voltage 20 V. On the other hand, the opposite effect where the polarity of the suppliedvoltage to the electromagnets results in a repulsive force shows an increase of the effective natural frequencyresulting in 21.4 Hz. These results are summarised in Table 1 , along with the calculated effective stiffnessconsidering the payload mass, and the equivalent damping ratio obtained from the curve fitting procedureusing the software MEScope.

4.2 Shock Response

This section presents the shock response of the experimental device for the electromagnetic different voltagesettings as considered before. The system was subjected to a base input in the form of a versed sine generatedwith a shaker controller LDS LASER USB then applied using a electrodynamic shaker LDS V721 in the

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Configuration fn (Hz) Stiffness k (N/m) Damping Ratio ζNo Electromagnets 17 741 0.012Magnets Off 16.9 732 0.042Magnets On Attractive 20V 10.7 293 0.088Magnets On Repulsive 20V 21.4 1175 0.028

Table 1: Measured parameters for the experimental rig corresponding to different voltage configurations.

horizontal position as before. The pulse is compensated by the controller using pre and post pulses to makesure the shaker dynamics do not alter the shape of the pulse. Four different pulse durations were considerednamely 5, 10, 15, and 20 milliseconds, and an amplitude of 5 g. These pulses were chosen so that therepresentative behaviour of the system could be characterised within the physical limits of the vibrationtesting system and the response of the device under test. An example of the pulse recorded on the shakertable is given in Fig. 3 showing a pulse of duration 20 milliseconds and amplitude of 5 g. The accelerationcontrol signal was measured at the base using a PCB accelerometer model 35C22, and on the suspendedmass using a KISTLER accelerometer.

Figure 3: Measurement of a typical versed sine pulse generated with the LASER USB controller showing thepre and post pulses used to compensate the shaker dynamics. This example has a duration of 20 millisecondsand amplitude of 5 g. Horizontal axis given in seconds and vertical axis in g.

The shock acceleration response results are presented in Fig. 4. The figures are divided in four subplots,each one for a particular pulse duration as follows (a) τ = 0.005s, (b) τ = 0.010s, (c) τ = 0.015s and(d) τ = 0.020s. The different situations are depicted by the different curves. i.e. the solid line representsthe electromagnets ON in attraction, dashed line for electromagnets ON in repulsion, dash dot line for theelectromagnets turned off and the doted line for the system with no electromagnets attached. The axes arenormalised, the time axis is normalised with respect to the pulse duration and the amplitude axis consideringthe value of gravity acceleration g as reference.

The Shock Response Spectra (SRS) is presented in Fig. 5. This figure explores the effect of the differentvoltage settings used in the experiments, as well as the different input amplitude levels considered before.The SRS is presented according to the shock response nomenclature, where the vertical axis gives the maxi-max acceleration response normalised by the maximum pulse amplitude. However, the horizontal axis givesthe input duration in milliseconds, since there is no actual value of the natural period to normalise against asthe system is nonlinear. The line styles are presented accordingly to the time responses presented before, i.e.solid line for the electromagnets ON in attraction, dashed line for electromagnets ON in repulsion, dash dotline for the electromagnets turned off, and the doted line for the system with no electromagnets attached.

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ν̈g

ν̈g

ν̈g

ν̈g

(a) (b)

(c) (d)

Figure 4: Shock Response of the experimental device subjected to a 5 g amplitude versed sine pulse fordifferent durations of the input pulse. (a) τ = 0.005s, (b) τ = 0.010s, (c) τ = 0.015s and (d) τ = 0.020s(— Electromagnets ON Attraction - - - Electromagnets ON repulsion - · - Electromagnets off, · · · NoElectromagnets attached.)

4.3 Discussion

By inspecting the results in Table 1, from the Frequency Response Function measurements, it can be seenthat the device is indeed able to reduce the dynamic stiffness when the electromagnets exert an attractionforce to the suspended magnet. The combination of the attraction force which can be seen as a negativestiffness, and the positive stiffness of the suspension provided by the nylon cables result in a reduced naturalfrequency. Depending upon the voltage applied the apparent natural frequency changes about to 50% from21.4 Hz to 10.7 Hz corresponding to the system with repulsive and attractive configurations respectively.This equates to an equivalent change of stiffness of approximately 75%. However, the behaviour of thesystem with no electromagnets attached is very similar compared to the situation when the electromagnetsare turned off. The effective natural frequency is almost the same, i.e. 17 Hz and 16.9 Hz respectively.

Regarding the actual shock response, the general trend is that the absolute acceleration of the system withlow dynamic stiffness system can be effectively reduced when the electromagnets are configured to providean attraction force. The amount of reduction in the response is considerable higher for the short pulses i.e 5,10 and 15 milliseconds, which is expected as shown in the theory. It is important to note that the responseof the low dynamic stiffness system no longer a becomes a sinusoid when compared to the normal systemin free vibration, this effect is thought to be due the nonlinearity of the magnetic forces involved. The othervoltage configurations show a very similar behaviour in both oscillating frequency and amplitude, followinga similar trend as the input amplitude increases, and have a higher response compared to the system in

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( ν̈)maxξ p

τ(ms)

Figure 5: Shock response spectra of the experimental device under versed sine excitation. Vertical axis is nor-malised considering the maximum pulse amplitude corresponding to each case and horizontal axis is givenin milliseconds. ( — Electromagnets ON Attraction - - - Electromagnets ON repulsion - · - Electromagnetsoff, · · · No Electromagnets attached.)

attraction unless the pulse is longer than 20 milliseconds, when the amplitude of the latter system increases.

The SRS plot shows a general overview of the behaviour of the system. In general, it can be seen that theshock response is greatly reduced when using a low dynamic stiffness configuration. Most important to noteis that the effective response when the magnets are in an attraction configuration is only reduced when thepulse duration is smaller than 15 milliseconds. Otherwise, when the pulse is of large duration, i.e. higher than20 milliseconds the response of the system is higher compared to the amplitude of the pulse. As observedin the time histories and explained before, other electromagnet configurations show a similar behaviour withhigher responses for shorter pulses.

5 Dry friction isolators

A common type of isolator used for vibration and shock isolation are the wire rope springs. These isolatorsare regarded as highly effective for extreme conditions found in military, naval and aerospace applications.These isolators are made up using a series of steel strands twisted around a core strand, and the resulting wirerope is arranged in a leaf or helical fashion. These isolators present a great capability for energy dissipationdue to the friction created between the wire strands as the cable twists when the isolator is loaded andunloaded. This behaviour is shown in Figure 6 (a). Figure 6(b) shows a commercially available isolatorAdvanced Antivibration Components model V10Z69-0937290.

An advantage of these isolators is that they can work in tension-compression, shear and torsion scenarios.Apart from the non linear Coulomb damping observed in these isolators, they also present nonlinear stiffnesscharacteristics. In this section, one sample of wire rope isolator is considered and studied experimentallyin order to obtain its damping properties and shock response. The results of the characterisation of wirerope isolators are presented as early results of an undergoing research project and thus are part of a work inprogress.

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(a) (b)

Figure 6: (a) Detail of the strands in a wire rope isolator. The lay direction of the wires is opposite to the di-rection of the rope coils, thus creating friction between strands when the isolator is loaded. (b) Commerciallyavailable isolator model Advanced Antivibration Components V10Z69-0937290

5.1 Damping measurement

The damping of the isolator was estimated measuring the hysteresis loops when the isolator was subjected toa low frequency harmonic force. The frequency of the sinusoid force considered was 5 Hz provided by usingan electrodynamic shaker LDS V408. One end of the isolator was attached to the shaker whilst the other endwas attached to a fixed wall.

Figure 7: Hysteresis loops for the isolator model Advanced Antivibration Components V10Z69-0937290.Each loop corresponds to a different amplitude of the shaker ( — 1 V RMS - - - 1.5 V RMS · · · 2.5 V RMS- · - 3 V RMS .)

The isolator was initially compressed and different values of the input force were considered by increasingthe gain in the amplifier. Thus, several hysteresis loops could be obtained. The acceleration and force datawas acquired with a PCB Piezotronics impedance sensor 288D01 through a DataPhsysics Quattro signalanalyzer. The acceleration signal was filtered and integrated twice to obtain the displacement of the isolator.The hysteresis loops are presented in Figure 7, where each loop corresponds to a different input force. i.e.the voltage of the signal supplied to the shaker was increased thus increasing input force.

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The loss factor of the system was estimated considering the area of the hysteresis loops, since it representsthe energy dissipated per cycle. A relationship is defined from the maximum energy the system can dissipateand the actual energy dissipated. The loss factor is calculated as:

η =Aloop

πFmaxDmax(1)

Where Dmax and Fmax are the major ands minor axis of the elipse respectively, and Aloop is the area of thehysteresis loop. As the displacement in the system is increased, i.e. higher input force, the area of the loopincreases thus increasing the effective energy dissipation. The fractional damping is presented in Figure 8,for the different values of the input signal supplied to the shaker. In general, as the input force increasesso does the damping, but there is a point in which the damping begins to decrease. It is believed that atafter certain deflection, the friction force in the wire strands decreases. This might be explained due to thenonlinear nature of the isolator, as the shape of the hysteresis loop changes for larger amplitudes,. i.e. thearea of the hysteresis loop is bigger but the ratio to the maximum energy that can be dissipated decreases, asstated by Equation 1.

Figure 8: Damping percent calculated from hysteresis loop for the wire rope spring considered. The hori-zontal axis represents the RMS voltage of input signal supplied to the shaker)

5.2 Shock Response

The SRS of the wire rope isolator was measured by applying a versed sine pulse to the base of the isolatorthrough the LDS V408 electrodynamic shaker. Acceleration signals were acquired on the base of the shakerand on top of the isolator then processed with a DataPhysics Quattro signal analyzer. Pulses of differentdurations were applied to get values of the period ratio τ

T of de 0.25, 0.5, 1, 1.5, 2, 3 and 4. These values arerepresentative of the different regions of the SRS. The isolator was loaded with different reference masses.Figure 9 shows the SRS plots for the different cases considered, including the system with no initial load.

A general trend can be observed regarding the mass loading. When the isolator has no mass, i.e. is initiallyundeformed, the system behaves closely to the linear mass spring system, showing the amplification andquasi static zones typically found in undamped linear isolators. As the system has no deformation, thedamping in the system is very small, because there is no relative motion between the wire strands, i.e. nofriction. However, when the system is loaded, the deflection in the system increases the amount of damping,effectively improving the isolation characteristics.

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Figure 9: Shock Response Spectra for the wire rope spring considered, for different values of the mass load.Horizontal axis represents period ratio τ

T and vertical axis represents maximal acceleration response in g (— No mass - - - 0.03439 kg · · · 0.5857kg.)

It can be seen how the effectiveness of the wire rope isolator is much higher specially as the load in theisolator increases. This statement is made based on comparison with the unloaded system, whose responseis very similar to the linear mass spring undamped isolator. It is also important to note that once the systemis loaded, the response is very similar regardless of the value of the mass attached, provided it is within thephysical limits of the isolator.

6 Concluding remarks and suggestions for future work

Two different approaches for shock isolation using nonlinear elements were introduced in this work. Thefirst approach considers an isolator with low dynamic stiffness . The concept is based on the shock isolationtheory stating that a soft support decreases the response to transient excitations for certain configurations.Experimental results were presented, using a prototype based on using magnetic forces in attraction to pro-vide negative stiffness and a positive stiffness element given by a mechanical suspension. Improvementsin shock isolation were observed when the electromagnets in the system were configured in attraction orsoftening for short duration pulses. In contrast, for longer pulses the response is actually amplified comparedto the reference system with no electromagnets attached. Since the system properties were estimated usinga linear approximation it remains an objective of future work to further investigate the nonlinear effects andphenomena intrinsic to the magnetic forces in the system. Moreover, it is important to consider differentinput amplitudes and validate with theoretical results.

The second approach is a dry friction isolator, or wire rope spring. These isolators are widely available ascomercial isolators, but little is known about its shock response. An insight into the damping characteristicsof these isolators was presented, estimating energy dissipation through the measurement of hysteresis loops.The shock response of the isolator was also presented, for pulses of different amplitudes. It was found howa preloaded isolator experiences a much better shock isolation compared with the linear model, however,when the equivalent damping of the system is small, i.e. low or no preload, the system approaches the linearmass spring model. It is suggested for further research to consider more isolator samples for analysis, aswell as apply different input amplitudes and investigate the nonlinear effects. Furthermore, the mathematical

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modelling of these isolators remain a future task, where a simplified system can be considered in order todevelop a model that can describe the shock response of the system. Another suggestion is to considernonlinear stiffness and damping in mathematical model, since the sire rope isolators presents nonlinearity inboth properties.

Acknowledgements

The authors would like to acknowledge the financial support of the Mexican Council for Science and Tech-nology, CONACyT, and the Ministry of Public Education, SEP-PROMEP for this project.

References

[1] Ayre, R.S. Engineering Vibrations , Mc. Graw Hill New York (2002).

[2] Harris, C.M., Persol A. G. Handbook of Shock and Vibration, Mc. Graw Hill New York (1958).

[3] Snowdon, J.C., Response of Nonlinear Shock Mountings to Transient Foundation Displacements, TheJournal of the Acoustical Society of America, Vol. 33 No.10, (1961), pp.1295-1304.

[4] Snowdon, J.C., Transient Response of Nonlinear Isolation Mountings to Pulselike Displacements, TheJournal of the Acoustical Society of America, Vol. 35, No. 3, (1963) pp.389-396.

[5] P. Alabuzhev, A.G., Eugene I. Rivin, L. Kim, G. Migirenko, V. Chon, P. Stepanov, Vibration Protectingand Measuring Systems with Quasi-Zero Stiffness., Taylor & Francis, (1989).

[6] Waters, T.P., Carrella, A., and Brennan, M.J., Static analysis of a passive vibration isolator with quasi-zero-stiffness characteristic, Journal of Sound and Vibration, Vol. 301 No. 3-5, (2007), pp.678–89,(2007).

[7] Carrella, A., Brennan, M.J., Waters, T.P. and Shin, K., On the design of a high-static-low-dynamicstiffness isolator using linear mechanical springs and magnets, Journal of Sound and Vibration, Vol.315, No. 3, (2008), pp. 712-20.

[8] Kovacic, I., Brennan, M.K., and Waters, T.P., A study of a nonlinear vibration isolator with a quasi-zerostiffness characteristic, Journal of Sound and Vibration, Vol. 315, No. 3, (2008), pp. 700-11.

[9] Zhou, N., Liu, K., A tunable high-static-low-dynamic stiffness vibration isolator, Journal of Sound andVibration, Vol. 329, No. 9, (2010), pp. 1254-1273.

[10] Xingtian, L., Xiuchang, H., Hongxing, H., Performance of a zero stiffness isolator under shock excita-tions, Journal of Vibration and Control, Article in Press, (2013).

[11] Mercer C.A., Rees P.L. (1971) An optimum shock isolator, Journal of Sound and Vibration, Vol. 18,No. 4, (1971), pp. 511-520.

[12] Ledezma, D.F., Ferguson, N.S., Brennan, M.J., Shock isolation using an isolator with switchable stiff-ness, Journal of Sound and Vibration, Vol. 330, No. 5, (2011), pp. 868-882.

[13] Ledezma, D.F., Ferguson, N.S., Brennan, M.J., An experimental switchable stiffness device for shockisolation, Journal of Sound and Vibration, Vol. 331, No. 23, (2012), pp. 4987-5001.

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