Shock isolation systems using non linear stiffness and damping

Transcription

1 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 Investigacion e Innovacion en Ingenieria Aeronautica, Nuevo Leon, Mexico. diego.ledezmard@uanl.edu.mx 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 experimentally, 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 levels 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

2 4112 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 frequency. Moreover, the isolation system could behave non-linearly due to the large deformations resulting from shocks. In this article, two approaches for shock isolation are discussed. The first one considers a nonlinear stiffness device following the idea of a low dynamic high static stiffness isolator. This strategy is sometimes called quasi zero stiffness, and usually involves the combination of a linear spring with a negative stiffness in order to get a very low stiffness value around the equilibrium point. Previous work regarding the use of such isolators 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 is characterised in terms of the frequency response and shock response for different situations. Then, the use of dry friction isolators is considered, based on wire rope springs, which are widely available as commercial off the shelf isolators. However, the properties and shock response of these devices has not been properly studied, and this paper is intended to give some insight into the damping quantification and shock response of 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 has been investigated recently as alternative means for passive vibration isolation. Particularly, the idea of low dynamic high static stiffness isolators is of interest, since it can provide excellent isolation at a particular equilibrium point where the dynamic stiffness of the system is very low [5]. Carella et al have studied the static 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]. Combination of mechanical springs and electromagnetic forces, have also been used to apply the concept of quasi zero stiffness to tunable vibration absorbers [9]. However, in the previous studies no shock response has been considered. Nevertheless, this idea presents great potential for shock isolation systems since one wants to ideally have a low frequency mount capable of high energy storage, but this is not always possible due to restrictions on space and supporting weight. A recent published work by Xingtian [10] shows theoretically that this strategy is beneficial for shock isolation in some situations, but might increase the amount of relative motion. It remains a further task to validate experimentally and investigate the practical implications of these systems in shock isolation. They found that a system with very low tangent stiffness has better shock isolation performance regarding absolute acceleration response, but the absolute displacement response is increased unless 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 an optimum shock isolator with great advantages over a common resilient mount. The device is passive in nature, but the friction force can be varied. For low frequency inputs, the device depends solely upon viscous damping given by a dashpot. However, in the case of high frequency and large amplitude inputs, the piston acts 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 compared with 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 or clearance, and the maximum acceleration, which is an indicator of the forces transmitted.

3 VIBRATION ISOLATION AND CONTROL 4113 ν max ξ p ν(t) ξ p ξ(t) τ T Figure 1: Example of a shock response spectra (SRS) for an undamped SDOF system, excited at the base by a versed sine pulse. Three typical regions are shown, namely the isolation, amplification and quasi-static regions. The variables υ and ξ represent the time histories of the response and the input pulse respectively, whilst ξ p 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 displacement pulse x(t). The response of the system is given as y(t) and it is normalized considering the maximum amplitude of the shock pulse x p as a reference. The horizontal axis is the period ratio, where τ is defined as the shock duration and T is the natural period of the system. When the pulse is of short duration compared to the natural period for instance, when τ T is less than 0.5, the shock is said to be impulsive and the response is smaller in magnitude than the excitation. For longer duration inputs, approximately similar to the natural period of the system the maximum response is larger than the amplitude of the input. Finally, for pulses of much longer duration compared to the natural period, the shock is applied relatively very slowly and it becomes a quasi-static response. In order to achieve shock isolation it is required to have flexible supports resulting in large relative displacements and static deformations resulting in a low natural frequency for the supported mass. 4 Non-linear stiffness experimental model The experimental device considers electromagnetic forces and mechanical stiffness to achieve a system with a very low natural frequency. The following sections describe the model, its properties and the shock response. The model presented here is very similar to a previously proposed system used for a different isolation strategy [12, 13]. For this work, the model was built again, maintaining the same dimensions, but reducing the weight of the supporting frame. The experimental device is shown in Fig. 2. The device includes two permanent, disk shaped neodymium magnets fitted inside an aluminium ring and suspended between two electromagnets using four tensioned nylon wires attached to the main frame. The nylon wires give positive stiffness in parallel with the stiffness provided by the magnets. The total mass of the permanent magnets and the aluminium ring was kg, which acted as the isolated mass (payload) in the experimental system. By configuring the DC voltage supplied to the electromagnets with the latter having opposite poles the magnetic force is essentially a negative stiffness force. The range of operation for the electromagnets is 0 to 24 DC volts, but a maximum of 20 V

4 4114 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 was used in the experiments to avoid overheating. The degree of the effective negative stiffness force can be controlled by moving the electromagnets closer to the suspended magnets. During the experiments the distance 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 an aluminum support disk and two permanent magnets, and two electromagnets in line with the suspended mass are mounted at a separation distance of 10mm. The setup is shown mounted on a shaker in the horizontal position. 4.1 Properties of the system The system was subjected to a random excitation test in order to estimate the equivalent natural frequency for a linear approximation and assess the effective stiffness change for different voltage configurations. The frequency response function (FRF) was obtained using a similar configuration to the setup used in the frequency sweept. However, in this test the amplitude was kept small and a random excitation was used in order to approximate the system to a linear behaviour. In this case, the frequency span of interest was 0 to 80 Hz, covering the first natural frequency of the system under different configurations. For the system with no electromagnets attached the effective natural frequency measured was 17 Hz. The system with electromagnets attached and turned off registered a natural frequency of 16.9 Hz. The case with electromagnets turned on and having an attraction force with respect to the suspended magnets reported a natural frequency is 10.7 Hz for a supply voltage 20 V. On the other hand, the opposite effect where the polarity of the supplied voltage to the electromagnets results in a repulsive force shows an increase of the effective natural frequency resulting in 21.4 Hz. These results are summarised in Table 1, along with the calculated effective stiffness considering the payload mass, and the equivalent damping ratio obtained from the curve fitting procedure using the software MEScope. 4.2 Shock Response This section presents the shock response of the experimental device for the electromagnetic different voltage settings as considered before. The system was subjected to a base input in the form of a versed sine generated with a shaker controller LDS LASER USB then applied using a electrodynamic shaker LDS V721 in the

5 VIBRATION ISOLATION AND CONTROL 4115 Configuration f n (Hz) Stiffness k (N/m) Damping Ratio ζ No Electromagnets Magnets Off Magnets On Attractive 20V Magnets On Repulsive 20V 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 make sure the shaker dynamics do not alter the shape of the pulse. Four different pulse durations were considered namely 5, 10, 15, and 20 milliseconds, and an amplitude of 5 g. These pulses were chosen so that the representative behaviour of the system could be characterised within the physical limits of the vibration testing system and the response of the device under test. An example of the pulse recorded on the shaker table is given in Fig. 3 showing a pulse of duration 20 milliseconds and amplitude of 5 g. The acceleration control signal was measured at the base using a PCB accelerometer model 35C22, and on the suspended mass using a KISTLER accelerometer. Figure 3: Measurement of a typical versed sine pulse generated with the LASER USB controller showing the pre and post pulses used to compensate the shaker dynamics. This example has a duration of 20 milliseconds and 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 represents the electromagnets ON in attraction, dashed line for electromagnets ON in repulsion, dash dot line for the electromagnets turned off and the doted line for the system with no electromagnets attached. The axes are normalised, the time axis is normalised with respect to the pulse duration and the amplitude axis considering the value of gravity acceleration g as reference. The Shock Response Spectra (SRS) is presented in Fig. 5. This figure explores the effect of the different voltage 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 maximax acceleration response normalised by the maximum pulse amplitude. However, the horizontal axis gives the input duration in milliseconds, since there is no actual value of the natural period to normalise against as the 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 dot line for the electromagnets turned off, and the doted line for the system with no electromagnets attached.

6 4116 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 (a) (b) ν g ν g t τ t τ (c) (d) ν g ν g t τ t τ Figure 4: Shock Response of the experimental device subjected to a 5 g amplitude versed sine pulse for different 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, No Electromagnets attached.) 4.3 Discussion By inspecting the results in Table 1, from the Frequency Response Function measurements, it can be seen that the device is indeed able to reduce the dynamic stiffness when the electromagnets exert an attraction force to the suspended magnet. The combination of the attraction force which can be seen as a negative stiffness, and the positive stiffness of the suspension provided by the nylon cables result in a reduced natural frequency. Depending upon the voltage applied the apparent natural frequency changes about to 50% from 21.4 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 the system with no electromagnets attached is very similar compared to the situation when the electromagnets are 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 with low dynamic stiffness system can be effectively reduced when the electromagnets are configured to provide an 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 response of the low dynamic stiffness system no longer a becomes a sinusoid when compared to the normal system in free vibration, this effect is thought to be due the nonlinearity of the magnetic forces involved. The other voltage configurations show a very similar behaviour in both oscillating frequency and amplitude, following a similar trend as the input amplitude increases, and have a higher response compared to the system in

7 VIBRATION ISOLATION AND CONTROL 4117 ( ν) max ξ p τ(ms) Figure 5: Shock response spectra of the experimental device under versed sine excitation. Vertical axis is normalised considering the maximum pulse amplitude corresponding to each case and horizontal axis is given in milliseconds. ( Electromagnets ON Attraction Electromagnets ON repulsion - - Electromagnets off, 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 the shock response is greatly reduced when using a low dynamic stiffness configuration. Most important to note is that the effective response when the magnets are in an attraction configuration is only reduced when the pulse duration is smaller than 15 milliseconds. Otherwise, when the pulse is of large duration, i.e. higher than 20 milliseconds the response of the system is higher compared to the amplitude of the pulse. As observed in the time histories and explained before, other electromagnet configurations show a similar behaviour with higher 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 isolators are 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 wire rope is arranged in a leaf or helical fashion. These isolators present a great capability for energy dissipation due to the friction created between the wire strands as the cable twists when the isolator is loaded and unloaded. This behaviour is shown in Figure 6 (a). Figure 6(b) shows a commercially available isolator Advanced Antivibration Components model V10Z 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 stiffness characteristics. In this section, one sample of wire rope isolator is considered and studied experimentally in order to obtain its damping properties and shock response. The results of the characterisation of wire rope isolators are presented as early results of an undergoing research project and thus are part of a work in progress.

8 4118 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 (a) (b) Figure 6: (a) Detail of the strands in a wire rope isolator. The lay direction of the wires is opposite to the direction of the rope coils, thus creating friction between strands when the isolator is loaded. (b) Commercially available isolator model Advanced Antivibration Components V10Z Damping measurement The damping of the isolator was estimated measuring the hysteresis loops when the isolator was subjected to a low frequency harmonic force. The frequency of the sinusoid force considered was 5 Hz provided by using an electrodynamic shaker LDS V408. One end of the isolator was attached to the shaker whilst the other end was attached to a fixed wall. Figure 7: Hysteresis loops for the isolator model Advanced Antivibration Components V10Z Each loop corresponds to a different amplitude of the shaker ( 1 V RMS V RMS 2.5 V RMS V RMS.) The isolator was initially compressed and different values of the input force were considered by increasing the gain in the amplifier. Thus, several hysteresis loops could be obtained. The acceleration and force data was acquired with a PCB Piezotronics impedance sensor 288D01 through a DataPhsysics Quattro signal analyzer. 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.

9 VIBRATION ISOLATION AND CONTROL 4119 The loss factor of the system was estimated considering the area of the hysteresis loops, since it represents the energy dissipated per cycle. A relationship is defined from the maximum energy the system can dissipate and the actual energy dissipated. The loss factor is calculated as: η = A loop πf max D max (1) Where D max and F max are the major ands minor axis of the elipse respectively, and A loop is the area of the hysteresis loop. As the displacement in the system is increased, i.e. higher input force, the area of the loop increases 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 increases so does the damping, but there is a point in which the damping begins to decrease. It is believed that at after certain deflection, the friction force in the wire strands decreases. This might be explained due to the nonlinear nature of the isolator, as the shape of the hysteresis loop changes for larger amplitudes,. i.e. the area of the hysteresis loop is bigger but the ratio to the maximum energy that can be dissipated decreases, as stated by Equation 1. Figure 8: Damping percent calculated from hysteresis loop for the wire rope spring considered. The horizontal 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 isolator through the LDS V408 electrodynamic shaker. Acceleration signals were acquired on the base of the shaker and on top of the isolator then processed with a DataPhysics Quattro signal analyzer. Pulses of different durations 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 are representative 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 initially undeformed, the system behaves closely to the linear mass spring system, showing the amplification and quasi static zones typically found in undamped linear isolators. As the system has no deformation, the damping in the system is very small, because there is no relative motion between the wire strands, i.e. no friction. However, when the system is loaded, the deflection in the system increases the amount of damping, effectively improving the isolation characteristics.

10 4120 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 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 kg kg.) It can be seen how the effectiveness of the wire rope isolator is much higher specially as the load in the isolator increases. This statement is made based on comparison with the unloaded system, whose response is very similar to the linear mass spring undamped isolator. It is also important to note that once the system is loaded, the response is very similar regardless of the value of the mass attached, provided it is within the physical 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. The first approach considers an isolator with low dynamic stiffness. The concept is based on the shock isolation theory 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 provide negative stiffness and a positive stiffness element given by a mechanical suspension. Improvements in shock isolation were observed when the electromagnets in the system were configured in attraction or softening for short duration pulses. In contrast, for longer pulses the response is actually amplified compared to the reference system with no electromagnets attached. Since the system properties were estimated using a linear approximation it remains an objective of future work to further investigate the nonlinear effects and phenomena intrinsic to the magnetic forces in the system. Moreover, it is important to consider different input amplitudes and validate with theoretical results. The second approach is a dry friction isolator, or wire rope spring. These isolators are widely available as comercial isolators, but little is known about its shock response. An insight into the damping characteristics of 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 how a 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 linear mass spring model. It is suggested for further research to consider more isolator samples for analysis, as well as apply different input amplitudes and investigate the nonlinear effects. Furthermore, the mathematical

11 VIBRATION ISOLATION AND CONTROL 4121 modelling of these isolators remain a future task, where a simplified system can be considered in order to develop a model that can describe the shock response of the system. Another suggestion is to consider nonlinear stiffness and damping in mathematical model, since the sire rope isolators presents nonlinearity in both properties. Acknowledgements The authors would like to acknowledge the financial support of the Mexican Council for Science and Technology, 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, The Journal of the Acoustical Society of America, Vol. 33 No.10, (1961), pp [4] Snowdon, J.C., Transient Response of Nonlinear Isolation Mountings to Pulselike Displacements, The Journal of the Acoustical Society of America, Vol. 35, No. 3, (1963) pp [5] P. Alabuzhev, A.G., Eugene I. Rivin, L. Kim, G. Migirenko, V. Chon, P. Stepanov, Vibration Protecting and 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 quasizero-stiffness characteristic, Journal of Sound and Vibration, Vol. 301 No. 3-5, (2007), pp , (2007). [7] Carrella, A., Brennan, M.J., Waters, T.P. and Shin, K., On the design of a high-static-low-dynamic stiffness isolator using linear mechanical springs and magnets, Journal of Sound and Vibration, Vol. 315, No. 3, (2008), pp [8] Kovacic, I., Brennan, M.K., and Waters, T.P., A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic, Journal of Sound and Vibration, Vol. 315, No. 3, (2008), pp [9] Zhou, N., Liu, K., A tunable high-static-low-dynamic stiffness vibration isolator, Journal of Sound and Vibration, Vol. 329, No. 9, (2010), pp [10] Xingtian, L., Xiuchang, H., Hongxing, H., Performance of a zero stiffness isolator under shock excitations, 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 [12] Ledezma, D.F., Ferguson, N.S., Brennan, M.J., Shock isolation using an isolator with switchable stiffness, Journal of Sound and Vibration, Vol. 330, No. 5, (2011), pp [13] Ledezma, D.F., Ferguson, N.S., Brennan, M.J., An experimental switchable stiffness device for shock isolation, Journal of Sound and Vibration, Vol. 331, No. 23, (2012), pp

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