- Email: [email protected]
An experimental switchable stiffness device for shock isolation
Journal of Sound and Vibration 331 (2012) 4987–5001
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
An experimental switchable stiffness device for shock isolation D.F. Ledezma-Ramirez a,n, N.S. Ferguson b, M.J. Brennan c a
´nica y Ele´ctrica, Av. Universidad s/n, San Nicola ´s de los Garza, Nuevo Leo ´noma de Nuevo Leo ´n, Facultad de Ingenierı´a Meca ´n 66451, Me´xico Universidad Auto Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK c Departamento de Engenharia Mecˆ anica, UNESP, Ilha Solteira, SP 15385-000, Brazil b
a r t i c l e i n f o
abstract
Article history: Received 12 August 2011 Received in revised form 15 June 2012 Accepted 19 June 2012 Handling Editor: D.J. Wagg Available online 21 July 2012
The development of an experimental switching stiffness device for shock isolation is presented. The system uses magnetic forces to exert a restoring force, which results in an effective stiffness that is used to isolate a payload. When the magnetic force is turned on and off, a switchable stiffness is obtained. Characterization of the physical properties of the device is presented. They are estimated in terms of the percentage stiffness change and effective damping ratio when switched between two constant stiffness states. Additionally, the setup is used to implement a control strategy to reduce the shock response and minimize residual vibration. The system was found to be very effective for shock isolation. The response is reduced by around 50 percent compared with passive isolation showing good correlation with theoretical predictions, and the effective damping ratio in the system following the shock was increased from about 4.5 percent to 13 percent. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction Shock is a common excitation that can be highly detrimental to sensitive objects and can also cause human discomfort. It is defined as a very short or transient excitation characterized by high displacements and accelerations that can potentially lead to large mechanical stresses and transmitted forces [1]. The process of shock isolation is based on energy storage by an elastic element normally requiring large deflections and the subsequent dissipation of the stored energy by some damping mechanism. This creates a compromise between the physical constraints or space available for the isolator and the effectiveness of it. Commonly, a shock isolator is a passive mount comprising some form of mechanical spring and/or viscoelastic elements. The analysis of shock isolation systems is often performed by considering single degree-of-freedom models with low damping under the effect of pulse input functions. Most of the fundamental theory of shock isolation largely in use today was developed in the 1950s and 1960s, for example, Ayre [2], Snowdon [3–5] and Eshleman and Rao [6]. The concept of the Shock Response Spectra (SRS) was developed and is one of the most important tools to evaluate the severity of impacts and selection of appropriate shock isolators. Snowdon was also one of the first researchers to incorporate nonlinear elements into shock isolators, and Eshlemann studied a number of devices such as ring springs, stranded wire springs, etc. More recently, researchers have also considered the further use of nonlinear elements for shock isolation [7–9]. Lately, there has been a growing interest in the use of semi-active isolation systems where the physical properties of the system can change in real time to provide better vibration isolation. Although these systems have been studied extensively for
n
Corresponding author. Tel.: þ52 81 83294020x5762; fax: þ 52 81 83320904. E-mail address: [email protected] (D.F. Ledezma-Ramirez).
0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.06.010
4988
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
harmonic and random vibration, there has been little work on shock isolation. Some switchable or semi-active damping strategies based on the skyhook damper concept have been studied [10], and Waters et al. [11] showed that reducing the damping to a lower value during a shock input can lead to better isolation performance. Switchable and variable stiffness strategies have also been investigated theoretically and experimentally. An important paper is one by Winthrop reviewing different methods to achieve variable stiffness [12]. A strategy to control transient vibrations was proposed by Onoda et al. [13] using an on/off control logic to provide energy dissipation in lightly damped systems, based on the idea of controlling structural vibration using variable stiffness as proposed by Chen [14]. Jabbari and Bobrow [15] and Leavitt et al. [16] devised a resetting technique, based on a switchable stiffness aiming to extract energy from a mechanical system while having a high stiffness value at all times. This concept has also been exploited in the field of tunable vibration absorbers. For instance, Zhou and Liu [17] have used an electromagnetic spring combined with a mechanical spring to develop a tunable isolator possessing nonlinear behavior. However, there are no results or investigations related to shock response in these studies. The concept of switchable stiffness has been investigated theoretically, by the authors of this current paper, as a means of energy dissipation in lightly damped systems, where it is difficult to implement another form of external damping (see for example [18] and the references therein). This is effectively a semi-active control strategy. The model presented in [18] comprises a mass supported by two springs, one of which can be disconnected. Switching in and out of the spring involves a two-stage control strategy; stiffness control during the shock to reduce the maximum response of the payload, and reduction of the residual vibration after the shock has occurred. The theoretical simulations presented demonstrate that it is possible to obtain better shock isolation by switching the stiffness in lightly damped systems. This paper is a continuation of the work in [18], describing the practical realization of a semi-active shock isolation system. The main motivation in developing such strategies is to improve methods of shock isolation. Potential engineering applications include the protection of sensitive electronic devices in harsh environments, for example in ships and military vehicles, low frequency and semi-active vibration isolators, aerospace structures and earthquake engineering. In this work, experimental results are presented, characterizing the system properties and demonstrating the switchable stiffness strategies and the improved shock response of such an isolation system. Prior to the description of the experimental work in later sections, some background information on the principles of shock isolation is given in Section 2, with special emphasis on semi-active shock isolation. 2. Brief overview of shock isolation theory 2.1. Shock isolation In the theory of shock isolation, the maximax response is defined as the maximum response to a particular shock pulse at any time. The value of this response depends mainly upon the duration and shape of the shock. Several ideal pulses can be considered such as half sine, rectangular, trapezoidal, etc., and more complex pulses can be used as well. Normally the duration of the pulse t is compared to the natural period of the system T in order to evaluate the maximax response. The theory has been extensively covered by Ayre [2]. To summarize a typical shock scenario, there are three zones in the shock response. For short pulses compared to the natural period, i.e. approximately when (t/T) o0.25, the shock is said to be impulsive, and the maximum response is smaller than the amplitude of excitation, i.e. the system is isolated from the shock input. For longer inputs (t/T)E1, the maximum response is larger than the amplitude of the input. For much longer duration pulses compared to the natural period, i.e. ðt=TÞ b0:25 the shock is applied relatively slowly and the input becomes quasi-static. From the practical point of view it is desirable to have a low natural frequency isolator in order to achieve isolation, thus requiring a flexible support resulting in large relative displacement and static deformation. 2.2. Shock isolation using switchable stiffness To overcome the compromise between minimizing the transmission of the shock through the isolator and minimizing the static deflection, a semi-active control method can be adopted, whereby the stiffness is switched to a lower value whilst the shock pulse is applied. This system was discussed in [18] and is outlined briefly here for the convenience of the reader. The model considered comprises a mass supported by two springs one of which can be disconnected. An undamped system is considered for simplicity. The switching of the spring is controlled by a control strategy, which is divided into two stages, namely the stiffness control during a shock, and the later stiffness switching during the residual vibration. The first part of the control strategy is to disconnect the stiffness Dk for the duration of the shock providing a softer support so that the maximum response is reduced. The equation of motion for the shock input is given by mn€ þkeffective n ¼ keffective xðtÞ
(1)
where keffective is the system stiffness, which can be either k or k Dk depending upon whether the secondary stiffness is connected or disconnected respectively as shown in Fig. 1. The term x(t) represents the shock excitation that is normally modeled as a pulse function such a half sine or a versed sine [2]. The displacement response of the system is given by n. The effect of reducing the stiffness is to change the effective period ratio t/T, of the system, and to shift the system from the amplification to the isolation region only whilst the shock is applied, maintaining a high stiffness otherwise. The strategy has
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4989
Fig. 1. Schematic of a sdof system with switchable stiffness under shock excitation x(t) applied to the base. The two stiffness in parallel have stiffness Dk and k Dk respectively and m is the mass with absolute displacement n.
Fig. 2. Effect of the switchable control strategy on the residual vibration: (a) phase plane plot, (b) time history of the displacement response (— high stiffness; - - - low stiffness).
been demonstrated to be capable of reducing the shock response in terms of the absolute displacement and acceleration responses, but the benefits on the relative response are small compared to the passive isolation model [18]. The second stage of the switching strategy involves the semi-active switching of the stiffness during the residual period, i.e. when the shock pulse ends and the system undergoes free vibration. The objective is to quickly dissipate the energy stored by the elastic element during the shock without adding an external damping mechanism. The same model is considered as before but now the control strategy is given by ( ) nn_ Z0 k keffective ¼ (2) kDk nn_ o0 where n_ is the velocity of the mass. The residual control strategy has to ensure that the amplitude of vibration decreases every cycle. The stiffness should be maximum and equal to k when the product nn_ is positive and minimum and equal to k Dk when nn_ is negative. When the displacement response satisfies the condition nn_ Z 0 the displacement n and the velocity n_ have the same sign. As a result the secondary spring, Dk, is disconnected when the absolute value of the displacement of the mass is a maximum. It is connected again when the absolute value of the velocity is maximum, when the system passes through its equilibrium position. The phase plane plot presented in Fig. 2(a) shows how the switching occurs effectively dissipating the energy at every stiffness reduction point. Fig. 2(b) depicts a time history corresponding to this example. A comprehensive study of this strategy is also presented in [18], concluding that a greater stiffness reduction leads to greater rate of reduction of the residual vibrations.
4990
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
Fig. 3. Passive and switched normalized displacement response. The switched response involves the implementation of both stages of the control strategy—during and after the shock pulse. The period ratio for the passive sytem is (t/T) ¼0.25 and a stiffness reduction of 50 percent was considered. The response is normalized by the maximum of the shock amplitude. The time axis is normalized with respect to the natural period of the system T (— switching system; and - - - passive system).
As an example of the control stages described above, a theoretical simulation is presented in Fig. 3, implementing both the stiffness reduction during the shock and the subsequent residual vibration control strategy. It is compared with that of a passive system, in which the stiffness has a fixed value of k, i.e. the high stiffness state. The response has been normalized by the maximum amplitude of the versed sine shock pulse. In this example there is a stiffness reduction of 50 percent and the period ratio for the system without control is ðt=TÞ ¼ 0:25. The benefits of the strategies acting together are clearly visible, achieving a reduction in the shock response for the absolute response, as well as a greater rate of reduction in the residual vibration.
3. Design of the experimental system The main objective of the work presented here was to design and implement a switchable stiffness device based on the theoretical studies in [18], briefly described above. The system was designed with the following considerations in mind: (a) to gain appreciable benefits in the shock response and the residual vibration suppression, the system had to be capable of achieving a high stiffness change—at least a factor of two, and (b) the stiffness change had to be rapid. Moreover, the system had to be lightly damped so that a significant change in the residual decay rate due to the switching could be clearly observed. It was also desirable that the system should behave as a single degree-of-freedom system, for which predictions had been previously obtained. To realize a system with these characteristics, an electromagnetic element was used to achieve a switchable stiffness. The fundamental idea is similar to the concept of magnetic levitation. If two electromagnets are aligned with their opposite poles facing, the resulting repulsive force can be switched on and off, to obtain a switchable stiffness element depending upon the voltage supplied to the electromagnets. This system is nonlinear because the repulsive force is not proportional to the separation of the magnets [19]. Also, the system needs to be physically stabilized by applying some stiffness constraints. The experimental system is shown in Fig. 4; a schematic of the system is given in Fig. 4(a) and a photograph of the system is shown in Fig. 4(b). The device included two permanent, disk shaped neodymium magnets which were suspended between two electromagnets. The permanent magnets were fitted inside an aluminum ring, which was suspended using four tensioned nylon wires attached to the main frame. The nylon wires acted as an additional permanent stiffness in parallel with the stiffness provided by the magnets. The total mass of the permanent magnets and the aluminum ring was 0.0753 kg, which acted as the isolated mass (payload) in the experimental system. It should be noted that although the system in Fig. 4 is shown in the vertical position, the experiments were made with the system in the horizontal position to avoid the effect of gravity. This is because the suspended weight could be closer to one of the electromagnets at some point due to the static deflection, resulting in an asymmetric system.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4991
Fig. 4. (a) Diagram of the switchable stiffness experimental system in the vertical position. The permanent magnets (1, 2) are suspended between two electromagnets (3, 4) using four wires (6, 7) that also join the magnet to the main frame (5). The permanent magnets are held by an aluminum disk (8). (b) Photograph of the rig.
When the electromagnets were turned off, the stiffness of the system was low and when they were supplied with a constant DC voltage of 12 V the stiffness was high. A Hameg triple voltage source HM7042-5 was used. There was a hardening effect when the electromagnets were on because of the nature of the repulsive force between the electromagnets and the permanent magnets. On the other hand, a softening effect exists when the electromagnets were off because of the attraction between the iron core of the electromagnets and the permanent magnets. This effect became more evident as the permanent magnets approached the electromagnets, due to the nonlinear nature of the magnetic forces. The electromagnets used were capable of working in the 0–24 V voltage range. However, to avoid excessive heating a voltage of only 12 V was applied continuously. 4. Measurement of the physical properties of the experimental system Some physical properties of the system were estimated by measuring the transmissibility of the experimental system using broadband random excitation from 0 to 1000 Hz applied to the base. The transmissibility was calculated using the accelerations of the base of the rig and the suspended mass. The resonance peak for different applied voltages was determined and the effective stiffness change was found from the measured natural frequency and the system mass. To check the linear behavior of the system a simple static test was undertaken. Transmissibility tests (described later) were also undertaken for different input amplitudes. The static deflection was measured using a dial gage and adding different values of mass to the suspended magnet for different settings of the electromagnets, i.e. off and on. As shown in Fig. 5 the force–displacement plots shows a good degree of linearity for the displacement of the mass in the range of 0–2 mm. The system response for the subsequent dynamic tests was limited to approximately 1 mm. A block diagram showing the setup of the equipment is shown in Fig. 6. Note that for the transmissibility tests the switching circuit was disconnected and the DC voltage source connected directly to the electromagnets. The shaker was
4992
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
Fig. 5. Measured force–deflection characteristics of the switchable stiffness system. ( þ magnets switched on, magnets switched off). The lines represent linear least squares fit to the data.
Fig. 6. Schematic diagram of the experimental setup. Accelerometers were attached to the suspended mass and to the base of the rig. A PC generated the excitation signal which was fed to the shaker via a power amplifier. For the transmissibility measurements the switching circuit was disconnected and the DC voltage supply was connected directly to the electromagnets.
supplied with a random noise signal generated by a Data Physics Mobilizer analyzer through a power amplifier. Two PCB teardrop miniature accelerometers type 352C22 were used to measure the acceleration of the base of the shaker and the suspended magnets respectively. The peak acceleration levels measured on the base were between 0.08g and 1g for the minimum and maximum amplifier gains used in the tests (approximately 0.777g rms). The transmissibility spanned 0–100 Hz with 1600 lines of resolution. Fig. 7 shows the measured transmissibility of the system, where the solid and dashed lines represent the off state and the on state with a supply voltage of 12 V respectively. This voltage was chosen in order to avoid overheating when higher voltages were applied. The linearity of the system was evaluated using the coherence function. The corresponding coherence plots are presented in Fig. 8. In general, the coherence was considered to be acceptable. However, there are certain frequencies at which the coherence falls, which are probably partly due to the small nonlinear effects of the magnetic forces. Overall, for frequencies close to the resonance frequencies, and for peak displacements of the suspended mass of approximately 0.8 mm, the system behaves broadly as a linear single-degree-of-freedom system. Thus, it was thought suitable for validating the theoretical studies presented previously [18,20].
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4993
Fig. 7. Magnitude of the transmissibility of the system. The solid line gives the transmissibility with system stiffness due to the nylon suspension alone (low stiffness state with electromagnetic support switched off) and the dotted line gives the transmissibility with the system stiffness provided by a combination of the elastic support and the electromagnetic support with a supply of 12 V (high stiffness condition).
Fig. 8. Coherence of the switchable stiffness system in the two stiffness states: (a) low stiffness and (b) high stiffness.
The equivalent stiffness reduction and damping ratio can be obtained from the transmissibility plots. The stiffness reduction factor s from the theoretical model is given by [20]
s ¼ 1
ooff oon
2
(3)
4994
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
where ooff and oon are the natural frequencies for the off and on states respectively. The effect of the magnetic force in changing the stiffness of the system is clearly visible in the transmissibility plots. The resonance peak shifted from 17.75 Hz to 12.75 Hz when the electromagnets were switched off. The percentage stiffness change was thus 48 percent. The approximate equivalent damping ratio z was calculated using the half-power bandwidth method using z ¼(o2 o1)/2on [3], where o2 o1 is the bandwidth and on the natural frequency of the system. As expected, the system had very low damping, corresponding to 0.045 and 0.034 when the electromagnets were turned on or off respectively. The higher value of damping when the electromagnets were turned on was most probably due to eddy currents caused by the conductive element moving through the magnetic field [19]. 5. Control of the shock response using switchable stiffness 5.1. Control of the response during the application of the shock pulse As mentioned previously, the switchable stiffness provides a soft support for the isolated mass during the time when the shock input is applied. The initial higher value of stiffness is subsequently recovered immediately after the shock pulse. The shock pulse was generated using an electrodynamic shaker. However, the generation of a symmetrical pulse was found to be difficult, because a shaker is a dynamic system that has a residual response after the shock pulse has been generated. An attempt was made to reduce the residual vibration by applying a method proposed by Shin and Brennan [21]. Two half sine pulses with different amplitudes were applied at different times. The response of the second pulse had the opposite phase of the response for the first pulse, canceling the residual vibrations, in principle at least. A pulse of 0.025 s duration was generated, which gave a period ratio of (t/T)¼0.443, considering the natural frequency of the supported magnet in its high stiffness state (17.75 Hz). Although it would have been desirable to generate pulses of a shorter duration than this, it was not possible because of the shaker dynamics. However, additional mass could be added to the shaker to generate pulses of longer duration. The responses of the system to the shortest duration shock pulse, for the high stiffness and a low stiffness settings (no switching), plus the shock pulse used for later tests are shown in Fig. 9. The input acceleration normalized by g is shown in Fig. 9(a), and the normalized accelerations of the suspended mass are shown for the high stiffness and the low stiffness states in Fig. 9(b) and (c) respectively. It can be seen in Fig. 9(a), that the method of generating a pulse without residual vibrations was not completely successful. An analog circuit was manufactured to generate signals to switch the stiffness in real time. This circuit uses the signal of the shock pulse acquired with an accelerometer and determines the first two minima points of the signal. As shown in Fig. 9(a), when the residual response is ignored, the effective shock pulse can be located between the first two minima.
Fig. 9. Passive shock response of the experimental rig: (a) shock pulse acceleration input x€ to the base, (b) high stiffness system response and (c) low stiffness system acceleration response n€ of the isolated mass. All acceleration time histories are normalized by g.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4995
What the circuit does is to detect a rise in the signal after the first minima and turns off the voltage supply, thus reducing the effective stiffness. After this event, the circuit detects when the signal rises again at the second minima and recovers the power supply, increasing the stiffness to its original value. A brief description of the circuit is presented in Appendix A. A block diagram showing the experimental configuration, which was used to test the switchable stiffness concept during the shock, is shown in Fig. 6. An example of the acceleration response time history of the system and the switching circuit is presented in Fig. 10. The shortest pulse of 0.025 s duration was used for this test and the normalized time history of the pulse is shown in Fig. 10(a). The normalized acceleration response of the passive system (high stiffness state only) is given in Fig. 10(b). Fig. 10(c) gives the normalized acceleration response when applying the switching strategy and Fig. 10(d) shows the voltage applied to the electromagnets. Examining the voltage time history, it can be seen that the system was initially in its high stiffness state as the voltage was set to 12 V, which then reduced to zero at the start of the shock as the system switched to its low stiffness state. At the end of the shock the voltage returned to its initial value such that the system was again in its high stiffness state. It should be noted that the DC voltage decreases exponentially after it is applied, probably due to the internal resistance of the voltage source. After the voltage is turned off a negative overshoot occurs. This phenomena is called ‘‘inductive kick’’ and occurs when the current is removed instantaneously thus the voltage tends to infinity [22]. Comparing the maximum peak amplitude of the passive system in its high stiffness state and the semi-active system, it is found a reduction in the peak response of about 53 percent. Following the stiffness recovery to its high state, however, the maximum response is slightly greater than the passive response. This part of the response is the residual vibration, which is examined in the next sub-section. There are some other interesting points to be observed from the experimental results. Firstly, it can be seen that the response amplitude of the suspended magnet rises in the first few cycles following the shock input, and then after reaching a maximum decays away. This behavior can also be seen in the response of the passive system. The reason for this can be attributed to the beating phenomena, as the natural frequency of the shaker (20 Hz) was very close to the natural frequency of the suspended magnet (17.75 Hz). Another point worth noting is the small jump in the response of the suspended magnet when the stiffness is recovered. This small jump occurs after the point when the maximum response is reached. It is due to the sudden voltage applied that transmits a small pulse and causes the suspended magnet to experience this small, but sudden motion. Finally, the voltage time history shows an overshoot when the voltage is removed and when it is applied again in the
Fig. 10. Stiffness switching strategy for adaptive shock response control: (a) shock acceleration input x€ , (b) passive acceleration response n€ , (c) switching acceleration response n€ and (d) the voltage applied to the electromagnets. All acceleration time histories are normalized by g.
4996
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
form of an ‘‘inductive kick’’ as discussed previously. This is because of the inductive nature of the electromagnetic circuit, which tends to oppose the change in current, causing transients when the current is forced to change instantaneously [22]. The relative and absolute displacement responses were also studied. The acceleration response is significant in the assessment of the transmitted forces and the estimation of the possible damage resulting from a shock. However, the relative displacement is another important parameter related to the space constraints of the isolation system. In the theoretical analysis presented in [18,20] it was observed that the use of the stiffness switching for shock control normally leads to negligible difference in the relative response unless the shock duration is short compared to the excitation pulse. Moreover, it is important to remember that the relative response is particularly important during the application of the shock because after the pulse the relative response is equal to the absolute residual response. For the experimental setup considered here, the pulse is short enough not to increase the relative response and some benefits could still be obtained. The absolute and relative displacements were obtained by numerical integration of the acceleration time series in MATLAB. Fig. 11 presents the relative displacement calculated from the experimental acceleration results. The absolute displacement response n for the switching system is presented in Fig. 11(a). For comparison, the absolute response of displacement n for the passive system is given in Fig. 11(b). In this case, taking into account the maximum response during the shock, a decrease of 40 percent in the absolute displacement response was observed by using the switchable stiffness. The absolute displacement responses presented in Fig. 11(a) and (b) are used to calculate the relative displacement as nrel ¼ n x and the results are presented in Fig. 11(c) for the switchable stiffness system, and in Fig. 11(d) for the passive system. Comparing the switching response with the passive response, very similar behavior can be seen. As expected, the relative response does not change very much, but the isolation performance is not adversely affected. For this case, a decrease of 14 percent in the relative response was found during the impulse input.
Fig. 11. Displacement response for the model with stiffness control during shock: (a) absolute displacement for switching system, (b) absolute displacement for passive system, (c) relative displacement for switching system and (d) relative displacement for the passive system. The vertical axis is given in meters and the horizontal axis in seconds.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4997
5.2. Control of the residual vibrations By using the acceleration and velocity of the isolated mass, rather than velocity and displacement, only a single integration needed to be performed, and the stiffness switching strategy according to Eq. (2), becomes ( ) n_ Z 0 k keffective ¼ (4) kDk nn_ o0 To implement control of the residual vibrations, a second control circuit was built. The description and behavior of this analog circuit is presented in Appendix A. The maximum permissible voltage in this circuit was 12 V, so it was not possible to achieve the maximum stiffness change obtained in the previous experiments. It was, however, possible to demonstrate the control law using 12 V, which gave a stiffness change of 48.4 percent. The system was subjected to a very short pulse compared to the natural period of the system (T¼0.01 s). The responses for the switching strategy are shown in Fig. 12. These plots include the acceleration response, as well as the voltage applied to the electromagnets. Fig. 12(a) depicts the free vibration acceleration response of the passive system in the high stiffness state, which is taken as a basis for comparison. The switching response is presented in Fig. 12(b) and the voltage applied to the electromagnets is given in Fig. 12(c). It can be seen from the voltage plot that the stiffness switched from a high state to a low state twice during each cycle of vibration. The overshoots or ‘‘inductive kick’’ can also be observed as before. The voltage supply turned off when the response was a maximum and turned on when the response passed through the static equilibrium position. It can also be seen that there were only four cycles of stiffness change. This was because the circuit was set to switch only when the input signals had a minimum voltage level of 0.7 V. However, even though the stiffness switched during the first two cycles of vibration, the vibration decayed at a much faster rate than the passive case, as can be seen by comparing Fig. 12(a) and (b). The equivalent damping ratio of the system was determined considering an approximation to the logarithmic decrement by measuring the
Fig. 12. Switching stiffness response for residual vibration suppression: (a) passive acceleration response for high stiffness state, (b) switching response and (c) the voltage supplied to the electromagnets. All acceleration time histories are normalized by g.
4998
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
Fig. 13. Comparison between the experimental response of the switching strategy for residual vibrations (bold line) and a simulation considering the parameters of the actual model (thin line). Acceleration is given in g, and time in seconds.
amplitudes of consecutive peaks in the time history, and was found to be 0.134 compared to a predicted damping ratio using the theory presented in [18] of 0.15. In order to correlate between the predicted and the experimental responses for the nylon suspension a comparison is given in Fig. 13, which presents the responses for the theoretical model and the experimental setup. For the simulation results, shown as a thin line, the physical parameters of the rig have been considered, and also the same initial conditions. When compared to the experimental response, which is shown as a bold line, there is a good agreement between theory and experiment.
5.3. Implementation of both control strategies The control circuits used to control the shock and residual vibrations were subsequently combined so that the stiffness could be adjusted to minimize the response during the shock input and to attenuate the resulting residual vibrations. The result of implementing the combined control strategy is shown in Fig. 14. Fig. 14(a) shows the normalized shock acceleration input to the base, Fig. 14(b) shows the acceleration response of the mass when there is no control action, i.e., the passive high stiffness case. Fig. 14(c) shows the acceleration response of the mass when the control system is implemented and Fig. 14(d) shows the voltage supplied to the electromagnets. The voltage time history shows that after the shock the stiffness remained in its low state, because the shock ended when the mass had maximum acceleration. At this point the residual switching logic meant there should have been low stiffness. Following this, the stiffness continued to switch between a low and a high state until the amplitude of the control signal was too small. Comparing Fig. 14(b) and (c), it is clear the two-stage stiffness switching control strategy has been effective in reducing the peak response and attenuating the residual vibrations. The results are comparable to when the control strategies were implemented separately as discussed in Sections 5.1 and 5.2.
6. Conclusions This paper has described the design of an experimental test-rig to demonstrate the effect of switching the suspension stiffness of a single degree-of-freedom system to attenuate the vibration due a to a shock input to the base. The suspension system used electromagnetics in parallel with a passive stiffness to realize a system with high and low stiffness states, in which the stiffness reduction was about 50 percent. An analog electrical circuit was used to implement a two-stage control strategy, the first stage reducing the response during the shock and the second stage attenuating the residual vibrations. In the experimental system the peak acceleration response of the mass was reduced by about 53 percent and the effective damping ratio in the system following the shock was changed from about 5 percent to 13 percent.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4999
Fig. 14. Experimental response of the switchable stiffness model for both stiffness control during the shock, and control for residual vibration: (a) shock acceleration input to the base, (b) passive high stiffness system acceleration response, (c) switching system response and (d) voltage supplied to the electromagnetic suspension. All acceleration time histories are normalized by g.
Acknowledgments The corresponding author would like to acknowledge the support provided by the Mexican Council for Science and Technology (CONACyT) and the Universidad Autono´ma de Nuevo Leon, as well as the Technical and administrative staff at the Institute of Sound and Vibration Research.
Appendix A In this work an analog circuit was used to process the acceleration signals measured on the payload and the base of the rig to perform the described switching strategies. A block diagram of the circuit is shown in Fig. A1(a) and the actual electrical circuit is shown in Fig. A1(b). The circuit analyzes the signal of the shock pulse acquired with an accelerometer depicted in Fig. A1(b) by Accelerometer SIGNAL IN and uses the first two minima points of the signal to determine the interval of the shock pulse. A rise in the signal after the first minima is detected and the voltage supplied to the electromagnets (Electromagnets SIGNAL OUT) is then turned off, thus reducing the stiffness. The circuit then detects when the signal rises again following the second minima and turns on the power supply, restoring the stiffness to its original value. This circuit could be used as a standalone unit or in conjunction with the residual control strategy circuit for the second stage of control. Additionally, the output voltage applied to the electromagnets was recorded. For the control of residual vibrations, a second circuit was used. The acceleration signal for the suspended mass is amplified and then split into two in order to have acceleration and velocity signals. Then both signals are multiplied and
5000
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
Fig. A1. Electrical circuit used for the switching experiments for control during the shock: (a) block diagram and (b) circuit diagram.
Fig. A2. Electrical circuit used for the residual control switching experiments: (a) block diagram and (b) circuit diagram.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
5001
the product compared with respect to zero. If the product is greater than zero, the voltage is set to zero. Otherwise, a voltage is applied to the electromagnets. A block diagram is shown in Fig. A2(a) whilst a circuit diagram is shown in Fig. A2(b). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
C.M. Harris, C.E. Crede, Shock and Vibration Handbook, McGraw-Hill, New York, 1996. R.S. Ayre, Engineering Vibrations, McGraw-Hill, New York, 1958. J.C. Snowdon, Vibration and Shock in Damped Mechanical Systems, Wiley and sons, New York, 1968. J.C. Snowdon, Response of nonlinear shock mountings to transient foundation displacements, Journal of the Acoustical Society of America 33 (10) (1961) 1295–1304. J.C. Snowdon, Transient response of nonlinear isolation mountings to pulse-like displacements, Journal of the Acoustical Society of America 35 (3) (1963) 389–396. R.L. Eshleman, P. Rao, Response of mechanical shock isolation elements to high rate input loading, Shock and Vibration Bulletin 40 (5) (1969) 217–234. S.N. Chandra, H. Hatwal, A.K. Mallik, Performance of non-linear isolators and absorbers to shock excitations, Journal of Sound and Vibration 227 (2) (1999) 293–307. M.S. Hundal, Impact absorber with linear spring and quadratic law damper, Journal of Sound and Vibration 48 (2) (1976) 189–193. N.C. Shekhar, H. Hatwal, A.K. Mallik, Response of non-linear dissipative shock isolators, Journal of Sound and Vibration 214 (4) (1998) 589–603. D.F. Ledezma, N.S. Ferguson, M.J. Brennan, Shock performance of different semiactive damping strategies, Journal of Applied Research and Technology 8 (2) (2010) 249–259. T.P. Waters, Y Hyun, M.J. Brennan, The effect of dual-rate suspension damping on vehicle response to transient road inputs, Transactions of the ASME, Journal of Vibration and Acoustics 131 (1) (2009) 011004. (8 pp.). M.F. Winthrop, W.P. Baker, R.G. Cobb, A variable stiffness device selection and design tool for lightly damped structures, Journal of Sound and Vibration 287 (4–5) (2005) 667–682. J. Onoda, T. Endo, H. Tamaoki, N. Watanabe, Vibration suppression by variable-stiffness members, AIAA Journal 29 (6) (1991) 977–983. J.C. Chen, Response of large spacecraft structures with stiffness control, Journal of Spacecrafts and Rockets 21 (5) (1984) 163–167. F. Jabbari, J.E. Bobrow, Vibration suppression with a resettable device, Journal of Engineering Mechanics 128 (9) (2002) 916–924. J.L. Leavitt, F. Jabbari, J.E. Bobrow, Optimal performance of variable stiffness devices for structural control, ASME Journal of Dynamic Systems, Measurement and Control 129 (2) (2007) 171–177. N. Zhou, K. Liu, A tunable high-static-low-dynamic stiffness vibration isolator, Journal of Sound and Vibration 329 (9) (2010) 1254–1273. D.F. Ledezma-Ramirez, N.S. Ferguson, M.J. Brennan, Shock isolation using an isolator with switchable stiffness, Journal of Sound and Vibration 330 (5) (2011) 868–882. F.A. Kip, Fundamentals of Electricity and Magnetism, McGraw-Hill, Kogakusha, Tokyo, 1969. D.F. Ledezma-Ramirez, Shock Isolation Using Switchable Stiffness, PhD Thesis, University of Southampton, 2008. K. Shin, M.J. Brennan., Two simple methods to suppress the residual vibrations of a translating or rotating flexible cantilever beam, Journal of Sound and Vibration 312 (1–2) (2008) 140–150. G. Rizzoni, Principles and Applications of Electrical Engineering, McGraw-Hill, Kogakusha, New York, 2004.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
An experimental switchable stiffness device for shock isolation D.F. Ledezma-Ramirez a,n, N.S. Ferguson b, M.J. Brennan c a
´nica y Ele´ctrica, Av. Universidad s/n, San Nicola ´s de los Garza, Nuevo Leo ´noma de Nuevo Leo ´n, Facultad de Ingenierı´a Meca ´n 66451, Me´xico Universidad Auto Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK c Departamento de Engenharia Mecˆ anica, UNESP, Ilha Solteira, SP 15385-000, Brazil b
a r t i c l e i n f o
abstract
Article history: Received 12 August 2011 Received in revised form 15 June 2012 Accepted 19 June 2012 Handling Editor: D.J. Wagg Available online 21 July 2012
The development of an experimental switching stiffness device for shock isolation is presented. The system uses magnetic forces to exert a restoring force, which results in an effective stiffness that is used to isolate a payload. When the magnetic force is turned on and off, a switchable stiffness is obtained. Characterization of the physical properties of the device is presented. They are estimated in terms of the percentage stiffness change and effective damping ratio when switched between two constant stiffness states. Additionally, the setup is used to implement a control strategy to reduce the shock response and minimize residual vibration. The system was found to be very effective for shock isolation. The response is reduced by around 50 percent compared with passive isolation showing good correlation with theoretical predictions, and the effective damping ratio in the system following the shock was increased from about 4.5 percent to 13 percent. & 2012 Elsevier Ltd. All rights reserved.
1. Introduction Shock is a common excitation that can be highly detrimental to sensitive objects and can also cause human discomfort. It is defined as a very short or transient excitation characterized by high displacements and accelerations that can potentially lead to large mechanical stresses and transmitted forces [1]. The process of shock isolation is based on energy storage by an elastic element normally requiring large deflections and the subsequent dissipation of the stored energy by some damping mechanism. This creates a compromise between the physical constraints or space available for the isolator and the effectiveness of it. Commonly, a shock isolator is a passive mount comprising some form of mechanical spring and/or viscoelastic elements. The analysis of shock isolation systems is often performed by considering single degree-of-freedom models with low damping under the effect of pulse input functions. Most of the fundamental theory of shock isolation largely in use today was developed in the 1950s and 1960s, for example, Ayre [2], Snowdon [3–5] and Eshleman and Rao [6]. The concept of the Shock Response Spectra (SRS) was developed and is one of the most important tools to evaluate the severity of impacts and selection of appropriate shock isolators. Snowdon was also one of the first researchers to incorporate nonlinear elements into shock isolators, and Eshlemann studied a number of devices such as ring springs, stranded wire springs, etc. More recently, researchers have also considered the further use of nonlinear elements for shock isolation [7–9]. Lately, there has been a growing interest in the use of semi-active isolation systems where the physical properties of the system can change in real time to provide better vibration isolation. Although these systems have been studied extensively for
n
Corresponding author. Tel.: þ52 81 83294020x5762; fax: þ 52 81 83320904. E-mail address: [email protected] (D.F. Ledezma-Ramirez).
0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.06.010
4988
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
harmonic and random vibration, there has been little work on shock isolation. Some switchable or semi-active damping strategies based on the skyhook damper concept have been studied [10], and Waters et al. [11] showed that reducing the damping to a lower value during a shock input can lead to better isolation performance. Switchable and variable stiffness strategies have also been investigated theoretically and experimentally. An important paper is one by Winthrop reviewing different methods to achieve variable stiffness [12]. A strategy to control transient vibrations was proposed by Onoda et al. [13] using an on/off control logic to provide energy dissipation in lightly damped systems, based on the idea of controlling structural vibration using variable stiffness as proposed by Chen [14]. Jabbari and Bobrow [15] and Leavitt et al. [16] devised a resetting technique, based on a switchable stiffness aiming to extract energy from a mechanical system while having a high stiffness value at all times. This concept has also been exploited in the field of tunable vibration absorbers. For instance, Zhou and Liu [17] have used an electromagnetic spring combined with a mechanical spring to develop a tunable isolator possessing nonlinear behavior. However, there are no results or investigations related to shock response in these studies. The concept of switchable stiffness has been investigated theoretically, by the authors of this current paper, as a means of energy dissipation in lightly damped systems, where it is difficult to implement another form of external damping (see for example [18] and the references therein). This is effectively a semi-active control strategy. The model presented in [18] comprises a mass supported by two springs, one of which can be disconnected. Switching in and out of the spring involves a two-stage control strategy; stiffness control during the shock to reduce the maximum response of the payload, and reduction of the residual vibration after the shock has occurred. The theoretical simulations presented demonstrate that it is possible to obtain better shock isolation by switching the stiffness in lightly damped systems. This paper is a continuation of the work in [18], describing the practical realization of a semi-active shock isolation system. The main motivation in developing such strategies is to improve methods of shock isolation. Potential engineering applications include the protection of sensitive electronic devices in harsh environments, for example in ships and military vehicles, low frequency and semi-active vibration isolators, aerospace structures and earthquake engineering. In this work, experimental results are presented, characterizing the system properties and demonstrating the switchable stiffness strategies and the improved shock response of such an isolation system. Prior to the description of the experimental work in later sections, some background information on the principles of shock isolation is given in Section 2, with special emphasis on semi-active shock isolation. 2. Brief overview of shock isolation theory 2.1. Shock isolation In the theory of shock isolation, the maximax response is defined as the maximum response to a particular shock pulse at any time. The value of this response depends mainly upon the duration and shape of the shock. Several ideal pulses can be considered such as half sine, rectangular, trapezoidal, etc., and more complex pulses can be used as well. Normally the duration of the pulse t is compared to the natural period of the system T in order to evaluate the maximax response. The theory has been extensively covered by Ayre [2]. To summarize a typical shock scenario, there are three zones in the shock response. For short pulses compared to the natural period, i.e. approximately when (t/T) o0.25, the shock is said to be impulsive, and the maximum response is smaller than the amplitude of excitation, i.e. the system is isolated from the shock input. For longer inputs (t/T)E1, the maximum response is larger than the amplitude of the input. For much longer duration pulses compared to the natural period, i.e. ðt=TÞ b0:25 the shock is applied relatively slowly and the input becomes quasi-static. From the practical point of view it is desirable to have a low natural frequency isolator in order to achieve isolation, thus requiring a flexible support resulting in large relative displacement and static deformation. 2.2. Shock isolation using switchable stiffness To overcome the compromise between minimizing the transmission of the shock through the isolator and minimizing the static deflection, a semi-active control method can be adopted, whereby the stiffness is switched to a lower value whilst the shock pulse is applied. This system was discussed in [18] and is outlined briefly here for the convenience of the reader. The model considered comprises a mass supported by two springs one of which can be disconnected. An undamped system is considered for simplicity. The switching of the spring is controlled by a control strategy, which is divided into two stages, namely the stiffness control during a shock, and the later stiffness switching during the residual vibration. The first part of the control strategy is to disconnect the stiffness Dk for the duration of the shock providing a softer support so that the maximum response is reduced. The equation of motion for the shock input is given by mn€ þkeffective n ¼ keffective xðtÞ
(1)
where keffective is the system stiffness, which can be either k or k Dk depending upon whether the secondary stiffness is connected or disconnected respectively as shown in Fig. 1. The term x(t) represents the shock excitation that is normally modeled as a pulse function such a half sine or a versed sine [2]. The displacement response of the system is given by n. The effect of reducing the stiffness is to change the effective period ratio t/T, of the system, and to shift the system from the amplification to the isolation region only whilst the shock is applied, maintaining a high stiffness otherwise. The strategy has
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4989
Fig. 1. Schematic of a sdof system with switchable stiffness under shock excitation x(t) applied to the base. The two stiffness in parallel have stiffness Dk and k Dk respectively and m is the mass with absolute displacement n.
Fig. 2. Effect of the switchable control strategy on the residual vibration: (a) phase plane plot, (b) time history of the displacement response (— high stiffness; - - - low stiffness).
been demonstrated to be capable of reducing the shock response in terms of the absolute displacement and acceleration responses, but the benefits on the relative response are small compared to the passive isolation model [18]. The second stage of the switching strategy involves the semi-active switching of the stiffness during the residual period, i.e. when the shock pulse ends and the system undergoes free vibration. The objective is to quickly dissipate the energy stored by the elastic element during the shock without adding an external damping mechanism. The same model is considered as before but now the control strategy is given by ( ) nn_ Z0 k keffective ¼ (2) kDk nn_ o0 where n_ is the velocity of the mass. The residual control strategy has to ensure that the amplitude of vibration decreases every cycle. The stiffness should be maximum and equal to k when the product nn_ is positive and minimum and equal to k Dk when nn_ is negative. When the displacement response satisfies the condition nn_ Z 0 the displacement n and the velocity n_ have the same sign. As a result the secondary spring, Dk, is disconnected when the absolute value of the displacement of the mass is a maximum. It is connected again when the absolute value of the velocity is maximum, when the system passes through its equilibrium position. The phase plane plot presented in Fig. 2(a) shows how the switching occurs effectively dissipating the energy at every stiffness reduction point. Fig. 2(b) depicts a time history corresponding to this example. A comprehensive study of this strategy is also presented in [18], concluding that a greater stiffness reduction leads to greater rate of reduction of the residual vibrations.
4990
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
Fig. 3. Passive and switched normalized displacement response. The switched response involves the implementation of both stages of the control strategy—during and after the shock pulse. The period ratio for the passive sytem is (t/T) ¼0.25 and a stiffness reduction of 50 percent was considered. The response is normalized by the maximum of the shock amplitude. The time axis is normalized with respect to the natural period of the system T (— switching system; and - - - passive system).
As an example of the control stages described above, a theoretical simulation is presented in Fig. 3, implementing both the stiffness reduction during the shock and the subsequent residual vibration control strategy. It is compared with that of a passive system, in which the stiffness has a fixed value of k, i.e. the high stiffness state. The response has been normalized by the maximum amplitude of the versed sine shock pulse. In this example there is a stiffness reduction of 50 percent and the period ratio for the system without control is ðt=TÞ ¼ 0:25. The benefits of the strategies acting together are clearly visible, achieving a reduction in the shock response for the absolute response, as well as a greater rate of reduction in the residual vibration.
3. Design of the experimental system The main objective of the work presented here was to design and implement a switchable stiffness device based on the theoretical studies in [18], briefly described above. The system was designed with the following considerations in mind: (a) to gain appreciable benefits in the shock response and the residual vibration suppression, the system had to be capable of achieving a high stiffness change—at least a factor of two, and (b) the stiffness change had to be rapid. Moreover, the system had to be lightly damped so that a significant change in the residual decay rate due to the switching could be clearly observed. It was also desirable that the system should behave as a single degree-of-freedom system, for which predictions had been previously obtained. To realize a system with these characteristics, an electromagnetic element was used to achieve a switchable stiffness. The fundamental idea is similar to the concept of magnetic levitation. If two electromagnets are aligned with their opposite poles facing, the resulting repulsive force can be switched on and off, to obtain a switchable stiffness element depending upon the voltage supplied to the electromagnets. This system is nonlinear because the repulsive force is not proportional to the separation of the magnets [19]. Also, the system needs to be physically stabilized by applying some stiffness constraints. The experimental system is shown in Fig. 4; a schematic of the system is given in Fig. 4(a) and a photograph of the system is shown in Fig. 4(b). The device included two permanent, disk shaped neodymium magnets which were suspended between two electromagnets. The permanent magnets were fitted inside an aluminum ring, which was suspended using four tensioned nylon wires attached to the main frame. The nylon wires acted as an additional permanent stiffness in parallel with the stiffness provided by the magnets. The total mass of the permanent magnets and the aluminum ring was 0.0753 kg, which acted as the isolated mass (payload) in the experimental system. It should be noted that although the system in Fig. 4 is shown in the vertical position, the experiments were made with the system in the horizontal position to avoid the effect of gravity. This is because the suspended weight could be closer to one of the electromagnets at some point due to the static deflection, resulting in an asymmetric system.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4991
Fig. 4. (a) Diagram of the switchable stiffness experimental system in the vertical position. The permanent magnets (1, 2) are suspended between two electromagnets (3, 4) using four wires (6, 7) that also join the magnet to the main frame (5). The permanent magnets are held by an aluminum disk (8). (b) Photograph of the rig.
When the electromagnets were turned off, the stiffness of the system was low and when they were supplied with a constant DC voltage of 12 V the stiffness was high. A Hameg triple voltage source HM7042-5 was used. There was a hardening effect when the electromagnets were on because of the nature of the repulsive force between the electromagnets and the permanent magnets. On the other hand, a softening effect exists when the electromagnets were off because of the attraction between the iron core of the electromagnets and the permanent magnets. This effect became more evident as the permanent magnets approached the electromagnets, due to the nonlinear nature of the magnetic forces. The electromagnets used were capable of working in the 0–24 V voltage range. However, to avoid excessive heating a voltage of only 12 V was applied continuously. 4. Measurement of the physical properties of the experimental system Some physical properties of the system were estimated by measuring the transmissibility of the experimental system using broadband random excitation from 0 to 1000 Hz applied to the base. The transmissibility was calculated using the accelerations of the base of the rig and the suspended mass. The resonance peak for different applied voltages was determined and the effective stiffness change was found from the measured natural frequency and the system mass. To check the linear behavior of the system a simple static test was undertaken. Transmissibility tests (described later) were also undertaken for different input amplitudes. The static deflection was measured using a dial gage and adding different values of mass to the suspended magnet for different settings of the electromagnets, i.e. off and on. As shown in Fig. 5 the force–displacement plots shows a good degree of linearity for the displacement of the mass in the range of 0–2 mm. The system response for the subsequent dynamic tests was limited to approximately 1 mm. A block diagram showing the setup of the equipment is shown in Fig. 6. Note that for the transmissibility tests the switching circuit was disconnected and the DC voltage source connected directly to the electromagnets. The shaker was
4992
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
Fig. 5. Measured force–deflection characteristics of the switchable stiffness system. ( þ magnets switched on, magnets switched off). The lines represent linear least squares fit to the data.
Fig. 6. Schematic diagram of the experimental setup. Accelerometers were attached to the suspended mass and to the base of the rig. A PC generated the excitation signal which was fed to the shaker via a power amplifier. For the transmissibility measurements the switching circuit was disconnected and the DC voltage supply was connected directly to the electromagnets.
supplied with a random noise signal generated by a Data Physics Mobilizer analyzer through a power amplifier. Two PCB teardrop miniature accelerometers type 352C22 were used to measure the acceleration of the base of the shaker and the suspended magnets respectively. The peak acceleration levels measured on the base were between 0.08g and 1g for the minimum and maximum amplifier gains used in the tests (approximately 0.777g rms). The transmissibility spanned 0–100 Hz with 1600 lines of resolution. Fig. 7 shows the measured transmissibility of the system, where the solid and dashed lines represent the off state and the on state with a supply voltage of 12 V respectively. This voltage was chosen in order to avoid overheating when higher voltages were applied. The linearity of the system was evaluated using the coherence function. The corresponding coherence plots are presented in Fig. 8. In general, the coherence was considered to be acceptable. However, there are certain frequencies at which the coherence falls, which are probably partly due to the small nonlinear effects of the magnetic forces. Overall, for frequencies close to the resonance frequencies, and for peak displacements of the suspended mass of approximately 0.8 mm, the system behaves broadly as a linear single-degree-of-freedom system. Thus, it was thought suitable for validating the theoretical studies presented previously [18,20].
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4993
Fig. 7. Magnitude of the transmissibility of the system. The solid line gives the transmissibility with system stiffness due to the nylon suspension alone (low stiffness state with electromagnetic support switched off) and the dotted line gives the transmissibility with the system stiffness provided by a combination of the elastic support and the electromagnetic support with a supply of 12 V (high stiffness condition).
Fig. 8. Coherence of the switchable stiffness system in the two stiffness states: (a) low stiffness and (b) high stiffness.
The equivalent stiffness reduction and damping ratio can be obtained from the transmissibility plots. The stiffness reduction factor s from the theoretical model is given by [20]
s ¼ 1
ooff oon
2
(3)
4994
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
where ooff and oon are the natural frequencies for the off and on states respectively. The effect of the magnetic force in changing the stiffness of the system is clearly visible in the transmissibility plots. The resonance peak shifted from 17.75 Hz to 12.75 Hz when the electromagnets were switched off. The percentage stiffness change was thus 48 percent. The approximate equivalent damping ratio z was calculated using the half-power bandwidth method using z ¼(o2 o1)/2on [3], where o2 o1 is the bandwidth and on the natural frequency of the system. As expected, the system had very low damping, corresponding to 0.045 and 0.034 when the electromagnets were turned on or off respectively. The higher value of damping when the electromagnets were turned on was most probably due to eddy currents caused by the conductive element moving through the magnetic field [19]. 5. Control of the shock response using switchable stiffness 5.1. Control of the response during the application of the shock pulse As mentioned previously, the switchable stiffness provides a soft support for the isolated mass during the time when the shock input is applied. The initial higher value of stiffness is subsequently recovered immediately after the shock pulse. The shock pulse was generated using an electrodynamic shaker. However, the generation of a symmetrical pulse was found to be difficult, because a shaker is a dynamic system that has a residual response after the shock pulse has been generated. An attempt was made to reduce the residual vibration by applying a method proposed by Shin and Brennan [21]. Two half sine pulses with different amplitudes were applied at different times. The response of the second pulse had the opposite phase of the response for the first pulse, canceling the residual vibrations, in principle at least. A pulse of 0.025 s duration was generated, which gave a period ratio of (t/T)¼0.443, considering the natural frequency of the supported magnet in its high stiffness state (17.75 Hz). Although it would have been desirable to generate pulses of a shorter duration than this, it was not possible because of the shaker dynamics. However, additional mass could be added to the shaker to generate pulses of longer duration. The responses of the system to the shortest duration shock pulse, for the high stiffness and a low stiffness settings (no switching), plus the shock pulse used for later tests are shown in Fig. 9. The input acceleration normalized by g is shown in Fig. 9(a), and the normalized accelerations of the suspended mass are shown for the high stiffness and the low stiffness states in Fig. 9(b) and (c) respectively. It can be seen in Fig. 9(a), that the method of generating a pulse without residual vibrations was not completely successful. An analog circuit was manufactured to generate signals to switch the stiffness in real time. This circuit uses the signal of the shock pulse acquired with an accelerometer and determines the first two minima points of the signal. As shown in Fig. 9(a), when the residual response is ignored, the effective shock pulse can be located between the first two minima.
Fig. 9. Passive shock response of the experimental rig: (a) shock pulse acceleration input x€ to the base, (b) high stiffness system response and (c) low stiffness system acceleration response n€ of the isolated mass. All acceleration time histories are normalized by g.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4995
What the circuit does is to detect a rise in the signal after the first minima and turns off the voltage supply, thus reducing the effective stiffness. After this event, the circuit detects when the signal rises again at the second minima and recovers the power supply, increasing the stiffness to its original value. A brief description of the circuit is presented in Appendix A. A block diagram showing the experimental configuration, which was used to test the switchable stiffness concept during the shock, is shown in Fig. 6. An example of the acceleration response time history of the system and the switching circuit is presented in Fig. 10. The shortest pulse of 0.025 s duration was used for this test and the normalized time history of the pulse is shown in Fig. 10(a). The normalized acceleration response of the passive system (high stiffness state only) is given in Fig. 10(b). Fig. 10(c) gives the normalized acceleration response when applying the switching strategy and Fig. 10(d) shows the voltage applied to the electromagnets. Examining the voltage time history, it can be seen that the system was initially in its high stiffness state as the voltage was set to 12 V, which then reduced to zero at the start of the shock as the system switched to its low stiffness state. At the end of the shock the voltage returned to its initial value such that the system was again in its high stiffness state. It should be noted that the DC voltage decreases exponentially after it is applied, probably due to the internal resistance of the voltage source. After the voltage is turned off a negative overshoot occurs. This phenomena is called ‘‘inductive kick’’ and occurs when the current is removed instantaneously thus the voltage tends to infinity [22]. Comparing the maximum peak amplitude of the passive system in its high stiffness state and the semi-active system, it is found a reduction in the peak response of about 53 percent. Following the stiffness recovery to its high state, however, the maximum response is slightly greater than the passive response. This part of the response is the residual vibration, which is examined in the next sub-section. There are some other interesting points to be observed from the experimental results. Firstly, it can be seen that the response amplitude of the suspended magnet rises in the first few cycles following the shock input, and then after reaching a maximum decays away. This behavior can also be seen in the response of the passive system. The reason for this can be attributed to the beating phenomena, as the natural frequency of the shaker (20 Hz) was very close to the natural frequency of the suspended magnet (17.75 Hz). Another point worth noting is the small jump in the response of the suspended magnet when the stiffness is recovered. This small jump occurs after the point when the maximum response is reached. It is due to the sudden voltage applied that transmits a small pulse and causes the suspended magnet to experience this small, but sudden motion. Finally, the voltage time history shows an overshoot when the voltage is removed and when it is applied again in the
Fig. 10. Stiffness switching strategy for adaptive shock response control: (a) shock acceleration input x€ , (b) passive acceleration response n€ , (c) switching acceleration response n€ and (d) the voltage applied to the electromagnets. All acceleration time histories are normalized by g.
4996
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
form of an ‘‘inductive kick’’ as discussed previously. This is because of the inductive nature of the electromagnetic circuit, which tends to oppose the change in current, causing transients when the current is forced to change instantaneously [22]. The relative and absolute displacement responses were also studied. The acceleration response is significant in the assessment of the transmitted forces and the estimation of the possible damage resulting from a shock. However, the relative displacement is another important parameter related to the space constraints of the isolation system. In the theoretical analysis presented in [18,20] it was observed that the use of the stiffness switching for shock control normally leads to negligible difference in the relative response unless the shock duration is short compared to the excitation pulse. Moreover, it is important to remember that the relative response is particularly important during the application of the shock because after the pulse the relative response is equal to the absolute residual response. For the experimental setup considered here, the pulse is short enough not to increase the relative response and some benefits could still be obtained. The absolute and relative displacements were obtained by numerical integration of the acceleration time series in MATLAB. Fig. 11 presents the relative displacement calculated from the experimental acceleration results. The absolute displacement response n for the switching system is presented in Fig. 11(a). For comparison, the absolute response of displacement n for the passive system is given in Fig. 11(b). In this case, taking into account the maximum response during the shock, a decrease of 40 percent in the absolute displacement response was observed by using the switchable stiffness. The absolute displacement responses presented in Fig. 11(a) and (b) are used to calculate the relative displacement as nrel ¼ n x and the results are presented in Fig. 11(c) for the switchable stiffness system, and in Fig. 11(d) for the passive system. Comparing the switching response with the passive response, very similar behavior can be seen. As expected, the relative response does not change very much, but the isolation performance is not adversely affected. For this case, a decrease of 14 percent in the relative response was found during the impulse input.
Fig. 11. Displacement response for the model with stiffness control during shock: (a) absolute displacement for switching system, (b) absolute displacement for passive system, (c) relative displacement for switching system and (d) relative displacement for the passive system. The vertical axis is given in meters and the horizontal axis in seconds.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4997
5.2. Control of the residual vibrations By using the acceleration and velocity of the isolated mass, rather than velocity and displacement, only a single integration needed to be performed, and the stiffness switching strategy according to Eq. (2), becomes ( ) n_ Z 0 k keffective ¼ (4) kDk nn_ o0 To implement control of the residual vibrations, a second control circuit was built. The description and behavior of this analog circuit is presented in Appendix A. The maximum permissible voltage in this circuit was 12 V, so it was not possible to achieve the maximum stiffness change obtained in the previous experiments. It was, however, possible to demonstrate the control law using 12 V, which gave a stiffness change of 48.4 percent. The system was subjected to a very short pulse compared to the natural period of the system (T¼0.01 s). The responses for the switching strategy are shown in Fig. 12. These plots include the acceleration response, as well as the voltage applied to the electromagnets. Fig. 12(a) depicts the free vibration acceleration response of the passive system in the high stiffness state, which is taken as a basis for comparison. The switching response is presented in Fig. 12(b) and the voltage applied to the electromagnets is given in Fig. 12(c). It can be seen from the voltage plot that the stiffness switched from a high state to a low state twice during each cycle of vibration. The overshoots or ‘‘inductive kick’’ can also be observed as before. The voltage supply turned off when the response was a maximum and turned on when the response passed through the static equilibrium position. It can also be seen that there were only four cycles of stiffness change. This was because the circuit was set to switch only when the input signals had a minimum voltage level of 0.7 V. However, even though the stiffness switched during the first two cycles of vibration, the vibration decayed at a much faster rate than the passive case, as can be seen by comparing Fig. 12(a) and (b). The equivalent damping ratio of the system was determined considering an approximation to the logarithmic decrement by measuring the
Fig. 12. Switching stiffness response for residual vibration suppression: (a) passive acceleration response for high stiffness state, (b) switching response and (c) the voltage supplied to the electromagnets. All acceleration time histories are normalized by g.
4998
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
Fig. 13. Comparison between the experimental response of the switching strategy for residual vibrations (bold line) and a simulation considering the parameters of the actual model (thin line). Acceleration is given in g, and time in seconds.
amplitudes of consecutive peaks in the time history, and was found to be 0.134 compared to a predicted damping ratio using the theory presented in [18] of 0.15. In order to correlate between the predicted and the experimental responses for the nylon suspension a comparison is given in Fig. 13, which presents the responses for the theoretical model and the experimental setup. For the simulation results, shown as a thin line, the physical parameters of the rig have been considered, and also the same initial conditions. When compared to the experimental response, which is shown as a bold line, there is a good agreement between theory and experiment.
5.3. Implementation of both control strategies The control circuits used to control the shock and residual vibrations were subsequently combined so that the stiffness could be adjusted to minimize the response during the shock input and to attenuate the resulting residual vibrations. The result of implementing the combined control strategy is shown in Fig. 14. Fig. 14(a) shows the normalized shock acceleration input to the base, Fig. 14(b) shows the acceleration response of the mass when there is no control action, i.e., the passive high stiffness case. Fig. 14(c) shows the acceleration response of the mass when the control system is implemented and Fig. 14(d) shows the voltage supplied to the electromagnets. The voltage time history shows that after the shock the stiffness remained in its low state, because the shock ended when the mass had maximum acceleration. At this point the residual switching logic meant there should have been low stiffness. Following this, the stiffness continued to switch between a low and a high state until the amplitude of the control signal was too small. Comparing Fig. 14(b) and (c), it is clear the two-stage stiffness switching control strategy has been effective in reducing the peak response and attenuating the residual vibrations. The results are comparable to when the control strategies were implemented separately as discussed in Sections 5.1 and 5.2.
6. Conclusions This paper has described the design of an experimental test-rig to demonstrate the effect of switching the suspension stiffness of a single degree-of-freedom system to attenuate the vibration due a to a shock input to the base. The suspension system used electromagnetics in parallel with a passive stiffness to realize a system with high and low stiffness states, in which the stiffness reduction was about 50 percent. An analog electrical circuit was used to implement a two-stage control strategy, the first stage reducing the response during the shock and the second stage attenuating the residual vibrations. In the experimental system the peak acceleration response of the mass was reduced by about 53 percent and the effective damping ratio in the system following the shock was changed from about 5 percent to 13 percent.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
4999
Fig. 14. Experimental response of the switchable stiffness model for both stiffness control during the shock, and control for residual vibration: (a) shock acceleration input to the base, (b) passive high stiffness system acceleration response, (c) switching system response and (d) voltage supplied to the electromagnetic suspension. All acceleration time histories are normalized by g.
Acknowledgments The corresponding author would like to acknowledge the support provided by the Mexican Council for Science and Technology (CONACyT) and the Universidad Autono´ma de Nuevo Leon, as well as the Technical and administrative staff at the Institute of Sound and Vibration Research.
Appendix A In this work an analog circuit was used to process the acceleration signals measured on the payload and the base of the rig to perform the described switching strategies. A block diagram of the circuit is shown in Fig. A1(a) and the actual electrical circuit is shown in Fig. A1(b). The circuit analyzes the signal of the shock pulse acquired with an accelerometer depicted in Fig. A1(b) by Accelerometer SIGNAL IN and uses the first two minima points of the signal to determine the interval of the shock pulse. A rise in the signal after the first minima is detected and the voltage supplied to the electromagnets (Electromagnets SIGNAL OUT) is then turned off, thus reducing the stiffness. The circuit then detects when the signal rises again following the second minima and turns on the power supply, restoring the stiffness to its original value. This circuit could be used as a standalone unit or in conjunction with the residual control strategy circuit for the second stage of control. Additionally, the output voltage applied to the electromagnets was recorded. For the control of residual vibrations, a second circuit was used. The acceleration signal for the suspended mass is amplified and then split into two in order to have acceleration and velocity signals. Then both signals are multiplied and
5000
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
Fig. A1. Electrical circuit used for the switching experiments for control during the shock: (a) block diagram and (b) circuit diagram.
Fig. A2. Electrical circuit used for the residual control switching experiments: (a) block diagram and (b) circuit diagram.
D.F. Ledezma-Ramirez et al. / Journal of Sound and Vibration 331 (2012) 4987–5001
5001
the product compared with respect to zero. If the product is greater than zero, the voltage is set to zero. Otherwise, a voltage is applied to the electromagnets. A block diagram is shown in Fig. A2(a) whilst a circuit diagram is shown in Fig. A2(b). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
C.M. Harris, C.E. Crede, Shock and Vibration Handbook, McGraw-Hill, New York, 1996. R.S. Ayre, Engineering Vibrations, McGraw-Hill, New York, 1958. J.C. Snowdon, Vibration and Shock in Damped Mechanical Systems, Wiley and sons, New York, 1968. J.C. Snowdon, Response of nonlinear shock mountings to transient foundation displacements, Journal of the Acoustical Society of America 33 (10) (1961) 1295–1304. J.C. Snowdon, Transient response of nonlinear isolation mountings to pulse-like displacements, Journal of the Acoustical Society of America 35 (3) (1963) 389–396. R.L. Eshleman, P. Rao, Response of mechanical shock isolation elements to high rate input loading, Shock and Vibration Bulletin 40 (5) (1969) 217–234. S.N. Chandra, H. Hatwal, A.K. Mallik, Performance of non-linear isolators and absorbers to shock excitations, Journal of Sound and Vibration 227 (2) (1999) 293–307. M.S. Hundal, Impact absorber with linear spring and quadratic law damper, Journal of Sound and Vibration 48 (2) (1976) 189–193. N.C. Shekhar, H. Hatwal, A.K. Mallik, Response of non-linear dissipative shock isolators, Journal of Sound and Vibration 214 (4) (1998) 589–603. D.F. Ledezma, N.S. Ferguson, M.J. Brennan, Shock performance of different semiactive damping strategies, Journal of Applied Research and Technology 8 (2) (2010) 249–259. T.P. Waters, Y Hyun, M.J. Brennan, The effect of dual-rate suspension damping on vehicle response to transient road inputs, Transactions of the ASME, Journal of Vibration and Acoustics 131 (1) (2009) 011004. (8 pp.). M.F. Winthrop, W.P. Baker, R.G. Cobb, A variable stiffness device selection and design tool for lightly damped structures, Journal of Sound and Vibration 287 (4–5) (2005) 667–682. J. Onoda, T. Endo, H. Tamaoki, N. Watanabe, Vibration suppression by variable-stiffness members, AIAA Journal 29 (6) (1991) 977–983. J.C. Chen, Response of large spacecraft structures with stiffness control, Journal of Spacecrafts and Rockets 21 (5) (1984) 163–167. F. Jabbari, J.E. Bobrow, Vibration suppression with a resettable device, Journal of Engineering Mechanics 128 (9) (2002) 916–924. J.L. Leavitt, F. Jabbari, J.E. Bobrow, Optimal performance of variable stiffness devices for structural control, ASME Journal of Dynamic Systems, Measurement and Control 129 (2) (2007) 171–177. N. Zhou, K. Liu, A tunable high-static-low-dynamic stiffness vibration isolator, Journal of Sound and Vibration 329 (9) (2010) 1254–1273. D.F. Ledezma-Ramirez, N.S. Ferguson, M.J. Brennan, Shock isolation using an isolator with switchable stiffness, Journal of Sound and Vibration 330 (5) (2011) 868–882. F.A. Kip, Fundamentals of Electricity and Magnetism, McGraw-Hill, Kogakusha, Tokyo, 1969. D.F. Ledezma-Ramirez, Shock Isolation Using Switchable Stiffness, PhD Thesis, University of Southampton, 2008. K. Shin, M.J. Brennan., Two simple methods to suppress the residual vibrations of a translating or rotating flexible cantilever beam, Journal of Sound and Vibration 312 (1–2) (2008) 140–150. G. Rizzoni, Principles and Applications of Electrical Engineering, McGraw-Hill, Kogakusha, New York, 2004.