- Email: [email protected]
HIGH-POWERED NON-LINEAR SHOCK DAMPER USING ER FLUID
HIGH-POWERED NON-LINEAR SHOCK DAMPER USING ER FLUID Kanjuro Makihara ∗ Junjiro Onoda ∗∗ Kenji Minesugi ∗∗ ∗
Department of Engineering, University of Cambridge Email: [email protected] ∗∗ Institute of Space and Astronautical Science, JAXA 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan
Abstract: We developed the novel control logic for a shock damper using particledispersion Electro-Rheological (ER) fluid. This is a high-powered means of shock attenuation for satellite instruments that are subjected to lift-off shock or pyrodevice ignition shock. The proposed method attenuates the shock so that the instrument’s acceleration does not exceed the critical value, even when the shock is too large to be accepted. In contrast to the conventional linear shock controls, the proposed shock control does not attempt to attenuate a small shock in order to prepare for attenuating a coming large shock. Keywords: Shock damper, Shock absorber, ER fluid
Nomenclature A c d f k1 , k2 m p pc x y y¨c z z β γ τ
: : : : : : : : : : : : : : : :
amplitude of acceleration viscous damping element elongation between points 2 and 4 Coulomb friction element spring elements mass of satellite instrument tensile load at points 1 and 4 critical value of p (≡ m¨ yc ) displacement of base displacement of satellite instrument critical value of acceleration y¨ relative displacement of y to x state vector (≡ [z, z] ˙ T) constant parameter (0 < β) constant parameter (0 ≤ γ ≤ 1) semi-sinusoidal shock duration 1. INTRODUCTION
On-board satellite instruments are often damaged by the shocks generated from rocket lift-
off, staging, or pyrodevice ignition often damage. The acceleration experienced by satellite instruments is proportional to the magnitude of the applied shock. Satellite instruments may be damaged when the acceleration generated by a shock exceeds a certain critical value. Therefore, acceleration is an important factor for safety design of satellite components and has drawn keen attention in operation. Satellite instruments can be protected from shock damage by mounting them on a shock damper. A great number of studies and attempts have been conducted on shock and vibration attenuation (see Rodden et al., 1986; Whorton et al., 1998; Krakov, 1999; Ping, 2003; Makihara et al., 2006). The Hubble Space Telescope (Rodden et al., 1986) is probably the most famous space structure that has a disturbance attenuation system essential to achieving its mission. Although these passive systems are very reliable, their attenuation performance is generally lower than those of active systems. Since a passive system cannot
adapt to environmental changes, its attenuation performance may deteriorate when design parameters change from their tuned values. Researchers have been working on various active systems to ensure high and robust attenuation performance. The Active Rack Isolation System (Whorton et al., 1998) is an active vibration attenuator that provides a microgravity environment aboard the International Space Station. Generally speaking, although such an active attenuation system usually provides satisfactory performance, it is costly and its reliability is reduced by unstable phenomena, such as spillover. To obtain a high level of attenuation performance without resorting to completely active systems, it may be desirable to use a semi-active attenuation system. It ensures stability even if the control system malfunctions, since it exploits passive mechanisms. Electro-Rheological (ER) fluid is an attractive means to implement a semi-active control for a shock damper, because its characteristics can be controlled simply according to the strength of the electric field applied to it (see Klass and Martinck, 1967; Block and Kelly, 1988; Onoda et al., 1997). There have been concentrated efforts to bring ER fluid devices out of the laboratory. In this paper, the primary purpose of the proposed shock attenuation method is to attenuate shocks so that the acceleration of the instruments does not exceed a critical value. So far, we only know of shock attenuation methods that attenuate a small shock and a large shock in an equal manner. In contrast to such conventional linear methods, the proposed method intends not to attenuate a small shock in order to prepare for attenuating a coming large shock. In other words, to attenuate a powerful shock, we accept the risk of limiting the performance in attenuating a small shock. This nonlinear shock control method enables the damper to attenuate a powerful shock effectively. In this paper, the proposed method is investigated from various viewpoints; not only its shock attenuation performance, but also its vibration suppression performance and the influence of the secondary shock induced by the frictional force modulation.
2. ER FLUID SHOCK DAMPER As shown in Fig. 1 the ER-fluid damper (Onoda et al., 1997) consists of two variable-volume chambers filled with ER fluid and a bottleneck connecting them. The damper is sealed with bellows, and the ER fluid is pressurized by a spring. The characteristics of the ER-fluid in the bottleneck vary according to the voltage applied to the electrode, thus varying the characteristics of the damper. To measure the characteristics of the damper, several
bellows electrode
ER-fluid bellows spring base
Fig. 1. ER fluid damper constant-rate extension test were performed while keeping the input voltage constant. Figure 2 shows
Fig. 2. Load-extension for ER damper (Onoda et al., 1997) the elongation-load relation measured while repeating the constant-rate extension and contraction at various constant input voltages in (Onoda et al., 1997). As the input voltage is higher, the hysteresis is larger. The plot obtained at each input voltage consists of high stiffness and low stiffness section forming a parallelogram. Using these date, we can model the damper with two spring element k1 and k2 , a viscous damping element c and a variable coulomb frictional element f . Figure. 3 shows variation of c, f for the particle-dispersion ER damper in (Onoda et al., 1997). We can see that the variation of f is small while the variation
of c is large. To investigate the performance of a
y
m 1 k1
ER fluid shock damper
2 3 f
c
4
k2 x
base
x ¨ Fig. 4. Model of shock damper
Fig. 3. c and f for particle-dispersion ER damper (Onoda et al., 1997) shock damper using particle-dispersion ER fluid, we consider the simple model with a variable friction element shown in Fig. 4. This model is consistent with the Bingham fluid characteristics of particle-dispersion ER fluid and is the same as the ER fluid damper model. The shock damper is located between the instrument and the satellite main structure, which is called the base. The shock damper model is composed of two spring elements k1 and k2 , a viscous damping element c, and a variable Coulomb friction element f . The element f can be controlled by the electric field applied to the ER fluid. The structural model of the satellite instrument is simplified, as long as the generality of its structural dynamics is not limited. The base receives a shock that is expressed by an acceleration x ¨. The equation of motion of the instrument can be written as m¨ z + p = −m¨ x.
(1)
When the tensile force at point 3 is smaller than f , the friction prevents the distance between points 3 and 4 from changing, i.e., d˙ = 0. Otherwise, the leak of friction causes a slip. This phenomenon can be expressed as ⎧ ⎨ (p − k2 d − f )/c when 0 < f < p − k2 d, d˙ = 0 when −f ≤ p − k2 d ≤ f, ⎩ (p − k2 d + f )/c when p − k2 d < −f < 0. (2) The spring constant k1 is dominated by the volume stiffness of the ER fluid, and k2 is dominated by the axial stiffness of the bellows structure. The tensile load at points 1 and 4 is p = k1 (z − d).
(3)
electric field, and other parameters, such as elongation or force, cannot be controlled as desired. In other words, we intend to attenuate shock and vibration only by modulating the value of f .
3.1 Shock Attenuation Logic The main purpose of the shock attenuation method is to attenuate shocks so that the acceleration of the instruments does not exceed a critical value, even when the input shock is too large to be accepted. For this purpose, the proposed method has an innovative control idea; it does not attempt to attenuate a small shock in order to preparing for attenuating a coming large shock. Specifically, when the shock is within a certain value, the damper does not respond to it and stores the shock energy, and when the shock becomes large, the damper will respond to it and release the stored energy. We assume that the instrument will be damaged when the absolute value of its acceleration |¨ y| exceeds a critical value y¨c (> 0). Since the acceleration is proportional to p, this assumption can be regarded as saying that |p| exceeds a critical value pc , where pc ≡ m¨ yc . Consequently, we control f so that |p| does not exceed pc . Our idea for a shock damper logic is to modulate f smoothly and to start this modulation well in advance before |p| reaches pc . While γpc ≤ |p| ≤ pc , we control f so that |p| reaches pc asymptotically, instead of using a quick modulation. p should thus follow −β(p + pc ) when p < 0, p˙ = (4) −β(p − pc ) when p > 0, where β is a constant parameter. To implement the control idea concerning p in Eq. (4), with Eq. (3), f should be controlled to satisfy
3. SHOCK DAMPER LOGIC
β d˙ = z˙ + [p − sgn(p)pc ]. k1
In this ER fluid shock damper model, only frictional force can be controlled directly with the
Under the relation of Eq. (2), a control scheme satisfying Eq. (5) should be established. When p − k2 d > 0, two cases can arise, i.e., the first and
(5)
the second cases in Eq. (2). We select the first case in order to adapt d˙ to the relation in Eq. (5), rather than simply setting d˙ to be 0. The control scheme is thus obtained as ¯ f = p − k2 d − h, (6) where
¯ ≡ cz˙ + cβ [p − sgn(p)pc ]. h (7) k1 After the consideration of the case that p − k2 d < 0, the control scheme concerning f becomes ¯ f = sgn(p − k2 d)(p − k2 d − h). (8) Since f is generally constrained to be between fmin and fmax , the aforementioned scheme can be implemented as ¯ f = R[sgn(p − k2 d)(p − k2 d − h)], (9) where R[f˜] is the following restriction function. ⎧ ⎨ fmin when f˜ < fmin ˜ R[f ] ≡ f˜ (10) when fmin ≤ f˜ ≤ fmax ⎩ fmax when fmax < f˜ 3.2 Vibration Control Logic The shock always induces subsequent vibrations, which need to be suppressed by the damper. We choose a vibration control method using a particle-dispersion ER fluid system. Equations (1) and (3) can be written as z˙ = Az + B1 d + B2 x ¨.
(11)
Equation (11) indicates that, if d can be controlled directly, the state of the system z can be controlled based on active control theory. The LQ control input d would be given by d = dT ≡ −Fz,
(12)
where F is the feedback gain. As described earlier, in our semi-active system, d cannot be modulated directly; only f can be modulated directly. Thus, we control f so that d becomes as large (i.e., positive) as possible when dT is positive, and as small (i.e., negative) as possible when dT is negative. Considering the combination of the polarities of dT and d˙ in Eq. (2), a control logic can be implemented as when when
dT (p − k2 d) > 0, f = fmin , dT (p − k2 d) ≤ 0, f = fmax . (13)
This linear vibration suppression logic is referred to as “conventional linear logic”.
3.3 Combined Logic To enable shock attenuation and vibration suppression with one shock damper, the two attenuation logics ought to be combined. Our idea is
that when γpc ≤ |p| the shock damper works to attenuate shock, and when |p| < γpc it works to suppress vibration. The shock attenuation logic in Eq. (9) and the linear vibration suppression logic in Eq. (13) are combined as follows: when γpc ≤ |p|, f = R[sgn(p − k2 d)(p − k2 d − ¯h)], when |p| < γpc , f = fmin if dT (p − k2 d) > 0, f = fmax if dT (p − k2 d) ≤ 0. (14) This is a combined control logic, and we refer to it as “shock damper logic”. Note that for powerful shock attenuation, the risk of limiting the vibration suppression performance must be accepted. In actual systems with ER fluid dampers, the tensile load on the damper p can be measured using a strain-gauge type load cell, and the total elongation of the damper z can be measured by the actual measurement setup in (Onoda et al., 1997). From Eq. (3), d can be estimated as p d=z− . (15) k1 4. NUMERICAL SIMULATION Table 1. Parameter used in simulations A c fmin , fmax k1 , k2 m w2 y¨c β γ τ
1.5 × 102 18.0 0.0, 1.0 × 102 1.0 × 104 , 1.0 × 103 1.0 1.0 9.0 × 101 1.0 × 103 0.95 5.0 × 10−2
m/s2 Ns/m N N/m kg m/s2 1/s s
4.1 Histories of Shock Control Numerical simulations using the ER fluid model shown in Fig. 4 were carried out. The shock applied to the base was assumed to be a semisinusoidal wave. The simulation parameters are listed in Table 1. Figure 5 shows a typical time history of the system with the proposed logic. As described earlier, the shock damper logic had two modes of shock control and vibration control, and its mode was switched according to the value of p. Just after the shock x¨ was applied, |¨ y | and |p| started to increase. When |p| reached γpc , the controller started to attenuate the shock by continuously modulating f . Consequently, the proposed shock damper kept |¨ y | within y¨c , as intended. While |p| was within γpc the controller worked in vibration suppression mode to suppress
Fig. 5. Time history of shock attenuation using 2 proposed shock damper logic (¨ yc = 90 m/s ) z. Figure 6 compares the time histories for systems with the shock damper logic, the conventional linear logic, and the optimal passive method. The optimal passive system was implemented with the ER fluid parameters without voltage input; c = 18 Ns/m and f = 0 N. The value of c was an optimal one that provided high performance over a wide range of shock durations τ . Only the proposed shock damper attenuated acceler2 ation to under an allowable value of 90 m/s . The conventional linear control system had the worst shock attenuation performance, reaching a 2 maximum acceleration of 162 m/s , but the best vibration suppression performance. The optimal passive system had the worst vibration suppression performance, although it had a better shock attenuation performance than the conventional linear control system. We can say that our shock damper system had the advantages of both systems. To assess the performance of the proposed
shock damper, a comprehensive investigation was conducted by applying various shock inputs. Figure 7 shows boundaries of τ −A combinations that satisfy the condition that the maximum value of |¨ y | is y¨c = 90 m/s2 . The boundary lines separate the allowable and unallowable combinations of A and τ . When the input shock whose τ − A combination existed within the allowable region, the maximum value of |¨ y | did not exceed y¨c and the satellite instrument was protected from damage. Figure 7 indicates that the proposed shock damper system had higher performance than the other two systems over a wide range of τ − A combinations. For example, when τ = 0.05 s, the shock damper constrained the maximum acceleration within the allowable range, even when A 2 was as large as 160 m/s . On the other hand, the conventional linear control system had an allowable shock acceleration A of only 57 m/s2 , and the passive system had an allowable shock acceleration of 130 m/s2 .
Fig. 7. Boundary of A− τ combinations for shocks whose maximum |¨ y | is y¨c (= 90 m/s2 )
4.2 Shock Response Spectrum
Fig. 6. Comparison of shock attenuations of control systems (¨ yc = 90 m/s2 )
There may be a possibility of a secondary shock induced by the quick modulation of f . In an actual satellite instrument, the mass denoted as m in Fig. 4 is composed of a great number of small components, such as electronic devices. Each component has its own resonance frequency, and due to this resonance, it might be damaged when it is excited by the secondary shock. The influence of the secondary shock can be evaluated in frequency domain by using the Shock Response Spectrum (SRS) index in (NASA-STD, 1999). Figure 8 shows the SRS index for each method by using the time history of y¨ in Fig. 6. Among the three methods, the proposed shock damper provided the smallest maximum SRS index. Thus, it gave the least influential secondary shock to the system,
which is reasonable when we think that the amplitude of y¨ for the shock damper was smaller than those of the other systems, as shown in Fig. 6. In conclusion, the proposed shock damper performed well in limiting the secondary shock in spite of the modulation of f . The influence of β on the secondary shock was small for this configuration.
Fig. 8. SRS index as a function of ω
4.3 Random-Wave Shock Input To guarantee reliability for a more realistic shock, the response of the shock damper should be evaluated against a more severe and complex shock condition than a semi-sinusoidal shock, for instance, against a random-wave shock input. By using an inverse FFT, a random-acceleration function was created. The power spectrum density (PSD) of the random acceleration was 0.1 m2 /s3 in the range of frequency from 60 to 350 rad/s, and its value was 0 over the rest of the frequency range. Figure 9 shows one example of time histories of each control system against a random-wave shock input x ¨. Only the proposed shock damper attenuated the instrument acceleration |¨ y | to within y¨c (= 2 90 m/s ). With the other two systems, the acceleration exceeded the critical value many times. It should be emphasized that the proposed shock damper worked as specified even when such a severe and complex shock was applied. 5. CONCLUSION We developed an innovative shock damper using particle-dispersion ER fluid for attenuating shocks experienced by satellite instruments. The main purpose of the shock damper is to attenuate shocks so that the acceleration of the instruments does not exceed a critical value, even when the input shock is too large to be accepted. In contrast to the conventional linear shock dampers, the proposed damper does not attempt to attenuate a small shock in order to prepare for attenuating a
Fig. 9. Time histories of shock damper experiencing random-wave shock input coming large shock. In comparison to a passive method and a conventional linear control, this proposed shock damper had the best shock attenuation performance. By using the SRS index, we confirmed that the proposed method causes no harmful secondary shocks that might be generated by quick modulation of frictional force. 6. REFERENCES Block, H. and Kelly, J. P. (1988). Electro-rheology, Journal of Physics D: Applied Physics, 21, 1661-1677. Klass, D. L. and Martinek, T. W. (1967). Electroviscous Fluids. I. Rheological Properties, Journal of Applied Physics, 38, 67-74. Krakov, M. S. (1999). Influence of Rheological Properties of Magnetic Fluid on Damping Ability of Magnetic Fluid Shock-Damper, Journal of Magnetism and Magnetic Materials, 201 , 368-371. Makihara, K., Onoda, J. and Minesugi, K. (2006). New Approach to Semi-Active Vibration Isolation to Improve The Pointing Performance of Observation Satellites, Smart Materials and Structures, 15, 342-350. Onoda, J., Oh, H-U. and Minesugi, K. (1997). Semiactive Vibration Suppression with ElectrorheologicalFluid Dampers, AIAA J. 35-12, 1844-1852. Ping, Y. (2003). Experimental and Mathematical Evaluation of Dynamic Behaviour of an Oil-Air Coupling Shock Damper, Mechanical Systems and Signal Processing, 17, 1367-1379. Pyroshock Test Criteria, NASA Technical Standard, (1999). NASA-STD-7003, May 18. Rodden, J. J., Dougherty. H. J., Rescheke, L. F., Hasha, M. D. and Davis, L. P. (1986). Line-of-Sight Performance Improvement with Reaction Wheel Isolation, Proc. Annual Rocky Mountain Guidance and Control Conf., Keynote, San Diego CA., pp. 71-84. Whorton, M. S., Eldridge, J. T., Ferebee, R. C., Lassiter, J. O. and Redmon, J. W. (1998). Damping Mechanisms for Microgravity Vibration Isolation, NASA-TM-1998206953.
Department of Engineering, University of Cambridge Email: [email protected] ∗∗ Institute of Space and Astronautical Science, JAXA 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan
Abstract: We developed the novel control logic for a shock damper using particledispersion Electro-Rheological (ER) fluid. This is a high-powered means of shock attenuation for satellite instruments that are subjected to lift-off shock or pyrodevice ignition shock. The proposed method attenuates the shock so that the instrument’s acceleration does not exceed the critical value, even when the shock is too large to be accepted. In contrast to the conventional linear shock controls, the proposed shock control does not attempt to attenuate a small shock in order to prepare for attenuating a coming large shock. Keywords: Shock damper, Shock absorber, ER fluid
Nomenclature A c d f k1 , k2 m p pc x y y¨c z z β γ τ
: : : : : : : : : : : : : : : :
amplitude of acceleration viscous damping element elongation between points 2 and 4 Coulomb friction element spring elements mass of satellite instrument tensile load at points 1 and 4 critical value of p (≡ m¨ yc ) displacement of base displacement of satellite instrument critical value of acceleration y¨ relative displacement of y to x state vector (≡ [z, z] ˙ T) constant parameter (0 < β) constant parameter (0 ≤ γ ≤ 1) semi-sinusoidal shock duration 1. INTRODUCTION
On-board satellite instruments are often damaged by the shocks generated from rocket lift-
off, staging, or pyrodevice ignition often damage. The acceleration experienced by satellite instruments is proportional to the magnitude of the applied shock. Satellite instruments may be damaged when the acceleration generated by a shock exceeds a certain critical value. Therefore, acceleration is an important factor for safety design of satellite components and has drawn keen attention in operation. Satellite instruments can be protected from shock damage by mounting them on a shock damper. A great number of studies and attempts have been conducted on shock and vibration attenuation (see Rodden et al., 1986; Whorton et al., 1998; Krakov, 1999; Ping, 2003; Makihara et al., 2006). The Hubble Space Telescope (Rodden et al., 1986) is probably the most famous space structure that has a disturbance attenuation system essential to achieving its mission. Although these passive systems are very reliable, their attenuation performance is generally lower than those of active systems. Since a passive system cannot
adapt to environmental changes, its attenuation performance may deteriorate when design parameters change from their tuned values. Researchers have been working on various active systems to ensure high and robust attenuation performance. The Active Rack Isolation System (Whorton et al., 1998) is an active vibration attenuator that provides a microgravity environment aboard the International Space Station. Generally speaking, although such an active attenuation system usually provides satisfactory performance, it is costly and its reliability is reduced by unstable phenomena, such as spillover. To obtain a high level of attenuation performance without resorting to completely active systems, it may be desirable to use a semi-active attenuation system. It ensures stability even if the control system malfunctions, since it exploits passive mechanisms. Electro-Rheological (ER) fluid is an attractive means to implement a semi-active control for a shock damper, because its characteristics can be controlled simply according to the strength of the electric field applied to it (see Klass and Martinck, 1967; Block and Kelly, 1988; Onoda et al., 1997). There have been concentrated efforts to bring ER fluid devices out of the laboratory. In this paper, the primary purpose of the proposed shock attenuation method is to attenuate shocks so that the acceleration of the instruments does not exceed a critical value. So far, we only know of shock attenuation methods that attenuate a small shock and a large shock in an equal manner. In contrast to such conventional linear methods, the proposed method intends not to attenuate a small shock in order to prepare for attenuating a coming large shock. In other words, to attenuate a powerful shock, we accept the risk of limiting the performance in attenuating a small shock. This nonlinear shock control method enables the damper to attenuate a powerful shock effectively. In this paper, the proposed method is investigated from various viewpoints; not only its shock attenuation performance, but also its vibration suppression performance and the influence of the secondary shock induced by the frictional force modulation.
2. ER FLUID SHOCK DAMPER As shown in Fig. 1 the ER-fluid damper (Onoda et al., 1997) consists of two variable-volume chambers filled with ER fluid and a bottleneck connecting them. The damper is sealed with bellows, and the ER fluid is pressurized by a spring. The characteristics of the ER-fluid in the bottleneck vary according to the voltage applied to the electrode, thus varying the characteristics of the damper. To measure the characteristics of the damper, several
bellows electrode
ER-fluid bellows spring base
Fig. 1. ER fluid damper constant-rate extension test were performed while keeping the input voltage constant. Figure 2 shows
Fig. 2. Load-extension for ER damper (Onoda et al., 1997) the elongation-load relation measured while repeating the constant-rate extension and contraction at various constant input voltages in (Onoda et al., 1997). As the input voltage is higher, the hysteresis is larger. The plot obtained at each input voltage consists of high stiffness and low stiffness section forming a parallelogram. Using these date, we can model the damper with two spring element k1 and k2 , a viscous damping element c and a variable coulomb frictional element f . Figure. 3 shows variation of c, f for the particle-dispersion ER damper in (Onoda et al., 1997). We can see that the variation of f is small while the variation
of c is large. To investigate the performance of a
y
m 1 k1
ER fluid shock damper
2 3 f
c
4
k2 x
base
x ¨ Fig. 4. Model of shock damper
Fig. 3. c and f for particle-dispersion ER damper (Onoda et al., 1997) shock damper using particle-dispersion ER fluid, we consider the simple model with a variable friction element shown in Fig. 4. This model is consistent with the Bingham fluid characteristics of particle-dispersion ER fluid and is the same as the ER fluid damper model. The shock damper is located between the instrument and the satellite main structure, which is called the base. The shock damper model is composed of two spring elements k1 and k2 , a viscous damping element c, and a variable Coulomb friction element f . The element f can be controlled by the electric field applied to the ER fluid. The structural model of the satellite instrument is simplified, as long as the generality of its structural dynamics is not limited. The base receives a shock that is expressed by an acceleration x ¨. The equation of motion of the instrument can be written as m¨ z + p = −m¨ x.
(1)
When the tensile force at point 3 is smaller than f , the friction prevents the distance between points 3 and 4 from changing, i.e., d˙ = 0. Otherwise, the leak of friction causes a slip. This phenomenon can be expressed as ⎧ ⎨ (p − k2 d − f )/c when 0 < f < p − k2 d, d˙ = 0 when −f ≤ p − k2 d ≤ f, ⎩ (p − k2 d + f )/c when p − k2 d < −f < 0. (2) The spring constant k1 is dominated by the volume stiffness of the ER fluid, and k2 is dominated by the axial stiffness of the bellows structure. The tensile load at points 1 and 4 is p = k1 (z − d).
(3)
electric field, and other parameters, such as elongation or force, cannot be controlled as desired. In other words, we intend to attenuate shock and vibration only by modulating the value of f .
3.1 Shock Attenuation Logic The main purpose of the shock attenuation method is to attenuate shocks so that the acceleration of the instruments does not exceed a critical value, even when the input shock is too large to be accepted. For this purpose, the proposed method has an innovative control idea; it does not attempt to attenuate a small shock in order to preparing for attenuating a coming large shock. Specifically, when the shock is within a certain value, the damper does not respond to it and stores the shock energy, and when the shock becomes large, the damper will respond to it and release the stored energy. We assume that the instrument will be damaged when the absolute value of its acceleration |¨ y| exceeds a critical value y¨c (> 0). Since the acceleration is proportional to p, this assumption can be regarded as saying that |p| exceeds a critical value pc , where pc ≡ m¨ yc . Consequently, we control f so that |p| does not exceed pc . Our idea for a shock damper logic is to modulate f smoothly and to start this modulation well in advance before |p| reaches pc . While γpc ≤ |p| ≤ pc , we control f so that |p| reaches pc asymptotically, instead of using a quick modulation. p should thus follow −β(p + pc ) when p < 0, p˙ = (4) −β(p − pc ) when p > 0, where β is a constant parameter. To implement the control idea concerning p in Eq. (4), with Eq. (3), f should be controlled to satisfy
3. SHOCK DAMPER LOGIC
β d˙ = z˙ + [p − sgn(p)pc ]. k1
In this ER fluid shock damper model, only frictional force can be controlled directly with the
Under the relation of Eq. (2), a control scheme satisfying Eq. (5) should be established. When p − k2 d > 0, two cases can arise, i.e., the first and
(5)
the second cases in Eq. (2). We select the first case in order to adapt d˙ to the relation in Eq. (5), rather than simply setting d˙ to be 0. The control scheme is thus obtained as ¯ f = p − k2 d − h, (6) where
¯ ≡ cz˙ + cβ [p − sgn(p)pc ]. h (7) k1 After the consideration of the case that p − k2 d < 0, the control scheme concerning f becomes ¯ f = sgn(p − k2 d)(p − k2 d − h). (8) Since f is generally constrained to be between fmin and fmax , the aforementioned scheme can be implemented as ¯ f = R[sgn(p − k2 d)(p − k2 d − h)], (9) where R[f˜] is the following restriction function. ⎧ ⎨ fmin when f˜ < fmin ˜ R[f ] ≡ f˜ (10) when fmin ≤ f˜ ≤ fmax ⎩ fmax when fmax < f˜ 3.2 Vibration Control Logic The shock always induces subsequent vibrations, which need to be suppressed by the damper. We choose a vibration control method using a particle-dispersion ER fluid system. Equations (1) and (3) can be written as z˙ = Az + B1 d + B2 x ¨.
(11)
Equation (11) indicates that, if d can be controlled directly, the state of the system z can be controlled based on active control theory. The LQ control input d would be given by d = dT ≡ −Fz,
(12)
where F is the feedback gain. As described earlier, in our semi-active system, d cannot be modulated directly; only f can be modulated directly. Thus, we control f so that d becomes as large (i.e., positive) as possible when dT is positive, and as small (i.e., negative) as possible when dT is negative. Considering the combination of the polarities of dT and d˙ in Eq. (2), a control logic can be implemented as when when
dT (p − k2 d) > 0, f = fmin , dT (p − k2 d) ≤ 0, f = fmax . (13)
This linear vibration suppression logic is referred to as “conventional linear logic”.
3.3 Combined Logic To enable shock attenuation and vibration suppression with one shock damper, the two attenuation logics ought to be combined. Our idea is
that when γpc ≤ |p| the shock damper works to attenuate shock, and when |p| < γpc it works to suppress vibration. The shock attenuation logic in Eq. (9) and the linear vibration suppression logic in Eq. (13) are combined as follows: when γpc ≤ |p|, f = R[sgn(p − k2 d)(p − k2 d − ¯h)], when |p| < γpc , f = fmin if dT (p − k2 d) > 0, f = fmax if dT (p − k2 d) ≤ 0. (14) This is a combined control logic, and we refer to it as “shock damper logic”. Note that for powerful shock attenuation, the risk of limiting the vibration suppression performance must be accepted. In actual systems with ER fluid dampers, the tensile load on the damper p can be measured using a strain-gauge type load cell, and the total elongation of the damper z can be measured by the actual measurement setup in (Onoda et al., 1997). From Eq. (3), d can be estimated as p d=z− . (15) k1 4. NUMERICAL SIMULATION Table 1. Parameter used in simulations A c fmin , fmax k1 , k2 m w2 y¨c β γ τ
1.5 × 102 18.0 0.0, 1.0 × 102 1.0 × 104 , 1.0 × 103 1.0 1.0 9.0 × 101 1.0 × 103 0.95 5.0 × 10−2
m/s2 Ns/m N N/m kg m/s2 1/s s
4.1 Histories of Shock Control Numerical simulations using the ER fluid model shown in Fig. 4 were carried out. The shock applied to the base was assumed to be a semisinusoidal wave. The simulation parameters are listed in Table 1. Figure 5 shows a typical time history of the system with the proposed logic. As described earlier, the shock damper logic had two modes of shock control and vibration control, and its mode was switched according to the value of p. Just after the shock x¨ was applied, |¨ y | and |p| started to increase. When |p| reached γpc , the controller started to attenuate the shock by continuously modulating f . Consequently, the proposed shock damper kept |¨ y | within y¨c , as intended. While |p| was within γpc the controller worked in vibration suppression mode to suppress
Fig. 5. Time history of shock attenuation using 2 proposed shock damper logic (¨ yc = 90 m/s ) z. Figure 6 compares the time histories for systems with the shock damper logic, the conventional linear logic, and the optimal passive method. The optimal passive system was implemented with the ER fluid parameters without voltage input; c = 18 Ns/m and f = 0 N. The value of c was an optimal one that provided high performance over a wide range of shock durations τ . Only the proposed shock damper attenuated acceler2 ation to under an allowable value of 90 m/s . The conventional linear control system had the worst shock attenuation performance, reaching a 2 maximum acceleration of 162 m/s , but the best vibration suppression performance. The optimal passive system had the worst vibration suppression performance, although it had a better shock attenuation performance than the conventional linear control system. We can say that our shock damper system had the advantages of both systems. To assess the performance of the proposed
shock damper, a comprehensive investigation was conducted by applying various shock inputs. Figure 7 shows boundaries of τ −A combinations that satisfy the condition that the maximum value of |¨ y | is y¨c = 90 m/s2 . The boundary lines separate the allowable and unallowable combinations of A and τ . When the input shock whose τ − A combination existed within the allowable region, the maximum value of |¨ y | did not exceed y¨c and the satellite instrument was protected from damage. Figure 7 indicates that the proposed shock damper system had higher performance than the other two systems over a wide range of τ − A combinations. For example, when τ = 0.05 s, the shock damper constrained the maximum acceleration within the allowable range, even when A 2 was as large as 160 m/s . On the other hand, the conventional linear control system had an allowable shock acceleration A of only 57 m/s2 , and the passive system had an allowable shock acceleration of 130 m/s2 .
Fig. 7. Boundary of A− τ combinations for shocks whose maximum |¨ y | is y¨c (= 90 m/s2 )
4.2 Shock Response Spectrum
Fig. 6. Comparison of shock attenuations of control systems (¨ yc = 90 m/s2 )
There may be a possibility of a secondary shock induced by the quick modulation of f . In an actual satellite instrument, the mass denoted as m in Fig. 4 is composed of a great number of small components, such as electronic devices. Each component has its own resonance frequency, and due to this resonance, it might be damaged when it is excited by the secondary shock. The influence of the secondary shock can be evaluated in frequency domain by using the Shock Response Spectrum (SRS) index in (NASA-STD, 1999). Figure 8 shows the SRS index for each method by using the time history of y¨ in Fig. 6. Among the three methods, the proposed shock damper provided the smallest maximum SRS index. Thus, it gave the least influential secondary shock to the system,
which is reasonable when we think that the amplitude of y¨ for the shock damper was smaller than those of the other systems, as shown in Fig. 6. In conclusion, the proposed shock damper performed well in limiting the secondary shock in spite of the modulation of f . The influence of β on the secondary shock was small for this configuration.
Fig. 8. SRS index as a function of ω
4.3 Random-Wave Shock Input To guarantee reliability for a more realistic shock, the response of the shock damper should be evaluated against a more severe and complex shock condition than a semi-sinusoidal shock, for instance, against a random-wave shock input. By using an inverse FFT, a random-acceleration function was created. The power spectrum density (PSD) of the random acceleration was 0.1 m2 /s3 in the range of frequency from 60 to 350 rad/s, and its value was 0 over the rest of the frequency range. Figure 9 shows one example of time histories of each control system against a random-wave shock input x ¨. Only the proposed shock damper attenuated the instrument acceleration |¨ y | to within y¨c (= 2 90 m/s ). With the other two systems, the acceleration exceeded the critical value many times. It should be emphasized that the proposed shock damper worked as specified even when such a severe and complex shock was applied. 5. CONCLUSION We developed an innovative shock damper using particle-dispersion ER fluid for attenuating shocks experienced by satellite instruments. The main purpose of the shock damper is to attenuate shocks so that the acceleration of the instruments does not exceed a critical value, even when the input shock is too large to be accepted. In contrast to the conventional linear shock dampers, the proposed damper does not attempt to attenuate a small shock in order to prepare for attenuating a
Fig. 9. Time histories of shock damper experiencing random-wave shock input coming large shock. In comparison to a passive method and a conventional linear control, this proposed shock damper had the best shock attenuation performance. By using the SRS index, we confirmed that the proposed method causes no harmful secondary shocks that might be generated by quick modulation of frictional force. 6. REFERENCES Block, H. and Kelly, J. P. (1988). Electro-rheology, Journal of Physics D: Applied Physics, 21, 1661-1677. Klass, D. L. and Martinek, T. W. (1967). Electroviscous Fluids. I. Rheological Properties, Journal of Applied Physics, 38, 67-74. Krakov, M. S. (1999). Influence of Rheological Properties of Magnetic Fluid on Damping Ability of Magnetic Fluid Shock-Damper, Journal of Magnetism and Magnetic Materials, 201 , 368-371. Makihara, K., Onoda, J. and Minesugi, K. (2006). New Approach to Semi-Active Vibration Isolation to Improve The Pointing Performance of Observation Satellites, Smart Materials and Structures, 15, 342-350. Onoda, J., Oh, H-U. and Minesugi, K. (1997). Semiactive Vibration Suppression with ElectrorheologicalFluid Dampers, AIAA J. 35-12, 1844-1852. Ping, Y. (2003). Experimental and Mathematical Evaluation of Dynamic Behaviour of an Oil-Air Coupling Shock Damper, Mechanical Systems and Signal Processing, 17, 1367-1379. Pyroshock Test Criteria, NASA Technical Standard, (1999). NASA-STD-7003, May 18. Rodden, J. J., Dougherty. H. J., Rescheke, L. F., Hasha, M. D. and Davis, L. P. (1986). Line-of-Sight Performance Improvement with Reaction Wheel Isolation, Proc. Annual Rocky Mountain Guidance and Control Conf., Keynote, San Diego CA., pp. 71-84. Whorton, M. S., Eldridge, J. T., Ferebee, R. C., Lassiter, J. O. and Redmon, J. W. (1998). Damping Mechanisms for Microgravity Vibration Isolation, NASA-TM-1998206953.