Multi-duct ER Dampers

Transcription

1 Multi-duct ER Dampers Henri P. Gavin Department of Civil & Environmental Engineering, Duke University, Durham NC 778, U.S.A. Accepted for Publication, Journal of Intelligent Material Systems and Structures, Keywords: electrorheological damper, electrorheological device, smart damping, rheological modeling Abstract This paper describes the design, construction, testing and modeling of controllable damping devices utilizing electro-rheological (er) materials. The rheological properties of er materials (yield stress and viscoelasticity) are extremely sensitive to electric fields. Modulations of the electric field in an electrorheological damper results in a corresponding change in device forces. The key feature of the er devices described in this paper is a set of multiple concentric annular ducts through which the er material flows. The annular ducts are formed by a set of concentric metallic tubes, which may be electrically charged with a high voltage potential, or electrically grounded. Three designs targeting different force levels (kn to 6kN), are designed, tested and modeled. The device analyses used in the design incorporate a simplified closed form relation for the flow behavior of er materials. Evolutionary and algebraic device models are fit to the measured force response of the devices over a range of harmonic frequencies and at zero and high electric field magnitudes. The evolutionary model is motivated by observations and simulations of er materials in simple shear, and the algebraic model is convenient for control and feedback linearization applications. The algebraic model is evaluated with data collected with random excitation and with switching electric fields. The three devices display different levels of preyield visco-elasticity. Friction, due to seals in the device, was not accounted for in the initial design, and affected the behavior of the device as tested. Introduction Electrorheological (er) fluids exhibit damping and stiffness properties which can be modulated by several orders of magnitude when subjected to strong electric fields. An increase in yield stress (from to 5 kpa) is characteristically observed when a high electric field (of 3 to 7 kv/mm) is applied. The change in material properties occurs within milliseconds and is completely reversible upon removal of the electric field. A potential application of er materials is controllable dampers for dynamically excited structures. The successful use of er materials in structural systems requires that the controllable properties of the materials be mechanically amplified in order to produce significant controllable forces and that the device forces can be modulated to a significant degree (a factor of or more) when the device undergoes characteristic motions. Because the range of adjustable forces is closely linked to the ratio of the finite field-controllable, and flow-independent yield stresses to the uncontrollable, and flow-dependent viscous stresses, it is desirable for controllable er dampers

2 MULTI-DUCT ER DAMPERS to operate at low flow rates. In addition to the range of available forces, er dampers should also perform well in other regards. Step-changes in the applied voltage should produce a rapid change in the device s forces, electrical energy requirements should be modest, and the forces in the device should be large enough to be effective for the intended application. In the present study, these goals are achieved by ensuring that the strain rates in er materials flowing through ducts (with large surface areas) are kept low. The er dampers described in this paper feature multiple concentric electrodes which are electrically in parallel, but may be hydraulically inter-connected in various configurations. The number of possible interconnections of N concentric ducts is (N ). Analyses of these various configurations is completed in closed form using an approximation to the Poiseuille flow of er materials. The devices are designed under the assumption of quasi-steady flow and are modeled with expressions incorporating the dynamic effects of pre-yield visco-elasticity and inertia. Background The sensitivity of rheological properties to electric fields in er suspensions results from the electrostatic polarization of particles ( to microns in diameter) dispersed in a dielectrically mismatched liquid. The inter-particle dipole-dipole interactions are attractive along the field direction and repulsive across the field. This anisotropic interaction energy gives an equilibrium arrangement of particles that is characterized by fibrated strands aligned with the field. Particulates that can be used as the dispersed phase in er suspensions are diverse and include alumina silicates, zeolites, sulfonated polymers, and carbonaceous particles (Block, et al. 99). At small dynamic strains (less than.) the shear stress is related to the shear strain via a complex modulus, at larger strains the material yields, and at intermediate strains, the material exhibits a combination of viscoelastic and yielding behavior (McLeish, et al. 99). The development of anhydrous er materials in the late 98 s enabled the wide-spread application of er materials. While the required electric fields are strong (kv/mm), the fields displacement currents are small (µamp/cm ) (Filisko et al. 99). Thus, er devices can be operated using low-power, high-voltage power sources. Under quasi-steady confined flow conditions, the shear stresses, τ, in an er suspension are resisted by a field-dependent yielding component τ y (E) and a temperature-dependent Newtonian viscous component η(θ) γ. The Bingham visco-plastic material model is commonly used to model er materials under quasi-steady flow, τ( γ, E, θ) = τ y (E)sgn γ + η(θ) γ, () and can be mechanistically represented by a dash-pot in parallel with a frictional element (Block et al. 988). The yield stress increases approximately with E and the viscosity is roughly fieldindependent and decreases with temperature (Jordan, et al. 989, Halsey, 99). Under steady shear flow the application of the electric field causes the shear stresses to increase by a factor of + τ y (E)/(η γ). Therefore, er devices designed to provide a wide range of adjustable forces have small viscous stresses (η γ) as compared to the yield stress (τ y ). Because yield stresses in er materials are limited, this is typically accomplished by using materials with low zero-field viscosity η and by designing devices in which the strain rates γ are low (Gavin, 998). Even though yield stresses in er materials are low as compared to stresses in structural members, er materials can be used in devices that exhibit large ranges of adjustable forces (tens or hundreds of kilo-newtons). Instead of forcing a viscous material to flow at high velocity through a relatively narrow orifice, the proposed device develops large forces by forcing the fluid to flow at a relatively low rates over a large surface area. In these devices, the duct walls are the same as

3 MULTI-DUCT ER DAMPERS 3 the electric field-generating electrodes. Energizing a large volume of er material is equivalent to providing a large surface area to develop large shear forces. To maintain a flow rate that is low enough so that the Newtonian stresses do not obfuscate the adjustable yield stresses (even at high damper piston velocities), the duct s cross sectional area can be adjusted with respect to the piston area. In the last decade multiple types of er devices have been developed for controllable damping applications. Wang et al. (999) have developed a squeeze-film damper using conical electrodes. An er damper designed using flow between multiple parallel plates was designed, tested, and modeled with a polynomial expansion (Gavin et al. 996, Gavin et al. 996b). This damper has a stroke of cm and had a controllable force range from. to.6 kn at. cm/sec and. Hertz when tested with a custom amorphous silicate-based ER material. Makris et al. (996a,b) developed an er device with multiple by-pass valves and presented a model for the device based on an isotropic continuum theory of er materials (Burton et al. 996). This damper has a stroke of cm and had a controllable force range from. to.3 kn at.3 cm/sec and. Hertz when tested with a zeolitebased ER material. Kamath and Werely (997a,b) developed, tested and modeled er dampers based on a dash-pot principle. This damper has a stroke of.3 cm and had a controllable force range from. to.8 kn at cm/sec and Hertz when tested with Lord Versa-flow ER-. A concentric electrode ER device was tested using ER fluids and viscous ER greases (Gordaninejad et al. 997, Marksmeier et al. 998). This damper has a stroke of cm and had a controllable force range from.6 to.7 kn at 3 cm/sec and Hertz when tested with the ER grease. An ER device with a single bypass valve was constructed and tested using commercial hydraulic components and a commercial ER fluid (Lindler et al. 999). This damper has a stroke of.5 cm and had a controllable force range from.7 to.6 kn at 6 cm/sec and. Hertz when tested with the a commercial ER fluid. Another long stroke ER damper with a single bypass valve was developed and tested with the same commercial ER material (Simms et al. 999). This damper has a stroke of.6 cm and had a controllable force range from. to 3.8 kn at cm/sec and Hertz. The simplicity of dampers utilizing er materials allows for controllable damping devices which do not require seals or valves. Such a device, based on the viscous shear-wall, has been implemented in a small-scale controllable vibration isolation system (Gavin ). While these devices embody the simplicity achievable with er materials, devices which can deliver kilo-newtons of electrically controllable forces at velocities of several cm/sec have not been widely demonstrated. In contrast to the by-pass or single duct style devices previously reported, the devices described in this paper consist of a set of concentric cylindrical electrodes and a concentric piston within the inner-most electrode (Gavin 998) A section of the device showing the arrangement of the ducts and piston is given in Figure. Alternating electrodes are grounded. The other electrodes are held at a common high voltage. In this figure, the black electrodes are grounded and the white electrodes are at high voltage. The outer electrode is the grounded device case. While these electrodes create a set of parallel capacitors, the flow gaps between the electrodes can be connected to form parallel ducts, ducts in series, or groups of parallel and series ducts. Apart from the nature of duct interconnections, design variables include the inter-electrode gap, h, the electrode thickness, w, the piston radius, R p, the piston shaft radius, R s, the device radius, R d, and the device length, L. A gap is defined by its inner and outer radii, i r, and o r The across-flow dimension of the duct, h, is o r i r. In Figure all ducts are hydraulically parallel. While this arrangement results in the largest range of adjustable forces, it does not produce the highest force magnitudes. Indeed, the flexibility of this type of device design lies in the multiple ways in which the set of concentric ducts communicate hydraulically with one another. Moreover, all of the ducts have large flow surfaces

4 MULTI-DUCT ER DAMPERS R o d r i r.. h w R p R s V p Figure. Arrangement of piston and ducts in the proposed er damper design. and cross-sections which are comparable to the piston area. This duct configuration enables devices with high (kn) ranges of controllable forces. The two objectives of this study are to experimentally verify relationships between the device geometry, the er material properties and the voltage-dependent force-velocity behavior of the device, and to evaluate mechanistic models for the behavior of these devices. These relationships make use of an approximation to the quasi-steady flow of er materials through annular ducts. A compact dynamic model motivated by extensive viscometric measurements and molecular dynamics simulation and a simple algebraic model are fit to data collected from the devices undergoing harmonic motions. In closing, the predictive ability of the algebraic model is illustrated using device forces generated with random motions and switching electric fields. Quasi-steady Flow of ER Materials A feature of primary importance in quasi-steady er Poiseuille flow of Bingham materials is the existence of a plug of non-shearing (semi-solid) material that translates through the duct without deformation. The steady flow of homogeneous er materials through rectangular ducts was cast into a non-dimensionalized cubic equation by Phillips (969). A closed form solution to this cubic was presented in 99 (Gavin et al. 996a). Small-scale, focused, experiments have shown that this cubic equation provides an upper bound to the design of er dampers (Gavin et al. 996b), and is valid for quasi-steady flow when the period of flow oscillation is long compared to the inertial response time of the er material (which can be controlled by the duct geometry). Particle concentration inhomogeneities in the vicinity of the duct walls can arise from inertial lift forces on the dispersed phase and prevent pressure gradients in er materials from reaching the levels predicted when assuming homogeneity (Gavin 997). Assuming that the yield stress increases quadratically with the electric field, that there are no dielectric discontinuities across the inter-electrode gap, and that fluid inertia forces are negligible, the pressure gradient (p = p/ x), in a homogeneous er material along an annular gap can be approximated by p p N +.7 τ y h + p N τ y p N h +.τ y ()

5 MULTI-DUCT ER DAMPERS 5 to within two percent of the true value of p /p N (Gavin et al. 996a). In the above expression p N is the pressure gradient of a Newtonian material of viscosity η flowing through the same gap at the same volumetric flow rate, Q. In a steady flow condition, the force in an er device increases by a factor of p /p N when the field is applied. The approximation for p given in equation is quadratic in the flow rate, Q. In simple geometries (all ducts in series, or all ducts in parallel), this does not present a great difficulty. Where a closed form solution cannot be obtained, a simple numerical algorithm is employed. In more complex geometries, (interconnected groups of parallel or series ducts) an approximation for p that is linear in Q allows a closed form solution for the device forces. p p N +. τ y h. (3) While equation 3 describes the exact root to the cubic er flow equation to within only percent, it is a better approximation to experimental data due to the inertial lift mechanisms and the ensuing particle concentration and yield stress inhomogeneities (Gavin 997). The Newtonian pressure gradient is proportional to Q (p N = QR), where R depends only on geometry and viscosity, R gd ( i r gd, o r gd, η) = 8η π o r gd i r gd (o r gd i r gd) log( o r gd / i r gd ). () In the analyses that follow, flow ducts are separated into groups. Adjacent ducts within a group may be hydraulically connected in series, or in parallel. Adjacent groups may be hydraulically connected in series or in parallel. The sub-scripts gd indicate that the sub-scripted quantity pertains to duct d within group g, where g G and d D g. There are D g ducts in group g. When the subscript g is omitted, G =. The values for the inner and outer radii can be derived from the geometry shown in Figure, i r gd = o r (g )D(g ) + d δ= d w gδ + h gδ (5) δ= and where o r D = R p. d d o r gd = o r (g )D(g ) + w gδ + h gδ (6) δ= δ= Analysis Assuming that the er material is incompressible, the total volumetric flow rate through all the ducts, Q T, is proportional to the piston velocity, V p, Q T = π(r p R s )V p. (7) The total force developed by the device is proportional to the total pressure drop along the network of ducts, p, F = π(r p R s )( p). (8) In the analyses which follow, p T is an effective total pressure gradient, such that p = p T L.

6 MULTI-DUCT ER DAMPERS 6 In a dynamic vibration control application, the electrically capacitive er device will be charged and discharged several times per second. Under these conditions, the power required to operate the er device is the capacitive energy of the device times the switching rate plus the product of device voltage and steady-state current draw. For er materials which draw negligible current, power required to repeatedly charge the device is much greater than the power required to maintain a constant electric field. The capacitive energy is the product of the the capacitance and the device voltage squared, and the capacitance of a multi-electrode device (as illustrated in Figure ) in Farads is: G D g o r gd + i r gd C = κɛ πl o r gd i. (9) r gd g= d= where ɛ is the permittivity of free space and κ is the dielectric constant of the material (κ ). Steel tubes are readily available in only discrete sizes of diameter and wall thickness, the following analysis will presume that all of the tubes have the same wall thickness, w, and that all of the ducts have the same across-flow dimension, h. Even within the constraints of limited tube availability and a fixed device size (diameter and length), the arrangement of the flow paths within the device leads to devices with a wide range of characteristics. While all of the ducts are electrically parallel, the flow paths can be arranged either in parallel, in series, or both. The number of possible hydraulic interconnections of N concentric ducts is (N ). Each configuration has distinctly different properties. These configurations will be analyzed in the context of several parallel groups of ducts in series. All ducts in parallel When all ducts are in parallel (case P), the total pressure gradient, p, is equal along every duct, and the total volumetric flow rate is the sum of the flow rates for each duct. In other words, p T = p d, and Q T = Q d. The Newtonian pressure gradient in case P is [ D p N = Q T R ] d. () d= For the non-zero field, each duct s flow is obtained from equation, which is quadratic in Q d. ( ) R dq τ y d + 3.7R d p R d Q d +.88 τ y.p τ y =. () h d h d h d The equation D d= Q d (p ) = Q T varies smoothly with p, and its root can be calculated numerically as follows. Given a trial value for the pressure gradient p i (at iteration i), the flow rate Q d in each duct d is obtained from equation. Every Q d which is positive and real is summed into the total flow, Q Ti, corresponding to p i. The pressure gradient is updated as follows: p i+ = (Q T Q i )(α + βp i), () until Q T Q i is sufficiently small. The parameters α and β are chosen to provide good numerical stability and convergence, and depend on the scale of the damper design. All ducts in series If all D ducts are in series (case S), the the flow in each duct is Q T. The total pressure drop is the sum of the pressure drops along each duct, or the net pressure gradient is the sum of the individual

7 MULTI-DUCT ER DAMPERS 7 pressure gradients. In other words, Q T = Q d and p T = p d. In case S, the net Newtonian pressure gradient is p N = Q T D d= R d, (3) and the pressure gradient due in a flowing homogeneous Bingham material is [ D p T = Q T R d +.7 τ ] y τ y Q T R d +. () h d h d Q T R d +.τ y d= Parallel groups of ducts in series In this configuration (case C), the total pressure drop across the piston is equal to the the pressure drop along each group of ducts. Mass conservation and an incompressibility assumption require that the sum of the volumetric flow through each group equals the total volumetric flow rate, Q T. Within each group, the sum of the individual duct pressure drops equals the total pressure drop and the volumetric flow through a duct equals the volumetric flow through that duct s group. These conditions give G Q T = Q gd, (5) and g= D g p T = p gd (6) d= Assuming a conservative linear approximation for the er pressure gradient (equation 3), the flow within group g of a set of parallel groups is D g Q g = R D g gd p T.τ y. (7) d= d= h gd The total flow is then: G D g Q T = R D g gd p T.τ y h gd. (8) g= d= d= This equation can be solved for the total effective pressure gradient: D p G g T = G D g D g QT +.τ y g= d= R gd g= d= h gd d= R gd. (9) The total effective pressure gradient due to a purely Newtonian flow is obtained by setting τ y = in the above equation, D p G g NT = Q T. () g= An important flow transient in er devices arises when the electric field is switched off. Nearfield repulsive forces immediately disrupt any fibrated micro-structure (Bonnecaze et al. 99). and d= R gd

8 MULTI-DUCT ER DAMPERS 8 the yield stress is lost in a matter of micro-seconds (Halsey et al. 99). (The zero-to-high voltage transition is longer due to longer fibril aggregation times.) The time required for the Newtonian flow profile to develop from a nearly plugged flow profile is t =. ρh η, () where ρ is the mass density of the er material (approximately. g/cc). (Gavin 996a). Design Four performance parameters are evaluated in order to compare competing er device designs: the maximum device force, F, at low piston velocities, the ratio of maximum to minimum device forces, p /p N, at high piston velocities, the electrical energy required to operate the damper, CE h, and the response time of the device due to fluid inertia, t. To illustrate the analysis outlined above, a device with the following geometry and material properties was investigated. The across flow dimension, h, and electrode thickness, w, were both fixed at.59 mm (/6 inch). The outer radius of the device, R d was fixed at 38. mm (.5 inches), and the length of the electrodes, L, was fixed at 5 mm ( inches). Devices with five ducts and seven ducts were analyzed. An er material with a yield stress, τ y of 3 kpa at 5 kv/mm and a zero-field viscosity of. Pa s was assumed in the initial design. The design variables are the arrangement of the ducts in the device, and the total number of ducts. Five-duct and seven-duct designs were evaluated. A device was determined to be feasible if the dynamic range, p /p N, and if the force levels are above.5 kn. was between and The wide range of device performances, attributed to different hydraulic designs is illustrated in Figure. Each point in this figure corresponds to a different design. The force, F, increases with velocity due to the Newtonian stresses. Therefore F was conservatively evaluated at V p = cm/sec. Likewise, p /p N decreases with velocity. So p /p N was conservatively evaluated at V p = 5cm/sec. The other performance variables are velocity-independent, but depend on geometry, viscosity, and density. The device with the greatest dynamic range corresponds to case P, and the device with the greatest force capacity corresponds to case S. Figure shows that devices with seven ducts provides an outstanding dynamic range but sacrifices the force level. The effect of grouping the ducts enhances the trade-off between dynamic range and force level. Many designs with five ducts in groups have an adequate force level and dynamic range. The three designs selected for fabrication all have five ducts. A design with all ducts in series, a design with all ducts in parallel, and a design with the inner-most three ducts in series and the outer two ducts in parallel were fabricated. The forces and dynamic ranges of these three designs are illustrated with arrows in figure. During the fabrication of the damper components, a new er material (TX-ER8) with a yield stress of kpa and a low plastic viscosity of.35 Pa s was developed. The five-duct designs were re-evaluated using the properties of TX-ER8. The corresponding performance increases are shown with circles in figure. The dramatic increase in the dynamic range is due more to a reduction in the zero-field force than an increase in the force at high voltage.

9 MULTI-DUCT ER DAMPERS 9 dynamic range, p /p N, at 5 cm/sec five ducts seven ducts five ducts with TX-ER8 P maximum damping force, F, at cm/sec (kn) C S Figure. Analytical Prediction of the Dynamic range vs. force for a multi-electrode er damper. While there is a trade-off between dynamic range time and force level, dampers with higher force levels have shorter response times. Because the response time is assumed to vary only with h, the manner in which the ducts are interconnected does not effect the inertial response time. Because all devices in this analysis have the same gap thickness h, the response time for every design is 5.5 milli-seconds. Like the response time, the energy requirement is not dependent on flow rate. Therefore, all the designs require similar amounts of energy. The capacitive stored energy of the five duct devices is only.35 Joules, and the capacitive stored energy of the seven duct devices is. Joules. Assuming that all of the dissipated mechanical energy is transferred to heat that does not dissipate, the temperature increase per cycle of the fluid within the ducts can reach to.5 degrees Celsius per cycle. Fabrication Details Three devices, equivalent in all aspects except for the hydraulic arrangement of the ducts, were fabricated. The overall dimensions and configuration of the devices is shown in Figure 3 and a photograph of the disassembled device (C) is given in figure. Each device has a through-piston construction. Because the piston must pump fluid from one reservoir, through the ducts to the fluid reservoir on the other side of the piston, the total number of ducts for the all-duct-in-series design must be odd. To retain similarity between the designs all designs make use of an odd number of ducts. Since the number of ducts is odd, and because the outer electrode of the outer-most duct is the grounded device case, the innermost electrode is charged to a high voltage potential. Therefore, to prevent electrification of the piston and the piston shaft, the piston is made of a dielectrically strong, non-conducting material. The outer-most tube carries high voltage. The voltage is introduced through an electrically insulated bolt that pierces the outer case. The piston is made of Delron, and Butyl seals were used around the piston rods. The tubes are separated by grooved Delron spacers which maintain uniform.6 mm concentric annular gaps. These grooved spacers are a central feature of this design: they electrically insulate the tubes from

10 MULTI-DUCT ER DAMPERS nylon bolt 8 3 screw Figure 3. Sectional view of a five-electrode er damper. Figure. Disassembled view of a multi-duct er device with tube configuration C. one-another and also allow for electrical contact between the grounded tubes, which all touch the case, and between the high voltage tubes, which are connected to the high voltage supply through the side of the case, but are electrically isolated from the case. Radial holes in these spacers align with radial holes in the tubes which allow for er material to flow between the ducts. The three high-voltage electrodes make contact with an insulating cup that has a conductive inner-surface. This cup gives electrical continuity between the high-voltage electrodes while isolating them from the grounded case. The two ground electrodes extend beyond the assembly, and make electrical contact with the grounded case. The case includes ports for accumulators and pressure sensors. er fluid enters the duct assembly through holes in the inner-most high-voltage electrode. It passes through the assembly and reaches the outer-most annular gap. From here it returns to the piston reservoir through a set of holes in the insulating spacers. For case P, the er fluid is allowed to commute with no resistance across the ducts at both ends of the device. This is accomplished by drilling holes in the tubes. For case S, the er fluid is forced to flow from duct to duct sequentially as it commutes from one side of the piston to the other. This is accomplished by drilling holes in alternating ends of alternating ducts. For case C, subsets of the two simpler designs, described above, are combined in series. Each device holds approximately one liter of er fluid. Prior to filling the devices, all portions of the device in contact with er material were carefully cleaned and thoroughly dried by heating them to 5 deg C for three hours. The devices were then filled with TX-ER8 and sealed rapidly to minimize the water absorption into the er fluid and onto the device surfaces. Air was bled from the device using a vacuum pump.

11 Testing MULTI-DUCT ER DAMPERS Each of the three devices were subjected to sinusoidal and filtered random motions and different levels of high-voltage. In many applications (Gavin ) it is convenient to control the properties of the device in an on-off fashion. For this reason we study the device at two discrete voltage levels: kv and 5 kv. Five kilovolts was the largest potential that could reliably be applied to the devices, which resulted in electric fields of approximately 3. kv/mm. Device forces and displacements were sampled at samples per second. The velocity limit of the loading system is 5 cm/s and the peak velocity during testing was limited to cm/s in order to avoid irregular behavior near the limit of the loading system. Device forces were measured at the device s stationary end to avoid the measurement of the inertial forces of the device and loading frame. The sinusoidal input motion changed slowly in frequency from.5 Hz to 5 Hz. The responses of the devices due to harmonic motions at kv and at 5 kv is compared in figures 6, 7 and 8. Of particular interest is the relative force capacity of the devices, their dynamic ranges, and the effect of visco-elasticity in their dynamic behavior. The force-velocity plots indicate that the device is primarily dissipative, as the force-velocity relationship is almost always bounded to the first and third quadrants. The small oscillations in the data are artifacts of the dynamics of the test fixture, and are not related to properties of the er material. According to the previous analyses, the P configuration should exhibit the greatest dynamic range () but with the smallest force capacity (.5 kn), and that the S configuration should display the greatest capacity (7 kn) but the smallest dynamic range (). The effect of seal and piston friction on the behavior of the devices was not included in these analyses. Despite great efforts to incorporate high-performance, low friction seals, the seal friction is particularly significant in the devices with lower force capacity, and degrades the dynamic range of those devices. When the device axis is horizontal, the device weight is supported by the piston and the piston rod seals, increasing the friction as compared to the device in a vertical configuration As tested, the parallelduct device has a force capacity of kn and a dynamic range of about at cm/sec, whereas the series-duct device has a dynamic range of.5 and a force capacity of 7.7 kn. In addition to the force capacity and the dynamic range, an additional feature of the hysteretic behavior of these devices arises from the presence of pre-yield visco-elasticity. In these devices, the inherent visco-elasticity of the er materials and the bulk modulus of the er fluid contribute to the pre-yield behavior of the devices. The low hydraulic stiffness of the S configuration is due to the long hydraulic column with a small cross sectional area created by the series connection of ducts. On the other hand, the high hydraulic stiffness of the P configuration is due to a shorter hydraulic column with a larger cross sectional area. The effect of the hydraulic stiffness of the device is revealed by the area of the hysteresis loop in the force-velocity plane. The hydraulic stiffness effects are not captured by figure but is nevertheless a property of the duct configuration. Modeling Frequency-Dependent Behavior The design of the multi-electrode er devices was carried out under the assumption of slowly varying flow and the analytical model used in the design was based upon the Bingham constitutive equation, which excludes pre-yield deformation. This approach provides a convenient tool for estimating relationships among the force capacity of the device, the velocity across the terminals of the device, the voltage level of the device, the device geometry and the er material properties.

12 MULTI-DUCT ER DAMPERS The Bingham constitutive model does not, however, capture the dynamic behavior of er materials and devices at low transient strain rates. Dynamic effects, arising from pre-yield visco-elasticity, bulk compressibility, and the built-up effects of the device s stiffness and inertia, are especially prominent when the device velocity changes sign. Pre-yield viscoelasticity of er materials is widely documented with simulation and measurement. Brooks et al. (986) compared complex modulus measurements obtained through shear wave observations to a Maxwell visco-elastic model Experiments conducted by Anaskin et al. (98) on an er shock absorber revealed significant stiffness effects at high electric fields and low vibration amplitudes. Modeling interactions between particles in the er suspension using a point-dipole approximation, and Stokes drag on each particle, McLeish et al. (99) simulated the low level oscillatory behavior of an er suspension and found that the visco-elasticity exhibited favors a Kelvin model when the fibrils are broken and a Voigt model when the fibers are intact. Further experiments by Otsubo et al. (99) demonstrate a strain-dependent complex modulus that is frequency independent at strains above.5. A mechanical analog proposed by Gamota and Filisko (99) captures many of these observed behaviors. Their model consists of a Zener viscoelastic element in series with a Bingham element. Detailed micro-structural simulations of Parthasarathy and Klingenberg (995) demonstrate Kelvin viscoelastic behavior at small strains. As in the experiments of Otsobo, the modulus loses frequency dependence at larger strains (Parthasarathy and Klingenberg 999). As the material starts to flow, its elasticity decreases rapidly (Klingenberg 993). Because of the complexity of the micro-structural behavior of er and mr materials and of built-up device effects, phenomenological descriptions are typically adopted in modeling the behavior of er devices. These models range from expansions of a Bingham-plastic model (Stanway, Sproston, and Stevens, 987), one-dimensional mechanical analogs for visco-elasticity, yielding, and viscosity (Gamota and Filisko, 99),(Kamath and Wereley, 997a,b), and ( Makris et al. 996a,b). The model proposed by Makris is developed from a constitutive material model which incorporates pre-yield elasticity and visco-plasticity. A model which utilizes the Bouc-Wen equation for the yielding component and linear elastic, or viscous elements for the built-up device effects was proposed by Spencer et al. (997). A model proposed by Powell (995) includes inertial effects by lumping mass in the model. Onada, et al. (997) places a spring in series with a parallel combination of a dash-pot, a spring, and a Coulomb friction element to model an er damper for space-truss vibration suppression. A recent mechanical analog proposed by Sims et al. () incorporates a series combination of a spring, a mass, and a nonlinear viscous element. Figure 5 illustrates the mechanical analog for the behavior of the multi-electrode er device. This model is a simplified version of the system proposed by Gamota et al. (99) It consists of a pair of Voigt viscoelastic elements connected by an inertial element that also resists motion through Coulomb friction. The spring and dash-pot k and c model pre-yield viscoelastic behavior, whereas components k and c model post-yield viscoelasticity. The inertia of the device and the fluid are lumped into the mass m, and the yield force f is the parameter most strongly affected by the device voltage. The plastic deformation x and its rate ẋ uniquely describe the state of the system. A minimal state-variable realization for this system is d dt [ x ẋ ] = [ ] [ ] x (k + k )/m (c + c )/m ẋ [ ] [ ] x + k /m c /m ẋ [ ] + f /m tanh(ẋ /V ref ). ()

13 k c. f (x ) MULTI-DUCT ER DAMPERS 3 m k c ^ f(t) x x Figure 5. Mechanical analog for the er device. ˆf = [ k c ] [ x ẋ ] [ + k c ] [ x ẋ ] (3) The displacement and velocity across the device, x = x + x, are the inputs to this system. The yielding mechanism is approximated by f tanh(ẋ /V ref ), where f is the value of the yield force and V ref is a reference velocity which governs the sharpness of the yield function. The hyperbolic tangent is the the only non-linearity in the model, and is explicitly separated from the dynamics of the system. Note that the pre-yield viscoelastic portion of this model retains elasticity in the zero frequency limit. At low strains it has a frequency independent storage modulus, and a loss modulus which increases monotonically with frequency. The dynamics matrix of this state variable representation are always asymptotically stable for positive-valued parameters. Furthermore, the dynamics are under-damped if c + c < (k + k ). () m The equations are well conditioned and can be solved using standard numerical techniques (Radhak et al. 993, Hindmarsh et al. 983). A second, algebraic model, also uses the hyperbolic tangent function to model the yielding effects. ( x f(x, ẋ, V ) = f (V ) tanh + ẋ ) + k x + c ẋ. (5) d v In this expression, the voltage-dependent yield force level f (V ), the post-yield stiffness, k, and the plastic viscosity, c have the same interpretation as in the dynamic model. The details of the behavior in the pre-yield region are captured by the parameters d and v in the argument of the hyperbolic tangent. As this model has no dynamic states, it misses some of the details of frequency dependent visco-elastic behavior of the device. Nonetheless, it provides a closed form solution for the device force. Furthermore, an inverse model for the behavior of the device, which is useful for feedback linearization, is easily derived. Assuming that the yield force follows a power-law relationship with the voltage f (V ) = αv n, and given the velocity and displacement across the device, the voltage V required to produce the desired damper force f is [ V (x, ẋ, f) = m f k x c ẋ α tanh(x/d + ẋ/v ) ] n (6)

14 MULTI-DUCT ER DAMPERS displacement (cm) velocity (cm/sec) displacement (cm) time (sec) velocity (cm/sec) time (sec) Figure 6. Case P, frequency-dependent behavior at and 3.kV/mm and.6 and 3.3 Hz. points: data, solid line: equation () dashed line: equation (5).

15 MULTI-DUCT ER DAMPERS displacement (cm) velocity (cm/sec) time (sec) displacement (cm) velocity (cm/sec) time (sec) Figure 7. Case S, frequency-dependent behavior at and 3.kV/mm and.6 and 3.3 Hz. points: data, solid line: equation () dashed line: equation (5).

16 MULTI-DUCT ER DAMPERS displacement (cm) velocity (cm/sec) displacement (cm) time (sec) velocity (cm/sec) time (sec) Figure 8. Case C, frequency-dependent behavior at and 3.kV/mm and.6 and 3.3 Hz. points: data, solid line: equation () dashed line: equation (5).

17 MULTI-DUCT ER DAMPERS 7 Table. Values of the identified parameters: 5 kv / kv Evolutionary Model Algebraic Model Series Parallel Mixed Series Parallel Mixed Units k.8/..5/../..3/../.8.9/. kn/cm c.39/../..767/ /.5./..3/.8 kn/cm/s f 6./..8/.9 3.6/. 6./.3.8/. 3.6/. kn m./../.3.8/ kn/cm/s k 6.8/3..76/.583.8/ kn/cm c.683/.5.3/.8.6/ kn/cm/s V r.9/..6/ cm/s d /. 7./6. 5.6/5. cm v /3.3.3/..53/. cm/s J For each of the two models, parameters were estimated which minimized the sum of the squared errors between the model and the data. [ P Pp= (f(t p ) ˆf(t p )) ] / J = [ P Pp= (f(t p )) ] / (7) The resulting parameters and the corresponding objective function value are given in table. The last row of the table gives the relative root mean squared error between the measured forces, f(t p ), and the modeled forces, ˆf(tp ), over data points p =... P. In figures 6, 7, and 8 the dynamic model is shown with solid lines, the algebraic model is shown with dashed lines, and the data is shown with points. For the dynamic model, a single set of model parameters fit the behavior of the device equally well at low frequencies and high frequencies. In fitting the algebraic model the errors at large displacements were weighted more heavily than the errors at small displacements. This weighting results in parameters that fit the low frequency, large displacement behavior well, but sacrifices fidelity at higher frequencies and smaller displacements. The effect of this is particularly evident in high frequency force-velocity plots of figure 7. The parameters which are least sensitive to the electric field are c and m. This is consistent with the Bingham model for er behavior, in which the viscosity of the dispersant is fieldindependent. The yield force is the most field-sensitive parameter, increasing by more than an order of magnitude for the P configuration. To test the response time of the device to transient voltages, the voltage was periodically switched while wide-band random motions were applied to the device. Experiments were conducted at kv, 5 kv, and with the voltage switching between and 5 kv periodically. To attenuate the high voltage transients in this experiment, Watt carbon composition resistors are placed in series with the device and power supply. This added resistance effectively lengthens the RC time constant of the system to about 3 milliseconds. The device voltage was measured along with the device s force and displacement and was used to simulate the device behavior with the algebraic device model. Figure 9 illustrates the hysteretic response, the voltage time history, and the force time histories corresponding to the 5 kv constant case and the switching case. The device achieves the constant-on force level within 3 milliseconds.

18 MULTI-DUCT ER DAMPERS (a) displacement (cm) 8 6 (b) velocity (cm/sec) time (sec) (c) voltage (kv) (d) time (sec) Figure 9. Response to random device motions and switching voltage, points: data, lines: equation 5. (a) force-displacement hysteresis, (b) force-velocity hysteresis, (c) force time history, (d) voltage time history.

19 Summary MULTI-DUCT ER DAMPERS 9 A design method for controllable damping devices featuring er materials was experimentally verified with three different embodiments of er devices with force capacities from kn to 8 kn. A set of annular ducts, created with concentric metallic tubes which also serve as electrodes, provide both large surface areas over which the er material flows at low rates and large volumes of er material subjected to the high electric field (3 kv/mm). Low flow rates enable large ranges of controllable forces (more than a factor of three, including seal friction, and at 5 cm/sec), and large duct surface areas enable high force levels (up to 8 kn in these dampers). The design method incorporates a simplified expression for the Poiseuille flow of er materials, and allows for convenient closed form expressions for the voltage-dependent force-velocity relationship for these devices. The dynamic behavior of these devices was modeled using an evolutionary model and an algebraic model. Three er dampers designed with this method were built and tested. The device parts were throughly dried by heating prior to filling and assembly. Dynamic measurements of the device forces from.5 to 5 Hertz illustrate the frequency dependence of the device behavior. This frequency dependence can be attributed to the hydraulic stiffness of the device and is more pronounced in the high-force devices, which had the lowest hydraulic stiffness. An evolutionary and an algebraic model were fit to the measured force, displacement and velocity records. The evolutionary model captures the frequency dependence well, whereas the frequency-independent algebraic model is not able to capture some details of the pre-yield behavior for the high-force device. Nonetheless, when the high-force device was tested with band-limited random motions and switching electric fields, the algebraic model provides a good approximation for the behavior of the high-force device. Friction in the piston rod seals increased the over-all force level of each device by about N, but decreased the range of available forces. This was most pronounced in the low-force device which would have had the greatest dynamic range (almost two orders of magnitude) if the seals had been perfectly frictionless. This study can be extended by including the seal friction in the analysis in order to improve the predictive capabilities of the design method described in this paper. Acknowledgements This material is based on work supported by the National Science Foundation under Award No s CMS-9699, CMS-9993, and by the Oak Ridge Associated Universities. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsors. We are indebted to Dr. R. Aizawa and Y. Funae of Nippon Shokkubai Co. Ltd., for supplying the TX-ER8 material used in this study. References Anaskin, I.F., V.K. Gleb, E.V. Korobko, B.P. Khizhinskii, and B.M. Khusid, (98). Effect of External Field On Amplitude-Frequency Characteristics of Electrorheological Damper, Journal of Engineering Physics, Vol. 6, pp Block H., and J.P. Kelly, (988). Review Article: Electro-rheology, J. Physics D: Applied Physics, Vol., pp Block, H., J.P. Kelly, A. Qin, and T. Watson, (99). Materials and Mechanisms in Electrorheology, Langmuir, Vol. 6, pp. 6.

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