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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2012, Article ID 826497, 11 pages doi:10.1155/2012/826497 Research Article 1 2 3 J. Christopherson, M. Mahinfalah, and Reza N. Jazar MTS Systems Corporation, 14000 Technology Drive, Eden Prairie, MN 55344-2290, USA Department of Mechanical Engineering, Milwaukee School of Engineering, Milwaukee, WI 53202, USA School of Aerospace, Mechanical, and Manufacturing Engineering, RMIT University, Melbourne, VIC 8083, Australia Correspondence should be addressed to M. Mahinfalah, email@example.com Received 28 June 2011; Accepted 14 September 2011 Academic Editor: Mohammad Tawﬁk Copyright © 2012 J. Christopherson et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Because of the density mismatch between the decoupler and surrounding ﬂuid, the decoupler of all hydraulic engine mounts (HEM) might ﬂoat, sink, or stick to the cage bounds, assuming static conditions. The problem appears in the transient response of a bottomed-up ﬂoating decoupler hydraulic engine mount. To overcome the bottomed-up problem, a suspended decoupler design for improved decoupler control is introduced. The new design does not noticeably aﬀect the mechanism’s steady-state behavior, but improves start-up and transient response. Additionally, the decoupler mechanism is incorporated into a smaller, lighter, yet more tunable and hence more eﬀective hydraulic mount design. The steady-state response of a dimensionless model of the mount is examined utilizing the averaging perturbation method applied to a set of second-order nonlinear ordinary diﬀerential equations. It is shown that the frequency responses of the ﬂoating and suspended decoupled designs are similar and functional. To have a more realistic modeling, utilizing nonlinear ﬁnite elements in conjunction with a lumped parameter modeling approach, we evaluate the nonlinear resorting characteristics of the components and implement them in the equations of motion. 1. Introduction and Statement of Problem allowed at low excitation frequencies, and low damping is allowed at increased excitation frequencies. Modern vehicles illustrate a trend toward lighter, higher per- Because there is a need for a vibration isolator that formance, aluminum-based engines thereby increasing the can exhibit a dual damping ratio that is dependent upon potential for vibration. The engine is the largest concentrated frequency the hydraulic engine mount was introduced. The mass in a vehicle and causes vibration if it is not properly hydraulic engine mount is a device that approximately provides the desired damping characteristics via the imple- isolated and constrained. The trend for many years to isolate vibrations was to simply connect the engine and frame by mentation of a mechanical switching mechanism known means of an engine mount made of elastomeric materials as the decoupler in conjunction with a narrow, highly such as rubber [1–3]. Modeling the rubber isolator by a linear restrictive ﬂuid path known as the inertia track [1, 2, system and considering a base excited single-degree-of- 5–9]. These two mechanisms act together, assuming an freedom system, we know in the frequency response curves of appropriately designed system, to provide a passive means of the acceleration transmitted to the isolated mass there exists variable damping dependent upon excitation characteristics 1/2 a crossing point at a frequency ratio value of ω/ω = 2 . More speciﬁcally, when a large pressure diﬀerential is in which all the curves representing systems with diﬀering imparted to the ﬂuid chambers, by means of a substantial damping ratios converge . This is a switching point for outside perturbation, the decoupler will bottom out in its systems where system behavior reverses dependent upon cage bounds and cause the pressure diﬀerential within the mount to be equalized via the inertia track. Due to the excitation frequency. This paradoxical behavior indicates that for optimum isolation of a structure from acceleration, inertia tracks dimensions it provides an increased damping hence force, a mount is needed in which high damping is coeﬃcient to the engine mount. However, when the external 2 Advances in Acoustics and Vibration The decoupler and its housing are shown in Figure 1(b). Large amplitude, low frequency excitations impart a signiﬁ- cant enough ﬂuid motion that the decoupler plate is forced to bottom out on its surrounding cage thereby forcing the ﬂuid Rubber Decoupler to ﬂow through the inertia track into the compliant lower Upper chamber chamber. The inertia track is a long, small-diameter tube that runs circumferentially around the engine mount providing a very restrictive ﬂow path between upper and lower chambers. Lower Due to the restrictive nature of the inertia track, an increased Inertia track chamber viscous damping coeﬃcient is realized for the system. This increased damping acts to reduce the acceleration transmissibility of the mount at low excitation frequencies. However, at increased excitation frequencies, the decoupler Bottom rubber plate does not bottom out on the cage bounds. Instead it compliance moves back and forth freely providing a relatively low ﬂow (a) restriction. Because the decoupler provides a low restriction to ﬂuid ﬂow, it becomes the preferred ﬂow path, and acts on Cage Plate reducing the damping coeﬃcient of the engine mount. This system works quite well and is in place on the large majority of automotive applications to date. It is analyzed, and modeled by researchers since 1980 from diﬀerent viewpoints. Adiguna and coworkers determined dynamic behavior of HEMs in time domain  and frequency domain , utilizing linear and nonlinear lumped models . The nonlinear function of the decoupler is successfully modeled, examined, and applied by Golnaraghi and Jazar [13, 14] utilizing a third-degree equation to describe a (b) nonlinear damping. Adapting their model, Christopherson Figure 1: (a) Typical hydraulic engine mount. (b) Typical decou- and Jazar [15, 16] optimized a sprung mass suspended by an pler mechanism. HEM and provided a design method. In the present investigation we study two common assumptions and explore their eﬀects in modeling and Decoupler Inertia track dynamics of hydraulic engine mounts. First assumption is that in lumped model of the system, the nonlinearities involved in elastomechanical parts are usually ignored and a linear behavior is assumed. Second assumption is that either in transient or steady-state responses it is assumed that the Decoupler support points decoupler is settled down in its neutral position exactly in the middle of the gap of decoupler duct. Therefore, two Figure 2: New decoupler mechanism. questions arise as to what are the eﬀects of nonlinearities involved in elastomechanical parts, and what happens on initial start up if the decoupler is bottomed up. perturbation is low in intensity, or at increased frequency, This investigation will utilize ﬁnite element analysis to the decoupler does not bottom out, and hence the inertia determine the mechanical behavior of the components, and track is eﬀectively short-circuited; therefore, due to the will employ perturbation analysis to determine transient and decoupler’s large dimensions, the system provides a low steady-state behaviors of the mount. damping coeﬃcient. Because of the density mismatch between the decoupler Figure 1(a) illustrates a schematic of a typical ﬂoating- and surrounding ﬂuid, the decoupler will ﬂoat, sink, twist, decoupler hydraulic engine mount (HEM) [9, 10]. The or stick to the cage bounds, assuming static conditions. The engine mount is named as such because the decoupler problem is what happens if the decoupler is in a nonop- “ﬂoats” freely inside of its housing. The basic premise of timum location for a given random or initial excitation to operation of the HEM is relatively straightforward. The provide either low damping by being open or closed to allow engine is supported by a rubber structure acting as the main for high damping. load carrying component, and a means by which to induce We introduce a supported decoupler mechanism illus- trated in Figure 2. By supporting the decoupler it is ensured ﬂuid motion within the engine mount . The ﬂuid motion induced in the engine mount, due to external excitation, is to be in a neutral location upon startup. However, the then forced through a system of passageways of inertia track trick to designing such a mechanism is to ensure that the nature of the support does not inﬂuence the previously and decoupler. The preferred pathway is dependent upon the nature of the excitation. mentioned steady-state operation of the mechanism, while Advances in Acoustics and Vibration 3 surpassing existing hydraulic engine mount benchmarks for (3) (5) performance. (4) Figure 3 illustrates a schematic of the proposed design intended to meet the aforementioned criteria. The mount utilizes the same decoupler mechanism (1) as illustrated in Figure 2. In addition, the mount does away with the traditional upper rubber structure common to practically every modern hydraulic engine mount. Instead the proposed mount makes use of a Belleville spring (2) to provide the primary axial stiﬀness and a thick circumferential rubber band (3) surrounding the upper structure of the mount to limit transverse motions of the mount. The volumetric compliance of the upper chamber of the mount is provided (2) through a relatively thick rubber chamber (4) which is (1) mechanically fastened to the upper moving head of the engine mount (5). The advantage of such a structure over Figure 3: Proposed hydraulic engine mount design. the traditional rubber structure is twofold. First, the stiﬀness of the engine mount is more tunable and is as simple as appropriate spring sizing as compared to complicated geo- still maintaining the advantage of the supported decoupler metrical designs required for the current rubber structure. in the initial transient response. Here the decoupler disk is Second, the damping of the system can be allowed to rest supported and forced to be in neutral position equidistance with the ﬂuid motion inside of the mount thereby allowing between either cage limit; however, the decoupler is not more precise tuning by means of inertia track and decoupler ﬁxed from motion. This design requires the decoupler to geometry . Such a method is far simpler than trying to be made of an elastomeric material to provide suﬃcient design an upper rubber structure with a speciﬁed amount of stiﬀness to keep the decoupler located during nonexcited hysteretic-type damping. (static) situations, and provide suﬃcient ﬂexibility to allow Figure 4 illustrates the three-dimensional representation normal operation during dynamic events. of the design. Here the decoupler geometry becomes clearer Up to the present, very few researchers have looked at in conjunction with the design of the upper structure. the start-up or transient behavior of the hydraulic mount Figure 5 provides a better illustration of the decoupler geom- with Adiguna et al. being among the few . However, etry required to achieve the aforementioned requirements. the behavior of the mount for a bottomed-up decoupler has As illustrated in Figure 5 the decoupler support tabs are never been investigated, although after a short period of time thinned regions with slots on either side to help them to not the decoupler recovers its purposed function and does what dramatically inﬂuence overall decoupler dynamics while still it is designed for. However, in the current ﬂoating-decoupler maintaining the required stiﬀness to ensure proper decoupler design the decoupler might simply sit against one of the position during static conditions. cage bounds while not excited, depending upon mounting conﬁgurations and density mismatch with the surrounding 3. Dynamic Parameters Evaluations ﬂuid, therefore causing the system to initially utilize only the inertia track. Therefore, after every excitation removal In every HEM there are two rubber-type components in the decoupler might sink or ﬂoat to create the problem upper and lower chambers to collect the moving ﬂuid. These again. With either condition it becomes apparent, after some rubbery components produce compliances of the system consideration, that because it is the decoupler that allows the which appear in the equations of motion. Besides the two mount to act as either a low damping or a high damping chambers, the suspended decoupler also show an elastic mechanism by means of its position, then the position of behavior. Utilizing FEM we show how to determine the the decoupler during the aforementioned excitations is quite elastic behavior of the decoupler, upper bellow, and lower important. collector compliances. To begin the analysis of the engine mount it is paramount that the necessary geometric and material parameters are 2. Suspended Decoupler HEM identiﬁed. To accomplish such, ﬁnite element analysis is Model Description utilized as a tool to provide knowledge of component load- Floating decoupler HEM is described in the literature very deﬂection relationships, volumetric expansion properties, well [5–16]. To be compared with ﬂoat type, here we describe and so forth. By creating a ﬁnite element model based upon a suspended decoupler HEM. Noting the disadvantages of the geometry illustrated in Figure 5, information regarding the ﬂoating decoupler compared to suspended decoupler the load-deﬂection behavior of the decoupler mechanism is mount, it seems advantageous to design a new mount readily obtainable. Figure 6 illustrates the discretized ﬁnite utilizing such a suspended decoupler mechanism. Such element model. The model was discretized using 10 node a mount should provide eﬀective isolation characteristics tetrahedralelementswithatotalof42,794 active degreesof through a broad frequency spectrum while maintaining or freedom for the model. 4 Advances in Acoustics and Vibration Figure 4: Model of the proposed hydraulic mount. (a) Decoupler plate (b) Housing of decoupler plate Figure 5: Decoupler geometry. surfaces (see Figure 7). The contact region at the decoupler support points was simulated using a rough-style interface between the two materials thereby allowing no slippage . Whereas the surfaces contacting after suﬃcient decoupler deformation were treated as frictionless thereby allowing relative motion between the two bodies. To simplify the analysis and determine the eﬀectiveness of the new design Figure 6: Discretized ﬁnite element model. compared to ﬂoating decoupler design, we ignore the ﬂuid- solid interaction as is done in modeling HEM [8–16]. Because the decoupler is to be made of an elastomeric To simulate the impact condition between the decou- material the three-parameter Mooney-Rivlin model, illus- pler and the surrounding cage bounds Lagrangian type- trated in (1)isutilized[18–20]. The three-parameter Moon- contact elements were imposed upon potential impacting ey-Rivlin model expresses the strain energy density as Advances in Acoustics and Vibration 5 Contact surfaces (a) Housing (b) Decoupler plate Figure 7: Contact element surfaces (not in scale). Table 1: Mooney-Rivlin constants. Parameter Value (MPa) c 4.838E − 01 c −9.456E − 02 c 1.235E − 02 a function of the material constants (c , c ,and c ), and 10 01 11 the ﬁrst two invariants (I and I ) of the Right Cauchy- 1 2 −50 Green deformation tensor [19–21]. The material constants that we adapted are shown in Table 1, and may be obtained from experiment by means of a least-squares curve-ﬁtting procedure [15, 22, 23], −100 −1 −0.50 0.51 −3 W = c (I − 3) + c (I − 3) + c (I − 3)(I − 3). (1) ×10 10 1 01 2 11 1 2 Deﬂection (m) Data To solve the ﬁnite element model an applied numerical Polynomial ﬁt solution method must be employed. Such an approach was required due primarily to two factors. First, the material Figure 8: Decoupler load-deﬂection relationship. for the decoupler is nonlinear and requires full geometric nonlinearity options to be utilized. Second, the contact Table 2: Material properties. between the rubber and metallic cage bounds is asymmetric noting the diﬀerences in material responses between the Component Young’s modulus (GPa) Poisson’s ratio two structures; therefore, the full Newton-Raphson approach Spring 207 0.30 must be employed to deal with the unsymmetrical nature of Upper structure 71 0.33 the assembled matrices . Spring support 71 0.33 Theﬂuidisassumed to be incompressiblecomparedto elastic and ﬂexible parts. To simulate ﬂuid-induced pressure an evenly distributed pressure of 20 kPa was assumed to a third-order polynomial, expressed in (2), approximates the one side of the entire exposed surface of the decoupler. data with reasonably good accuracy with E = 8.0246 N, To constrain the entire assembly from motion the lower surface of the cage was ﬁxed in all degrees of freedom. In order to obtain information regarding the load-deﬂection f (x ,Δ) = E . (2) d d 1 relationship of the supported decoupler the applied pressure was resolved into a force component by multiplying the area Consider the upper structure of the engine mount upon which the pressure was applied. The corresponding with material properties shown in Table 2. The structure is deﬂection measurement was taken in the vertical direction considered as a whole because of the nonlinearity inherent from the center node (exposed due to symmetry conditions) in the load-deﬂection relationship of the spring, but also of the decoupler disk. The results of the ﬁnite element the material nonlinearity of the rubber components. Because analysis are illustrated in Figure 8. of the nonlinearity, the principle of superposition is not Notice from Figure 8 that even after the decoupler applicable; therefore, the stiﬀness of the upper structure will impacts the cage bounds the disk, it continues to displace be modeled by one nonlinear spring element (as compared with a corresponding increase in applied load due to the elas- to multiple springs in parallel). Figure 9 illustrates the tic nature of the decoupler material. In addition, notice that model geometry and corresponding ﬁnite element mesh Force (N) 6 Advances in Acoustics and Vibration Applied axial displacement Fixed constraints Symmetry conditions Figure 9: Upper structure model and meshed geometry. Next, consider the upper bellows and its corresponding volumetric compliance. The corresponding ﬁnite element model is illustrated in Figure 11. The mesh consisted of 8 node quadrilateral-type elements with 768 total degrees of freedom. The analysis allowed for ﬁnite strains to account for the hyperelastic behavior of the rubber upper compliance, and therefore required solution by the Newton- Raphson approach. The ﬁnite element model illustrated in Figure 11 was constrained from motion on the bottom and top surfaces while an evenly distributed pressure was applied on the internal surface of the upper compliance to simulate ﬂuid pressure. Figure 12 illustrates the volume-pressure relationship for Deﬂection (mm) the upper bellows structure. Note the relative linearity of the relationship; therefore, by using a least-squares ﬁt of a linear Figure 10: Load-deﬂection relationship. line to the data results in a line with a slope of 2.457E − 09 m /N, which corresponds to the volumetric compliance of the upper bellows structure. Determination of the lower chamber volumetric compli- which consisted of 20 node hexahedral elements and 10 ance is accomplished much the same as for the upper cham- node tetrahedral elements with a total of 71,211 degrees ber. Figure 13 illustrates the ﬁnite element model for the of freedom. Additionally, contact surfaces were speciﬁed lower chamber. However, to model the rubber compliance everywhere metallic components are in contact or were shell elements were utilized noting the constant thickness of to contact. Bonded-type contact surfaces were speciﬁed the part, and the large deformations this structure is intended everywhere that elastomeric materials were in contact with to undergo. In addition, due to the large deformations metallic components as the design intent was to have said expected the rubber compliance has the tendency to buckle metallic components bonded to the rubber parts as a part of outwards. This buckling is diﬃcult to model using solid the manufacturing process. hexahedral elements noting such deformations can result The ﬁnite element model was constrained on the lower in unacceptable element shapes and potentially inaccurate surface with load being applied in the form of a speciﬁed solutions; therefore, 4 node shell elements were employed displacement in the axial direction on the opposing surface. as such deformations do not necessarily cause such element In addition, ﬁxed constraints were applied to the lower and shape problems [24, 25]. Figure 14 illustrates the results of outer surfaces of the surrounding rubber component as the analysis of the model illustrated in Figure 13. illustrated in Figure 9. Notice the behavior of the lower compliance illustrated To solve the ﬁnite element model a nonlinear simulation in Figure 14 is also nonlinear; however, it appears approxi- was utilized allowing for ﬁnite strains. Figure 10 illustrates mately bilinear. After closer investigation the initial portion the resulting load-deﬂection relationship obtained from the of the volume-pressure curve represents the chambers initial ﬁnite element analysis. Because of nonlinearity, a third- expansion until it contacts the surrounding structural walls. degree polynomial is ﬁt to the data. Equation (3) is the result At the point where contact between the two bodies initiates of a least-squares curve ﬁt to the ﬁnite element results. In this the slope of the volume-pressure curve drastically changes equation, the input deﬂection x has units of mm, indicating a less compliant structure. It is the slope of this segment of line that is used to approximate the volumetric 3 2 F = 5.2605x − 63.907x + 526.73x. (3) compliance of the lower structure noting the small amount Load (N) Advances in Acoustics and Vibration 7 Fix Fixeed d cco onst nstrraints aints Ap pplied plied p prressur essuree Figure 11: Upper compliance model and meshed geometry. 3.05E−03 cause such an operating point shift. Table 3 introduces the complete compliment of hydraulic engine mount parameters 3E−03 . 2.95E−03 2.9E−03 4. Mathematical Analysis 2.85E−03 By introducing the support to the decoupler the momentum 2.8E−03 balance equation for the decoupler exhibits a restoring 2.75E−03 force term. Additionally, the nonlinear damping term ﬁrst 0 102030405060708090 100 introduced by Golnaraghi and Jazar is utilized [13, 14]. Pressure (kPa) However, to fully describe system dynamics, the inertia Figure 12: Volume-pressure relationship (upper compliance). track momentum equation is needed along with the ﬂuid continuity equations [10–14, 26, 27]. In this commonly accepted modeling, the ﬂuid-solid interaction is ignored, Table 3: Hydraulic mount parameters. Property Value Unit d M x¨ + B + E x˙ + f (x ,Δ) = A (P − P ),(4) d d d d d d d 1 2 A 8.107E − 05 m A 5.026E − 06 m M x¨ + B x˙ = A (P − P ),(5) i i i i i 1 2 A 3.663E − 03 m B 1.031E − 04 Ns/m ˙ ˙ A x˙ − y˙ = A x˙ + A x˙ + C P − P ,(6) p i i d d 1 1 atm B 3.257 Ns/m ˙ ˙ A x˙ + A x˙ = C P − P . (7) B 0.50E + 02 Ns/m d d i i 2 2 atm C 2.457E − 09 m /N 5 Equations (4)and (5) are momentum balance of the ﬂuid C 2.674E − 09 m /N mass in decoupler canal and inertia track, while the (5)and k 5.2605 N/mm (6) are continuity equations for upper and lower chambers, k −32.344 N/mm respectively. Utilizing (4) through (7) results in the following k 334.22 N/mm equations of motion which describe the internal dynamics of M 5.220E − 04 kg the hydraulic mount: M 1.354E − 03 kg E 0.50 — Mq ¨ + Cq˙ + Kq + f = f,(8) E 8.0246 N where, Δ 0.5 mm Y 1.0 mm ⎡ ⎤ ⎡ ⎤ M 0 d ⎢ B + E 0⎥ ⎣ ⎦ M = , C = ⎣ ⎦ , 0 M 0 B of ﬂuid pressure required to move the system operating point ⎡ ⎤ into this region. Such an assumption in regards to the system A KA A K d i operating point being located in said region can be validated ⎣ ⎦ K = , by noting that the static load of the engine is suﬃcient to A AKA K d i Volume (m ) 8 Advances in Acoustics and Vibration Figure 13: Lower compliance model and meshed geometry. 3.5E−04 Introducing the small parameter ε as a measure of the nonlinearity, the following nondimensional parameters are 3E−04 instituted: 2.5E−04 ε = a, εd = ζ , εd = ζ , d d i i 2E−04 (14) εq = e, εg = fY , ν = . 1.5E−04 0 5 101520253035 40 Pressure (kPa) Using the parameters in (14) the equations of motion from (11)and (12) are now expressed: Figure 14: Volume-pressure relationship (lower compliance). y + ε d + qy y + g + y + εy = εg sin(wτ), d d d i d d d ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ (15) ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ 1 A x A X d d f = f (x ,Δ) , f = sin(ωt), q = , y + εd y + νy + ενy = ενg sin(wτ). d d d i d i i i ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ 0 1 A x i i To obtain a solution in the frequency domain for (15) the −1 −1 K = C + C . 1 2 averaging method is employed by introducing an assumed (9) solutions in the following form [28, 29]: In order to make the analysis general, the following nondi- y = r (τ) sin τ + φ (τ) , mensional parameters are introduced: d d d (16) y = r (τ) sin τ + φ (τ) , (17) A K i i i 2 d τ = Ωt, Ω = , x = Δy , d d (10) y = r (τ) cos τ + φ (τ) , (18) d d x = Δy , x = Δy. i i ( ) ( ) y = r τ cos τ + φ τ . (19) i i Using the parameters in (10), (8) is now expressed in the following nondimensional forms: Expressing the ﬁrst derivatives as in (18)and (19) requires two constraint equations to maintain the validity of ( ) y + ζ + ey y + g y + y + ay = fY sin wτ , the solution, d d d d i d d d (11) r (τ) sin τ + φ (τ) + r φ (τ) cos τ + φ (τ) = 0, d d d d d a a fa (20) (12) y + ζ y + y + y = Y sin(wτ), i d i i i r (τ) sin τ + φ (τ) + r φ (τ) cos τ + φ (τ) = 0. i i i i i m m m where, Now the second-time derivatives can be obtained directly from (18)and (19): ω A M i i w = , a = , m = , f = , Ω A M C KA d d 1 d y = r (τ) cos τ + φ (τ) − r (τ) 1+ φ (τ) d d d d d E B B d i e = , ζ = , ζ = , d i M Ω M Ω M Ω d d i × sin τ + φ (τ) , (21) f (x ,Δ) d d y = r (τ) cos τ + φ (τ) − r (τ) 1+ φ (τ) i i i i i g y = . d d ΔΩ M ( ) (13) × sin τ + φ τ . Volume (m ) Advances in Acoustics and Vibration 9 Equations (16) through (19)and (21) are now substituted 10 directly into the equations of motion and utilized in con- junction with (20) to transform the second-order diﬀerential equations in (15) into a system of four ﬁrst-order diﬀerential equations. After extracting the slow terms of the resulting ﬁrst-order diﬀerential equations, averaging over one period of oscillation, the equations of motion in terms of the ﬁrst- order diﬀerential equations are obtained: r =− g sin wτ − τ − φ + d r , (22) d d d d ε qr 3 d r φ = sr − g cos wτ − τ − φ + νr + , (23) d d d d d 2 4 r =− d r + νg sin wτ − τ − φ , (24) 0 30 60 90 120 150 i i i εν Suspended decoupler r φ = g cos wτ − τ − φ + r , (25) i i i Free decoupler where, Figure 15: Decoupler frequency response. s = . (26) A KΔ In order for equations (22) through (25) to be useable in 100 a frequency domain consider the following transformation to allow conversion of (22) through (25) to an autonomous system of equations: wτ − τ − φ = ψ , d d (27) wτ − τ − φ = ψ . i i Utilizing (27) in equations (22) through (25) and noting that for steady-state conditions to prevail the time derivatives must vanish results in implicit frequency response functions for the system, ⎛ ⎞ 2 2 2 r 4εsr +8+4εν + εqr − 8w d r d d d d ⎝ ⎠ + = 1, 0 0.6 1.2 1.8 2.4 3 g 4εg 2 2 Suspended decoupler d r r (εν − 2w +2) i i i + = 1. Free decoupler νg ενg (28) Figure 16: Inertia track frequency response. Equations (28) are identical to the frequency response functions obtained in [13, 16] for a ﬂoating decoupler mount if s is allowed to equal zero thereby validating the solution noting that the only diﬀerence mathematically between the Figure 16 illustrates the inertia track frequency response two systems is the sy term. function for the mount obtained from the averaging solution above in conjunction with the solution from  indicating no appreciable diﬀerence or eﬀect on its behavior due to the 5. Dynamic Responses decoupler modiﬁcation. Figure 15 illustrates the frequency response function for both Noting that the supported decoupler design is based on the supported decoupler introduced in this investigation the initial transient response of the system, consider the and the unsupported decoupler model from . It is seen force transmitted through the engine mount due to a 1 mm there is no discernible diﬀerence between the two models pulse input held for a period of 0.1 seconds. To calculate the indicating that by supporting the decoupler disk the overall force transmitted through the engine mount, consider the function of the mechanism was not substantially aﬀected in following equation developedin and illustrated here to its steady-state response. describe the response to a step input. The transmitted force r d i 10 Advances in Acoustics and Vibration This information was then readily utilized by the lumped parameter modeling approach utilized by practically all researchers investigating hydraulic engine mounts. Using the lumped parameter model, the frequency response of the system was investigated utilizing averaging method and compared to previously published results describing ﬂoating-decoupler-type mounts with excellent agreement. The agreement between the two models indicated that by supporting the decoupler on thin, low-stiﬀness tabs, the overall steady-state response of the system is practically unaﬀected. Additionally, by using numerical analysis to −500 determine the transient response of the system, the sup- ported decoupler substantially improves the engine mounts’ response to sudden excitations. Future work must be about −1000 optimizing the supported decoupler design illustrated in this 0 0.02 0.04 0.06 0.08 0.1 investigation utilizing the RMS optimization method. Time (s) Suspended decoupler Abbreviations Free decoupler A:Area Figure 17: Transmitted force. B: Equivalent viscous damping coeﬃcient C: Volumetric compliance E: Nonlinear decoupler damping is the mount dynamic including the nonlinear stiﬀness of the coeﬃcient upper rubber, E : Nonlinear decoupler force coeﬃcient 3 2 f:Force f (t) = k x(t) + k x(t) + k x(t) + B x˙(t) + A P (t). T 1 2 3 r p 1 K = 1/C +1/C : Inverse sum of compliances 1 2 (29) k : Upper rubber load-deﬂection Determining the solution to the equations of motion in coeﬃcient k : Upper rubber load-deﬂection (8) numerically allows determination of the pressure term in (29) by means of numerical integration of the continuity coeﬃcient equations, thereby allowing determination of the transmitted k : Upper rubber equivalent stiﬀness force by means of (29). M:Mass Figure 17 illustrates the force transmitted through the P: Pressure Q: Flow rate engine mount for the supported decoupler mount and the free decoupler mount. In this analysis we assumed the R: RMS of acceleration transmissibility initial condition of the ﬂoating decoupler to be bottom- t:Time x: Position up position. Therefore, the supported decoupler mount transmits substantially less force (∼200 N) at startup as y: Excitation compared to the free decoupler mount. In addition, the Δ:Gapsize ω: Excitation frequency maximum amplitude of force transmitted via the supported decoupler mount is 716 N whereas the maximum amplitude ξ: Damping ratio of force transmitted via the free decoupler mount is 1.060E + ω : Natural frequency 03 N. The supported decoupler mount provides a reduction r: Nondimensional amplitude in peak amplitude on startup of 32.5% over the free w: Nondimensional frequency I , I : Tensor invariants decoupler mount thereby indicating the eﬀectiveness of the 1 2 supported decoupler design. In addition, Figures 16 and c , c , c : Material constants 10 01 11 17 illustrate that by utilizing the supported decoupler the W: Strain energy density. steady-state dynamics of the engine mount are not eﬀected Subscripts in a measurable amount; therefore, the supported decoupler design has been shown to be superior in improving overall system dynamics and mount isolation characteristics. i: Inertia track d:Decoupler p:Piston 6. Conclusion r:Rubber This study has introduced a decoupler design motivated by 1: Upper chamber the desire to improve upon the current ﬂoating-decoupler 2: Lower chamber design. Using nonlinear ﬁnite elements, information in atm: Atmosphere regards to the structural elastic behavior was obtained. T: Transmitted. f (N) T Advances in Acoustics and Vibration 11 References  M. M. Attard and G. W. Hunt, “Hyperelastic constitutive modeling under ﬁnite strain,” International Journal of Solids  W. C. Flower, “Understanding hydraulic mounts for improved and Structures, vol. 41, no. 18-19, pp. 5327–5350, 2004. vehicle noise, vibration and ride qualities,” SAE Technical  W. B. Shangguan and Z. H. Lu, “Experimental study and Paper Series 850975, 1985. simulation of a hydraulic engine mount with fully coupled  M. Bernuchon, “A new generation of engine mounts,” SAE ﬂuid—structure interaction ﬁnite element analysis model,” Technical Paper Series 840259, 1984. 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