Linear analysis of automotive hydro-mechanical mount with emphasis on decoupler characteristics

Journal of Sound and Vibration (1992) 158(2), 219-243

L I N E A R ANALYSIS OF A U T O M O T I V E H Y D R O - M E C H A N I C A L M O U N T W I T H EMPHASIS ON DECOUPLER CHARACTERISTICS R. SINGH, G. KIM AND P. V. RAVINDRA Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210-1107, U.S.A. (Received 26 June 1990, and in final form 25 June 1991)

A hydro-mechanical mount for an automotive engine can provide superior stiffness and damping characteristics which may vary with frequency and excitation amplitude. In order to better understand such a passive device, a linear time-invariant model with lumped mechanical and fluid elements is proposed and validated by comparing dynamic stiffness spectra predictions with experimental data over the frequency range 1-50 Hz. Special emphasis has been placed on the modeling of both free and fixed type decouplers. Most, if not all, of the prior models have been shown to be the special cases of the proposed fourdegree-of-freedom system model. Several reduced or simplified forms of this model have been examined, and parametric design studies have been carried out. The performance of a typical hydro-mechanical mount has also been compared with the conventional rubber mount for the resonance control and vibration isolation characteristics. Inappropriateness of the low frequency model beyond 100 Hz is identified. Other limitations associated with the linear models are identified and future research issues have been discussed briefly.

1. INTRODUCTION 1.1. ENGINE MOUNTS An automotive engine-body-chassis system is typically subjected to unbalanced engine forces in the frequency range 25-250 Hz, uneven firing forces especially at the idling speeds below 25 Hz, and road excitation primarily below 30 Hz [1-3]. Since the design trends have been towards compact and efficient automobiles, engine to frame weight ratio and engine force densities have increased. Consequently, recent research and development efforts have been focused on improving engine mounting technology to achieve better vibration isolation, smooth vehicle ride and noise reduction [1-20]. An engine mount must satisfy two essential but conflicting criteria. First, it should be stiff and highly damped to control the idle shake and engine mounting resonance over 530 Hz. Also, it must be able to control, like a shock absorber, the motion resulting from quasi-static load conditions such as travel on bumpy roads, abrupt vehicle acceleration or deceleration, and braking and cornering. Second, for a small amplitude excitation over the higher frequency range, a compliant but lightly damped mount is required for vibration isolation and acoustic comfort, like a conventional rubber mount. A conventional rubber mount cannot satisfy both requirements simultaneously as the lumped stiffness kr and the viscous damping coefficient br in the shear mode are nearly invariant with excitation amplitude and frequency over the concerned excitation range (1-250 Hz) of vehicle systems. Thus, a compromise between resonance control and isolation is inevitably needed. The mount is typically optimized for placement, orientation, kr and b, [4]. 219 0022-460x/92/200219 + 25 $08.00/0 © 1992 AcademicPress Limited

220

R. S I N G H E T A L .

To meet both performance requirements, hydro-mechanical mounts have been designed recently and employed in a few cases [1-20]. Such a mount can provide improved stiffness and damping characteristics which vary with frequency and excitation amplitude. It is conceptually the best passive mount known at present, even though it is not fully understood [21]. Furthermore, tunable hydro-mechanical mount concepts, which can provide optimal dynamic properties for given vehicle, engine performance requirements or operating conditions, need to be developed [8, 10-15, 22]. Accordingly, a long-term fundamental research project on the analysis, design and control of such a mount has been undertaken by the authors. The scope of this paper is, however, limited to the linear time-invariant (LTI) analysis of a passive hydro-mechanical device with emphasis on decoupler characteristics. 1.2.

LITERATURE REVIEW

Bernuchon [1 ] reports that a hydro-mechanical engine mount with a fixed-type decoupler improves ride comfort and reduces noise level by 5 dB. He proposes simplified mechanical and fluid models for the mount without carrying out any analysis. Corcoran and Ticks [2] model several hydro-mechanial mounts as a linear single-degree-of-freedom (SDOF) viscoelastic system with a rubber mass m , , stiffness kr, effective fluid stiffness and damping. They explain the characteristics and operating principles of the mount, and suggest the best location for it on a vehicle, considering that the mount exhibits the maximum damping when actuated through its axis. Through extensive experiments with such a mount, they have achieved a noticeable reduction (5 dB) in the vehicle acceleration and noise levels on a simulated bumpy road, compared with the conventional rubber mount. They have also investigated the shake spectrum during idle, especially around the resonance at 10 Hz, and found the acceleration significantly reduced. Muzechuk [5] explains several issues associated with manufacturing, such as the sealing technique and the fluid filling process. According to him, complete fluid filling is critical as underfills and small leaks affect the mount properties. Flower [6] presents mechanical and fluid analogs of hydro-mechanical mount designs, which explain some of the features of the inertia track and decoupler. He demonstrates experimentally the superiority of the hydro-mechanical mount over the rubber mount. Sugino and Abe [7] have developed an equivalent mechanical model which does not include the fluid element concept. They also report a reduction in the vertical acceleration level with a tuned device. Ushijima et al., [8] suggest, without any analytical treatment, a parallel connection between mechanical and fluid elements in order to investigate the fluid resonance effect at higher frequencies. Helber et al. [9] propose a new passive stiffnessswitching mount for machinery isolation by using a dilatant fluid which has variable viscosity shear rates. They employ a SDOF system model in the frequency domain to examine its properties including the switching effect. Some progress has been made on the design of semi-active or active hydro-mechanical engine mounts to overcome the limitations of the passive hydro-mechanical device. Mizuguchi et al. [10] develop an electronically controlled engine mount that incorporates alternating orifices switched by a solenoid-operated rotary valve. Shoureshi et al. [11-13] present an adaptive device and comment on its optimal tuning, which includes an openloop type fluid injection system with a hydraulic servo-valve controlled by an on-board microprocessor. They investigate the feasibility of an active system employing a closedloop control strategy, but it seems to be uneconomical due to substantial power consumption [14]. A semi-active hydro-mechanical engine mount utilizing an air-spring concept has also been developed [3]. Duclos [15] and Ushijima et al. [8] examine the feasibility of tunable mounts by using electro-rheological fluid.

221

HYDRO-MECHANICAL MOUNT

A semi-active mount concept has been analyzed extensively by Shoureshi et al. [ 11-14]. They have shown reasonable agreement between their predictions, based on the bond graph modeling approach, and measured data. However, the mount presented in this paper is quite different as evident from the next section. Accordingly, such a passive device warrants a detailed analysis.

2. P R O B L E M F O R M U L A T I O N 2.1. EXAMPLE CASE: PASSIVE MOUNT

The passive hydro-mechanical mount chosen for the analytical study is illustrated in Figure 1. It consists of two fluid chambers filled with water and anti-freezing mixture, which communicate through one or more orifices and an inertia track or a damping channel. This fluid mixture is not only inexpensive but it has appropriate dynamic and thermal properties needed for isolation and control [21]. The upper chamber is bounded

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(b) Figure !. Hydro-mechanical mount example case: (a) Physical system with a free dccoupler, and lumping schcm©; (b) physical system with a fixed dccoupler.

222

R. S I N G H E T A L

by a steel casing and an elastomeric spring which supports the engine weight, taken to act uniaxially through the mount axis. The lower chamber is enclosed in a thin compliant rubber bellow which acts as an accumulator when the fluid is forced to flow through the communicating orifices. The lower chamber is separated from the upper chamber by a rubber or fabric diaphragm, called here the "decoupler", which is bounded on both sides by perforated plates or "snubbers" to allow controlled deflection. Thus, the components which differentiate the hydro-mechanical mount from the rubber mount are two fluid chambers, decoupler, inertia track and rubber bellow. All of the hydro-mechanical mounts reported in the literature are conceptually similar but differ essentially in the design of the following: (i) inertia track which can by cylindrical [l, 5] or spiral [2], and (ii) decoupler which can be a fixed or free type [ 1, 8]. A typical device with a free decoupler and cylindrical inertia track is shown in Figure l(a), and the device with a fixed decoupler is illustrated in Figure 1(b). Since the upper chamber of the passive mounts shown in Figure 1 experiences sub-atmospheric or vacuum pressures, it must be assembled in a fluid bath to avoid air entrapment. 2.2. DECOUPLER CHARACTERISTICS When the elastomeric spring moves relative to the frame, fluid pumping action occurs and the fluid is forced to flow back and forth between two chambers mainly through the inertia track. Furthermore, a fraction of the displaced fluid is accommodated by the decoupler, depending upon its compliance Cdin the fixed type, or upon its diaphragm area Ad and free travel gap Ad between the decoupler and snubber plate in the free type. For instance, if no fluid flow through the inertia track is desired up to an input engine displacement X,,, then the gap is chosen to be Ad= 2ArX,n/,4d, where Jr is the equivalent piston area of rubber. The same concept can be applied to the design of the fixed decoupler for which Ca can be made to be a non-linear function of chamber volumes or pressures. In general, vibration isolation is required for small engine displacement (say 4-0.1 ram), mainly over the high frequency range (50-250 Hz). When the decoupler accommodates all of the displaced fluid, there is no fluid flow through the inertia track. Then the hydromechanical mount acts almost like the conventional rubber mount below 120 Hz, but quite differently at higher frequencies. Also, it should be noted that the rubber can be designed to be softer than the conventional case, thus yielding better vibration and acoustic isolations. Conversely, motion control is required for a large engine displacement (say 4-1.0 mm) over the low frequency range (1-50 Hz). This is accomplished by forcing the fluid to flow through the inertia track. Thus the hydro-mechanical mount is an amplitude-sensitive device. 2.3. SCOPE AND OBJECTIVES A passive hydro-mechanical mount exhibits non-linearities associated with fluid chamber compliances, decoupler stiffness and orifice resistances. Nonetheless, an attempt will be made to first develop an LTI model of the system valid over 1-250 Hz. Only vertical motion is considered here, even though other engine motions are possible in practice. Furthermore, excitation is assumed to take the same form as that employed in the industrial testing of an engine mount [23]: sinusoidally varying engine displacement in the vertical direction of amplitude 1.0 mm over lower frequencies (1-50 Hz), and of amplitude 0. l mm over higher frequencies (50-250 Hz), since the engine excitation is typically small at higher frequencies. Accordingly, fluid and mechanical parameters will be given by their nominal time-invariant values about the operating point. Specific objectives are as follows: (i) to develop a comprehensive lumped parameter LTI dynamic model of Figure l, and examine

HYDRO-MECHANICAL

MOUNT

223

various forms of this model; (ii) to solve the governing equations in the frequency domain and predict performance indices such as the dynamic stiffness magnitude, loss angle, damping ratio and force transmissibility; (iii) to validate the proposed models by comparing the predicted frequency spectra with experimental data; (iv) to examine the applicability b f each model and gain an improved understanding of the device including the decoupler mechanisms; and (iv) to conduct various parametric design studies.

3. DEVELOPMENT OF MATHEMATICAL MODEL 3.1. L U M P E D E L E M E N T S A N D A S S U M P T I O N S A lumped parameter model is developed considering several control volumes as in Figure l(a) [24, 25]. One inertia lump has been chosen for each of the upper (~/~-l) and lower (~/~2) chambers. The vehicle frame is assumed to be fixed. The excitation is applied by the engine through F'(t) = F + F(t) or xt(t) = ~ + x(t), where P is the engine weight or preload acting on the rubber, ~ is the corresponding static displacement, and F(t) and x(t) are fluctuating components. Typically, x(t) =x~ sin cot, where xa is the displacement amplitude and co is the excitation circular frequency. The Voigt model is assumed for rubber, for which kr and b, are assumed to be invariant with co and x'(t) [26]. The effective mass of rubber in the shear mode is represented as mr, which is typically extremely small compared with the engine mass mE. The upper chamber control volume ( # l ) is lumped into a fluid mass ml =pVi, or inertance Ii = ml/A~, and compliances C,~ and C,2, as shown in Figure 2(a), where p is the fluid density and V~ is the volume of the upper chamber. C~, incorporates the series connection of the compliances of the liquid and the rubber container, which is approximated as a solid cylinder fixed at all surfaces except the face contacting the fluid: Cl,~Arlr(l - 3 v 2 - 2v3)/((1 - v2)Er)+ VI/(2BI), where lr is the thickness, v is the Poisson ratio, E~ is the Young's modulus of rubber, and B/is the bulk modulus of the fluid. It is assumed that there is no entrapped air inside the control volume; the compliance is very sensitive to the amount of entrapped air [4, 24]. Furthermore, C~2 incorporates a series connection of liquid and decoupler compliances Ca: C~2,~VI/(2BI)+Ca, where Ca is assumed to be the equivalent linear compliance for a free or fixed type decoupler. Similarly, the lower chamber fluid volume ( ~ 2 ) is lumped into a fluid mass m2=pV2, or I2=m2/A 2, and compliances C21 and C22. Approximating the rubber bellow as a hemispherical membrane, we obtain C21 = C22~{(~d4/16t2E~)+(V2/B,)}/2, where d2 is the diameter of the lower chamber, t2 is the thickness of the rubber bellow, and V2 is the volume of the lower chamber. Since the rubber is more compliant than the fluid, further approximation is possible: Cii ~,A,lr(1 - 3v 2 - 2v3)/((1 - v2)E,), Cl2 ~ Ca, and C2, = C22~ rcd~/32t2Er. Fluid flow through the decoupler (-~d) is modeled using an equivalent orifice of linear fluid resistance gd and negligible fluid inertance as shown in Figure 2(a). The inertia track (-;~-i) is modeled as consisting of a lumped fluid resistance R,.= 128plj/~td~, and an inertance ^ 2 element I~= plJA~=m~/Ae, where p is the viscosity of fluid, l~ is the effective length, d~ is the hydraulic diameter, and A~is the cross-sectional area of the inertia track. Non-linearities arising due to pressure-flow relationships of the entry and exit orifices of the inertia track and decoupler are neglected in this model. The expressions given above include many approximations. Consequently, experiments may have to be carried out to find the exact values. Nonetheless, we have found that the empirical values of parameters are not much different from those calculated using these expressions.

224

R. SINGH E T A L .

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3.2. MATHEMATICAL MODEL I

As shown in Figure 2(a), the proposed Jumped parameter model consists of six fluid control volumes, and three lumped masses mr, m2 and m,. It has four dynamic degrees of freedom: x'(t), xi(t), q[(t) and x[(t). The governing equation of motion for rubber, from the Voigt model, is

F'( t) - k,x'( t) - b,fc'( t) - A j p l l ( t) - p , , , ) = m,Oi'( t).

(1)

Here, p~ mis the absolute fluid pressure at the upper end of ~ I. Application of the momentum balance to the fluid mass m~ = I~A2,, control volumes # i and 4/~-d,yields the following equations:

[pil(t)-pi2(t)]A,=mnTii(t),

pi2(t)-p~(t)=Rdq~t)=Ilil~+

[p~21( t) -pt22( t)]A, = m2~(t).

Riq~(t), (2--4)

HYDRO-MECHANICAL MOUNT

225

Continuity arguments applied to the compliant volume chambers yield: A,[~t(t) -.~[ (t)] = C, I/~ i(t),

(5, 6)

A , ~ ( t ) - q'a(t) - q[(t) = Ci2/~2(t),

q'd(t) + q~(t) -- d~YCt2(t) = C21/~[l(t),

,4,~t2(t) = C22/i[2(t).

(7, 8)

In the LTI analysis, Cjl, C,2, C2, and C22 are assumed to be nominal time-invariant values. The mean value of each variable associated with the static load, given by a bar over the symbol, can be obtained from the following: F = F ' ( t ) - F(t) = kr$ + A,(filj --p~,,), .~r ~

=

fill :ffl 2 :if21 =/~22~--ff,

qi = qd = 0,

.4r[Xt(t) -- X(t)] = [Cll + CI2 + C21 + C22]ff,

.4r~, =

[CI2 + C21 + C221P,

A,~2 =

C22P.

Eliminating the static components of the variables from equations (1)-(8), similar equations will be obtained with only time varying or dynamic parts with respect to the static equilibrium condition in terms of F(t), x(t), p,l(t), etc. In Figure 2(a), F'r(t) represents the total transmitted force to the fixed frame, the dynamic component of which is typically measured in the experiment. Consider the hydro-mechanical mount with a fixed decoupler where qd(t)= O. Eliminating the internal variables ptl(t), pl2(t), p21(t) and p22(t) from equations (1)-(8), and by defining m i -_A J i 2, b i = A r R2i , kll = Ar2/ Cl l , kl2 = A2 / CI2 , k21 = .4,2/ C2, k22 = Ar2/ C22 and Ycj(t)=qi(t)/A~, we obtain the following four equations in terms of the time-varying variables: F( t) - kll[x( t) -- xl (t)] -- b~Yc(t) - k , x ( t) = m ~ ( t), kl , [x( t) - xl (t)] - kl 2[xl ( t) - xi( t) ] = ml ~, ( t), - bi Yci(t) + k l 2[xl (t) - xi( t)] - k2 i[xi(t) - x2(t)] = Mi xi(t),

k2j [x;(t) - x2(t)] - kE2X2(t) = m2Jc½(t).

(9a-d)

From the above equations, an analogous mechanical system is developed in Figure 2(b). This analog shows the operating principle of the hydro-mechanical mount. For a small x(t), kl2 becomes soft with the decoupler in the free travel mode: namely, the fluid damping b~ associated with the inertia track is decoupled from x(t). Thus the engine is supported only by k~ and b,. However, if x ( t ) is large, kl2 becomes stiff with the decoupler motion restrained by the snubbers; i.e., ba is coupled to x(t). Thus the system is highly damped since b~>>b~. Observe that m; is given by/,- multiplied by A,2, not by A~. Since mi>>m~, so that m~ is of the same order of magnitude as m e , the inertia track performs as an inertiaaugmented damping channel. Note that the analog mechanical system does not represent exactly the same dynamic system. For instance, the forces caused by p~ i(t) and p~2(t) are directly transmitted to the vehicle frame from the hydro-mechanical mour~t, unlike the mechanical system of Figure 2(b). ' 4. LOW FREQUENCY ANALYSIS 4.1. DEVELOPMENTOF MODEL II Since the impedances iwlt and icol2 of the lumped fluid masses are negligible at frequencies, a two-degree-of-freedom model can be developed as in Figure 3(a). CI = Cl, + CI2, C2 = C2, + (722, ptl(t) ~-.pl2(t) = p f f t ) , and p21(t),~p22(t)=p2(t). equations (1)-(8), we obtain the following equations for a free decoupler in terms

lower Here, Using of the

226

R. SINGH E T A L

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Ft(t)

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I I

2

p:l r ), c,

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2

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xt( t )

"m=~q

r~__-,--_~le¢

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(b) Figure 3. Mathematical model II: (a) Schematic with a free decoupler; (b) analogous mechanical system with a fixed decoupler.

time-varying variables:

F(t) - krx(t)

-

-

b ~ ( t ) - A~p, (t) = m~5~(ty),

pj (t) - p 2 ( t ) = l,~l,(t) + R,q,(t),

p,( t) -p2(t) = Raqa( t), A ~ ( t ) - qi(t) - qd(t) = C , ~ (t),

q~(t) + qd(t) = C2/~2(t).

(10a-e)

Eliminating the internal variables p,(t) and pz(t) from equation (10), and by defining m i = Aili, 2 b i _- A r R2 j , k, =A~/C, and k2=A~/C2 for a device with a fixed decoupler, we obtain two governing equations, corresponding to the analogous mechanical system of Figure 3(b) :

5Ci(t)=qi(t)/Ar,

F( t) - brYc(t) - ki [x(t) - x,(t)] - krx( t) = mrS'(t), - bike(t) + k I[x(t) - x;(t) ] - k2xi(t) = m ~ ( t ).

( 11 a, b )

Equation (10) can be transformed to the Laplace domain with zero initial conditions. From the block diagram shown in Figure 4(a), the hydro-mechanical mount is essentially a feedback system, and the fluid part acts as a passive controller. It is observed from Figure 4(b) that the feedback loop is a (second order)/(second order) system. Considering the real engine mass, this feedback system has been found to be absolutely stable based

HYDRO-MECHANICAL

227

MOUNT

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Figure 4. Blockdiagrams for ModelII given by equation (10).

on the Routh stability criterion; this has also been confirmed by applying the root locus and Nyquist analyses [27]. A closed-loop transfer function is obtained as follows:

F

- (s) = x

a4=m,CiC2Rdll,

ot4s4+ot3s3+a2s2+ot|s+Oto ~2s ~ + ~ l s + ~o

a3=brC1C2Rdl~ + m,CiC2RaRi + m,li( Cl + C2),

¢t2= mr( R~+ Rd )( Cj + (?2)+ b, Ci C2RaR, + b,l~(C~ + C2) + A2CzRdlI + k, Ci C2Rdli, aI=b,.(RI+Rd)(CI+C2)+k, CIC~RaRt+k,Ii(G+C2) + A,.I~ 2 + ArC2RdR~, 2 ao = k,.(Rd+ R~)(G + C2) + A2,(Rd+ R~), fl2--CIC2RdIj,

fll=CIC2RdRi+Ii(Cl+C2),

flo=(Ra+Rs)(Cl+C2). (12a-i)

228

R. S I N G H E T A L .

4.2. TRANSMITTED FORCE F r o m the free b o d y diagrams shown in Figures 5 and 6 for the static equilibrium, we obtain

fi=(~'-krYc+A~po,,,)/A,,

P=kr~:+A,(fi-p,,,,)=Pr.

(13, 14)

F o r the d y n a m i c case shown in Figure 7,

F'r( t) = k S ( t ) + b,Yc'(t) + d~(p~ ( t) -p'~( t) ) + Ar(p'~( t) -p~,,,).

Potm

N P

b,I"

k,

"/'/> y f f f f ~ f f f f f f ~ f f f f ~

~.,,'.,~

Frome

Figure 5. Free body diagram of the upper rubber at static equilibrium.

Figure 6. Force transmission at static equilibrium. xt( t )

pit( t )

' g~g~W,

VO,g~

Figure 7. Force transmission for the dynamic case with excitation x'(t).

(15)

HYDRO-MECHANICAL MOUNT

229

Using equations (13)-(15) and by defining Fr(t) = Fir(t) - P and p, (t) =p] (t) - p ,

Fr( t) = k~x( t) + b~( t) + pt ( t)A,.

(l 6)

The cross-point dynamic stiffness K(s) is now given as

K(s)

=

Fr (s) - t~3s3+ a2s2 + a,s + ao x fl2s2+fl,s+flo '

(! 7)

where a3 = b,C~C2RaI~ and ~2 = C~C2RSb,R~ + krI,) + b,I~(C, + C2) + A,C2RdI~, 2 while other parameters are given by equation (12). Observe that K(s) is a system of (third order)/ (second order). The sinusoidal dynamic stiffness K(ico) is given by substituting s = io~ into equation (17): K(i~) = IK(iat)le'** ~') = IK(ico)l cos ~x(co) + ilK(io)l sin ~bx(co),

(18)

where IK(kol and ~bx(co) are the magnitude and phase angle of K(io~). 4 . 3 . R E D U C E D - O R D E R F O R M O F M O D E L II

Since the influence of b, over 1-50 Hz is negligible, assume that b, = 0. Noting further that C2>IOOC~, therefore, K(s) can be simplified to a (second order)/(second order) system:

K ( s ) , . ~ v ( ~ + 2 ~ t s + 1"~1(~+2(2s+ 1)

(19)

where ? = k , + [A~/(CI + C2)] is the static stiffness. Other system parameters are given in Table 1 for both fixed and free decouplers. Here ( is the damping ratio, and co, is the natural frequency. When we set Ra~ oo in the free decoupler expression, we obtain the results of the fixed decoupler, since Razor corresponds to qa(t)--}O. Note that c0,2 of the fixed decoupler is equal to the natural frequency of a SDOF Helmholtz resonator with compliance CI and inertance Ii. The typical frequency response for both fixed and free

TABLE 1 System parameters of equation (19) Fixed decoupler (Rd--}O0)

""' /,,<

,,, ,42,+ k,C,)

Free decoupler (finite Ra)

/(R,+R~

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~

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'

R. S1NOH ET AL.

230

decouplers with x , = 1.0 mm is plotted in Figure 8, where f = co/2z, and the computed values of their system parameters are listed in Table 2. Note thai the peak loss angle frequency f~ is located between co,l and co,2. The difference in the frequency responses can be attributed to qd(t). It is seen from Table 2 and Figure 8 that qa(t) increases (~ and (2, and shifts co,l and c0,2, while decreasing [g(ico)[ and ~b~(co) for the free decoupler, compared to the fixed decoupler. From the reliability viewpoint, the free decoupler may be favorable because the fixed decoupler diaphragm may undergo a plastic deformation. Further research should investigate thoroughly the overall effect of qd(t) on the transmissibility of engine excitation to the chassis. TABLE 2

System parameters predicted by equation (19) with xo = I.O mm Fixed decoupler (Rd-'~ 00) Free decoupler (finite Rd)

Parameter

eo.~2/2z

6'0 Hz 11'3 Hz 0'66 O' 35

o~.2/2z (i (2

4.4.

7' I Hz 13'3 Hz 0"69 O' 54

STATE SPACE FORMULATION

Equation (10) can be expressed in a state space form with state vector {X}T= Ix ~ p, P2 q;J, input vector {U} =F(t), and output vector { Y} =x(t)" { Y} = [C] {X} + [D]A U}

{X = [A] {X} + [B]{ U},

[A] =

0

1

0

0

0

-k,/m, 0 0 0

-br/m,. Ar/G 0 0

-Ar/m,. - 1/GRd 1/C2Rd 1/I,

0 1/CIRd -1/C2Ra -1/I,

0 - 1/G 1/C2 -R,/I,.

[B]r = [0 1/m, 0 0 0],

[ C ] = [ 1 O0 0 0], ~\\

i

I

f

I

L

r

r

[O] = [0],

(20a-f)

~n2

,,

II 80

~

I|ll

6O 40

3 .#.

2o

0 0

t[:I io

20

30 I (HZ)

40

50

Figure 8. Frequency response predicted by equation (16) for fixed ( - - -) and free (

) decouplers.

231

HYDRO-MECHANICAL M O U N T

Using equation (20), we can obtain the frequency responses of all variables, such as p,(t), p2(t), qt(t) and qa(t), in addition to K(ito). The typical magnitude and phase spectra are shown in Figures 9 and 10. Note that [p2(iro)l is considerably smaller than Ip,(iro)l, and that qa(t) is the same order of magnitude as qM). The spectral shapes of Ip~(ico)l and Iqa(iro)l are similar to that of IK(iro)l. Note that Ipl(iro)[ shows the upper chamber experiencing vacuum pressures for the upward engine motion because it is the harmonic component with respect to the initial static pressure (~0.116 MPa) for a free decoupler. EXPERIMENT A schematic of the experiment setup used for static and dynamic tests on a hydromechanical mount is illustrated in Figure ll. The computer controlled testing machine is a servo-hydraulic system operating under closed-loop control. The almost frequencyinvariant k, and br are determined by the static test. For a given preload F = 1200 N, 4.5.

(b)

(o) 0.20 0 O. =E

0.15

--

OqO

I

I

I

'°°I

~

'

,

,

50

-

o.os

I

I

L

'

'

'

0

120

~

9c

'

I

'

i _,oor ',,

"o ~

'

6C

3.

I

I I0

20

30

40

50

O

10

20

30

40

50

f (Hz) Figure 9. Pressure response spectra predicted by Model II for fixed ( - - - ) and free ( p,(ie)); (b) p2(ia)). (a) 0.0006

I

I

) decouplers: (a)

Cb) I

0.0003[

I

I

I

i

I

I

I

I

I

10

2O

30

40

/" \ ~E 0.0004 v

~

-~ 0,0002

~.~ 0.0001

0.0000

I

I

1

0,0002

0.0000

I

...$

.=

=,

-,oo/

0

J

10

I

20

-I----/--J 30

40

0

,50

5O

f (Hz)

Figure 10. Inertia track and decoupler flow response spectra predicted by Model 1! for fixed (- - -) and free ) dccouplers. Note that qa(t)=0 for a fixed decoupler. (a) qi(ioJ); (b) qa(im).

232

R. S I N G H E T A L .

Static load cell, F= x(t) x(t)

Hydraulic actuator Hydro-mechanical mount Dynamic load cell, FT(!)

L_.~J conditioner Signal N t

Signal conditioner

Digital data acquisition and processing system

Figure 11. Schematic of the experiment set-up.

corresponding to a part of the engine weight, a displacement excitation x(t)= x, sin cot is applied to the mount as discussed earlier in section 2.3. Digital data acquisition and processing techniques are employed to estimate K(ico). In the experiment, Fr(t) is typically measured using a dynamic load cell in order to avoid the inertia effect of mr. Note that equation (12) has terms associated with mr, but there is no mr term in equation (17)~ Since the measured force signal Fr(t) exhibits non-linear characteristics, fundamental harmonic components of x(t) and Fr(t) at co are estimated, both analytically and experimentally [23], as xl(t)=x~l sin cot and Fr~(t)=Fat sin (cot+~b~), using Fourier analysis, and definitions IK(ico)l =Fo,/x,, and ~bjr(co)= ~ are introduced. The appropriateness of this procedure is not discussed here, as it is beyond the scope of this paper. 4.6. RESULTS

The Voigt model for rubber, and the proposed mathematical models I and II, are verified by using the results of an experiment shown in Figure 11. The rubber mount case is used for draining fluid from the hydro-mechanical mount. Since kr # kr(co ; x) and br # b,(co; x) from the Voigt model, K(ico)=kr+icobr in Figure 12, where br and kr are nominal values.

2ii r.5

6O

~

2O 0

0

---,

--

I0

--

-I-

----

20 30 f (Hz)

-r" ----

40

J I

50

Figure 12. K(it0) of rubber simulated by using a hydro-mechanical mount without fluid: - - . Voigt model.

experiment;

HYDRO-MECHANICAL

233

MOUNT

The Voigt model represents the dynamic response of rubber reasonably well. The experimental results show that br and k, vary with co slightly. For instance, kr = 1.06/~, at 50 Hz, and k, = 1.5/~, at 250 Hz, where/~, is the static spring constant. Reference [29] also shows this trend. Thus the Voigt model for rubber is adequate for our LTI analysis. Dynamic stiffness spectra as predicted by Models I and II, and by the reduced-order formulation of equation (19), are compared with measured results in Figure 13. Fluid parameters of Figure 2(a) are estimated based on section 3.1. However, given the complex geometry and many simplifying assumptions, a minor adjustment in the numerical data set has been made by comparing Model I or II with measured K(i~o). No further perturbations in numerical values are then made. We observe an excellent agreement even though the simulation is limited to a linear analysis. Furthermore, the reduced-order formulation shows a close agreement with the full-order formulation given by equation (17) and the experiment. Thus, the hydro-mechanical mount can be represented as a (second order)/ (second order) system for a high amplitude excitation over 1-50 Hz. The reason is that the effect of the third zero, say the third root of the numerator of equation (17), which is located around 1100 Hz, is negligible over the low frequency range. I

I

I

I

I

I

1

l

I

I

I

I

I 10

I 20

I 30

I 40

3

80

so 40

3

zo 0

0

50

f (Hz)

Figure

13. D y n a m i c

s t i f f n e s s w i t h x o = 1.0 m m :

- -

, experiment;

---,

Models

I a n d I1; - - - - - ,

equation

(16). When xa is reduced to 0.1 mm, C~ increases because the decoupler is essentially in the free travel mode. Corresponding spectra are plotted in Figure 14. Model I or II predicts the experimental results reasonably well. The spectra are similar to those of the rubber mount shown in Figure 12. Hence, the hydro-mechanical mount behaves like the rubber mount for a low amplitude engine excitation at lower frequencies. 5. HIGH FREQUENCY DYNAMIC RESPONSE The high frequency dynamics can be examined from the measured K(ito) given in Figure 15 for a free decoupler. Observe that IK(ito)l of the hydro-mechanical mount becomes large as the frequency is increased beyond 120 Hz, unlike the rubber mount. It is because the fluid inertia force is transmitted to the fixed frame. This problematic aspect of the passive device deteriorates the vibration isolation properties. In fact, at higher frequencies,

234

ET AL.

R. SINGH

2.0

I

i

I

50

40

'5I

"~

1.0

0.5

'K 0 6O

4

~

2o 0

0

20

50

f (Hz)

Figure 14. Dynamic stiffness with Xo=0.1 m m : - - -, experiment; - - --, Model I and 1I.

I

,.~

I

I

I0

s

0 150

I

I

I00

150

I

IO0 "o so

4 0 50

f

T. . . . 200

250

(Hz)

Figure 15. Measured dynamic stiffness at higher frequencies: - - , with fluid; - - -, without fluid. practical hydro-mechanical mounts have been found to be worse than the conventional rubber mount [21]. The experimental result shows that ~ c o ) exceeds 90 degrees, which implies that the mounting system cannot be considered a SDOF system at higher frequencies. Further research is required to resolve the high frequency dynamics issues. 6. ANALYSIS OF PRIOR MODELS Most, if not all, of the mathematical models available in the literature can be shown to be the special cases of Model I or II. For instance, consider the viscoelastic model given by Corcoran and Ticks [2]. It is a simplified form of Model II, where the fluid part is defined by a global spring stiffness and a viscous damping coefficient. Clark [17] also explains low frequency behavior analytically by using a mechanical model with a spring and damper. He represents K(ico) as a (first order)/(first order) system. However, since the augmented inertial effect of the inertia track is significant, K(ita) must be expressed at

235

HYDRO-MECHANICAL MOUNT

least as a (second order)/(second order) system, as discussed earlier in sections 4.1-4.3. Flower [6] proposes simplified fluid and mechanical analogs similar to Model II, but without any analytical study. Sugino and Abe [7] develop an analogous mechanical model without utilizing the concept of fluid parameters. Shroureshi et al. [I 1-13] include fluid parameters similar to Model II, but their adaptive laboratory device does not include a decoupler. Ushijima et al. [8] graphically propose a higher order mechanical analog with three lumped fluid masses in order to explain the fluid resonance effect over the higher frequency range, but they do not present any mathematical details. 7. PERFORMANCE STUDIES AT LOWER FREQUENCIES 7. I. VOIGT-TYPEMODEL The hydro-mechanical mount for lower frequencies is represented in Figure 16 by the SDOF Voigt-type model with equivalent ke(co) and be(c0). However, these parameters are highly frequency-variant, unlike those for a generic rubber mount:

~

EnginIe

Ft(t}

mE

--~xt(t)

Hydromeehonifcol mount Figure 16. Equivalent SDOF systems using Voigt-type model.

K(ico) = ( Fr/x)(ico ) = ke( ca ) + ibe(co)c0.

(21)

Using equation (18), we obtain ke(co) = IK(ica)l cos ~bk(CO),

be(co)= (IK(ico)l sin ~k(CO))/CO,

(22)

The frequency-variant damping ratio (e(a~) for me can be given as

ee(co) - 2

be(o~) ~

tan ~bk(CO)~/[K(ico)[cos ~bk(CO) 2(o V ,n-E

(23)

In Figure 17 are shown ke(ta), be(co) and (e(co) for a typical device with a free decoupler and for mE = 122 kg, along with a comparison to a rubber mount. All parameters have been calculated using the measured [K(ico)l and ~r(co) of Figures 12 and 13. Observe that ke(to) and be(co) of the hydro-mechanical mount are higher than those seen for the rubber mount. Note that this hydro-mechanical engine-mounting system yields (e--0.6 at the idle engine shake frequency of 8 Hz; this value is deemed as very suitable for controlling shock pulses [20] as well as reducing the resonance amplitude. However at higher frequencies, since the engine-mounting system should be represented as a multi-degree-of-freedom (MDOF) system due to fluid inertia effect, Figure 16 is obviously not applicable. 7.2. PARAMETRIC DESIGN STUDIES Several parametric studies have been conducted using Model If. For instance, the effect of Ct on K(ico) and (e(co) is illustrated in Figure 18, where Ci is taken to be a nominal

236

ET AL.

R. SINGH

80o[ 600~-

I8r

l

I

i

I

'

'

'

'

J

t

4

Z

4

2 )

~

0-8

--

J.-

I

J

I

l

I

I

0-6 O.a! 0.2 l

" ~ L .

I0

0

~

20

I

30

40

50

f (Hz)

Figure 17. Mechanical

properties for Figure

16 (

,

hydro-mechanical

mount;

--

-,

rubber mount).

dimensionless value for the example case. In practice, CI is given by the free travel gap as well as by the membrane stiffness of the decoupler. If the decoupler is fixed, CI is determined solely by the membrane stiffness. A softer upper chamber with a high C, will accommodate a larger portion of the displaced fluid and hence reduce @(t). This results I

I /

-x

=

/

v

I

.o/

/_\,

} ~ol- /

a 7"° /

I

|

\\

,

4

,

,

,

|

/',,, ~

0.4

°7o 0

ro

20

50

40

50

f (Hz)

Figure 18. Effect of G on K(ioJ) and (,(o), predicted by Model

I1; - - -, 0.5G

;

, I'0Cj ; -----,

2"0CI.

HYDRO-MECHANICAL

237

MOUNT

in decreased damping effect of the inertia track, so that IK(ico)l and 0~c(co)become smaller. Analytically, this can be explained considering co.I and c0.2 of Table 1 and Figure 8. Note that IK(ico)l and 0k(co) vary significantly depending on the difference between co.2 and co.,. Since Cj mostly affects co.2 rather than to.i, IK(ico)l and 0K(co) change noticeably according to a variation in C~. The effect of C~ on f $ , corresponding to the peak loss angle 0*, is shown in Figure 19, where Cm= 1.0 is the nominal dimensionless value for the example case. The trends approximate the functional relationships of C? ~/2, which is noted from c o . 2 o c ~ given in Table 1. From Figures 20 and 21, we can examine the effect of inertia track geometry li and A/ on K(ico) and ~'e(CO), where 1;= 1.0 and A~= 1.0 are the nominal dimensionless values. I0

5-I

I

i

I

•

i

I t

J

L 2

~ 7o 6o 5O

5O

0

3

Ci F i g u r e 19. Effect o f CI o n f ~ a n d ~ c , predicted b y M o d e l [I. 5

I

I

1

I

I

I

I

I

I

I

I

I

I

o 8o

/-\\

~ 6o

I i

-~

3

0'8

I

[

~ \

O.E

3

I

0.4

\

0

I

I0

I

2O

310

f (Hz)

't 40

50

F i g u r e 20. Effect o f l i o n K(ico), predicted b y M o d e l 1I: - - -, 0-5It ; - - ,

1.01,; . . . .

,2.0/i.

238

R. S I N G H E T A L . I

I

I

I

|

I

I

I

|

I

I

.1--~

I00

I

-

/ \

60 4o

2.0

I ~

A

-

!~.

r,6 ,.c

0.5 I0

2O 30 f (Hz)

40

50

Figure 21. Effect of A~ on K(ito) and (,(to), predicted by Model II: . . . . , 0 . 5 A , , - - .

1.0A, ; . . . .

,2.0A,.

Referring to section 3.1, Rt and Iiocl~, 6ocAi -~ and R~ocA; 2. From Table 1, it is noted that l; and A; influence both co,i and o~,2, unlike Q . Thus, the effects of l; and A; on [K(ito)[ are not as significant as that of C,. Note, however, that the variation of l; is quite effective for the frequency-tuning process, while virtually holding the peak damping ratio (,*. The (,(to) spectra do not show any (* for 0.SA;. The predictedf~ is compared with measured results in Figure 22. Observe that f~ decreases somewhat exponentially as li is increased, and that f ~ increases almost linearly as A; is increased. Discrepancies between predicted and measured data are most likely related to the assumptions made earlier. A non-linear analysis is expected to resolve some of these discrepancies. The usual industrial practice is to tune f~ to a resonant frequency of the mounting system. It is seen from @x(co) and (,(co) spectra that this practice seems reasonable for most parameter variations, except 2 . 0 Q , 2-01; and 0.SA~. Thus, for a more accurate design, the peak damping ratio frequency f~', corresponding to the maximum system damping (,*, must be tuned to a resonant frequency.

30 L I

I

i

I

1

I

i

I

*

! '~*

A,.=I.O

I0

I

0

i I/ I" I I

I

2 6

I

I I

3

4

2

,6

0

i I

J [

J

2

~

I I

I I

3

4

5

,4,

Figure 22. Effect of inertia track geometry on f ~ : + , experiment; - o - , Model II.

239

HYDRO-MECHANICAL MOUNT

7.3. TRANSMISSIBILITY Model II yields the following expression.

Fr($)= ~'3S3 + ~'2S2 + '~1$1 + ' ~ F ~434+a3s3+a2s2+a2sl+a o ' ~3=brCtC2Rali,

~,t = a l ,

Ao= ao,

2,2 = b,C, C2Ran, + brI,( C, + C2) + A2C2RdI, + k,C, C2RdIl. di4 = (me + mr) CmC2 Rdl~

(24a-f)

Substituting s = ico into equation (24a), the sinusoidal force transmissibility is

Tr( co) = J(Fr/ F)(ico )l .

(25)

From the experimental viewpoint, we develop the following alternative estimation scheme for Tr, given the equivalent SDOF system of Figure 16: Tr(CO) = ~ / ( i

l+(b'(co)co/ke(co))2 _ (o)/O)o(O)))2)2+ (be(co)co/k,(co))2,

,~/ke(co) co0(co)= ¥ me+m,.'

(26)

where coo(co) is the frequency-variant natural frequency of the engine-mounting system. Since me>>m,, in practice, coo(co)~x/ke(co)/me. Predictions obtained by using equations (25) and (26) are shown by one solid line in Figure 23, on a logarithmic scale since they are virtually the same expressions analytically. Hence, even though Figure 23 plots Tr on the assumption of a fixed frame, the equivalent SDOF system concept is useful for estimating T~-. Results for two cases of rubber mount are also illustrated in Figure 23, where rubber is represented as the Voigt model with frequency-invariant kr and br. A rubber mount with very low damping (( = 0.01) represents the hydro-mechanical mount without fluid, so that the engine is supposed to be supported only by rubber. We can observe a peak resonance at coo= 6.4 Hz. When we increase ( to 0-6 in order to duplicate the resonance control feature of the hydro-mechanical mount, it deteriorates the high frequency attenuation property. l0

?,

I

~

[

\

0.01

I I0

1

I

.............

I 20

I 30 f (Hz)

Figure 23. Comparison of force transmissibility spectra: - - - - , rubber (~'=0.01); . .. rubber (~'=0.6).

I 40

50

, Hydro-mechanical (equations (21) and (22));

TRANSMISSIBILITY The results for a free decoupler, the effects of is, At, Cj and Ra on TF(co) being plotted, are shown in Figure 24. Note that b, At and Rd primarily influence the lower frequency

7.4. DESIGN FOR LOW

240

R. SINGH 20

/~ r

f

I

E T AI.. 3F~---

1(o )

!

I

----

I

i

/b, I

I'g

i

I.C

[

O'g

2,C

J

I

I

_.'h

l

l

J

~ 2(

I

I

I

IO

20

50

F5 I-0

I.(

0.5

O. ~

0

O

~0

20

30

40

C

50

40

50

f (Hz)

Figure 24. Force transmissibilityspectra for a free decoupler. (a) - - - , 0.51j; - - , 1-01~;- - - - 2.01, , 2.0A~. (c) - - - , 0"5(71; - - , 1.0C, ; - - - - - , 2-0Q. (d) - - -, 0'5Ra;

(b) - - - , 0 " 5 A i ; - - , I ' 0 A j ; - - - - --, l ' 0 R a ; - - - - - , 2"ORd.

characteristics. Conversely, CI influences the high frequency attenuation property, but it has no effect on lower frequencies. We can observe the existence of an optimum set of mount parameters for which TF will be a minimum. Thus, high ]K(ico)l and large ~r(to) are not always desirable from the Tr viewpoint. The hydro-mechanical mount chosen for the example case (li = 1.0, Ai= 1.0, CI = 1-0 and R d = l ' 0 ) is fairly close to the optimum device. However, the optimum parameters of a mount may also depend on the corresponding me and other vehicle parameters. The effect of mount parameters for a fixed decoupler is examined in Figure 25. The high frequency attenuation property is influenced by Cj rather than by li and Ai, as in the free decoupler case. From Figure 25(a), we observe that the inertia track length should be 0.55/~ in order to obtain a smaller TF; i.e., it should be shorter than that for the free decoupler. The effect of the variation in Ai is more significant than the free decoupler case as seen in Figure 25(b). It is interesting to note that a variation in Cj only shifts the second peak of Tr in Figure 23(c); recall from section 7.2 that the effect of C) on ]K(i~)] was primarily to shift c0,2. Also, a variation on li shifts the second peak while influencing the magnitude of the first peak of Tr, as noted in Figure 25(a). A comparison of Figures 24 and 25 shows that the peaks of Tr(co) are widely separated for the fixed decoupler. Also, Tr(o~) is smaller for the free decoupler since qd(t) # 0; here qa(t) increases system damping, similar to the effect of internal leakage in hydraulic servo-systems [27]. 8. CONCLUDING REMARKS In this paper, several LTI models of a passive hydro-mechanical mount with both fixed and free decouplers have been developed and validated by comparing predictions with the measured cross-point dynamic stiffness spectra over 1-50 Hz. This device has been shown to be an absolutely stable feedback system, and it can be represented by a reduced (second order)/(second order) system. The frequency-tuning procedure has been clarified by employing an equivalent frequency-variant damping ratio for a given engine mass. Through parametric design and transmissibility studies, it has been found that an optimum set of mount parameters exists for an engine-mounting SDOF system. The proposed LTI models are not valid over higher frequencies because of increased fluid inertia effects.

HYDRO-MECHANICAL I

51

241

MOUNT I

I

I

I

1o1

/

I

0 5

I

I

I

I

(b)

4

2 /i //\

./,\

. ,

7,.0

I

A

/'~

I0

I

/",

I

I

,,

20 50 f (Hz)

¢cl

40

50

Figure 25. Force transmissibility spectra for a fixed decoupler. (a) - - - , 0.4It; ,0.551j; - - - - - , 0.81, (b) - - - , 0.5A,; - - - , 1.0A~; - - - - - , 2.0A,; with 0.551,. (c) - - - , 0.5CI ; - - -, 1.0c~ ; - - - - - , 2-0c~; with 0.55Â,. Research is under way to resolve several issues including high frequency dynamics and non-linear response.

ACKNOWLEDGMENTS The authors would like to thank Teledyne Industries for supporting this study under its research assistance program. We acknowledge Karl Winkler, William Cummins and James Vance of Teledyne Monarch C o m p a n y for providing experimental data. Thanks are also due to The Ohio State University Transportation Research Endowment Program (Honda) for sponsoring related research.

REFERENCES 1. M. BERNUCnON 1984 SAE Technical Paper Series 840259. A new generation of engine mounts. 2. P. E. CORCORAN and G. H. TIcKs 1984 SAE Technical Paper Series 840407. Hydraulic engine mount characteristics. 3. J. P. WEST 1987 Automotive Engineer 12, 17-19. Hydraulically-damped engine mounting. 4. D. M. FORD 1985 SAE Technical Paper Series 850976. An analysis and application of a decoupled engine mount system for idle isolation. 5. R. A. MUZEeNUK 1984 SAE Technical Paper Series 840410. Improved engine isolation. 6. W. C. FLOWER 1985 SAE Technical Paper Series 850975. Understanding hydraulic mounts for improved vehicle noise, vibration and ride qualities. 7. M. SUG~NO and E. ABE 1986 SAE Technical Paper Series 861412. Optimum application for hydroelastic engine mount. 8. T. USHIJIMA, K. TAKANO and H. KAJIMA 1988 SAE Technial Paper Series 880073. High performance hydraulic mount for improving vehicle noise and vibration.

242

R. S I N G H E T A L .

9. R. HELBER, F. DONKER and R. BUNG 1990 Journal of Sound and Vibration 138, 47-57. Vibration attenuation by passive stiffness switching mounts. 10. M. MIZUGUCHI,T. SUDA, S. CHIKAMORIand K. KOBAYASHI1984 S,4E Technical Paper Series 840258. Chassis electronic control systems for the Mitsubishi 1984 Galant. 11. P. L. GRAF and R. SHOURESHI 1988 Journal of Dynamic Systems, Measurement, and Control 110, 422-429. Modeling and implementation of semi-active hydraulic mounts. (See also ,4SME paper 87- lYA / DSC-28.) 12. R. SHOURESHI, P. L. GRAY, S. BOUCHILLON,T. KNUREK and R. W. STEVENS 1988 S,4E Technical Paper Series 880075. Open-loop versus closed-loop control for hydraulic engine mounts. 13. S. BOUCHTLLON,R. SHOURESHI,T. KNUREK and P. SCHILKE1989 SAE Technical Paper Series 891160. Optimal tuning of adaptive hydraulic engine mounts. 14. P. L. GRAF, R. SHOURESm,J. STARKEY,D. NOVOTNYand R. W. STEVENS1988 SAE Technical Paper Series 880074. Active frame vibration control for automotive vehicles with hydraulic engine mounts. 15. T. G. DucLos 1987 SAE Technical Paper Series 870963. An externally tunable hydraulic mount which uses electro-rheological fluid. 16. G. EBERHARD and J. HE~TZIG 1984 Continental Gummi-Werke AG Report. Hydraulically damped mounts for vibration and acoustic isolation. 17. M. CLARK 1985 SAE Technical Paper Series 851650. Hydraulic engine mount isolation. 18. ESCAN CORP. 1984 Hydromounts. 19. H. YOSHIDA(et al.) 1986 United States Patent 4573656. Fluid-sealed engine mounting. 20. K. KADOMATSU 1989 S,4E Technical Paper Series 891138. Hydraulic engine mount for shock isolation at acceleration on the FWD cars. 21. K. WINKLER,W. CUMMINSand J. VANCE 1989-90 Private Communication. 22. R. M. GOODALLand W. KORTOM 1983 Vehicle System Dynamics 12, 225-257. Active controls in ground transportation--a review of the state-of-the-art and future potential. 23. CARL SCHENCKAG CORP. 1985 Software Instruction Manual for Schenck Hydropuls-System. 24. R. SINGH 1989 Notes on ME675 Course: Design of FluidPower Systems. Department of Mechanical Engineering, The Ohio State University. 25. E. O. DOEBELIN 1972 System Dynamics: Modeling and Response. Columbus: Merrill. 26. B. J. LAZAN 1968 Damping of Materials and Members in Structural Mechanics. Oxford: Pergamon Press. 27. E. O. DOEBELIN 1985 Control System Principles and Design. New York: John Wiley. 28. R. SINGH 1990 Notes on ME850 Course: Dynamics of High Speed Machinery. Department of Mechanical Engineering, The Ohio State University. 29. W. P. FLETCHERand A. N. GENT 1957 British Journal of Applied Physics 8, 194--201. Dynamic shear properties of some rubber-like materials.

APPENDIX A: LIST OF SYMBOLS Ar

A~

be b, Ca Ci, Cll, Ci2

C2, C21, C22 f

f~ ft F

FT F' i

I, k,

equivalent piston area of rubber cross-sectional area of inertia track frequency-variant equivalent damping coefficient damping coefficient of rubber deeoupler compliance lumped compliances of upper chamber lumped compliances of lower chamber frequency (Hz) frequency corresponding to peak loss angle frequency corresponding to peak damping ratio = F ' - F, alternating force of engine excitation transmitted force static load total force

=4-~T inertance of inertia track frequency-variant equivalent spring constant of a mount

HYDRO-MECHANICAL M O U N T kr

K lK(iw)l 1, mE mr mr mls ~ 2

pl,pll,p12 P2, P21, P22 Paml qd, qt

Rd, R, S

t Tr X Xa

O3 COn

243

spring constant of rubber F/x(to), dynamic stiffness dynamic stiffness magnitude inertia track length engine mass A2,I~, analogous fluid mass associated with the inertia track equivalent mass of rubber in shear mode lumped fluid mass of upper and lower chambers perturbation pressures in upper chamber control volumes static pressure perturbation pressures in lower chamber control volumes atmospheric pressure fluid flow rate through decoupler and inertia track fluid resistances of decoupler and inertia track Laplace transformation variable time sinusoidal force transmissibility displacement amplitude of excitation displacement coefficients of transfer function loss angle or dynamic stiffness phase circular frequency (tad/s) natural frequency (rad/s) damping ratio frequency-variant equivalent damping ratio peak damping ratio

Superscripts static component of variable total variable APPENDIX B: MOUNT DATA

Rubber. Upper rubber and bottom chamber diaphragm: duro 51. Decoupler: duro 70. /~, ~ 200 N / m m . Fluid. 50%-50% mixture by volume of anti-freezer (glycol) and water. /u= 3.8 x 10 -3 N.s/m2; p = 1059 kg/m 3. Geometry. A , = 5 0 cm2; di= 5.27 ram; Ii=25 cm; Aa=23 cm2; Ad=0"7 mm.