Journal of Sound and Vibration and Singh_JSV_2012.pdffoundation for the powertrain mounting system. In real-life vehicle systems, the base ﬂexibility may have a signiﬁcant effect

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Journal of Sound and Vibration

Journal of Sound and Vibration 331 (2012) 1498–1518

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Improved torque roll axis decoupling axiom for a powertrainmounting system in the presence of a compliant base

Jin-Fang Hu a,b, Rajendra Singh b,n

a School of Machinery and Automobile Engineering, Hefei University of Technology, Hefei 230009, Chinab Smart Vehicle Concepts Center and Acoustics and Dynamics Laboratory, The Ohio State University, Columbus, OH 43210, USA

a r t i c l e i n f o

Article history:

Received 7 June 2011

Received in revised form

30 September 2011

Accepted 24 November 2011

Handling Editor: S. Ilankoof a compliant base on torque roll axis decoupling is first analytically examined in terms

Available online 20 December 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.11.022

esponding author. Tel.: þ1 614 292 9044; fa

ail address: [email protected] (R. Singh).

a b s t r a c t

The existing torque roll axis decoupling theories for powertrain mounting systems

assume a rigid foundation, thus ignoring dynamic interactions between the powertrain

and other sub-systems. To overcome this deficiency, a coupled mounting system

problem is formulated based on the linear time-invariant system theory. The influence

of eigensolutions and frequency responses. Then, a new analytical axiom is proposed

based on decoupling indices as well as given the properties of the coupling matrix. Five

examples are chosen to examine frequency and time domain responses given the

torque excitation along the crankshaft axis. To satisfy the new condition, the mounting

system is redesigned in terms of the stiffness rates, mount locations, and orientation

angles. The results show that the torque roll axis of the redesigned powertrain

mounting system is indeed decoupled in the presence of a compliant base (given

oscillating or impulsive torque excitation). Finally, eigensolutions are validated by using

published data.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

There is considerable interest in designing the mounting system of vehicle powertrains (or comparable prime movers)to ideally decouple the rotational and translational motions of a powertrain from the torque roll axis direction [1–4]. Jeongand Singh [3] have examined prior torque roll axis (TRA) decoupling design practices and proposed the necessary axioms;for instance, they suggested the roll mode assignment based on an eigenvalue problem for a proportionally dampedsystem consisting of multiple, arbitrarily located mounts. This work has been extended by Park and Singh [4] for anon-proportionally damped system by proposing two eigenvalue problems that must be concurrently satisfied. Completedecoupling between roll and other motions is achieved by using their methods. However, both articles [3,4] assume a rigidfoundation for the powertrain mounting system. In real-life vehicle systems, the base flexibility may have a significanteffect on engine vibration and the forces transmitted to the vehicle body, especially when some vibration modes of thesub-frame, body, or suspension system are excited. This issue has been discussed by Lee et al. [5], Ashrafiuon [6],and Ashrafiuon and Nataraj [7] who claim that the base compliance should be incorporated into analytical or

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x: þ1 614 292 3163.

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dx.doi.org/10.1016/j.jsv.2011.11.022

J.-F. Hu, R. Singh / Journal of Sound and Vibration 331 (2012) 1498–1518 1499

computational models when the excitation frequencies are close to the natural frequencies of body. Earlier, Den Hartog [8]had suggested that the rigid foundation assumption is not justified when the foundation weight is less than the powertrainitself.

Bessac and Guyader [9] have proposed a coupling matrix concept, and the interactions between linear sub-systems arequantified via coupling eigenproperties. Further, Courteille et al. [10] have analytically examined the coupling mechanismsbetween the powertrain and vehicle chassis. Nevertheless, no prior study has examined the TRA decoupling conditions fora powertrain mounting system in the presence of a compliant base. This is the main goal of this work, and as such it wouldextend prior articles [3–10] by proposing new axioms that would account for dynamic interactions between thepowertrain and coupled sub-systems.

2. Problem formulation

Fig. 1 illustrates a coupled powertrain mounting system on a compliant base. Assuming small motions, the powertrainis modeled as a rigid body of time-invariant inertia matrix Mp of dimension 6 [3–7,10–13]. The powertrain is supported by3 or 4 mounts on a compliant base that is defined as a discrete system of dimension Nb. Although various subsets of thecompliant base can be expressed, such as a rigid body only or massless elastic connections, it is assumed that its inertiamatrix Mb and stiffness matrix Kb exist. Each powertrain mount element is described by a set of three tri-axial springelements, and their stiffness values are assumed to be spectrally invariant and insensitive to the excitation amplitude.Like Park and Singh [4], the following three coordinate systems are used to describe the powertrain mounting system:inertial coordinates (XYZ)g, torque roll axis coordinates (XYZ)TRA, and local mount coordinates (XYZ)mi, where i¼1,y,n,where n is the number of powertrain mounts (assumed to be arbitrarily located). The powertrain’s inertial coordinatesystem is assumed to be consistent with that of the compliant base. The powertrain motion is defined by a generalizedvector qpðtÞ ¼ ½Xp Yp Zp yp

X ypY yp

Z �T ðtÞ that includes three translational displacements qp

t ðtÞ and three angulardisplacements qp

yðtÞ of the center of gravity (CG). In this analysis, only the torque excitation Tyy(t) along the powertraincrankshaft axis is considered. First, it is assumed to be harmonic, though the emphasis is placed on the lower frequencyregime (up to 50 or 100 Hz). Since the rigid body modes of the powertrain tend to dominate, a torque impulse is appliedfor transient analysis.

The specific objectives of this paper are as follows: (1) develop improved mathematical models of the coupled systemof Fig. 1 and study the influence of the compliant base on the TRA decoupling of a powertrain mounting system;(2) extend the theory proposed by Bessac and Guyader [9] to study the eigenproperties of the coupling matrix, andpropose a new condition based on the properties of the coupling matrix; (3) propose frequency domain decoupling indicesand a new analytical condition that should yield an uncoupled powertrain motion in the presence of a compliant base;(4) provide illustrative examples in frequency and time domains, as well as a comparison of eigensolutions with publisheddata [5,12].

Fig. 1. Powertrain mounting system with a compliant base.

J.-F. Hu, R. Singh / Journal of Sound and Vibration 331 (2012) 1498–15181500

3. Analysis of uncoupled sub-systems and coupled system

3.1. Powertrain mounting sub-system with a rigid base

Like Jeong and Singh [3] and Park and Singh [4], the powertrain is assumed to be connected to the ground by 3 or 4mounts which could be arbitrarily located at any exterior point and oriented at any angle as shown in Fig. 2(a).The stiffness matrix Kmi of the ith mount is written in the local coordinate system (XYZ)mi:

Kmi ¼

kpi 0 0

0 kqi 0

0 0 kri

0B@

1CA: (1)

Here, kp is the principal compressive stiffness, and kq and kr are the principal shear stiffness. Both matrices aretransformed and expressed with respect to the global coordinate system (XYZ)g using a rotational matrix Hg,mi whichconsists of directional cosines of (XYZ)mi with respect to (XYZ)g. The resulting matrix Kg,mi is

Kg,mi ¼Hg,miKmiHTg,mi ¼

kxxi kxyi kxzi

kxyi kyyi kyzi

kxzi kyzi kzzi

0B@

1CA: (2)

Based on the rigid base assumption, the translational displacements at the ith mount in terms of the powertrainmotions are calculated as qp

i,tðtÞ ¼ qpt ðtÞþqp

yðtÞ � rpi . Here, rp

i ¼ ½rpxi rp

yi rpzi�

T is the position vector of the ith mount (elasticcenter) with respect to the center of gravity of the powertrain. Then, the translational displacements vector can be

Fig. 2. Blocked uncoupled sub-systems: (a) typical blocked uncoupled powertrain mounting sub-system with a rigid base and (b) blocked uncoupled

base mounting sub-system.


expressed by a position matrix, Ppi , as follows:

qpi,tðtÞ ¼ Pp

i qpðtÞ, (3a)

in which

Ppi ¼

I Epi

h i, I¼

1 0 0

0 1 0

0 0 1

264

375, Ep

i ¼

0 rpxi �rp

yi

�rpzi 0 rp

xi

rpyi �rp

xi 0

2664

3775: (3b2d)

The reaction forces fpi,tðtÞ and moments fp

i,yðtÞ as applied by the ith mount on the powertrain body are calculated as

fpi,tðtÞ ¼�Kg,miq

pi,tðtÞ ¼�Kg,miPi

pqpðtÞ, (4)

fpi,yðtÞ ¼ rp

i � fpi,tðtÞ ¼ EpT

i fpi,tðtÞ ¼�EpT

i Kg,miPipqpðtÞ: (5)

Combining Eqs. (4) and (5), the governing equations of the undamped powertrain mounting system with a rigid baseare as follows:

Mp €qpðtÞ�

Xn

i ¼ 1

PpT

i fpi,tðtÞ ¼ fp

ðtÞ: (6)

In order to avoid boundless responses at the resonant frequencies, assume that the damping ratio zp is 3–5 percent.A proportional viscous damping matrix formulation, Cp

¼ apMpþbpKp (with ap and bp calculated from at least 2 modal

damping ratios zp) is used, where Kp¼Pn

i ¼ 1PpT

i Kg,miPpi is the mount system stiffness matrix calculated from Eqs. (4) and

(6). Then, the governing equations of the motion of the powertrain sub-system (of Fig. 2(a)) can be formulated in matrixform, where fp(t) is the generalized external force/torque vector:

Mp €qpðtÞþCp _qp

ðtÞþKpqpðtÞ ¼ fpðtÞ: (7)

3.2. Blocked compliant base mounting sub-system (without any coupling)

Fig. 2(b) shows a blocked uncoupled compliant base structure of dimension Nb with the powertrain mounts. Thegeneralized displacement of the base system is qbðtÞ ¼ ½qb

1ðtÞ � � � qbNb ðtÞ�

T , where the translational displacements qbi,tðtÞ at

the ith mount are given by the following expression:

qbi,tðtÞ ¼ Pb

i qbðtÞ: (8)

Here, Pbi is the position matrix of the ith mount with respect to the compliant base sub-system, and it is determined by the

type of base. Translational reaction forces fbi,tðtÞ on the compliant base due to the elastic constraint forces exerted by the ith

mount are as follows:

fbi,tðtÞ ¼�Kg,miq

bi,tðtÞ ¼ �Kg,miPi

bqbðtÞ: (9)

Thus the governing equations of the blocked, undamped compliant base with powertrain mounts sub-system (ofFig. 2(b)) are:

Mb €qbðtÞþKbqbðtÞ�

Xn

i ¼ 1

PbT

i fbi,tðtÞ ¼ fb

ðtÞ: (10)

Combining Eqs. (9) and (10), assume that the damping ratio zb is 3–5 percent, and introduce a proportional viscousdamping matrix, Cb

¼ abMbþbb½ðKbþKb1

Þ� (with ab and bb calculated from at least 2 modal damping ratios zb), whereKb1¼Pn

i ¼ 1PbT

i Kg,miPbi is the stiffness matrix of the compliant base as induced by the powertrain mounts. The governing

equations of motion are formulated in matrix form, where fb(t) is the external force/torque vector:

Mb €qbðtÞþCb _qb

ðtÞþKbqbðtÞþKb1qbðtÞ ¼ fbðtÞ: (11)

3.3. Coupled powertrain mounting system with a compliant base

As shown in Fig. 1, the mounts connect the powertrain and compliant base sub-systems. Accordingly, the translationalforces fp

i,tðtÞ on the powertrain and fbi,tðtÞ on the compliant base as applied by the ith mount are calculated as follows:

fpi,tðtÞ ¼�Kg,miq

pi,tðtÞþKg,miq

bi,tðtÞ ¼�Kg,miPi

pqpðtÞþKg,miPibqbðtÞ, (12)

fbi,tðtÞ ¼�Kg,miq

bi,tðtÞþKg,miq

pi,tðtÞ ¼�Kg,miPi

bqbðtÞþKg,miPipqpðtÞ: (13)


Thus the equations of the undamped coupled system are expressed in matrix form as

Mp €qpðtÞþKpqpðtÞ�KpbqbðtÞ ¼ fp

ðtÞ, (14)

Mb €qbðtÞþKbqbðtÞþKb1qbðtÞ�KbpqpðtÞ ¼ fb

ðtÞ: (15)

Here, Kpb and Kbp are the coupling stiffness matrices as described by the following:

Kpb¼Xn

i ¼ 1

PpT

i Kg,miPbi , Kbp

¼Xn

i ¼ 1

PbT

i Kg,miPpi : (16a,b)

Further, it can be shown that the reciprocity applies, i.e. Kpb¼KbpT

. Combining Eqs. (14) and (15), and again assumingthe damping ratio zc is 3–5 percent, and a proportional viscous damping matrix Cc, Cc

¼ acMcþbcKc(with ac and bc

calculated from at least 2 modal damping rations zc), the coupled system is now represented in matrix form, wherefcðtÞ ¼ ½fp

ðtÞ fbðtÞ�T is the total external force/torque vector:

Mc €qcðtÞþCc _qc

ðtÞþKcqcðtÞ ¼ fcðtÞ, (17a)

in which,

Mc¼

Mp 0

0 Mb

" #, Kc

¼Kp

�Kpb

�Kbp KbþKb1

" #: (17b,c)

Comparing Eqs. (7), (11) and (17a), we find that the physical system can be treated in terms of two uncoupledsub-systems only when Kpb

¼KbpT

¼ 0.

4. Eigenvalue problems and frequency responses of the coupled system

4.1. Direct solution of the coupled system

Assuming fc(t)¼0 and ignoring Cc, the pre-multiplication of Eq. (17a) by Mc�1

leads to:

€qcðtÞþMc�1

KcqcðtÞ ¼ 0: (18)

Assume the harmonic solution is qcðtÞ ¼ uchðtÞ ¼ ucHeiðotþfÞ, where uc ¼ ½up ub�T is the modal vector, h(t) is harmonicsolution in terms of amplitude H, phase f and frequency o (rad/s). Now reformulate Eq. (18) in the eigenvalue form, wherelc¼oc2

, oc is the natural frequency, and I is the identity matrix :

9Mc�1

Kc�lcI9¼ 0, lcMcuc ¼Kcuc: (19a,b)

Next, determine the frequency responses ~qcðoÞ of coupled system from Eq. (17a) as

~qcðoÞ ¼

~qpðoÞ

~qbðoÞ

" #¼

~Kp�o2Mp

� ~Kpb

� ~Kbp ~K

b�o2Mb

" #�1fpðoÞ

fbðoÞ

" #: (20)

Here, ~Kp¼Kp

þ joCp and ~Kb¼Kb

þKb1þ joCb are the complex valued stiffness formulations of the powertrain and

compliant base sub-systems, respectively, and ~Kpb

and ~Kbp

are the complex valued coupling stiffness matrices. Thefrequency responses of uncoupled blocked sub-systems can be calculated using Eqs. (7) and (11) as shown below:

~qpuðoÞ ¼ ð ~K

p�o2Mp

Þ�1fpðoÞ, ~qb

uðoÞ ¼ ð ~Kb�o2Mb

Þ�1fbðoÞ: (21a,b)

In order to better understand the coupling phenomenon, the frequency responses of coupled system of Eq. (20) areexpressed in the form of a spectrally varying coupling matrix ~DðoÞ and Eqs. (21a,b), this technique is well known andapplied to similar problems [9–11] as

~qpðoÞ

~qbðoÞ

" #¼

~qpuðoÞ~qb

uðoÞ

" #þ ~DðoÞ

~qpðoÞ

~qbðoÞ

" #: (22)

Here,

~DðoÞ ¼O ð ~K

p�o2Mp

Þ�1 ~K

pb

ð ~Kb�o2Mb

Þ�1 ~K

bpO

24

35: (23)

Each coupling matrix term of dimension Nbþ6 represents interactions between the powertrain and the compliant base.

Now, apply only the harmonic torque fpðtÞ ¼ Fp ejot to the powertrain sub-system. The resulting complex-valued frequency

responses of the coupled system are as follows from Eq. (22):

~qpðoÞ ¼ ~qp

uðoÞþð ~Kp�o2Mp

Þ�1 ~K

pb~qbðoÞ, (24a)


~qbðoÞ ¼ ð ~Kb

�o2MbÞ�1 ~K

bp~qpðoÞ: (24b)

Combine the above equations to yield the frequency responses of the powertrain mounting sub-system as

~qpðoÞ ¼ ½I�ð ~Kp

�o2MpÞ�1 ~K

pbð ~K

b�o2Mb

Þ�1 ~K

bp��1 ~qp

uðoÞ: (25)

The above equation suggests that even if no excitation is applied to the base sub-system, the base still affects theresponses of the powertrain sub-system. Thus the compliant base cannot be ignored in the dynamic design process.

4.2. Solution by using eigenvalues and eigenvectors of coupling matrix

The eigenvalues of the coupling matrix ~DðoÞ quantify the coupling strength, and the associated coupling eigenvectors

indicate the modal transmission path [9]. Corresponding to a pair of coupling eigenvalues ~lþ

D ðoÞ ¼ � ~l�

D ðoÞ, a pair of coupling

eigenvectors exists, and their properties are expressed as: ~lþ

D ðoÞ ) ½ ~jpDðoÞ ~jb

D ðoÞ�T , ~l�

D ðoÞ ) ½ ~jpDðoÞ � ~jb

DðoÞ�T . Here,

~jpDðoÞ and ~jb

DðoÞ are the associated coupling eigenvectors of ~lDðoÞ for the powertrain and base sub-systems. The natural

frequencies and modes of both uncoupled powertrain and compliant base sub-systems and the coupled system can be

calculated from ~DðoÞ as demonstrated below. The determinant and eigenvalue problems of ~DðoÞ are written using Eq. (23),where l is the eigenvalue index:

detð ~DðoÞÞ ¼ detð ~KpbÞdetð ~K

bpÞ

detð ~Kp�o2Mp

Þdetð ~Kb�o2Mb

Þ

¼YNbþ6

l ¼ 1

~lDlðoÞ, (26)

ð�o2MpþKpÞ9 ~jp

DlðoÞ9¼1

9 ~lDlðoÞ9Kpb9 ~jb

DlðoÞ9, (27a)

ð�o2MbþKbþKb1

Þ9 ~jbDlðoÞ9¼

1

9 ~lDlðoÞ9Kbp9 ~jp

DlðoÞ9: (27b)

Eq. (26) suggests that detð ~DðoÞÞ ð ~lDla0Þ is boundless at the resonant frequencies ou of uncoupled sub-systems. Define

the highest coupling eigenvalues as ~lmax

D ðoÞ; note that ~lmax

D ðoÞ becomes infinite at ou, and thus the resonant peaks are

smoothed by increasing damping in the components connections [10]. Meanwhile, from Eqs. (27a) and (27b), it is found

that at natural frequencies opu of the uncoupled powertrain mounting sub-system, the associated coupling eigenvectors

9 ~jbDðo

puÞ9� 0,9 ~jp

DðopuÞ9� up

u, where upu are the eigenvectors of the uncoupled powertrain mounting sub-system which are

obtained from the direct solution. Accordingly, at the natural frequencies obu of the uncoupled compliant base mounting

sub-system, 9 ~jpDðo

buÞ9� 0, 9 ~jb

DðobuÞ9� ub

u is obtained. On the other hand, at the natural frequencies oc of the coupled

system, 9 ~lmax

D ðocsÞ9� 1, where s is the modal index of the coupled system, s¼1,y,Nb

þ6. Thus, Eq. (27a) and (27b) are

reformulated in matrix form as

oc2s

Mp 0

0 Mb

" # ~jpDlðo

csÞ

�� ~jb

DlðocsÞ

�� 24

35¼ Kp

�Kpb

�Kbp KbþKb1

" # ~jpDlðo

cs Þ

�� ~jb

Dlðocs Þ

�� 24

35 (28)

Comparing Eq. (28) with (19b), we observe that the coupling eigenvectors 9 ~jpDðo

cÞ9 and 9 ~jbDðocÞ9, as calculated from

~DðocÞ, are exactly the same as up and ub obtained from the direct solution.

4.3. Vibration responses by using eigenvalues and eigenvectors of coupling matrix

The coupling eigenvectors ½ ~jpDðoÞ ~jb

DðoÞ�T associated with the highest coupling eigenvalue ~l

max

D ðoÞ controls theinteractions between sub-systems. The solution of coupled system can now be decomposed in terms of the uncoupledsub-system responses and the coupling eigenfunctions [9] as

~qpðoÞ

~qbðoÞ

" #¼

~qpuðoÞ~qb

uðoÞ

" #þ ~l

max

D ðoÞ~abðoÞ ~jp

DðoÞ~apðoÞ ~jb

DðoÞ

24

35: (29)

Here, ~apðoÞ and ~ab

ðoÞ are the eigenfactors. Using Eqs. (22) and (29), the ~apðoÞ and ~ab

ðoÞ are found to be

~apðoÞ ¼

~apuðoÞþ ~l

max

D ðoÞ ~abuðoÞ

1� ~lmax

D ðoÞ2, ~ab

ðoÞ ¼~ab

uðoÞþ ~lmax

D ðoÞ ~apuðoÞ

1� ~lmax

D ðoÞ2(30a,b)


Here, ~apuðoÞ and ~ab

uðoÞ are the eigenfactors of the uncoupled sub-systems, and they are described as follows:

~apuðoÞ ¼

ðð ~Kb�o2Mb

Þ�1 ~K

bp~qp

uðoÞÞj~l

max

D ðoÞð ~jbDðoÞÞj

, ~abuðoÞ ¼

ðð ~Kp�o2Mp

Þ�1 ~K

pb~qb

uðoÞÞn~l

max

D ðoÞð ~jpDðoÞÞn

: (31a,b)

Here, ð ~jpDðoÞÞn and ð ~jb

DðoÞÞj are the nth and jth associated coupling eigenvectors of ~lmax

D ðoÞ for powertrain and basesub-systems, respectively, n¼1,y,6, j¼1,y,Nb. Substitution of Eqs. (30a,b) and (31a,b) into Eq. (29) leads to the followingexpression for the coupled system response:

~qpðoÞn

~qbðoÞj

24

35¼ ~qp

uðoÞn~qb

uðoÞj

24

35þ 1

1� ~lmax

D ðoÞ2ðð ~K

p�o2Mp

Þ�1 ~K

pb~qb

uðoÞÞnðð ~K

b�o2Mb

Þ�1 ~K

bp~qp

uðoÞÞj

24

35

þ~l

max

D ðoÞ1� ~l

max

D ðoÞ2

ðð ~Kb�o2Mb

Þ�1 ~K

bp~qp

uðoÞÞjð ~jp

DðoÞÞn

ð ~jbDðoÞÞj

ðð ~Kp�o2Mp

Þ�1 ~K

pb~qb

uðoÞÞnð ~jb

DðoÞÞjð ~jp

DðoÞÞn

2664

3775: (32)

Here, ~qpðoÞn and ~qp

uðoÞn are the responses of the nth dimension of the coupled and uncoupled powertrain sub-systems,respectively, and ~qb

ðoÞj and ~qbuðoÞj are the responses of the jth dimension of the coupled and uncoupled base sub-systems,

respectively. Eq. (32) clearly illustrates dynamic interactions within the coupled system as derived from the eigenpro-perties of ~DðoÞ.

5. New TRA decoupling axiom

The torque roll axis (TRA) is uniquely defined by the inertial matrix M of an unconstrained rigid body (with smallmotions) when the torque Ta is applied along the crankshaft axis [3] as follows where g is an arbitrary scalar constant:

qTRA ¼ gM�1Ta: (33)

Two analytical methods will now be used to establish a new TRA decoupling axiom in the presence of a compliant base.

5.1. Method I: TRA decoupling axiom based on the TRA decoupling index

Axiom: If one of the vibration modes of the coupled system is in the torque roll axis direction, the TRA decoupling iscompletely achieved.

Proof: Apply the harmonic torque on the powertrain sub-system as fpðtÞ ¼ Fp ejot . Eq. (17a) is now represented in the

frequency domain (o) as follows where fcðtÞ ¼ ½fp

ðtÞ 0�T :

�o2Mc ~qcðoÞþ joCc ~qc

ðoÞþKc ~qcðoÞ ¼ fc

ðoÞ: (34)

The forced harmonic response of the coupled system (of dimension Nbþ6) is expressed by ~qc

ðoÞ ¼PNb

þ6r ¼ 1

~dc

r ðoÞucr ,

where ucr ¼ ½u

pr ub

r �T are the eigenvectors, ~d

c

r ðoÞ are the modal coordinates, and r is the modal index (r¼1,y,Nbþ6). Using

the orthogonal properties, where r and s are distinct modes, ucT

r Mcucs ¼ drs, ucT

r Kcucs ¼ kc

rdrs, and kcr ¼oc2

r mcr , where

mcr ¼ ucT

r Mcucr , cc

r ¼ ucT

r ðacMcþbcKc

Þucr ¼ acmc

rþbckcr , and f c

r ðoÞ ¼ ucT

r fcðoÞ, we get

�o2 ~dc

r ðoÞþ joðacþbcoc2

r Þ~d

c

r ðoÞþ ~dc

r ðoÞoc2

r ¼ f cr ðoÞ=mc

r : (35)

Then the modal coordinates ~dc

r ðoÞ are calculated from Eq. (35):

~dc

r ðoÞ ¼f c

r ðoÞ=mcr

�o2þ joðacþbcoc2

r Þþoc2

r

: (36)

Next, new frequency domain decoupling indices are proposed (in percent) as follows, where ~qcðoÞn is the harmonic

response of the coupled system in nth dimension:

GnðoÞ ¼9 ~qcðoÞn9PNbþ6

n ¼ 1 9 ~qcðoÞn9¼

PNbþ6

r ¼ 1~d

c

r ðoÞðucr Þn

�� PNbþ6

n ¼ 1

PNbþ6

r ¼ 1~d

c

r ðoÞðucr Þn

�� 100%: (37)

Eq. (37) shows that when Gn¼100 percent, the motion would exist only in the nth dimension as it is completelydecoupled with other motions. In order to achieve perfect TRA decoupling, one of the coupled modes uc

s is assumed to beparallel to the torque roll axis direction and it is defined as qc

TRA ¼ ½qpTRA 0�T . Relate qp

TRA and ups using scalar constants g

and r:

qpTRA ¼ gMp�1

Fp, ups ¼ rqp

TRA, ucs ¼ ½u

ps 0�T : (38a2c)


Combining Eqs. (38a), (38b) and (38c), the harmonic response ~qcr ðoÞ of the coupled system at the rth mode is

~qcr ðoÞ ¼

ucT

r Mcucs

grmcr ð�o2þ joðacþbcoc2

r Þþoc2

r Þuc

r ¼drs

grmcr ð�o2þ joðacþbcoc2

r Þþoc2

r Þuc

r : (39)

Eq. (39) suggests that the above is achieved only when ~qcsðoÞa0 and ~qc

r ðoÞ ¼ 0; thus according to Eq. (37), the TRAdecoupling index is 100 percent at any frequency.

As stated above, when one of the coupled modes is parallel to the torque roll axis direction, perfect TRA decoupling willbe achieved. Assume that only the harmonic torque has been applied to the powertrain sub-system and no externalexcitation is applied to the compliant base sub-system. Further let one of the coupled modes uc

r be in the torque roll axis

direction as ucr ¼ qc

TRA ¼ ½qpTRA 0�T , where qp

TRA ¼ ½0 0 0 0 1 0�T in the TRA coordinate system. The corresponding

eigenvalue problem is reformulated as KcqcTRA ¼ lcMcqc

TRA from Eq. (19b). Thus the new TRA decoupling axiom for the

powertrain mounting sub-system in the presence of a compliant base is given by the following set of equations:

KPqPTRA ¼ lPMPqP

TRA, (40)

KbpqpTRA ¼ 0: (41)

Eq. (40) is the eigenvalue problem using the stiffness matrix of the uncoupled powertrain mounting sub-system. This isindeed the same axiom that is followed by the traditional TRA decoupling method with a rigid base as originally proposedby Jeong and Singh [3]. This implies that when only the harmonic torque is applied to the powertrain sub-system, the

Table 1Natural frequencies of Example 1 with three calculation sets for the coupled system including results given by Sirafi and Qatu [13].

Dominant mode Uncoupled sub-systems, ou (Hz) Coupled system, oc (Hz)

Direct solution

Powertrian

with mounts

Sub-frame

with bushings

Sub-frame with mounts

and bushings

Proposed method Results of Sirafi

and Qatu [13]

Coupling matrix Direct solution

Xp 6.4 N/A N/A 6.0 6.0 5.8

Yp 5.7 5.7 5.7 5.4

Zp 8.3 7.1 7.1 7.0

ypX

11.3 10.5 10.5 10.2

ypY

9.8 9.0 9.0 8.8

ypZ

7.7 7.6 7.6 7.5

Xb N/A 55.1 57.2 57.2 57.2 57.3

Yb 123.2 124.2 124.2 124.2 124.2

Zb 57.3 59.0 59.1 59.1 59.1

ybX

64.0 65.2 65.2 65.2 65.2

ybY

69.4 71.6 71.6 71.6 71.8

ybZ

97.9 98.7 98.7 98.7 98.6

Fig. 3. Powertrain mounting system with a rigid sub-frame supported by 4 bushings.

Fig. 4. Eigenvalues of the coupling matrix ~DðoÞ. Arrows indicate the natural frequencies (in Hz) with respect to the resonant peaks of 9 ~lmax

D ðoÞ9.

Table 2Selected eigenvectors of Example 1 including a comparison of results for coupled system.

Displacement Uncoupled powertrain sub-system Coupled system

9.8 Hz mode 9.0 Hz mode

Coupling matrix method Direct solution Coupling matrix method Direct solution

Xp 0.10 0.10 0.10 0.10

Yp�0.01 �0.01 �0.01 �0.01

Zp 0.03 0.03 0.03 0.03

ypX

0.07 0.07 0.08 0.08

ypY

1.00 1.00 1.00 1.00

ypZ

0.14 0.14 0.02 0.02

Xb 0.00 N/A 0.01 0.01

Yb 0.00 0.00 0.00

Zb 0.00 0.00 0.00

ybX

0.00 0.00 0.00

ybY

0.00 0.04 0.04

ybZ

0.00 0.10 0.10

Table 3Natural frequencies of Examples 2 and 3.

Dominant mode Example 2, or (Hz) Example 3, or (Hz)

Ratio¼kmount/kbushing¼0.1 Ratio¼kmount/kbushing¼10

Base with finite mass Rigid base In series Mb� 10�3Mb

example 1In series Mb

� 10�3Mbexample 1

Xp 0.1 6.4 9.5 6.1 3.0 2.7

Yp 0.2 5.7 6.1 5.5 1.9 1.9

Zp 0.1 8.3 6.9 7.1 2.2 1.6

ypX

12.4 11.3 8.6 10.7 2.7 6.1

ypY

3.0 9.8 4.5 9.1 1.4 3.4

ypZ

2.7 7.7 7.7 7.5 2.4 4.7

Xb 14.6 N/A N/A N/A N/A N/A

Yb 11.0

Zb 18.2

ybX

1.0

ybY

21.9

ybZ

18.7



powertrain motion exists only in the TRA direction. However, when a compliant base exists, Eqs. (40) and (41) must beconcurrently satisfied. Otherwise, the base motions/responses are not zero even if no external excitation is applied to thebase sub-system, and they will affect the responses of the powertrain sub-system. Eq. (41) mainly eliminates the effect ofthe compliant base on the powertrain sub-system, which means the compliant base system must not exert any constraintforce on the powertrain sub-system. Eqs. (40) and (41) imply that when only the harmonic torque is applied to thepowertrain sub-system, no force is induced within the coupling stiffness Kbp. The powertrain motion exists only in the TRA

Fig. 5. Simplified powertrain sub-system with focalized mounting system (Example 4).

Fig. 6. Effect of a compliant base on the frequency response functions for Example 4: (a) Xp(o); (b) Yp(o); (c) Zp(o); (d) ypX ðoÞ; (e) yp

Y ðoÞ; (f) ypZ ðoÞ.

Key: , with a rigid base; , with a compliant base (sub-frame and bushings sub-system).


direction and the base motions are zero. Thus, a complete TRA decoupling of the motions is achieved. From the dynamicdesign perspective, Eq. (41) also indicates that only the position matrix Pb

i of the ith mount (i¼1,y,n) with respect to thecompliant base must be known and designed. This method is very promising for practical mounting system design,because it is difficult to posses all of the information regarding the sub-frame and chassis systems in the early stages ofvehicle development.

5.2. Method II: TRA decoupling axiom based on coupling matrix ~D

If the base sub-system does not interact with the powertrain sub-system, the two formulations from Eq. (32) must beconcurrently satisfied:

ð ~Kp�o2Mp

Þ�1 ~K

pb~qb

uðoÞ ¼ 0 (42a)

ð ~Kb�o2Mb

Þ�1 ~K

bp~qp

uðoÞ ¼ 0: (42b)

Assume there is no external excitation in the compliant base sub-system; this implies that ~qbuðoÞ ¼ 0. Thus Eq. (42a) is

automatically satisfied. Yet another way Eq. (42b) can be satisfied is when ~Kbp~qp

uðoÞ ¼ 0. On the other hand, if theuncoupled powertrain mounting sub-system satisfies the traditional TRA decoupling condition, as given by~qp

uðoÞ ¼ ~dp

r ðoÞqpTRA, then Kpqp

TRA ¼ lcMpqpTRA. This leads to Kbpqp

TRA ¼ 0 since Cc is a proportional viscous damping matrix.This is indeed the same condition as Eq. (41) which uses the TRA decoupling index method. Finally, according to Eqs. (23)

Fig. 7. Effect of a compliant base on the impulsive responses for Example 4: (a) Xp(t); (b) Yp(t); (c) Zp(t); (d) ypX ðtÞ; (e) yp

Y ðtÞ; (f) ypZ ðtÞ. Key: , with a

rigid base; , with a compliant base (sub-frame and bushings sub-system).

Fig. 8. Frequency response functions for Example 4 with a redesinged mounting system based on the new axiom: (a) Xp(o); (b) Yp(o); (c) Zp(o);

(d) ypX ðoÞ; (e) yp

Y ðoÞ; and (f) ypZ ðoÞ. Key: , original with a rigid base; , new design with a compliant base (sub-frame and bushings sub-system).

Table 4Redesigned mount parameters for Examples 4 and 5 to illustrate the new TRA decoupling axiom in the presence of the compliant base.

Mount parameters Mount #

1 2 3 4

Example 4Stiffness (N mm�1) kp 440 440 440 440

kq 336 336 336 336

kr 506 506 506 506

Location (mm) rpx �198 198 �198 198

rpy 128 128 �128 �128

rpz �73.5 73.5 �73.5 73.5

rbx

�236 �203 134 101

rby

278 278 �278 �278

rbz

�44.4 �44.4 �44.4 �44.4

Example 5Stiffness (N mm�1) kp 240 240 240 240

kq 196 196 196 196

kr 176 176 176 176

Location (mm) rpx 231 �231 231 �231

rpy �913 �834 854 893

rpz 110 �110 �138 �138

Orientation (deg.) a 44 44 44 44



and (42b), the new axiom can also be expressed as follows:

KpqpTRA ¼ lcMpqp

TRA, (43)

~Dj5ðoÞ ¼ 0, j¼ 1,2,. . .,Nbþ6: (44)

Eq. (44) suggests that one of the coupling eigenvalues is ~lDðoÞ ¼ 0, and the associated coupling eigenvectors are~jp

DðoÞ ¼ qpTRA and ~jb

DðoÞ ¼ 0. This implies that there must be no energy exchange between the two sub-systems in the TRAdirection. Since the coupling matrix ~DðoÞ is formulated in the frequency domain, this method is very useful whenspectrally varying mount properties must be incorporated [14].

6. Coupled system eigenvalue problems (Examples 1, 2 and 3)

6.1. Example 1

As shown in Fig. 3, the compliant base in this case is composed of a rigid sub-frame (of dimension 6) and 4 elastomeric

bushings. The position matrix PbiðkÞ for the ith mount or kth bushing with respect to the sub-frame is written as follows

where rbiðkÞ ¼ ½

rbxiðkÞ rb

yiðkÞ rbziðkÞ �

T is the position vector of the ith mount or kth bushing with respect to the center of gravity

Fig. 9. Impulsive responses of Example 4 with redesifned mounting system based on the new axiom: (a) Xp(t); (b) Yp(t); (c) Zp(t); (d) ypX ðtÞ; (e) yp

Y ðtÞ; and

(f) ypZ ðtÞ. Key: , original design with a rigid base; , new design with a compliant base (sub-frame and bushings sub-system).


of the sub-frame

PbiðkÞ ¼ I Eb

iðkÞ

h i, Eb

iðkÞ ¼

0 rbxiðkÞ �rb

yiðkÞ

�rbziðkÞ 0 rb

xiðkÞ

rbyiðkÞ �rb

xiðkÞ 0

26664

37775: (45a,b)

The stiffness matrix Kb of the sub-frame and bushings is expressed by Kb¼P4

k ¼ 1Kbk , where Kb

k is the stiffness matrix ofthe kth bushing:

Kbk ¼ PbT

k kbkPb

k , kbk ¼

kbxk 0 0

0 kbyk 0

0 0 kbzk

0BBB@

1CCCA: (46a,b)

Sample natural frequencies of the uncoupled sub-systems and coupled system are shown in Table 1; this datasetcorresponds to the nominal values reported by Sirafi and Qatu [13]. The natural frequencies of the powertrain sub-systemin the current coupled system are slightly higher than those predicted by Sirafi and Qatu [13]. This is because the proposedmethod treats the sub-frame as a rigid body instead of the finite element method used by Sirafi and Qatu [13]. The resultsof Table 1 also show that the natural frequency range of the uncoupled sub-frame with bushings is much higher than theuncoupled powertrain mounting sub-system. However, since the mass of the sub-frame (mb

¼52.1 kg) is about 1/4 of thepowertrain mass (mp

¼214 kg), the base indeed affects the natural frequencies of the uncoupled powertrain mountingsub-system as suggested by Den Hartog [8]. Nevertheless, the dynamic coupling is not very strong as the natural frequencyranges of the two uncoupled sub-systems do not overlap.

The highest eigenvalues 9 ~lmax

D ðoÞ9 of ~DðoÞ are shown in Fig. 4. These are compared with the direct solution in Table 1;

the frequencies o with respect to the resonant peaks of 9 ~lmax

D ðoÞ9 are the same as ou which are obtained from the direct

Fig. 10. Torque roll axis decoupling indices for Example 4: (a) Xp(o); (b) Yp(o); (c) Zp(o); (d) ypX ðoÞ; (e) yp

Y ðoÞ; and (f) ypZ ðoÞ. Key: , with a rigid

base; , with a compliant coupled base ; , with a compliant decoupled base.


solution of the uncoupled sub-systems. The frequencies corresponding to 9 ~lmax

D ðoÞ9¼ 1 are almost the same as oc, which

are predicted by the direct solution of the coupled system. For the sake of illustration, consider the dominant ypY mode for

both uncoupled sub-systems and coupled system. Predicted eigenvectors are compared in Table 2. Observe that thecoupling matrix method yields the same eigenvectors as the direct solution. Since the uncoupled sub-systems are weaklycoupled, the eigenvectors of the powertrain in the uncoupled and coupled systems are almost the same.

6.2. Examples 2 and 3

Two additional examples are shown below to further relate the coupled system with two uncoupled sub-systems.In Example 2, it is assumed that the sub-frame is a rigid body. To simulate this, modify the stiffness matrix of Example 1as: Kb

example 2 � 10�3Kbexample 1. The mass matrix Mb of the base is expressed as follows:

Mb¼

mb 0 0 0 0 0

0 mb 0 0 0 0

0 0 mb 0 0 0

0 0 0 Ibxx �Ib

xy �Ibxz

0 0 0 �Ibxy Ib

yy �Ibyz

0 0 0 �Ibxz �Ib

yz Ibzz

0BBBBBBBBBB@

1CCCCCCCCCCA: (47)

Here, m is the sub-frame mass, and I is its moment of inertia. Of course when Mb-N, the rigid base assumption canbe made.

Table 5Natural frequencies of Example 5 (as shown in Fig. 11). Results for uncoupled chassis sub-systems are compared with those reported by Lee et al. [5].

Mode

index

Uncoupled

powertrain, opu (Hz)

Uncoupled chassis,

obu (Hz) by our method

Uncoupled chassis by

Lee et al. [5], obu (Hz)

Coupled system, oc

(Hz) by our method

1 6.0 1.7 1.7 1.6

2 6.8 2.4 2.4 2.7

3 8.2 3.4 3.4 2.8

4 12.0 5.1 5.0 5.3

5 16.5 8.6 8.5 6.7

6 26.4 10.2 10.2 8.0

7 N/A 10.3 10.3 8.6

8 13.8 14.0 8.9

9 14.3 14.4 10.2

10 15.0 14.8 10.3

y y y y

31 194.6 194.6 60.1

y y

37 194.9

Fig. 11. Powertrain mounting system in a simplified vehicle model (Example 5).


In Example 3, the sub-frame is assumed to be massless but supported by four discrete bushing elements. This issimulated as: Mb

example 3 � 10�3Mbexample 1.

The natural frequencies of the uncoupled sub-systems and coupled system (for Examples 2 and 3) are shown in Table 3.The results of Example 2 show that the natural frequencies of both powertrain and rigid sub-frame change significantly in

Fig. 12. Effect of a compliant base on the frequency response functions for Example 5: (a) Xp(o); (b) Yp(o); (c) Zp(o); (d) ypX ðoÞ; (e) yp

Y ðoÞ; (f) ypZ ðoÞ.

Key: , with a rigid base; , with a compliant base (flexible chassis sub-system).

Table 6Eigenvectors of the uncoupled sub-systems and coupled system using 2 methods for Example 5.

Displacement Uncoupled powertrain sub-system Coupled system

12.0 Hz mode 11.1 Hz mode

Coupling matrix method Direct solution Coupling matrix method Direct solution

Xp 0.00 0.00 �0.01 �0.01

Yp 0.00 0.00 0.00 0.00

Zp 0.00 0.00 �0.01 �0.01

ypX

0.00 0.00 �0.02 �0.02

ypY

1.00 1.00 �1.00 �1.00

ypZ

0.00 0.00 0.01 0.01

qb1

0.00 N/A �0.01 �0.01

qb2

0.00 0.00 0.00

qb3

0.00 0.04 0.04

qb4

0.00 0.00 0.00

qb5

0.00 �0.03 �0.03

qb6

0.00 0.01 0.01


the coupled system (when compared with the results of Example 1), especially for the X, Y, Z, yY , and yZ modes. Thisimplies that a strong coupling exists between the powertrain and compliant base in these directions. Note that the resultsof Example 3 do not correlate well with the simple formula of springs in series (in one direction only). The chief reason isthat there are 4 multidimensional mounts and 4 bushings in the coupled system, and it is not clear which mount andbushing pairs should be in series; in fact, there are 4 choices of each mount in any given direction. Using the proposedmethod, it is observed that when the stiffness ratio (kmount/kbushing) is low, weak coupling is found and thus the naturalfrequencies of the powertrain are virtually unchanged. Conversely when the stiffness ratio is high, strong coupling isevident and that would significantly alter the natural frequencies of powertrain in the coupled system.

7. Frequency and impulse responses of a focalized mounting system (Example 4)

A simplified powertrain with the focalized mounting system [4] is considered next, as shown in Fig. 5. It is the mostdesired case for a mounting system (when connected to the ground) as it would yield a complete TRA decoupling given thetorque excitation. The rigid sub-frame supported by 4 bushings is considered next and connected to a focalized mountingsystem (like Fig. 3). The powertrain mounting system parameters are: mass mp

¼214 kg; moment of inertia (kg m2)IpXX ¼ 15:32, Ip

YY ¼ 6:6, IpZZ ¼ 12:76; inertia product (kg m2) Ip

XY ¼ IpXZ ¼ Ip

YZ ¼ 0; stiffness kp¼8.4�105 N m�1; stiffness rateratio Lk(kp/kq)¼2.5; and mount orientation a¼ 303. Assuming 4 identical bushings, the sub-frame and bushingsparameters are: mass mb

¼52.1 kg; moment of inertia (kg m2) IbXX ¼ 3:8, Ib

YY ¼ 1:6, IbZZ ¼ 3:1; inertia product (kg m2)

IbXY ¼ Ib

XZ ¼ IbYZ ¼ 0; the bushing stiffness (N m�1) kb

x ¼ 1:2� 106, kby ¼ 1:0� 106, and kb

z ¼ 9� 105.Frequency response functions for this example, given harmonic excitation along the crank axis, are calculated for

both rigid and compliant bases, respectively, as shown in Fig. 6. Next, an impulse torque is applied about the crankshaft

Fig. 13. Effect of a compliant base on the impulsive responses for Example 5: (a) Xp(t); (b) Yp(t); (c) Zp(t); (d) ypX ðtÞ; (e) yp

Y ðtÞ; and (f) ypZðtÞ. Key: ,

with a rigid base; , with a compliant base (flexible chassis sub-system).


axis direction and the resulting transient responses are shown in Fig. 7, again for both cases. Observe that a completedecoupling of motions is achieved (only the pitch motion yp

Y exists) by the focalized mounting scheme for the rigidbase case. However, vibrations are coupled with an introduction of sub-frame and bushings system as shown inFigs. 6 and 7.

Now the mounting system of Example 4 is redesigned by selecting the appropriate mount parameters to illustratethe effectiveness of the new TRA decoupling axiom in the presence of a compliant base; redesigned mount properties suchas stiffness rates and locations (with respect to the center of gravity of the powertrain and sub-frame) are listed in Table 4.Observe a complete decoupling of the motions in Figs. 8 and 9 in both frequency and time domains as shown. For the sakeof comparison, decoupled responses with a rigid base are also depicted. Note that the magnitude of the pitch motionincreases whenever the TRA decoupling scheme (with a sub-frame and bushings sub-system) is applied, and its naturalfrequency of the pitch mode (with a compliant base) is slightly lower than the rigid base case, as expected.

Fig. 10 shows the spectral contents of the TRA decoupling indices GTRA for rigid and compliant bases. This analysisshows that the TRA decoupling index in the torque roll axis direction is 100 percent at any frequency with a rigid base withthe original design, but with an introduction of the sub-frame and bushings sub-system, the GTRA varies with frequencyas shown in Fig. 10. However, the TRA decoupling index is again 100 percent with the redesigned mount parameters(of Table 4).

8. Vehicle model study (Example 5)

As shown in Fig. 11, the compliant vehicle base is now described by a 31 degree of freedom vehicle chassis systemmodel with body, suspension system, and axle-tire sub-systems. The body itself is described by 9 discretized masses, with3 translations per lump (X, Y, Z). Only the vertical motions of axle-wheel components are however considered to primarilydescribe low frequency modes.

Fig. 14. Torque roll axis decoupling indices for Example 5: (a) Xp(o); (b) Yp(o); (c) Zp(o); (d) ypX ðoÞ; (e) yp

Y ðoÞ; and (f) ypZ ðoÞ. Key: , with a rigid

base; , with a compliant coupled base ; , with a compliant decoupled base.


Assume mounts 1 and 2 are connected to the body mass m2; their translational displacement vector in the chassis

sub-system is expressed as: qb1,tðtÞ ¼ qb

2,tðtÞ ¼ Pb1i qbðtÞ. Likewise, mounts 3 and 4 connected to body mass m3; their trans-

lational displacement vector is: qb3,tðtÞ ¼ qb

4,tðtÞ ¼ Pb2i qbðtÞ. Rewrite the expressions in Eq. (16a) as: Kpb

¼P

i ¼ 1,2PpT

i Kg,miPb1i þP

i ¼ 3,4 PpT

i Kg,miPb2i . Here, Pb1

i ¼ I O1� �

and Pb2i ¼ O2 I O3

� �, O1 is a null matrix (3�28), O2 is a null matrix (3�3) and O3

is a null matrix (3�25) as well. Using the TRA decoupling axiom [3], powertrain mounting system parameters are selectedas follows: mass mp

¼211.5 kg; moment of inertia (kg m2) IpXX ¼ 16:75, Ip

YY ¼ 8:26, IpZZ ¼ 17:03; inertia products (kg m2)

IpXY ¼�1:77, Ip

XZ ¼�4:82, IpYZ ¼�1:25; stiffness kp¼8.4�105 N m�1; stiffness rate ratio Lk(kp/kq)¼2.5; and mount orientation

angle a¼ 303.First, the compliant chassis model is verified by comparing its natural frequencies with the finite element results of

Lee et al. [5]. As shown in Table 5, a good agreement is seen between the proposed analytical method and the eigenvaluesreported by Lee et al. [5]. The vehicles modes have a significant effect on the natural frequencies of the powertrain sub-system since their natural frequencies overlap. One particular mode (pitch in the vehicle coordinate system) calculated byusing the direct and coupling matrix methods is compared in Table 6; observe that the coupling matrix method yields thesame eigenvectors as the direct solution for both uncoupled and coupled systems. The eigenvectors of the coupled systemillustrate some coupling though the vehicle pitch remains a primary mode.

Frequency and impulsive responses along with TRA decoupling indices are shown next in Figs. 12–14. Theseindicate that a complete decoupling is achieved by the original TRA decoupling scheme for the rigid base, but thepowertrain motions are coupled again with the vehicle chassis sub-system as illustrated in Fig. 14. Significant effect on theengine vibration is seen in Fig. 12, especially when the flexible modes of the vehicle body are excited. However, likeExample 4, the mounting system is redesigned (stiffness rates, locations with respect to the center of gravity of thepowertrain, and orientation of mounts as listed in Table 4). A complete decoupling of the motions is achieved again, asshown in Figs. 14–16. Now motion exists only in the pitch direction (in the vehicle coordinate system) with the redesignedsystem.

Fig. 15. Frequency response functions for Example 5 with a redesinged mounting system based on the new axiom: (a) Xp(o); (b) Yp(o); (c) Zp(o);

(d) ypX ðoÞ; (e) yp

Y ðoÞ; and (f) ypZ ðoÞ. Key: , original with a rigid base; , new design with a compliant base (flexible chassis sub-system).

Fig. 16. Impulsive responses of Example 5 with redesifned mounting system based on the new axiom: (a) Xp(t); (b) Yp(t); (c) Zp(t); (d) ypX ðtÞ; (e) yp

Y ðtÞ;

(f) ypZ ðtÞ. Key: , original design with a rigid base; , new design with a compliant base (flexible chassis sub-system).


9. Conclusion

In this paper, the powertrain mounting system problem as originally proposed by Jeong and Singh [3] has beenreformulated to study the effects of the compliant base on the TRA decoupling of a powertrain mounting system. Themodal properties and frequency responses of the coupled system have been derived from the eigenproperties of thecoupling matrix. A new analytical axiom for the TRA decoupling has been proposed based on the frequency decouplingindices and a spectral coupling matrix, but the latter method should be more useful when spectrally varying mountproperties are considered. Five examples are chosen to illustrate the proposed theory and to gain some insights ofeigensolutions. The complete decoupling of the motions is achieved by using the new condition. The eigensolutions of twoexamples are also compared with published data [5,13] and predictions are satisfactory. Modal, frequency, transientresponses clearly highlight interactions; these are then confirmed by TRA decoupling indices.

Chief contribution of this article is to extend the prior theories [3–4,9] by proposing new analytical conditions andeigenproperties of coupling matrix. The results of this study should lead to an examination of both motion control and vibrationisolation issues at lower frequencies. Complications arising from spectrally varying and amplitude-sensitive stiffness anddamping properties of the engine mounts, as well as of sub-frame bushings, should be examined in future studies. Yet anotherarea is the application of screw theory [2,15] to achieve decoupling in the presence of a compliant chassis.

Acknowledgments

We are grateful to the member organizations of the Smart Vehicle Concepts Center (www.SmartVehicleCenter.org) andthe National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc)

www.SmartVehicleCenter.org




www.nsf.gov/eng/iip/iucrc





for supporting this work. The China Scholarship Council (CSC) (www.cscse.edu.cn) is also acknowledged for supportingthis work.

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Journal of Sound and Vibration and Singh_JSV_2012.pdffoundation for the powertrain mounting system. In real-life vehicle systems, the base ﬂexibility may have a signiﬁcant effect