A method for calculating the displacements of a powrtrain and mounts in a powertrain mounting system...

A method for calculating the displacements of apowertrain and mounts in a powertrain mountingsystem including torque struts under quasi-static loadsWen-Bin Shangguan1,2* and Da-Ming Chen1

1School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, Guangdong,

People’s Republic of China2State Key Laboratory of Automobile Dynamic Simulation, Jilin University, Changchun, Jilin, People’s Republic of China

The manuscript was received on 11 July 2011 and was accepted after revision for publication on 7 September 2011.

DOI: 10.1177/0954407011424816

Abstract: A modelling and calculation method for obtaining the displacements of the power-train’s centre of gravity (CG) and mounts in a powertrain mounting system (PMS) includingthe torque struts under a static or quasi-static load is presented. A new mount model that ismodelled as three-dimensional (3D) springs with both a translational stiffness along each axisof its local coordinate system (LCS) and a rotational stiffness around each axis of its LCS ispresented. Also a torque strut is a generic construction and is regarded as a massless rigidbody with a mount at each of its two ends.

The equation and the solution method for estimating the displacements of the powertrain’sCG and the mounts under a quasi-static load are developed when torque struts exist in a PMS.An example is given of the calculation and comparison of the displacements of the power-train’s CG and mounts in a generic PMS including a torque strut using both the method pro-posed in this paper and the conventional method published in the referenced papers.

Keywords: automotive powertrain mounting system, torque strut, displacements of the

powertrain’s centre of gravity and mounts, mount model with both a translational and a

rotational stiffness

1 INTRODUCTION

As transversely mounted and front-wheel-drive

powertrains are widely used for cars [1–3], a torque

strut is frequently used in this configuration of a

powertrain mounting system (PMS) [4]. The config-

uration of a transversely mounted powertrain and

the structure of a torque strut are shown in Figs 1(a)

and 1(b) respectively. A torque strut has two

mounts: mount 1 is connected with the powertrain,

and mount 2 is connected with the body or cradle.

The two mounts can be rubber mounts or hydraulic

mounts [5].

A torque strut is mainly used to control the rota-

tional motion of the powertrain around the crank-

shaft direction when the powertrain bears a large

torque in the crankshaft direction or a large impact

load in the vehicle’s longitudinal direction [4]. The

advantage of a torque strut is that it provides linear

stiffness only in the direction of P1P2, and torsional

stiffness around the mount axis, where the points P1

and P2 are the centres of mount 1 and mount 2

respectively. Since the torque strut in a PMS is usu-

ally installed in a horizontal plane (the line P1P2 is

in horizontal plane), the vertical stiffness that a tor-

que strut provides to a PMS is greatly decreased,

thus enhancing the isolation of a PMS.

In designing a PMS, two basic strategies are used

frequently. One strategy is to move the six natural

frequencies of the powertrain rigid body modes

away from the frequencies of the input sources,

and to decouple the powertrain vibration in six

*Corresponding author: School of Mechanical and Automotive

Engineering, South China University of Technology, Guangzhou,

Guangdong 510641, People’s Republic of China.

email: [email protected]

634

Proc. IMechE Vol. 226 Part D: J. Automobile Engineering

directions for each vibration mode in the best way

possible [6–12]. Many methods can be used to

design mounts to meet the requirement of this strat-

egy [6–12]. The other strategy is to determine the

force versus displacement relation of a mount in its

local coordinate system (LCS) to limit the motion of

a powertrain under an external static or quasi-static

load [12–14]. Since the relation between the force

and the displacement of a mount in its LCS is usu-

ally non-linear, it is simplified as piecewise linear in

five ranges [13]. The stiffness and corresponding

displacement ranges in the force versus displace-

ment curve of a mount can be easily implemented

by the structural design of a mount.

One of the present authors and a co-worker [13]

developed a method for calculating the displace-

ments by simplifying the non-linear relation of the

force versus the displacement as piecewise linear in

five ranges. Swanson et al. [14] presented a method

to calculate the displacements of the powertrain’s

CG and mounts if the force versus displacement

relation of a mount in its LCS is linear and thus the

stiffness is constant. In reference [13], the mount is

modelled as three-dimensional (3D) springs with

only a translational stiffness along each axis of its

LCS. Also the assumption is made that there is no

torque strut in the PMS. The displacements can also

be estimated using the method proposed in refer-

ence [13] if a torque strut installed in a PMS is sim-

plified as one conventional mount that is connected

with the powertrain. However, this simplification

causes errors if the motion of the powertrain is not

small. When the powertrain undergoes a large

motion, the torque strut in a PMS can provide tor-

sional stiffness to a powertrain.

This study is the extension of previous work [13].

An enhancement of this paper with respect to refer-

ence [13] is that a new mount model is presented

and a torque strut is included in a PMS when calcu-

lating the displacements of the powertrain’s CG and

mounts. The new mount model includes the rota-

tional stiffness around each axis of its LCS. The tor-

que strut is a generic construction that makes no

simplification and is regarded as a massless rigid

body with two mounts at its two ends.

The relation between the force (or torque) and

the displacement (or angle) of a mount in its LCS is

also simplified in this paper as piecewise linear in

five ranges, as in reference [13]. The equation and

the solution method for estimating the displace-

ments of the powertrain’s CG and mounts under a

quasi-static load are developed when torque struts

exist and the new mount model is utilized in a PMS.

An example is given to calculate the displacements

of the powertrain’s CG and mounts in a generic

PMS including a torque strut under some typical

and extreme load cases using the two different

methods. It is concluded that the displacements and

reaction forces of a mount are different when the

powertrain bears a large torque around the crank-

shaft direction or a large load along the vehicle’s

longitudinal direction (the impact load case) using

different methods. Only the rotational stiffness of

mount 1 on the torque strut connected with the

powertrain needs to be considered when calculating

the displacements under extreme loads.

2 MODELLING OF A POWERTRAIN MOUNTING

SYSTEM

A sketch of a PMS including a torque strut is shown

in Fig. 2. The PMS has a transverse powertrain (an

engine and a transmission), two mounts (an engine

mount and a transmission mount), and a torque

strut. There are two mounts, namely mount 1 and

(a) Powertrain mounting system (b) Torque strut

Eng mountTrans mount

Torque strut

Fig. 1 The configuration of (a) a powertrain mounting system with (b) a torque strut

A method for calculating the displacements of a powertrain and mounts in a powertrain mounting system 635


mount 2, on the torque strut. Mount 1 is connected

with the powertrain, and mount 2 is connected with

the body or cradle.

A right-hand powertrain coordinate system (PCS),

denoted OpXpYpZp, is built to describe the move-

ment of the powertrain, with its origin Op located at

the powertrain’s CG under static equilibrium. Here,

the static equilibrium is defined as the position of

powertrain at rest under its dead weight. The three

orthogonal coordinate axes are set with the Xp and

Yp axes parallel to the horizontal plane, with the Zp

axis normal to the plane, and where the positive

direction of the Xp axis points to the rear of the vehi-

cle. Also a right-hand torque strut coordinate system

(SCS), denoted OsXsYsZs, is established to describe

the movement of the torque strut, with its origin Os

located at the torque strut’s CG under static equili-

brium of the powertrain. The Xs, Ys, and Zs axes are

parallel to the Xp, Yp, and Zp axes respectively.

The powertrain is modelled as a rigid body with

six degrees of freedom, including three translation

and three rotation motions along and around each

axis in PCS. Therefore the movement of the power-

train’s CG in the PCS can be described as

fqpg= (xp, yp, zp, ap, bp, gp)T ð1Þ

where xp, yp, and zp are the translational displace-

ments along each axis respectively of the PCS, and

ap, bp, and gp are the rotational angles around each

axis respectively of the PCS.

The mass of a torque strut can be ignored in

comparison with the mass of the powertrain.

Therefore the torque strut is modelled as a massless

rigid body. The movement of a torque strut CG (the

midpoint of the line P1P2 is usually regarded as the

CG of a torque strut) in its SCS can be described as

fqsg= (xs, ys, zs, as, bs, gs)T ð2Þ

where xs, ys, and zs are the translational displace-

ments along each axis respectively of its SCS and as,

bs, and gs are the rotational angles around each axis

respectively of its SCS.

The number of mounts excluding the mounts on

the torque strut is assumed to be Nm in a PMS. An

LCS, denoted omiumivmiwmi (i = 1, 2, ., Nm) is built

for each mount, with its origin omi at the elastic cen-

tre of the mount, and the three orthogonal coordi-

nate axes are expressed as umi, vmi, and wmi

respectively. The number of torque struts in a PMS

is assumed to be Ns. Two LCSs, denoted

osjm1usjm1vsjm1wsjm1 and osjm2usjm2vsjm2wsjm2 (j = 1,

2, ., Ns), are built for mount 1 and mount 2 respec-

tively on the torque strut j, with the origins osjm1

and osjm2 at the elastic centres of mount 1 and

mount 2 respectively.

Usually a mount is modelled as 3D linear springs

in its LCS [1–15]. In this paper, a mount is modelled

as 3D springs with both a translational stiffness

along and a rotational stiffness around each axis of

its LCS and is shown in Fig. 3. The translational stiff-

ness along the axes of its LCS are represented by ktu,

ktv, and ktw respectively. The rotational stiffness

around the axes of the mount’s LCS are represented

by kru, krv, and krw respectively.

The relation between the force (or torque) and

the displacement (or angle) of a mount in one direc-

tion of its LCS is usually non-linear and is simplified

as piecewise linear in five ranges in these studies

[13]. Figure 4 shows the piecewise linearity in five

ranges. The force (or torque) versus displacement

(or angle) relation in Fig. 4 is described as

F = kx + D ð3Þ

Fig. 3 A mount model

Fig. 2 A PMS including a torque strut (top view) (Eng,engine; Trans, transmission)

636 Wen-Bin Shangguan and Da-Ming Chen


where k and D are the stiffness (translational or

rotational) and the correction term respectively, and

their values can be obtained using the formulae in

Table 1 when the displacement x is determined.

3 EQUATIONS OF THE POWERTRAIN–TORQUE

STRUT SYSTEM

3.1 Equation of the powertrain

The quasi-static load including the force and the

torque applied to the powertrain’s CG in the PCS is

assumed to be {F}, and it is a (6 3 1) vector. The dis-

placements of a powertrain’s CG and of the CG of

torque strut j under the load {F} are {qp} and {qsj}

respectively. If the displacements of the power-

train’s CG is {qp} in the PCS, then the displacements

of mount i in PCS are calculated from [13, 14]

Upmi

� �f

pmi

� �8<:

9=;=

I, ~Rpmi

T

0, I

24

35fqpg ð4Þ

where the superscript p denotes that the values are

defined in the PCS, T denotes the transpose of a

matrix, Upmi

� �and f

pmi

� �are the translational dis-

placements and the rotational angles respectively of

mount i in the PCS, I is a (3 3 3) unity matrix, and 0

is a (3 3 3) zero matrix.~Rp

mi in equation (4) is a (3 3 3) skew-symmetric

matrix of the position vector Rpmi for mount i in the

PCS, i.e. if Rpmi is given by

Rpmi = x

pmii + y

pmi j + z

pmik ð5Þ

Then the skew-symmetric matrix ~Rpmi is expressed

as [13]

~Rpmi =

0 �zpmi y

pmi

zpmi 0 �x

pmi

�ypmi x

pmi 0

266664

377775 ð6Þ

where xpmi, y

pmi, and z

pmi are the coordinates of mount

i in the PCS.

The displacements of mount i in its LCS are then

obtained using

�Umi

� ��fmi

� �� =

Ami½ �, 00, Ami½ �

� �I, ~Rp

mi

T

0, I

� �fqpg ð7Þ

where �Umi

� �and �fmi

� �are the translational displa-

cements of mount i along each axis and the rota-

tional angles of mount i around each axis

respectively of its LCS, and [Ami] is a (3 3 3) transfor-

mation matrix from the PCS to the LCS of mount i.

The transformation matrix can be easily formed

from the orientation of a mount with respect to the

PCS, which can be defined using Euler angles, direc-

tion cosines, etc. [14].

The forces applied to mount i in the PCS due to

the movement of the powertrain are

Fpmi

� �M

pmi

� �8<:

9=;=

Kpmit

� , 0

0, Kpmir

� 24

35 I, ~R

pmi

T

0, I

24

35fqpg

+Ami½ �T, 0

0, Ami½ �T

24

35fDmig ð8Þ

where Fpmi

� �and M

pmi

� �are the force vector (3 3 1)

and the torque vector (3 3 1) respectively applied to

mount i in the PCS, where

½K pmit�= ½Ami�T½Kmit�½Ami� ð9Þ

and

½K pmir�= ½Ami�T½Kmir�½Ami� ð10Þ

Fig. 4 Force versus displacement relation of a mountin one direction of its LCS

Table 1 Values for k and D

x range k D

Range 1 ( –N, a] k1 – k1a + k2(a – b) + k3bRange 2 (a, b] k2 –k2b + k3bRange 3 (b, c] k3 0Range 4 (c, d] k4 k3c – k4cRange 5 (d, + N) k5 k3c – k4c + k4d – k5d



are the translational stiffness matrix and rotational

stiffness matrix respectively of mount i in the PCS,

and where [Kmit] = diag(kmitu, kmitv, kmitw) and

[Kmir] = diag(kmiru, kmirv, kmirw) are the translational

stiffness matrix and rotational stiffness matrix

respectively of mount i in its LCS.

fDmig in equation (8) is the correction vector of

mount i in its LCS and is expressed as

fDmig= (Dmitu, Dmitv, Dmitw, Dmiru, Dmirv, Dmirw)T

ð11Þ

where the value of each component in fDmig for

mount i is a correction term for different force ver-

sus displacement relations along or around its LCS.

The values of the components in [Kmit], [Kmir], and

fDmig can be obtained with the formulae in Table 1

according to the values of the components in �Umi

� �and �fmi

� �.

The powertrain is moved under the forces and

moments applied to the powertrain’s CG, and so the

location of mount i with respect to the PCS is chan-

ged. The new position vector R�pmi of mount i in the

PCS is estimated using

R�pmi = R

pmi + ½I, ~RpT

mi�fqpg ð12Þ

However, the mount location has usually been

regarded as fixed in previous studies [13, 14]. This

will cause calculation errors of the displacements if

the motion of a powertrain is large.

According to equation (8), the reaction forces and

moments of mount i applied to the powertrain in

the PCS in terms of the displacements of the power-

train’s CG are

RFpmi

� �RM

pmi

� �( )

=I, 0

~R�pmi, I

� � � Fpmi

� �� M

pmi

� �( )

=� K

pmit

� , K

pmit

� ~R

pmi

� ~R�pmi Kpmit

� , ~R�pmi K

pmit

� ~Rp

mi � Kpmir

� " #

qp

� ��

Ami½ �T, 0

~R�pmi Ami½ �T, Ami½ �T

" #fDmig

ð13Þ

The displacements of mount 1 on torque strut j

in the PCS are

Upsjm1

n of

psjm1

n o8><>:

9>=>;=

I, ~Rpsjm1

T

0, I

" #fqpg �

I, ~Rssjm1

T

0, I

" #fqsjg

ð14Þ

where Upsjm1

n oand f

psjm1

n oare the translational

displacements of mount 1 along each axis and the

rotational angles of mount 1 around each axis

respectively of the PCS. The superscript s denotes

that the values are defined in the SCS; ~Rssjm1 is a

(333) skew-symmetric matrix of the position vector

Rssjm1 for mount 1 in the SCS.

The displacements of mount 1 in its LCS are then

written as

�U sjm1

� �fsjm1

n o8<:

9=;=

Asjm1

� , 0

0, Asjm1

� " #

I, ~Rpsjm1

T

0, I

" #fqpg

�Asjm1

� , 0

0, Asjm1

� " #

I; ~Rssjm1

T

0, I

" #fqsjg

ð15Þ

where �U sjm1

� �and �fsjm1

n oare the translational dis-

placements and rotational angles respectively of

mount 1 in its LCS, and [Asjm1] is the transformation

matrix from the PCS to the LCS of mount 1.

The forces applied to mount 1 owing to the

movement of the powertrain and the torque strut j

in the PCS are

Fpsjm1

n oM

psjm1

n o8><>:

9>=>;=

Kpsjm1t

h i, 0

0, Kpsjm1r

h i264

375 I, ~Rp

sjm1

T

0, I

" #fqpg

�K

psjm1t

h i, 0

0, Kpsjm1r

h i264

375 I, ~Rs

sjm1T

0, I

" #fqsjg

+Asjm1

� T, 0

0, Asjm1

� T

" #fDsjm1g

ð16Þ

where Fpsjm1

n oand M

psjm1

n oare the force vector

(3 3 1) and torque vector (3 3 1) respectively of

mount 1, and fDsjm1g is the correction vector of

mount 1 in its LCS. The expression for the transla-

tional stiffness matrix ½K psjm1t� of mount 1 and the

rotational stiffness matrix ½K psjm1r� of mount 1 in the

PCS are

½K psjm1t�= ½Asjm1�T½Ksjm1t�½Asjm1� ð17Þ

and

½K psjm1r�= ½Asjm1�T½Ksjm1r�½Asjm1� ð18Þ

respectively where [Ksjm1t] = diag(ksjm1tu, ksjm1tv,

ksjm1tw) and [Ksjm1r] = diag(ksjm1ru, ksjm1rv, ksjm1rw)

are the translational stiffness matrix and



rotational stiffness matrix respectively of mount 1 in

its LCS.

The values of the components in [Ksjm1t], [Ksjm1r],

and fDsjm1g can be obtained using the formulae in

Table 1 according to the values of the components

in �Usjm1

� �and �fsjm1

n o.

The powertrain is moved under the action of {F},

and so the location of mount 1 on torque strut j is

also changed. The new position vector R�psjm1 of

mount 1 in the PCS is calculated from

R�psjm1 = R

psjm1 + ½I, ~Rp

sjm1

T�fqpg ð19Þ

The reaction forces and moments of mount 1 to

the powertrain in the PCS are

RFpsjm1

n oRM

psjm1

n o8><>:

9>=>;=

I, 0~R�psjm1, I

" # � Fpsjm1

n o� M

psjm1

n o8><>:

9>=>;

=� K

psjm1t

h i, K

psjm1t

h i~Rp

sjm1

� ~R�psjm1 Kpsjm1t

h i, ~R�psjm1 K

psjm1t

h i~Rp

sjm1 � Kpsjm1r

h i264

375 qp

� �

�� K p

sjm1t

h i, K p

sjm1t

h i~Rs

sjm1

� ~R�psjm1 Kpsjm1t

h i, ~R�psjm1 K

psjm1t

h i~Rs

sjm1 � Kpsjm1r

h i264

375 qsj

� �

�Asjm1

� T, 0

~R�psjm1 Asjm1

� T, Asjm1

� T

24

35fDsjm1g

ð20Þ

Since the reaction forces and moments applied to

the powertrain from the mounts connected with the

powertrain, and the external load {F} applied to the

powertrain are in equilibrium status, the static equi-

librium equation of the powertrain can be obtained

and is expressed as

PNm

i = 1

RFpmi

� �PNm

i = 1

RMpmi

� �8>>><>>>:

9>>>=>>>;

+

PNs

j = 1

RFpsjm1

n oPNs

j = 1

RMpsjm1

n o8>>><>>>:

9>>>=>>>;

+ fFg= f0g

ð21Þ

3.2 Equation for the torque strut

Since each axis in the SCS of torque strut j is parallel

to the corresponding axis of the PCS, and according

to equation (16), the forces F ssjm1

n o, Ms

sjm1

n on oT

applied to mount 1 in the SCS of torque strut j are

expressed as

F ssjm1

n oM s

sjm1

n o8<:

9=;= �

Fpsjm1

n oM

psjm1

n o8<:

9=; ð22Þ

where the superscript s denotes that the values are

defined in the SCS.

When the displacements of torque strut j CG in

its SCS are fqsjg, the new position vector R�ssjm1 of

mount 1 on torque strut j in the SCS is

R�ssjm1 = Rssjm1 + ½I, ~Rs

sjm1T�fqsjg ð23Þ

The reaction forces and moments of mount 1

applied to torque strut j in the SCS are

RF ssjm1

n oRM s

sjm1

n o8><>:

9>=>;=

I, 0~R�ssjm1, I

" # � F ssjm1

n o� M s

sjm1

n o8><>:

9>=>;

=� K

psjm1t

h i, K

psjm1t

h i~Rs

sjm1

� ~R�ssjm1 Kpsjm1t

h i, ~R�ssjm1 K

psjm1t

h i~Rs

sjm1 � Kpsjm1r

h i264

375 qsj

� �

�� K

psjm1t

h i, K

psjm1t

h i~Rp

sjm1

� ~R�ssjm1 K psjm1t

h i, ~R�ssjm1 K p

sjm1t

h i~Rp

sjm1 � K psjm1r

h i264

375 qp

� �

+Asjm1

� T, 0

~R�ssjm1 Asjm1

� T, Asjm1

� T

24

35fDsjm1g

ð24Þ

Following the same procedures described above

to obtain the reaction forces and moments of mount

1 to torque strut j, the reaction forces and moments

of mount 2 to torque strut j in the SCS are

RF ssjm2

n oRM s

sjm2

n o8><>:

9>=>;=

� K ssjm2t

h i, K s

sjm2t

h i~Rs

sjm2

� ~R�ssjm2 K ssjm2t

h i, ~R�ssjm2 K s

sjm2t

h i~Rs

sjm2 � K ssjm2r

h i24

35 qsj

� �

� Asjm2

� T, 0

~R�ssjm2 Asjm2

� T, Asjm2

� T

" #fDsjm2g ð25Þ

where

½K ssjm2t�= ½Asjm2�T½Ksjm2t�½Asjm2� ð26Þ

and

½K ssjm2r�= ½Asjm2�T½Ksjm2r�½Asjm2� ð27Þ



are the translational stiffness matrix and rotational

stiffness matrix respectively of mount 2 in the SCS

of torque strut j, [Asjm2] is the transformation matrix

from the SCS of torque strut j to the LCS of mount

2, [Ksjm2t] and [Ksjm2r] are the translational stiffness

matrix and rotational stiffness matrix respectively of

mount 2 in its LCS and they are diagonal matrices,

Rssjm2 is the initial position vector of mount 2 in the

SCS of torque strut j, R�ssjm2 is the new position vector

of mount 2 in the SCS of torque strut j if the torque

strut is moved owing to the movement of power-

train, and fDsjm2g is the correction vector of mount

2 in its LCS.

Since torque strut j is assumed to be a rigid body

without mass, and it is in equilibrium under the

reaction forces and moments of the two mounts at

its two ends, thus the equation for torque strut j in

its SCS is then written as

RF ssjm1

n oRM s

sjm1

n o8<:

9=;+

RF ssjm2

n oRM s

sjm2

n o8<:

9=;= 0f g ð28Þ

3.3 Equation of the powertrain–torque strut

system

By combining the equilibrium equations for the

powertrain, i.e. equation (21), and for torque strut j,

i.e. equation (28), the equilibrium equation of the

powertrain–torque strut system is then expressed as

PNm

i = 1

RFpmi

� �PNm

i = 1

RMpmi

� �8>>><>>>:

9>>>=>>>;

+

PNs

j = 1

RFpsjm1

n oPNs

j = 1

RMpsjm1

n o8>>><>>>:

9>>>=>>>;

+ fFg

RF ssjm1

n oRM s

sjm1

n o8<:

9=; +

RF ssjm2

n oRM s

sjm2

n o8<:

9=;

8<:

9=;

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

= 0f g

ð29Þ

3.4 Solution of equations

Equation (29) is non-linear and a discrete Newton

iterative algorithm is used to solve it. The variables

in equation (29) are the displacement {qp} of the CG

for the powertrain and the displacement {qsj} of tor-

que strut j. For a non-linear equation, F(X) = 0,

where X are unknown variables, the discrete

Newton iterative algorithm is used to estimate its

solution. The relation for unknown variables in the

(m + 1)th iteration and mth iteration is

xm + 1 = xm � J(xm, hm)�1 � F(xm) ð30Þ

where

J(X , h) =

f1(X + h1e1)�f1(X )h1

� � � f1(X + hnen)�f1(X )hn

..

. . .. ..

.

fn(X + h1e1)�fn(X )h1

� � � fn(X + hnen)�fn(X )hn

2664

3775 ð31Þ

where X is the vector that has n unknown variables,

h is the increment step vector of numerical differen-

tiation given by h = (h1, h2, ., hn)T, and e is the

unity vector where the kth component of ek (k = 1, 2,

., n) is one and the rest of the components are

zero.

Since the values of the components in the stiff-

ness matrix and the correction vector depend on the

magnitude of the external load, an iterative algo-

rithm must be used to determine the displacements

of the powertrain and the torque struts. The steps

are described as follows.

Step 1: Calculate the displacements of each mount

in its LCS under the weight of the powertrain. The

displacements of each mount in its LCS should be in

linear range (k3), and {D} in equation (3) should be

zero under the weight of the powertrain. The displa-

cements of the CG for the powertrain and the torque

struts are obtained by forming equation (29) and

then solving the equation. The displacements of each

mount in its LCS are calculated using equations (4),

(7), (14), and (15).

Step 2: Move the origin of the force (or torque)

versus displacement (or angle) in Fig. 4 for each

mount in its LCS to the point where the displace-

ments of a mount in its LCS equals that obtained in

step 1.

Step 3: Calculate the displacements of the CG for

the powertrain and the torque struts by solving

equation (29) when an external load is applied to

the powertrain’s CG, assuming that the displace-

ments of each mount in its LCS are in the linear

range.

Step 4: Calculate the displacements of each

mount in its LCS using equations (4), (7), (14), and

(15).

Step 5: Check the displacements of each mount

obtained in step 4 to see whether they are in the lin-

ear range. If they are, output the displacements of

each mount and then stop. The displacements of

each mount in its LCS under the load case are a

summation of the calculated displacements in step

4 and the displacements estimated in step 1. If they

are not in the linear range, record the ranges where

the displacements for each mount in its LCS are

located, and then go to step 6.

Step 6: Construct the new stiffness matrix and the

correction vector of each mount from the range data

recorded in step 5 and the formulae in Table 1.



Step 7: Estimate the displacements of the CG for

the powertrain and the torque struts by solving the

re-formed equation (29). Obtain the displacements

of each mount in its LCS using equations (4), (7),

(14), and (15).

Step 8: Check the displacements of each mount in

its LCS obtained in step 7 to see which ranges of the

displacements are located, and record the ranges of

the displacements for each mount in its LCS. If the

ranges are the same as in step 5, output the calcu-

lated results and then stop; otherwise, go to step 6.

In some loads, the iteration procedures stated above

may not converge owing to the inappropriate values

of the stiffness and tuning points. Therefore, an

interrupt program should be included in the pro-

gram designed.

4 APPLICATION EXAMPLES

4.1 Parameters of a powertrain mounting system

A generic PMS consisting of two mounts and a tor-

que strut is shown in Fig. 2. The powertrain has a

mass of 66.48 kg. The location of the powertrain’s

CG in the vehicle coordinate system (VCS) is

(1203.96, 31.09, 373.2) mm. The locations of each

mount in the VCS are listed in Table 2. Direction

angles are used to represent the orientation of a

mount in its LCS with respect to the PCS and are

listed in Table 3. The static stiffness (translational

and rotational) and the coordinates of tuning points

for each mount in three directions of its LCS are

listed in Tables 4 and 5 respectively. It is seen that

the translational and rotational stiffness of mount 1

on the torque strut are much larger than those of

the other mounts.

4.2 Calculation results

In the conventional calculation method, the torque

strut is simplified as a mount located at the position

of mount 1, as shown in Fig. 5. The translational

stiffness of the equivalent mount in the u direction

(line P1P2 in Fig. 1) is obtained by adding the trans-

lational stiffness of mount 1 and mount 2 in the u

direction using the principle of adding the stiffness

of two springs arranged in a series. The translational

stiffness of the simplified mount in the v and w

directions of its LCS are very small compared with

those in the u direction, and they are usually less

than 15 N/mm from experimental data. In the

Table 4 Translational static stiffness and x coordinates of the tuning points for each mount

Mount LCS of a mount Stiffness for a piecewise linear relation (N/mm) x coordinate of the tuning points (mm)

k1 k2 k3 k4 k5 a b c d

Engine u 500 250 70 250 500 –8 –3 3 8v 15 15 15 15 15 –6 –3 3 6w 800 300 60 300 800 –14 –9 2 5

Transmission u 800 250 60 250 800 –6 –3 3 6v 800 250 60 250 800 –6 –3 3 6w 800 300 85 300 800 –12 –9 0 2

Mount 1 u 8000 3000 1500 3000 8000 –3.5 –2 2 3.5v 250 250 250 250 250 –6 –4 4 6w 8000 3000 1500 3000 8000 –3.5 –2 2 3.5

Mount 2 u 800 500 90 500 800 –13 –4 3 13v 500 300 120 300 500 –13 –4 4 13w 500 40 40 40 500 –8 –6 6 8

Table 3 Mount orientations

Mount LCS of a mount Orientations in the PCS (deg)

X Y Z

Engine u 0 90 90v 90 0 90w 90 90 0

Transmission u 0 90 90v 90 0 90w 90 90 0

Mount 1 u 0 90 90v 90 0 90w 90 90 0

Mount 2 u 0 90 90v 90 0 90w 90 90 0

Table 2 Mount locations

Mount Coordinates in the VCS (mm)

X Y Z

Engine 1179.3 374.5 529.3Transmission 1178.5 –206.5 520.6Mount 1 1403.5 –0.9 165.7Mount 2 1570.5 –0.9 183.2



conventional calculation method the rotational stiff-

ness of the mounts are ignored [13, 14].

To find the difference between calculating the

displacements by the methods proposed in this

paper and calculating the displacements by the con-

ventional method, two typical loads and four

extreme loads applied to the powertrain’s CG are

used to calculate the displacements of the power-

train’s CG and mounts. Also the calculated reaction

forces of each mount are given.

4.2.1 Two typical loads

The first typical load is a g force applied to the

powertrain’s CG in the lateral direction of the PCS

(Yp axis in PCS) which is equal to 3 mg, where m is

the powertrain mass and g is the acceleration due to

gravity. The second typical load also consists of g

forces applied to the powertrain’s CG in the vertical

direction Zp and the lateral direction Yp simultane-

ously, which are equal to 3.5 mg and 2 mg respec-

tively. The values of two typical loads are listed in

Table 6. The calculated displacements of the power-

train’s CG and mounts, and the reaction forces of

the mounts, using different torque strut models, are

listed in Tables 7 to 10.

From Tables 7 and 8, it is seen that the displace-

ments of the powertrain’s CG are almost the same

under the two typical loads using different torque

strut models. It is seen from Tables 9 and 10 that

the displacements and reaction forces of the engine

mount and the transmission mount are different

using different torque strut models. However, the

difference can be ignored from the viewpoint of

engineering application in designing a PMS. In addi-

tion, in a generic torque strut model, the displace-

ments and reaction forces of mount 1 and mount 2

Table 5 Rotational static stiffness and x coordinates of the tuning points for each mount

Mount LCS of a mount Stiffness for a piecewise linear relation (N m/deg) x coordinate of the tuning points (deg)

k1 k2 k3 k4 k5 a b c d

Engine u 0.3 0.3 0.3 0.3 0.3 –20 –10 10 20v 0.2 0.2 0.2 0.2 0.2 –20 –10 10 20w 0.3 0.3 0.3 0.3 0.3 –20 –10 10 20

Transmission u 0.3 0.3 0.3 0.3 0.3 –20 –10 10 20v 0.2 0.2 0.2 0.2 0.2 –20 –10 10 20w 0.3 0.3 0.3 0.3 0.3 –20 –10 10 20

Mount 1 u 7 5 3 5 7 –10 –5 5 10v 1.5 1.5 1.5 1.5 1.5 –20 –10 10 20w 7 5 3 5 7 –10 –5 5 10

Mount 2 u 0.3 0.3 0.3 0.3 0.3 –20 –10 10 20v 0.2 0.2 0.2 0.2 0.2 –20 –10 10 20w 0.3 0.3 0.3 0.3 0.3 –20 –10 10 20

Fig. 5 The simplification of a torque strut

Table 7 Displacements of the powertrain’s CG in the

PCS under typical load 1

Torque strutmodel

Translationaldisplacement (mm)

Rotationalangle (deg)

xp yp zp ap bp gp

Simplified 0.01 9.60 0.01 1.05 –0.01 0.02Generic 0.08 9.94 –0.14 1.11 0.01 0.08

Table 8 Displacements of the powertrain’s CG in the

PCS under typical load 2

Torque strutmodel

Translationaldisplacement (mm)

Rotationalangle (deg)

xp yp zp ap bp gp

Simplified 0.02 6.82 7.66 0.24 –0.24 –0.01Generic –0.26 6.99 7.67 0.26 –0.27 0.01

Table 6 Two typical loads in the PCS

Load case Force (N) Torque (N m)

Fx Fy Fz Mx My Mz

1 0 1954.5 0 0 0 02 0 1303.0 2280.26 0 0 0



can be calculated simultaneously whereas, in the

conventional calculation method, a result is pro-

vided only for the simplified mount and cannot be

used directly in designing a torque strut.

From Tables 9 and 10, it is concluded that the

powertrain is in equilibrium under the external load

and the reaction forces of the engine mount, the

transmission mount, and mount 1. It is also shown

that the powertrain–torque strut system is in equili-

brium under the external load and the reaction

forces of the engine mount, the transmission

mount, and mount 2 on the torque strut. The equili-

brium for the powertrain and powertrain–torque

strut system validates the developed equations and

calculation method of obtaining the displacements

and reaction forces presented in this paper.

4.2.2 Extreme load cases

The first extreme load is a g force applied to the

powertrain’s CG in the lateral direction Yp of the

PCS and is equal to 5 mg. The second extreme load

is a g force applied to the powertrain’s CG in the

vertical direction Zp of the PCS and is equal to 5 mg.

The third extreme load is a g force applied to the

powertrain’s CG in the longitudinal direction Xp of

the PCS and is equal to 11 mg; this load case is

called longitudinal impact. The fourth extreme load

is a maximum torque load applied around the

crankshaft of powertrain (the Yp axis of the PCS)

and is equal to 15 times the maximum engine tor-

que. The values of the four extreme loads applied to

the powertrain’s CG are listed in Table 11. The cal-

culated displacements of the powertrain’s CG and

mounts, and the reaction forces of the mounts, are

listed in Tables 12 to 16 using different torque strut

models.

It is seen from Table 12 that, when an extreme

load in a lateral or vertical direction is applied to the

powertrain, the displacements of the powertrain’s

CG are approximately equal using different torque

strut models and calculation methods. However,

when the powertrain is applied with a longitudinal

impact load or a large torque, the translational

movements of the powertrain’s CG in the Xp and Zp

directions obtained by the proposed method in this

paper are larger than those obtained by the conven-

tional method. Therefore it is concluded that, when

the powertrain is applied with a longitudinal impact

load or a large torque, the torque strut cannot be

simplified as one mount in order to obtain the

results with high calculation precision.

Table 9 Displacements and reaction forces of the mounts under typical load 1

Mount Torque strut model Displacement in the LCS (mm) Reaction force in the PCS (N)

u v w Fx Fy Fz

Engine Simplified 0.42 6.74 1.84 –29.6 –101.1 –110.5Generic –0.25 6.88 2.17 17.23 –103.10 –171.82

Transmission Simplified 1.24 6.90 –8.87 –74.7 –1651.4 753.70Generic 0.74 7.04 –9.13 –44.2 –1763.0 802.90

Torque strut Simplified –1.16 13.47 –0.55 104.3 –202.0 8.30Mount 1 –0.02 0.35 –0.01 26.99 –88.36 20.39Mount 2 –0.30 0.74 –0.51 26.99 –88.36 20.39

Table 10 Displacements and reaction forces of the mounts under typical load 2


u v w Fx Fy Fz

Engine Simplified –0.04 6.17 4.55 2.60 –92.60 –883.50Generic –0.93 6.25 4.78 64.88 –93.80 –955.36

Transmission Simplified 0.51 6.21 2.02 –30.70 –1095.70 –619.70Generic –0.62 6.29 2.07 36.9 –1164.8 –654.90

Torque strut Simplified –0.31 7.65 8.37 28.10 –114.70 –125.60Mount 1 0.07 0.18 0.01 –101.79 –44.45 –18.47Mount 2 1.13 0.37 0.46 –101.79 –44.45 –18.47

Table 11 Four extreme loads in the PCS

Load case Force (N) Torque (N m)

Fx Fy Fz Mx My Mz

Lateral load 0 3257.5 0 0 0 0Vertical load 0 0 3257.5 0 0 0Longitudinal impact load 7166.6 0 0 0 0 0Large torque 0 0 0 0 2009 0



It is seen from Tables 13 and 14 that the displace-

ments and reaction forces of the engine mount and

the transmission mount obtained by the conven-

tional method using the simplified toque strut

model or the method proposed in this paper are

almost the same from the viewpoint of engineering

application when the powertrain is applied with an

extreme load in the lateral or vertical direction.

Instead, from Tables 15 and 16 it is seen that, under

the two extreme loads of longitudinal impact and

Table 12 Displacements of the powertrain’s CG in the PCS under extreme loads

Load case Torque strut model Translational displacement (mm) Rotational displacement (deg)

xp yp zp ap bp gp

Lateral load Simplified 0.07 11.76 –0.18 1.27 0.02 0.13Generic 0.15 12.06 –0.33 1.31 0.03 0.19

Vertical load Simplified –0.01 0.30 8.52 0.18 –0.43 –0.09Generic –0.35 0.40 8.52 0.18 –0.51 –0.12

Longitudinal impact load Simplified 8.27 0.06 0.03 0.02 –0.36 –0.17Generic 9.40 0.17 –2.70 0.07 –0.47 –0.13

Large torque Simplified –0.55 –0.04 –0.24 –0.02 3.37 0.02Generic –1.52 –0.12 2.52 –0.07 3.83 0.07

Table 13 Displacements and reaction forces of the mounts under a lateral load


u v w Fx Fy Fz

Engine Simplified –0.13 8.25 2.98 8.84 –123.74 –412.95Generic –0.76 8.41 3.17 53.04 –126.17 –472.25


Torque strut Simplified –1.13 16.81 –0.96 102.04 –252.10 14.38Mount 1 –0.02 0.46 –0.01 25.66 –115.46 21.65Mount 2 –0.29 0.96 –0.54 25.66 –115.46 21.65

Table 14 Displacements and reaction forces of the mounts under a vertical load


u v w Fx Fy Fz

Engine Simplified –0.08 –0.15 4.96 5.93 2.18 –1007.30Generic –0.86 –0.05 5.04 60 0.7 –1054.60

Transmission Simplified –0.35 –0.12 3.06 20.90 7.04 –1449.73Generic –1.87 –0.02 3.15 112.1 1 –1523.4

Torque strut Simplified 0.30 0.61 9.93 –26.82 –9.21 –148.96Mount 1 0.11 0.01 0.02 –172.17 –1.66 –28.04Mount 2 1.91 0.01 0.70 –172.17 –1.66 – 28.04

Table 15 Displacements and reaction forces of the mounts under a longitudinal impact load


u v w Fx Fy Fz

Engine Simplified 8.88 0.09 –4.48 –1900.98 –1.41 268.64Generic 9.03 0.02 –6.80 –1975.04 –0.24 407.8


Torque strut Simplified 8.26 –0.49 1.27 –2901.61 7.34 –18.98Mount 1 1.80 –0.01 0.27 –2694.4 2 –402.3Mount 2 7.85 –0.02 8.16 –2694.4 2 –402.3



large torque, the reaction forces of the engine

mount and the transmission mount in the u direc-

tion of their LCS calculated using different torque

strut models and methods are almost the same.

However, a difference between the forces in the w

direction exists.

In all the four load cases, the reaction forces of

the torque strut obtained using different torque

strut models are quite different, and this demon-

strates that the method developed in this paper

should be used to calculate the displacements and

reaction forces of torque strut if a torque strut exists

in a PMS and if the powertrain bears an extreme

load.

4.3 Effect of the rotational stiffness in the calcula-

tion results

One enhancement of this study in modelling a

mount is to consider the rotational stiffness.

Usually, the rotational stiffness of mount 1 on a tor-

que strut connected with a powertrain is much

larger than those of the other mounts in a PMS, as

shown in Table 5. To verify the effect of the rota-

tional stiffness on the calculation results, three set

of parameters are used. The first set of parameters

(parameters I) is when the rotational stiffness of all

mounts are considered, the second set of para-

meters (parameters II) is when only the rotational

stiffness of mount 1 on the torque strut is included,

and the third set of parameters (parameters III) is

when the rotational stiffness of all mounts are

ignored.

When the powertrain is applied with a large tor-

que load and a longitudinal impact load as listed in

Table 11, the displacements of the powertrain’s CG

and mounts, and the reaction forces of the mounts,

using the three set of parameters mentioned above

are given in Tables 17 to 19.

It is seen from Tables 17 to 19 that the calculated

displacements of the powertrain’s CG and mounts,

and the reaction forces of mounts, are almost the

same using parameters I and parameters II.

However, using parameters III the results are differ-

ent from those obtained using parameters I and

parameters II. Therefore it is concluded that the

rotational stiffness of mount 1 on a torque strut

should be included when it is much larger than

those of the other mounts using the model devel-

oped in this study.

5 CONCLUSION

A modelling and calculation method for estimating

the displacements of the powertrain’s CG and

mounts, and the reaction forces of mounts, in a

PMS including torque struts under a quasi-static

load is developed. In the model, a powertrain is

modelled as a rigid body, and a torque strut is mod-

elled as a rigid body without a mass. Also a new

mount model is proposed, and it is modelled as 3D

Table 16 Displacements and reaction forces of the mounts under a large torque load


u v w Fx Fy Fz

Engine Simplified 9.05 0.01 –3.37 –1984.19 –0.22 202.46Generic 8.65 0.03 –0.58 –1785.9 –0.41 34.9

Transmission Simplified 9.37 0.01 –3.17 –3623.82 –0.63 269.78Generic 8.92 0.02 0.12 –3264.6 –1.01 –35

Torque strut Simplified –13.94 –0.06 –11.95 5608.01 0.85 179.27Mount 1 –2.68 –0.01 –0.43 5050.5 1.5 651.7Mount 2 –13.24 –0.01 –8.66 5050.5 1.5 651.7

Table 17 Displacements of the powertrain’s CG in the PCS under extreme loads

Load case Parameters Translational displacement (mm) Rotational displacement (deg)

xp yp zp ap bp gp

Large torque Parameters I –1.52 –0.12 2.52 –0.07 3.83 0.07Parameters II –1.53 –0.12 2.52 –0.07 3.83 0.07Parameters III –1.54 –0.13 2.42 –0.09 3.81 0.06

Longitudinal impact load Parameters I 9.40 0.17 –2.70 0.07 –0.47 –0.13Parameters II 9.40 0.18 –2.72 0.08 –0.47 –0.13Parameters III 9.36 0.08 –2.85 0.07 –0.49 –0.14



springs with both a translational stiffness along each

axis and a rotational stiffness around each axis of its

LCS. The equation and the solution method for esti-

mating the displacements of the powertrain’s CG

and mounts under a quasi-static load is developed

when a torque strut exists in a PMS. An example is

given to calculate the displacements of the power-

train’s CG and mounts in a generic PMS including a

torque strut under some typical and extreme load

cases using the two different torque strut models.

From the calculations, the following conclusions are

found.

1. The calculated displacements of the power-

train’s CG are almost the same under typical

loads in the lateral and the vertical directions

using different torque strut models and calcula-

tion methods.

2. The displacements and the reaction forces of the

mounts excluded the mounts on the torque

struts obtained by different torque strut models,

and the calculation methods are different under

typical load cases. However, the difference can

be ignored from the viewpoint of engineering

application in designing a PMS.

3. When an extreme longitudinal impact load or a

large torque around the crankshaft is applied to

the powertrain, the translational displacements

of the powertrain’s CG calculated by the method

proposed in this paper are larger than those

obtained by the conventional method. Also the

estimated displacements and reaction forces of

mounts are different using different torque strut

models. Therefore it is concluded that, when the

powertrain is applied with an extreme longitudi-

nal impact load or a large torque, the torque

strut cannot be simplified as one mount.

4. The displacements and reaction forces of mount

1 and mount 2 on a torque strut can be calcu-

lated simultaneously when no simplification is

made to a torque strut. When in the conven-

tional calculation method a torque strut is sim-

plified as one mount, the displacements and

reaction forces are only provided for a simplified

mount, and the results cannot be used directly

in designing a torque strut.

Table 18 Displacements and reaction forces of the mounts under a large torque load

Mount Parameters Displacement in the LCS (mm) Reaction force in the PCS (N)

u v w Fx Fy Fz

Engine Parameters I 8.65 0.03 –0.58 –1785.9 –0.41 34.9Parameters II 8.66 0.03 –0.60 –1787.70 –0.46 35.82Parameters III 8.69 0.08 –0.87 –1803.84 –1.16 51.91

Transmission Parameters I 8.92 0.02 0.12 –3264.6 –1.01 –35Parameters II 8.92 0.02 0.11 –3268.44 –1.17 –33.34Parameters III 8.97 0.06 –0.02 –3305.74 –3.84 1.60

Mount 1 Parameters I –2.68 –0.01 –0.43 5050.5 1.5 651.7Parameters II –2.69 –0.01 –0.43 5056.14 1.62 649.03Parameters III –2.70 –0.02 –0.40 5109.58 4.99 598

Mount 2 Parameters I –13.24 –0.01 –8.66 5050.5 1.5 651.7Parameters II –13.25 –0.01 –8.66 5056.14 1.62 649.03Parameters III –13.31 –0.04 –8.56 5056.14 1.62 649.03

Table 19 Displacements and reaction forces of the mounts under a longitudinal impact load

Mount Rotational stiffness Displacement in the LCS (mm) Reaction forces in the PCS (N)

u v w Fx Fy Fz

Engine Parameters I 9.03 0.02 –6.80 –1975.04 –0.24 407.8Parameters II 9.03 0.02 –6.82 –1976.74 –0.25 409.40Parameters III 9.07 –0.05 –7.06 –1994.72 0.78 423.37

Transmission Parameters I 7.96 0.03 –7.6 –2496.6 –1.75 646.1Parameters II 7.96 0.03 –7.64 –2498.52 –1.84 649.54Parameters III 7.99 –0.04 –7.84 –2520.68 2.31 666.55

Mount 1 Parameters I 1.80 –0.01 0.27 –2694.4 2 –402.3Parameters II 1.79 –0.01 0.27 –2691.23 2.09 –407.44Parameters III 1.77 0.01 0.29 –2651.10 –3.09 –438.42

Mount 2 Parameters I 7.85 –0.02 8.16 –2694.4 2 –402.3Parameters II 7.84 –0.02 8.17 –2691.23 2.09 –407.44Parameters III 7.76 0.03 8.24 –2651.10 –3.09 –438.42



5. When the rotational stiffness of mount 1 on the

torque strut is much larger than those of the

other mounts, only the rotational stiffness of

mount 1 needs to be considered in the

calculation.

FUNDING

This work was supported by the the NaturalScience Foundation of China [project nos50575073 and 50975091] and the Foundation ofNational Automobile Dynamic SimulationLaboratory in Jilin University [grant no. 2010-06].

� Authors 2011

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