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Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 7089–7108

http://d0022-46

n CorrE-m

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

Critical examination of isolation system design paradigms fora coupled powertrain and frame: Partial torque roll axisdecoupling methods given practical constraints

Jared Liette, Jason T. Dreyer, Rajendra Singh n

Acoustics and Dynamics Laboratory, Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus,OH 43210, USA

a r t i c l e i n f o

Article history:Received 29 January 2014Received in revised form17 June 2014Accepted 7 August 2014
Handling Editor: S. Ilanko
However, critical examination shows that the derivation does not always lead to a

Available online 12 September 2014

x.doi.org/10.1016/j.jsv.2014.08.0080X/& 2014 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ1 614 292 9044.ail address: [email protected] (R. Singh).

a b s t r a c t

The torque roll axis motion decoupling concept is analytically and computationallystudied in a realistic coupled powertrain and frame system using discrete, proportionallydamped linear models. Recently, Hu and Singh (2012 [1]) (Journal of Sound and Vibration331 (2012) 1498–1518) proposed new paradigms to fully decouple such a system.

physically realizable system, as each powertrain mount is not referenced to a singlelocation. This deficiency is overcome by deriving mount compatibility conditions toensure realistic mount positions which are incorporated into proposed decouplingconditions. It is mathematically shown that full decoupling is not possible for a practicalsystem, and therefore partial decoupling paradigms are pursued. Powertrain mountdesign using only the decoupled powertrain achieves better decoupling than minimizingconditions for the coupled system using a total least squares method. Further decouplingis obtained through frame isolation design using a decoupled frame model such that thetorque roll mode is dominant over the frequency range considered. Other methods forlimiting frame coupling are also briefly discussed.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, Hu and Singh [1] proposed new design paradigms to account for low frequency dynamic interactions between apowertrain mounting system and coupled compliant base sub-systems, successfully decoupling all rotational andtranslation motions from the powertrain torque roll axis (TRA). These paradigms are defined for a discrete, proportionallydamped system as an eigenvalue problem using modal analysis:

KqTRA ¼ λTRAMqTRA: (1)

Here, qTRA ¼ qTRAg1

� �TO1x6

� �T

, λTRA is the eigenvalue for the TRA roll mode qTRAg1 ¼ f0 0 0 1 0 0 gT , On�n is a n�n

null matrix, M is the inertia matrix of the powertrain and coupled sub-systems, K is the coupled stiffness matrix,

qTRA ¼ qTRAg1

� �TO1�6

� �T

, and superscript T indicates a transpose. If K is designed such that qTRAg1 is a natural mode of the

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J. Liette et al. / Journal of Sound and Vibration 333 (2014) 7089–71087090

system, then the response to the engine excitation shows up only at qTRAg1 , regardless of the excitation frequency

characteristics [1,2]. While this criterion is mathematically correct and complete, the derivation of K in [1] does not alwayslead to a physically realizable system. This results in mount locations that are not suitable (or even possible) from theengineering design perspective. Namely, each mount coupling the powertrain to the compliant base is referenced to twodifferent locations: from the powertrain to the mount and from the compliant base to the mount. If caution is not taken, thelocations do not coincide, and a physical isolation system cannot be constructed. Such is the case in Hu and Singh [1], andthis issue must be addressed. Therefore, the chief goal of this paper is to extend the results from the research conducted byHu and Singh [1] to include the physical constraints needed for a real-world design with partial decoupling. This is done byincluding compatibility conditions in the derivation of K for Eq. (1) such that mounts are always referenced to a singlelocation, thereby allowing for a physically realizable mounting system. It will also be shown that the conditions imposed byEq. (1) in the presence of physical constraints result in a system that cannot be fully decoupled. Accordingly, methods forpartial decoupling of the TRA direction must be pursued without imposing severe burden on the isolation system design.

In prior studies, Jeong and Singh [2] propose TRA decoupling paradigms assuming a rigid foundation for a powertrainmounting system and a discrete, proportionally damped model. Arbitrary mount locations are assumed, and an eigenvalueproblem similar to Eq. (1) is derived. These paradigms are extended by Park and Singh [3] to a non-proportionally dampedmodel, where two eigenvalue problems must be concurrently satisfied. It is suggested that sub-system dynamics cannot beneglected in a real-life vehicle, where the foundation dynamics (e.g. powertrain frame or cradle) may have a significanteffect on the dynamic powertrain response [4–7], especially when frame natural frequencies and the excitation frequencieslie in the same regime. For instance, Lee et al. [4] calculate several modes in the 1–10 Hz range for an uncoupled model ofthe frame connected to the vehicle body and tires. Hu and Singh [1] calculate modes up to 125 Hz for the uncoupled frameonly, which are excited by higher orders of the engine torque excitation. Also considering that the weight of the frame isgenerally less than the powertrain itself [8], there is a need to properly include the frame dynamics for TRA decoupling.Siraif and Qatu [7] conclude as such by comparing modal results of a powertrain model to that of full vehicle experimentaldata, and inclusion of the frame dynamics drastically improves results. Sub-structuring methods [9–11] and mode shifting[12] may be used to limit the modal coupling, but these are outside the scope of this paper.

′′′′

′′

′′′

′′

′

′

′′

′′

′′

′′′′

′′

′′

′′

′

′′

′

Fig. 1. Example case with schematics and coordinate systems for (a) coupled powertrain and frame system, (b) powertrain sub-system, and (c) frame sub-system. Key: —, inertial coordinates (Γ0

gj); - - -, mount coordinates (Γ″mi); and � � � , TRA coordinates (ΓTRA).

J. Liette et al. / Journal of Sound and Vibration 333 (2014) 7089–7108 7091

2. Problem formulation

The TRA is defined as an axis around which rotation occurs when a torque pulse is exerted on a free rigid body (engineand transmission combined) about an arbitrary direction [2], often deviating from the crankshaft or inertial axes by up to251 in many practical cases [13]. The vector dictating the TRA orientation is determined by the rigid body inertia matrix only,though approximations such as connecting engine and transmission centers of gravity are suggested to estimate this axis[14]. When constraints (engine mounts) are added to the free rigid body, conditions similar to Eq. (1) must be satisfied suchthat rotation about the TRA is also a natural mode of the system [2]. A conceptual model of the coupled powertrain andframe system is shown in Fig. 1(a) to illustrate a practical TRA orientation, where Γ0

gj are inertial Cartesian coordinates (x0, y0, z0)at the center of gravity (c.g.) of the jth rigid body, Γ″

mi are principal elastic Cartesian coordinates (x″, y″, z″) of the ith mount(diagonal stiffness matrices), and ΓTRA are Cartesian coordinates (x, y, z) with the TRA as the x-axis. It is assumed that all Γ0

gj areparallel with a vertical axis and an axis along the driveline.

Analysis of the system in Fig. 1(a) requires transformation of all excitation and dynamic reaction forces into a singlecoordinate system. Specifically, the system should be analyzed in ΓTRA to better facilitate derivation of the decouplingconditions in Eq. (1). The scope of this paper is limited to proportionally damped, discrete linear time-invariant systemswith small motions (as also assumed by Hu and Singh [1]), and necessary transformations are easily implemented.Additionally, the scope is limited to TRA dynamic decoupling methods of a realistic powertrain and frame mounting systemwith asymmetric inertia matrices. Jeong and Singh [2] consider both symmetric and asymmetric inertia matrices, where theformer simplifies model formulation. However, it is not realistic given the typical powertrain and frame geometries. Even ifthe powertrain is assumed to have a rigid foundation, Kim [15] argued that full decoupling is not possible for completelyarbitrary mount locations. If all powertrain mounts are somehow located in the so-called mounting plane (x and x″ areparallel), partial decoupling of powertrain motions are achieved even before seeking the appropriate mount locations [1,2].Therefore, the partial decoupling paradigms examined in this paper assume all powertrain mounts are located in the parallelx and x″ mounting plane. Specific objectives include the following: (1) provide a mathematical proof that full TRAdecoupling is not possible for a physically realizable powertrain and coupled frame system; (2) propose powertrain andframe mount design paradigms to enhance the partial decoupling of the powertrain TRAwithout imposing severe burden onthe isolation system design; and (3) examine alternative isolation system design methods to further improve decoupling.

3. Analytical model

3.1. Powertrain sub-system

Like Hu and Singh [1], the individual models for the decoupled powertrain and frame sub-systems are formulated first.

Each sub-system is assumed to have six degrees of freedom with three translations ε¼ εx εy εz� �T and three rotations

θ¼ θx θy θz� �T in a generalized displacement vector q¼ εT θTn oT

. The mathematical model is formulated for N

arbitrarily located powertrain mounts, though it is assumed that four elastomeric mounts are used (like real-life cases inmany vehicles). A discrete model of the decoupled powertrain sub-system is shown in Fig. 1(b) where Tx0 ðtÞ is a harmonic

torque applied about the x0-axis (driveline), F0g1ðtÞ ¼ 0 0 0 Tx0 ðtÞ 0 0n oT

is the external force vector, M0g1 ¼

diag M0εg1 M0θ

g1

n o� �is the powertrain inertia matrix in Γ0

gj with diag( ) as a diagonal matrix operator:

M0εg1 ¼ diag m1 m1 m1

� �� ; M0θ

g1 ¼Ix01 � Ix0y

0

1 � Ix0z0

1

� Ix0y0

1 Iy01 � Iy0z0

1

� Ix0z0

1 � Iy0z0

1 Iz01

2664

3775 (2a,b)

wherem is the mass and I is the inertia, rg1;mi ¼ rxg1;mi ryg1;mi rzg1;mi

n oTis the position vector from the powertrain c.g. to the

ith mount elastic center in ΓTRA, and K″mi ¼ diag kx″mi ky″mi kz″mi

n o� �is the ith mount stiffness matrix in Γ″

mi. For a typical

elastic mount, torsional stiffnesses are negligible [14]. Thus, only translational stiffnesses are included in K″mi.

System matrices M0g1, F

0g1 tð Þ, and K″mi should be transformed into ΓTRA prior to deriving equations of motion. The TRA

direction is defined by Jeong and Singh [2] in Γ0g1 as

q0TRAg1 ¼ 0 0 0 ρ011

g1 ρ021g1 ρ031

g1

n oT(3)

where ρ0g1 ¼ b M0θ

g1

h i�1and b is a normalizing constant for the first column of M0

g1θ. Next,

ℜ0 ¼νxx

0νxy

0νxz

0

νyx0

νyy0

νyz0

νzx0

νzy0

νzz0

264

375 (4)


is defined as an orthonormal rotation matrix between q0TRAg1 and qTRA

g1 where ν are normalized directional cosines [2]. Finally,Π0 ¼ diag ℜ0 ℜ0� ��

is defined as the needed transformation matrix, and the powertrain inertia matrix in ΓTRA isMg1 ¼Π0M0

g1Π0T while Fg1ðtÞ ¼Π0F0g1ðtÞ. A similar transformation Kmi ¼ℜ″miK

″miℜ

″Tmi is done where

ℜ″mi ¼

cos φymi cos φ

zmi cos φx

mi sin φzmiþ sin φx

mi sin φymi cos φ

zmi sin φx

mi sin φzmi� cos φx

mi sin φymi cos φ

zmi

� cos φymi sin φz

mi cos φxmi cos φ

zmi� sin φx

mi sin φymi sin φz

mi sin φxmi cos φ

zmiþ cos φx

mi sin φymi sin φz

mi

sin φymi � sin φx

mi cos φymi cos φx

mi cos φymi

2664

3775(5)

and φmi ¼ φxmi φy

mi φzmi

n oTare Euler angles from Γ″

mi to ΓTRA [3]. All φmi are fixed geometric angles with no relevance to

the dynamic rotations of the system. Large φmi therefore do not violate any small motions or linear system assumptions. In

general, Kmi ¼KTmi is a fully populated 3�3 matrix. Both the reaction force Rε

g1;mi and reaction moment Rθg1;mi exerted by

each mount must be accounted for as

Rεg1;mi ¼ �Kmi I Eg1;mi

h iqg1; (6)

Rθg1;mi ¼ rg1;mi � Rε

g1;mi ¼ ETg1;miR

εg1;mi ¼ �ET

g1;miKmi I Eg1;mi

h iqg1; (7)

respectively, where

Eg1;mi ¼0 rzg1;mi �ryg1;mi

�rzg1;mi 0 rxg1;mi

ryg1;mi �rxg1;mi 0

2664

3775 (8)

is a skew symmetric matrix [2]. Combining these reaction forces as Rg1;mi ¼ Rεg1;mi; Rθ

g1;mi

h iresults in the total resistance

applied by each mount, and

Kg1 ¼ ∑N

i ¼ 1Kg1;mi ¼ ∑

N

i ¼ 1

Kmi KmiEg1;mi

KmiEg1;mi� T ET

g1;miKmiEg1;mi

24

35 (9)

is the total stiffness contribution from all mounts on the powertrain with Kg1 ¼KTg1. Finally,

Mg1 €qg1ðtÞþCg1 _qg1ðtÞþKg1qg1ðtÞ ¼ Fg1ðtÞ (10)

are the powertrain equations of motion in ΓTRA, where Cg1 is a viscous (proportional) damping matrix.

3.2. Frame sub-system and coupled system

The decoupled frame sub-system discrete model is shown in Fig. 1(c), and parameters also appearing in Fig. 1(b) remain

the same. Additional matrices include M0g2 ¼ diag M0ε

g2 M0θg2

n o� �as the frame inertia in Γ0

g2 and K″bi ¼

diag kx″bi ky″bi kz″bin o� �

as the ith frame mount stiffness in Γ″bi. New position vectors in ΓTRA are rg2;mi ¼

rxg2;mi ryg2;mi rzg2;mi

n oTfrom the frame c.g. to the ith powertrain mount elastic center and rg2;bi ¼ rxg2;bi ryg2;bi rzg2;bi

n oT

from the frame c.g. to the ith frame mount elastic center. The mathematical model is formulated in the same manner as the

powertrain sub-system such that Mg2 ¼Π0M0g2Π0T and Kbi ¼ℜ″biK″biℜ″biT in ΓTRA. No external forces excite the frame such

that Fg2ðtÞ ¼O6x1. Also,

Kg2;m ¼ ∑N

i ¼ 1Kg2;mi ¼ ∑

N

i ¼ 1

Kmi KmiEg2;mi

KmiEg2;mi� T ET

g2;miKmiEg2;mi

24

35 (11)

where rg2;mi replace rg1;mi in the skew symmetric matrix formulation of Eq. (8), and

Kg2;b ¼ ∑N

i ¼ 1Kg2;bi ¼ ∑

N

i ¼ 1

Kbi KbiEg2;bi

KbiEg2;bi� T ET

g2;biKbiEg2;bi

24

35: (12)

The total stiffness matrix is Kg2 ¼Kg2;mþKg2;b, and the equations of motion

Mg2 €qg2ðtÞþCg2 _qg2ðtÞþKg2qg2ðtÞ ¼O6x1 (13)


are in ΓTRA where Cg2 is a viscous proportional damping matrix. While Kg1 and Kg2;m both arise due to the powertrainmounts, each involves different position vectors: rg1;mi from the powertrain to the mount and rg2;mi from the frame tothe mount.

Coupling between the sub-systems comes through the powertrain mounts. Namely, Rεg1;g2;mi ¼ �Rε

g2;mi ¼Kmi I Eg2;mi

h iqg2 is the translational reaction force on the powertrain due to the frame motion. A rotational reaction

Rθg1;g2;mi ¼ �rg1;mi � Rε

g2;mi also occurs relative to the powertrain c.g., resulting in a coupling stiffness matrix:

�Kg1;g2 ¼ � ∑N

i ¼ 1Kg1;g2;mi ¼ � ∑

N

i ¼ 1

Kmi KmiEg2;mi

KmiEg1;mi� T ET

g1;miKmiEg2;mi

24

35: (14)

Reciprocity exists, and Kg2;g1 ¼KTg1;g2 is derived in the same manner. Equations of motion for the coupled system in ΓTRA are

written compactly in matrix form as

M €q tð ÞþC _qðtÞþKq tð Þ ¼ F tð Þ (15a)

with FðtÞ ¼ FTg1ðtÞ O1x6

n oT, qðtÞ ¼ qT

g1ðtÞ qTg2ðtÞ

n oT, C¼ μ1Mþμ2K as a viscous (proportional) damping matrix with

Rayleigh coefficients μ1 and μ2,

M¼Mg1 00 Mg2

" #; K¼

Kg1 �Kg1;g2

�Kg2;g1 Kg2

" #: (15b,c)

The values of μ1 ¼ μ2 ¼ 0:001 are chosen so as to yield roughly 15 percent modal damping. Even though Eq. (15a–c) areessentially the same as Eqs. (17a–c) in [1], their derivation is included for the sake of completeness and clarity in latersections. The nomenclature is slightly different, with a focus on the different coordinate systems and position vectorsreferencing the mounts.

4. Conditions for full decoupling

4.1. Compatibility conditions for a realizable system

The axioms derived by Hu and Singh [1] in Eq. (1) are expanded using the expressions in Eq. (15), resulting in two sets ofconditions, denoted as (i) and (ii), that must be satisfied:

Kg1qTRAg1 ¼ λTRAMg1qTRA

g1 ; (16a)

Kg2;g1qTRAg1 ¼O6�1: (16b)

Condition (i), from Eq. (16a), is identical to that derived by Jeong and Singh [2] for a powertrain with a rigid foundation.These matrices are further expanded into six equations per condition. The first three equations for each condition areidentical; and

∑N

i ¼ 1kxzmir

yg1;mi�kxymir

zg1;mi

� �¼ 0; ∑

N

i ¼ 1kyzmir

yg1;mi�kymir

zg1;mi

� �¼ 0;

∑N

i ¼ 1�kzmir

yg1;miþkyzmir

zg1;mi

� �¼ 0 (17a–c)

∑N

i ¼ 1kymi rzg1;mi

� �2þkzmi ryg1;mi

� �2�2kyzmir

yg1;mir

zg1;mi

�¼ λTRAIx1; (17d)

∑N

i ¼ 1kxzmir

yg1;mir

zg1;miþkyzmir

xg1;mir

zg1;mi�kzmir

xg1;mir

yg1;mi�kxymi rzg1;mi

� �2 �

¼ �λTRAIxy1 ; (17e)

∑N

i ¼ 1kxymir

yg1;mir

zg1;miþkyzmir

xg1;mir

yg1;mi�kymir

xg1;mir

zg1;mi�kxzmi ryg1;mi

� �2 �

¼ �λTRAIxz1 ; (17f)

∑N

i ¼ 1kymir

zg1;mir

zg2;miþkzmir

yg1;mir

yg2;mi�kyzmir

yg1;mir

zg2;mi�kyzmir

zg1;mir

yg2;mi

� �¼ 0; (18a)

∑N

i ¼ 1kxzmir

yg1;mir

zg2;miþkyzmir

zg1;mir

xg2;mi�kzmir

yg1;mir

xg2;mi�kxymir

zg1;mir

zg2;mi

� �¼ 0; (18b)

Fig. 2. Impractical mount locations for an example in Hu and Singh [1] referenced to (a) powertrain sub-system and (b) frame sub-system. Key: ●, mount #1; ◊,mount #2; *, mount #3; and ○, mount #4.


∑N

i ¼ 1kxymir

zg1;mir

yg2;miþkyzmir

yg1;mir

xg2;mi�kymir

zg1;mir

xg2;mi�kxzmir

yg1;mir

yg2;mi

� �¼ 0 (18c)

are nine independent equations that must be satisfied for complete TRA decoupling. Eqs. (17a–f) are from condition (i), andEq. (18a–c) are the last three equations of condition (ii).

Using these equations, Hu and Singh [1] successfully designed a computational powertrain systemwith a compliant base(frame) that is fully decoupled in the TRA direction, denoted as Example 4 in [1]. The locations of the mount elastic centersrelative to both the powertrain and frame are given, reproduced graphically in Fig. 2. Note that the x and y axes of Fig. 2 areinverted from those in [1] to match the right-hand coordinate system of Fig. 1. Both rigid bodies are assumed to berectangular prisms with characteristic lengths ℓ calculated from the provided inertias as

ℓxj

� �2ℓyj

� �2ℓzj

� �2� �T

¼ 12mj

0 1 11 0 11 1 0

264

375

�1 IxjIyjIzj

8>><>>:

9>>=>>;¼ 6

mj

�1 1 11 �1 11 1 �1

264

375

IxjIyjIzj

8>><>>:

9>>=>>;; (19)

and some position vector rg1;g2 ¼ rxg1;g2 ryg1;g2 rzg1;g2n oT

is assumed between the powertrain and frame centers of

gravity (it was not considered in [1]). In order for the system to be realizable, the elastic center of each mount must existin only one physical location. Fig. 2 shows mount #2 on the þy side of both the powertrain and frame. However, mount#1 is on the þy side of the powertrain and the –y side of the frame. Therefore, there is no realistic rg1;g2 which can fixthe elastic center of both mount #1 and #2 to a single location, and the system cannot be physically constructed.Mathematically,

rg1;g2 ¼ rg1;mi�rg2;mi; (20)

and rg1;g2 ¼ 38 �150 �29� �T mm is needed for mount #1 while rg1;g2 ¼ 401 �150 118

� �T mm is needed for mount #2.The same issue exists for mounts #3 and #4. To ensure that the system is realizable, the three compatibility conditions in Eq. (20)must be mathematically incorporated into the TRA decoupling conditions.

4.2. Analysis of decoupling conditions

To implement the needed compatibility conditions, Eq. (20) is rearranged as rg2;mi ¼ rg1;mi�rg1;g2 and used in decouplingcondition (ii), from Eq. (16b). After rearranging terms, the resulting decoupling equations for condition (ii) are

rzg1;g2∑Ni ¼ 1 kymir

zg1;mi�kyzmir

yg1;mi

� �þryg1;g2∑

Ni ¼ 1 kzmir

yg1;mi�kyzmir

zg1;mi

� �

¼∑Ni ¼ 1 kyg;mi rzg1;mi

� �2þkzg;mi ryg1;mi

� �2�2kyzg;mir

yg1;mir

zg1;mi

�; (21a)


rzg1;g2∑Ni ¼ 1 kxzmir

yg1;mi�kxymir

zg1;mi

� �þrxg1;g2 ∑

N

i ¼ 1kyzmir

zg1;mi�kzmir

yg1;mi

� �

¼ ∑N

i ¼ 1kxzmir

yg1;mir

zg1;miþkyzmir

xg1;mir

zg1;mi�kzmir

xg1;mir


� �2 �

; (21b)

ryg1;g2 ∑N

i ¼ 1kxymir

zg1;mi�kxzmir

yg1;mi

� �þrxg1;g2 ∑

N

i ¼ 1kyzmir

yg1;mi�kymir

zg1;mi

� �

¼ ∑N

i ¼ 1kxymir

yg1;mir

zg1;miþkyzmir

xg1;mir

yg1;mi�kymir

xg1;mir


� �2 �

: (21c)

All condition (i) expressions from Eq. (16a) appear in these new equations. Define

P1 ¼ ∑N

i ¼ 1kxzmir

yg1;mi�kxymir

zg1;mi

� �; P2 ¼ ∑

N

i ¼ 1kyzmir

yg1;mi�kymir

zg1;mi

� �

P3 ¼ ∑N

i ¼ 1kzmir

yg1;mi�kyzmir

zg1;mi

� �(22a–c)

as summations from condition (i), used to re-write Eq. (21a–c) as

P¼0 P3 �P2

�P3 0 P1

P2 �P1 0

264

375; P

rxg1;g2ryg1;g2rzg1;g2

8>><>>:

9>>=>>;¼

Ix1� Ixy1� Ixz1

8><>:

9>=>;λTRA: (23a,b)

Eqs. (17a–c) define P1 ¼ P2 ¼ P3 ¼ 0 nominally. Thus, P¼O3x3, and no non-trivial solution exists for rg1;g2. A trivial solutionmay be achieved when λTRA ¼ 0 (rg1;g2 is then arbitrary) or when rg1;g2-71. However, neither results in a physicallyrealizable dynamic system, and it is concluded that full decoupling of a realistic powertrain and frame system in the TRAdirection is not possible. This conclusion extends to any rigid body source and coupled receiver mass system, each with 6degrees of freedom and connected by multiple structural paths. Complete motion decoupling of the source mass roll motioncannot be achieved with practical path location constraints, and consequently partial decoupling must be pursued. Onesimple method is to assign small values to P1, P2, and P3 such that rg1;g2 has a non-trivial solution. However, P is a n�nskew-symmetric matrix where n is odd; and detðPÞ ¼ 0 regardless of the values of P1, P2, and P3. Other partial decouplingmethods are therefore investigated in the next section.

5. Design paradigms for partial decoupling

5.1. Coupled system with arbitrary mount placement (Example I)

In the mounting plane, φymi ¼φz

mi ¼ 0 and each mount is only rotated φxmi from Γ″

mi to ΓTRA, as illustrated in Fig. 3. Thissimplifies ℜ″

mi and results in

Kmi ¼kxmi 0 00 kymi kyzmi

0 kyzmi kzmi

2664

3775¼

kx″mi 0 0

0 ky″mi cos2φx

miþkz″mi sin2φx

mi kz″mi�ky″mi

� �sin φx

mi cos φxmi

0 kz″mi�ky″mi

� �sin φx

mi cos φxmi ky″mi sin

2φxmiþkz″mi cos

2φxmi

266664

377775 (24)

Fig. 3. Rotation of mounts from the mount coordinates (Γ″mi) to the TRA coordinates (ΓTRA) in the mounting plane.

Table 1Powertrain and frame parameters that (a) remain the same for all examples and (b) are specific to Example I.

(a) Common for all examples

Parameter Powertrain Frame

Mass (kg) m1 ¼ 73:2 m2 ¼ 23:9Inertia (kg m2)

M0θg1 ¼

1:94 �0:129 �0:415�0:129 3:43 0:073�0:415 0:073 3:39

264

375 M0θ

g2 ¼3:65 0 0:0270 0:678 0

0:027 0 4:24

264

375

Stiffness (N mm�1) K″mi ¼ diag 336 336 840

� �� K″

bi ¼ 2K″mi

Position (mm) rg1;g2 ¼ �166 54 �548� �T –

(b) Parameters of Example I (baseline design)

Mount #

i¼1 i¼2 i¼3 i¼4

Powertrainφmi (deg) 30 0 0

� �T �30 0 0� �T 30 0 0

� �T �30 0 0� �T

r0g1;mi (mm) �316 �197 �282� �T �316 197 �282

� �T 316 �197 �282� �T 316 197 �282

� �T

Frameφbi (deg) �6:9 0:2 2:0

� �T �6:9 0:2 2:0� �T �6:9 0:2 2:0

� �T �6:9 0:2 2:0� �T

r0g2;bi (mm) �283 �673 �100� �T �283 673 �100

� �T 283 �673 �100� �T 283 673 �100

� �T


for each mount. To provide sufficient static support,��φx

mi

��401 is needed since the z direction does not correspond withthe vertical direction for a realistic powertrain. Stiffness components kxymi ¼ kxzmi ¼ 0 in the mounting plane, and Eq. (17a) isnull with zero on the left and right hand sides. Only five independent equations now exist for decoupling condition (i)from Eq. (16a).

Example I (the baseline case) places both the powertrain and frame mounts at the corners of the respective rigid bodieswithout any specific design considerations, though the powertrain mounts remain in the mounting plane and the framemounts are vertically oriented (Γ″

bi ¼Γ0gj). Additionally, the powertrain mounts are assumed to be identical with

φxm1 ¼ �φx

m2 ¼φxm3 ¼ �φx

m4 such that kxm1 ¼ kxm2 ¼ kxm3 ¼ kxm4 while all kxmi, kymi, and kzmi are respectively equal for each

mount. The frame mounts are also assumed identical and aligned with Γ0gj such that K″

bi ¼K″bi. All Kbi ¼Π0K0

biΠ0T are equaland fully populated. Realistic stiffness and inertia parameters for both the powertrain and frame in Example I are listed inTable 1(a) along with r0g1;g2. These parameters remain the same throughout all examples. Specific φmi, φbi, r

0g1;mi, and r0g2;bi are

listed in Table 1(b) for Example I, and these design parameters will be re-selected in later sections. Recall,r0g2;mi ¼ r0g1;mi�r0g1;g2 and is thus not tabulated, and rn ¼ℜ0r0n for all position vectors where n is a general index.

5.2. Coupled system mount design using only the powertrain sub-system (Example II)

Since full decoupling is not possible, alternate strategies must be pursued. Example II designs the powertrain mountpositions using only decoupling condition (i), from Eq. (16a). The frame and its mounts are thus not considered for thedesign process, though frame rigid body dynamics are included in the system response computation. This design process isanalogous to assuming the powertrain is attached to a rigid foundation, as analyzed by Jeong and Singh [2]. However, thesolution strategy employed is different than in [2], attempting here to create a solution more suited for design purposes andrealizable systems. Like in Jeong and Singh [2], powertrain mounts are identical and assumed in the mounting plane withdefined stiffness and orientation values. The same values used in Example I are employed for both the powertrain and framemounts and inertias. The same r0g2;bi are also used. However, r0g1;mi and λTRA are assumed arbitrary, resulting in 13 totalparameters that must be defined. Only five can be solved from condition (i), while realistic values are selected for the rest.Jeong and Singh [2] obtain a closed form solution to calculate four locations and λTRA by assuming some symmetry in themount locations. Conversely, the following procedure assumes no relationships and utilizes both analytical and computa-tional methods to obtain a reasonable solution. It is also relatively easy to interchange which locations are selected andwhich are solved.

Define dimensionless locations γnmi40 to attempt to place mounts near the four corners of the powertrain and to ensurethe powertrain mass is well distributed:

rxg1;mi ¼γxmir

xg1;m0 for i¼ 1;2

�γxmirxg1;m0 for i¼ 3;4

(; ryg1;mi ¼ ð�1Þiþ1γymir

yg1;m0; rzg1;mi ¼ γzmir

zg1;m0: (25a–c)


Here, rg1;m0 ¼ rxg1;m0 ryg1;m0 rzg1;m0

n oTare equal to the location of mount #1 in Example I, referencing the improved

decoupling design to the baseline. Eqs. (17b–f) are expanded as

kym1rzg1;m0 γzm1þγzm2þγzm3þγzm4

� �kyzm1ryg1;m0 γym1þγym2þγym3þγym4

� ¼ 0; (26a)

kzm1ryg1;m0 �γym1þγym2�γym3þγym4

� þkyzm1rzg1;m0 γzm1�γzm2þγzm3�γzm4

� ¼ 0; (26b)

kym1 rzg1;m0

� �2γzm1

� 2þ γzm2

� 2þ γzm3

� 2þ γzm4

� 2� �þkzm1 ryg1;m0

� �2γym1

� 2þ γym2

� 2þ γym3

� 2þ γym4

� 2� ��2kyzm1r

yg1;m0r

zg1;m0 γym1γ

zm1þγym2γ

zm2þγym3γ

zm3þγym4γ

zm4

� ¼ λTRAIx1; (26c)

kyzm1rzg1;m0 γxm1γ

zm1�γxm2γ

zm2�γxm3γ

zm3þγxm4γ

zm4

� �kzm1r

yg1;m0 γxm1γ

ym1�γxm2γ

ym2�γxm3γ

ym3þγxm4γ

ym4

� ¼ �λTRAIxy1rxg1;m0

; (26d)

kyzm1ryg1;m0 γxm1γ

ym1þγxm2γ

ym2�γxm3γ

ym3�γxm4γ

ym4

� �kym1r

zg1;m0 γxm1γ

zm1þγxm2γ

zm2�γxm3γ

zm3�γxm4γ

zm4

� ¼ �λTRAIxz1rxg1;m0

(26e)

with all mounts identical. If γxmi are defined parameters, Eq. (26a,b,d,e) are linear with respect to γymi and γzmi. Thus, the

solution strategy is to solve for four γymi and γzmi from a linear set of equations in terms of λTRA and use these expressions in

Eq. (26c) to solve λTRA. Specifically, parameters γxmi, γym1, γ

ym3, γ

zm1, and γzm3 are fixed; and the solved parameters to satisfy

condition (i) are γym2, γym4, γ

zm2, γ

zm4 and λTRA. These are summarized in Table 2 for all examples considered, along with the

corresponding solution method.Define known constants σn;p

m ¼ knm1rpg1;m0 where n and p are general indices and re-write Eq. (26a,b,d,e) as Aγγ¼ Yγ in

compact matrix form. Here, γ is the vector of unknowns, Aγ is the system matrix, and Yγ is the output vector. This isexpanded as

�σyz;ym �σyz;y

m σy;zm σy;z

m

σz;ym σz;y

m �σyz;zm �σyz;z

m

σz;ym γxm2 �σz;y

m γxm4 �σyz;zm γxm2 σyz;z

m γxm4

σyz;ym γxm2 �σyz;y

m γxm4 �σy;zm γxm2 σy;z

m γxm4

266664

377775

γym2

γym4

γzm2

γzm4

8>>>><>>>>:

9>>>>=>>>>;

¼

σyz;ym γym1þγym3

� �σy;zm γzm1þγzm3

� σz;ym γym1þγym3

� �σyz;zm γzm1þγzm3

� �λTRAIxy1

rxg1;m0þσz;y

m γxm1γym1�γxm3γ

ym3

� þσyz;zm �γxm1γ

zm1þγxm3γ

zm3

� �λTRAIxz1

rxg1;m0þσy;z

m γxm1γzm1�γxm3γ

zm3

� þσyz;ym �γxm1γ

ym1þγxm3γ

ym3

�

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

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

(27)

and the equations are easily manipulated to instead solve for γym1, γym3, γ

zm1, or γ

zm3 in γ. Analytical methods could be

employed to explicitly solve for γ as linear functions of λTRA and define needed conditions for all γnmi40. However, a

Table 2Fixed mount locations, solved mount locations, and solution methods for all examples analyzed.

Example Powertrain mount locationsa Frame mount locationsa Solution method

Fixed Solved Fixed Solved

I γym1, γym2, γ

ym3, γ

ym4

– γyb1, γyb2, γ

yb3, γ

yb4

– Mounts at inertia corners

γzm1, γzm2, γ

zm3, γ

zm4 – γzb1, γ

zb2, γ

zb3, γ

zb4 –

II γym1, γym3 γym2, γ

ym4 γyb1, γ

yb2, γ

yb3, γ

yb4

– Condition (i) for powertrain

γzm1, γzm3 γzm2, γ

zm4 γzb1, γ

zb2, γ

zb3, γ

zb4 –

III – γym1, γym2, γ

ym3, γ

ym4 γyb1, γ

yb2, γ

yb3, γ

yb4

– Minimize conditions (i) and (ii)

γzm1 γzm2, γzm3, γ

zm4 γzb1, γ

zb2, γ

zb3, γ

zb4 –

IV γym1, γym3 γym2, γ

ym4 γyb1, γ

yb3 γyb2, γ

yb4

Condition (i) for powertrain and frame

γzm1, γzm3 γzm2, γ

zm4 γzb1, γ

zb3 γzb2, γ

zb4

a For all examples, γxmi and γxbi are fixed, and λTRA is solved.


derivation of rather lengthy expressions would not yield tractable solutions. Instead, computational methods and a symbolictoolbox [16] are preferred to solve for γðλTRAÞ, incorporate them into Eq. (26c), solve for λTRA using the quadratic equation,and finally obtain numerical values for γ using the calculated λTRA.

0 50-50

-40

-30

-20

-10

0

10

dB re

f. 1

mm

0 50-50

-40

-30

-20

-10

0

10

0 50-50

-40

-30

-20

-10

0

10

0 50-50

-40

-30

-20

-10

0

10

Freq [Hz]

dB re

f. 1

deg

0 50-50

-40

-30

-20

-10

0

10

Freq [Hz]0 50

-50

-40

-30

-20

-10

0

10

Freq [Hz]

Fig. 4. Powertrain displacement magnitude spectra for Examples I and II in the ΓTRAcoordinate system: (a)��εxg1��, (b) ��εyg1��, (c) ��εzg1��, (d) ��θxg1��, (e) ��θyg1��, and

(f)��θzg1��. Key: , Example I and - - -, Example II.

Table 3Mount locations and ΩTRA for (a) powertrain of Examples II and III and (b) frame of Example IV with and without q0TRA

g1 ¼ q0TRAg2 .

Mount #

i¼1 i¼2 i¼3 i¼4

(a) Examples II and IIIExample IIΩTRA=2π ¼ 33:2 Hzγ0xmi (–) 1.00 1.15 1.21 1.04

γ0ymi (–) 1.00 0.35 0.59 0.14

γ0zmi (–) 1.00 �0.15 0.61 �0.78

Example IIIΩTRA=2π ¼ 32:4 Hzγ0xmi (–) 1.24 1.09 1.12 1.01

γ0ymi (–) 0.94 �0.01 �0.08 �0.02

γ0zmi (–) 1.38 �0.71 �0.05 �0.06

(b) Example IVExample IV: q0TRA

g1 aq0TRAg2

ΩTRA=2π ¼ 155 Hzγ0xbi (–) 1.00 1.64 1.19 1.07

γ0ybi (–) 1.00 0.54 0.67 0.81

γ0zbi (–) 1.00 �10.92 0.17 0.38

Example IV: q0TRAg1 ¼ q0TRA

g2

ΩTRA=2π ¼ 155 Hzγ0xbi (–) 1.00 1.16 1.19 0.60

γ0ybi (–) 1.00 0.69 0.67 0.65

γ0zbi (–) 1.00 0.56 0.17 �11.10


All parameters (both selected and solved for) are listed in Table 3(a) for Example II with ΩTRA ¼ffiffiffiffiffiffiffiffiffiλTRA

pas the TRA roll

mode resonance frequency. Note, not all γzmi40. This is not strictly required for a well-distributed mass as it is the verticalcoordinate; and γxmi40, γymi40 are more pertinent. Further, selecting all γxmiZ1 helps ensure that the mounts are locatedoutside the powertrain body. The resulting displacement magnitude spectra for qg1 are shown in Fig. 4 for both Examples Iand II. Note that the dynamic range (60 dB) is very large, and anything beyond that would represent the computationalnoise floor. Overall, the design in Example II significantly reduces the magnitudes, particularly in the translational directions.The TRA roll motion in

��θxg1

�� is dominant over most of the frequency range, but there is still finite motion in the��εyg1�� and��εzg1�� directions due to the frame coupling. Selecting different γnmi values or solving for alternate parameters will change the

magnitude spectra, though only by 2–3 dB if λTRA is similar. Example II is always an improved design. Akanda and Adulla [17]have applied optimization techniques for mount placement and design in a fixed powertrain system, and such methodscould also be used on the coupled system design of Example II to further enhance the TRA decoupling.

5.3. Minimization of decoupling conditions for coupled system design (Example III)

Example III attempts to improve on Example II (results using only condition (i), from Eq. (16a)) by also includingcondition (ii), from Eq. (16b). Since both conditions cannot be simultaneously satisfied, minimization of the decouplingequations is pursued using a total least squares method [18]. Define parameters

P4 ¼ ∑N

i ¼ 1kymi rzg1;mi

� �2þkzmi ryg1;mi

� �2�2kyzmir

yg1;mir

zg1;mi

�; (28a)

P5 ¼ � ∑N

i ¼ 1kxzmir

yg1;mir

zg1;miþkyzmir

xg1;mir

zg1;mi�kzmir

xg1;mir


� �2 �

; (28b)

P6 ¼ � ∑N

i ¼ 1kxymir

yg1;mir

zg1;miþkyzmir

xg1;mir

yg1;mi�kymir

xg1;mir


� �2 �

(28c)

from Eq. (17d–f) in condition (i). All powertrain mounts are assumed to be in the mounting plane; and P2 ¼ 0, P3 ¼ 0,

P4 ¼ λTRAIx1, P5�λTRAIxy1 ¼ 0, P6�λTRAIxz1 ¼ 0, P3rxg1;g2þP5 ¼ 0, P2rzg1;g2þP3ryg1;g2þP4 ¼ 0, and P2rxg1;g2þP6 ¼ 0 are the eight

necessary decoupling conditions. With rxg1;mi as defined parameters, only P4 is nonlinear with respect to ryg1;mi and rzg1;mi.

Thus, P4 ¼ λTRAIx1 is left as a constant; and the remaining decoupling conditions are defined as residuals ψ1 ¼ P2, ψ2 ¼ P3,

ψ3 ¼ P5�λTRAIxy1 , ψ4 ¼ P6�λTRAIxz1 , ψ5 ¼ P2rzg1;g2þP3ryg1;g2þP4, ψ6 ¼ P3rxg1;g2þP5, and ψ7 ¼ P2rxg1;g2þP6. With ψ¼

ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7

n oTas the vector of residuals and equal weighting, J ¼ψTψ=2¼ΣN

p ¼ 1ðψ2pÞ=2 is the

objective function to minimize with Jp ¼ψ2p=2.

The equations are again analyzed in terms of γ40 where

P2 ¼ kyzm1ryg1;m0 γym1þγym2þγym3þγym4

� �kym1rzg1;m0 γzm1þγzm2þγzm3þγzm4

� ; (29a)

P3 ¼ kzm1ryg1;m0 γym1�γym2þγym3�γym4

� �kyzm1rzg1;m0 γzm1�γzm2þγzm3�γzm4

� ; (29b)

P5 ¼ �kyzm1rxg1;m0r

zg1;m0 γxm1γ

zm1�γxm2γ

zm2�γxm3γ

zm3þγxm4γ

zm4

� þkzm1r

xg1;m0r

yg1;m0 γxm1γ

ym1�γxm2γ

ym2�γxm3γ

ym3þγxm4γ

ym4

� ; (29c)

P6 ¼ �kyzm1rxg1;m0r

yg1;m0 γxm1γ

ym1þγxm2γ

ym2�γxm3γ

ym3�γxm4γ

ym4

� þkym1r

xg1;m0r

zg1;m0 γxm1γ

zm1þγxm2γ

zm2�γxm3γ

zm3�γxm4γ

zm4

� (29d)

after expanding the summations. Then, γ¼ γym1 γym2 γym3 γym4 γzm2 γzm3 γzm4

n oTand λTRA are defined as the

parameters to solve such that conditions (i) and (ii) are minimized. Fixed parameters are γzm1 and γxm1. The function J is

minimized with respect to γ such that ∂J=∂γn ¼ΣNp ¼ 1ð∂Jp=∂γnÞ ¼ 0 for seven total equations to solve γ, and then P4 ¼ λTRAIx1 is

used to solve λTRA. Equations from ∂J=∂γn ¼ΣNp ¼ 1ð∂Jp=∂γnÞ ¼ 0 are written compactly as Aγγ¼ Yγ , and the full derivation and

definition of Aγγ¼ Yγ is shown in Appendix A. Computational methods and a symbolic toolbox [16] are again utilized to

solve for γ and λTRA, following the same methods as in Example II.The resulting displacement magnitude spectra for qg1 are shown in Fig. 5 for Examples II and III. Parameters for Example

III are selected such that ðλTRAÞII � ðλTRAÞIII, and the two examples produce a nearly identical roll motion jθxg1j spectra which

dominates over most of the frequency range. Overall, Example II is the better design, achieving lower magnitude levels onall motions other than the desired jθx

g1j. This is somewhat unexpected, as the frame dynamics are neglected entirely.

0 50-50

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-20

-10

dB re

f. 1

mm

0 50-50

-40

-30

-20

-10

0 50-50

-40

-30

-20

-10

0 50-50

-40

-30

-20

-10

Freq [Hz]

dB re

f. 1

deg

0 50-50

-40

-30

-20

-10

Freq [Hz]0 50

-50

-40

-30

-20

-10

Freq [Hz]

Fig. 5. Powertrain displacement magnitude spectra for Examples II and III in the ΓTRAcoordinate system: (a)��εxg1��, (b) ��εyg1��, (c) ��εzg1��, (d) ��θxg1��, (e) ��θyg1��, and

(f)��θzg1��. Key: - - -, Example II and ..., Example III.

-5000

500 -2000

200-400

-200

0

200

400

y (mm)x (mm)

z (m

m)

mount #1

mount #2mount #4

mount #3

Fig. 6. Comparison of powertrain mount locations relative to the powertrain center of gravity for Examples I, II, and III. Key: , Example I (baseline); ,Example II; and �, Example III.


However, decoupling conditions (i) and (ii) from Eq. (16a,b) are highly contradictive of each other, and while minimizing theeffect of both is mathematically viable, the residuals are still significantly large in the minimized state. It is thus concludedthat little can be done with the powertrain mounts to counteract the reactive motion from the frame coupling. However,adequate decoupling is still achieved with only condition (i) considered for a realistic system.

All parameters (both selected and solved) are listed in Table 3(a) for Example III, and γxmiZ1 are again selected to locatethe mounts outside the powertrain body. As in Example II, not all γzmi40, though this is still not required for a welldistributed mass. Additionally, not all γymi40. Locations γym2, γ

ym3, and γym4 are very close to zero; and the mass may not be

fully supported. A graphical representation of the mount locations is shown in Fig. 6 for Examples I, II, and III; and littlesupport for the mass is indeed provided in Example III for the þy portion of the mass. This further concludes that Example IIis the superior design. Interestingly, mounts #3 and #4 are now located in approximately the same location for Example III,approaching a more common 3-point mounting system instead of a historical 4-point scheme.

6. Design of frame mounts to improve partial decoupling

6.1. Alignment of powertrain and frame TRA axes

Further improvement in the partial TRA decoupling of Example II is possible through an appropriate design of the framemounts. Only the powertrain mount properties are included in decoupling conditions (i) and (ii) from Eq. (16a,b), and the


frame mount properties have thus far been unaltered from Example I (placed at corners of the frame with K″bi ¼K0

bi). Oneapproach could be to limit the modal coupling between the frame and powertrain through the mount design [9–12], but thisis outside the scope of the paper. Instead, a TRA decoupling type approach is utilized. Consider a powertrain system wherethe roll motion θx

g1 dominates. This motion induces reaction forces and moments Rg1;mi from the powertrain mounts alongsome line of action, which are also transmitted to the frame. Likewise, motions from the frame transmit forces Rg2;g1;mi to thepowertrain. If the frame motion is also dominated by θx

g2, the transmitted forces align with Rg1;mi, and the resulting force isalong the same line of action as Rg1;mi. Thus, the powertrain motion is still dominated by θx

g1. Frame motions other than θxg2

alter the Rg2;g1;mi line of action and induce other unwanted motions in the powertrain.It is therefore desired that the frame motion is dominated by θx

g2, essentially designing the mounts using TRA decouplingconditions. Ideally, the TRA of both the powertrain and frame are parallel. This is not likely for a realistic system, thoughnecessary design modifications to the frame inertia could be practical. The TRA direction is defined in Eq. (3) as

q0TRAgj ¼ 0 0 0 ρ011

gj ρ021gj ρ031

gj

n oT, and conditions can be derived such that q0TRA

g1 and q0TRAg2 align. The simplest solution

is to designM0θg1 ¼ τM0θ

g2 where τ is a dimensionless scaling constant, though this is also not realistic given the vastly different

geometries of typical powertrains and frames. Instead, ρ011g1 ρ021

g1 ρ031g1

n o¼ ρ011

g2 ρ021g2 ρ031

g2

n ois directly enforced. A

detailed derivation found in Appendix B results in

Iy0

g1Iz0g1� Iy

0z0

g1

� �2¼ τ Iy

0

g2Iz0g2� Iy

0z0

g2

� �2 �

; (30a)

Ix0z0g1 Iy

0z0

g1 þ Ix0y0

g1 Iz0g1 ¼ τ Ix

0z0g2 Iy

0z0

g2 þ Ix0y0

g2 Iz0g2

� �; (30b)

Ix0y0

g1 Iy0z0

g1 þ Ix0z0g1 Iy

0

g1 ¼ τ Ix0y0

g2 Iy0z0

g2 þ Ix0z0g2 Iy

0

g2

� �(30c)

as the mathematical conditions to align q0TRAg1 and q0TRA

g2 . Three parameters can be solved from Eqs. (30a–c), and many

different solution sets are possible. An example set τ Ix0y0

g2 Ix0z0g2

n ois taken with the parameters solved from Eqs. (30a),

(30b), and (30c), respectively. This results in τ¼ 4:04, Ix0y0

g2 ¼ 0:024 kg m2, and Ix0z0g2 ¼ 0:516 kg m2 where Ix0z

0g2 significantly

differs from the original frame. Altering the frame geometry to match these inertias is a design specific procedure, althoughit may not always be feasible. Additional sets and analysis are discussed in Appendix B.

6.2. Full system frame mount design using only the frame sub-system (Example IV)

The frame is assumed to have the original inertia matrix defined in Table 1(a). Thus q0TRAg1 aq0TRA

g2 , and the effect of aligningthese axes is examined later in this section. Design of the coupled system is still not feasible due to the mount compatibilityconditions, and the decoupled frame system of Fig. 1(c) is considered instead. The TRA decoupling conditions (with theframe mounts in the mounting plane and φx

bi ¼ 101) are

∑N

i ¼ 1kybir

zg2;bi�kyzbi r

yg2;bi

� �¼ �K24

g2;m; ∑N

i ¼ 1�kzbir

yg2;biþkyzbi r

zg2;bi

� �¼ �K34

g2;m; (31a,b)

∑N

i ¼ 1kybi rzg2;bi

� �2þkzbi ryg2;bi

� �2�2kyzbi r

yg2;bir

zg2;bi

�¼ λTRAIx2�K44

g2;m; (31c)

∑N

i ¼ 1kyzbi r

xg2;bir

zg2;bi�kzbir

xg2;bir

yg2;bi

� �¼ �λTRAIxy2 �K54

g2;m; (31d)

∑N

i ¼ 1kyzbi r

xg2;bir

yg2;bi�kybir

xg2;bir

zg2;bi

� �¼ �λTRAIxz2 �K64

g2;m (31e)

where Kng2;m are known from Example II. Define dimensionless locations γnbi40 similar to Eq. (25) for the frame mounts

with rxg2;bi ¼ γxbirxg2;b0 for i¼ 1;2; rxg2;bi ¼ �γxbir

xg2;b0 for i¼ 3;4; ryg2;bi ¼ ð�1Þiþ1γybir

yg2;b0; and rzg2;bi ¼ γzbir

zg2;b0. Here, rg2;b0 ¼

rxg2;b0 ryg2;b0 rzg2;b0n oT

is equal to the location of frame mount #1 in Example II (same as I and III), referencing the

improved decoupling design to the baseline. For the Example IV frame mount design, parameters γxbi, γyb1, γ

yb3, γ

zb1, and γ

zb3 are

fixed; and the solved parameters to satisfy condition (i) are γyb2, γyb4, γ

zb2, γ

zb4 and λTRA (summarized in Table 2). Eq. (31a,b,d,e)


are re-written in matrix form as

�σyz;yb �σyz;y

b σy;zb σy;z

b

σz;yb σz;y

b �σyz;zb �σyz;z

b

σz;yb γxb2 �σz;y

b γxb4 �σyz;zb γxb2 σyz;z

b γxb4σyz;yb γxb2 �σyz;y

b γxb4 �σy;zb γxb2 σy;z

b γxb4

2666664

3777775

γyb2γyb4γzb2γzb4

8>>>><>>>>:

9>>>>=>>>>;

¼

�K24g2;mþσyz;y

b γyb1þγyb3� �σy;z

b γzb1þγzb3�

�K34g2;mþσz;y

b γyb1þγyb3� �σyz;z

b γzb1þγzb3�

�λTRAIxy2 �K54g2;m

rxg2;b0

þσz;yb γxb1γ

yb1�γxb3γ

yb3

� þσyz;zb �γxb1γ

zb1þγxb3γ

zb3

� �λTRAIxz2 �K64

g2;mrxg2;b0

þσy;zb γxb1γ

zb1�γxb3γ

zb3

� þσyz;yb �γxb1γ

yb1þγxb3γ

yb3

�

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

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

(32)

with σn;pb ¼ knb1r

pg2;b0, and the same solution procedure for Example II is repeated.

All frame mount parameters (both selected and solved) andΩTRA are listed in Table 2(b) for Example IV with q0TRAg1 aq0TRA

g2 .

Frame mount locations for Examples II and IV are shown in Fig. 7. All γxbiZ1 and γybi40 for a well-distributed mass withmounts outside the body, and all γzmi40 except for γzm2, which is a very high negative number. This places mount #2 wellabove the frame and is not a realistic mounting position. The resulting displacement magnitude spectra for qg1 are shown in

Fig. 8 for both Examples II and IV with q0TRAg1 aq0TRA

g2 . The TRA roll motion spectra��θx

g1

�� remains the same in both cases, and

the frame mount design of Example IV has successfully reduced the prominent coupling peak in��εyg1�� by 6 dB. Now,

��θxg1

�� is

-5000

500 -10000

1000-500

0

500

1000

1500

y (mm)x (mm)

z (m

m)

mount #1

mount #2

mount #4

mount #3

Fig. 7. Comparison of frame mount locations relative to the powertrain center of gravity for Examples II and IV. Key: , Example II and , Example IV withq0TRAg1 aq0TRA

g2 .

0 50-50

-40

-30

-20

-10

dB re

f. 1

mm

0 50-50

-40

-30

-20

-10

0 50-50

-40

-30

-20

-10

0 50-50

-40

-30

-20

-10

Freq [Hz]

dB re

f. 1

deg

0 50-50

-40

-30

-20

-10

Freq [Hz]0 50

-50

-40

-30

-20

-10

Freq [Hz]

Fig. 8. Powertrain displacement magnitude spectra for Examples II and IV in the ΓTRAcoordinate system: (a)��εxg1��, (b) ��εyg1��, (c) ��εzg1��, (d) ��θxg1��, (e) ��θyg1��, and (f)��θzg1��. Key: - - -, Example II and -·-·-, Example IV with q0TRA

g1 aq0TRAg2 .


the dominant motion throughout the entire frequency range considered. Coupling in the��εxg1��, ��θy

g1

��, and ��θzg1

�� directionshave increased due to q0TRA

g1 aq0TRAg2 , but not to significant magnitude levels. Results may improve if λTRA of the frame mount

design matched that of the powertrain mount design. However, components in Kg2 ¼Kg2;mþKg2;b are much larger than

0 50-80

-60

-40

-20

0

dB re

f. 1

mm

0 50-80

-60

-40

-20

0

0 50-80

-60

-40

-20

0

0 50-80

-60

-40

-20

0

Freq [Hz]

dB re

f. 1

deg

0 50-80

-60

-40

-20

0

Freq [Hz]0 50

-80

-60

-40

-20

0

Freq [Hz]

Fig. 10. Powertrain displacement magnitude spectra for Examples I and IV in the ΓTRAcoordinate system: (a)��εxg1��, (b) ��εyg1��, (c) ��εzg1��, (d) ��θxg1��, (e) ��θyg1��, and (f)��θzg1��. Key: , Example I and , Example IV with q0TRA

g1 ¼ q0TRAg2 .

0 50-50

-40

-30

-20

-10

dB re

f. 1

mm

0 50-50

-40

-30

-20

-10

0 50-50

-40

-30

-20

-10

0 50-50

-40

-30

-20

-10

Freq [Hz]

dB re

f. 1

deg

0 50-50

-40

-30

-20

-10

Freq [Hz]0 50

-50

-40

-30

-20

-10

Freq [Hz]

Fig. 9. Powertrain displacement magnitude spectra for Example IV in the ΓTRAcoordinate system: (a)��εxg1��, (b) ��εyg1��, (c) ��εzg1��, (d) ��θxg1��, (e) ��θyg1��, and (f)

��θzg1��.Key: -·-·-, Example IV with q0TRA

g1 aq0TRAg2 and , Example IV with q0TRA

g1 ¼ q0TRAg2 .


those in Kg1 ¼Kg1;m due to stiff frame mounts and the combined effect of both frame (Kg2;b) and powertrain (Kg2;m) mounts.

The resulting ΩTRA ¼ 155 Hz is thus high, and realistic frame mount locations cannot be selected to lower it to say

ΩTRA ¼ 33 Hz.

Next, the TRA axes are aligned in Example IV with τ¼ 4:04, Ix0y0

g2 ¼ 0:024 kg m2, and Ix0z0

g2 ¼ 0:516 kg m2. The resulting

frame mount locations for q0TRAg1 ¼ q0TRA

g2 are shown in Table 2(b), and the qg1 displacement magnitude spectra for Example IV

with and without q0TRAg1 ¼ q0TRA

g2 are shown in Fig. 9. Mount locations are similar for both cases with mount #4 now being way

above the frame body in the q0TRAg1 ¼ q0TRA

g2 case (mount #2 in q0TRAg1 aq0TRA

g2 case). This may cause a feasibility issue, though the

frame geometry could be specially designed on a case by case basis. The new design with q0TRAg1 ¼ q0TRA

g2 lowers the coupling

peak in��εyg1�� by an additional 5 dB for a total reduction of 11 dB from the Example II design. Additionally, couplings in the��εxg1��, ��θyg1

��, and ��θzg1

�� directions are decreased, with motion in the��εzg1�� direction being similar to Example II. Note, solving

different parameters to satisfy Eq. (30) has a minimal effect on γnmi and qg1. Examples I and IV (with q0TRAg1 ¼ q0TRA

g2 ) are

compared in Fig. 10 to show the drastic improvement made over the original mounting system, and��θx

g1

�� is now thedominant motion.

7. Alternate design methods and practical constraints

This paper has focused on one particular mounting system design for reduced noise and vibration. Other strategies maybe pursued (possibly in conjunction with TRA decoupling), and a few are discussed briefly in this section. One simplestrategy is to add more damping (say via mounts or structural damping treatments on the frame) to lower peak motions.This is demonstrated in Fig. 11 for Example II with C elements being four times as large. Though the roll mode peak λTRA is nolonger prominent, the

��θxg1

�� motion now dominates over the entire frequency range. This solution assumes the addition ofevenly distributed damping over the entire system, which is costly and adds significant mass (often unattractive options inpractice). Localized damping patches may be effective if proper design procedures and mathematical analyses are utilized.Another relatively straightforward solution is to stiffen the frame mounts, increasing frame rigidity and shifting the coupledmodes to higher frequencies. This is also demonstrated in Fig. 11 for Example II where K″

bi elements are an order ofmagnitude higher. Here, the desired roll motion remains while the coupled motions are greatly reduced. Such a solutionmay not be satisfactory for vibration and structureborne isolation needs (for acoustic comfort) where compliant mounts areessential. Tuned hydraulic mounts with spectrally-varying properties are then a potential solution [19]. Adjusting K″

bi orother stiffness components is a strategy congruent with sub-structuring [9–11] and mode shifting [12] methods, operatingin the frequency and modal domains, respectively. Altering mass components is also an option, though financial and

0 50-50

-40

-30

-20

-10

dB re

f. 1

mm

0 50-50

-40

-30

-20

-10

0 50-50

-40

-30

-20

-10

0 50-50

-40

-30

-20

-10

Freq [Hz]

dB re

f. 1

deg

0 50-50

-40

-30

-20

-10

Freq [Hz]0 50

-50

-40

-30

-20

-10

Freq [Hz]

Fig. 11. Powertrain displacement magnitude spectra for Examples II in the ΓTRA coordinate system with damping and stiffness modifications: (a) εxg1

�� ,(b) εyg1

�� , (c) εzg1

�� , (d) θxg1

�� , (e) θyg1

�� , and (f) θzg1

�� . Key: - - -, Example II; ·· ··, Example II with 4C; and , Example II with 10K″bi .


practical limitations exist: more mass entails higher production costs and worse fuel economy while less mass generallycreates higher vibration levels and becomes an issue for crash worthiness testing.

A more design specific strategy could be to place powertrain mounts at rigid connection points on the frame, where littlemode participation exists up to a prescribed frequency. A simplifying assumption made in the frame model is that it behavesas a single rigid body at low frequencies. Depending on the design, a more realistic frame model could either consider acontinuous system [20] or one with multiple discrete masses [4]. In this case, hard points or anti-nodes could be optimalplaces to attach powertrain mounts, as minimal frame deflection occurs. Incorporating these desired locations into the TRAdecoupling condition (i) would create a complication. Another design specific strategy could be excitation shaping or inputreduction (as observed by the powertrain inertial body), focusing on control of the source as opposed to the paths(powertrain mounts) or receiver (frame). Methods such as designing torque balance shafts [8], increasing the number ofengine cylinders [14], and controlled fuel injection [21,22] are viable options. Also, a multidimensional dynamic vibrationabsorber could be attached to the powertrain or frame to reduce motions at selected resonance peaks, though fundamentalresearch is needed to explore such an approach.

8. Conclusion

The paradigms (leading to an eigenvalue problem KqTRA ¼ λTRAMqTRA) recently proposed by Hu and Singh [1] for adiscrete, proportionally damped coupled powertrain and frame system are critically examined. It has been found that thederivation of K in [1] neglects the need for a physically realizable system. Namely, each powertrain mount is referenced totwo different locations: from the powertrain to the mount and from the frame to the mount. Since these locations do notcoincide in [1], the system cannot be constructed, and this deficiency in the decoupling analysis is overcome by deriving andimplementing mount compatibility conditions into the derivation of K such that mounts are always referenced to a singlelocation. Further, it is mathematically proven that a non-trivial solution does not exist for KqTRA ¼ λTRAMqTRA whencompatibility conditions are implemented. Full decoupling of the powertrain TRA is not possible for a physically realizablecoupled powertrain and frame system.

Thus, partial decoupling powertrain mount design paradigms which do not impose severe burden on the isolationsystem design are proposed via realistic example cases. One paradigm minimizes the needed decoupling conditions for thecoupled system using a total least squares method. This results in a dominant TRA roll motion

��θxg1

�� over much of the

frequency range considered (up to 70 Hz) but with finite coupling motions in the��εyg1�� and ��εzg1�� translational directions. This

occurs because the decoupling conditions are highly contradictive of each other, and the residuals are still significantly largein the minimized state. A second paradigm considers the decoupled powertrain only, neglecting frame coupling. Throughcareful mount parameter selection, a nearly identical roll motion

��θxg1

�� as in the first paradigm occurs but with reduced

coupling motion. This results in an improved design, and��θx

g1

�� now dominates over nearly the entire frequency range.Decoupling is further improved by implementing similar frame mount and inertia paradigms. Frame mounts are designedwith the decoupled frame model only, neglecting powertrain coupling, and the frame inertia is modified such that thepowertrain and frame TRA axes are aligned. Coupled translational motions are further reduced, and

��θxg1

�� now dominatesover the entire frequency range. Here, an important contribution immerges. Design of the frame mounts and inertia forsuperior TRA partial decoupling appears to be an additive process to the powertrain design, not a simultaneous one. Such aresult is not intuitive and only concluded through the rigorous analysis of various paradigms conducted in this paper.

Alternative isolation system design methods to limit powertrain and frame coupling are also briefly discussed, and these may beimplemented independently or simultaneously with the TRA decoupling paradigms examined. Such methods include adding moredamping, increasing the frame mount stiffnesses, placing powertrain mounts at hard points or anti-nodes on the frame, excitationshaping, multidimensional dynamic vibration absorbers, mode shifting, and sub-structuring. Limitations of this article include theneglect of mount rotational stiffness effects, material frequency dependence, nonlinear effects due to mount properties or finiteamplitudes, driveline shaft influence, vehicle body and tire dynamics, and flexural mode participation.

Acknowledgment

We acknowledge the member organizations such as Hyundai Motor Company (R&D Division) of the Smart VehicleConcepts Center (www.SmartVehicleCenter.org) and the National Science Foundation Industry/University CooperativeResearch Centers program (www.nsf.gov/eng/iip/iucrc) for supporting this work.

Appendix A. Least squares minimization of Example III

The function J ¼ψTψ=2¼ ð1=2Þ∑Np ¼ 1ðψ2

p=2Þ defined in Section 5.3 is minimized with respect to γ such that ∂J=∂γn ¼ΣN

p ¼ 1ð∂Jp=∂γnÞ ¼ 0 for seven total equations. Considering one ∂Jp=∂γn at a time,

∂J1∂γn

¼ P2∂P2

∂γn;

∂J2∂γn

¼ P3∂P3

∂γn;

∂J3∂γn

¼ P5�λTRAIxy1� �∂P5

∂γn;

∂J4∂γn

¼ P6�λTRAIxz1� �∂P6

∂γn(A1-4)

www.SmartVehicleCenter.org

www.nsf.gov/eng/iip/iucrc


∂J5∂γn

¼ rzg1;g2P2�ryg1;g2P3þP4

� �rzg1;g2

∂P2

∂γn�ryg1;g2

∂P3

∂γn

�; (A-5)

∂J6∂γn

¼ �rxg1;g2P3þP5

� ��rxg1;g2

∂P3

∂γnþ∂P5

∂γn

�;

∂J7∂γn

¼ rxg1;g2P2þP6

� �rxg1;g2

∂P2

∂γnþ∂P6

∂γn

�(A6-7)

are the resulting partial derivatives after applying the chain rule and simplifying expressions. Summing over p andcombining terms gives ∂J=∂γn ¼ ςn0þP2ςn2þP3ςn3þP5ςn5þP6ςn6 where

ςn0 ¼ λTRAIx1 rzg1;g2∂P2

∂γn�ryg1;g2

∂P3

∂γn� Ixy1

Ix1

∂P5

∂γn� Ixz1

Ix1

∂P6

∂γn

�; (A-8)

ςn2 ¼ 1þ rxg1;g2� �2

þ rzg1;g2� �2

�∂P2

∂γn�ryg1;g2r

zg1;g2

∂P3

∂γnþrxg1;g2

∂P6

∂γn; (A-9)

ςn3 ¼ �ryg1;g2rzg1;g2

∂P2

∂γnþ 1þ rxg1;g2

� �2þ ryg1;g2� �2

�∂P3

∂γn�rxg1;g2

∂P6

∂γn; (A-10)

ςn5 ¼ �rxg1;g2∂P3

∂γnþ2

∂P5

∂γn; ςn6 ¼ rxg1;g2

∂P2

∂γnþ2

∂P6

∂γn; (A11-12)

and partial derivatives in ς are known constants calculated from Eq. (29a–d):

∂P2=∂γym1 ∂P3=∂γ

ym1 ∂P5=∂γ

ym1 ∂P6=∂γ

ym1

∂P2=∂γym2 ∂P3=∂γ

ym2 ∂P5=∂γ

ym2 ∂P6=∂γ

ym2

∂P2=∂γym3 ∂P3=∂γ

ym3 ∂P5=∂γ

ym3 ∂P6=∂γ

ym3

∂P2=∂γym4 ∂P3=∂γ

ym4 ∂P5=∂γ

ym4 ∂P6=∂γ

ym4

∂P2=∂γym2 ∂P3=∂γ

ym2 ∂P5=∂γ

ym2 ∂P6=∂γ

ym2

∂P2=∂γym3 ∂P3=∂γ

ym3 ∂P5=∂γ

ym3 ∂P6=∂γ

ym3

∂P2=∂γym4 ∂P3=∂γ

ym4 ∂P5=∂γ

ym4 ∂P6=∂γ

ym4

26666666666664

37777777777775¼

σyz;ym σz;y

m rxg1;m1σz;ym �rxg1;m1σ

yz;ym

σyz;ym �σz;y

m �rxg1;m2σz;ym �rxg1;m2σ

yz;ym

σyz;ym σz;y

m rxg1;m3σz;ym �rxg1;m3σ

yz;ym

σyz;ym �σz;y

m �rxg1;m4σz;ym �rxg1;m4σ

yz;ym

�σy;zm σyz;z

m rxg1;m2σyz;zm rxg1;m2σ

y;zm

�σy;zm �σyz;z

m �rxg1;m3σyz;zm rxg1;m3σ

y;zm

�σy;zm σyz;z

m rxg1;m4σyz;zm rxg1;m4σ

y;zm

2666666666666664

3777777777777775

: (A-13)

The equation An1γym1þAn2γ

ym2þAn3γ

ym3þAn4γ

ym4þAn5γzm2þAn6γzm3þAn7γzm4 ¼ Yn is derived from ∂J=∂γn ¼ 0 by substituting

in all P expressions and rearranging where

An1 ¼ σyz;ym ςn2þσz;y

m ςn3þσz;ym rxg1;m1ςn5�σyz;y

m rxg1;m1ςn6; (A-14)

An2 ¼ σyz;ym ςn2�σz;y

m ςn3�σz;ym rxg1;m2ςn5�σyz;y


An3 ¼ σyz;ym ςn2þσz;y

m ςn3þσz;ym rxg1;m3ςn5�σyz;y


An4 ¼ σyz;ym ςn2�σz;y

m ςn3�σz;ym rxg1;m4ςn5�σyz;y


An5 ¼ �σy;zm ςn2þσyz;z

m ςn3þσyz;zm rxg1;m2ςn5þσy;z


An6 ¼ �σy;zm ςn2�σyz;z

m ςn3�σyz;zm rxg1;m3ςn5þσy;z


An7 ¼ �σy;zm ςn2þσyz;z

m ςn3þσyz;zm rxg1;m4ςn5þσy;z


Yn ¼ σy;zm ςn2þσyz;z

m ςn3þσyz;zm rxg1;m1ςn5�σy;z

m rxg1;m1ςn6� �

γzm1�ςn0: (A-21)


Finally, the seven equations for the minimized cost function are constructed as

A11 A12 A13 A14 A15 A16 A17

A21 A22 A23 A24 A25 A26 A27

A31 A32 A33 A34 A35 A36 A37

A41 A42 A43 A44 A45 A46 A47

A51 A52 A53 A54 A55 A56 A57

A61 A62 A63 A64 A65 A66 A67

A71 A72 A73 A74 A75 A76 A77

2666666666664

3777777777775

γym1

γym2

γym3

γym4

γzm2

γzm3

γzm4

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

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

¼

Y1

Y2

Y3

Y4

Y5

Y6

Y7

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

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

; (A22)

written compactly as Aγγ¼ Yγ .

Appendix B. Alignment of powertrain and frame TRA

To align q0TRAg1 and q0TRA

g2 , define components ρ011gj , ρ

021gj , and ρ031

gj in an analytical manner as

ρ011gj det M0θ

gj

� �¼ Iy0gjI

z0gj� Iy0z

0

gj

� �2; (B1)

ρ021gj det M0θ

gj

� �¼ Ix0z

0gj Iy0z

0

gj þ Ix0y0

gj Iz0gj; (B2)

ρ031gj det M0θ

gj

� �¼ Ix0y

0

gj Iy0z0

gj þ Ix0z0

gj Iy0gj (B3)

from ρ0gj ¼ M0θ

gj

h i�1. Modify the normalizing constant b¼ 1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ011gj

� �2þ ρ021

gj

� �2þ ρ031

gj

� �2r

as

b¼det M0θ

gj

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ011gj det M0θ

gj

� �� 2þ ρ021

gj det M0θgj

� �� 2þ ρ031

gj det M0θgj

� �� 2r (B4)

for ρ0gj ¼ b M0θ

gj

h i�1, and normalized components

ρ0ngj ¼

ρ0ngjdet M0θ

gj

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρ011gj det M0θ

gj

� �� 2þ ρ021

gj det M0θgj

� �� 2þ ρ031

gj det M0θgj

� �� 2r (B5)

in q0TRAgj are fully defined. Here,

ρ011g1 ρ021

g1 ρ031g1

n o¼ ρ011

g2 ρ021g2 ρ031

g2

n o(B6)

is the most general design solution with highly nonlinear (and difficult to solve) equations. Instead, define the relationshipsρ0ng1det M0θ

g1

� �¼ τρ0n

g2det M0θg2

� �which satisfy Eq. (B6). Now,

Iy0g1Iz0g1� Iy0z

0

g1

� �2¼ τ Iy0g2I

z0g2� Iy0z

0

g2

� �2 �

; (B7)

Ix0z0

g1 Iy0z0

g1 þ Ix0y0

g1 Iz0g1 ¼ τ Ix0z0

g2 Iy0z0

g2 þ Ix0y0

g2 Iz0g2� �

; (B8)

Ix0y0

g1 Iy0z0

g1 þ Ix0z0

g1 Iy0g1 ¼ τ Ix0y0

g2 Iy0z0

g2 þ Ix0z0

g2 Iy0g2� �

(B9)

are the mathematical conditions to align q0TRAg1 and q0TRA

g2 .Three parameters can be solved from the conditions of Eqs. (B7)–(B9), and two example sets are selected:

τ Ix0y0

g2 Ix0z0g2

n oand Iy

0z0

g2 τ Ix0z0g2

n owith parameters solved from Eqs. (B7)–(B9), respectively. The former results in

τ¼ 4:04, Ix0y0

g2 ¼ 0:024 kg m2, and Ix0z0g2 ¼ 0:516 kg m2 where Ix

0z0g2 significantly differs from the original frame. The latter results

in Iy0z0

g2 ¼ 0:195 kg m2, τ¼ 4:10, and Ix0z0g2 ¼ 0:509 kg m2 where both Iy

0z0

g2 and Ix0z0g2 significantly differ from the original frame.

Altering frame geometry to match these inertias is a design specific procedure, though it may not always be feasible. Also,

different sets of parameters can be solved to produce more realistic results. For instance if Ix0g2, I

y0

g2, and Iz0g2 take nominal

values, alternate sets τ Iy0z0

g2 Ix0z0g2

n o¼ 4:10 0:195 1:769� �

and Iy0z0

g2 Ix0z0g2 τ

n o¼ 0:195 0:509 4:10� �

can be solved.

The latter is identical to the Iy0z0

g2 τ Ix0z0g2

n oset. Possible sets are limited as Eqs. (B7)–(B9) are nonlinear with respect to the

unknown parameters. For instance, Ix0y0

g2 ¼ Iy0z0

g2 ¼ 0 originally, and a denominator is zero for some sets if either Ix0y0

g2 or Iy0z0

g2 is


taken as a nominal value (no non-trivial solution exists). Additional sets are possible if Ix0g2, I

y0

g2, or Iz0g2 are considered as

variables instead of Ix0y

0

g2 , Ix0z0g2 , Iy

0z0

g2 , or τ.

References

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