Finite Elements in Analysis and Design - Dr. Amr M. …ele f d e2g¼fF ele2gð9Þ where dynamic stiffness ½KD ele2¼½K ele2 o 2M ele2 and fd e2g¼fd jr d skg T. The dynamic stiffness

ARTICLE IN PRESS

Finite Elements in Analysis and Design 46 (2010) 685–697

Contents lists available at ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

� Corr

E-m

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

Mechanical filtering characteristics of passive periodic engine mount

Woojin Jung a, Zheng Gu b, A. Baz b,�

a Agency of Defence Development, P.O. Box 18, Jinhae, Gyeongnam 645-600, South Koreab Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA

a r t i c l e i n f o

Article history:

Received 6 April 2009

Accepted 14 March 2010Available online 10 April 2010

Keywords:

Periodic engine mount

Finite element analysis

Transfer matrix approach

Experimental validation

4X/$ - see front matter & 2010 Elsevier B.V. A

016/j.finel.2010.03.007

esponding author. Tel.: +1 301 405 5216; fax

ail address: [email protected] (A. Baz).

a b s t r a c t

The transmission of automotive engine vibrations to the chassis is isolated using a new class of mounts

which rely in their operation on optimally designed and periodically distributed viscoelastic inserts. The

proposed mount acts as a mechanical filter for impeding the propagation of vibration within specific

frequency bands called the ‘‘stop bands’’. The spectral width of these bands is enhanced by making the

viscoelastic inserts operate in a shear mode rather than compression mode. The theory governing the

operation of this class of periodic mounts is presented using the theory of finite elements combined

with the transfer matrix approach. The predictions of the performance of the mount are validated

against the predictions of the commercial finite element code ANSYS and against experimental results

obtained from prototypes of plain and periodic mounts. The obtained results demonstrate the feasibility

of the shear mode periodic mount as another means for blocking the transmission of vibration over a

broad frequency band. Extending the effective width of the operating frequency bands of this class of

mount through active control means is the ultimate goal of this study.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

A periodic structure consists of an assembly of identicalelements connected in a repeated manner [1]. Examples of thesestructures can be found in many engineering applications such assatellite solar panels, wings and fuselages of aircraft, petroleumpipe-lines, railway tracks, submarines and many others.

In these structures, waves can propagate in some frequencybands called ‘‘pass bands’’ and are attenuated in others called‘‘stop bands’’ [2–7]. Excellent reviews on the state-of-the-art havebeen given by Mead [7] and by Mester and Benaroya [8], whereextended lists of references can be found. Since then, studies ofthe characteristics of periodic structures and their applications inengineering have been extensively investigated including passiveand active periodic structures [9–17].

In this paper, the emphasis is placed on the development of ashear mode passive periodic engine mounts in order to effectivelyisolate the transmission of vibration from the engine to thechassis. This class of mounts is radically different from otherconventional types of engine mounts such as the passive rubbermounts [18] and hydraulic engine mount [19,20] which aregenerally effective at narrow frequency ranges. It is equally aseffective as other types of active engine mounts [21,22] which canoperate over broader frequency ranges but at the expense of theclassical complexity and reliability.

ll rights reserved.

: +1 301 405 8331.

The effectiveness of the presented periodic engine mount stemsfrom its unique design that relies in its operation on optimallydesigned and periodically distributed viscoelastic inserts in orderto generate broad band filtering characteristics. Such characteristicsenable the mount to completely block the propagation of thevibration rather than attenuating it. The spectral width of theoperating band is enhanced by making the viscoelastic insertsoperate in a shear mode rather than compression mode used in thepassive periodic mount of Asiri [23].

The paper is accordingly organized in five sections. In Section 1, abrief introduction is given. In Section 2, a mathematical model of theshear mode periodic mount is presented and the equations ofmotion are derived from the finite element approach and then thetransfer matrix is obtained. The basic filtering characteristics ofthese mounts are outlined in Section 3. Section 4 demonstrates theexperimental results and numerical analysis using ANSYS. Section 5summarizes the obtained results and conclusions reached.

2. Mathematical modeling of passive periodic mounts

2.1. Overview

Fig. 1a shows a schematic drawing of the shear mode passiveperiodic mount which is made of identical cells in the longitudinaldirection. Each cell can be divided into four elements as shown inFig. 1b. These elements are numbered 1,2,3, and 4 from the left tothe right. The dynamic behavior of element 2 is dominated by theshear of the viscoelastic layer while elements 1, 3, and 4

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dx.doi.org/10.1016/j.finel.2010.03.007

mailto:[email protected]

ARTICLE IN PRESS

Nomenclature

A Areab widthE Young’s modulus of elasticityF Longitudinal traction forceG2 Shear modulus of viscoelasticityh1,h2,h3 Thickness of core, viscoelastic material and outer

layerKij The element of dynamic stiffness matrix of sub-cellLi the length of sub-cellm Mass per unit lengthNi Shape function of sub-cellT Kinetic energyT Transfer matrixu Longitudinal deflectionV Potential energyW External workai Coefficient of exponential shape function

a Propagation attenuation factorb Propagation phase angleg Shear strainl Eigenvalues of the transfer matrixdij Longitudinal deflection vectorr Mass densitym Propagation parametero Frequency (rad/s)

Subscripts

jr, sk nodes of element of core and outer layer, respectively.x Partial differential with respect to x

t Partial differential with respect to t

Superscripts

�1 Matrix inverse

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685–697686

experience longitudinal loading. The transfer matrix of the unitcell is derived by applying the finite element approach along withthe appropriate boundary conditions.

2.2. The transfer matrix of element 2

2.2.1. Main assumptions

(1)
the shear strains in the metal core are negligible; (2) the longitudinal stresses in the viscoelastic layer are negligible;
Unit Cell

Base Metal Viscoelas

Unit Cell

Periodic

h3

L1 L2

L4L3

h1

h2

Element 1Element 2

Element 3 Element 4

h

Element 1

i

[T1]

Undeflected

Fig. 1. Schematic drawing of shear mode periodic mount and main p

(3)

tic MMou

j

aram

the transverse displacements of all the points on any crosssection of the periodic mount are not considered;

(4)
the metal core and outer layers are assumed to be elastic anddissipate no energy;
(5)
the viscoelastic layer is linearly viscoelastic.
2.2.2. Kinematic relationships

The deflected configuration is shown in Fig. 2 where A1,E1,h1

denote the cross section area, Young’s modulus and thickness ofcore. A2,G2,h2,g are the cross section area, shear’s modulus,

aterial

Unit Cell Unit Cell

nt

Element 2 Element 3 Element 4

k m

n

[T2] [T3] [T4]

k m

Deflected

eters: (a) periodic mount; (b) undeflected and (c) deflected.

ARTICLE IN PRESS

γ

h3

r

[T2]

Element 2

j

k

k

s

s

Fj

Fk

Fk

h2

h1

A2,G2, L2

A3,G3, L2

A1,E1, L2

Fig. 2. Deflected configuration of Element 2.

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685–697 687

thickness and shear strain of viscoelastic material. Also, A3,E3,h3

define the cross section area, Young’s modulus and thickness ofouter layers.

From the geometry of Fig. 2, the shear strain g in theviscoelastic material is given by

g¼ ðujr�uskÞ=h2 ð1Þ

where ujr and usk are the longitudinal deflections of the core andouter layer, respectively. Also, h2 defines the thickness ofviscoelastic layer.

2.2.3. Energies of the element 2

Potential energies. The potential energies, V1 and V2, associatedwith the longitudinal extension of the core/outer layers and theshear of viscoelastic layers are given by

V ¼ V1þV2 ¼

Z Z ZV

1

2ðsjrejrþsskeskþt2gÞdV

¼1

2E1A1

Z L2

0

@ujr

@x

� �2

dxþ1

2E3A3

Z L2

0

@usk

@x

� �2

dx

þ1

2G2A2

Z L2

0g2 dx ð2Þ

where A1 ¼ h1b, A2 ¼ 2h2b, A3 ¼ 2h3b with b denoting the width ofthe core and the outer layer. Also, t2, G2 denote shear stress andstorage shear modulus of the viscoelastic layer. L2 is the length ofelement 2. Subscripts jr, sk denote the nodes of the core and outerlayer, respectively.

Kinetic energies. The kinetic energy T associated with thelongitudinal deflection ujr and usk is given by

T ¼1

2r1A1

Z L2

0

@ujr

@t

� �2

dxþ1

2r3A3

Z L2

0

@usk

@t

� �2

dx ð3Þ

where r1, r3 are the densities of the core and outer layer,respectively.

2.2.4. Finite element model of element 2

The displacements of core and outer layer can be described bythe following shape functions:

ujrðxÞ ¼ ½Nj Nr �uj

ur

( )¼ ½Nj Nr�fdjrg,

uskðxÞ ¼ ½Ns Nk�us

uk

( )¼ ½Ns Nk�fdskg ð4Þ

where uj,ur ,Nj,Nr are nodal displacements and shape functions at

nodes j, r in core, as are us,uk,Ns,Nk at nodes s, k of outer layer.From Eqs. (1) and (4), the shear strain g becomes,

g¼ ðujr�uskÞ=h2 ¼ ð1=h2ÞðNjujþNrur�Nsus�NkukÞ

ð5Þ

ujr, usk in Eq. (4) can be also rewritten as

ð6Þ

The potential energy of Eq. (2) can be evaluated using Eq. (6) asfollows,

ð7Þ

where

One can also obtain mass matrix of element 2 using the kineticenergies,

ð8Þ

The equation of motion for element 2 can be obtained as follows:

½Mele2�f€de2gþ½Kele2�fde2g ¼ fFele2g or ½KD

ele2�fde2g ¼ fFele2g ð9Þ

where dynamic stiffness ½KDele2� ¼ ½Kele2�o2Mele2� and fde2g ¼ fdjr

dskgT . The dynamic stiffness matrix of element 2 can be given by

ð10Þ

ARTICLE IN PRESS

Table 1Geometric properties.

Length (mm) Thickness (mm) Width (mm)

L1 4.76 h1 3.17 b 25.4

L2 17.46 h2 3.18, 8, 15

L3 4.76

L4 4.76 h3 3.18


where

Kab ¼

Z L2

0½E1A1ðNa,xNb,xÞþgðNaNbÞ�o2r1A1ðNaNbÞ�dx;

a¼ j,r; b¼ j,r

Kjs ¼�

Z L2

0½gðNjNsÞ�dx, Kjk ¼�

Z L2

0½gðNjNkÞ�dx,

Krs ¼�

Z L2

0½gðNrNsÞ�dx

Krk ¼�

Z L2

0½gðNrNkÞ�dx, Ksj ¼�

Z L2

0½gðNsNjÞ�dx,

Ksr ¼�

Z L2

0½gðNsNrÞ�dx

Kkj ¼�

Z L2

0½gðNkNjÞ�dx, Kkr ¼�

Z L2

0½gðNkNrÞ�dx

Kcd ¼

Z L2

0½E3A3ðNc,xNd,xÞþgðNcNdÞ�o2r3A3ðNcNdÞ�dx;

c¼ s,k; d¼ s,k

g ¼ ðG2A2=h22Þ

2.2.5. The transfer matrix of element 2

The transfer matrix of element 2 can be obtained by usingdynamic stiffness matrix of Eq. (10) and applying the freeboundary condition at ur, us:

ð11Þ

Rearranging Eq. (11) leads to the following equation:

Kele211 Kele2

12

Kele221 Kele2

22

" #uj

uk

( )¼

Fj

Fk

( )ð12Þ

From Eq. (12) and Fig. 2, one can establish the transfer matrix [T2]:

ujþ1

Fjþ1

( )¼

uk

�Fk

( )¼ ½T2�

uj

Fj

( )ð13Þ

where [T2] is the transfer matrix of element 2. Combining Eq. (12)and (13) yields the transfer matrix [T2]:

ð14Þ

or

ð15Þ

where

Kele211 ¼ ðKjjÞþðKjsÞCjþðKjrÞDj, Kele2

12 ¼ ðKjkÞþðKjsÞCkþðKjrÞDk

Kele221 ¼ ðKkjÞþðKksÞCjþðKkrÞDj, Kele2

22 ¼ ðKkkÞþðKksÞCkþðKkrÞDk

Cj ¼�½ðKssÞ�ðKsrÞðKrrÞ�1ðKrsÞ�

�1½ðKsjÞ�ðKsrÞðKrrÞ�1ðKrjÞ�

Ck ¼�½ðKssÞ�ðKsrÞðKrrÞ�1ðKrsÞ�

�1½ðKskÞ�ðKsrÞðKrrÞ�1ðKrkÞ�

Dj ¼ ½�ðKrrÞ�1ðKrjÞ�ðKrrÞ

�1ðKrsÞCj�

Dk ¼ ½�ðKrrÞ�1ðKrkÞ�ðKrrÞ

�1ðKrsÞCk�

2.3. The transfer matrices of elements 1, 3 and 4

Elements 1,3 and 4 can be regarded as one-dimensional rodwith different cross section areas, therefore, their potential andkinetic energies have the similar form:

Vi ¼1

2EiAi

Z Li

0

@ui

@x

� �2

dx, ð16Þ

Ti ¼1

2riAi

Z Li

0

@ui

@t

� �2

dx, i¼ 1,3,4 ð17Þ

Let u1, u2 be the axial displacement of two nodes of one element.Also Let N1ðxÞ,N2ðxÞ be the shape function of u1, u2. The deflectionvariation of one-dimensional rod can be derived as

uiðxÞ ¼ ½N1ðxÞ N2ðxÞ�u1

u2

( )¼ ½N�½de� ð18Þ

where fdeg ¼ fu1 u2gT . One can evaluate potential energies and

kinetic energies as follows:

Vi ¼1

2EiAi

Z Li

0

@ui

@x

� �T @ui

@x

� �dx

¼1

2deg

T EiAi

Z Li

0

@N1

@x

@N1

@x

@N1

@x

@N2

@x@N2

@x

@N1

@x

@N2

@x

@N2

@x

2664

3775dx

0BB@

1CCA de

� �8>><>>:

¼1

2deg

T ½kia� de

� �nð19Þ

Ti ¼1

2riAi

Z Li

0

@ui

@t

� �T @ui

@t

� �dx

¼1

2_deg

T ðriAi

Z Li

0

N1N1 N1N2

N2N1 N2N2

" #dxÞ _de

n o¼

1

2_deg

T ½mia�

_de

n on(

ð20Þ

The equation of motion for one element can be obtained asfollows:

½mia�f€degþ½k

ia�fdeg ¼ fFeg or ½ki

a�o2mi

a�fdeg ¼ fFeg ð21Þ

From Eq. (21), the dynamic matrix of elements 1,3, and 4 are given by

½kiD� ¼

EiAi

R Li

0

@N1

@x

@N1

@x�o2 ri

Ei

� �ðN1N1Þ

� �dx EiAi

R Li

0

@N1

@x

@N2

@x�o2 ri

Ei

� �ðN1N2Þ

� �dx

EiAi

R Li

0

@N2

@x

@N1

@x�o2 ri

Ei

� �ðN2N1Þ

� �dx EiAi

R Li

0

@N2

@x

@N2

@x�o2 ri

Ei

� �ðN2N2Þ

� �dx

26664

37775

ð22Þ

From Eq. (21), the equation of motion of an element can be expressedas follows:

½kiD�

ui1

ui2

( )¼

Fi1

Fi2

( )or

ki11 ki

12

ki21 ki

22

" #ui

1

ui2

( )¼

Fi1

Fi2

( )ð23Þ

ARTICLE IN PRESS

Attenuation[α=real(μ)]

Frequency (Hz)

Mag

nitu

de(d

B)

by Exponential Shape Function(h2=3.18mm)by Linear Shape Function(h2=3.18mm)

Mag

nitu

de(d

B)

attenuation at h2 = 3.18mm

attenuation at h2 = 8mm



Frequency (Hz)

Mag

nitu

de(d

B)

by Exponential Shape Function(h2=8mm)by Linear Shape Function(h2=8mm)

Mag

nitu

de(d

B)


-2

0

2

4

6

8

10

12

-202468

1012141618

-5

0

5

10

15

20

Frequency (Hz)

Mag

nitu

de(d

B)


Mag

nitu

de(d

B)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

Fig. 3. The propagation constant and determinant of [T] for passive periodic shear mode

at h2¼3.18 mm; (b) determinant of [T] at h2¼3.18 mm; (c) attenuation at h2¼8 mm; (d)

of [T] at h2¼15 mm.

Table 2Physical properties.

Material Density (kg m�3) Modulus (MPa)

Aluminum 2700 70 000a

Viscoelastic layer 1200 15+0.0ib

a Young’s modulus.b Complex shear modulus(G(1+Zi), Z¼0.0).


The transfer matrix [Ti] between (i)th element and (i+1)thelement has the following relationship,

uiþ11

Fiþ11

( )¼

ui2

�Fi2

( )¼ ½Ti�

ui1

Fi1

( )ð24Þ

Combining Eqs. (23) and (24) gives the transfer matrix as follows:

ð25Þ

Determinant of Transfer Matrix

Frequency (Hz)

by Exponential Shape Function(h2=3.18mm)by Linear Shape Function(h2=3.18mm)

determinant of [T] at h2 = 3.18mm

determinant of [T] at h2 = 8mm



Frequency (Hz)



0.9980

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

1.0020

0.9980

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

1.0020

0.000.250.500.751.001.251.501.752.002.252.50

Frequency (Hz)


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

mount using exponential and linear shape functions for element 2: (a) attenuation

determinant of [T] at h2¼8 mm; (e) attenuation at h2¼15 mm and (f) determinant

ARTICLE IN PRESS


2.4. The transfer matrix of the passive periodic mount

Now, the transfer matrix of unit cell can be computed as

Tcell ¼ ½Telement4� � ½Telement3� � ½Telement2� � ½Telement1� ð26Þ

and for the complete periodic mount

T¼ ðTcellÞNcell ð27Þ

where Ncell is the number of cells in the passive periodic mount.Thus, all the information about the propagation characteristics isgiven by the eigenvalues l of the transfer matrix T:

l¼ em ¼ eaþbi ð28Þ

F.E.M(h2=3.18mm)Analytic(h2=3.18mm)

F.E.M(h2=8mm)Analytic(h2=8mm)



Frequency (Hz)

Mag

nitu

de(d

B)

Mag

nitu

de(d

B)





Frequency (Hz)

Mag

nitu

de(d

B)

Mag

nitu

de(d

B)


-2

0

2

4

6

8

10

12

-202468

1012141618

-5

0

5

10

15

20

25

Frequency (Hz)

Mag

nitu

de(d

B)

Mag

nitu

de(d

B)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

Fig. 4. The propagation constant and determinant of ½T�for passive periodic shear mode

determinant of [T] at h2¼3.18 mm; (c) attenuation at h2¼8 mm; (d) determinant of [T] a

where m is the propagation constants, a and b are calledattenuation factor and phase angle and represent the real andimaginary portion of the propagation constant. Also, one anotherimportant characteristic of the transfer matrix T is

Determinant of ½T� ¼ 1 ð29Þ

This can be proved using Eq. (15) and the symmetry of thedynamic stiffness matrix:

ð30Þ

F.E.M(h2=3.18mm)Analytic(h2=3.18mm)




Frequency (Hz)determinant of [T] at h2 = 3.18mm




Frequency (Hz)


0.9980

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

1.0020

0.9980

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

1.0020

3.00

0.000.250.500.751.001.251.501.752.002.25

Frequency (Hz)0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

mount using F.E.M and analytical method [28]: (a) attenuation at h2¼3.18 mm; (b)

t h2¼8 mm; (e) attenuation at h2¼15 mm and (f) determinant of [T] at h2¼15 mm.

ARTICLE IN PRESS


Eq. (30) can be used effectively for checking the accuracy oftransfer matrix T.

3. Performance of passive periodic mount

3.1. Shape function of element 2 and elements 1, 3, 4

The one-dimensional rod element has two nodes and onedegree of freedom at each node, the axial displacement can berepresented by exponential function which is derived from theequation of longitudinal vibration in a rod and also is suitable inthe higher frequency range:

uiðxÞ ¼ Ae�jkixþBejkix ¼ ½e�jkix ejkix�A

B

� ð31Þ

where k2i ¼ ðri=EiÞo2, o is the exciting frequency(rad/s). Applying

boundary conditions uið0Þ ¼ u1 at x¼0 and uiðLiÞ ¼ u2 at x¼Li andsolving A, B yield

A

B

� �¼

1

ðejkiLi�e�jkiLi Þ

ejkiLi �1

�e�jkiLi 1

" #u1

u2

( )

¼ aiejkiLi �1

�e�jkiLi 1

" #u1

u2

( )ð32Þ

Substituting Eq. (32) into Eq. (31) leads to

uiðxÞ ¼ ai½ejkiðLi�xÞ�e�jkiðLi�xÞ ejkix�e�jkix�

u1

u2

( )

¼ ½NpðxÞ NqðxÞ�u1

u2

( )¼ ½N�fdeg ð33Þ

From Eq. (33), Nj,Nr ,Ns,Nk of element 2 and N1, N2 of element1,3,4 are expressed as

NpðxÞ ¼ ap½ejkpðLp�xÞ�e�jkpðLp�xÞ�, p¼ j,s,1

NqðxÞ ¼ aq½ejkqx�e�jkqx�, q¼ r,k,2 ð34Þ

In addition to the exponential shape functions of Eq. (34),linear shape functions for element 2 are also used to obtain the

L4

L3

h3 h1h2

L1

L2

periodic shear mode mountwith viscoelastic material

Wc = L2

equivalent compremount with viscoel

h4

Fig. 5. Drawing of the cell of the passive periodic she

mechanical filtering characteristics of passive periodic shearmode mount.

uiðxÞ ¼ ½1 x�A

B

� �¼ ½1 x�

1 0

�1=Li 1=Li

" #u1

u2

( )

¼Li�x

Li

x

Li

� u1

u2

( )¼ ½Nm Nn� de

� �¼ ½N� de

� �ð35Þ

From Eq. (35),

NmðxÞ ¼ 1�ðx=LmÞ, m¼ j,s

NnðxÞ ¼ ðx=LnÞ, n¼ r,k ð36Þ

3.2. Materials

The passive periodic mount is made of two materials, one isaluminum, and the other is rubber as shown in Fig. 1. Thegeometric and physical properties of them are given in Tables 1and 2.

3.3. The propagation of waves in passive periodic mount

3.3.1. The comparison between exponential and linear shape

function

Fig. 3 shows comparisons between the filtering characteristicsof the passive periodic shear mode mount with four cells whenthe shape function of element 2 is exponential and linear. FromFig. 3, it is evident that there is no difference between exponentialand linear shape function of element 2. In this paper, exponentialshape function is used for calculation of the propagationcharacteristics.

3.3.2. Comparison between F.E.M and analytic approach

Fig. 4 displays comparisons between the filtering characteristicof the passive periodic shear mount with four cells as predicted bythe F.E.M and analytical method suggested by the authors [24].Accordingly, the F.E.M will be used to calculate the propagationcharacteristics of the passive periodic shear mount.

L4

L3

h3 h1h2

L1

L2

Lc

hc = 2h2

ssion modeastic material

unifrom mount withall aluminum

h4

ar, equivalent compression and uniform mounts

ARTICLE IN PRESS


3.3.3. Comparison between shear and compression mount

Fig. 5 shows unit cells of the passive periodic shear mount,equivalent compression and uniform mount, respectively. Thedimensions of the equivalent passive periodic compression mountare determined to maintain the same dimension of theviscoelastic material as the shear mount and have the samecross section of the aluminum parts in both mounts, namely by,

hc ¼ 2h2, Wc ¼ L2, Lc ¼ h1ðL1þL2Þþ2h3ðL2þL3Þþh4L4 ð37Þ

Fig. 6 shows the attenuation factor of the propagationconstant, respectively, for the shear, compression, and uniform

-2

0

2

4

6

8

10

12

14

16

Frequency (Hz)

Mag

nitu

de(d

B)

Shear mount(h2=3.18mm)Comp. mount(hc=2*h2)Uniform mount(Aluminum)

Mag

nitu

de(d

B)




-202468

1012141618

Frequency (Hz)

Mag

nitu

de(d

B)

Shear mount(h2=8mm)Comp. mount(hc=2*h2)Uniform mount(Aluminum)

Mag

nitu

de(d

B)




-5

0

5

10

15

20

0Frequency (Hz)

Mag

nitu

de(d

B)


Mag

nitu

de(d

B)

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

Fig. 6. The propagation constant and determinant of ½T�for passive periodic shear

h2¼3.18 mm; (b) determinant of [T] at h2¼¼3.18 mm; (c) attenuation at h2¼¼8

(f) determinant of [T] at h2¼15 mm.

mount configurations. It can be seen that the compression mountis more effective than the shear mount when the thickness of theviscoelastic layer is small (Fig. 6a). However, increasing thethickness of the viscoelastic layer makes the passive periodicshear mount exhibit broader stop band characteristics than thecompression mount (Fig. 6c). It should also be noted that stopbands are not observed over the entire frequency range for theuniform aluminum mount. This result emphasizes that thetransmission of the vibration along the passive periodic mountis blocked over certain frequency bands by virtue of theperiodicity effect.

determinant of [T ] at h2 = 3.18mm


0.9980

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

1.0020

Frequency (Hz)


determinant of [T ] at h2 = 8mm

determinant of [T ] at h2 = 15mm


0.9980

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

1.0020

Frequency (Hz)



0.000.250.500.751.001.251.501.752.002.252.50

Frequency (Hz)


0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

mode, equivalent compression mode and uniform mounts: (a) attenuation at

mm; (d) determinant of [T] at h2¼8 mm; (e) attenuation at h2¼15 mm and

ARTICLE IN PRESS


Fig. 7 displays a numerical comparison between thetransmissibility of the passive periodic shear, equivalentcompression and uniform mode mounts. It can be seen that asignificant attenuation of vibration transmission occurs over thezones of the stop bands. More importantly, it can be seen that theshear mode mount, with thicker viscoelastic layers, is moreeffective than the compression mode mount. The reverse is truefor thinner viscoelastic layers.

Vibration Isolation(Ft/Fo)

-300

-250

-200

-150

-100

-50

0

50

100

Frequency (Hz)

Mag

nitu

de(d

B)


vibration isolation at h2 =3.18mm

vibration iso

Vibration

-300

-250

-200

-150

-100

-50

0

50

100

150

0Fre

Mag

nitu

de(d

B)

500 1000 1500 2000 2500

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

Fig. 7. Theoretical amplitude of the vibration isolation of periodic and uniform mount

(c) vibration isolation at h2¼15 mm.

uniform mount

Fig. 8. Experimental models of uniform and passive periodic mounts:

4. Experimental performance of periodic mount

In order to validate the predictions of theoretical model, aseries of experiments are performed. Two experimental proto-types of mounts are designed and manufactured. One prototype isused to measure the amplitude of the transfer function thepassive periodic mount, the other is used to determine thetransfer function of a conventional non-periodic mount. Fig. 8

Vibration Isolation(Ft/Fo)

-300

-250

-200

-150

-100

-50

0

50

100

Frequency (Hz)

Mag

nitu

de(d

B)


vibration isolation at h2 =8mm

lation at h2 =15mm

Isolation(Ft/Fo)

quency (Hz)


3000 3500 4000 4500 5000 5500 6000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

s: (a) vibration isolation at h2¼3.18 mm; (b) vibration isolation at h2¼8 mm and

passive periodic mount(shear mode)

(a) uniform mount and (b) passive periodic mount (shear mode).

ARTICLE IN PRESS

Fig. 11. Amplitude of the experimental transfer function.


shows schematic drawings of the two prototypes and Fig. 9 showsa photograph of the experimental periodic mount. It can be seenthat the prototype with four passive periodic mounts is used tomeasure the vibration transmission from the upper plate which isexcited by a shaker. Each periodic mount is made of four cells.Piezoelectric accelerometers (PCB Model 303A3) are placed at theends of the passive periodic mount. An accelerometer is used tomeasure the acceleration produced by the shaker at the top of themount while the other accelerometer is used to captureacceleration transmitted to the bottom of the mount. Aspectrum analyzer (ONO SOKKI Model CF910) is used to recordthe output signals of the accelerometers.

The predictions of the developed model are also validatedagainst the predictions of the commercially available finiteelement package ANSYS. Fig. 10 displays ANSYS finite elementmodel of the passive periodic mount. Also, the predictions of theANSYS model are validated against the experimental results.

Fig. 11 shows a comparison between the amplitude of theexperimental transfer functions relating the input excitation ofthe top end of the mount to transmitted acceleration to its otherend. Fig. 12 shows the corresponding numerical transfer functionas obtained by using ANSYS. It can be clearly seen that the stop

Fig. 9. The experimental passive periodic shear mode mount.

Fig. 10. ANSYS finite element model of the passive periodic shear mode mount.

Fig. 12. Amplitude of the ANSYS transfer function.

bands cover the whole frequency range from the low frequenciesto high frequencies. The attenuation of the vibration transmissionis obvious and effective over the entire frequency range.

It is also evident that the experimental results are inagreement with the prediction of theoretical model in Section 3and those obtained from numerical analysis using ANSYS.

5. Conclusions

In this study, a passive periodic engine mount with periodicviscoelastic inserts is presented. A theoretical model is developed todescribe the dynamics of wave propagation in the passive periodicmount. The model is derived using the theory of finite elements. Acell of the passive periodic mount is divided into four elements, thetransfer matrix formulation for each element is given. The overalltransfer matrix of unit cell is obtained by multiplying the transfermatrices of the four elements composing the cell. The mechanicalfiltering characteristics of wave propagation in four series cells thusare analyzed by the transfer matrix formulation.

ARTICLE IN PRESS


Numerical examples are given to illustrate the effectiveness ofthis class of periodic mounts. The experiments are performed tovalidate the predictions of the theoretical model. Both thetheoretical and experimental results show that the passiveperiodic mount exhibit stop bands covering a broad frequencyrange.

The presented engine mounts can find many appli-cations in gearbox support struts, engine mounts of auto-mobiles and aircraft as well as underwater vehicles. Thedevelopment of active prototypes of the shear mode periodicmount presented here is a natural extension of the presentwork.

Appendix A. Element of matrix in Eq. (10)

A.1. Exponential shape function

Exponential shape function can be Eq. (A.1):

Nj ¼ a1½ejk1ðL2�xÞ�e�jk1ðL2�xÞ�, Nj,x ¼ a1ð�jk1Þ½e

jk1ðL2�xÞ þe�jk1ðL2�xÞ�,

Nr ¼ a1ðejk1x�e�jk1xÞ, Nr,x ¼ a1ðjk1ejk1xþ jk1e�jk1xÞ ¼ a1ðjk1Þðe

jk1xþe�jk1xÞ

Ns ¼ a3½ejk3ðL2�xÞ�e�jk3ðL2�xÞ�, Ns,x ¼ a3ð�jk3Þ½e

jk3ðL2�xÞ þe�jk3ðL2�xÞ�,

Nk ¼ a3ðejk3x�e�jk3xÞ, Nk,x ¼ a3ðjk3ejk3xþ jk3e�jk3xÞ ¼ a3ðjk3Þðe

jk3xþe�jk3xÞ

a1 ¼1

ðejk1L2�e�jk1L2 Þ, a3 ¼

1

ðejk3L2�e�jk3L2 Þ, k2

i ¼ ðri=EiÞo2 ðA:1Þ

Inserting exponential shape function in (A.1) into Eq. (10) gives

Kjj ¼

Z L2

0½E1A1ðNj,xNj,xÞþgðNjNjÞ�o2r1A1ðNjNjÞ�dx

¼E1A1

L2

� �ð1�e�j4k1L2 Þ

ð1�e�j2k1L2 Þ2

" #ðjk1L2Þþga2

1 �2L2�j1

2k1

� �ðej2k1L2�e�j2k1L2 Þ

�

Kjr ¼

Z L2

0½E1A1ðNj,xNr,xÞþgðNjNrÞ�o2r1A1ðNjNrÞ�dx¼�

E1A1

L2

� �2e�j3k1L2 ð�1þej2k1L2 Þ


" #ðjk1L2Þþga2

1 ðejk1L2þe�jk1L2 ÞL2þ j

1

k1

� �ðejk1L2�e�jk1L2 Þ

�

Kjs ¼�g

Z L2

0½ðNjNsÞ�dx¼

ga1a3ðjÞ1

k1þk3

� �ðejðk1þk3ÞL2�e�jðk1þk3ÞL2 Þ�

1

k1�k3

� �ðejðk1�k3ÞL2�e�jðk1�k3ÞL2 Þ

� , k1ak3

gðaaÞ2 2L2þ j1

2ka

� �ðej2kaL2�e�j2kaL2 Þ

� , k1 ¼ k3 ¼ ka

8>>><>>>:

Kjk ¼�g

Z L2

0½ðNjNkÞ�dx¼

ga1a3ðjÞ2k1

ðk21�k2

3Þ

!½ðejk1L2�e�jk1L2 Þ�ðejk3L2�e�jk3L2 Þ�, k1ak3

gðaaÞ2 �ðejkaL2þe�jkaL2 ÞL2�j1

ka

� �ðejkaL2�e�jkaL2 Þ

� , k1 ¼ k3 ¼ ka

8>>>><>>>>:

Krj ¼

Z L2

0½E1A1ðNr,xNj,xÞþgðNrNjÞ�o2r1A1ðNrNjÞ�dx¼�

E1A1

L2



" #ðjk1L2Þþga2


1

k1


�

Krr ¼

Z L2

0½E1A1ðNr,xNr,xÞþgðNrNrÞ�o2r1A1ðNrNrÞ�dx¼

E1A1

L2

� �ð1�e�j4k1L2 Þ


" #ðjk1L2Þþga2

1 �2L2�j1

2k1


�

Krs ¼�

Z L2

0½gðNrNsÞ�dx¼

ga1a3ðjÞ2k3

ðk21�k2

3Þ

!" #½ðejk1L2�e�jk1L2 Þ�ðejk3L2�e�jk3L2 Þ�


ka


� 8>>>><>>>>:

Krk ¼�

Z L2

0½gðNrNkÞ�dx¼

ga1a3ðjÞ1

k1þk3

� �½ejðk1þk3ÞL2�e�jðk1þk3ÞL2 ��

1

k1�k3

� �½ejðk1�k3ÞL2�e�jðk1�k3ÞL2 �

� , k1ak3

gðaaÞ2 2L2þ j1

2ka


� , k1 ¼ k3 ¼ ka

8>>><>>>:

ARTICLE IN PRESS


Ksj ¼�

Z L2

0½gðNsNjÞ�dx¼

ga1a3ðjÞ1

k1þk3

� �ðejðk1þk3ÞL2�e�jðk1þk3ÞL2 Þ�

1

k1�k3

� �ðejðk1�k3ÞL2�e�jðk1�k3ÞL2 Þ

� , k1ak3

gðaaÞ2 2L2þ j1

2ka


� , k1 ¼ k3 ¼ ka

8>>><>>>:

Ksr ¼�

Z L2

0½gðNsNrÞ�dx¼

ga1a3ðjÞ2k3

ðk21�k2

3Þ

!" #½ðejk1L2�e�jk1L2 Þ�ðejk3L2�e�jk3L2 Þ�, k1ak3


ka


� , k1 ¼ k3 ¼ ka

8>>>><>>>>:

Kss ¼

Z L2

0½E3A3ðNs,xNs,xÞþgðNsNsÞ�o2r3A3ðNsNsÞ�dx¼

E3A3

L2

� �ð1�e�j4k3L2 Þ


" #ðjk3L2Þþga2

3 �2L2�j1

2k3


�

Ksk ¼

Z L2

0½E3A3ðNs,xNk,xÞþgðNsNkÞ�o2r3A3ðNsNkÞ�dx¼�

E3A3

L2



" #ðjk3L2Þþga2


1

k3


�

Kkj ¼�

Z L2

0½gðNkNjÞ�dx¼

ga1a3ðjÞ2k1

ðk21�k2

3Þ

!½ðejk1L2�e�jk1L2 Þ�ðejk3L2�e�jk3L2 Þ�, k1ak3


ka


� , k1 ¼ k3 ¼ ka

8>>>><>>>>:

Kkr ¼�

Z L2

0½gðNkNrÞ�dx¼

ga1a3ðjÞ1

k1þk3

� �½ejðk1þk3ÞL2�e�jðk1þk3ÞL2 ��

1

k1�k3

� �½ejðk1�k3ÞL2�e�jðk1�k3ÞL2 �

� , k1ak3

gðaaÞ2 2L2þ j1

2ka


� , k1 ¼ k3 ¼ ka

8>>><>>>:

Kks ¼

Z L2

0½E3A3ðNk,xNs,xÞþgðNkNsÞ�o2r3A3ðNkNsÞ�dx¼�

E3A3

L2



" #ðjk3L2Þþga2


1

k3


�

Kkk ¼

Z L2

0½E3A3ðNk,xNk,xÞþgðNkNkÞ�o2r3A3ðNkNkÞ�dx¼

E3A3

L2

� �ð1�e�j4k3L2 Þ


" #ðjk3L2Þþga2

3 �2L2�j1

2k3


�

where

a1 ¼ 1=ðejk1L2�e�jk1L2 Þ, a3 ¼ 1=ðejk3L2�e�jk3L2 Þ, aa ¼ 1=ðejkaL2�e�jkaL2 Þ

k1 ¼ ðr1=E1Þo2, k3 ¼ ðr3=E3Þo2, g ¼ ðG2A2=h22Þ

A.2. Linear shape function

Linear shape function can be Eq. (A.2):

Nj ¼L2�x

L2, Nj,x ¼�

1

L2, Nr ¼

x

L2, Nr,x ¼

1

L2

Ns ¼L2�x

L2, Ns,x ¼�

1

L2, Nk ¼

x

L2, Nk,x ¼

1

L2ðA:2Þ

Inserting exponential shape function in (A.2) into Eq. (10) yields

Kjj ¼ ð1=L2ÞðE1A1þ2pG2A2Þ�ðo2=3ÞðM1Þ, Kjr ¼ ð1=L2Þð�E1A1þpG2A2Þ�ðo2=6ÞðM1Þ

Kjs ¼�ð1=L2Þð2pG2A2Þ, Kjk ¼�ð1=L2ÞðpG2A2Þ

Krj ¼ ð1=L2Þð�E1A1þpG2A2Þ�ðo2=6ÞðM1Þ, Krr ¼ ð1=L2ÞðE1A1þ2pG2A2Þ�ðo2=3ÞðM1Þ

Krs ¼�ð1=L2ÞðpG2A2Þ, Krk ¼�ð1=L2Þð2pG2A2Þ

Ksj ¼�ð1=L2Þð2pG2A2Þ, Ksr ¼�ð1=L2ÞðpG2A2Þ

Kss ¼ ð1=L2ÞðE3A3þ2pG2A2Þ�ðo2=3ÞðM3Þ, Ksk ¼ ð1=L2Þð�E3A3þpG2A2Þ�ðo2=6ÞðM3Þ

Kkj ¼�ð1=L2ÞðpG2A2Þ, Kkr ¼�ð1=L2Þð2pG2A2Þ

Kks ¼ ð1=L2Þð�E3A3þpG2A2Þ�ðo2=6ÞðM3Þ, Kkk ¼ ð1=L2ÞðE3A3þ2pG2A2Þ�ðo2=3ÞðM3Þ

ARTICLE IN PRESS


where,

p¼1

6

L2

h2

� �2

, M1 ¼ r1A1L2 ¼ r1ðbh1ÞL2, M3 ¼ r3A3L2 ¼ r3ð2bh3ÞL2

Appendix B. Element of matrix in Eq. (23)

ki11 ¼ EiAi

Z Li

0

@N1

@x

@N1

@x�o2 ri

Ei

� �ðN1N1Þ

� �dx¼

EiAi

Li

� �ðjkLiÞ

ð1�e�j2kLi Þ2

" #ð1�e�j4kLi Þ

ki12 ¼ EiAi

Z Li

0

@N1

@x

@N2

@x�o2 ri

Ei

� �ðN1N2Þ

� �dx¼

EiAi

Li

� �ðjkLiÞ

ð1�e�j2kLi Þ2

" #½�2e�j3kLi ð�1þej2kLi Þ�

ki21 ¼ EiAi

Z Li

0

@N2

@x

@N1

@x�o2 ri

Ei

� �ðN2N1Þ

� �dx¼

EiAi

Li

� �ðjkLiÞ

ð1�e�j2kLi Þ2

" #½�2e�j3kLi ð�1þej2kLi Þ�

ki22 ¼ EiAi

Z Li

0

@N2

@x

@N2

@x�o2 ri

Ei

� �ðN2N2Þ

� �dx¼

EiAi

Li

� �ðjkLiÞ

ð1�e�j2kLi Þ2

" #ð1�e�j4kLi Þ

or

References

[1] A. Baz, Active control of periodic structures, ASME Journal of vibration andAcoustics 123 (2001) 472–479.

[2] D.J. Mead, Free wave propagation in periodically supported, infinite beams,Journal of Sound and Vibration 11 (1970) 181–197.

[3] D.J. Mead, Vibration response and wave propagation in periodic structures,ASME Journal of Engineering for Industry 21 (1971) 783–792.

[4] D.J. Mead, A general theory of harmonic wave propagation in linear periodicsystems with multiple coupling, Journal of Sound and Vibration 27 (1973)235–260.

[5] D.J. Mead, Wave propagation and natueal modes in periodic systems: mono-coupled systems, Journal of Sound and Vibration 40 (1975) 1–18.

[6] D.J. Mead, A new method of analyzing wave propagation in periodicstructures; applications to periodic timoshenko beams and stiffened plates,Journal of Sound and Vibration 104 (1986) 9–27.

[7] D.J. Mead, Wave propagation in continuous periodic structures: researchcontributions from southampton, 1964–1995, Journal of Sound and Vibration190 (1996) 496–524.

[8] S. Mester, J. Benaroya, Periodic and near periodic structures: review, Shockand Vibration 2 (1995) 69–95.

[9] M. Ruzzene, A. Baz, Active control of wave propagation in periodic fluid-loaded shells, Smart Material and Structure 10 (2001) 893–906.

[10] G. Solaroli, Z. Gu, A. Baz, M. Ruzzene, Wave propagation in periodic stiffenedshells: spectral finite element modeling and experiments, Journal ofVibration and Control 9 (2003) 1057–1081.

[11] O. Thorp, M. Ruzzene, A. Baz, Attenuation of wave propagation in fluid-loadedshells with periodic shunted piezoelectric rings, Smart Material and Structure14 (2005) 594–604.

[12] M. Ruzzene, A. Baz, Control of wave propagation in periodic composite rodsusing shape memory inserts, Journal of vibration and acoustics, Transactionsof ASME 122 (2000) 151–159.

[13] O. Thorp, M. Ruzzene, A. Baz, Attenuation and localization of wavepropagation in rods with periodic shunted piezoelectric patches, SmartMaterial and Structure 10 (2001) 979–989.

[14] M. Toso, A. Baz, Wave propagation in periodic shells with tapered wallthickness and changing material properties, Shock and Vibration 11 (2004)411–432.

[15] A. Singh, D.J. Pines, A. Baz, Active/passive reduction of vibration of periodicone-dimensional structures using piezoelectric actuators, Smart Material andStructure 13 (2004) 698–711.

[16] D. Richards, D.J. Pines, Passive reduction of gear mesh vibration using aperiodic drive shaft, Journal of Sound and Vibration 264 (2003) 317–342.

[17] M. Tawfik, J. Chung, A. Baz, Wave attenuation in periodic helicopter blades,in: Jordan International Mechanical Engineering Conference (JIMEC), Amman,Jordan, 26–28 April 2004.

[18] E.I. Rivin, Passive engine mount-some directions for further development.SAE Paper 850481, 1985.

[19] T. Ushijima, K. Takano, H. Kojoma, High performance hydraulic mount forimproving vehicle noise and vibration. SAE Paper 880073, 1988.

[20] R. Singh, G. Kim, R.V. Ravinda, Linear analysis of automotive hydro-mechanical mount with emphasis on decoupler characteristics, Journal ofSound and Vibration 158 (1992) 219–243.

[21] H. Matsuoka, T. Mikasa, H. Nemoto, NV countermeasure technology for acylinder-on-demand engine-development of active control engine mount,SAE Paper 2004-01-0413, 2004.

[22] L.R. Miller, M. Ahmadian, C.M. Nobles, D.A. Swanson, Modeling andperformance of an experimental active vibration isolator, Journal of Vibrationand Acoustics 117 (1995) 272–278.

[23] S. Asiri, Vibration attenuation of automotive vehicle engine using periodicmounts, International Journal of Vehicle Noise and Vibration 3 (2007) 302–315.

[24] L. Zheng, W.J. Jung, Z. Gu, A. Baz, Passive periodic engine mount, in:Proceedings of the ASME 2008 International Design Engineering TechnicalConferences & Computers and Information in Engineering Conference(IDETC/CIE), New York, NY, USA, August 3–6 2008.

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Finite Elements in Analysis and Design - Dr. Amr M. …ele f d e2g¼fF ele2gð9Þ where dynamic stiffness ½KD ele2¼½K ele2 o 2M ele2 and fd e2g¼fd jr d skg T. The dynamic stiffness