MechFilteringCharacteristicsPassiveperiodicEngineMount.pdf

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& 2010 Elsevier B.V. All rights reserved.

1. Introduction

f anannerineeriselageses andropagattens on thster anfound

operate over broader frequency ranges but at the expense of the

2.1. Overview

Fig. 1a shows a schematic drawing of the shear mode passive

ARTICLE IN PRESS

Contents lists available at ScienceDirect

.e

Finite Elements in An

Finite Elements in Analysis and Design 46 (2010) 685697shear of the viscoelastic layer while elements 1, 3, and 4E-mail address: [email protected] (A. Baz).classical complexity and reliability. periodic 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 the

0168-874X/$ - see front matter & 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.nel.2010.03.007

Corresponding author. Tel.: +1 3014055216; fax: +1 3014058331.generally effective at narrow frequency ranges. It is equally aseffective as other types of active engine mounts [21,22] which canchassis. 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 are

2. Mathematical modeling of passive periodic mountsthe characteristics of periodic structures and their applications inengineering have been extensively investigated including passiveand active periodic structures [917].

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 the

motion are derived from the nite element approach and then thetransfer matrix is obtained. The basic ltering 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.A periodic structure consists oelements connected in a repeated mstructures can be found in many engsatellite solar panels, wings and fupipe-lines, railway tracks, submarin

In these structures, waves can pbands called pass bands and arestop bands [27]. Excellent reviewbeen given by Mead [7] and by Meextended lists of references can beassembly of identical[1]. Examples of theseng applications such asof aircraft, petroleummany others.ate in some frequencyuated in others callede state-of-the-art haved Benaroya [8], where. Since then, studies of

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 ltering 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 ve sections. In Section 1, abrief introduction is given. In Section 2, a mathematical model of theshear mode periodic mount is presented and the equations ofMechanical ltering characteristics of p

Woojin Jung a, Zheng Gu b, A. Baz b,

a Agency of Defence Development, P.O. Box 18, Jinhae, Gyeongnam 645-600, South Korb Department of Mechanical Engineering, University of Maryland, College Park, MD 20

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

a b s t r a c t

The transmission of autom

which rely in their operati

proposed mount acts as a

frequency bands called th

viscoelastic inserts operat

operation of this class of

with the transfer matrix

against the predictions of

obtained from prototypes

of the shear mode periodi

broad frequency band. Ex

mount through active con

journal homepage: wwwUSA

e engine vibrations to the chassis is isolated using a new class of mounts

n optimally designed and periodically distributed viscoelastic inserts. The

chanical lter for impeding the propagation of vibration within specic

top bands. The spectral width of these bands is enhanced by making the

a shear mode rather than compression mode. The theory governing the

odic mounts is presented using the theory of nite elements combined

roach. The predictions of the performance of the mount are validated

commercial nite element code ANSYS and against experimental results

lain and periodic mounts. The obtained results demonstrate the feasibility

ount as another means for blocking the transmission of vibration over a

ing the effective width of the operating frequency bands of this class of

means is the ultimate goal of this study.sive periodic engine mount

lsevier.com/locate/finel

alysis and Design

ARTICLE IN PRESS

experience longitudinal loading. The transfer matrix of the unitcell is derived by applying the nite 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;

(3) 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 deected conguration is shown in Fig. 2 where A1,E1,h1denote the cross section area, Youngs modulus and thickness of

Nomenclature

A Areab widthE Youngs 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 deectionV Potential energyW External workai Coefcient of exponential shape function

a Propagation attenuation factorb Propagation phase angleg Shear strainl Eigenvalues of the transfer matrixdij Longitudinal deection 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 xt Partial differential with respect to t

Superscripts

1 Matrix inverse

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697686Unit Cell Unit Cell(2) the longitudinal stresses in the viscoelastic layer are negligible;Base Metal ViscoelasPeriodic

h3

L1 L2L4L3

h1h2

Element 1Element 2

Element 3 Element 4

h

Element 1

i

[T1]

Undeflected

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

j

aramcore. A2,G2,h2,g are the cross section area, shears modulus,

Unit Cell Unit Cellaterialnt

Element 2 Element 3 Element 4

k m

n

[T2] [T3] [T4]k m

Deflected

eters: (a) periodic mount; (b) undeected and (c) deected.

ARTICLE IN PRESS

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697 687thickness and shear strain of viscoelastic material. Also, A3,E3,h3dene the cross section area, Youngs modulus and thickness ofouter layers.

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

g ujrusk=h2 1where ujr and usk are the longitudinal deections of the core andouter layer, respectively. Also, h2 denes 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 V1V2 Z Z Z

V

1

2sjrejrsskeskt2gdV

12E1A1

Z L20

@ujr@x

2dx 1

2E3A3

Z L20

@usk@x

2dx

12G2A2

Z L20

g2 dx 2

where A1 h1b, A2 2h2b, A3 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 deection ujr and usk is given by

T 12r1A1

Z L20

@ujr@t

2dx 1

2r3A3

Z L20

@usk@t

2dx 3

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

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. Deected conguration of Element 2.2.2.4. Finite element model of element 2

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

ujrx Nj Nr uj

ur

( ) Nj Nrfdjrg,

uskx Ns Nkus

uk

( ) Ns Nkfdskg 4

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,

g ujrusk=h2 1=h2NjujNrurNsusNkuk5

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

6

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

7where

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

8

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

Mele2f de2gKele2fde2g fFele2g or KDele2fde2g fFele2g 9where dynamic stiffness KDele2 Kele2o2Mele2 and fde2g fdjrdskgT . The dynamic stiffness matrix of element 2 can be given by

10

ARTICLE IN PRESS

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 1

2EiAi

Z Li0

@ui@x

2dx, 16

Ti 1

2riAi

Z Li0

@ui@t

2dx, i 1,3,4 17

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697688where

Kab Z L20

E1A1Na,xNb,xgNaNbo2r1A1NaNbdx;a j,r; b j,r

Kjs Z L20

gNjNsdx, Kjk Z L20

gNjNkdx,

Krs Z L20

gNrNsdx

Krk Z L20

gNrNkdx, Ksj Z L20

gNsNjdx,

Ksr Z L20

gNsNrdx

Kkj Z L20

gNkNjdx, Kkr Z L20

gNkNrdx

Kcd Z L20

E3A3Nc,xNd,xgNcNdo2r3A3NcNddx;c s,k; d s,k

g G2A2=h22

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:

11

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

Kele211 Kele212

Kele221 Kele222

" #uj

uk

( )

Fj

Fk

( )12

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

uj1Fj1

( )

uk

Fk

( ) T2

uj

Fj

( )13

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

14or

15

where

Kele211 KjjKjsCjKjrDj, Kele212 KjkKjsCkKjrDk

Kele221 KkjKksCjKkrDj, Kele222 KkkKksCkKkrDk

Cj KssKsrKrr1Krs1KsjKsrKrr1Krj

Ck KssKsrKrr1Krs1KskKsrKrr1Krk

Dj Krr1KrjKrr1KrsCj

1 1Dk Krr KrkKrr KrsCkLet u1, u2 be the axial displacement of two nodes of one element.Also Let N1x,N2x be the shape function of u1, u2. The deectionvariation of one-dimensional rod can be derived as

uix N1x N2xu1

u2

( ) Nde 18

where fdeg fu1 u2gT . One can evaluate potential energies andkinetic energies as follows:

Vi 1

2EiAi

Z Li0

@ui@x

T @ui@x

dx

12

degT EiAiZ Li0

@N1@x

@N1@x

@N1@x

@N2@x

@N2@x

@N1@x

@N2@x

@N2@x

2664

3775dx

0BB@

1CCA de

8>>>:

12

degT kia de n 19

Ti 1

2riAi

Z Li0

@ui@t

T @ui@t

dx

12

_degT riAiZ Li0

N1N1 N1N2

N2N1 N2N2

" #dx _den o

12

_degT mia _den on(

20

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

miaf degkiafdeg fFeg or kiao2miafdeg fFeg 21

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

kiD EiAi

R Li0

@N1@x

@N1@x

o2 riEi

N1N1

dx EiAi

R Li0

@N1@x

@N2@x

o2 riEi

N1N2

dx

EiAiR Li0

@N2@x

@N1@x

o2 riEi

N2N1

dx EiAi

R Li0

@N2@x

@N2@x

o2 riEi

N2N2

dx

26664

37775

22From Eq. (21), the equation of motion of an element can be expressedas follows:

kiDui1

ui2

( )

Fi1

Fi2

( )or

ki11 ki12

ki21 ki22

" #ui1

ui2

( )

Fi1

Fi2

( )23

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

ARTICLE IN PRESS

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

ui11Fi11

( )

ui2

Fi2

( ) Ti

ui1

Fi1

( )24

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

25

Attenuation[=real()]

Frequency (Hz)

Mag

nitu

de(d

B)

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

Determinant of Transfer Matrix

Frequency (Hz)

Mag

nitu

de(d

B)

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

attenuation at h2 = 3.18mm determinant of [T] at h2 = 3.18mm

attenuation at h2 = 8mm determinant of [T] at h2 = 8mm

attenuation at h2 = 15mm


Frequency (Hz)

Mag

nitu

de(d

B)

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


Frequency (Hz)

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)



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

1.752.002.252.50

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

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 modeat h23.18mm; (b) determinant of [T] at h23.18mm; (c) attenuation at h28mm; (d)of [T] at h215mm.

Table 2Physical properties.

Material Density (kgm3) Modulus (MPa)

Aluminum 2700 70000a

Viscoelastic layer 1200 15+0.0ib

a Youngs modulus.b Complex shear modulus(G(1+Zi), Z0.0).

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697 689determinant of [T] at h2 = 15mm

0.000.250.500.751.001.251.50

Frequency (Hz)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 h28mm; (e) attenuation at h215mm 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 Telement4 Telement3 Telement2 Telement1 26

and for the complete periodic mount

T TcellNcell 27

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:

l em eabi 28

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 T 1 29

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

30

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

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

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

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

F.E.M(h2=15mm) F.E.M(h2=15mm)


Frequency (Hz)

Mag

nitu

de(d

B)


Frequency (Hz)

Mag

nitu

de(d

B)

attenuation at h2 = 3.18mm determinant of [T] at h2 = 3.18mm

attenuation at h2 = 8mm determinant of [T] at h2 = 8mm


Frequency (Hz)

Mag

nitu

de(d

B)


Frequency (Hz)

Mag

nitu

de(d

B)


-2

0

2

4

6

8

10

12

-202468

1012141618

25Determinant of Transfer Matrix

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

3.002.25

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

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

ode

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697690Analytic(h2=15mm)

attenuation at h2 = 15mm

-5

0

5

10

15

20

Frequency (Hz)

Mag

nitu

de(d

B)

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

Fig. 4. The propagation constant and determinant of Tfor passive periodic shear m

determinant of [T] at h23.18mm; (c) attenuation at h28mm; (d) determinant of [T] aAnalytic(h2=15mm)

determinant of [T] at h2 = 15mm

0.000.250.500.751.001.251.501.752.00

Frequency (Hz)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 h23.18mm; (b)

t h28mm; (e) attenuation at h215mm and (f) determinant of [T] at h215mm.

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:

uix AejkixBejkix ejkix ejkixA

B

31

where k2i ri=Eio2, o is the exciting frequency(rad/s). Applyingboundary conditions ui0 u1 at x0 and uiLi u2 at xLi andsolving A, B yield

A

B

1ejkiLiejkiLi

ejkiLi 1ejkiLi 1

" #u1

u2

( )

ejkiLi 1" #

u1( )

mechanical ltering characteristics of passive periodic shearmode mount.

uix 1 xA

B

1 x

1 0

1=Li 1=Li

" #u1

u2

( )

LixLi

x

Li

u1

u2

( ) Nm Nn de

N de 35

From Eq. (35),

Nmx 1x=Lm, m j,s

Nnx x=Ln, n r,k 36

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.

preoel

h4

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697 691periodic shear mode mountwith viscoelastic material

equivalent commount with visc ai ejkiLi 1 u2 32

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

uix aiejkiLixejkiLix ejkixejkixu1

u2

( )

Npx Nqxu1

u2

( ) Nfdeg 33

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

Npx apejkpLpxejkpLpx, p j,s,1

Nqx aqejkqxejkqx, q r,k,2 34In 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

Wc = L2Fig. 5. Drawing of the cell of the passive periodic she3.3. 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 ltering 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 ltering 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

h4ar, 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 2h2, Wc L2, Lc h1L1L22h3L2L3h4L4 37Fig. 6 shows the attenuation factor of the propagation

constant, respectively, for the shear, compression, and uniform

mount congurations. 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

-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)


0.9980

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

1.0020

Frequency (Hz)

Mag

nitu

de(d

B)


attenuation at h2 = 3.18mm

determinant of [T ] at h2 = 8mm attenuation at h2 = 8mm

-202468

1012141618

Frequency (Hz)

Mag

nitu

de(d

B)

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


0.9980

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

1.0020

Frequency (Hz)

Mag

nitu

de(d

B)





10

15

20

e(dB

)



1.501.752.002.252.50

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

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

ear

8

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697692attenuation at h2 = 15mm

-5

0

5

0Frequency (Hz)

Mag

nitu

d

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

Fig. 6. The propagation constant and determinant of Tfor passive periodic shh23.18mm; (b) determinant of [T] at h23.18mm; (c) attenuation at h2

(f) determinant of [T] at h215mm.determinant of [T ] at h2 = 15mm

0.000.250.500.751.001.25

Frequency (Hz)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 h28mm; (e) attenuation at h215mm 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 asignicant 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.

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)



-300

-250

-200

-150

-100

-50

0

50

100

Frequency (Hz)

Mag

nitu

de(d

B)


vibration isolation at h2 =3.18mm vibration isolation at h2 =8mm

vibration isolation at h2 =15mm


-300

-250

-200

-150

-100

-50

0

50

100

150

0Frequency (Hz)

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 60000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

Fig. 7. Theoretical amplitude of the vibration isolation of periodic and uniform mounts: (a) vibration isolation at h23.18mm; (b) vibration isolation at h28mm and(c) vibration isolation at h215mm.

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697 693uniform mountFig. 8. Experimental models of uniform and passive periodic mounts:passive periodic mount(shear mode) (a) uniform mount and (b) passive periodic mount (shear mode).

ARTICLE IN PRESS

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 niteelement package ANSYS. Fig. 10 displays ANSYS nite 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

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 nite 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 mechanicalltering characteristics of wave propagation in four series cells thus

Fig. 11. Amplitude of the experimental transfer function.

Fig. 12. Amplitude of the ANSYS transfer function.

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697694Fig. 9. The experimental passive periodic shear mode mount.Fig. 10. ANSYS nite element model of the passive periodic shear mode mount. are analyzed by the transfer matrix formulation.

ARTICLE IN PRESS

Krs Z L2

gNrNsdxga1a3j k21k23

e e e e

jka

>>>>:

Krk Z L20

gNrNkdxga1a3j

1

k1k3

ejk1k3L2ejk1 k3L2

gaa2 2L2 j1

2ka

ej2kaL2ej2kaL2

, k

8>>>>>:L2

1

k1k3

ejk1k3L2ejk1k3L2

, k1ak3

k3 kaNumerical 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 nd 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 a1ejk1L2xejk1L2x, Nj,x a1jk1ejk1L2x ejk1L2x,

Nr a1ejk1xejk1x, Nr,x a1jk1ejk1x jk1ejk1x a1jk1ejk1xejk1x

Ns a3ejk3L2xejk3L2x, Ns,x a3jk3ejk3L2x ejk3L2x,

Nk a3ejk3xejk3x, Nk,x a3jk3ejk3x jk3ejk3x a3jk3ejk3xejk3x

a1 1

ejk1L2ejk1L2 , a3 1

ejk3L2ejk3L2 , k2i ri=Eio2 A:1

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

Kjj Z L20

E1A1Nj,xNj,xgNjNjo2r1A1NjNjdx

E1A1L2

1ej4k1L2 1ej2k1L2 2

" #jk1L2ga21 2L2j

1

2k1

ej2k1L2ej2k1L2

Kjr Z L20

E1A1Nj,xNr,xgNjNro2r1A1NjNrdxE1A1L2

2ej3k1L2 1ej2k1L2

1ej2k1L2 2

" #jk1L2ga21 ejk1L2 ejk1L2 L2 j

1

k1

ejk1L2ejk1L2

Kjs gZ L20

NjNsdxga1a3j

1

k1k3

ejk1 k3L2ejk1 k3L2 1

k1k3

ejk1k3L2ejk1k3L2

, k1ak3

gaa2 2L2 j1

2ka

ej2kaL2ej2kaL2

, k1 k3 ka

8>>>>>:

Kjk gZ L20

NjNkdxga1a3j

2k1k21k23

!ejk1L2ejk1L2 ejk3L2ejk3L2 , k1ak3

gaa2 ejkaL2 ejkaL2 L2j1

ka

ejkaL2ejkaL2

, k1 k3 ka

8>>>>>>>:

Krj Z L20

E1A1Nr,xNj,xgNrNjo2r1A1NrNjdxE1A1L2

2ej3k1L2 1ej2k1L2

1ej2k1L2 2


1

k1

ejk1L2ejk1L2

Krr Z L20

E1A1Nr,xNr,xgNrNro2r1A1NrNrdxE1A1L2

1ej4k1L2 1ej2k1L2 2

" #jk1L2ga21 2L2j

1

2k1

ej2k1L2ej2k1L2

2k3 !" #

jk1L2 jk1L2 jk3L2 jk3L28>>>

ARTICLE IN PRESS

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697696Kkr 2

0gNkNrdx

k1k3 k1k3gaa2 2L2 j

1

2ka

ej2kaL2ej2kaL2

, k1 k3 ka>>>:

Kks Z L20

E3A3Nk,xNs,xgNkNso2r3A3NkNsdxE3A3L2

2ej3k3L2 1ej2k3L2

1ej2k3L2 2


1

k3

ejk3L2ejk3L2

Kkk Z L20

E3A3Nk,xNk,xgNkNko2r3A3NkNkdxE3A3L2

1ej4k3L2 1ej2k3L2 2

" #jk3L2ga23 2L2j

1

2k3

ej2k3L2ej2k3L2

where

a1 1=ejk1L2ejk1L2 , a3 1=ejk3L2ejk3L2 , aa 1=ejkaL2ejkaL2

k1 r1=E1o2, k3 r3=E3o2, g G2A2=h22

A.2. Linear shape function

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

Nj L2xL2

, Nj,x 1

L2, Nr

x

L2, Nr,x

1

L2

Ns L2xL2

, Ns,x 1

L2, Nk

x

L2, Nk,x

1

L2A:2

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

Kjj 1=L2E1A12pG2A2o2=3M1, Kjr 1=L2E1A1pG2A2o2=6M1

Kjs 1=L22pG2A2, Kjk 1=L2pG2A2

Krj 1=L2E1A1pG2A2o2=6M1, Krr 1=L2E1A12pG2A2o2=3M1

Krs 1=L2pG2A2, Krk 1=L22pG2A2

Ksj 1=L22pG2A2, Ksr 1=L2pG2A2

Kss 1=L2E3A32pG2A2o2=3M3, Ksk 1=L2E3A3pG2A2o2=6M3

Kkj 1=L2pG2A2, Kkr 1=L22pG2A2Ksj Z L20

gNsNjdxga1a3j

1

k1k3


k1k3

ejk1k3L2ejk1k3L2

, k1ak3

gaa2 2L2 j1

2ka

ej2kaL2ej2kaL2

, k1 k3 ka

8>>>>>:

Ksr Z L20

gNsNrdxga1a3j

2k3k21k23

!" #ejk1L2ejk1L2 ejk3L2ejk3L2 , k1ak3


ka

ejkaL2ejkaL2

, k1 k3 ka

8>>>>>>>:

Kss Z L20

E3A3Ns,xNs,xgNsNso2r3A3NsNsdxE3A3L2

1ej4k3L2 1ej2k3L2 2

" #jk3L2ga23 2L2j

1

2k3

ej2k3L2ej2k3L2

Ksk Z L20

E3A3Ns,xNk,xgNsNko2r3A3NsNkdxE3A3L2

2ej3k3L2 1ej2k3L2

1ej2k3L2 2


1

k3

ejk3L2ejk3L2

Kkj Z L20

gNkNjdxga1a3j

2k1k21k23

!ejk1L2ejk1L2 ejk3L2ejk3L2 , k1ak3


ka

ejkaL2ejkaL2

, k1 k3 ka

8>>>>>>>:

Z L ga1a3j 1


ejk1k3L2ejk1k3L2

, k1ak3

8>>>

ARTICLE IN PRESS

where,

p 16

L2h2

2, M1 r1A1L2 r1bh1L2, M3 r3A3L2 r32bh3L2

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

Z L 2

" #1ej4kLi

j3kLi j2kLi

2

1

W. Jung et al. / Finite Elements in Analysis and Design 46 (2010) 685697 697Journal of Sound and Vibration 104 (1986) 927.[7] D.J. Mead, Wave propagation in continuous periodic structures: research

contributions from southampton, 19641995, Journal of Sound and Vibration190 (1996) 496524.

[8] S. Mester, J. Benaroya, Periodic and near periodic structures: review, Shockand Vibration 2 (1995) 6995.

[9] M. Ruzzene, A. Baz, Active control of wave propagation in periodic uid-loaded shells, Smart Material and Structure 10 (2001) 893906.

[10] G. Solaroli, Z. Gu, A. Baz, M. Ruzzene, Wave propagation in periodic stiffenedshells: spectral nite element modeling and experiments, Journal ofVibration and Control 9 (2003) 10571081.

[11] O. Thorp, M. Ruzzene, A. Baz, Attenuation of wave propagation in uid-loadedshells with periodic shunted piezoelectric rings, Smart Material and Structure14 (2005) 594604.

[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) 151159.

[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) 219243.

[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) 272278.

[23] S. Asiri, Vibration attenuation of automotive vehicle engine using periodicmounts, International Journal of Vehicle Noise and Vibration 3 (2007) 302315.

[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 36 2008.ki21 EiAiZ Li0

@N2@x

@N1@x

o2 riEi

N2N1

dx EiAi

Li

jkLi1ej2kLi 2

" #

ki22 EiAiZ Li0

@N2@x

@N2@x

o2 riEi

N2N2

dx EiAi

Li

jkLi1ej2kLi 2

" #

or

References

[1] A. Baz, Active control of periodic structures, ASME Journal of vibration andAcoustics 123 (2001) 472479.

[2] D.J. Mead, Free wave propagation in periodically supported, innite beams,Journal of Sound and Vibration 11 (1970) 181197.

[3] D.J. Mead, Vibration response and wave propagation in periodic structures,ASME Journal of Engineering for Industry 21 (1971) 783792.

[4] D.J. Mead, A general theory of harmonic wave propagation in linear periodicsystems with multiple coupling, Journal of Sound and Vibration 27 (1973)235260.

[5] D.J. Mead, Wave propagation and natueal modes in periodic systems: mono-coupled systems, Journal of Sound and Vibration 40 (1975) 118.

[6] D.J. Mead, A new method of analyzing wave propagation in periodicstructures; applications to periodic timoshenko beams and stiffened plates,ej3kLi 1ej2kLi

ej4kLi

[13] O. Thorp, M. Ruzzene, A. Baz, Attenuation and localization of wavepropagation in rods with periodic shunted piezoelectric patches, SmartMaterial and Structure 10 (2001) 979989.

[14] M. Toso, A. Baz, Wave propagation in periodic shells with tapered wallthickness and changing material properties, Shock and Vibration 11 (2004)411432.

[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) 698711.

[16] D. Richards, D.J. Pines, Passive reduction of gear mesh vibration using aperiodic drive shaft, Journal of Sound and Vibration 264 (2003) 317342.

[17] M. Tawk, J. Chung, A. Baz, Wave attenuation in periodic helicopter blades,in: Jordan International Mechanical Engineering Conference (JIMEC), Amman,Jordan, 2628 April 2004.

[18] E.I. Rivin, Passive engine mount-some directions for further development.SAE Paper 850481, 1985.k12 EiAi0 @x @x

oEi

N1N2 dx Li 1ej2kLi 22e 1e ki11 EiAii

0

@N1@x

@N1@x

o2 riEi

N1N1 dxEiAiLi

jkLi1ej2kLi

i

Z Li @N1 @N2 2 ri EiAi jkLi" #

Mechanical filtering characteristics of passive periodic engine mountIntroductionMathematical modeling of passive periodic mountsOverviewThe transfer matrix of element 2Main assumptionsKinematic relationshipsEnergies of the element 2Finite element model of element 2The transfer matrix of element 2

The transfer matrices of elements 1, 3 and 4The transfer matrix of the passive periodic mount

Performance of passive periodic mountShape function of element 2 and elements 1, 3, 4MaterialsThe propagation of waves in passive periodic mountThe comparison between exponential and linear shape functionComparison between F.E.M and analytic approachComparison between shear and compression mount

Experimental performance of periodic mountConclusionsElement of matrix in Eq. (10)Exponential shape functionLinear shape function

Element of matrix in Eq. (23)References

MechFilteringCharacteristicsPassiveperiodicEngineMount.pdf

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