FINITE ELEMENT ANALYSIS OF

VISCOELSTIC SANDWICH CANTILEVER BEAM

A

PROJECT REPORT

Submitted in partial fulfillment of the

Requirement for the award of the

Fellowship of

SUMMER INTERNSHIP PROGRAMME-2010

by

Pulkit Sharma

Under the supervision of

Prof. H. Roy Prof. T. Roy

Department of Mechanical Engineering

National Institute of Technology

Rourkela-769 008, Orissa, India

ACKNOWLEDGEMENT

Looking back on my summer internship programme, I realized that I have gone through a great

program which strengthened my academic knowledge and gave me a broader scope of what the

Robotic discipline really is. Needless to say, I have faced a number of situations that seem to

hard to overcome. However, I was lucky enough to have great faculty members and friends who

always are ready to help me out.

First of all, I deeply thank my God for that He always listens to me whenever I am in any kind of

difficult situations.

I feel a great thankfulness for my faculty advisors, Prof. H.Roy and Prof. T.Roy. Their precious

guidance and support always encouraged me in the right direction of my research goal. It would

not have been possible for me to finish my work without his kind helps and wise suggestions

through the summer internship programme.

I am also obliged to Prof. Bidyadhar Subudhi, chairman of SIP-2010 for giving me this

opportunity to participate in this programme and providing all the necessary facilities.

The help and cooperation received from the friend-circle, staff of CAD Laboratory, staff of

Department of Mechanical Engineering is thankfully acknowledged.

Pulkit Sharma

(Summer Internship fellow-2010)

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ABSTRACT

In this swiftly growing world where technology has affected all the aspects of human life with

the advancements in material technology and evolution of new lighter materials there is drastic

reduction in mass of structures which is the need of the hour. But the other aspect is also worth

taking into account, structures and machineries like buildings, monuments, aircrafts, automobiles

etc., with the reduction in mass are more prone to vibrate with resonating frequency hence large

amplitudes due to either natural vibrations or the vibrations produced by powerful engines.

Hence structural damping plays an important role in their designing. In the present work, Finite

element analysis is conducted on viscoelastic sandwich cantilever beam to study the effects of

embedded viscoelastic layer on the response of beam to static, transient and harmonic loading

conditions by CAM package ANSYS. The sandwich beam under consideration is composed of

Aluminium as face material and Polyvinyl chloride (PVC) as sandwich material, wherein PVC

has frequency and temperature dependent elastic modulus and loss factor. The obtained response

clearly indicates the damping effects of sandwich layer. The transient response decays with time

unlike in the case of an undamped beam and for harmonic analysis; amplitude at natural

frequency is reduced to a finite value which is comparable with amplitudes of other frequencies

unlike in the case of undamped beam where response at resonance is infinite.

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CONTENTS

Certificate i

Acknowledgment ii

Abstract iii

Contents iv

Chapter 1: Introduction 1

1.1 Structural Damping 1

1.2 Sandwich Concept 2

1.3 Viscoelastic damping 3

1.4 Finite Element Analysis for thin damped sandwich beams 4

1.5 Literature survey 4

1.6 Objective of the present work 5

Chapter 2: Material Modelling 6

2.1 Aluminium 6

2.2 Poly vinyl chloride (PVC) 6

Chapter 3: Element 8

3.1 Tetrahedral geometry 9

3.2 Special Features of SOLID186 10

Chapter 4: Results and discussions 11

4.1 Cantilever beam 11

4.2 Solid model 11

4.3 Static analysis 13

4.4 Modal analysis 13

4.5 Dynamic analysis 16

4.6 Transient analysis 17

Chapter 5: Conclusions 18

REFRENCES 19

LIST OF FIGURES

Fig. 1 Sandwich plate 2

Fig. 2 Cyclic stress and strain curves for different materials 3

Fig. 3 Viscoelastic models 4

Fig. 4 SOLID186 Structural Solid Geometry 8

Fig. 5 10 Node Tetrahedral Element 9

Fig. 6 Solid Model of viscoelastic constrained layer cantilever beam 11

Fig. 7 Deflection of Sandwich cantilever beam under static load 12

Fig. 8 Deflection of Al cantilever beam under static loading 13

Fig. 9 Mode shapes of cantilever sandwich beam 14

Fig. 10 Frequency response for sandwich and Al cantilever beam 16

for frequency range 0-100 Hz

Fig. 11 Nature of transient force applied 17

Fig. 12 Transient response of sandwich beam 17

LIST OF TABLES

Table 1 Typical properties of Aluminium 6

Table 2 Frequency dependent elastic modulus and loss factor for PVC 7

Table 3 Dimensions of sandwich cantilever beam 13

Table 4. Natural frequencies of sandwich beam 15

Chapter 1

Introduction

1.1 Structural damping

Structural damping can be defined as the process by which a structure or structural component

dissipates Mechanical energy or transfers it to connected structures or ambient media. These

mechanisms have the effect of controlling the amplitude of resonant vibrations and modifying

wave attenuation and sound transmission properties, increasing structural life through reduction

in structural fatigue.

The effect of vibrations on structures and machineries can be devastating. With the

advancements in material technology and due to economic constraints emphasis is laid on the

light-weight structures and machineries which in turn make them more prone to resonance.

Hence damping plays a decisive role in designing of bridges, engine mounts, and machine

components such as rotating shafts, component vibration isolation, novel spring designs which

incorporate damping without the use of traditional dashpots or shock absorbers, and structural

supports.

Passive damping treatments are widely used in engineering applications in order to reduce

vibration and noise radiation. Passive layer damping can be implemented as free and constrained

layer damping. Constrained layer damping is the most common form of damping treatment,

where the damping layer deforms in shear mode, thus dissipating energy in a more efficient way.

1

1.2 Sandwich concept (Constrained layer damping)

Sandwich is built up of three elements-

Two face

Core

Joints

Face Plates

Joints

Fig 1. Sandwich plate Core

The aim is to use the material with the maximum of efficiency. The two faces are placed at a

distance from each other to increase the moment of inertia, and thereby the flexural rigidity,

about the neutral axis of the structure. A sandwich beam of the same width and weight as a solid

beam has a remarkably higher stiffness because of its higher moment of inertia.

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1.3 Viscoelastic damping

A viscoelastic material is characterized by possessing both viscous and elastic behavior.

What this means exactly is best illustrated in Figure 2, which shows how various types of

materials behave in the time domain. For a slab of material with a cross-sectional area, A,

and a thickness, T, subject to cyclic loading, F(t), the corresponding response is given by the

displacement function, x(t).

Fig 2. Cyclic stress and strain curves for different materials

Unlike many other damping mechanism most homogeneous isotropic materials exhibit

damping behavior which depends strongly upon temperature and frequency, but linear with

respect to vibration amplitude, at least within limits. Viscoelastic materials, such as

amorphous polymers, semi crystalline polymers, and biopolymers, can be modeled in order

to determine their stress or strain interactions as well as their temporal dependencies. These

models, which include the Maxwell model, the Kelvin-Voigt model, and the Standard Linear

Solid Model, are used to predict a material's response under different loading conditions.

Viscoelastic behavior has elastic and viscous components modeled as linear combinations

of springs and dashpots, respectively. Each model differs in the arrangement of these

elements.

3a. Maxwell model 3b. Kelvin-Voigt model

3c. Standard linear solid model

Fig 3. Viscoelastic models

3

1.4 Finite Element Analysis for thin damped sandwich beams

Finite element analysis has emerged as a very efficient tool for solving complex problem in the

field of design engineering. The experimental procedure is a very tedious task and lots of

assumption must be taken care off for precision of the work and using finite element method we

can reduce this complexity of the problem and get rid of calculations. In this report a finite

element analysis has been done for both undamped and damped sandwich structures and

frequency response for the same has been shown.

1.5 Literature survey

It is evident that vibration and noise are matter of concern for industries like aeronautical and

automobile. Lot of research in this area have resulted in control of these external unwanted

effects and hence influencing the efficiency of the system [4, 6]. Passive damping methods have

provided an efficient and cost-effective solution for overcoming this problem. The steel industry

proposes damped sandwich sheets in which a thin layer of viscoelastic material is sandwiched

between two elastic face layers.

The fundamental work in this field was pioneered by Ross, Kerwin and Ungar (RKU) [7], who

used a three-layer model to predict damping in plates with constrained layer damping treatments.

Kerwin [5] was the first to present a theoretical approach of damped thin structures with a

constrained viscoelastic layer. He stated that the energy dissipation mechanism in the constrained

core is attributable to its shear motion. He presented the first analysis of the simply supported

sandwich beam using a complex modulus to represent the viscoelastic core. Several authors

DiTaranto [3], Mead and Markus extended Kerwin’s work using his same basic assumptions.

DiTaranto proposed an exact sixth-order theory for the unsymmetrical three-layer beam, and this

was subsequently refined [7-10].

Analysis of Transient response of a viscoelastic sandwich structure was studied in detail by

Barkanov and useful contributions in the field came from N Al- Huniti and M Meunier.

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Diab Sandwich Handbook prepared by Divinycell, Kelgecell and ProBalsa[4] provides detailed

calculations involved in sandwich structures including struts, beams, panels.

1.6 Objective of the present work

This report provides a final summary of the progress made over the past two months on the study

of passive viscoelastic constrained layer cantilever beam, specifically applied to high stiffness

structural members.

The main focus of this dissertation is to study the response of a viscoelastic constrained layer

cantilever beam under the application of static, harmonic and transient excitation and compare

the damping properties with orthodox cantilever beam of the same dimensions using finite

element method package ANSYS.

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Chapter 2

Material Modeling

The viscoelastic sandwich beam is comprised of face plates of Aluminium and viscoelastic core of PVC (polyvinyl chloride) binded together. The properties of the materials are mentioned below -

2.1 Aluminium

Aluminium is remarkable for the metal's low density and for its ability to resist corrosion due to

the phenomenon of passivation. Structural components made from aluminium and its alloys are

vital to the aerospace industry and are very important in other areas of transportation and

building. Its reactive nature makes it useful as a catalyst or additive in chemical mixtures,

including ammonium nitrate explosives, to enhance blast power.

Table 1. Typical properties of Aluminium

S.no. Property Value

1. Young’s modulus 7.13 x 1010

N/m2

2. Poisson’s ratio 0.3

3. Density 2750 Kg/m3

2.2 Polyvinyl chloride (PVC)

Polyvinyl chloride is a well known versatile commodity thermoplastic, whose production and

consumption worldwide is second to other plastics. PVC is generally known to have the

advantages of low ingredient cost, wide processing versatility, high decorative potential and is

used to manufacture various types of products ranging from highly rigid to very flexible.

PVC also has remarkable property of a viscoelastic material i.e. frequency and temperature

dependent elastic modulus and loss factor.

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Table 2. Frequency dependent elastic modulus and loss factor for PVC

Frequency (Hz) Young’s modulus (N/m2) Loss factor

30 2.38e7 0.38

70 2.83e7 0.43

120 3.12e7 0.48

200 3.32e7 0.53

240 3.36e7 0.55

308 3.38e7 0.57

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Chapter 3

Element

The element utilized in the present case is SOLID186. It is a higher order 3-D 20-node solid

element that exhibits quadratic displacement behavior. The element is defined by 20 nodes

having three degrees of freedom per node: translations in the nodal x, y, and z directions. The

element supports plasticity, hyperelasticity, creep, stress stiffening, large deflection, and large

strain capabilities. It also has mixed formulation capability for simulating deformations of nearly

incompressible elastoplastic materials, and fully incompressible hyperelastic materials.

SOLID186 is available in two forms:

Structural Solid (KEYOPT (3) = 0, the default)

Layered Solid (KEYOPT(3) = 1)

Fig. 4 SOLID186 Structural Solid Geometry

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3.1 Tetrahedral geometry

In the present work tetrahedral geometry of SOLID186 element is used for meshing of beam.

The following are its salient features-

Fig.5 10 Node Tetrahedral Element

The resulting effective shape functions are:

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3.2 Special Features of SOLID186

Plasticity

Hyperelasticity

Viscoelasticity

Viscoplasticity

Creep

Stress stiffening

Large deflection

Large strain

Initial stress import

Automatic selection of element technology

Birth and death

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Chapter 4

Results and discussion

4.1 Cantilever beam

A cantilever is a beam supported on only one end. The beam carries the load to the support

where it is resisted by moment and shear stress. Cantilever construction allows for overhanging

structures without external bracing. Cantilevers can also be constructed with trusses or slabs. In

the following section various analysis are conducted on viscoelastic sandwich cantilever beam

(Al-PVC-Al) and simultaneously compared with Aluminium cantilever beam of same

dimensions.

4.2 Solid Model

Fig 6. Solid Model of viscoelastic constrained layer cantilever beam

Table 3. Dimensions of sandwich cantilever beam

Properties Dimensions (mm)

Length 500

Width 50

Thickness (each layer) 5

4.3 Static Analysis

Force applied at the tip, F = 10N (negative Y direction)

Fig 7. Deflection of Sandwich cantilever beam under static load

Fig 8. Deflection of Al cantilever beam under static loading

4.3 Modal Analysis

The following are the natural frequencies and the corresponding mode shapes of the viscoelastic

sandwich cantilever beam.

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First mode Second mode

Third mode Fourth mode

Fig 9. Mode shapes of cantilever sandwich beam

Top View Front View

Front View Isometric View

Isometric View

Fifth mode

Table 4. Natural frequencies of sandwich beam

S.no Natural Frequency(Hz)

1 30

2 35

3 139

4 184

5 318

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4.4 Dynamic Analysis

Fig 10. Frequency response for sandwich and Al cantilever beam for frequency range 0-100 Hz

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4.5 Transient Analysis

Fig 11. Nature of transient force applied

Fig. 12 Transient response of sandwich beam

17

F0 = 10N

tf = 3 sec

dt = 0.001sec

Chapter 5

Conclusion

The analysis clearly shows the effects of introducing the viscoelastic material in the cantilever

beam and the following conclusions can be drawn from it –

1. The static deflection in case of sandwich beam and Al beam are almost equal. Hence

damping has no significant effect in case of static loading.

2. The response for transient loading clearly indicates that the amplitude of oscillations gets

reduced with time giving the importance of damping.

3. The comparison of frequency response for harmonic loading between sandwich beam and Al

beam shows the effect of damping at resonance wherein the amplitude at natural frequency for

Al beam is infinite whereas for sandwich beam is finite and comparable to the amplitudes at

other frequencies.

Furthermore, the sandwich beam can be analyzed for explicit dynamic loading using

ANSYS/LS-DYNA package. Also using piezoelectric material in place of viscoelastic material

can give us

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REFRENCES

1. Asnani N.T., 1971, “Vibration Analysis of Multi-layered Beams with Constrained

Viscoelastic Layers”, Ph.D. Thesis, IIT Delhi.

2. Clarence W. de, Silva, 2000, Vibration: Fundamentals and Practice, Boca Raton, FL: CRC

Press, cop.

3. DiTaranto, R. A., 1965, “Theory of Vibratory Bending for Elastic and Viscoelastic

Layered Finite Length Beams,” ASME J. Appl. Mech., 87, pp. 881–886.

4. Jones, D. I. G. 2001. Handbook of Viscoelastic Vibration Damping. West Sussex,

England: John Wiley and Sons, LTD.

5. Kerwin, E.M. 1959. Damping of flexural waves by a constrained viscoelastic layer.

Journal of the Acoustical Society of America, 31(7), 952-962.

6. Nashif A.D., Jones D.I.G. and Henderson J.P., 1985, Vibration Damping ,Wiley, New

York.

7. Ross, D., Ungar, E., and Kerwin, E., 1959, “Damping of Flexural Vibrations by

Means of Viscoelastic Laminate,” Structural Damping, ASME, New York.

8. Singiresu S.Rao., 2004, The finite element method in engineering ,Heinemann-Butterworth

9. Yan, M.J., and Dowell, E.H. 1972. Governing equations for vibrating constrained layer

damping of sandwich beams and plates. Transactions of the ASME, Journal of

Applied Mechanics, 94, 1041-1047.

10. Yu, Y. Y., 1962, “Damping of Flexural Vibrations of Sandwich Plates,” J. Aerosp.

Sci., 29, pp. 790–803.

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