Wavepropagation and Resonance in Honeycomb Panels...

Wavepropagation and Resonance in HoneycombPanels including Cavity Dynamics

A Project Report

Submitted in partial fulfilment of the

requirements for the Degree of

Master of Engineering

in

Faculty of Engineering

by

Sudheesh Kumar C.P.

Department of Aerospace Engineering

Indian Institute of Science

BANGALORE – 560 012

July 2009

TO

My Son

Acknowledgements

I would like to express my sincere and deep gratitude to Prof. D.Roy Mahapatra for his

efficient guidance and constant motivation throghout the project without which it would

have been a tough task to complete it successfully and time-bound. His confidence

and courage to go beyond the comfort zone to take new challenges in the emerging

areas of research have always taken me by surprise. His enthusiasm and commitment to

research have always been motivating. I would also like to extend my gratitude to all

the Professors who taught subjects of different kinds many of which were very relevant

to this work.

I express my sincere gratitude to Keshava for his great help in finishing the experi-

mental part of this work successfully. I also thank all labmates namely Kannan, Abishek,

Nibir, Vadiraja, Tulseeram, Vivek, Aheesh, Indrajith, Renukanand, Rejin and Sandeep

for their right ideas and appropriate suggestions which helped me a lot to clear the hur-

dles on my way to the completion of this project. I would also like to thank Narendar, my

class mate and extend special thanks to my friends Binoj and Saji for giving a pleasant

and nice atmosphere during my two years stay in the campus.

Sudheesh Kumar C.P.

Dept. of Aerospace Engg.

IISc.

i

ii

Publications From This Project

1. “Wave Propagation in Honeycomb Panels including Cavity Dynamics”, C.P. Sud-

heesh Kumar and D. Roy Mahapatra, Journal of Sound and Vibration (submitted).

2. “An Experimental Study on the Noise Attenuation Characteristics of Patterned

Cellular Panels ”, C.P.Sudheesh Kumar, S. Keshava Kumar and D. Roy Mahapatra,

National Conf. on MEMS, Smart Structures and Materials, October 14-16, 2009,

Calcutta (submitted).

Abstract

Dynamic response of a honeycomb panel including the dynamics of its cell cavities is

analyzed in this paper. The structure of the panel is composed of a sequence of unit cell

repeating along the length and width of the panel. To characterize the effective dynamic

stiffness, the honeycomb cell walls are modeled as rectangular plate elements subjected

to fluid dynamic loading. Wave dispersion in the cell wall is analyzed. A finite ele-

ment model is developed to evaluate the structural resonance of the honeycomb panels.

The interpolation function represents a Fourier spectral basis space derived directly by

solving the governing equations for the cavity pressure couple wave motion in the cell

walls and their in-plane motion due to the unit cell structure of the entire honeycomb

panel. The developed finite element approach gives a very high scalability to solve wave

propagation and resonance problems in large structures involving honeycomb cells and

others. Frequency responses due to air-filled cavities are compared with those in vacuum.

An experiment has been conducted to study the noise transmission loss characteristics

of patterned cellular solids and to determine the effect of parametric variation on the

frequency band-gaps. These results can further be used to develop a noise control strategy

while designing sandwich panels for different applications over a broad band of frequen-

cies.

iii

Contents

Acknowledgements i

Abstract iii

1 Introduction 2

1.1 Honeycomb structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Honeycomb Core . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Cell Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.5 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.6 Merits and Demerits . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Literature Review 12

2.1 Scope of the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Modeling of cell wall assembly in 3D 15

3.0.1 Governing equation for out-of-plane motion . . . . . . . . . . . . 173.0.2 Governing equation for in-plane motion . . . . . . . . . . . . . . . 203.0.3 Fluctuation of cell cavity pressure . . . . . . . . . . . . . . . . . . 21

4 Wave Propagation in Honeycomb Cells 24

4.1 Flexural waves in the wall of the unit hexagonal cell . . . . . . . . . . . . 244.2 In-plane waves in the cell wall . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Spectral FEM based Analysis of Honeycomb Panels 30

5.1 Frequency Domain Spectral Finite Element Formulation . . . . . . . . . 305.1.1 Dynamic stiffness of a wall of the unit hexagonal cell . . . . . . . 315.1.2 Transformation of nodal vectors from cell wall co-ordinate to hon-

eycomb panel coordinate . . . . . . . . . . . . . . . . . . . . . . . 365.2 Consistent foce vector due to dynamic pressure . . . . . . . . . . . . . . 395.3 Evaluation of Dynamic Stiffness Matrix and Dynanamic Response of the

whole Honeycomb panel . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

iv

CONTENTS v

5.4 Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.5 Simulation Results and Discussions . . . . . . . . . . . . . . . . . . . . . 46

6 Experimental Studies using Patterned Honeycomb Panels 48

6.1 Aircraft Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Noise Attenuation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2.1 Passive Noise Control . . . . . . . . . . . . . . . . . . . . . . . . . 506.2.2 Active Noise Control Strategies . . . . . . . . . . . . . . . . . . . 516.2.3 Transmission Loss Characteristics of Panels . . . . . . . . . . . . 52

6.3 Determination of Noise Reduction in Patterned Cellular Panels . . . . . . 556.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 586.3.2 Test panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.3.3 Tests of cellular panels and Results . . . . . . . . . . . . . . . . . 60

7 Conclusions and Future Scopes 65

Bibliography 67

List of Tables

5.1 Connectivity table for elements of honeycomb unitcell Ref.fig. 5.2 . . . . 37

vi

List of Figures

1.1 Honeycomb Panel Construction . . . . . . . . . . . . . . . . . . . . . . . 31.2 Efficiency of Sandwich Structures . . . . . . . . . . . . . . . . . . . . . . 41.3 Characteristics of Typical Honeycomb Core Materials . . . . . . . . . . . 51.4 Honeycomb Core Terminology . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Cost Vs Performance for Core Materials . . . . . . . . . . . . . . . . . . 71.6 Strength and Stiffness of Various Core Materials . . . . . . . . . . . . . . 8

3.1 (a) Schematic diagram of a Honeycomb panel (b) Honeycomb cell withwalls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Descritization of honeycomb core configuration in XY plane: (a) Honeycombcore configuration, its unit cell with dotted rectangle, (b) Unit cell further

descritized into its individual plates indicated with numbers inside the circle. . 163.2 Honeycomb cell wall subjected to dynamic pressure. . . . . . . . . . . . . 17

4.1 A projection of the representative wall, shown in Fig. 3.2, in XZ planeunder distributed load (pressure). . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Effect of dynamic viscosity µ on the wavenumber dispersion behaviour ofaluminium wall of honeycomb cell filled with air. . . . . . . . . . . . . . 26

4.3 Effect of dynamic viscosity µ on the wavenumber dispersion behaviour forthe aluminium plate element, shown in Fig. 4.1, in water. . . . . . . . . 27

5.1 (a) Shape functions for the representative plate element shown in Fig. 4.1.(b),(c) and (d) Dynamic behaviour of shape functions N1 ,N2 and N3respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Unit cell of honeycomb panel, the circled numbers represents element num-bers and remaining plain numbers represent global node numbers, smallnumbers 1 and 2 on the element 2 represent local node numbering.x̄,ȳrepresents global coordinate system and x,y represents local coordinatesystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Frequency response of the unit hexagonal cell (Fig. 5.2), in air withdynamic viscosity and in vacuum, due to a combined periodic load of1µN(axial and transverse) and 1µNm (moment) applied at nodes 5 and6, (a) in-plane response u2, u5 and b) flexural response w2 ,w5 at nodes 2and 5 respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

vii

LIST OF FIGURES viii

5.4 (a), (b) Frequency response of the unit hexagonal cell (see Fig. 5.2) dueto a periodic load of 1µN applied in the X− direction, at nodes 5 and 6,and the effect of dynamic viscosity on the displacements at a) node 2 andb) node 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.5 (a), (b) Frequency response of the unit hexagonal cell (Fig. 5.2) in air withdynamic viscosity and in vacuum, due to a periodic load of 1µN appliedin the X− direction at nodes 5 and 6, a) flexural response w2, w5 and b)in-plane response u2, u5 at nodes 2 and 5 respectivley. . . . . . . . . . . . 43

5.6 Frequency Response of the unit hexagonal cell (Fig. 5.2) due to a unitperiodic load of 1µN applied in the Z− direction at nodes 5 and 6, (a)in-plane response u2, u5 and (b) flexural response w2, w5 at nodes 2 and5respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.7 Effect of wall thickness h on the frequency response of the unit cell (seeFig. 5.2) at (a) node 2 and (b) node 5 due to a combined unit periodicload of 1µN (axial and transverse) and 1µNm (moment) acting at nodes5 and 6. h1 = 0.00025mm and h2 = 0.00050mm. . . . . . . . . . . . . . . 45

5.8 Effect of wall thickness h on the frequency response of the unit hexagonalcell ( see Fig. 5.2) at (a) node 2 and (b) node 5 due to a flexural periodicload of 1µN applied in Z-direction at nodes 5 and 6. h1 = 0.00025mmand h2 = 0.00050mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.1 Typical single panel transmission loss as a function of frequency: (a)isotropic panel characterized by a single critical frequency; (b) orthotropicpanel characterized by a critical frequency range. . . . . . . . . . . . . . 56

6.2 Experimental arrangement: (a) Photograph of the arrangement and (b)Schematic representation. . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3 Cavity dynamics effect on noise transmission in bare honeycomb panels . 596.4 Test panels: (a) Bare honeycomb panel (b) Line fill pattern (c) Diagonal

fill pattern and (d) Floral fill pattern . . . . . . . . . . . . . . . . . . . . 616.5 Comparison plot for transmission loss of different materials . . . . . . . . . . 626.6 Noise Reduction for Different Filler Patterns . . . . . . . . . . . . . . . . . . 636.7 (a) Spatial noise attenuation characterstic (b) Mic position reference figure

(plan view) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

LIST OF FIGURES 1

keywords

Cellular Composite, Wave Dispersion, Cavity Dynamics, Fluid-structure interaction,

Resonance, Dynamic Shape Functions, Spectral Finite Element.

Chapter 1

Introduction

Cellular composite materials with various types of microstructures [1] are potential candi-

dates for applications in vibro-acoustic and aero-acoustic disturbance suppressions. The

basic principle by which these composite enhance the disturbance suppression charac-

teristics is the following. Most sandwich structures have a light-weight core in the form

of corrugated, open or close cell geometry encased between two face sheets.

Numerous studies in the past and present reveal the facts that for a high-perfomance

civil-aircraft designs it is essential that one goes for light weight and high-temperature

composite materials for structural applications. This natrually reduces the fuel consump-

tion and minimises the direct operating and maintenance costs. Sandwich structures

have wide range of applications mainly in aerospace and automobile structures, because

they offer great energy absorption and increase the moment of inertia without increas-

ing the weight much. Sandwich panels are typically used for their structural, electrical,

insulation, and/or energy absorption characteristics .A sandwich panel primarily con-

sists of face sheets and core with the facesheets carrying the bending loads and core the

shear loads. Aluminium, glass, carbon, or aramind are the commonly used materials for

facesheets whereas core materials inlcude metallic and non-metallic honeycomb, balsa

wood, open and closed cell foams.

2

Chapter 1. Introduction 3

1.1 Honeycomb structures

1.1.1 Honeycomb Core

Figure 1.1: Honeycomb Panel Construction

The details of a typical honeycomb core panel are shown in Fig. 1.1. Typical

facesheets include aluminium, glass, aramind, and carbon. Structural film adhesives

are normally used to bond the facesheets to the core. It is important that the adhesive

provide a good fillet at the core-to-skin interface.

Honeycomb structures are composed of plates or sheets that form the edges of unit

cells. These can be arranged to create triangular, square, hexagonal or related shapes.

Their unit cells are repeated in two dimensions to create a cellular solid. One of the

manufacturing methods used to create hexagonal honeycomb leads to a doubling in

wall thickness of every other web which results in anisotropic mechanical behavior. All


Figure 1.2: Efficiency of Sandwich Structures

honeycombs are closed cell structures. The bending loading is mainly carried by the face

layers (e.g. metal or reinforced polymers). Transverse shear loading is carried by the

core and requires a material with sufficient shear stiffness and shear strength. Classical

core materials are paper or aluminium honeycombs.

Doubling the thickness increases the stiffness over 7X with only a 3 weight gain, while

quadrupling thickness increase stiffness over 37X with only a 6 weight gain. A cost versus

performance comparison is given in Fig. 1.5. Note that, in general, the honeycomb cores

are more expensive than the foam cores but offer superior performance. This explains

why many commercial applications use foam cores, while aerospace applications use the

higher performance but more expensive honeycombs. A relative strength and stiffness

comparison of different core materials is given in Fig. 1.6.

Foam core sandwich assemblies can be bonded together with supported film adhesives,

but the more common case is either to use liquid/paste adhesives or to do wet lay-up of

the skin plies directly on the foam surface.


Figure 1.3: Characteristics of Typical Honeycomb Core Materials

1.1.2 Cell Terminology

Although there are a variety of cell configurations available, the three most prevalent

are hexagonal, flexible-core, and overexpanded core. Hexagonal core is, by far, the most

commonly used core configuration. It is available in aluminium and all non-metallic

materials. Typical honeycomb core terminology is given in Fig. 1.4.

The honeycomb itself can be manufactured from aluminium, glass fabric, aramind

paper, aramind fabric, or carbon fabric. Honeycomb manufactured for use with organic

matrix composites is bonded together with adhesive, called the node bond adhesive. The

”L” direction is the core ribbon direction and is stronger than the width(node bond) or

”W” direction. The thickness is denoted by ’t’ and the cell size is the dimension across


Figure 1.4: Honeycomb Core Terminology

the cell as shown in the Fig. 1.4. Hexagonal core is structurally very efficient, and can

even be made stronger by adding longitudinal reinforcement(reinforced hexagonal) in

the ”L” direction along the nodes in the ribbon direction.

The main disadvantage of the hexagonal configuration is limited formability; alu-

minium hexagonal core is typically rolled formed to shape, while non-metallic hexagonal

core must be heated formed. Flexible-core was developed to provide much better forma-

bility. This configuration provides for exceptional formability on compound contours

without cell wall buckling. It can be formed around tight radii in both the ”L” and the

”W” directions. Another configuration with improved formability is overexpanded core.

This configuration is hexagonal core that has been over-expanded in the ” W” direction,

providing a rectangular configuration that facilitates forming in the ”L” direction. The

”W” direction is about twice the ”L” direction. This configuration, as compared to

regular hexagonal core, increases the ”W” shear properties but decreases the ”L” shear

properties.


Figure 1.5: Cost Vs Performance for Core Materials

1.1.3 Properties

The comparative properties of some of the commercial honeycomb cores are given in

Table. Aluminium honeycomb has the best combination of strength and stiffness. The

higher performance aerospace grades are 5052-H39 and 5056-H39, while the commercial

grade is 3003 aluminium. Cell sizes range from 1/16 to 3/8 in. but 1/8 and 3/16 in. are

the ones most frequently used for aerospace applications. Glass fabric honeycomb can

be made from either a normal bi-directional glass cloth or a bias weave(±45) cloth.


((a) (b)

Figure 1.6: Strength and Stiffness of Various Core Materials

1.1.4 Processing

The major processing before bonding are perimeter trimming, mechanical or heat form-

ing, core splicing, core potting, contouring, and cleaning.

Trimming

This is the process of cutting the honeycomb to dimensions with tools like serrated knife,

razor blade knife, band saw and die.

Forming

Metallic hexagonal honeycomb can be rolled or brake formed into curved parts. The

brake forming method crushes the cell walls and densify the inner radius. Non-metallic

honeycomb can be heat formed to obtain curved parts.

Splicing

This is the method of splicing together the smaller pieces or different desnisties of core

to form the finished part. This is done with a foaming adhesive when large pieces of core

are required or when strength requirements dictate different densities.


Potting

Potting compounds are frequently required for fitting attachments where fasteners must

be put through the honeycomb assembly.

Machining

Machining is done to get the thickness to some contour. Valve stem cutters are used to

accomplish this.

Cleaning and drying

It is cleaned by solvent vapor degreasing. This is required before bonding. It is to be

thoroughly dried prior to bonding as the large surface are can cause the absorption of

moisture. So bonding is to be done soon after drying. Honeycomb bonding: The pressure

selection is the important consideration during the bonding. The pressure should be high

enough to push the parts together, but not be so high that there is danger of crushing

or condensing the core. The allowable pressure depends on both the core density and

the part geometry. Common bonding pressure can range anywhere between 15 and 50

psi for honeycomb assemblies.

1.1.5 Features

The main features of honeycomb panel for aerospace structural applications are

• Low weight/High Strength

• Elevated use temperatures

• High thermal conductivity

• Flame resistant

• Excellent moisture and corrosion resistance

• Fungi Resistance


1.1.6 Merits and Demerits

The advantage of honeycomb core is that it does offer superior performance compared

to other sandwich cores. Note that aluminium core has the best combination of strength

and stiffness, followed by the non-metallic honeycombs, and then polyvinyl chloride

(PVC) foam. The demerit about honeycomb core is that it is expensive and difficult to

fabricate complex assemblies, and the in-service experience, particular with aluminium

honeycomb, has not always been good. It can also be very difficult to make major repairs

to honeycomb assemblies.

1.1.7 Applications

Honeycomb panels are widely used in aerospace, marine and automobile industries. A

few of the specific applications are mentioned below.

• Aircraft engine structures

• Aircraft doors and hatches

• Aircraft floor panels

• Helicopter blades

• Racing car structures and aero foils

• Building cladding panels

• Commercial vehicle panels

• Railway floors, doors, interiors, fairing panels

• Boat hulls and interior panels and furniture

• Racing car chassis and body panels

• Cable car structural panels


• Specialist flight cases

• Energy absorbers

1.2 Organization of the thesis

The thesis is organized in six chapters including this introduction.

• Chapter 1, the current chapter, gives an introduction of the cellular solids, theirproperties and applications.

• In Chapter 2, a breif description on the developments in the area of cellular solidsand the scope of the present work are given.

• In Chapter 3, the mathematical formulation of the governing equations for in-planeand out-of plane motion of the thin rectangular plate element of the unit hexagonal

cell of the honeycomb are presented.

• Chapter 4 presents the wavepropogation analysis to study the dispersion behaviourof the fluid-interacted plate element with and without the inclusion of dynamic

viscosity of the fluid.

• In Chapter 5, dynamic homogenization of the honeycomb unit cell and modelorder reduction to determine the response of the whole panel using spectral FEM

are described.

• Chapter 6 describes the experimental studies conducted to study the passive noisesuppression characteristics of pattern-filled aluminium honeycomb panels.

• Chapter 7 includes the conclusions and future scope of the project.

Chapter 2

Literature Review

The cellular composites have microstructures, which deform very differently than the

elastic continua. Due to the geometric features at multiple scales, the microstructure

in a cellular composite shows certain resonance characteristics (especially when the cell

walls / ligaments are highly flexible). The resonance could originate either due to a single

cell resonance or due to a network of cells behaving as coupled resonator array. Incident

disturbances with their characteristic power spectral densities get transmitted through

the cellular composite layers along the thickness direction and or in the laminate plane

as a complicated function of frequency, wavelength and disturbance amplitude. The cell

walls are modeled as rectangular plate elements subjected to fluid dynamic loading to

characterize the effective dynamic stiffness. Therefore it is essential that we do a review

of the works related to the vibration of thin plates reported till date.

In the vibration analysis of thin plates, it is the flexural mode that is of great practical

importance because its natural frequencies are prone to external excitation whereas in-

plane vibrations are a matter of concern generally at higher frequencies. There are several

literature which covers various related topics. A transfer function method was used to

determine the dynamic mechanical properties of beams by Park [2] with 2 DOFs at each

node without considering the in-plane motion. Since very high strain energy is involved

in the deformation, in-plane vibrations occur at high frequencies. So this plays a role

significantly when it comes to the case of structures that vibrate at high frequencies. This

12

Chapter 2. Literature Review 13

analysis has been done by Hyde et.al. [3]. Dynamic response and vibrational power

flow characteristics of isolated as well as coupled finite thin plates have been studied

by Farag et.al. [4]. A modal receptance method was used throughout their analaysis,

and the frequency response was expressed in the form of receptance functions for both

flexural and in-plane vibrations. The receptance functions were then used in the coupling

of two plates at an arbitrary angle. In ref. [5] both the flexural and axial modes have

been considered in the analysis to determine the flexural energy transmission through

structural junctions of connected plates. Alhough several works have been reported in

literature on the in-plane and out-of-plane vibration of thin plates and connected plates,

no work has been reported on the combined effect of both on structures which interact

with compressible and viscous fluids. An efficient computational tool has been presented

by Mahapatra et.al [6] to study the dynamic response and the effect of excitation

frequencies on the propagation of coupled axial-flexural waves. In-plane and flexural

vibrations of rectangular plates have been discussed in ref. [7-23]. A mathematical

model has been developed for a fluid-coupled rectangular plate by Kerboua et.al [8]. A

dynamic stiffness approach has been used by Bercine et.al [9] to study in-plane vibrations.

Analysis of flexural vibration of a combination of plates has been reported in refs. [11, 25].

An overview of the developments in the area of structural accoustics and vibration has

been given in ref. [12]. Even in the studies of fluid-structure interaction with in-plane

motion conducted so far, the fluid (air) was assumed invscid though it is not actually so.

2.1 Scope of the project

We can see from the literatures that though many works have been reported on in-plane

and out-of-plane vibration of thin plates and connected plates, no work has been reported

on the combined effect of both on structures which interact with fluids. Also, in all the

above-mentioned works, the fluid (air) was assumed inviscid and incompressible though

it is not so especially at higher frequencies. Therefore, the main objectives of this project

are

Chapter 2. Literature Review 14

• To develop a mathematical model for the unit hexagonal cells of honeycomb panelsto predict the dynamic response.

• To study the dispersion behaviour of the fluid-interacted cell walls of honeycombpanels.

• To determine the resonance in honeycomb panels due to the dynamic pressure inthe cell cavities using Frequency Domain Spectral FEM.

• To study the noise attenuation characteristics and explore the band-gap phe-nomenon of patterned honeycomb panels.

Chapter 3

Modeling of cell wall assembly in 3D

The honeycomb panel considered here features a three dimensional arrangement of core

with cell walls. A detailed configuration is shown schematically in Fig. 3.1. Fig. 3.1(a)

shows the panel with global coordinate system. Various loading arrangements are also

shown in this figure. Fig. 3.1(b) shows the core honeycomb configuration. When the

panel is subjected to dynamic vibratory or accoustic load ( axial force F (ω), transverse

force V (ω), or moment M(ω)) the vibration of the cell walls are coupled with the cavity

pressure oscillations. We shall consider this effect in the formulation. In Fig. 3.1(b), one

of the faces of a honeycomb cell is marked as the ‘representative cell wall’, which will

be referred while deriving the governing equation of motion for the cell walls. Fig. 3.2

shows the represnetative cell wall in the local coordinate system x′, y′, z′ where Vxz and

Vyz are the shear forces on x′ normal and y′ normal faces, Mxx and Myy are the bending

moments about x′ and y′, respectively.

15

Chapter 3. Modeling of cell wall assembly in 3D 16

x

y

z

xzxx vMM yy

(a)(b)

vyx

Representative wallx

y

/

/

z/ Local coordinate

system

t

l

h

Figure 3.1: (a) Schematic diagram of a Honeycomb panel (b) Honeycomb cell with walls.

1

2 3

4

5

6

7

8 9

10

2

1 3

4

5

6

78

9

(a)

(b)

Unit cell

X

Z

c

c

1

c2 3i

i

i

e

e

e

e

e e e M

r

r

r1

2

3

r4

Figure 3.3: Descritization of honeycomb core configuration in XY plane: (a) Honeycomb

core configuration, its unit cell with dotted rectangle, (b) Unit cell further descritized into its

individual plates indicated with numbers inside the circle.

In Fig. 3.1(b), one of the faces of a honeycomb cell is marked as the ‘representative

face’, which will be referred while deriving the governing equation of motion for the cell

walls. Fig. 3.2 shows the represnetative face in the local coordinate system x′, y′, z′ where

Vxz and Vyz are the shear forces on x′ normal and y′ normal faces, Mxx and Myy are the

bending moments about x′ and y′ respectively. A vibrating flat plate usually exhibits


x

z p(x,y,t)

/

/

/

y

M

M

Vyy

xx

Vxz

yz

Figure 3.2: Honeycomb cell wall subjected to dynamic pressure.

three types of mode, namely bending (out-of-plane), longitudinal and shear modes (in-

plane modes). Among these three modes, the bending mode, also referred to as the

flexural mode, is the predominant one. But in the case of a series of connected plates

a strongly excited out-of-plane mode in one plate causes in-plane and flexural waves in

the adjacent plates. The flexural waves at some junctions are partly transmitted to in-

plane waves at other junctions and vice versa. This effect becomes significant at higher

frequencies.

3.0.1 Governing equation for out-of-plane motion

In order to determine the dynamic mechanical properties of the honeycomb cell, we

use classical plate theory, which is ideally suited for thin plates as is the case of cells

with walls in the present problem. The honeycomb cells considered here have very thin

walls (≈ 0.25 mm for a cell size of 7 mm and panel thickness of 6 mm) and the cellwall thickness h is much smaller compared to the wavelength λ which is appropriate for

shear deformation and rotational inertia to be neglected [12]. Also, it is assumed that

the walls are homogeneous, isotropic, elastic and of uniform thickness. According to

Kirchoff’s thin plate theory, the displacements for flexural motion can be approximated


as

ū(x, y, z, t) = −z∂w∂x

(x, y, t) , v̄(x, y, z, t) = −z∂w∂y

(x, y, t) ,

w̄(x, y, z, t) = w(x, y, t). (3.1)

Here we assume (x, y, z) ≡ (x′, y′, z′) (See Fig.3.1) The in-plane axial strains and theshear strains corresponding to the above deformations are

ǭxx =∂ū

∂x= −z∂

2w

∂x2, ǭyy =

∂v̄

∂y= −z∂

2w

∂y2, ǭzz = 0 ,

γ̄xy =∂ū

∂x+

∂v̄

∂x= −2z ∂

2w

∂x∂y, γ̄xz =

∂ū

∂z+

∂w̄

∂x= 0 ,

where ū, v̄, w̄ are the cell wall displacements in x, y, and z− directions respectively, andz is the distance from the neutral axis in the thickness direction of the wall.

γ̄yz =∂v̄

∂z+

∂w̄

∂y= 0. (3.2)

Substituting these strains into the Hooke’s law for plane stress condition gives

σ̄xx =−Ez1 − ν2

[

∂2w

∂x2+ ν

∂2w

∂y2

]

, σ̄yy =−Ez1 − ν2

[

∂2w

∂y2+ ν

∂2w

∂x2

]

,

σ̄xy = −2Gz[

∂2w

∂x∂y

]

. (3.3)

where E, G are the Young’s modulus and the modulus of rigidity, ρ is the density, ν is

the Poisson’s ratio and h is the thickness of the aluminium cell wall. The strain energy

can be expressed as

U =1

2

∫

V

[σ̄xxǭxx + σ̄yy ǭyy + σ̄xyγ̄xy] dV . (3.4)


Substituting for the stresses and strains in the above strain energy equation described

by Eq. (3.4) and integrating with respect to the thickness in the z direction, one has

U =1

2D

∫

[

(

∇2w)2

+ 2(1 − ν)[

(

∂2w

∂x∂y

)2

− ∂2w

∂x2∂2w

∂y2

]]

dxdy .

The kinetic energy of the plate is expressed as

T =1

2ρh

∫ ∫

[

ẇ2]

dxdy , (3.5)

where D = Eh3/(1 − ν2). The kinetic energy of the plate is expressed as

T =1

2ρh

∫ ∫

[

ẇ2]

dxdy , (3.6)

where we have assumed that the bending rotational inertia is negligible. The work done

is expressed as

V = −∫ ∫

p(x, y)wdxdy − M bxx∂w

∂x− V bxzw + M byy

∂w

∂y− V byzw , (3.7)

where the distributed load p is due to dynamic pressure in the cell cavities and M b, V b

are the bending moments and shear forces at the isolated cell boundaries and they are

dynamically equilibriated by the forces from the neighboring cells. Using Hamilton’s

principle, the governing equation of the cell wall is obtained as

D∇2∇2w + ρh∂2w

∂t2= p(x, y) . (3.8)

D∇2∇2w + ρh∂2w

∂t2+ ηh

∂w

∂t= p(x, y) . (3.9)

where we have added a term involving viscous damping factor η per volume. By per-

forming integration by parts we get the boundary conditions

w = w(prescribed) or Vxz = −D[

∂3w

∂x3+ (2 − ν) ∂

3w

∂x∂y2

]

,


Vyz = −D[

∂3w

∂y3+ (2 − ν) ∂

3w

∂y∂x2

]

;

∂w

∂x= θx(prescribed) or Mxx = D

[

∂2w

∂x2+ ν

∂2w

∂y2

]

;

∂w

∂y= θy(prescribed) or Myy = D

[

∂2w

∂y2+ ν

∂2w

∂x2

]

. (3.10)

In the boundary conditions described by Eq. (3.10) the Poisson’s ratio ν enters the

natural boundary conditions (bending moment and shear force) and couples the varia-

tions in x to those in y.

3.0.2 Governing equation for in-plane motion

Gibbs and Craven [18] have shown by using an energy flow method that once a flexural

wave impinges on a junction, some parts are converted into an in-plane wave and it

can travel through several junctions before getting converted back into a flexural wave.

Similar effect in the velocity field in context of connected beams have been discussed

by Mahapatra et.al [6]. Here also, it would be important to include the in-plane waves

while modeling the connected walls of the honeycomb cells.

The stress-strain relations for plane stress case are given by

σ̄xx =E

1 − ν2 [ǫxx + νǫyy] , σ̄yy =E

1 − ν2 [ǫyy + νǫxx] . (3.11)

Substituting these in the strain energy expression (4) yields

U =Eh

2(1 − ν2)

∫ ∫[

ǫ2xx + ǫ2yy + 2νǫxxǫyy +

(1 − ν)2

γ2xy

]

dxdy .

In terms of displacement fields this becomes

U =Eh

2(1 − ν2)

∫ ∫

[

(

∂u

∂x

)2

+

(

∂v

∂y

)2

+ 2ν

(

∂u

∂x

) (

∂v

∂y

)

+(1 − ν)

2

(

∂u

∂y+

∂v

∂x

)2]

dxdy .


Integration by parts (assuming there is no displacement in the y-direction, i.e, v=0)

gives the governing equation as

∂2u

∂x2+

(1 − ν)2

∂2u

∂2y− (1 − ν

2)ρ

E

∂2u

∂t2= 0 (3.12)

with the boundary condition

Nxx =Eh

(1 − ν2)∂u

∂x. (3.13)

Next we incorporate the effect of dynamic pressure fluctuation on the cell wall vibration.

This is described in the following section.

3.0.3 Fluctuation of cell cavity pressure

The frequency response of the fluid coupled wall would be different from that in vacuum.

As the acoustic waves pass through the cells they impart momentum to the fluid (air)

in the cell cavity causing a dynamic pressure fluctuation. Hence, to design the cellular

structure in general, which interact with fluids, and to predict the natural frequencies

and mode shapes of such structures more accurately, one needs to include the cavity

dynamics.

Let us consider that a single wall of a honeycomb cell as considered in sec.2 is im-

mersed in fluid. The constitutive relations for fluid in the frequency spectrum gives the

expression for the dynamic pressure acting normal to the cell wall. Here, we restrict the

plate motion only in x and z-directions and consequently set v = 0. This assumption is

more accurate when face sheets are bonded to both sides of the honeycomb panel. As a

result, the momentum conservation equation will have only u and w terms as follows

∂σxx∂x

+∂σxz∂z

= ρaü ,∂σxz∂x

+∂σzz∂z

= ρaẅ , (3.14)

where ρa is the density of the accoustic medium, σxx, σzz are the normal stresses and σxz


is the shear stress. The constitutive behaviour of the fluid is described by

σxx = −p + 2µ∂u̇

∂x, σzz = −p + 2µ

∂ẇ

∂z, σxz = µ

(

∂u̇

∂z+

∂ẇ

∂x

)

,

p = −B(

∂u

∂x+

∂w

∂z

)

,

where µ is the dynamic viscosity, B is the bulk modulus. The frequency domain coun-

terpart of the above equations can be written as

∂σ̂xx∂x

+∂σ̂xz∂z

= −ρaω2û ,∂σ̂xz∂x

+∂σ̂zz∂z

= −ρaω2ŵ , (3.15)

σ̂xx = −p̂ + 2µiω∂û

∂x, σ̂zz = −p̂ + 2µiω

∂ŵ

∂z, σ̂xz = µωi

(

∂û

∂z+

∂ŵ

∂x

)

, (3.16)

where

p̂ = p̃e−i(kxx+kyy+kzz)eiωt , û = ũe−i(kxx+kyy+kzz)eiωt , ŵ = w̃e−i(kxx+kyy+kzz)eiωt

ω is angular frequency, i =√−1, kx, ky, kz are cartesian components of wave vector

and p̃, ũ , w̃ are the amplitudes of pressure and cell wall displacements in the x and z

-directions, respectively.

At the fluid-wall interface, for continuity to be satisfied, uf=up and wf=wp, where uf ,

wf refer to the fluid displacements and up and wp the wall displacements. Substituting

Eq. (3.15) in Eq. (3.16) and simplifying, we get the expression for dynamic pressure as

p̂ =

[

iρaω2

kzf+

µωk2xkzf

]

ŵ , (3.17)

where kx is the wall wavenumber in x-direction and kzf is the wavenumber in the fluid

normal to the wall (Fig. 3.2) i.e, in the z-direction . Thus, the dynamic pressure is a


function of the transverse displacement amplitude of the fluid-wall interface and the wall

wavenumber. But, in the absence of viscosity, it is independent of the wall wavenumber

and depends only on the flexural displacement ŵ.

Let p̂1, p̂2 be the pressures acting on the two sides of cell wall, which could be the

case when dissimilar fluids/viscoelastic fillers are used in the honeycomb cell. Then the

net pressure ∆p̂ = p̂1 + p̂2. But in the present case, since the cavities are filled with air,

p̂1 = p̂2. Therefore,

∆p̂ = 2

[

iρaω2

kzf+

µωk2xkzf

]

ŵ (3.18)

Thus, the wall vibration in the out-of-plane direction is mainly due to this distributed

load. This expression for dynamic pressure, being function of the plate flexural dis-

placement ŵ, can be substituted in the governing equation (3.8) to solve for the plate

displcement in the transverse direction.

Chapter 4

Wave Propagation in Honeycomb

Cells

4.1 Flexural waves in the wall of the unit hexagonal

cell

In the last chapter we found that the flexural vibration of the cell element is mainly

due to the dynamic pressure which is a function of plate flexural wavenumbers and the

trasnsverse displacement of the fluid-plate interface.

Figure 4.1: A projection of the representative wall, shown in Fig. 3.2, in XZ plane underdistributed load (pressure).

Thus, the plate vibration in the out-of-plane direction is mainly due to this distributed

load. This is substituted in the governing equation Eq. (3.8), the frequency domain

spectral form of which is obtained as

24

Chapter 4. Wave Propagation in Honeycomb Cells 25

D∇4ŵ + (iω)2ρhŵ + (iω)ηhŵ = ∆p̂(x, y) . (4.1)

Substituting for ∆p̂ from Eq.( 3.18) in the above equation and simplifying, we get the

characteristic dispersion equation,

D(k2x + k2y)

2 −[

2µω

kzf

]

k2x +

[

(iω)2ρh + (iω)ηh − 2iρaω2

kzf

]

= 0 , (4.2)

where ky= πn/Ly is the modal wavenumber along the y- direction with n as the mode

number.

The accoustic wavenumber ka in the fluid, is related to the Cartesian components of

the fluid wavenumber as k2xf + k2yf + k

2zf = k

2a, where kxf = πm/Lx, kyf = πn/Ly where

ka is the accoustic wavenumber (longitudinal) in the fluid. The accoustic wavenumbers

for plane waves propagating in the fluid are [13]

ka = ω

√

ρaB + 2µiω

, kas = ω

√

ρaµiω

, (4.3)

Here, the square of shear mode wavenumber k2as, being entirely imaginary, the shear

response will get diffused and localised to the boundaries or where the disturbance is

initiated and hence can be neglected. The longitudinal mode is non-dispersive when

the viscosity of the fluid is negligible. Taking ky = 0 in Eq.( 4.2), the case when whole

honeycomb core is relatively thin and bonded to relatively stiff face sheets on both sides

of the panel and solving for kx gives 4 roots given by,

k1,2,3,4 =

√

µω

kzfD± 1

D+ D

(

ω2ρh − iωηh + 2iρaω2

kzf

)

. (4.4)

These wavenumbers are plotted for different frequencies in Fig. 4.2. Wavenumber dis-

persions shown for the cell filled with water in Fig. 4.3. This result shows the dispersion

behaviour of the cell wall and the viscosity effect in the two different fluids (air and


water). This reveals that in the case of cell wall with air, the dynamic viscosity slightly

decreases the wavenumber for the first flexural wave mode upto a particular frequency

at which there is a sharp rise in the flexural wavenumber followed by usual trend. But,

in the case of water, there is a considerable reduction in the wall flexural wavenumber

due to dynamic viscosity µ upto a particular frequency beyond which it doesnt cause

any further change in the wavenumber compared to that in vacuum and air. Here also,

a sudden rise in the wavenumber at a particular frequency can be seen. The dispersion

curves without viscosity given in ref. [13] exactly matches with the one obtained here.

0 2000 4000 6000 8000 10000−300

−200

−100

0

100

200

300

Frequency [Hz]

Wav

enum

ber

K x [m

−1]

k1 without µ

k1 with µ

k2 with µ

k2 without µ

k1 with or

without µ

k2 with or

without µ

Re [Kx]

Im [Kx]

Figure 4.2: Effect of dynamic viscosity µ on the wavenumber dispersion behaviour ofaluminium wall of honeycomb cell filled with air.

The relation between the wavenumber k and frequency and ω is called the spectrum

relation and is fundamental to the spectral analysis of waves. Figures show the spectrum

relations for the cell element of the hexagonal honeycomb and the effect of dynamic

viscosity effect. The dispersion curves are also given. The relation between phase speed

and frequency is called the dispersion relation. When the phase speed is constant with

respect to frequency the signal is said to be non-dispersive and it will maintain its

superposed shape.

The phase velocity (or phase speed) of a wave is the rate at which the phase of the

wave propagates in space. This is the speed at which the phase of any one frequency

component of the wave travels. For such a component, any given phase of the wave (for


0 5 10 15

x 104

−3000

−2000

−1000

0

1000

2000

3000

Frequency [Hz]

Wav

enum

ber

K x [

m−1]

with µ

without µRe [ k

1]

Im [ k1]

0 5 10 15

x 104

−3000

−2000

−1000

0

1000

2000

3000

Frequency [Hz]

Wav

enum

ber

[m−1]

Re [k2]

Im [ k2]

with µ

without µ

(a) (b)

Figure 4.3: Effect of dynamic viscosity µ on the wavenumber dispersion behaviour forthe aluminium plate element, shown in Fig. 4.1, in water.

example, the crest) will appear to travel at the phase velocity. The phase speed is given

in terms of the wavelength λ and period T as

vp =λT.

Or, equivalently, in terms of the wave’s angular frequency ω and wavenumber k by

vp =ωk.

The group velocity of a wave is the velocity with which the overall shape of the wave’s

amplitudes known as the modulation or envelope of the wave propagates through space.

If ω is directly proportional to k, then the group velocity is exactly equal to the phase

velocity. Otherwise, the envelope of the wave will become distorted as it propagates.

4.2 In-plane waves in the cell wall

In-plane motion of the plate

As in the case of plate flexural displacement, to get the in-plane displacement of the plate

û in x-direction, one has to solve the governing equation for in-plane motion( Eq. 3.12),


the frequency domain spectral form of which is

(−ikx)2 +1 − ν

2(−iky)2 −

ρ(1 − ν2)E

(iω)2 = 0 (4.5)

⇒ kx =√

ρω2(1 − ν2)E

−(1 − ν)k2y

2(4.6)

Again, as in the previous case, when ky = 0, the expression for wavenumber due to

in-plane vibration is

kp =

√

ρω2(1 − ν2)E

(4.7)

Here, the notation kp is used just to distinguish the in-plane wavenumber in x-direction

from the flexural wavenumber component in the same direction. Once the wavenumber

is obtained, the solution can be written as

u(x, t) =∑

n

∑

m

[

E ′e−ikpx + F ′e+ikpx]

e−ikyxeiωt , (4.8)

where E ′ and F ′ are the spatially dependent Fourier coefficients for the forward-moving

and backward-moving waves. After integrating over the depth of the wall to get average

effect in y-direction and converting into the frequency domain form, we get the in-plane

displacement of the wall as

û(x, ω) = Ee−ikpx + Fe+ikpx, (4.9)

where E and F are the new coefficients obtained by multiplying E ′and F ′ by βi. where

β = (e−ikyL − 1)/(Lyky) . Thus, the flexural displacement ŵ of the plate is mainly dueto a differential pressure ∆p̂ developed due to cavity dynamics and acting normal to

the wall whereas the in-plane displacement û is due to the flexural displacement of the

connected walls.



where E and F are the new coefficients obtained by multiplying E ′and F ′ by βi. Now, we

have the flexural displacement ŵ and in-plane displacement û of the cell wall as functions

of the wall wavenumbers for the corresponding modes. Since the wavenumbers depend

on frequency, the displacements are also functions of frequency. We will find these

displacements in the next chapter by dynamic analysis using Spectral Finite Element

Method.

Chapter 5

Spectral FEM based Analysis of

Honeycomb Panels

In this section, we find the dynamic response of the honeycomb panel using a method

called Frequency Domain Spectral FEM. The frequency response of the honeycomb panel

is studied by determining the dynamic stiffness of a unit cell first and assembling them

to get the global stiffness for the whole panel. Since the wavelength is short at higher

frequencies FEM is not a perfect method to study the behaviour of the structure. So a

Frequency Domain Spectral FEM is used here to analyse the dynamic response.

5.1 Frequency Domain Spectral Finite Element For-

mulation

This is a matrix methodology for use on computers to efficiently handle wave propogation

problems in structures with complicated boundaries and discontiunuities. This is very

similar to the finite element method with the only difference that the matrix is in the

frequency domain. This allows the inertia of the distributed mass to be described exactly

and hence the spectral formulation of the elements exactly describe the wave propogation

dynamics.

The main difference between these two methods is that the standard FEM demands

30

Chapter 5. Spectral FEM based Analysis of Honeycomb Panels 31

the discritization of the structure into a number of smaller elements at higher frequencies

to get accurate results whereas mesh refinement is not required in the case of Frequency

Domain Spectral FEM. This approach provides the advantages of both the spectral

analysis method and finite element method.

5.1.1 Dynamic stiffness of a wall of the unit hexagonal cell

Here, we will find the nodal displacements of the unit hexagonal cell from the force-

stiffness relation of the element. Each element of the cell has 3 Degrees Of Freedom

(D.O.F.) u and w along x-axis and z-axis directions and θ about y-axis directions in the

local co-ordinate systems. The flexural displacement w is obtained from the solution of

the governing equation Eq. (4.2)

w(x, y, t) =∑

n

∑

m

[

A′e−ik1x + B′e−ik2x + C ′e+ik1x + D′e+ik2x]

e−ikyxeiωt ,

where A′, B′, C ′ and D′ are spatially dependent Fourier coefficients corresponding to a

forward-moving wave, a backward-moving wave and their respective evanescent waves.

So, ŵ is a function of both x and y. Consider the solution as a plane wave in x modified

in y to eliminate y dependency and to get an average effect along the y-direction. To get

this, the expression is integrated over the depth of the plate in the y-direction.

w(x, y, t) =1

Ly

∫ Ly

0

(

A′e−ik1x + B′e−ik2x + C ′eik1x + D′eik2x)

e−ikyyeiωtdy

Upon integration we get

w(x, y, t) = i(


eiωtγ .

Frequency domain spectral form of this can be written as

ŵ(x, y, ω) = i(


γ ,


where γ = (e−ikyL − 1)/(Lyky) .This can be further simplified to the form

ŵ(x, y, ω) =(

Ae−ik1x + Be−ik2x + Ceik1x + Deik2x)

, (5.1)

where A,B,C and D are the coefficients obtained by multiplying A′, B′, C ′ and D′ by γi.

This gives the solution for flexural-displacement in the frequency domain spectral form.

As in the case of plate flexural displacement, the in-plane displacement of the plate u

in x-direction is of the following form

u(x, t) =∑

n

∑

m

[

E ′e−ikpx + F ′e+ikpx]

e−ikyxeiωt , (5.2)

where E ′ and F ′ are the spatially dependent Fourier coefficients for the forward-moving

and backward-moving waves. After integrating over the depth of the plate to get average

effect in the y-direction and converting into the frequency domain spectral form, we get

the in-plane displacement of the plate as


where E and F are the new coefficients obtained by multiplying E ′and F ′ by γi.

The axial and flexural displacement fields can be written in the matrix form as

{

û

ŵ

}

2×1

= [P ]{a} , (5.4)

where

[P ] =

[

e−ikpx 0 0 e+ikpx 0 0

0 e−ik1x e−ik2x 0 e+ik1x e+ik2x

]

, (5.5)

and

{ a } = {E A B F C D }T . (5.6)


The displacement interpolation functions given by Eqs. (5.1) and (5.3) as functions of

x only. We consider a representative wall of the honeycomb cell as shown in Fig. 3.1(b).

Let us introduce two nodes at the wall edges x = 0 and x = L (where L=4 mm in the

present problem) in the local coordinate system shown in Fig. 4.1. The nodal degrees of

freedom are taken as u1, w1, θ1 and u2, w2, θ2 at the nodes 1 and 2 respectively as shown

in Fig. (4.1). The nodal degrees of freedom in the frequency domain spectral form are

û1 = û(0, ω), ŵ1 = ŵ(0, ω), θ̂1 = θ̂(0, ω), û2 = û2(L, ω), ŵ2 = ŵ2(L, ω), θ̂2 = θ̂2(L, ω), .

Expressing the above nodal degrees of freedom in the matrix form, we get

û1

ŵ1

θ̂1

û2

ŵ2

θ̂2

=

1 0 0 1 0 0

0 1 1 0 1 1

0 −ik1 −ik2 0 ik1 ik2e−ipL 0 0 eipL 0 0

0 e−ik1L e−ik2L 0 eik1L eik2L

0 −ik1e−ik1L −ik2e−ik2L 0 ik1eik1L ik2eik2L

E

A

B

F

C

D

(5.7)

Using Eq.( 5.4) and Eq.( 5.7)we get the following expression

{

û

ŵ

}

= [P ][G]−1{d̂} (5.8)

where [P ][G]−1 is the element interpolation function matrix [N ], G is the 6×6 matrix inEq.( 5.7) and d̂ is the nodal displacement vector. The above expression can be written

as

{

û

ŵ

}

=

[

N̂1 0 0 N̂4 0 0

0 N̂2 N̂3 0 N̂5 N̂6

]

û1

ŵ1

θ̂1

û2

ŵ2

θ̂2

, (5.9)

where N̂1, N̂4 are the frequency dependent interpolation functions for the in-plane


displacements (û1, û2) and N̂2, N̂5 are those corresponding to the transverse displace-

ments (ŵ1, ŵ2) whereas N̂3 and N̂6 refer to the slopes (θ̂1, θ̂2) at the element nodes 1

and 2 respectively. These shape functions are plotted with respect to the wall coordinate

x/Lx and shown in Fig. 5.1. From this figure, one can observe that all shape functions

are having values 1 at their respective nodes and 0 at the other nodes, thus satisfying the

essential boundary conditions at the nodal points. Since these interpolation functions

depend on frequency they exhibit a change in the shape with frequencies. The curves (see

Fig. 5.1(b),(c)and (d)) show the dynamic behaviour of the shape functions at different

frequencies.

Now, we have the displacement field in-terms of element nodal degrees of freedom and

element interpolation functions. To get the dynamic stiffness matrix, we impose natural

boundary conditions (force and bending moment) at each node and in three degrees of

freedom. Assuming nodal forces and moments at each of the two nodes as F1, V1, M1

and F2, V2, M2, respectively. Since the displacement function is function of only one

space dimension in x, the nodal forces and moments can be written as

F̂xx = C∂ [̂N ]

∂x{d̂} , V̂xz = −D

∂3 ˆ[N ]

∂x3

{

d̂}

, M̂xx = D∂2 ˆ[N ]

∂x2

{

d̂}

, (5.10)

where C = Eh/(1 − ν2), F̂1, V̂1, M̂1 are the axial force, shear force and bending mo-ment, respectively, at node 1 (i.e., x = 0), which can be evaluated from the following

expressions.

F̂1 = C

[

∂N̂1∂x

∂N̂4∂x

]

{d̂} ,

V̂1 = −D[

∂3N̂2∂x3

∂3N̂3∂x3

∂3N̂5∂x3

∂3N̂6∂x3

]

{d̂} ,

M̂1 = D

[

∂2N̂1∂x2

∂2N̂2∂x2

∂2N̂3∂x2

∂2N̂4∂x2

]

{d̂} .

similarly we can get for F̂2 ,V̂2, and M̂2 at x=L. Presenting these equations in a matrix

form with appropriate notations, we get,


0 1 2 3 4

x 10−3

−0.2

0

0.2

0.4

0.6

0.8

1

x/L

Sha

pe F

unct

ions

N4

N5

N2

N1

N6x200

N3x200

0 1 2 3 4

x 10−3

−2

−1

0

1

2

x/L

Sha

pe fu

nctio

n,N

1

10kHz400kH800kHz1.5MHz

(a) (b)

0 1 2 3 4

x 10−3

−6

−4

−2

0

2

4

6

Frequency [Hz]

Sha

pe fu

nctio

n,N

2

10kHz400kHz800kHz1.5MHz

0 1 2 3 4

x 10−3

−1

−0.5

0

0.5

1x 10

−3

x/L

Sha

pe fu

nctio

n,N

3

10kHz400kHz1.5MHz

(c) (d)

Figure 5.1: (a) Shape functions for the representative plate element shown in Fig. 4.1.(b),(c) and (d) Dynamic behaviour of shape functions N1 ,N2 and N3 respectively.

F̂1

V̂1

M̂1

F̂2

V̂2

M̂2

6×1

=

k11 k12 k13 k14 k15 k16

k21 k22 k23 k24 k25 k26

k31 k32 k33 k34 k35 k36

k41 k42 k43 k44 k45 k46

k51 k52 k53 k54 k55 k56

k61 k62 k63 k64 k65 k66

6×6

d̂1

d̂2

d̂3

d̂4

d̂5

d̂6

6×1

, (5.11)

where k11 = C(∂N̂1/∂x), k14 = C(∂N̂4/∂x), k12 = k13 = k15 = k16 = 0, k22 =

−D(∂3N̂2/∂x3), k23 = −D(∂3N̂3/∂x3), k25 = D(∂3N̂5/∂x3), k26 = D(∂3N̂6/∂x3), k21 =k24 = 0, k32 = −D(∂2N̂2/∂x2), k33 = −D(∂2N̂3/∂x2), k35 = −D(∂2N̂5/∂x2), k36 =


−D(∂2N̂6/∂x2) and k31 = k34 = 0. Similarly all other coefficients can be written. Thismatrix form can be written in the compact form using notations as

[

K̂(ω)]{

d̂(ω)}

={

f̂(ω)}

(5.12)

where [K̂] is the dynamic stiffness matrix and {f̂} denotes the nodal force vector of acell wall element.

5.1.2 Transformation of nodal vectors from cell wall co-ordinate

to honeycomb panel coordinate

Once the stiffness matrix for an element is evaluated, we need to assemble the matrices

for all the elements after transforming them into the global co-ordinate system to get the

dynamic stiffness for the entire unit hexagonal cell given in 5.2. The assembling of the

individual element stiffness matrices is implemented by creating global stiffness matrix

accommodating all degrees of freedom in the unit cell, this is of the order of 30×30. Thereare 10 nodes in the unit cell and each node has three degrees of freedom. The global

stiffness matrix of the unit cell is first initialized to zero. Next the stiffness matrices of

individual elements are inserted into the global stiffness matrix according to its global

node numbering. For example, element 2 has node numbers 3 and 1, the first node

number 3 leads to global degrees of freedom of 5 and 6, similarly second node number 1

leads to global degrees of freedom 1 and 2. Therefore according to these global degrees of

freedom numbers the transformed stiffness matrix of element 2 is inserted into the global

stiffness matrix. Similarly all the elements’ transformed stiffness matrices are assembled

into the global stiffness matrix. At the common nodes there will be contribution from

more than one elements so these are added together, for example at node 3 there will

be contribution form elements 1,2 and 3, so all these terms are added. Thus we obtain

dynamic stiffness matrix of a unit cell of honeycom panel.


1

2 3

4

5

6

7

89

10

1

3

4 6

7

8

9

l

5 x

z

2α

2

1

x−

−y

Figure 5.2: Unit cell of honeycomb panel, the circled numbers represents element num-

bers and remaining plain numbers represent global node numbers, small numbers 1 and 2

on the element 2 represent local node numbering.x̄,ȳ represents global coordinate system

and x,y represents local coordinate system.

We have already evaluated, in the previous section, the dynamic stiffness matrix of

one face of the honeycomb unit cell. By using this we can determine dynamic stiffness

matrix of the honeycomb unit cell. The schematic diagram of the honeycomb unit cell

is shown in the Fig. (5.2), it consists of 9 elements (circled numbers in the figure) and

10 nodes. The connectivity table of these elements according to global numbering is

given in the following table. In the above table l is the length of the element, A is

Table 5.1: Connectivity table for elements of honeycomb unitcell Ref.fig. 5.2

Element No. Global nodal dof Geometric properties Orientation1 2 3 A, l=L/2 α = 02 3 1 A, l=L/2 α = 603 3 4 A, l=L α = 3004 5 4 A, l=L/2 α = 2405 4 6 A, l=L α = 06 6 7 A, l=L/2 α = 3007 6 8 A, l=L α = 608 8 9 A, l=L/2 α = 09 8 10 A, l=L/2 α = 120


cross-sectional are and α is the orientation of the element. α is calculated according to

local node numbering.

After evaluating stiffness matrix for each of these element in the local coordinate

system according to their geometry, we transform them into global coordinate system

by using transformation matrix. Once we get the stiffness matrix for all the elements we

can assemble them according to the connectivity of the elements.

The transformation matrix is evaluated using the following relation between local

co-ordinate and global co-ordinate system.

ū1

w̄1

θ̄1

ū2

w̄2

θ̄2

=

cos α sin α 0 0 0 0

− sin α cos α 1 0 0 00 0 1 0 0 0

0 0 0 cos α sin α 0

0 0 0 − sin α cos α 00 0 0 0 0 1

u1

w1

θ1

u2

w2

θ2

. (5.13)

Here nodal degrees of freedom with a bar on top indicates them as global representation.

We can find the dynamic stiffness matrix of all elements of the unit cell (see Fig. (5.2))

by using the relations given in Eq. (5.11). These stiffness matrices of all the elements

can be transformed to the global coordinate system X̄Ȳ Z̄ (see Fig. (5.2)) by multiplying

with transformation matrix given in Eq. (5.13). All these element stiffness matrices in

the global coordinate system are assembled according to the global nodal numbering of

respective element as shown in the connectivity table. 5.1.2. The unit hexagonal cell has

9 elements and 10 nodes and each node has 3 degrees of freedom. Therefore the size

of the global stiffness matrix is 30 × 30, and force and displacement vectors are of size30 × 1 each.


5.2 Consistent foce vector due to dynamic pressure

Since the dynamic pressure due to fluid in the cell this is acting through the thickness

of the wall in the z-direction, this can be taken as distributed load. For the dynamic

analysis, the equivalent forces and moments at the nodes 1 and 2 are required to be

calculated. These are determined as follows. The flexural vibration of the cell walls are

mainly due to the differential pressure ∆p(x) acting on respective walls. The work done

by the distributed load ∆p̂(x) is given by

∫ Lx

0

∆p̂(x)Lyŵ(x)dx = { d̂ }T∫ Lx

0

[N̂(x)]T ∆p̂(x)Lydx = { d̂ }T { f̂ } (5.14)

where {f̂} is the consistent nodal force vector, {d̂} is the displacement vector and [N̂(x)]is the shape function corresponding to w(x). From Eq.( 5.9), we have

[N(x)] = [ 0 N̂2 N̂3 0 N̂5 N̂6 ] (5.15)

Substituting N(x) from Eq. (5.9) and ∆p̂ from Eq. (3.18) in Eq. (5.14) we get the

consistent nodal load vector as

{ f̂ } =

0

a1ŵ1 + b1θ̂1 + c1ŵ2 + d1θ̂2

a2ŵ1 + b2θ̂1 + c2ŵ2 + d2θ̂2

0

a3ŵ1 + b3θ̂1 + c3ŵ2 + d3θ̂2

a4ŵ1 + b4θ̂1 + c4ŵ2 + d4θ̂2

(5.16)

where

a1 = Lyγ

∫ Lx

0

N̂21 (x)dx , b1 = Lyγ

∫ Lx

0

N̂1(x)N̂2(x)dx ,

a2 = Lyγ

∫ Lx

0

N̂1(x)N̂2(x)dx , b2 = Lyγ

∫ Lx

0

N̂22 (x)dx ,

a3 = Lyγ

∫ Lx

0


∫ Lx

0

N̂2(x)N̂3(x)dx ,


a4 = Lyγ

∫ Lx

0


∫ Lx

0

N̂2(x)N̂4(x)dx ,

c1 = Lyγ

∫ Lx

0

N̂1(x)N̂3(x)dx , d1 = Lyγ

∫ Lx

0

N̂1(x)N̂4(x)dx ,

c2 = Lyγ

∫ Lx

0


∫ Lx

0

N̂2(x)N̂4(x)dx ,

c3 = Lyγ

∫ Lx

0

N̂23 (x)dx , d3 = Lyγ

∫ Lx

0

N̂3(x)N̂4(x)dx ,

c4 = Lyγ

∫ Lx

0


∫ Lx

0

N̂24 (x)dx ,

and γ = [iρaω2/kzf + µωk

2x/kzf ].

Thus, the nodal loads are obtained as functions of nodal displacements and interpo-

lation functions. Integrations of interpolation functions are done using symbolic manip-

ulation in MAPLE. Now, we have the foce-displacement relation for a single wall of the

unit cell as

k11 k12 k13 k14 k15 k16

k21 k22 k23 k24 k25 k26

k31 k32 k33 k34 k35 k36

k41 k42 k43 k44 k45 k46

k51 k52 k53 k54 k55 k56

k61 k62 k63 k64 k65 k66

6×6

û1

ŵ1

θ̂1

û2

ŵ2

θ̂2

6×1

=

0

a1ŵ1 + b1θ̂1 + c1ŵ2 + d1θ̂2

a2ŵ1 + b2θ̂1 + c2ŵ2 + d2θ̂2

0

a3ŵ1 + b3θ̂1 + c3ŵ2 + d3θ̂2

a4ŵ1 + b4θ̂1 + c4ŵ2 + d4θ̂2

6×1

(5.17)

To analyze the frequency response of the flexural mode of a cell wall, Eq. 5.17 is solved,

as an example, by fixing one end of the wall, and by applying a load of 1 µN at the

other end in the Z−direction. Frequency response shown is shown in Fig. ?? whichreveals the effect of dynamic pressure on the amplitudes of vibration. First curve shows

the response when the wall is in air and the second one shows the dynamic behaviour

when it is in vacuum. It clearly shows that the cavity dynamics does play a role on the

dynamic response. Similar to the assembly of stiffness matrices explained in the previous


section, the load vector can also be assembled to get the force-displacement relationship

for the whole unit cell in the global co-ordinate system.

Now to get the dynamic response of the unit cell at any nodal point on the boundary

first we fix some nodes of the cell Fig. 5.2(a) and load is applied at some other nodes as

explained in the following section. To get the axial response in the y-direction, we fix

nodes 1 and 10 and a load of 1 µN is applied in the y-direction at each node 5 and 7.

Similarly, to get the shear response of the cell in the x-direction, it is fixed at the same

nodes 1 and 10 and an axial force of 1 µN is applied at nodes 5 and 7 in the x-direction.

These reponses are plotted with respect to frequency in Fig. 5.5 The third plot shows

the combined effect of axial load, transverse load as well as moment on the FRF of the

unit cell.

102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

u2 in vacuum

u5 in vacuum

u2 in air

u5 in air

102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

w2 in vacuum

w5 in vacuum

w2 in air

w5 in air

(a) (b)

Figure 5.3: Frequency response of the unit hexagonal cell (Fig. 5.2), in air with dynamicviscosity and in vacuum, due to a combined periodic load of 1µN(axial and transverse)and 1µNm (moment) applied at nodes 5 and 6, (a) in-plane response u2, u5 and b)flexural response w2 ,w5 at nodes 2 and 5 respectively.


102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

w2 in air (viscous)

w2 in vacuum

w2 in air (invscid)

102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

w5in air (viscous)

w5 in vacuum

w5 in air (inviscid)

(a) (b)

Figure 5.4: (a), (b) Frequency response of the unit hexagonal cell (see Fig. 5.2) due toa periodic load of 1µN applied in the X− direction, at nodes 5 and 6, and the effect ofdynamic viscosity on the displacements at a) node 2 and b) node 5.

5.3 Evaluation of Dynamic Stiffness Matrix and Dy-

nanamic Response of the whole Honeycomb panel

The procedure for finding global stiffness matrix for honeycomb unit cell is described

in the previous section. By using the stiffness matrix of the unit cell we can find the

stiffness matrix of whole honeycomb panel shown in Fig. 3.1. For an illustration let us

assume the panel is composed of nc unit cells along X− axis direction and nr cells alongthe Z− axis direction in the Fig. 3.1. We have to assemble the stiffness matrices forall these cells. Since all the cells exactly resemble the unit cell we have considered, the

stiffness matrix for all the cells is same and this can be evaluated through the procedure

explained in the last section.

From the descritization figure (Fig. 3.1) we can see that for each cell assembled in

the X−axis direction there is only one common node. Therefore for nc cells there willbe (nc − 1) common nodes.From the same figure 3.1, for each element connected inthe Z− direction there are two common nodes (c2 and c3 in the Fig 3.1). Thereforefor nr cells in a column there will be 2(nr − 1) number of common nodes. Thus thetotal number of common nodes is 2(nr − 1)+(nc − 1). Since each cell has 10 nodes,


102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

w

2 in air

w5 in air

w2 in vacuum

w5 in vacuum

102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

u2 in air

u5 in air

u2 in vacuum

u5 in vacuum

(a) (b)

Figure 5.5: (a), (b) Frequency response of the unit hexagonal cell (Fig. 5.2) in air withdynamic viscosity and in vacuum, due to a periodic load of 1µN applied in the X−direction at nodes 5 and 6, a) flexural response w2, w5 and b) in-plane response u2, u5at nodes 2 and 5 respectivley.

for nr rows and nc columns we have a total of 10nrnc nodes and each node has 3 de-

grees of freedom. Therefore the size of global stiffness matrix of whole honeycomb core is

[3 × 10nrnc − 3 × (2(nr − 1) + (nc − 1))]×[3 × 10nrnc − 3 × (2(nr − 1) + (nc − 1))] Thusby assembling the honeycomb unit cell in X− and Z− direction we get the assembledglobal stiffness matrix of complete honeycomb beam. Representation of whole honey-

comb beam nodal loads, stiffness matrix and nodal degrees of freedom is shown below

[

K̄] {

d̄}

={

F̄}

, (5.18)

Here [K̄] represents the global stiffness matrix of complete honeycomb panel. F̄ is the

global load matrix and d̄ is the global nodal degrees of freedom matrix.

5.4 Model Order Reduction

A reduced order model derived from finite element model is very useful for the efficient

handling of the descritized computational model. The methods and applications of

reduced order models in high frequency vibrations have been reported in ref. [10].


102

103

104

105

10−25

10−20

10−15

10−10

10−5

10−15

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

u

2 in air

u5 in air

u2 in vacuum

u5 in vacuum

102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

w

2 in air

w5 in air

w2 in vacuum

w5 in vacuum

(a) (b)

Figure 5.6: Frequency Response of the unit hexagonal cell (Fig. 5.2) due to a unit periodicload of 1µN applied in the Z− direction at nodes 5 and 6, (a)in-plane response u2, u5and (b) flexural response w2, w5 at nodes 2 and5 respectively.

Hence, we reduce the size of the global stiffness matrix into only those contain external

nodal degrees of freedom (indicated in the Fig. 3.3 with e). This process is called dynamic

condensation. Assume the internal nodal degrees of freedom as slave degrees of freedom

(d̄s) and external nodal degrees of freedom as master degrees of freedom (d̄m). Similarly,

nodal loads corresponding to slave degrees of freedom are represented with F̄s and nodal

loads corresponding to master degrees freedom are represented with F̄m. With these

notations Eq. (5.18) can be rewritten as

{

K̄mm K̄ms

K̄sm K̄ss

} {

d̄m

d̄s

}

=

{

F̄m

F̄s

}

(5.19)

Above Eq. (5.19) can be written as two individual equations, one is for master loads

(Fm) and other for slave loads (Fs), these are given below

Kmmdm + Kmsds = Fm , (5.20)

Ksmdm + Kssds = Fs . (5.21)


102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

w2 in air, h=h

2

w2 in vacuum, h=h2

w2 in air, h=h

1

w2 in vacuum, h=h1

102

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

u2 in air, h=h

2

u2 in vacuum, h=h

2

u2 in air, h=h

1

u2 in vacuum, h=h

1

(a) (b)

Figure 5.7: Effect of wall thickness h on the frequency response of the unit cell (seeFig. 5.2) at (a) node 2 and (b) node 5 due to a combined unit periodic load of 1µN(axial and transverse) and 1µNm (moment) acting at nodes 5 and 6. h1 = 0.00025mmand h2 = 0.00050mm.

The Eq. (5.21) can be written as

⇒ ds = −K−1ss (Fs − Ksmdm)

plugging this ds into Eq. (5.20) will results in,

Fm =(

Kmm − KmsK−1ss Ksm)

dm + KmsK−1ss KsmFs .

Thus whole system of equations in the Eq. (5.18) is condensed to only master degrees of

freedom, this is equal to the external nodal degrees of freedom (External nodal degrees

of freedom are shown in the Fig. 3.1(a) with a symbol ’e’). From the Fig. 3.1(a) for each

column (along Z direction) there are 4 external nodes (two on the top edge of the beam

and two on the bottom edge of the plate) and for each row (along X direction) there

are 2 external nodes (one on the left side of the beam and the other on the right side of

the beam for the configuration shown in figure 3.3)(a). Therefore the total number of

master degrees of freedom for a beam consisting of nc columns and nr rows of honeycomb

unit cell is (4 ∗ nc + 2 ∗ nr) ∗ 3 and therefore the size of the condensed dynamic stiffness


103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

u2 in vacuum, h=h

2

u2 in air, h=h

2

u2 in vacuum, h=h1

u2 in air, h=h

1

103

104

105

10−25

10−20

10−15

10−10

10−5

100

Frequency [Hz]

Nod

al d

ispl

acem

ents

[m]

u5 in vacuum, h= h

2

u5 in air, h=h

2

u5 in vacuum, h= h

1

u5 in air, h=h1

(a) (b)

Figure 5.8: Effect of wall thickness h on the frequency response of the unit hexagonalcell ( see Fig. 5.2) at (a) node 2 and (b) node 5 due to a flexural periodic load of 1µNapplied in Z-direction at nodes 5 and 6. h1 = 0.00025mm and h2 = 0.00050mm.

matrix is ((4 ∗ nc + 2 ∗ nr) ∗ 3) × ((4 ∗ nc + 2 ∗ nr) ∗ 3). The final condensed system ofequations are represented as

[Ke] {de} = {Fe} . (5.22)

Using this system of equations by applying proper boundary conditions we can find

the dynamic response of the beam at any boundary point. The assembled matrices

are straight forward to deal with using symbolic manipulation packages like MAPLE.

Further numerical calculations are done using a MATLAB code.

5.5 Simulation Results and Discussions

The effect of dynamic pressure on the resonance behaviour of the honeycomb cell is

shown in Fig. 5.3 and Fig. 5.5. From these plots one can understand very clearly that

the dynamic loading reduces the resonance in the low frequency range but increases

drastically as the frequency increases. This may be due to the inertia of the fluid mass

present in the cell cavity which offers resistance to vibration at low frequenies. But

as the frequency goes higher the fluid mass gain momentum and start impinging on


the walls and attribute to the increase in the resonance after a certain frequency limit.

Here, we find that this limit is nearly 1kHz which is nothing but that corresponds to the

point of intersection between the two curves -first one representing resonance of the cell

with dynamic loading and the second corresponding to the resonance in the absence of

dynamic pressure ie,in vacuum.

The plots shown in Fig. 5.4 explain influence of dynamic viscosity on the dynamic

response of the cell. The dynamic viscosity is found signifcant only in the lower frequency

range and a sudden drop in the nodal displacement is observed at a frequency of nearly

10kHz. The two curves, one with viscosity and the other without viscosity, clearly

show this behaviour. The difference between the two curves narrows down as frequency

increases and beyond a certain value there is found to be no difference between the two

resonance natures-with viscosity and without viscosity.

The effect of thickness on the FRF of the cell is also shown in Fig. 5.8 and Fig. 5.7.

These are plotted with flexural and combined (axial, flexural and moments) loads. As

the thickness vary a change in amplitude of the nodal displacements and a shift in the

natural frequencies are observed.

Chapter 6

Experimental Studies using

Patterned Honeycomb Panels

Analytical formulation and simulation has shown that there exists a locally resonant

band-

Wavepropagation and Resonance in Honeycomb Panels...

Documents

Transcript of Wavepropagation and Resonance in Honeycomb Panels...