Lecture Notes in Mechanical Engineering · 2020. 4. 4. · Free Vibration Analysis of Laminated Composite Plates and Shells Subjected to ... by Tension and Shear Cracks in RCC Beams

Lecture Notes in Mechanical Engineering

B. N. SinghArnab RoyDipak Kumar Maiti Editors

Recent Advances in Theoretical, Applied, Computational and Experimental MechanicsProceedings of ICTACEM 2017

Lecture Notes in Mechanical Engineering

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B. N. Singh • Arnab Roy •Dipak Kumar MaitiEditors

Recent Advancesin Theoretical, Applied,Computationaland Experimental MechanicsProceedings of ICTACEM 2017

123

EditorsB. N. SinghDepartment of Aerospace EngineeringIndian Institute of Technology KharagpurKharagpur, West Bengal, India

Arnab RoyDepartment of Aerospace EngineeringIndian Institute of Technology KharagpurKharagpur, West Bengal, India

Dipak Kumar MaitiDepartment of Aerospace EngineeringIndian Institute of Technology KharagpurKharagpur, West Bengal, India

ISSN 2195-4356 ISSN 2195-4364 (electronic)Lecture Notes in Mechanical EngineeringISBN 978-981-15-1188-2 ISBN 978-981-15-1189-9 (eBook)https://doi.org/10.1007/978-981-15-1189-9

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https://doi.org/10.1007/978-981-15-1189-9

Contents

Multi-scale Simulation of Elastic Waves Containing HigherHarmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Ambuj Sharma, Sandeep Kumar and Amit Tyagi

Effect of Skewness on Random Frequency Responsesof Sandwich Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13R. R. Kumar, Vaishali, K. M. Pandey and S. Dey

Design and Simulation of 3-DoF Strain Gauge Force Transducer . . . . . 21Ankur Jaiswal, H. P. Jawale and Kshitij Shrivastava

A Micromechanical Study of Fibre-Reinforced Compositeswith Uncertainty Quantification and Statically Equivalent RandomFibre Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37S. Koley, P. M. Mohite and C. S. Upadhyay

Free Vibration Analysis of Laminated Composite Plates and ShellsSubjected to Concentrated Mass at the Centre . . . . . . . . . . . . . . . . . . . . 49Arpita Mandal, Chaitali Ray and Salil Haldar

Buckling Analysis of Thick Isotropic Shear Deformable Beams . . . . . . . 59Kedar S. Pakhare, Rameshchandra P. Shimpi and P. J. Guruprasad

Spectral Finite Element for Dynamic Analysis of PiezoelectricLaminated Composite Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Namita Nanda

Determination of Interlaminar Stress Componentsin a Pretwisted Composite Strip by VAM . . . . . . . . . . . . . . . . . . . . . . . 81Santosh B. Salunkhe and P. J. Guruprasad

A Study on Wrinkling Characteristics of NBR Material . . . . . . . . . . . . 109Vaibhav S. Pawar, Rajkumar S. Pant and P. J. Guruprasad

v

First Ply Failure Study of Laminated Composite Conoidal ShellsUsing Geometrically Nonlinear Formulation . . . . . . . . . . . . . . . . . . . . . 119Kaustav Bakshi and Dipankar Chakravorty

Analysis of Transformed Sixth-Order Polynomial for the ContractionWall Profile by Using OpenFOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133R. Lakshman and Ranjan Basak

Fatigue Life Assessment of an Existing Railway Bridge in IndiaIncorporating Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Mrinal Chanda, Kishore Chandra Misra and Soumya Bhattacharjya

Numerical Simulation of Acoustic Emission Waveforms Generatedby Tension and Shear Cracks in RCC Beams . . . . . . . . . . . . . . . . . . . . 155Arun Roy, Paresh Mirgal and Sauvik Banerjee

Applicability of Tricycle Modelling in the Simulation of AircraftSteering System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171S. Sathish, L. Suryanarayanan, J. Jaidev Vyas and G. Balamurugan

Free Vibration and Stress Analysis of Laminated Box Beamwith and Without Cut-Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185Raj B. Bharati, Prashanta K. Mahato, E. Carrera, M. Filippi and A. Pagani

Free Vibration Analysis of the Functionally Graded Porous CircularArches in the Thermal Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Mohammad Amir and Mohammad Talha

Vibration Response of Shear Deformable Gradient Platewith Geometric Imperfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Ankit Gupta and Mohammad Talha

Characterization of 2D Nanomaterials for Energy Storage . . . . . . . . . . 221Akarsh Verma and Avinash Parashar

Cold Expansion of Elongated Hole: A Realistic Finite ElementSimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227S. Anil Kumar and N. C. Mahendra Babu

Effect of Module on Wear Reduction in High Contact Ratio SpurGears Drive Through Optimized Fillet Stress . . . . . . . . . . . . . . . . . . . . 239R. Ravivarman, K. Palaniradja and R. Prabhu Sekar

Force Estimation on a Clamped Plate Usinga Deterministic–Stochastic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Akash Shrivastava and Amiya R. Mohanty

vi Contents

Dynamic Analysis of Composite Cylinders Using 3-D DegeneratedShell Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261Pratik Tiwari, Dipak Kumar Maiti and Damodar Maity

A New Hybrid Unified Particle Swarm Optimization Techniquefor Damage Assessment from Changes of Vibration Responses . . . . . . . 277Swarup K. Barman, Dipak Kumar Maiti and Damodar Maity

Semi-active Control of a Three-Storey Building Structure . . . . . . . . . . . 297P. Chaudhuri, Damodar Maity and Dipak Kumar Maiti

A Direction-Based Exponential Crossover Operator for Real-CodedGenetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311Amit Kumar Das and Dilip Kumar Pratihar

Axial Deformation Characteristics of Graphene-Sonicated Vinyl EsterNanocomposites Subjected to High Rate of Loading . . . . . . . . . . . . . . . 325B. Pramanik, P. R. Mantena and A. M. Rajendran

State Estimation Using Filtering Methods Applied for AircraftLanding Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339P. S. Suresh, Niranjan K. Sura and K. Shankar

Numerical Solution of Steady Incompressible Flow in a Lid-DrivenCavity Using Alternating Direction Implicit Method . . . . . . . . . . . . . . . 353Banamali Dalai and Manas Kumar Laha

Stagnation and Static Property Correlations for EquilibriumFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365Shubham Maurya and Aravind Vaidyanathan

CFD Simulation of Hypersonic Shock Tunnel Nozzle . . . . . . . . . . . . . . . 381Jigarkumar Sura

A DNS Study of Bulk Flow Characteristics of a Transient DiabaticPlume that Simulates Cloud Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387Samrat Rao, G. R. Vybhav, P. Prasanth, S. M. Deshpandeand R. Narasimha

Transverse-Only Vibrations of a Rigid Square Cylinder . . . . . . . . . . . . 397Subhankar Sen

Steady Flow Past Two Square Cylinders in Tandem . . . . . . . . . . . . . . . 407Deepak Kumar, Kumar Sourav and Subhankar Sen

A Robust and Accurate Convective-Pressure-Split ApproximateRiemann Solver for Computation of Compressible HighSpeed Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415Sangeeth Simon and J. C. Mandal

Contents vii

Numerical Investigation of Flow Through a Rotating, Annular,Variable-Area Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425Palak Saini, Sagar Saroha, Shrish Shukla and Sawan S. Sinha

Development of M–DSMC Numerical Algorithmfor Hypersonic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437G. Malaikannan and Rakesh Kumar

Fluid–Structure Interaction Dynamics of a Flexible Foilin Low Reynolds Number Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449Chandan Bose, Sunetra Sarkar and Sayan Gupta

viii Contents

About the Editors

B. N. Singh is a HAL Chair Professor and former Head of Aerospace EngineeringDepartment and currently Dean (HR) and Registrar at the Indian Institute ofTechnology (IIT) Kharagpur, India. Prof. Singh has more than 25 years of teachingand research experience. His work focuses on aerospace composite structures andits uncertainty quantification and he has developed several stochastic and deter-ministic mathematical models and their applications in aerospace structural com-ponents made of smart composites. Prof. Singh has published more than 130 papersin reputed journals and more than 85 conference papers.

Arnab Roy is a Professor in the Department of Aerospace Engineering at IITKharagpur. His research areas include the development of high accuracy NavierStokes solvers for LES and DNS for incompressible flows- studies on bluff bodies,jets and associated instabilities, reaction control system jets in supersonic crossflow, low Reynolds number airfoils for fixed wing UAV/ MAV applications andParticle Image Velocimetry for flapping wing aerodynamic studies. He has pub-lished 3 book chapters, 38 research papers in reputed journal and conferences.

Dipak Kumar Maiti is Professor and Former Head, Department of AerospaceEngineering at IIT Kharagpur, prior to which he has worked at ADA Bangalore andDepartment of Aerospace Engineering, IIT Bombay. His research interest includeaeroelastic analysis of smart lifting surface, analysis of smart landing gear, struc-tural health monitoring and FE analysis of smart multidirectional composites. Hehas authored over 80 international journal papers, over 70 conference papers and 50technical reports related to various projects.

ix

Multi-scale Simulation of Elastic WavesContaining Higher Harmonics

Ambuj Sharma, Sandeep Kumar and Amit Tyagi

Abstract Numerical simulation of wave propagation is essential to understand thephysical phenomenonof thewide variety of practical problems.However, the require-ment of minimum grid point density per wavelength limits the computational stabil-ity, convergence, and accuracy of simulation of engineering application by numericalmethod. The purpose of this paper is to provide an improved framework for simula-tion of linear and nonlinear elastic wave propagation and guided-wave-based damageidentification techniques feasible in the context of online structural healthmonitoring(SHM). Nonstandard wavelet-based multi-scale operator developed by using finiteelement discretization is used to represent waves. The proposed masking eliminatesthe requirement of a very large number of nodes in finite element method neces-sary for the propagation of such waves. The method is also useful in the situationwhere higher harmonics of propagating waves are ignored due to very high computa-tional cost. Thewavelet-based finite element scheme achieves an excellent numericalsimulation and expresses an applicability for the guided waves’ study.

Keywords Nonstandard wavelet operator · Structural health monitoring ·Multi-scale simulation · Higher harmonics · Lamb waves

1 Introduction

Wave propagation can be characterized by the localized region of the sharp gradientof field variable which changes its locations in spacewith time. This gives permissionto recognize the unusual nature that could be suitable for ultrasonic nondestructive

A. Sharma (B)Mechanical Engineering Department, VIT-AP, Amaravati, Indiae-mail: [email protected]

S. Kumar · A. TyagiMechanical Engineering Department, IIT (BHU), Varanasi, Indiae-mail: [email protected]

A. Tyagie-mail: [email protected]

© Springer Nature Singapore Pte Ltd. 2020B. N. Singh et al. (eds.), Recent Advances in Theoretical, Applied, Computationaland Experimental Mechanics, Lecture Notes in Mechanical Engineering,https://doi.org/10.1007/978-981-15-1189-9_1

1

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2 A. Sharma et al.

testing techniques. Guided-wave-based nondestructive techniques offer to evaluatethe integrity of critical structures and to find out damage position, shape, and size [1,2]. Several numerical techniques have been proposed to analyze the wave equations.Due to their relatively simple mathematical expressions and the possibility to beapplied to the very large class of engineering problems, finite difference [3], boundaryelement [4], and finite element [2] based methods have been used by many authorsfor the simulation of guided Lamb waves. Finite element method (FEM) [2, 5] isa widespread numerical method used to simulate elastic guided wave propagationproblem. Finite difference method (FDM) has also been used for the study of wavesimulation and damage interaction by several researchers. Although FDM schemesare well situated for wave propagation in homogeneous media, however, the majorlimitation of the FDM schemes is that stiffness jumps due to continuously changingphysical properties cause stability problems [6]. Furthermore, boundaries as wellas discontinuities between different types of media lead to fairly accurate solutionsand can generate severe errors [7]. With this in mind, more recently, Delsanto et al.have proposed the local interaction simulation approach (LISA) in combination withthe sharp interface model [7]. Recently, customized elements and geometric multi-scale finite element method have been introduced to analyze various types of wavepropagation problems [8]. The finite element method, which has been preferredfor elastic wave propagation, is not suitable to simulate nonlinear waves or higherharmonics of propagating waves. A drastic increase of nodes for the simulation ofnonlinear wave problems demands some necessary alteration in FEM which mustbe numerically efficient and straightforward.

In recent years, wavelet-based numerical methods gain much attention for solv-ing partial differential equations. The major advantage of this approach allows oneto examine a problem in different resolutions, simultaneously. In addition, wavelet-based schemes are efficient in problems comprising singularities and sharp transitionsin solutions for limited zones of a computation domain. Initially, Beylkin et al. [9]employed the study of numerical computation based on wavelet. Several mathe-maticians and scientists have established the superiority of wavelet-based methodsfor solving elliptic partial differential equations [10, 11]. The adaptivity of waveletsis one of the leading advantages for the implementation of wavelets in numericalanalysis [12, 13]. Liandrat and Tchamitchian have solved regularized 1D Burgers’equation by using spatial wavelet approximation [14]. Later, Beylkin and Keiser[15], Vasilyev and Bowman [16], and Kumar and Mehra [17] have developed differ-ent wavelets-based algorithms and tested on 1D Burgers’ and advection–diffusionequations. Researchers have increased the usage of wavelets for solution of par-tial differential equations (PDEs) after the development of the lifting scheme bySwelden [17] and stable completion by Carnicer et al. [18]. A review of wavelettechniques for the solution of PDEs has been presented by Dahmen [19]. How-ever, a very few researchers have applied the wavelet-based method for analyzingwave propagation problem. Hong and Kennett proposed wavelet-based method forthe numerical modeling and simulation of elastic wave propagation in 2D media[20]. Chen and Yang et al. presented the wave motion analysis of short wave inone-dimensional structures [21]. Mitra and Gopalakrishnan proposed wavelet-based

Multi-scale Simulation of Elastic Waves Containing … 3

spectral finite element method (WSFEM) for simulation of elastic wave propagationin one- and two-dimensional situations [22]. In the literature, some researchers haveused wavelets as basis function to solve PDEs but most researchers have applied thewavelet-based adaptive technique in finite difference schemes. These papers havepresented the adaptive method for propagation of a single wave, but there is a needfor different algorithms for more than one waves propagating with different veloc-ities. Generation of higher harmonics due to material nonlinearity is not addressedin these papers.

Multi-scale modeling is one possible solution for higher harmonics in wave prop-agation simulation. Wavelet-based multi-scale method leads to fast and locally adap-tive algorithms. The compactly supported refinable basis functions aremain potentialadvantage of the wavelet [10, 11]. However, these methods are unable to competewith conventional finite elementmethod. In this paper, proposed technique is inspiredby the interesting paper by Krysl et al. [23].

This paper presents multi-scale adaptive approach for solving the wave propaga-tion problem. In the proposed wavelet-based technique, FEM is preferred due to itscapability to handle complex boundary and loading conditions instead of any othermethods. This multi-scale transformation hierarchically filters out the less significantfrequencies and offers an operative framework to retain the necessary frequencies ofthe wave. In this procedure, the finest level of the coefficient matrix is calculated oncefor the whole domain while the adaptively compressed coefficient matrix, which isvery small compared to complete coefficient matrix, is used in everymarching step ofthe solution. This paper is presenting wavelet-based nonstandard operator to improvefinite element simulation of linear and nonlinear wave propagation in a large struc-ture. We use nonstandard operator because it is more efficient than standard operator[24]. This will not only be useful to the structural health monitoring, but it can alsobe used where waves with higher harmonics move at different group velocities. Asimple description of the nonstandard operator along with necessary algorithm andmathematical comments is provided to remove an execution headache connectedwith adaptive grid techniques. The algorithm is applied to 2D plane strain problem,but it is general and independent of domain dimensions.

2 Mathematical Formulation

2.1 Lamb Waves

In an elastic medium, elastic waves are defined as propagating disturbances thattransport energy without any material transfer. Elastic waves of plane strain thatexist in free plates are called Lamb waves. For an orthotropic and symmetrical plate,particle motion is often outlined through the elemental elastodynamic differentialequation of wave

4 A. Sharma et al.

∂l(Cklmn∂nwm) = ρ∂2t wk, (k, l,m, n = 1, 2). (1)

Substituting stress relation in governing equations and Lamb wave can beexpressed as

C2L∂2u

∂x2+ (C2L − C2T )

∂2v

∂x∂y+ C2T

∂2u

∂y2+ fx = ∂

2u

∂t2, (2(a))

C2L∂2v

∂y2+ (C2L − C2T )

∂2u

∂x∂y+ C2T

∂2v

∂x2+ fy = ∂

2v

∂t2. (2(b))

C2L = λ+2μρ and C2T = μρ are longitudinal velocity and shear velocity, respectively,where λ = Eν

(1+ν)(1−2ν) and μ = E2(1+ν) are Lamé constants, E is Young’s modulus,and ν is Poisson ratio. The 2D plane strain problem is discretized into the set of finiteelement equations as

[K ][u] + [M][ü] = 0, (3)

where [u] and [ü] are unknown coefficient vectors. [K ] and [M] are global stiffnessand mass matrix, respectively.

2.2 Multi-scaling Using Wavelets

The idea of multi-scale exploration is to interpolate an unknown field at a coarse levelwith the assistance of supposed scaling functions.Any enhancement to initial approx-imation is accomplished by adding “details” rendered by basis functions referred toas wavelets. Amulti-scaling analysis forms a sequence of closed subspaces to satisfycertain self-similarity relations as well as completeness and regularity relations. Thebasis functions in Wj are called wavelet functions. Wavelet functions are symbol-ized by ψ j,k . These scaling and wavelet functions are employed for wavelet-basedmulti-scaling. A function f ∈ L2(R) is approximated by its projection P j f ontothe space Vj and the projection of f on Wj as Q j f , we have

P j f = P j−1 f + Q j−1 f. (4)

If the coefficient vector of P j f (or coefficients of scaling functions) is C j ={C j,0, . . . ,C j,v( j)}T and coefficient vector of Q j f (or coefficients of some wavelets)is d j = {d j,0, . . . , d j,w( j)}T, then we can write wavelet transform as

C j = [Tj ][C j−1d j−1

]. (5)


Thematrix [Tj ] is used to achieve next higher level by transforming scaling and detailcoefficients of Vj−1 and Wj−1 spaces, respectively. In this paper, B-spline waveletand Daubechies (D4) wavelet [25] are used for wave propagation.

2.3 Nonstandard Multi-scale Decomposition of FiniteElement Matrix

Two observations can be made while solving some PDEs using the wavelet bases: (i)In theoretical terms, most of the available wavelet methods have stable Riesz basisand better condition number than FEM or FDM. (ii) But in practical applications,wavelet methods are not yet ready to compete with the traditional FEM approach.One important reason is while the FEM can always produce a sparse matrix withmore regular sparsity patterns, use of wavelet bases does not produce such sparsematrices. But the combination of wavelets with other methods, such as FDM, FEM,and recently SEM [22], show good results. Here, we have used FEM discretizationto derive a sparse matrix, as the FEM remains the most versatile tool to solve PDEs.

Let us consider a continuous wave field u(x, y) and v(x, y) for a source of excita-tion over 2D homogeneous medium. The approximation of the continuous wave fieldon the discrete domain is denoted by u j and v j . It represents the discrete wave fieldthat is obtained with a classic time–space finite element method for a sufficiently finediscretization of Vj ⊂ R2. The 2Dwavelet transform cascades projections of the dis-crete wave field over different approximation grids V1, V2, V3, . . . , Vj of increasingresolution.

In this multi-scale algorithm, we used NS operator proposed by Beylkin [26]. Tothe best of authors’ knowledge, no one researcher has used NS operator in wavelet–FEM coupling or wavelet–FDMcoupling. It has been proved by Beylkin [26, 27] thatNS operator is more efficient than the standard form of operator used by most of theresearchers. In this paper, we have used NS operator in two-dimensional wavelet–finite element coupling technique. The finite element equations for the transientproblem, Eq. 3, can be expressed in the expanded form as

⎡⎣

[k juu

] [k juv

][k jvu

] [k jvv

]⎤⎦[ u j

v j

]=

⎡⎣

[f ju

][f jv

]⎤⎦. (6)

We can apply the wavelet transformation on the field variables of both thedirections:

6 A. Sharma et al.

[ [T T

] [0][0] [T T]

]⎡⎣[k juu

] [k juv

][k jvu

] [k jvv

]⎤⎦[ [T ] [0][0] [T ]

]⎡⎢⎢⎣

[d1

][u1

][e1

][v1

]

⎤⎥⎥⎦ =

[ [T T

] [0][0] [T T]

]⎡⎣[f ju

][f jv

]⎤⎦.

(7)

The organization of the matrix after three-level transform of nonstandard formcan be extended and expressed in the new notations as [24]

(8)

In order to solve it, Gines et al. [24] proposed nonstandard LU decomposition.

3 Results and Discussion

Elastic waves have been employed for identification of damage in the thin wall struc-tures such as plates and pipes [5]. Guided Lamb waves are excited in the structuresthrough narrowband burst signals. In order to evaluate the performance of wavelet-based multi-scale method. We considered an example in which a 50 × 50 mm2homogeneous, isotropic aluminum plate with a density of 2700 kg/m3. The simula-tion of Lamb wave in this plate with 400 kHz central frequency is presented in Fig. 1.Contour plots of the displacement in the x-direction at three different time instantsare well depicted in this figure.

Nonlinear Lamb wave is more sensitive to small-scale damage identification.However, the investigations of the higher harmonics in propagating Lamb waves


X

Y

0 0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05

8.5E-128E-127.5E-127E-126.5E-126E-125.5E-125E-124.5E-124E-123.5E-123E-122.5E-122E-121.5E-121E-125E-13

X

Y

0 0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05


(a) 8 sµ (b) 16 sµ

(c) 24 sµX

Y

0 0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05


Fig. 1 Contour plots of the displacement in the x-direction at three different time frames forisotropic plate

were ignored due to high computational cost. To see the efficiency of the wavelet-based method, higher harmonics are added in the Lamb wave and propagation ofwaves is observed. This study uses the following actuation function with 400 kHzcentral frequency:

E(t) ={

fo sin(�t) ∗ (sin(0.1�t))2 + 0.1 fo sin(2 ∗ 5�t) ∗ (sin(�t))2, t < 10π/

�

0, otherwise,

where � is the frequency of excitation and Eo is the maximum amplitude. Theexcitation signal on a plate with a higher harmonics is shown in Fig. 2. An aluminumplate of 200 mm length and 2 mm thickness is used in the analysis. Poisson’s ratio =

8 A. Sharma et al.

Time ( S)0 5 10 15 20 25 30 35

Am

plitu

de

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 2 Excitation signal for Lamb wave with higher harmonics

0.3, density= 2700 kg/m3, andYoung’smodulusE = 69GPa are assumed asmaterialproperties. The Lamb wave in this material has longitudinal velocity CL = 5299 m/sand transverse velocity CT = 3135 m/s. The waves are actuated by employing pinforces applied to the left boundary of the plate. The excitation forces are parallel tothe longitudinal (propagating) direction. In-phase pin forces are applied to the topand bottom edge nodes of the plate for excitation of fundamental symmetric (S0)modes, and the antisymmetric modes are propagated by imposing out-of-phase pinforces. In this paper, we considered the cases in which the pure S0 mode is excited.Ten cycles Hanning-window actuation is given through excitation function to delivera limited cycle sinusoidal tone burst.

Higher frequency wave propagation problems demand enormous computerresources because of very large number of time integration steps and highly densemesh. Generally, in the case of Lamb wave, 20 elements per wavelength are requiredbut this is not sufficient for higher harmonic simulation. Figure 3 depicts themeasurednodal displacement response of time-domain signals obtained using FEM simulationof the plate with 40, 80, and 120 elements per wavelength. It can be observed thathigher harmonics are not properly visible in the response of the plate with 40 ele-ments per wavelet. On the other hand, as shown in the same figure, higher harmonicsare visible for 80 elements per wavelength.


0 10 20 30

Am

plitu

de10-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Time ( S)

Fig. 3 Comparison of response of plate for 40, 80, and 120 elements per wavelength

In the present analysis, B-spline and Daubechies (D4) wavelet are used to estab-lish robustness and sensitivity ofwavelet-basedwave propagationmethod. To capturehigher harmonics in the plates, FEM uses 17,080 uniformly distributed nodes whilehalf of the FEM nodes are required after application of one level of wavelet trans-form. Nodal displacement response of plate received from B-spline and D4 wavelettransform at level 1 along with FEM results is demonstrated in Fig. 4. It establishesgood agreement between conventional finite element and proposed wavelet-basedmethod. It can be observed that B-spline wavelet produces response close to FEMresults, while there is some deviation in the results of D4 wavelet. Further, we exam-ined wavelet-based method at various levels of wavelet transform to find the levelup to which this method can work efficiently. These results show some attenuationbut wavelets are not eliminating higher frequency components of waves which areimportant in many analyses.

10 A. Sharma et al.

0 10 20 30 40

Am

plitu

de10 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

FEMB-splineD4

Time ( S)

Fig. 4 Comparison of higher harmonic Lamb waves plates response at wavelet transform level 1

4 Conclusion

The spatial derivative operators in the wave equations are handled using multi-resolution transforms in a physical domain. We presented a wavelet-based frame-work to reduce the size of global stiffness matrix of finite element analysis whichis becoming too large in the case of nonlinear wave propagation problem. Wavelet-based method is not only to develop the compressed stiffness matrix, but also topropagate higher harmonics of waves using least number of nodes and able to reducethe computational cost significantly.Without disturbing the programming advantagesof FE regarding the implementation of boundary conditions and efficient numericalintegration of interpolation functions, wavelet-based methods are able to reduce thesize ofmatrix asmuch as one by a sixteenth of original FEmatrix. These fundamentalcharacteristics show that the wavelet-based method can be utilized for more complexwave propagation problems.


References

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Effect of Skewness on RandomFrequency Responses of Sandwich Plates

R. R. Kumar, Vaishali, K. M. Pandey and S. Dey

Abstract This study presents the effect of skewness in natural frequency responsesof sandwich plates. The free vibration analysis is carried out by using higher orderzigzag theory (HOZT) considering random input parameters. It satisfies the trans-verse shear stress continuity condition and the transverse flexibility effect. The in-plane displacement throughout the thickness is assumed to vary cubically whiletransverse displacement is considered to vary quadratically within the core and con-stant at top and bottom plates. An efficient C0 stochastic finite element approach isdeveloped for the implementation of proposed plate theory in the random variablesurrounding. Compound stochastic effect of all input parameters is presented for thedifferent degrees of skewness in sandwich plates. Intensive Monte Carlo simulation(MCS) is employed for solving the stochastic-free vibration equations and statis-tical analysis is conducted for illustration of the results. The present algorithm forsandwich plate is validated with previous literatures and it is found to be in goodagreement.

Keywords Monte Carlo simulation · Sandwich plate · Natural frequency · Higherorder zigzag theory · Skewness

1 Introduction

A sandwich plate is a multilayered plate having two face sheets and a core embed-ded in between them through adhesive. The face sheets are relatively thin but of highstrength and stiffness material, whereas the core is made up of relatively thick andlower density material. The high specific strength and stiffness of sandwich struc-tures make them suitable for crucial engineering applications like automobile, civilconstruction, aerospace, and marine industries. Sandwich plates are widely used indesign and construction of aerospace craft. In such application, these materials aresubjected to wide environmental changes such as pressure, temperature, density, and

R. R. Kumar · Vaishali (B) · K. M. Pandey · S. DeyDepartment of Mechanical Engineering, National Institute of Technology, Silchar, Silchar, Indiae-mail: [email protected]

© Springer Nature Singapore Pte Ltd. 2020B. N. Singh et al. (eds.), Recent Advances in Theoretical, Applied, Computationaland Experimental Mechanics, Lecture Notes in Mechanical Engineering,https://doi.org/10.1007/978-981-15-1189-9_2

13

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14 R. R. Kumar et al.

humidity. This inevitable change in surrounding affects the vibrational response ofthe structure. Therefore, it is essential to include the actual operating condition inorder to get the changes in vibration characteristic of sandwich plate. The vibra-tion response of the aerospace craft is usually carried out in atmospheric conditionrather than actual unevenly varying condition for the cause of convenience. Thus, it isessential to consider the material and geometric uncertainty in order to accommodateabove-mentioned environmental changes as well as other inaccuracies occurring dur-ing design and fabrication of the sandwich plate. The cost-effective sandwich panelrequires sandwich core of low-cost material which exhibits better weight sensitive-ness aswell. The development and automation in production processesmake possiblethe production of low-cost sandwich panel. The sandwich panel is not considered forlow-cost application due to insufficient knowledge about their cost-saving potential.The manufacturing of such sandwich structure always experiences spatial variabil-ity due to manufacturing imperfections and other inaccuracies. Moreover, dynamicbehavior of sandwich structure possesses high statistical variation due to unavoidableskewness occurring during complex fabrication processes. Various interdependentparameters influencing the properties are core thickness, number of face sheet layer,face sheet and corematerial properties, and topology of core. Because of these param-eters, uncertain responses can be seen and the system properties become inevitablyrandom in nature. So to have a safe and realistic design, we must not neglect theseinherent uncertainties. This cannot be obtained through usual deterministic approach.So, to incorporate the source uncertainties in design and analysis of the mechanicalsystem, it is required to quantify the present uncertainties.

Recently, Grover et al. [1] worked on sandwich plate and studied the parametricuncertainties influencing the deflection statics and after that they also ensured thevalidity by comparing it with that of Monte Carlo simulation having finite elementsolution. Earlier, Aguib et al. [2] worked onmagnetorheological elastomer core sand-wich beam. The proposed structure was directly applied to civil engineering. Nayaket al. [3] studied the free vibration response on sandwich plates in damped randomenvironment. Jin et al. [4] studied the natural frequency response by considering a vis-coelastic core sandwich beam. For honeycomb sandwich beams, Debruyne et al. [5]analyzed the design parameters’ variability. The compressibility effect in transversedirection is studied using laminate mechanics by Plagianakos and Papadopoulos [6]and Liu [7] carried out the analytical study on sensitivity analysis for natural frequen-cies and their mode shapes. SFEMwas furthermore studied by Gadade et al. [8]. Thevibration characteristic was studied by Scarpa and Tomlinson [9] on regular hexag-onal honeycomb cells and re-entrant auxetic honeycomb cells. Later, spectral finiteelement method was used by Ruzzene [10]. By using this method, we can accuratelyevaluate the acoustic properties of honeycomb. An optimized study of truss-coresandwich panel was done by Denli and Sun [11]. A similar study was also presentedby Franco et al. [12]. With recent advancement in finite element software, for exam-ple, ABAQUS and ANSYS, the efficiency and accuracy have greatly increased. Thestudy of stochastic natural frequency including the effect of noise was done by Dey

Effect of Skewness on Random Frequency Responses … 15

Fig. 1 Skewed plate

et al. [13]. Recently, stochastic analysis is carried out by Kumar et al. [14–19], Karshet al. [20–26], and Mukhopadhyay et al. [27, 28]. Most of the research is carriedout by using deterministic approach, whereas few researchers focused on stochasticapproach.

Here, the effect of skewness (Fig. 1) on natural frequency response, having takeninto consideration the compound variation of all input parameters, is studied. There-after, this paper is presented as: Theoretical formulation is described in Sect. 2,result and discussion are illustrated in Sect. 3, whereas conclusion and future scopeare presented in Sect. 4.

2 Theoretical Formulation

The strain–displacement equation [29] can be shown as

{ε̄(�)} =[∂u(�)

∂x

∂v(�)

∂y

∂w(�)

∂z

∂u(�)

∂x+ ∂v(�)

∂y

∂u(�)

∂z

+∂w(�)∂x

∂v(�)

∂z+ ∂w(�)

∂x

], (1)

i.e., {ψ(�)} = [a(�)]{ψ(�)},

where [a] is unit step function. The equation for generalized displacement vector isgiven as

{s(�)} =n∑

k=1ζi (�)si (�), (2)

where {s} = {U0V0W0θxθyUuVuWuUlVlWl}T. Equation (1) is used to give strainvector equation

{ψ(�)} = [a(�)]{s(�)}. (3)


The strain–displacement matrix can be represented as [a]. The dynamic equilib-rium equation for natural frequency analysis is written by using Hamilton’s principleas

[r(�)]{_s} = λ2[m(�)]{_s}, (4)

where [r(�)] is the random natural frequency. The global mass matrix [m(�)] is

[m(�)] =nu+nl∑k=1

˚ρk(�)[n]T[ j]T[n][ j]dxdydz =

¨[n]T[k(�)][n]dxdy, (5)

whereρk(�) is stochastic mass density of kth order, [j] is of the order of 3X11, and[n] is the shape function matrix. The equation for stiffness matrix [k(�)] is given as

[k(�)] =nu+nl∑kl

ρk(�)[ j]T[ j]dz. (6)

For storing the global stiffness in one array, we have used the skyline technique.For getting static solution, Gaussian decomposition scheme is used and for freevibration analysis simultaneous iteration technique is used.

3 Results and Discussion

Here, HOZT is applied to a sandwich plate (simply supported boundary condition)of length (l) = 10 cm, width (b) =10 cm, and thickness (t) = 1 cm to demonstratethe proposed finite element (FE) model. The present model is having eight-layersymmetric cross-ply laminate having core thickness of 0.8 and face sheet thicknessof 0.1 with equal layers on both sides of core. Here, the first, second, and thirdnatural frequencies without any skewness are compared with 15°, 30°, 45°, and 60°skewed plates. The material properties considered for the present analysis are shownin Table 1.

Based on the present model, the natural frequencies for the first mode obtainedfor 30° and 45° skew angles and the results of Wang et al. [30] and Kulkarni andKapuria [31] are tabulated in Table 2. Such a small deviation between various resultsobtained for natural frequencies is shown in Table 2, which can justify the accuracyand applicability of HOZT.

It is evident from Fig. 2 that with increase in skew angle, fundamental and thirdnatural frequencies initially increase for (φ) = 15° and 30°, decrease for (φ) =45°, and then again increase up to maximum for (φ) = 60°, whereas second naturalfrequency initially increases for (φ) = 15° and 30° and decreases for (φ) = 45° and60°. The mean value of second natural frequency for (φ) = 45° and 60° remains


Table 1 Material properties Core Face sheet

E1 (GPa) 0.5 38.6

E2 = E3 (GPa) 0.5 8.27G12 = G13 (GPa) 0.4 4.14G23 (GPa) 0.2 1.656

υ12 = υ13 = υ23 = υ32 0.27 0.26υ21 = υ31 0.006 0.006P (kg/m3) 1000 2600

Table 2 Natural frequencyof (0°/90°/0°/90°) sandwichplate

Skew angle(φ) (°)

Present study Wang et al.[30]

Kulkarni andKapuria [31]

30 1.8889 1.9410 1.9209

45 2.5806 2.6652 2.6391

almost same which lies in between φ = 15° and φ = 30°. This corroborates the factobtained vide probability density function (PDF) plots.

4 Conclusions

Based on higher order zigzag theory (HOZT), the accuracy and applicability ofthe proposed finite element model for free vibration analysis of sandwich platesare studied. The novelty of the present study includes the skewness effect on freevibration of sandwich plates. The natural frequency of sandwich plate is comparedwith that of skewed sandwich plate by means of probability density function (PDF)plots. The first, second, and third natural frequencies of unskewed sandwich plates arecompared with plates having skewness of 15°, 30°, 45°, and 60°. It is observed thatthe unavoidable source uncertainties cause significant deviation of natural frequencyfrom the mean deterministic value. Therefore, it is of utmost importance to considerthe effect of skewness and source uncertainty in design and analysis of sandwichplate and other complex structures for safe and realistic design. Based on theseobservations, the present work can be extended to deal withmore complex structures.


Fig. 2 Random natural frequency (rad/s) of sandwich plates for a first, b second, and c third naturalfrequencies with skew angle, (ϕ) = 0°, 15°, 30°, 45°, and 60°


Acknowledgements The first and second authors would like to acknowledge the financial supportreceived from MHRD GOI during this research work.

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