soundandvibration_android插入usb之后振动资源-CSDN文库

vibration

4星 · 超过85%的资源需积分: 12 200 浏览量 2014-03-01 18:20:07 上传评论收藏 1.41MB PDF 举报

资源推荐

资源详情

资源评论

Free vibration analysis of a cracked shear deformable beam

on a two-parameter elastic foundation using a lattice

spring model

M. Attar

, A. Karrech

, K. Regenauer-Lieb

c,d

School of Mechanical Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

School of Earth and Environment, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

CSIRO Earth Science and Resource Engineering, Kensington, WA 6151, Australia

article info

Article history:

Received 23 August 2013

Received in revised form

30 October 2013

Accepted 2 November 2013

Handling Editor: L.G. Tham

Available online 5 December 2013

abstract

The free vibration of a shear deformable beam with multiple open edge cracks is studied

using a lattice spring model (LSM). The beam is supported by a so-called two-parameter

elastic foundation, where normal and shear foundation stiffnesses are considered.

Through application of Timoshenko beam theory, the effects of transverse shear

deformation and rotary inertia are taken into account. In the LSM, the beam is discretised

into a one-dimensional assembly of segments interacting via rotational and shear springs.

These springs represent the flexural and shear stiffnesses of the beam. The supporting

action of the elastic foundation is described also by means of normal and shear springs

acting on the centres of the segments. The relationship between stiffnesses of the springs

and the elastic properties of the one-dimensional structure are identified by comparing

the homogenised equations of motion of the discrete system and Timoshenko beam

theory.

The effects of the transverse open cracks are modelled by increasing the flexibility of

the rotational springs in the discrete model at crack locations. In this manner, the cracked

section is modelled by a massless rotational spring combined in series with the rotational

spring that represents the flexural stiffness at that point. Numerical examples are

provided to show the versatility and convergence of the present method, and to

investigate the effects of geometrical and physical parameters on free vibration of a

cracked beam. An analytical approach is also developed based on the transfer matrix

method (TMM) to examine and validate the results obtained by the LSM against the

corresponding analytical solution.

1. Introduction

The problem of dynamic analysis of a one-dimensional structure supported by a foundation is applicable in several areas

of engineering. It plays a crucial role in specialised disciplines such as offshore, structural, foundation and railway

Contents lists available at ScienceDirect

journal h omepage: www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

http://dx.doi.org/10.1016/j.jsv.2013.11.013

Corresponding author. Tel.: þ61 420816454.

E-mail addresses: mostafa@mech.uwa.edu.au, mostafa.attar@gmail.com (M. Attar), ali.karrech@gmail.com, ali.karrech@uwa.edu.au,

ali_karrech@yahoo.fr (A. Karrech), klaus.regenauer-lieb@uwa.edu.au (K. Regenauer-Lieb).

Journal of Sound and Vibration 333 (2014) 2359 – 2377

engineering. There have been a number of attempts in the literature to develop approximate and exact methods for

investigating the dynamic behaviour of beam-like structures (e.g. pipelines or piles) resting on elastic foundations. In order

to achieve this objective, different beam theories (e.g. Euler–Bernoulli, Rayleigh or Timoshenko) and foundation models

have been utilised.

To model elastic support effects, the simplest way is to idealise the foundation by using Winkler linear springs [1]. These

springs are closely spaced longitudinally, and create a supporting reaction proportional to the lateral deflection of the beam.

Over the past few decades, several researchers have considered this type of foundation for beams to perform static and

dynamic analysis (e.g. [2–5]). The disadvantage of the Winkler model is that it considers foundation elements mutually

independent, and overlooks the role of cohesion among the particles of the subgrade medium. To overcome this limitation, a

two-parameter model was proposed by Pasternak [6]. This type of foundation can capture the shear interaction between the

Winkler spring elements as well. Therefore, the foundation shear modulus is introduced as the second stiffness parameter of

the substrate. This is to create a shear force proportional to the curvature of the beam, which is in fact the reaction force of

the support shear layer [7].

A thorough understanding of the mechanical behaviour of elastically supported structures can be achieved by the

development of numerical schemes which can properly deal with complex problems. A pioneering numerical approach in

this context has been the finite element technique to address the static and dynamic response of a beam interacting with

various types of two-parameter foundations [8,9]. Employing Hamilton's principle, Matsunaga [10] proposed the method of

power series expansion of displacement components. Chen [11] developed the differential quadrature element method

(DQEM) capable of evaluating the vibration response of non-prismatic beams. A combination of the differential quadrature

and finite element methods was introduced by Malekzadeh and Karami [12] to avoid the possible shear locking

phenomenon which may appear in the conventional shear deformable finite elements. The transfer displacement function

method (TDFM) has been implemented by Ma et al. [13] to conduct a static analysis of an infinite beam supported by a

tensionless (unilateral) Pasternak foundation.

Although a considerable amount of attention has been paid to the dynamic analysis of beam-like mechanical members

on various types of foundations, very little work has been done in the literature that addresses the dynamic behaviour of

damaged structures interacting with elastic supports. Nevertheless, beam-like in-service structures may contain imperfec-

tions such as cracks, so the predictive simulation of their dynamic behaviour is a mandatory step to ensure safety and

reliability. Papadopoulos and Dimarogonas [14] suggested a promising approach for simulating an open non-propagating

surface crack in a beam as a source of local flexibility which can be formulated based on the fundamentals of fracture

mechanics. For the most general loading conditions, the local flexibility can be characterised by means of a 6 6 matrix,

where the matrix components depend on the depth of crack. In the case of bending vibration, the cracked section may be

simulated by a massless rotational spring representing the local stiffness reduction due to the cracked section [15].

Considerable research has been carried out on the prediction of dynamic responses of cracked structural elements by

expressing the influence of cracks in terms of local flexibilities and applying different beam theories [16–22]. In the context

of structural damage detection, a non-destructive analytical algorithm has been presented by Attar [23] in terms of the

changes in the dynamic characteristics of a Euler–Bernoulli beam due to the multiple damages. This method has been

applied to estimate the crack sizes and locations in a stepped beam by employing natural frequencies of the system as

input data.

The eigenvalue problem of a single-cracked Euler–Bernoulli beam resting on a Winkler foundation was first treated by

Hsu [24] using the differential quadrature method (DQM). The same method has been employed by Matbuly et al. [25] to

obtain free vibration frequencies of cracked functionally graded beams on the Pasternak-type foundation based on the

Euler–Bernoulli theory. Yan et al. [26] studied the case of a single-damaged Timoshenko beam made of functionally graded

materials (FGMs) and subjected to a lateral force moving at constant velocity. Limitations to present studies remain

concerning their ability to furnish a numerical model for a beam with shear deformation effect and rotary inertia which is

weakened by multiple cracks and supported by a two-parameter foundation.

The objective of the present paper is to develop a lattice spring model (LSM) for free vibration analysis of the cracked

Timoshenko beam perfectly bonded to a two-parameter (Pasternak-type) elastic support. By employing the LSM, the

distribution of matter is discretised into a number of discrete units connected via springs at a microscale. Thus, elastic

properties of the solid at macroscale are constructed from the interactions between these small units. The LSM, firstly

introduced by Hrennikoff [27] to solve continuum elasticity problems with Poisson's ratio of ν

¼1/3, has recently attracted

substantial attention by many researchers for simulating fracture mechanics problems [28,29]. This model can be easily

implemented to simulate fractured materials by removing the connecting springs of the lattice model.

For this study, the LSM fundamentals have been employed to treat the beam-like structure as a chain of cubic discrete

units (segments) interacting through shear and rotational springs which represent shear and bending stiffnesses of the

beam. One major advantage of this method is that the effects of Winkler and Pasternak elastic foundations may be

characterised in terms of normal and shear springs connected to the segments as well. Moreover, the local stiffness

reduction due to the surface damages can be easily modelled by increasing the rotational spring flexibility at cracked

sections. Therefore, the matrix system of equations for the whole model may be derived by Euler–Lagrange's equations and

used for conducting dynamic analysis. The proposed LSM has been checked by comparing the computed results with closed-

form analytical solutions and pre-existing literature. Finally, it must be noted that the application of this study is not only

limited to free vibration analysis of the beam supported by a linear elastic foundation. The developed LSM will hopefully

M. Attar et al. / Journal of Sound and Vibration 333 (2014) 2359–23772360

spur novel applications including variable elastic foundations, tensionless foundations and cracked beams traversed by

moving loads.

The present paper is divided into the following sections. In Section 2, the governing equations of motion are derived.

Section 3 describes the details of the proposed LSM. The cracked beam model used in analytical and numerical simulations

is explained in Section 4. The numerical results are shown and discussed in Section 5, and concluding remarks and

recommendations for future studies are presented in Section 6.

2. Governing equations

Consider a straight homogeneous beam of length L, cross-sectional area A, thickness b and width c. This beam is

supported by a two-parameter elastic foundation (Pasternak-type) and also elastically restrained at both ends, as depicted in

Fig. 1. The beam is securely attached to the foundation, so its supporting reaction can be delineated by the Pasternak-type

equation as pðx; tÞ¼K

wK

ð∂

w=∂x

Þ, where p is the foundation reaction per unit length (N/m), w is the transverse

displacement of the beam (m), K

is the Winkler modulus (N/m

) and K

is the Pasternak shear modulus (N) of the elastic

foundation (corresponding to the normal and shear interactions of the beam and foundation). It should be noted that the

numerical values of the Pasternak foundation parameters can be estimated using analytical methods. Girija Vallabhan and

Das [30] proposed an analytical technique to use Young's modulus and Poisson's ratio of the subgrade medium to calculate

two elastic parameters of the foundation (K

and K

Based on the Timoshenko beam theory, displacement field (u

) of the point (x,y,z) on the beam cross section are

defined as

ðx; y; z; tÞ¼uðx; tÞþzφðx; tÞ

ðx; y; z; tÞ¼0

ðx; y; z; tÞ¼wðx; tÞ

(1)

where u and w are respectively the x- and z-components of displacement vector of the point (x,0,0) on the centroidal axis of

the beam, φ is the angle of rotation (about the y-axis) of the beam cross section and t is time. The x-coordinate is taken along

the length of the beam, the y-coordinate is along the width and the z-coordinate is along the thickness, as shown in Fig. 1.

Hamilton's principle may be used to derive the governing differential equations of motion and corresponding boundary

conditions as

ðT

þR

Π

Þ dt ¼0 (2)

where δ denotes the variation symbol, T

is the kinetic energy of the beam, R

is the virtual work done by external forces

exerted on the beam in a time interval ½0; t

 and Π

is the total potential energy including the strain energy of the beam and

potential energy of the elastic foundation and elastic end supports.

Applying Hamilton's principle and replacing the first variation of kinetic energy, virtual work done by external forces and

total potential energy into Eq. (2), the governing equations of motion are derived as

∂Q

∂x

K

wþK

∂

∂x

þq m

∂

∂t

¼0 (3)

∂M

∂x

Q þ

μ m

∂

∂t

¼0 (4)

where m

ρ dA ¼ ρA and m

ρz

dA ¼ρI are the inertial terms, I ¼

dA is the second moment of the cross-

sectional area about the y-axis, M ¼EI∂φ=∂x is the internal bending moment, Q ¼κ

GAðφþ∂w=∂xÞ is the internal shear force,

E is Young's modulus, G is the shear modulus and κ

is the Timoshenko shear correction factor. The associated boundary

conditions are also obtained as

Q þ

Q K

w ¼0orw ¼w

at x ¼0 (5)

Q þ

Q þK

w ¼0orw ¼w

at x ¼L (6)

M K

φ ¼0orφ ¼φ

at x ¼0 (7)

Fig. 1. Sketch of a beam fully supported by a two-parameter elastic foundation and elastically restrained boundary conditions.

M. Attar et al. / Journal of Sound and Vibration 333 (2014) 2359–2377 2361

M þK

φ ¼0orφ ¼φ

at x ¼L (8)

It should be noted that

Q ¼ K

∂w=∂x in Eqs. (5) and (6) is the internal shear force associated with the Pasternak shear layer

[31,8], while Q is the shear force in the beam due to the bending. It means that the total transverse shear force (S) at each

cross section is the resultant (sum) of Q and

Q , as it is depicted in Fig. 2. It is worth mentioning that the role of internal shear

force associated with the shear layer has been overlooked in most of previous studies [25,26] for cracked beams perfectly

bonded to the Pasternak-type elastic foundation.

By substituting the internal bending moment and shear force into Eqs. (3) and (4), the governing equations of motion of

the Timoshenko beam on the two-parameter foundation can be derived in terms of displacements as

∂φ

∂x

∂

∂x



K

wþK

∂

∂x

þq ¼m

∂

∂t

(9)

∂

∂x

κ

GA φþ

∂w

∂x



μ ¼m

∂

∂t

(10)

3. Lattice spring model

3.1. Discrete equations of motion in the LSM

The LSM is in fact a discrete representation of the continuum matter in which material is discretised into a network of

discrete units interacting via springs. The elastic properties of the springs between the neighbouring units in small scale

define the elastic characteristics of the solid at large scale. Therefore, due to the discrete nature of the LSM, it can be

considered as a mesoscopic model of the media, but the mechanical behaviour of the continuum at large scale can be

accurately reproduced from the resultant response of the lattice network. Depending on the underlying structure of matter,

the type of connecting springs and particles degrees of freedom in the LSM may be different. However, the common feature

of all LSM is describing the continuous elastic medium by a discrete assembly of the particles and constructing a system of

equations in which unknown values are the displacements of the nodes.

In this paper, we take advantage of the LSM principles and divide the continuous Timoshenko beam into a one-

dimensional chain of periodic cubic segments, as illustrated in Fig. 3. By neglecting the axial displacement effects, we only

need shear and rotational springs between segments to model shear and flexural stiffnesses of the beam properly. All

segments are identical with mass m and connected by shear (K

) and rotational (K

) springs. The total number of the

Fig. 2. Transverse shear force contributions of the bending theory and foundation shear layer.

Fig. 3. One-dimensional LSM representation of the Timoshenko beam supported by the elastic foundation.

M. Attar et al. / Journal of Sound and Vibration 333 (2014) 2359–23772362

剩余18页未读，继续阅读

评论收藏

内容反馈

六道2天下

2015-07-21

这是一篇声音与振动的模块杂志，刚好在学NI的SV模块，顺便学习一下

gaobuliao

粉丝: 0
资源: 1

sound and vibration

最新资源

sound and vibration

LabVIEW声音和振动工具包用户手册.rar_individualkw5_labview_labview声振_声振_声音与振动

Sound and Structure Vibration

基于声音与振动工具包的振动测量分析.zip_基于声音与振动工具包的振动测量分析_声音 分析_声音分析_振动_振动测量

LabVIEW声音和振动工具包用户手册

NI声音和振动工具包使用说明

NI Sound and Vibration Measurement Suite 7.0 keygen

Sound and Vibration Toolkit user manual

NI Sound and Vibration Measurement Suite 7.0 以及2009 keygen

light-and-sound

Play sound

motion-sound

vibration analysis

the-sound

SoundandStructuralVibrationRadiation,transmissionandresponse.pdf

sound collection

javascript_sound

linux_sound

sound-classification:4组

sound-walker

imit_sound.c

sound_recognize

direct sound

sound-board

vibration_matlab_vibration_

mechanical vibration

sound-separation

sound-space

免费下载Navicat15安装包+工具+教程.zip

cmu 15445 2023spring project0

最新资源

基于声音与振动工具包的振动测量分析.zip_基于声音与振动工具包的振动测量分析_声音分析_声音分析_振动_振动测量