FREE VIBRATION ANALYSIS OF AN UN-CRACKED AND CRACKED CANTILEVER BEAM A PROJECT REPORT Submitted By Arkanu Chaudhury, Jayanta Kumar Saha, Washim Younus Mallick & Tridib Paul

In partial fulfillment for the award of the degree Of

BACHELOR IN TECHNOLOGY In MECHANICAL ENGINEERING

MEGHNAD SAHA INSTITUTE OF TECHNOLOGY, KOLKATA

MAULANA ABUL KALAM AZAD UNIVERSITY OF TECHNOLOGY, WEST BENGAL Formerly known as

(WEST BENGAL UNIVERSITY OF TECHNOLOGY)

MAY 2017

MAULANA ABUL KALAM AZAD UNIVERSITY OF TECHNOLOGY, WEST BENGAL

BONAFIDE CERTIFICATE Certified that this project report “FREE VIBRATION ANALYSIS OF UN-CRACKED AND CRACKED CANTILEVER BEAM USING ANSYS” is the bonafide work of “ARKANU CHAUDHURY, JAYANTA KUMAR SAHA, WASHIM YOUNUS MALLICK & TRIDIB PAUL” who carried out the project under my supervision.

……………………………………………….

…………………………………………..

SIGNATURE

SIGNATURE

GOUTAM LAHA

MALAY QUILA

(HEAD OF THE DEPARTMENT)

(ASSISTANT PROFESSOR)

MECHANICAL ENGINEERING

MECHANICAL ENGINEERING

MSIT, KOLKATA

MSIT, KOLKATA

ACKNOWLEDGEMENT We avail this opportunity to express my profound sense of sincere and deep gratitude to those who have played an indispensable role in the accomplishment of the project work given to us by providing their willing guidance and help. We would like to thank Mr. Malay Quila, Asst. Professor for his kind and continual support and constructive suggestion given during the entire course of this project. We are blissful to express my deep sense of gratitude to him for his constant interaction, expert guidance and valuable suggestions that helped us to complete this project successfully. We would also like to thank Mr. Goutam Laha (Head Of Department Of Mechanical Branch) for allowing us to use lab facilities, the staff of the lab and mechanical faculty members for their help and support.

Name of the Students: ARKANU CHAUDHURY JAYANTA KUMAR SAHA WASHIM YOUNUS MALLICK TRIDIB PAUL

TABLE OF CONTENTS Abstract

i

List of Tables

ii

List of Figures

iii

List of Symbols

v Page No

1. Introduction………………………………………………………………………………….1 2. Literature Survey…………………………………………………………………………..4 3. Vibration……………………………………………………………………………………….6 3.1 Classification Of Vibration……………………………………………………….6 3.2 Importance Of Vibration………………………………………………………..7 4. Euler-Bernoulli Beam Theory…………………………………………………………9 4.1 Mathematical Formulation……………………………………………………...9 5. Theoretical Analysis Of Free Vibration Of Cantilever Beam………….12 6. Theoretical Results And Calculations……………………………………………14 6.1 Specifications Of The Beam…………………………………………………….15 6.2 Beam Modeling Using FEM Method…………………………………..…..16 7. Mode Shapes Of The Cantilever Beam………………………………………….17 7.1 Mode Shapes In ANSYS…………………………………………………………...18 7.1.1 Mode 1………………………………………………………………………18 7.1.2 Mode 2………………………………………………………………………19

7.1.3 Mode 3……………………………………………………………………….20 7.1.4 Mode 4……………………………………………………………………….21 7.1.5 Mode 5……………………………………………………………………….22 7.1.6 Mode 6……………………………………………………………………….23 8. Crack In Beams……………………………………………………………………………24 8.1 Classification Of Crack………………………………………………………….24 8.2 Crack Detection……………………………………………………………………25 9. Crack Modeling and Parametric Studies of The Beam In ANSYS…..26 9.1 Crack Modeling…………………………………………………………………….26 9.2 Parametric Studies of The Beam……………………………………….….27 9.2.1 Data Tables For Varying Crack Positions And Crack Depths………………………………………………………………28 9.2.2 Comparison Of Natural Frequency Ratio And Related Graphs………………………………………………………….31 10. Observations………………………...…………………………………………………..37 CONCLUSION…………………………………………………………………………..…..39 REFERENCES………………………………………………………………………………...40

ABSTRACT The presence of cracks causes changes in the physical properties of a structure which introduces flexibility, and thus reducing the stiffness of the structure with an inherent reduction in modal natural frequencies. Consequently it leads to the change in the dynamic response of the beam. This paper focuses on the theoretical analysis of transverse vibration of a CANTILEVER BEAM and investigates the mode shape frequency. All the theoretical values are analyzed with the numerical method by using ANSYS software and co relate the theoretical values with the numerical values to find out percentage error between them. Also in this paper, a model for free vibration analysis of a beam with an open edge crack has been presented. Variations of natural frequencies due to crack at various locations and with varying crack depths have been studied. A parametric study has been carried out. The analysis was performed using ANSYS software.

i

LIST OF TABLES Table.1 Value of Roots Of Equation cosh(βL)*cos(βL) = -1 Table.2 Theoretical Frequencies for the cantilever beam for 6 modes Table.3 Natural Frequencies for the cantilever obtained by FEM method Table.4 Percentage Error in the Frequencies obtained by FEM method Table.5 Freqs. For Cracked beam with crack at ζc (xc/L) =0.2 H(a/h)= 0.1 Table.6 Freqs. For Cracked beam with crack at ζc (xc/L) =0.2 H(a/h)= 0.3 Table.7 Freqs. For Cracked beam with crack at ζc (xc/L) =0.2 H(a/h)= 0.5 Table.8 Freqs. For Cracked beam with crack at ζc (xc/L) =0.3 H(a/h)= 0.1 Table.9 Freqs. For Cracked beam with crack at ζc (xc/L) =0.3 H(a/h)= 0.3 Table.10 Freqs. For Cracked beam with crack at ζc (xc/L) =0.3 H(a/h)= 0.5 Table.11 Freqs. For Cracked beam with crack at ζc (xc/L) =0.5 H(a/h)= 0.1 Table.12 Freqs. For Cracked beam with crack at ζc (xc/L) =0.5 H(a/h)= 0.3 Table.13 Freqs. For Cracked beam with crack at ζc (xc/L) =0.5 H(a/h)= 0.5 Table.14 Variation Of Natural Freq. Ratio with ζc an d H (MODE 1) Table.15 Variation Of Natural Freq. Ratio with ζc an d H (MODE 2) Table.16 Variation Of Natural Freq. Ratio with ζc an d H (MODE 3) Table.17 Variation Of Natural Freq. Ratio with ζc an d H (MODE 4) Table.18 Variation Of Natural Freq. Ratio with ζc an d H (MODE 5) Table.19 Variation Of Natural Freq. Ratio with ζc an d H (MODE 6) ii

LIST OF FIGURES  Fig.1 A cantilever beam with transverse load P at the free end  Fig.2 A Beam under Transverse Vibration  Fig.3 Free body Diagram of a section of a beam under transverse vibration  Fig.4 Mode Shapes Of A Fixed-Free Cantilever Beam  Fig.5 Mode shape of a cantilever beam in MODE 1 (isometric view)  Fig.6 Mode Shape of a cantilever beam in MODE 1 (x-y plane)  Fig.7 Mode shape of a cantilever beam in MODE 2 (isometric view)  Fig.8 Mode Shape of a cantilever beam in MODE 2 (x-y plane)  Fig.9 Mode shape of a cantilever beam in MODE 3 (isometric view)  Fig.10 Mode Shape of a cantilever beam in MODE 3 (x-y plane)  Fig.11 Mode shape of a cantilever beam in MODE 4 (isometric view)  Fig.12 Mode Shape of a cantilever beam in MODE 4 (x-y plane)  Fig.13 Mode shape of a cantilever beam in MODE 5 (isometric view)  Fig.14 Mode Shape of a cantilever beam in MODE 5 (x-y plane)  Fig.15 Mode shape of a cantilever beam in MODE 6 (isometric view)  Fig.16 Mode Shape of a cantilever beam in MODE 6 (x-y plane)  Fig.17 A V-shaped Open Surface Transverse Crack As Shown In ANSYS  Fig.18 V-shaped edge crack with a 0.5 mm width on the top surface of the beam  Fig.19 A cantilever beam with a crack  Fig.20 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 1)  Fig.21 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 2)  Fig.22 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 3) iii

 Fig.23 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 4)  Fig.24 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 5)  Fig.25 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 6)

iv

LIST OF SYMBOLS     

M bending moment V shear force f external force per unit length w displacement E Young’s Modulus of Elasticity  I Moment of inertia  ρ density of the material of the beam  A cross section area of the beam  A1, B, C, C1, C2, C3 and C4 positive constants  ω natural frequency  L length of beam  b breadth of beam  h height of beam  m mass per unit length of beam  µ Poisson’s ratio  xn Mode Shape  xc distance of crack from fixed end  a perpendicular depth of crack tip from the top surface  ζc Crack Position ratio (xc/L)  H crack depth ratio (a/H)  (ωc/ω) Natural Frequency Ratio

v

1

CHAPTER 1 INTRODUCTION Most of the members of engineering structures operate under loading conditions, which may cause damages or cracks in overstressed zones. The presence of cracks in a structural member, such as a beam, causes local variations in stiffness, the magnitude of which mainly depends on the location and depth of the cracks. The presence of cracks causes changes in the physical properties of a structure which in turn alter its dynamic response characteristics. The monitoring of the changes in the response parameters of a structure has been widely used for the assessment of structural integrity, performance and safety. Irregular variations in the measured vibration response characteristics have been observed depending upon whether the crack is closed, open or breathing during vibration. Modal analysis is the study of the dynamic properties of systems in the frequency domain. A typical example would be testing structures under vibrational excitation. Modal analysis is the field of measuring or calculating and analyzing the dynamic response of structures and/or fluids or other systems during excitation. Examples would include measuring the vibration of a car's body when it is attached to an electromagnetic shaker, analysis of unforced vibration response of vehicle suspension, or the noise pattern in a room when excited by a loudspeaker. Modern day experimental modal analysis systems are composed of: 1) Sensors such as transducers (typically accelerometers, load cells), or noncontact via a Laser Vibrometer, or stereo photogrammetric cameras

2

2) Data acquisition system and an analog-to-digital converter front end (to digitize analog instrumentation signals) 3) Host PC (personal computer) to view the data and analyze it.

The goal of our modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of a cantilever beam during free vibration. A cantilever is a beam anchored at only one end. The beam carries the load to the support where it is forced against by a moment and shear stress. Cantilever construction allows for overhanging structures without external bracing. Cantilevers can also be constructed with trusses or slabs.

Fig.1 A cantilever beam with transverse load P at the free end

Main advantages are:



Building out from each end enables construction to be done with little disruption to navigation below The span can be greater than that of a simple beam, because a beam can be added to the cantilever arms

3





Because the beam is resting simply on the arms, thermal expansion and ground movement are fairly simple to sustain Cantilever arms are very rigid, because of their depth.

Main disadvantages are



Like beams, they maintain their shape by the opposition of large tensile and compressive forces, as well as shear, and are therefore relatively massive Truss construction is used in the larger examples to reduce the weight.

Vibration principles are the inherent properties of the physical science applicable to all structures subjected to static or dynamic loads. All structures again due to their rigid nature develop some irregularities in their life span which leads to the development of crack. The problem on crack is the basic problem of science of resistance of materials. Considering the crack as a significant form of such damage, its modelling is an important step in studying the behavior of damaged structures. Knowing the effect of crack on stiffness, the beam or shaft can be modelled using either Euler-Bernoulli or Timoshenko beam theories. In our project we use the Euler-Bernoulli theorem as the basis of our theoretical analysis. The beam boundary conditions are used along with the crack compatibility relations to derive the characteristic equation relating the natural frequency, the crack depth and location with the other beam properties. Our research is based on effect of crack location and depth on natural frequencies and mode shapes of the beam.

4

CHAPTER 2 LITERATURE SURVEY The vibration behavior of cracked structures has been investigated by many researchers. The majority of published studies assume that the crack in a structural member always remains open during vibration. However, this assumption may not be valid when dynamic loadings are dominant. In such case, the crack breathes (opens and closes) regularly during vibration, inducing variations in the structural stiffness. These variations cause the structure to exhibit non-linear dynamic behavior. Christides and Barr [1] developed a one-dimensional cracked beam theory at same level of approximation as Bernoulli-Euler beam theory. Liang, Choy and Jialou Hu [3] presented an improved method of utilizing the weightless torsional spring model to determine the crack location and magnitude in a beam structure. Dimaragonas [4] presented a review on the topic of vibration of cracked structures. His review contains vibration of cracked rotors, bars, beams, plates, pipes, blades and shells. Shen and Chu [5] and Chati, Rand and Mukherjee [6] extended the cracked beam theory to account for opening and closing of the crack, the so called “breathing crack” model. Kisa and Brandon [7] used a bilinear stiffness model for taking into account the stiffness changes of a cracked beam in the crack location. They have introduced a contact stiffness matrix in their finite element model for the simulation of the effect of the crack closure which was added to the initial stiffness matrix at the crack location in a half period of the beam vibration. Saavedra and Cuitino [8] and Chondros , Dimarogonas and Yao [9] evaluated the additional flexibility that the crack generates in its vicinity using fracture mechanics theory. Zheng et

5

al [10] the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An overall additional flexibility matrix, instead of the local additional flexibility matrix, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix.

Zsolt huszar [11] presented the quasi periodic opening and closings of cracks were analyzed for vibrating reinforced concrete beams by laboratory experiments and by numeric simulation. The linear analysis supplied lower and upper bounds for the natural frequencies. Owolabi, Swamidas and Seshadri [12] carried out experiments to detect the presence of crack in beams, and determine its location and size. Behzad, Ebrahimi and Meghdari [14] developed a continuous model for flexural vibration of beams with an edge crack perpendicular to the neutral plane. The model assumes that the displacement field is a superposition of the classical Euler-Bernoulli beam's displacement and of a displacement due to the crack. Shifrin [16] presented a new technique is proposed for calculating natural frequencies of a vibrating beam with an arbitrary finite number of transverse open cracks. Most of the researchers studied the effect of single crack on the dynamics of the structure. A local flexibility will reduce the stiffness of a structural member, thus reducing its natural frequency. Thus most popular parameter applied in identification methods is change in natural frequencies of structure caused by the crack. In this report, the natural frequencies of cracked and un-cracked beams have been calculated using Finite element software ANSYS and up to sixth mode natural frequency graph is presented.

6

CHAPTER 3 Vibration Vibration are time dependent displacements of a particle or a system of particles w.r.t an equilibrium position. If these displacements are repetitive and their repetitions are executed at equal interval of time w.r.t equilibrium position the resulting motion is said to be periodic.

3.1 Classification of vibration Vibration can be classified in several ways. Some of the important classification are as follows:  Free and forced vibration: If a system, after an internal disturbance, is left to vibrate on its own, the ensuing vibration is known as free vibration. No external force acts on the system. The oscillation of the simple pendulum is an example of free vibration. If a system is subjected to an external force (often, a repeating type of force), the resulting vibration is known as forced vibration. The oscillation that arises in machineries such as diesel engines is an example of forced vibration. If the frequency of the external force coincides with one of the natural frequencies of the system, a condition known as resonance occurs, and the system undergoes dangerously large oscillations. Failures of such structures as buildings, bridges, turbines and airplane have been associated with the occurrence of resonance.

7

 Un-damped and damped vibration: If no energy is lost or dissipated in friction or other resistance during oscillation, the vibration is known as un-damped vibration. If any energy lost in this way, however, it is called damped vibration. In many physical systems, the amount of damping is so small that it can be disregarded for most engineering purposes. However, consideration of damping becomes extremely important in analyzing vibratory system near resonance.  Linear and nonlinear vibration: If all the basic components of vibratory system—the spring, the mass and the damper—behave linearly, the resulting vibration is known as linear vibration. If, however, any of the basic components behave non- linearly, the vibration is called non-linear vibration.

3.2 IMPORTANCE OF VIBRATION  The increasing demands of high productivity and economical design led to higher operation speeds of machinery and efficient use of materials through light weight structures. Theses makes the trend of resonance conditions more frequent the periodic measurement of vibrations characteristic of machinery and structures become essential to ensure adequate safety margins. Any observed shift in the natural frequencies or other vibration characteristics will indicate either failure or a need for maintenance of the machine.

8

 The measurement of the natural frequencies of the structure or machine is useful in selecting the operational speed of nearby machinery to avoid resonant conditions.  The theoretically computed vibration characteristics of a machine or structure may be different from the actual values due to the assumptions made in the analysis.  In many applications survivability of a structure or machine in a specified vibration environment is to be determined. If the structure or machine can perform the expected task even after completion of testing under the specified vibration environment, it is expected to survive the specified conditions.  Continuous systems are often approximated as multi-degree of freedom systems for simplicity .If the measured natural frequencies and mode shapes of a continuous system are comparable to the computed natural frequencies and mode shapes of the Multi-degree of freedom model, then the approximation will be proved to be a valid one.  The measurement of the input and the resulting output vibration of a system helps in identifying the system in terms of its mass stiffness and damp.  The information about ground vibration due to earthquakes, fluctuating wind velocities on structures, random variation of ocean waves and road surface roughness are in the design of structures, machines oil platforms and vehicle suspensions systems.

9

CHAPTER 4 Euler Bernoulli Beam Theory Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case for small deflections of a beam that is subjected to lateral loads only. It is thus a special case of Timoshenko beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution. 4.1 Mathematical Formulation

Fig.2 A Beam under Transverse Vibration

Consider a long slender beam as shown in figure 1 subjected to transverse vibration. The free body diagram of an element of the beam is shown in the figure 2. Here, M(x,t) is the bending moment, V(x,t) is the shear force, f(x,t) and is the external force per unit length of the beam. Since the inertia force acting on the element of the beam is ρA(x)dx

𝜕2 𝑤(𝑥,𝑡) 𝜕𝑡 2

10

Fig.3 Free body Diagram of a section of a beam under transverse vibration

Balancing the forces in z direction gives -(V+ dV) + f(x,t)dx + V = ρA(x)dx

𝜕2 𝑤(𝑥,𝑡) 𝜕𝑡 2

where ρ is the mass density and A(x) is the cross sectional area of the beam. The moment equation about the y axis leads to (M+dM) – (V+dV)dx +f(x,t)dx

𝑑𝑥 2

– M= 0

By writing 𝜕𝑉

dV =

𝜕𝑥

dx and

dM =

𝜕𝑀 𝜕𝑥

dx

and disregarding terms involving second powers in dx , the above equations can be written as -

𝜕𝑉(𝑥,𝑡) 𝜕𝑥

+ f(x,t) = ρA(x) 𝜕𝑀(𝑥,𝑡) 𝜕𝑥

By using the relation V = -

𝜕𝑀 𝜕𝑥

𝜕2 𝑀(𝑥,𝑡) 𝜕𝑥 2

𝜕2 𝑤(𝑥,𝑡) 𝜕𝑡 2

– V(x,t)= 0

from above two equations + f(x,t) = ρA(x)

𝜕2 𝑤(𝑥,𝑡) 𝜕𝑡 2

11

From the elementary theory of bending of beam, the relationship between bending moment and deflection can be expressed as M(x,t) = EI(x)

𝜕2 𝑤(𝑥,𝑡) 𝜕𝑥 2

where E is the Young’s Modulus and I(x) is the moment of inertia of the beam cross section about the y axis. Inserting above two equations, we obtain the equation of the motion for the forced transverse vibration of a non-uniform beam: 𝜕2

2

𝜕𝑥 2

2

𝜕 𝑤(𝑥,𝑡) [EI(x)𝜕 𝑤(𝑥,𝑡) ] +ρA(x) = f(x,t) 𝜕𝑥 𝜕𝑥 2

2

For a uniform beam above equation reduces to 4

2

𝑤(𝑥,𝑡) 𝑤(𝑥,𝑡) [ EI(x)𝜕 𝜕𝑥 ]+ρA(x)𝜕 𝜕𝑥 = f(x,t) 4

2

For free vibration, f(x,t) =0 , and so the equation of motion becomes 4 2 𝜕 𝑤(𝑥,𝑡) 𝑐 𝜕𝑥 4

+

𝜕2 𝑤(𝑥,𝑡) 𝜕𝑥 2

=0

where c=√

𝐸𝐼 𝜌𝐴

Since the equation of motion involves a second order derivative with respect to time and a fourth order derivative with respect to x , two initial equations and four boundary conditions are needed for finding a unique solution for 𝑤(x,t). Usually,the values of transverse displacement and velocity are specified as 𝑤0 (𝑥 ) and 𝑤0′ (𝑥 ) at t=0, so the initial conditions become: 𝑤(x,t=0) = 𝑤0 (𝑥 ) 𝜕𝑤(𝑥,𝑡=0) 𝜕𝑡

= 𝑤0′ (𝑥 )

12

CHAPTER 5 Theoretical Analysis Of Free Vibration Of Cantilever Beam

The free vibration solution can be found using the method of separation of variables as W(x,t) = W(x)T(t) Substituting this equation in the final equation of motion and rearranging leads to 𝑐2

𝑑4 𝑊(𝑥)

𝑊(𝑥)

𝑑𝑥 4

=-

1 𝑑2 𝑇(𝑡)

𝑇(𝑡)

𝑑𝑡 2

= a = ω2

where a = ω2 is a positive constant. Above equation can be written as two equations: 𝑑4 𝑊(𝑥)

𝑑𝑥 4 2 𝑑 𝑇(𝑡) 𝑑𝑡 2

where,

ω2

– β4W(x) =0

+ ω2T(t) = 0 𝜌𝐴𝜔2

β= 2 = 𝑐 𝐸𝐼 The solution to time dependent equation can be expressed as T(t) = A1 cos 𝜔𝑡 + B sin 𝜔𝑡 where, A and B are constant that can be found from the initial conditions. For the solution of displacement dependent equation we assume, W(x) = Cesx where C and s are constant, and derive the auxiliary equation as S1,2 = ± β, s1,2 = ± iβ Hence the solution of the equation becomes W(x) = C1eiβx + C2e-iβx + C1eβx + C2e-βx where C1, C2, C3 and C4 are constant.

13

Above equation can also be expressed as: W(x) = C1 cosh βx + C2 sinh βx + C3 cos βx + C4 sin βx At w(0) = 0, At w’(0)=0, At w’’(L)=0, At w’’’(l) =0, From Eq.(1) From Eq.(2) From Eq.(3) From Eq.(4) From Eq.(5) From Eq.(6) Hence,

C1 + C3 = 0 C2 + C4 = 0 C1 cosh βL + C2 sinh βL- C3 cos βL - C4 sin βL=0 C1 sinh βL + C2 cosh βL+ C3 sin βL - C4 cos βL=0 C 1 = - C3 C 2 = - C4 C1 cosh βL + C2 sinh βL+ C1 cos βL + C2 sin βL=0 C1 sinh βL + C2 cosh βL- C1 sin βL + C2 cos βL=0 𝑐1 𝑐2 𝑐1 𝑐2

-

==

sinh 𝛽(𝐿)+sin 𝛽(𝐿) cosh 𝛽(𝐿)+cos 𝛽(𝐿)

cosh 𝛽(𝐿)+cos 𝛽(𝐿) sin 𝛽(𝐿)−sinh 𝛽(𝐿)

sinh 𝛽(𝐿)+sin 𝛽(𝐿) cosh 𝛽(𝐿)+cos 𝛽(𝐿)

=

cosh 𝛽(𝐿)+cos 𝛽(𝐿) sin 𝛽(𝐿)−sinh 𝛽(𝐿)

And finally we get cosh 𝛽(𝐿) ∗ cos 𝛽(𝐿) = −1 This transcendental equation has an infinite number of solution for βi=1,2,3…n Corresponding giving an infinite number of frequencies, Hence, ωi = (βiL)2 √

𝐸𝐼 𝜌𝐴𝐿4

(1) (2) (3) (4)

(5) (6)

14

CHAPTER 6 Theoretical Results And Calculations We get the transcendental equation valid for a cantilever beam which is cosh(βL) * cos(βL) = -1 from the theoretical analysis we performed before. This transcendental equation has an infinite number of solution for 𝛽𝑖 = 1,2,3….n We use the MATLAB software to solve this equation and finally get the value of the roots of the equation as: 𝜷𝒊 𝑳 1.8750 4.6941 7.8547 10.9955 14.1374 17.2791

Roots(i) 1 2 3 4 5 6 Table.1

Now putting the value of these roots of the equation in the formula we obtained before i.e. ω = 𝛽2 √

𝐸𝐼 𝜌𝐴

= (𝛽𝑙)2 √

𝐸𝐼 𝜌𝐴𝑙 4

(i)

we can get the mode shape frequencies of the concerned beam .

15

6.1 Specifications Of The Beam

Type Of Beam = Cantilever i.e. fixed at one end and free at the other end Material Of The Beam = Mild Steel Length Of The Beam(L) = 0.5m Width Of The Beam(b) = 0.045m Height Of The Beam(h) = 0.005m Moment Of Inertia(I) = 4.6875 * 10−10 𝑚4 Young’s Modulus(E) = 200 GPa Mass per unit length(m) = 0.30375 kg Density Of The Material Of The Beam(ρ) = 7800 kg/𝑚3 Poisson’s Ratio(µ) = 0.3 Using the required parameters and putting the values in the formula (i) we get the mode shape frequencies of a cantilever beam for 6 modes as follows: Mode

Frequency in Hz

1 2 3 4 5 6

16.357 102.525 287.068 562.543 929.973 1389.26 Table.2

16

6.2 Beam Modeling Using FEM Method Now a cantilever beam according to the specifications mentioned in the previous page is modeled by FEM method using Software package ANSYS. After modeling the beam we generate the mode shape results for the beam up to 6 modes and noted the value in the following table: Mode

Frequency in Hz

1 2 3 4 5 6

16.464 103.12 288.69 565.78 935.46 1397.6 Table.3

The percentage of error between the theoretical and experimentally obtained results are shown in Table.4

Mode 1 2 3 4 5 6

Theoretical Frequency Natural Frequency in Hz from ANSYS in Hz 16.357 16.464 102.525 103.12 287.068 288.69 562.543 565.78 929.973 935.46 1389.26 1397.6 Table.4

Percentage Error (%) 0.65 0.58 0.57 0.58 0.59 0.61

17

CHAPTER 7 Mode Shapes Of The Cantilever Beam In the previous section, the theoretical and experimental frequencies of the cantilever beam are obtained and compared. In this chapter the mode shapes of the cantilever beam as obtained in our experiment are demonstrated using FEM software package ANSYS. The graph of the mode shapes of a cantilever beam with mode shape 𝒙𝒏 as y-axis and distance from the fixed end (x/l) as x-axis is as follows:

Fig.4 Mode Shapes Of A Fixed-Free Cantilever Beam

18

7.1 Mode Shapes In ANSYS The cantilever beam is modeled in ANSYS as per specifications and the mode shape results are created in order to obtain the mode shapes

7.1.1 Mode 1

Fig.5 Mode shape of a cantilever beam in MODE 1 (isometric view)

Fig.6 Mode Shape of a cantilever beam in MODE 1 (x-y plane)

19

7.1.2 Mode 2

Fig.7 Mode shape of a cantilever beam in MODE 2 (isometric view)

Fig.8 Mode Shape of a cantilever beam in MODE 2 (x-y plane)

20

7.1.3 Mode 3

Fig.9 Mode shape of a cantilever beam in MODE 3 (isometric view)

Fig.10 Mode Shape of a cantilever beam in MODE 3 (x-y plane)

21

7.1.4 Mode 4

Fig.11 Mode shape of a cantilever beam in MODE 4 (isometric view)

Fig.12 Mode Shape of a cantilever beam in MODE 4 (x-y plane)

22

7.1.5 Mode 5

Fig.13 Mode shape of a cantilever beam in MODE 5 (isometric view)

Fig.14 Mode Shape of a cantilever beam in MODE 5 (x-y plane)

23

7.1.6 Mode 6

Fig.15 Mode shape of a cantilever beam in MODE 6 (isometric view)

Fig.16 Mode Shape of a cantilever beam in MODE 6 (x-y plane)

24

CHAPTER 8 Crack in Beams A crack in a structural member introduces local flexibility that would affect vibration response of the structure. This property may be used to detect existence of a crack together its location and depth in the structural member. The presence of a crack in a structural member alters the local compliance that would affect the vibration response under external loads. 8.1 Classification of Crack Based on geometries, cracks can be broadly classified as follows: Transverse crack: These are cracks perpendicular to beam axis. These are the most common and most serious as they reduces the cross section as by weaken the beam .They introduce a local flexibility in the stiffness of the beam due to strain energy concentration in the vicinity or crack tip. Longitudinal cracks: These are cracks parallel to beam axis. They are not that common but they pose danger when the tensile load is applied at right angles to the crack direction i.e. perpendicular to beam axis. Open cracks: These cracks always remain open .They are more correctly called “notches”. Open cracks are easy to do in laboratory environment and hence most experimental work is focused on this type of crack

Fig.17 A V-shaped Open Surface Transverse Crack As Shown In ANSYS

25

Breathing crack: These are cracks those open when the affected part of material is subjected to tensile stress and close when the stress is reversed. The component is most influenced when under tension. The breathing of crack results in non‐linearity in the vibration behavior of the beam. Most theoretical research efforts are concentrated on “transverse breathing” cracks due to their direct practical relevance. Slant cracks: These are cracks at an angle to the beam axis, but are not very common. Their effect on lateral vibration is less than that of transverse cracks of comparable severity. Surface cracks: These are the cracks that open on the surface .They can normally be detected by dye‐penetrates or visual inspection. Subsurface cracks: Cracks that do not show on the surface are called subsurface crack. Special techniques such as ultrasonic, magnetic particle, radiography or shaft voltage drop are needed to detect them. Surface cracks have a greater effect than subsurface cracks in the vibration behavior of shafts. In our project, we deal with a V-shaped Open Surface Transverse Crack or a V-shaped Notch on the top surface of the Cantilever Beam. 8.2 Crack detection: Detection of crack in a beam is performed in two steps. First, the finite element model of the cracked cantilever beam is established. The beam is discretized into a number of elements, and the crack position is assumed to be in each of the elements. Next, for each position of the crack in each element, depth of the crack is varied. Modal analysis for each position and depth is then performed to find the natural frequencies of the beam and variation in frequency is noted.

26

CHAPTER 9 Crack Modeling and Parametric Studies of The Beam In ANSYS 9.1 Crack Modeling Cracks in the beam create changes in geometrical properties so it becomes complex to study the effect of cracks in the beam. The crack modeling has been very important aspect. The analysis has been done using finite element method. FEM software package ANSYS has been used. Cracked beam has been modeled and free vibration analysis has been performed considering geometric and material non linearity. The crack is considered to be an open edge notch. Crack with a 0.5 mm width on the top surface of the beam has been modeled. It is assumed that crack have uniform depth across the width of the beam.

Fig.18 V-shaped edge crack with a 0.5 mm width on the top surface of the beam

27

9.2 Parametric Studies of The Beam The effects of the crack on natural frequency of a cantilever steel beam were investigated for various crack depths and crack locations.

Fig.19 A cantilever beam with a crack Properties: Width of the beam = 0.045 m Depth of the beam = 0.005 m Length of the beam = 0.5 m Elastic modulus of the beam (E) = 200GPa Poisson’s Ratio = 0.3 Density = 7800 kg/𝑚3 Natural frequencies of the cantilever beam for the first, second, third, fourth, fifth and sixth modes of vibration have been studied. For simplicity, the following dimensionless quantities are introduced: Crack location ratio 𝜻𝒄 = (𝒙𝒄 /L) Crack depth ratio H= (a/h)

28

9.2.1 Data Tables For Varying Crack Positions And Crack Depths 1.1 Cracked beam with crack at ζc (xc/L) =0.2 H(a/h)= 0.1 Mode 1 2 3 4 5 6

Frequency 16.453 103.12 288.64 565.53 935.09 1397.4 Table.5

1.2 Cracked beam with crack at ζc (xc/L) =0.2 Mode 1 2 3 4 5 6

H(a/h)= 0.3 Frequency 16.343 103.1 288.1 562.61 930.71 1395

Table.6

1.3 Cracked beam with crack at ζc (xc/L) =0.2 Mode 1 2 3 4 5 6

H(a/h)= 0.5 Frequency 16.028 103.08 286.6 554.76 919.92 1390.9

Table.7

29

2.1 Cracked beam with crack at ζc (xc/L) =0.3 Mode 1 2 3 4 5 6

H(a/h)= 0.1 Frequency 16.457 103.11 288.56 565.67 935.4 1396.9

Table.8

2.2 Cracked beam with crack at ζc (xc/L) =0.3 Mode 1 2 3 4 5 6

H(a/h)= 0.3 Frequency 16.382 102.99 287.03 564.42 934.61 1388.6

Table.9

2.3 Cracked beam with crack at ζc (xc/L) =0.3 Mode 1 2 3 4 5 6

H(a/h)= 0.5 Frequency 16.19 102.66 283.13 561.66 933.32 1369.4

Table.10

30

3.1 Cracked beam with crack at ζc (xc/L) =0.5 Mode 1 2 3 4 5 6

H(a/h)= 0.1 Frequency 16.462 103.06 288.69 565.47 935.45 1396.9

Table.11

3.2 Cracked beam with crack at ζc (xc/L) =0.5 Mode 1 2 3 4 5 6

H(a/h)= 0.3 Frequency 16.437 102.4 288.68 561.86 935.13 1387.7

Table.12

3.3 Cracked beam with crack at ζc (xc/L) =0.5 Mode 1 2 3 4 5 6

H(a/h)= 0.5 Frequency 16.376 100.81 288.66 553.68 935.01 1368.3

Table.13

31

9.2.2 Comparison Of Natural Frequency Ratio And Related Graphs In order to compare the behavior of the vibrating beam at different crack positions and varying crack depths at a particular mode, we calculate the natural frequency ratio which is equal to the natural frequency of the cracked beam to the frequency of the un-cracked beam at a particular mode and compare this ratio at different conditions. MODE 1 Crack Position ζc (xc/L)

0.2

0.3

0.5

Crack Depth Ratio H(a/h) Natural Frequency Ratio (ωc/ω)

Un-Cracked Beam 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5

1.0000 0.9993 0.9926 0.9735 0.9995 0.9950 0.9833 0.9998 0.9983 0.9946

Table.14

Fig.20 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 1)

32

MODE 2 Crack Position ζc (xc/L)

0.2

0.3

0.5

Crack Depth Ratio H(a/h) Natural Frequency Ratio (ωc/ω)

Un-Cracked Beam 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5

1.0000 1.0000 0.9998 0.9996 0.9999 0.9987 0.9955 0.9994 0.9930 0.9775

Table.15

Fig.21 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 2)

33

MODE 3 Crack Position ζc (xc/L)

0.2

0.3

0.5

Crack Depth Ratio H(a/h) Natural Frequency Ratio (ωc/ω)

Un-Cracked Beam 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5

1.0000 0.9998 0.9979 0.9927 0.9995 0.9942 0.9807 1.0000 0.9999 0.9998

Table.16

Fig.22 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 3)

34

MODE 4 Crack Position ζc (xc/L)

0.2

0.3

0.5

Crack Depth Ratio H(a/h) Natural Frequency Ratio (ωc/ω)

Un-Cracked Beam 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5

1.0000 0.9995 0.9943 0.9805 0.9998 0.9975 0.9927 0.9994 0.9930 0.9786

Table.17

Fig.23 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 4)

35

MODE 5 Crack Position ζc (xc/L)

0.2

0.3

0.5

Crack Depth Ratio H(a/h) Natural Frequency Ratio (ωc/ω)

Un-Cracked Beam 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5

1.0000 0.9996 0.9949 0.9833 0.9999 0.9990 0.9977 0.9999 0.9996 0.9995

Table.18

Fig.24 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 5)

36

MODE 6 Crack Position ζc (xc/L)

0.2

0.3

0.5

Crack Depth Ratio H(a/h) Natural Frequency Ratio (ωc/ω)

Un-Cracked Beam 0.1 0.3 0.5 0.1 0.3 0.5 0.1 0.3 0.5

1.0000 0.9998 0.9981 0.9952 0.9994 0.9935 0.9798 0.9994 0.9929 0.9790

Table.19

Fig.25 Graph of Natural frequency Ratio Vs. Crack Depth Ratio (Mode 6)

37

CHAPTER 10 Observations In the graphs plotted above we observe the following things: We observe that a presence of crack reduces the natural frequency of a beam in comparison to the un-cracked beam which increase the chances of resonance occurring in a structure which is very harmful for any structure or mechanism. The mode by mode observations are given below: Mode 1: Here the frequency variation is highest when the crack is closest to the fixed end and reduces as the crack position goes away from the fixed end Mode 2: Here the frequency variation is highest when the crack is farthest to the fixed end and reduces as the crack position comes closer to the fixed end Mode 3: Here the frequency variation is highest when the crack is at distance of ζc= 0.3 from the fixed end and lowest when it is farthest from the fixed end. Mode 4: Here the frequency variation is highest when the crack is at distance of ζc= 0.5 from the fixed end and lowest when the crack is at distance of ζc= 0.3. When the crack is at distance of ζc= 0.2 the variation in frequency is almost similar to that when the crack is at distance of ζc= 0.5. Mode 5: Here the frequency variation is highest when the crack is closest to the fixed end and reduces as the crack position goes away from the fixed end

38

Mode 6: Here the frequency variation is highest when the crack is farthest to the fixed end and reduces as the crack position comes closer to the fixed end So we can say that the drop in natural frequency for a cracked beam does not only depend on the crack position but it also depends on the mode number in which the beam is vibrating. Also from the graph we can easily say that when the crack positions are constant i.e. at particular crack location, the natural frequencies of a cracked beam are inversely proportional to the crack depth irrespective of its mode number.

39

CONCLUSION Presence of cracks reduces the stiffness of the beam, thus causing the natural frequency of that beam to drop considerably which increases the chances of resonance occurring there which is very harmful for any structure. So detection of crack in a beam is very important in order to avoid any unfortunate hazardous situations. Comparing the natural frequencies of the cracked beam to the un-cracked one as performed by us in this project is quite an effective way for crack detection. We can conclude that natural frequencies of a beam changes substantially due to the presence of cracks depending upon location and size of cracks. When the crack positions are constant i.e. at particular crack location, the natural frequencies of a cracked beam are inversely proportional to the crack depth.

And finally we can conclude that the change in natural frequencies of a beam is not only a function of crack depth, and crack location, but also of the mode number.

40

References [1]. J.P. Chopade , R.B. Barjibhe, journal of Free Vibration Analysis of Fixed Free Beam with Theoretical and Numerical Approach Method (Vol. 2 Issue 1 February 2013). [2]. Gawali A.L. and Sanjay C. Kumawat , journals of Vibration Analysis of Beams (Volume 1, Issue 1, 2011) [3]. Papadopoulos C.A., Dimarogonas A.D. (1987) Journal of Sound and Vibration, 117(1), 81-93. [4]. Zheng D.Y., Fan S.C. (2003) Journal of sound and vibration, 267, 933954. [5]. Christides S., Barr A.D.S. (1984) Journal of mechanical science, 26(11/12), 639-648. [6]. Shen M.H.H., Pierre C. (1990) Journal of sound and vibration, 138(1), 115-134. [7]. Shen M.H.H., Pierre C. (1994) Journal of sound and vibration, 170(2), 237-259. [8]. Kocat¨urk T., S¸ims¸ek M. (2005) Sigma J. Eng. and Nat. Sci. 1, 108122. [9]. Kocat¨urk T., S¸ims¸ek M. (2005) Sigma J. Eng. and Nat. Sci., 3, 79-93. [10]. Lee J., Schultz W.W. (2004) J. Sound Vib., 269, 609-621 [11]. S.Christides, A.D.S. Barr. "One- dimensional theory of cracked Bernoulli-Euler beam." Journal of Mechanical Science 26, 1984: 639-648.

41

[12]. Robert Liang, Fred K. Choy, Jialou Hu. "Detection of cracks in beam structure using measurment of natural frequencies." Journal of the Franklin Institute, Vol.328, 1991: 505-518. [13]. D.Dimarogonas, Andrew. "Vibration of cracked structures: Astate of the art review." Engineering Fracture Mechanics, Vol.55, 1996: 831-857. [14]. [M. H. H. Shen, Y. C. Chu. "Vibration of beams with fatigue cracks." Computers and Structures 45, 1992: 79-93. [15]. M. Chati, R. Rand, S. Mukherjee. "Modal analysis of cracked beam." Journal of Sound and Vibration 207, 1997: 249-270. [16]. M. Kisa, B. Brandon. "The effects of closure of cracks on the dynamics of cracked cantilever beam." Journal of Sound and Vibration, 238(1), 2000: 1-18. [17]. P.N. Saavedra, L.A.Cuitino. "Crack detection and vibration behavior of cracked beams." Computers and Structures 79, 2001: 1451-1459. [18]. T. G. Chondros, A. D. Dimorogonas, J. Yao. "Vibration of a beam with a breathing crack." Journal of Sound and Vibration 239(1), 2001: 57-67. [19]. D.Y. Zheng. "Free Vibration Analysis Of A Cracked Beam By finite Element Method." Journal of Sound and Vibration, 2004: 457-475 [20]. D.Y. Zheng. "Free Vibration Analysis Of A Cracked Beam By finite Element Method." Journal of Sound and Vibration, 2004: 457-475. [21]. Huszar, Zsolt. "Vibration of cracked reinforced and prestressed concrete beams Architecture and Civil Engineering, Vol. 6, 2008: 155164

42

[22]. G.M.Owolabi, Swamidas, SeshadrI " Crack detection in beams using changes in frequencies and amplitudes of frequency response functions." Journal of Sound and Vibration, 265(2004) 1-22 [23]. M.Behzad, Ebrahimi, A.Meghdari. "A continuous vibration theory for beams with a vertical edge crack." Mechanical Engineering, Vol. 17, 2010: 194-204. [24]. E. I. Shifrin. “Natural Frequencies Of A Beam With An Arbitrary Number Of Cracks.” Journal of Sound and Vibration, 222(3), 1999: 409– 423