In this chapter, the theory related to vibration, crack and the Linear Elastic Fracture Mechanics (LEFM) are presented. Then the attention is given to the mathematical formulation of a cracked uniform cantilever beam. The presence of crack reduces the local stiffness matrix which alters the dynamic response of the system.
2. Theory
2.1 Concept of vibration
Vibration definition
Most of the human activities like hearing, seeing, breathing etc are due to vibration. Hearing involves vibration of the eardrum, seeing is associated with vibration of light waves,breathing is based on the periodic motion of lungs. For the safe design, construction and operation of variety of machines and structures, an understanding of the vibratory behavior of structural
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a) Longitudinal or Rectilinear vibration: When the particles of the body move parallel to the axis of the body, it is called longitudinal vibration
b) Lateral or Transverse vibration: When the particles of the body move perpendicular to the axis of the body is called transverse vibration.
c) torsional vibration ()
3. Linear and Non-linear vibration:If the components of vibratory system( spring, mass, behave in linear way, then the vibration is known to be Linear vibration. Principle of superposition is valid here.
When the components vibrate in non-linear way, then it is called non-linear vibration.
2.2 History and Importance of vibration
The equation of motion for the transverse vibration of thin beams was derived by Daniel Bernoulli in 1735. The first solutions of the equation for various support conditions were given by Euler in 1744, which is known to be Euler-Bernoulli or Thin beam theory. Rayleigh included the effect of inertia and presented a beam theory. The improved theory by including the effect of rotary inertia and shear deformation known as Timoshenko or thick beam theory was presented by Stephen Timoshenko in
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2.3 Theory of crack
Crack can be defined as a line on the surface of a structure along which splitting takes place without breaking apart. Due to the presence of crack, the vibration response of the structure is affected. This property is utilized for detection of the location and depth of crack in the member.
Classification of cracks based upon their geometries:
1. Transverse cracks:
These are most common and very serious because the presence of this type of cracks reduces the cross sectional area that weakens the beam.
2. Longitudinal cracks:
These cracks act parallel to the beam axis. To the right angles of crack direction, when tensile loading is applied, they are very dangerous..
3. Open cracks:
These cracks remain open. More precisely they are known as notches. These are very easily adaptable for the lab environment. So, mostly for the experimental purpose, open cracks are preferred.
4. Breathing crack :
These cracks depend on the nature of the stresses acting on the material, if tensile stresses act , the cracks open. If the stresses are reversed, the cracks close. When cracks are open, its more critical. Due to the breathing of crack, the vibration response of the beam is
The three most important properties of a wave are the wavelength, the amplitude, and the frequency. The wavelength is the distance from one point on a wave to the next identical point on the next wave. The amplitude is the distance from a waves rest position to either the crest or trough of the wave. The frequency is a rate which represents the amount of times a wave repeats
Wave properties: Frequency: How many waves go past a point in one second; measured in hertz (Hz). The higher the frequency, the more energy in the wave. 1 Ask the children to play 'verbal tennis' in partners on words that describe sounds (eg loud, quiet, vibration, pitch). One child says one word related to sound, then their partner says another and so on. 2 Ask the children what things make sounds.
In "Passage 1: Sound is All around Us, Sound is Energy" Jason Torres claims that sound is a form of waves that vibrate in one directions called longitudinal waves. "The distance between two adjacent compressions, two adjacent rabout the longitudinal arefactions, or a single compression and rarefaction combined is a wavelength of a sound wave. " This is Jason Torres ' definition of Longitudinal Waves. The wave measures the sound and pitch. This pitch and sound can differ due to the extent of energy/waves released.
Your incisors are important for allowing you to easily bite into foods, and they play a critical role in determining the quality of your smile. Unfortunately, it is fairly common for individuals to crack these teeth. This is often due to an injury, but it can also be the result of extensive decay to the tooth. When this problem arises, it can present some cosmetic and functional challenges for the patient. By having these two questions answered, you should find yourself better able to have this type of dental injury addressed.
However, if the crack extends across your entire windshield, they will probably have to replace your entire windshield. Scratch Windshield Repair On your windshield, a scratch generally only appears on the surface of your glass and does not go deeper. Although you may not get a ticket for scratched glass, it could still impair your vision and compromise your safety. Generally, an auto glass repair shop should be able to fix the scratch without replacing your windshield in no time at all.
Mass vs Tangential Velocity K.Kirtanaa, Ms. Perez, November 14, 2016 Research Question: What is the effect of increasing mass on tangential velocity? Introduction: The experiment explores the relationship between the independent variable and the dependent variable. The independent variable is what you change in an experiment.
The mechanical vibrations transmitted through a medium; solid, liquid or a gas generates a sound wave. The sound is an energy, which deflects the particles of a medium in the same direction, and itself travel as a longitudinal waveform with areas of compression and rarefactions. The image generated by ultrasound is essentially based on similar principle by means of a machine, which includes an ultrasound transducer and a screen. The transducer contains a piezoelectric crystal made up of lead zirconite, which generates an ultrasound beam on applying an electrical current to the transducer. These signals travel through a medium in the body and after striking with various tissues, return back to the transducer.
First of all, what type of wave is light and sound? Light and sound are not the same types of wave. There are 2 types of waves and light is a transverse wave while sound is a longitudinal wave. Source D states, “Longitudinal waves are waves whose direction of vibration is the same as their direction of travel…. Transverse wave is a moving wave that consists of oscillations perpendicular to the direction of the energy transfer.”
Structural Crack - Shear Crack and Extra Compression Loading Crack Figure 1. Shear Crack Figure 1[1] is an example of shear crack found at Swinburne
Some types include Dry Friction; which resists the motion of two solid surfaces. Fluid Friction; which
I had first formal exposure to civil engineering in course “Architecture 1”. This course provided me the fundamental concepts of civil architectural principles such as designing procedure, architectural component’s layouts, functional spaces as well as environmental impacts. To strengthen my concepts I pursued the courses “Strength of Materials”, “Structural Mechanics” which gave me essential knowledge about the external and internal forces occuring in simple structural elements under action of various types of loading as well as fundamental concepts in structural analysis such as modellisation, classification of structures. I gained valuable exposure to the analysis of simple structures submitted to compound loadings by the application of the principle of superposition for such cases as: unsymmetric bending, combined bending and tension or compression, combined bending and torsion, general compound loading. Futhermore, I was fascinated by the force method and displacement method for analysis of statically and kinetically indeterminate structures.
Bernoulli’s theorem is a special application of the laws of motion and energy. The principle equation describes the pressure measured at any point in a fluid, which can be a gas or a liquid, to the density and the velocity of the specified flow. The theorem can be explained by the means of imagining a particle in a cylindrical pipe. If the pressure on both sides of the particle in the pipe is equal, the particle will be stationary and in equilibrium.
The purpose of this lab was to observe the phenomenon of resonance in an open ended cylindrical tube and use resonance to determine the velocity of sound in air at regular temperatures. A hypothesis for this lab was that if the frequency of the tuning fork increased, then the length of the tube to achieve resonance will increase because of the high amplitude of the vibration. The resonance of the open-closed tube was found through these steps. First, the tube was filled with water with an inner tube inside the outer tube. The water temperature was measured.
Table 1: 2024T351 Specimen Experimental Data The experimental ultimate tensile strength of 65,507.15 Psi is relatively close to the typical tensile strength of 64,000 Psi with 2.35 percent error. The experimental young's modulus of 10,644,380 Psi is close to the standard elastic modulus of 10,600,000 Psi with 0.42 percent error. Using the graphs, the yield stress was found using a 0.2% offset. The yield stress was found to be about 50,000 Psi, far from the standard 42,000 Psi. This resulted in a 19.05 percent error.
that zone is more subject to seismic disasters. Tectonic, volcanic, collapse and explosion, are the four different types of earthquakes. A tectonic earthquake is one that occurs when the earth 's crust breaks due to geological forces on rocks and adjoining plates that cause physical and chemical changes. A volcanic earthquake is when a volcano erupts and shakes the plates.