What is Modal Analysis

Vibration can occur in any physical system. Properties of the system are the frequencies at which vibration occurs and the modal shapes which the vibrating system assumes, these properties can be analytically determined using Modal Analysis.

Even though Vibration modes analysis is a critical element of a design it is often ignored. Structural elements like complex steel floor systems can be predominantly subjected to noticeable vibration. Natural vibration modes in structural works and mechanical support systems will cut down equipment life, Premature or unexpected failure may occur due to vibration modes this may result in hazardous situations. Potential for failure or damage due to rapid stress cycles of vibration can be assessed by detailed fatigue analysis.

Detailed seismic study also needs an understanding of the natural vibration modes of a system, because during seismic activity the large amount of energy acting on a system varies with frequency.

The fundamental vibration mode shapes and corresponding frequencies are resolved by detailed modal analysis. This can be comparatively simple for basic elements of a simple system and extremely complex when qualifying a complicated mechanical device or a complicated structure exposed to periodic loading. These systems necessitate exact determination of natural frequencies and mode shapes by means of techniques such as Finite Element Analysis.

What is Hooke’s Law?

Hooke’s law is named after the seventeenth century physicist Robert Hooke who discovered it in 1660 (18 July 1635 - 3 March 1703).

Deformation (change of shape) of a solid is caused by a force that can either be compressive or tensile when applied in one direction (plane). Compressive forces try to compress the object (make it smaller or more compact) while tensile forces try to tear it apart. We can study these effects by looking at what happens when you compress or expand a spring. Hooke’s Law describes the relationship between the force applied to a spring and its extension.

Hooke’s Law - the relationship between extension of a spring and the force applied to it.

Deviation from Hooke’s Law
We know that if you have a small spring and you pull it apart too much it stops ’working’. It bends out of shape and loses its springiness. When this happens Hooke’s Law no longer applies, the spring’s behaviour deviates from Hooke’s Law. Depending on what type of material we are dealing the manner in which it deviates from Hooke’s Law is different. We give classify materials by this deviation. The following graphs show the relationship between force and extension for different materials and they all deviate from Hooke’s Law. Remember that a straight line show proportionality so as soon as the graph is no longer a straight line, Hooke’s Law no longer applies.

Brittle material :
This graph shows the relationship between force and extension for a brittle, but strong material.
Note that there is very little extension for a large force but then the material suddenly fractures. Brittleness is the property of a material that makes it break easily without bending. Have you ever dropped something made of glass and seen it shatter? Glass does this because of its brittleness.
Plastic material :
Here the graph shows the relationship between force and extension for a plastic material. The material extends under a small force but it does not fracture.

Ductile material :
In this graph the relationship between force and extension is for a material that is ductile. The material shows plastic behavior over a range of forces before the material finally fractures.
Ductility is the ability of a material to be stretched into a new shape without breaking.
Ductility is one of the characteristic properties of metals.

A good example of this is aluminium, many things are made of aluminium. Aluminium is used for making everything from cool drink cans to aeroplane parts and even engine blocks for cars. Think about squashing and bending a cool drink can. Brittleness is the opposite of ductility.

When a material reaches a point where Hooke’s Law is no longer valid, we say it has reached its limit of proportionality. After this point, the material will not return to its original shape after the force has been removed. We say it has reached its elastic limit.

Properties and characteristics of Wave

 Contents :
1.0 Definition of Wave
2.0 Characteristics of Waves
2.1 Amplitude
2.2 Wavelength
2.3 Period
2.4 Frequency
2.5 Speed
3.0 Types of Waves
3.1 Transverse waves
3.2 Longitudinal wave
4.0 Properties of Waves
4.1 Reflection
4.2 Refraction
4.3 Interference
4.4 Standing Waves
4.5 Diffraction
4.5.1 Huygen's Principle
4.6 Dispersion
1.0 Definition of waves :
Waves are disturbances which propagate (move) through a medium, Light is a special case, it exhibits wave-like properties but does not require a medium through which to Propagate. Waves occur frequently in nature. The most obvious examples are waves in water, on a dam, in the ocean, or in a bucket. Waves can be viewed as transfer energy rather than the movement of a particle. Particles form the medium through which waves propagate but they are not the wave. Waves in water consist of moving peaks and troughs. A peak is a place where the water rises higher than when the water is still and a trough is a place where the water sinks lower than when the water is still. A single peak or trough we call a pulse. A wave consists of a train of pulses.

So waves have peaks and troughs. The following diagram shows the peaks and troughs on a wave.

If we look very carefully we notice that the height of the peaks above the level of the still water is the same as the depth of the troughs below the level of the still water. The size of the peaks and troughs is the same.

2.0 Characteristics of Waves:
2.1 Amplitude :

The characteristic height of a peak and depth of a trough is called the amplitude of the wave. The vertical distance between the bottom of the trough and the top of the peak is twice the amplitude. We use symbols agreed upon by convention to label the characteristic quantities of the waves. Normally the letter A is used for the amplitude of a wave. The units of amplitude are metres (m).

2.2 Wavelength :
Look a little closer at the peaks and the troughs. The distance between two adjacent (next to each other) peaks is the same no matter which two adjacent peaks you choose. So there is a fixed distance between the peaks.

Looking closer you'll notice that the distance between two adjacent troughs is the same no matter which two troughs you look at. But, more importantly, it is the same as the distance between the peaks. This distance which is a characteristic of the wave is called the wavelength.
Waves have a characteristic wavelength. The units are metres (m).

The wavelength is the distance between any two adjacent points which are in phase. Two points in phase are separate by an integer (0,1,2,3,...) number of complete wave cycles. They don't have to be peaks or trough but they must be separated by a complete number of waves.

2.3 Period :
The time between two adjacent peaks is same and also the time between two adjacent troughs always the same, no matter which two adjacent troughs you pick. The time you have been measuring is the time for one wavelength to pass by. We call this time the period and it is a characteristic of the wave.

Waves have a characteristic time interval which we call the period of the wave and denote with the symbol T. It is the time it takes for any two adjacent points which are in phase to pass a fixed point. The units are seconds (s).

2.4 Frequency :
There is another way of characterising the time interval of a wave. We timed how long it takes for one wavelength to pass a fixed point to get the period. We could also turn this around and say how many waves go by in 1 second.

We can easily determine this number, which we call the frequency and denote f. To determine the frequency, how many waves go per second, we work out what fraction of a waves goes by in 1 second by dividing 1 second by the time it takes T. The unit of frequency is the Hz.
Waves have a characteristic frequency. F=1/T

2.5 Speed :
Now if you are watching a wave go by you will notice that they move at a constant velocity. The speed is the distance you travel divided by the time you take to travel that distance. This is excellent because we know that the waves travel a distance equal to wavelength in a time T. This means that we can determine the speed.

There are a number of relationships involving the various characteristic quantities of waves.
A simple example of how this would be useful is how to determine the velocity when you have the frequency and the wavelength. We can take the above equation and substitute the relationship between frequency and period to produce an equation for speed of the form

3.0 Types of Waves :
We agreed that a wave was a moving set of peaks and troughs and we used water as an example. Moving peaks and troughs, with all the characteristics we described, in any medium constitute a wave. It is possible to have waves where the peaks and troughs are perpendicular to the direction of motion, like in the case of water waves. These waves are called transverse waves.

There is another type of wave called a longitudinal wave and it has the peaks and troughs in the same direction as the wave is moving. The question is how do we construct such a wave?

An example of a longitudinal wave is a pressure wave moving through a gas. The peaks in this wave are places where the pressure reaches a peak and the troughs are places where the pressure is a minimum.

4.0 Properties of Waves :
We have discussed some of the simple characteristics of waves that we need to know. Now we can progress onto some more interesting and, perhaps, less intuitive properties of waves.

4.1 Reflection :
When waves strike a barrier they are reflected. This means that waves bounce off things. Sound waves bounce off walls, light waves bounce off mirrors, radar waves bounce off planes and it can explain how bats can fly at night and avoid things as small as telephone wires. The property of reflection is a very important and useful one.

When waves are reflected, the process of reflection has certain properties. If a wave hits an obstacle at a right angle to the surface then the wave is reflected directly backwards. If the wave strikes the obstacle at some other angle then it is not reflected directly backwards. The angle that the wave arrives at is the same as the angle that the reflected wave leaves at. The angle that waves arrives at or is incident at equals the angle the waves leaves at or is reflected at.
4.2 Refraction :
Sometimes waves move from one medium to another. The medium is the substance that is carrying the waves. In our first example this was the water. When the medium properties change it can affect the wave.

Let us start with the simple case of a water wave moving from one depth to another. The speed of the wave depends on the depth. If the wave moves directly from the one medium to the other then we should look closely at the boundary. When a peak arrives at the boundary and moves across it must remain a peak on the other side of the boundary. This means that the peaks pass by at the same time intervals on either side of the boundary. The period and frequency remain the same! But we said the speed of the wave changes, which means that the distance it travels in one time interval is different i.e. the wavelength has changed. Going from one medium to another the period or frequency does not change only the wavelength can change.

Now if we consider a water wave moving at an angle of incidence not 90 degrees towards a change in medium then we immediately know that not the whole wave front will arrive at once. So if a part of the wave arrives and slows down while the rest is still moving faster before it arrives the angle of the wave front is going to change. This is known as refraction. When a wave bends or changes its direction when it goes from one medium to the next. If it slows down it turns towards the perpendicular.
If the wave speeds up in the new medium it turns away from the perpendicular to the medium surface.
When you look at a stick that emerges from water it looks like it is bent. This is because the light from below the surface of the water bends when it leaves the water. Your eyes project the light back in a straight line and so the object looks like it is a different place. 4.3 Interference :
If two waves meet interesting things can happen. Waves are basically collective motion of particles. So when two waves meet they both try to impose their collective motion on the particles. This can have quite different results.

If two identical (same wavelength, amplitude and frequency) waves are both trying to form a peak then they are able to achieve the sum of their efforts. The resulting motion will be a peak which has a height which is the sum of the heights of the two waves. If two waves are both trying to form a trough in the same place then a deeper trough is formed, the depth of which is the sum of the depths of the two waves. Now in this case the two waves have been trying to do the same thing and so add together constructively. This is called constructive interference.

If one wave is trying to form a peak and the other is trying to form a trough then they are competing to do different things. In this case they can cancel out. The amplitude of the resulting wave will depend on the amplitudes of the two waves that are interfering. If the depth of the trough is the same as the height of the peak nothing will happen. If the height of the peak is bigger than the depth of the trough a smaller peak will appear and if the trough is deeper then a less deep trough will appear. This is destructive interference.
4.4 Standing Waves :
When two waves move in opposite directions, through each other, interference takes place. If the two waves have the same frequency and wavelength then a specific type of constructive interference can occur: standing waves can form.

Standing waves are disturbances which don't appear to move, they look like they stay in the same place even though the waves that from them are moving.

4.5 Diffraction :
One of the most interesting, and also very useful, properties of waves is diffraction. When a wave strikes a barrier with a hole only part of the wave can move through the hole. If the hole is similar in size to the wavelength of the wave diffractions occurs. The waves that comes through the hole no longer looks like a straight wave front. It bends around the edges of the hole. If the hole is small enough it acts like a point source of circular waves. This bending around the edges of the hole is called diffraction. To illustrate this behavior we start by with Huygen's principle.

4.5.1 Huygen's Principle :
Huygen's principle states that each point on a wave front acts like a point source or circular waves. The waves emitted from each point interfere to form another wave front on which each point forms a point source. A long straight line of points emitting waves of the same frequency leads to a straight wave front moving away.

4.6 Dispersion :
Dispersion is a property of waves where the speed of the wave through a medium depends on the frequency. So if two waves enter the same dispersive medium and have different frequencies they will have different speeds in that medium even if they both entered with the same speed.

What the atom is made up of?

The Greek word atom means indivisible. The discovery of the fact that an atom is actually a complex system and can be broken in pieces was the most important step and pivoting point in the development of modern physics.

It was discovered by Rutherford in 1911, that an atom consists of a positively charged nucleus and negative electrons moving around it. At first, people tried to visualize an atom as a microscopic analog of our solar system where planets move around the sun. This naive planetary model assumes that in the world of very small objects the Newton laws of classical mechanics are valid. This, however, is not the case.

The microscopic world is governed by quantum mechanics which does not have such notion as trajectory. Instead, it describes the dynamics of particles in terms of quantum states that are characterized by probability distributions of various observable quantities.
For example, an electron in the atom is not moving along a certain trajectory but rather along all imaginable trajectories with different probabilities. If we were trying to catch this electron, after many such attempts we would discover that the electron can be found anywhere around the nucleus, even very close to and very far from it. However, the probabilities of finding the electron at different distances from the nucleus would be different. What is amazing: the most probable distance corresponds to the classical trajectory!

You can visualize the electron inside an atom as moving around the nucleus chaotically and extremely fast so that for our \mental eyes" it forms a cloud. In some places this cloud is denser while in other places more thin. The density of the cloud corresponds to the probability of finding the electron in a particular place. Space distribution of this density (probability) is what we can calculate using quantum mechanics. Results of such calculation for hydrogen atom are shown in Fig. As was mentioned above, the most probable distance (maximum of the curve) coincides with the Bohr radius.

Quantum mechanical equation for any bound system (like an atom) can have solutions only at a discrete set of energies E1;E2;E3 : : : , etc. There are simply no solutions for the energies E in between these values, such as, for instance, E1 < E < E2. This is why a bound system of microscopic particles cannot have an arbitrary energy and can only be in one of the quantum states. Each of such states has certain energy and certain space configuration, i.e. distribution of the probability.

A bound quantum system can make transitions from one quantum state to another either spontaneously or as a result of interaction with other systems. The energy conservation law is one of the most fundamental and is valid in quantum world as well as in classical world. This means that any transition between the states with energies Ei and Ej is accompanied with either emission or absorption of the energy ¢E = jEi ¡ Ej j. This is how an atom emits light.
Electron is a very light particle. Its mass is negligible as compared to the total mass of the atom. For example, in the lightest of all atoms, hydrogen, the electron constitutes only 0.054% of the atomic mass. In the silicon atoms that are the main component of the rocks around us, all 14 electrons make up only 0.027% of the mass. Thus, when holding a heavy rock in your hand, you actually feel the collective weight of all the nuclei that are inside it.

How Fibre Optics works?

Water can be directed from one place to another by confining it within a pipe. In the same way light can be directed from one place to another by confining it within a single glass Fibre.

The light is kept within the fibre by total internal reflection. The amount of light which can be carried by a single Fibre is very small so it is usual to form a light tube tapping a few thousand Fibres together. On great advantage of such a light tube is flexibility; it can be ties in knots and still function. However since total internal reflection only occurs when light is going from a medium to a less dense medium, it is necessary to coat each fibre with glass of a lower refractive index. Otherwise light would leak from one fibre at their points of contact.

Light tube can be used to bring light from a lamp to an object, thus illuminating the object. A second light tube can then used to carry light from the illuminated object to an observer, thus enabling the object to be seen and photographed. The procedure has been used to photograph the digestive system the reproductive system and many other parts of the human body. In the case of the light tube carrying light from the object to the observer, it is vital that the individual fibres in the tube do not cross each other, otherwise the image will become garbled. Like radio waves, light waves are electromagnetic. However, their shorter wavelength and higher frequency means that a single light beam can carry far more telephone conservations at one time compared with a radio wave.

In the case of long fibre cables it would be necessary to incorporate a device to boost the intensity of the light to make up for losses due to absorption. Nevertheless the system has great potential for the communication industry, including the possibility of transmitting pictures over long distances.

The reason why light bends when going from one medium to another is because of the change of velocity.