Frequency of Free Damped Vibrations (Viscous Damping)

          We have already know that the motion of a body is resisted by frictional forces. In vibrating systems, the effect of friction is referred to as "Damping". The Damping provided by fluid resistance is known as viscous damping.

          We have also discussed that in damped vibrations, the amplitude of the resulting vibration gradually diminishes. This is due to the reason that a certain amount of energy is always dissipated to overcome the frictional resistance. The resistance to the motion of the body is provided partly by the medium in which the vibration takes place and partly by the internal friction, and in some cases partly by a dash pot or other external damping device.

          Consider a vibrating system, as shown in Fig., in which a mass is suspended from one end of the spiral spring and the other end of which is fixed. A damper is provided between the mass and the rigid support.

Let          m = Mass suspended from the spring,

                s = Stiffness of the spring, 

                x = Displacement of the mass from the mean position at time t, 

               δ = Static deflection of the spring

                 = m.g/s, and

              c = Damping coefficient or the damping force per unit velocity.

          Since in viscous damping, it is assumed that the frictional resistance to the motion of the body is directly proportional to the speed of the movement, therefore

          Damping force or frictional force on the mass acting in opposite direction to the motion of the mass

                 = c . dx/dt

Accelerating force on the mass, acting along the motion of the mass

                = m . d² x/dt² 

and spring force on the mass. acting in opposite direction to the motion of the mass,

               = s.x

Therefore the equation of motion becomes

This is differential equation of second order. Assuming a solution of the form x = e^kt , where 'k' is a constant to be determined. Now the above differential equation reduces to

The two roots of the equation are

          The most general solution of the differential solution (i) with its right hand side equal to zero has only complementary function and it is given by

Where C_1 and  C₂ are two arbitrary constants which are to be determined from the initial conditions of the motion of the mass.

          It may be noted that the roots K₁ and K₂ may be real, complex conjugate (imaginary) or equal. We shall now discuss the three cases as below:

Case 1: When the roots are real (overdamping)

If ,, then the roots K₁ and K₂ are equal and negative. This is a case of Over damping or Large damping and the mass moves slowly to the equilibrium position. This motion is known as aperiodic, when the roots are real, the most general solution of the differential equation is  

Note: In actual practice, the overdamped vibrations are avoided.

Case  2: When the roots are complex conjugate (underdamping)

If , then the radical (i.e., the term under the square root) becomes negative.

The two roots K₁  and K₂ are then known as complex conjugate. This is a most practical case of damping and it is known as Under damping or Small damping. the roots are 

          We see from equation (iv), that the motion of the mass is simple harmonic whose circular damped frequency is  ω_d and the amplitude ae^-at  which diminishes exponentially with time as shown in Fig., Though the mass eventually returns to its equilibrium position because of its inertia, yet it overshoots and the oscillations may take some considerable time to die away. 

Note: When no damper is provided in the system, then C = 0. Therefore the frequency of the undamped vibration, 

It is the same as discussed under free-vibrations.

Case 3. When the roots are equal (critical damping)

  then the radical (i.e., the term under the square root) becomes zero and the two roots K₁  and K₂ are equal. This is a case of Critical damping. In other words, the critical damping is said to occur when frequency of damped vibration ( f_d) is zero (i.e motion is aperiodic). This type of damping is also avoided because the mass moves back rapidly to its equilibrium position, in the shortest possible time. 

          For critical damping, equation (ii) may be written as

          Thus the motion is again aperiodic. The critical damping coefficient (C_C) may be obtained by substituting C_C for C in the condition for critical damping, i.e.,

          The critical damping coefficient is the amount of damping required for a system to be critically damped.

Terms used in Automatic Control of Systems

Terms used in Automatic Control of Systems

The following terms are generally used in automatic control of systems: 

1. Command:  The result of the act of adjustment, i.e. closing a valve, moving a lever, pressing buttons etc., is known as Command.
2. Response: The subsequent result of the system to the command is known as response.
3. Process control: The automatic control of variables like  change in pressure, temperature and speed etc., in machine is termed as process control.
4. Process controller: The device which controls a process is called a process control.
5. Kinetic control: The automatic control of the displacement or velocity or acceleration of a member of a machine is called as Kinetic control.
6. Regulator: The device used to keep the variables at a constant desired value is called as regulator.
7. Feed back: It is defined as measuring the output of the machine for comparison with the input to the machine.
8. Error detector: A differential device used to measure the actual controlled quantity and to compare it continuously with the desired value is called an error detector. It is also called deviation sensor.
9. Transducer: It is a device to change a signal which is in one physical form to a corresponding signal in another physical form. The example of transducer are a loudspeaker (because it converts electrical signal into a sound) and photo-electric cell (because it converts a light signal into an electric signal). Similarly, the primary elements of all the many different forms of thermometers are transducers.
10. Amplification: It is defined as increasing the amplitude of the signal without affecting its waveform. For example, an error detector itself has insufficient power output to actuate the correcting mechanism and hence the error signal has to be amplified. This is generally done by employing mechanical or hydraulic or pneumatic amplifying elements like levers, gears etc.,

Types of Automatic Control System

Types of Automatic Control System

The automatic control systems are classified into two types they are:

1) Open loop (or) Unmonitored system

2) Closed loop (or) monitored system 

1. Open-loop or unmonitored system: When the input to a system is independent of the output from the system, then the system is called an open-loop or unmonitored system. It is also called calibrated system. Most measuring instruments are open-loop control systems, as for the same input signal, the readings will depend upon things ambient temperature and pressure. Following are the examples of open-loop system:

• A simple Bourdon tube pressure gauge commonly used for measuring pressure.
• A simple carburattor in which the air-fuel ratio adjusted through venturi remains same irrespective of load conditions.
• In traffic lights system, the timing of lights is present irrespective of intensity of traffic.

2. Closed-loop or monitored system: When output of a system is measured and is continuously compared with the required value, then it is known as closed-loop or monitored system. In this system, the output is measured and through a feedback transducer, it is sent to an error detector which detects any error in the output from the required value thus adjusting the input in a way to get the required output. Following are the examples of a closed-loop system:

(a) In a traffic control system, if the flow of traffic is measured either by counting the number of vehicles by a person or by counting the impulses due to the vehicles passing over a pressure pad and then setting the time of signal lights.

(b) In a thermostatically controlled water heater, whenever the temperature of water heater rises above the required point, the thermostat senses it and switches the water heater off so as to bring the temperature down to the required point. Similarly, when the temperature falls below the required point, the thermostat switches on the water heater to raise the temperature of water to the required point.

Thermostatically controlled water heater

Block Diagrams

Block Diagrams

Fig. Block diagram of a single Carburettor.

          The block diagrams are used to study the automatic control system in simplified way. In this, the functioning of a system is explained by the inter connected blocks where each block represents a labelled rectangle and is thought of as a block box with a definite function. These blocks are connected to other blocks by lines with arrow marks in order to indicate the sequence of events that are taking place. It may also show how the system operates, what are its inputs and outputs at various stages, and how the energy, information, and/or materials flow through it. Above Fig., shows the diagram of a simple carburattor. The reduction of a control system to a block diagram greatly facilitates the analysis of the system performance or response.

Lag in Response

Lag in Response

          We know that response is the subsequent result of the system to the command. In any control system, there is a delay in response (output) due to some inherent cause and it becomes difficult to measure the input and output simultaneously. This delay in response is termed as lag in response. For example, in steam turbine, with the sudden decrease in load, the hydraulic relay moves in the direction to close the valve. But unless the piston valve ports are made with literally zero overlap, there would be some lag in operation, since the first movement of the piston valve would not be sufficient to open the ports. This lag increases the probability of unstable operation.

Damping

Damping

          When torque is applied in a system in a direction opposite to its motion, it is known as Damping. In case of coulomb damping, the opposition is constant and, thus there will be a constant difference between the input and the output under steady conditions. In the viscous damping provided by dashpot, the opposition is proportional to the relative velocity. As the relative velocity is zero in the steady state, the damping is also zero.

NON-DESTRUCTIVE TEST

                      Non-Destructive tests (NDT) may be defined as those in which the test specimen would not damage such that it is rendered useless for future for which it was originally meant.

                       While studying various mechanical tests in previous sections, we have noted the effects of cracks and flaws. These should be detected at the early stages and the component replaced otherwise disaster will result. One can detect all microscopic flaws by NDT. NDT is the the method of detection and measurement of properties or condition of material, structures, machines without damaging (or) destroying their operational capabilities. Examples of NDT are: radiography, magnetic particle inspection, ultrasonic test, penetrating liquid method, electrical method, damping. All NDTs are used to detect various types of flaws on the surface of material or internal inclusions of impurities and these techniques are also very useful during preventing maintenance and repair. There are few techniques which do not require any special apparatus and are quite simple to handle and only a moderate skill being required. Some of the applications of NDTs are detecting: (i) surface cracks (ii) material composition (iii) internal inclusions (iv) internal voids and discontinuities and (v) condition of internal stress.

Now, we describe the various methods used for Non-destructive testing are as follows:

• X-Ray radiography
• Gamma radiography
• Magnetic particle inspection
• Ultrasonic testing
• Liquid penetration test
• Electrical method
• Damping

X-RAY RADIOGRAPHY

           Radiography technique is based on exposing the components to short wavelength radiations in the form of X-Rays, Gamma rays and radio-isotope welds. This method is used to check internal cracks, shrinkage cavities, slag inclusions, defects in materials and welds. These defects are of special importance designed to withstand high temperature and pressure employed in power plants, atomic reactors, pressure vessels and oil refining equipment; because they cause stress concentration which may frequently lead to part failure. Nowadays, radiography techniques are finding more extensive applications in the field of physical metallurgy and in the treatment of various diseases.

           Rays are absorbed by the materials through which they are passed in the proportion of their density. The rays, after passing through the components, show a picture on a fluorescent screen or on a photographic plate. The cracks, blow holes and cavities appear lighter, where as inclusions of impurities appear darker than the metal component. Developed photographic film show lighter and darker areas to represent the radiograph of defects in the component.

                  In X-Ray radiography, the portion of the casting where defects are suspected is exposed to X-Rays emitted from the X-ray tube. A cassette containing X-ray film is placed behind and in contact with the casting perpendicular to rays. X-rays after allowing through the blow hole in a casting, will be absorbed to lesser extent than X-rays which allowed to pass through sound metal, therefore, film appears to be more dark where defects are in line of X-ray beam. The exposed and developed X-ray film showing light and dark areas is termed as Radiograph. X-rays are useful only for small thickness material as their penetration power is less than that of gamma rays.

GAMMA RADIOGRAPHY

                The principle of detecting defects is same as X-ray radiography. Gamma rays are emitted during the disintegration of radio active material and X-rays are electromagnetic radiation.Gamma rays have shorter wavelengths and are more penetrating than X-rays. The source for gamma radiations is usually the radioactive isotopes of cobalt-60 enclosed in a special container or capsule. Gamma rays radiography give better results for thicker materials. Now a days cheap radioisotopes are available and this test can be performed in a very short time and therefore this method is becoming more popular. However, there are some limitations of this method, e.g., Handling of radioisotopes and precautions required.

Gamma-ray radiography differs from X-ray radiography in the following aspects:

1. The apparatus for Gamma-ray radiography is very simple and less costly than X-ray unit.
2. Unlike X-rays, Gamma rays from its source are emitted in all directions, therefore, a number of separate castings having cassette containing film, fastened to the back of each casting, are disposed in a circle around the equipment placed in a central position. This way, many castings can be radiographed simultaneously and overnight exposures may be taken without continues supervision.
3. X-rays are better than gamma rays for detecting small defects in casting sections less than about 50 mm.
4. Gamma rays are used for detecting defects in castings thicker than those inspected by X-rays.
5. X-ray method is much more rapid than Gamma-ray method, it requires seconds or minutes instead of hours.    

MAGNETIC PARTICLE INSPECTION

                 This test is generally used to locate cracks and surface defects in a wide range of products. But in particular, it is employed to detect fatigue cracks at points of local high tensile strengths. The name "Magnaflux" is commonly associated with this process. This method is a relatively simple and easy technique. It is almost free from any restriction as to size, shape, composition and heat treatment of ferromagnetic substance.

              This method is restricted to magnetic materials e.g., iron, cobalt, nickel. etc. This test is based on the principle that if there is flaw in the magnetic material through which a magnetic field is allowed to pass, the lines of magnetic force or flux will be distorted near the flaw and lines of magnetic flux will be uniform for magnetic materials which are defect free.

             This test is performed by magnetising the substance and the immersing the test piece in a bath of kerosene oil containing iron oxide powder. One can also use the coloured power. If a crack or void lies across the path of the magnetic flux, each side of the crack or void becomes a magnetic pole which attracts iron powder. The accumulation of iron dust on the crack portion of the sample reveal the crack. This test can detect both internal and external defects. One can detect the cracks caused by quenching fatigue failure in welding, blow holes in castings and grinding operations by this method.

ULTRASONIC TESTING

                       The sound waves whose frequency is above the upper pitch limit of the human ear are called Ultrasonics. Ultrasonic testing and inspection is one of the most useful non-destructive methods in metal testing. Rail roads, water pipes, boilers, air craft parts of forged materials, etc., are tested for cracks, inclusions or other internal discontinuities.

                      Ultrasonic waves are usually generated by the piezoelectric effect which converts electrical energy to mechanical energy. A quartz crystal is used for the purpose.

• The surface of casting to be inspected by ultrasonics is made fairly smooth by machining otherwise ultasonic waves can't be efficiently transmitted from the probe into the casting. Before transmitting ultrasonic waves, an oil or water film is provided between the probe and the casting surface; this ensures proper contact between them and better transmission of waves from the probe into the surface of the object to be tested.
• For carrying out the operation, ultrasonic wave is introduced into the metal and the time intervals between transmission of the outgoing and reception of incoming signals are measured with a cathode ray oscilloscope (CRO). The time base of CRO is so adjusted that the full width of the trace represents the section being examined.
• As the wave is sent from the transmitter, if the test piece is free from cracks, or flawless, then it reflects ultrasonic waves without distortion. If there are any flaws in the specimen, the time taken by the ultrasonic waves will be less as the reflection of these waves will be from flaw points and not from the bottom of the specimen. Cathode ray oscilloscope (CRO) is used to receive the sound signals,whose time base circuit is connected to it. Knowing the time interval between the transmission of the sound pulse and the reception of the echo signal, we can calculate the depth of the crack.

ADVANTAGES :

1. It involves low cost and high speed of operation.
2. This method is more sensitive than radiography.
3. Big castings can be symmetrically scanned for initial detection of major defects.
4. Depth of penetration for flaw detection or measurement is superior to the other methods.
5. Only single sided access is required.
6. It provides flaw distance information.
7. The minimum flaw size which can be detected is equals to about 0.1% of the distance from the probe to the detect.

DISADVANTAGES :

1. This method of inspection is sensitive to surface roughness.
2. Thin parts may be difficult to inspect.
3. Linear defects oriented parallel to the sound beam can go undetected.
4. Reference standards are often needed.

LIQUID PENETRATION TEST

                    This test is employed for detection of small defects which are very small to detect with naked eye. This test is used to detect surface cracks or flaws in non-ferrous metals. The test specimen is first thoroughly cleaned and dried before the test. A liquid penetrant is applied to the surface; This test employs a visible colour-contrast dye penetrant technique for the detection of open surface flaws in metallic and non-metallic objects. The penetrants are applied by dipping or brushing over the surface of material to be inspected. The excess penetrant is then washed or cleaned. Absorbent powder is then applied to absorb the penetrants in the cracks, voids which reveals the flaws. This test reveals flaws such as shrinkage cracks, porosity, fatigue cracks, grinding cracks, forging cracks, seams, heat treatment cracks and leaks etc., on casting, welding, machined parts, cutting tools, pipes and tubes.

                  If the fluorescent penetrant is used the developed surface must be examined under ultra violet light to see the presence of defects. This technique is used for non-porous and non-absorbent materials. Care may be taken to clean the surface so that it is free from dust, scale, etc, to have better results. penetrants are highly toxic and flammable and hence proper precautions should be taken both during use and of storage of penetrants.

              Liquid penetrant tests are simple, versatile, portable and inexpensive. The results are easy to interpret but only surface faults can be detected. If a permanent record is required a photograph or videotape or inspectors report may be kept. The use of laser scanners and digital control allows this process to be used as a mass production technique.

ELECTRICAL METHOD (OR) EDDY CURRENT TEST

                   The electrical method consist in measuring the electrical resistance of the material and then to note the variation in the electrical resistance. The variation is co-related to the physical defect. A number of electrical methods have been employed in non-destructive inspection and testing of machinery and wide variety of metallic material and parts for dimensional inaccuracies and physical defects.

                   A crack detector operates on the principle that if a crack occurs anywhere within the piece it interfaces with the flow of electric current through the metal, increasing its overall resistance. This holds true regardless of the shape of piece. Operation of the instrument consists of accurately measuring the electrical resistance of some critical machine part between two definitely established contact points, usually chosen at extreme opposite ends and of repeating the measurement at regular intervals. When successive measurements show the increase in resistance at a progressive rate, a fatigue fracture is beginning to propagate and the part should be removed from the surface.

ADVANTAGES:

1. Detects surface and near surface defects.
2. Test probe doesn't need to contact the part.
3. Method can be used for more than flaw detection.
4. Minimum part preparation is required. 

 DIS ADVANTAGES:

• Only conductive materials can be inspected.
• Ferromagnetic materials require special treatment to address magnetic permeability.
• Depth of penetration is limited.
• Flaws that lie parallel to the inspection probe coil winding direction can go undetected.
• Surface finish and roughness may interfere.
• Reference standards are needed for setup.

DAMPING

                    As the Measurement of damping can give information on the origin of defect such as forming of quenching cracks. For Example, an increase in damping was found in steel specimen which had been quenched. It is possible to determine the position of a crack in a cylindrical piece. When a solid specimen vibrates, its free oscillations decay when isolated from their environment. Some of the energy is always convert into heat. The various mechanisms by which this transfer of energy occurs are collectively termed as internal friction.