Intro to Vibrations

8/10/2019 Intro to Vibrations

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UNFUNDAMENTALS OF VIBRATI

FREE VIBRATIONS OF S

LE

R B KARTHIK A

DEPARTMENT OF AERONAUTICAL EN

GITAM UN


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INNARDS (PART I)

1. History 2 Hours2. Importance 1 Hour

3. Fundas 2 Hours

4. Classification 1 Hour

5. Analysis Procedures 3 Hours6. Spring Elements 2 Hours

7. Inertia and damping Elements 2 Hours

8. Harmonic Analysis 2 Hours


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INNARDS (PART II)

1. Free Vibration Undamped (Translational) 2 Hours

2. Free Vibration Undamped (Torsional) 2 Hours

3. Stability Conditions 1 Hour

4. Raleighs Energy Method 2 Hours

5. Free Vibration with Viscous Damping 2 Hours6. Free Vibration with Coulomb damping 2 Hours

7. Free Vibration with Hysteretic damping 2 Hours


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HISTORY OF VIBRATIONS

Strings (Music)

1. Indian Mythology2. Egypt (Nanga)3. Pythagoras4. Aristotle, Aristoxenus and

Euclid5. Vitruvius (Acoustics

Properties of Theatre)


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Worlds First seismograph

developed in China during 132AD.

Zhang Heng is that place.


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Laws of Vibrating String

Galileo Simple pendulum, resonance Mersenne father of acoustics Hooke Relation between pitch and frequency Sauveur Modes shapes and nodes, harmonics



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IMPORTANCE OF VIBRATIONS

Aloha Airlines Flight 243 Eschede Train disaster


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TERMINOLOGY USED

AMPLITUDE: Maximum displacement of anoscillation from an equilibrium (zero level)position.

TIME PERIOD: The smallest interval ofwhich a system undergoing oscillation retuthe state it was in at a time arbitrarily chosthe beginning of the oscillation.

FREQUENCY: is the number of occurrenrepeating event per unit time.

DOF: The minimum number of coordinatesrequired to determine completely the positionsof all parts of a system at any instant of time.


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TYPES OF SYSTEMS

Discrete and Continuous Systems: Finite DOF Discrete or Lumped parameters

Infinite DOF Continuous or distributed Systems

CLASSIFICATION OF VIBRATIONS

Free Vibrations : After an initial disturbance, system is left on it s own. Forced Vibrations : Subjected to external force. Undamped systems : If no energy is lost or dissipated from the system. Damped Systems : If energy is lost due to friction or other resistance. Linear Vibrations : System behaves linearly. Non- Linear Vibrations : System behaves non-linearly. Deterministic Vibration : The magnitude of excitation is known at any given of time. Random Vibrations : The magnitude cannot be determined or is random at any time.


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VIBRATION ANALYSIS PROCEDURE

Mathematical Modelling

Derivation of governing equations Solution of the governing equations Interpretation of Results


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SPRING ELEMENTS

F = KX

Linear Springs Non - Linear Springs


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Problem 1:

A milling machine weighing 1000 kgs is supported on a rubber mount. The deflection The force deflection isgiven by

F = 2000 X* + 200(X*) 3 . Determine the equivalent linearized spring constant of the rubber mount at it s staticequilibrium position.

Problem 2Show that the stiffness for a bar element is given AE/L

Problem 3

Determine stiffness for cantilever with end loading as shown in Fig 1. The mass acting is 30 kg while thlength of the beam being 5 mt with cross sectional area of 10 mm 2 . The beam is made up of MS.

Fig 1: Cantilever with end loading


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COMBINATION OF SPRINGS

Springs in Parallel

Springs in Ser


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Prob 4: Determine the equivalent torsionalspring constant of the propeller shaft

CLUE:

Use Polar Moment of Inertia, Shear Modulus


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Prob 5: Find out the equivalent springconstant for the system.

CLUE:

K for cantileverK for wire


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Prob 6. Find the equivalent spring constantfor the system shown.


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INERTIA & DAMPING ELEMENTS

Mass or Inertia Elements are those which lose/gain kinetic energy whenever there is a velocity change. These arto be rigid bodies.

Damping: The mechanism by which the vibrational energy is gradually converted into heat or sound.

VISCOUS Damping : If the vibration system is in fluid medium such as Air, Water, Oil or Gas.

Coulomb or Dry Friction Damping : The damping force is constant in magnitude but opposite in directsystem.

Material or solid or hysteretic Damping : When a body is deformed, energy is absorbed or dissipated. Thi

internal friction between planes. When this happens, the stress-strain diagram shows hysteresis loop.


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Prob 7 :

Consider two plates separated by a distance of h with a fluid whose viscosity is . Derive anexpression for damping constant when one plate is moving with a velocity of V relative toother.

Clue: Shear stress and shear force.

Intro to Vibrations

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