In the Name of
Allah, the Gracious,
the Merciful
Engineering
Dynamics
By
Muhammad Ishfaq Khan
Lecture 1
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Course Content B-Tech
• Vibration:
– Introduction,
– types of vibration, basic concepts and terminologies
• Revision of matrix algebra:
– Characteristic equations, solution methods, free vibration, equation
of motion, energy methods, series and parallel combination,
viscously damped free vibration, logarithmic decrement, coulomb
damping.
• Harmonically excited vibration:
– Forced harmonic vibration, rotating unbalanced, and vibration
measuring instruments.
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• Mechanical system:
– Gravity pendulum, spring-mass vibrating system, compound gravity
pendulum, stiffness, potential energy, stability, torsional pendulum, free
vibration and resonance.
• Physical stiffness of elements:
– Rods, beams, stiffness coefficients as matrix elements.
• Eigen value analysis of:
– Two degree of freedom problems, three degree of freedom problems,
– determination of modes of vibration.
• Rayleigh’s method and its applications
• Books Recommended:
1. Vibration theory and applications – William, T. and Thompson.
2. Fundamentals of mechanical vibrations by Kelly.
3. Mechanical Vibration by S. S. Rao
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Lab Outline
• 1. Demonstration on simple spring mass
system and related calculation.
• 2. Demonstration simulation of beams and
rod vibration.
• 3. Demonstration on damped vibration and
analysis.
• 4. Computer simulation of whole range of
vibrating components.
• 5. Application of computer for solving set
of
equations.
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History
• first musical instruments
• 4000 B.C. with 1st Musical Instument
• Pythagoras’s Monotone (582 507 B.C.)
• 1st Scientific Investigation!
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• Zhang Heng measured earthquakes
• Time of Occurrence & Direction
• Galileo Galilei (1564 1642) From a Lamp in a Church
in Pisa:
• Time Period was Independent of Amplitude
• Frequency of vibration on the length of a simple pendulum
• Phenomenon of sympathetic vibrations i.e. resonance
• Newton’s 2nd Law
• J.B. Fourier Transformation
• Charles Coulomb Discovered the period of oscillation is
independent of the angle of twist
• E. F. F. Chladni (1756 1824) mode shapes of a Plate
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Importance of Vibrations
• Tacoma Narrows bridge during windinduced vibration
• Aero planes Vanished due to fatigue caused
by wind on the wing and chassis vibrations
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Introduction
• What is Vibration?
– Any motion that repeats itself after an interval
of time is called vibration or oscillation.
– Any movement w.r.t. reference position
•
•
•
•
•
Vibration of road surface!
Earth Quakes
Excited Tuning Fork
Speaking
Singing
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Basic Concepts & Terminologies
• Means for Storing Potential Energy
– Spring or Elasticity
• a means for storing kinetic energy
– mass or inertia
• a means by which energy is gradually lost
– damper
• the transfer of its potential energy to kinetic energy and of
kinetic energy to potential energy, alternately!
• some energy is dissipated in each cycle of vibration due to
damping
• An external source if a state of steady vibration is to be
maintained for damped system
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• Bob of mass m be released at angle θ
• At 1, K.E. 0, P.E. = mgl(cosθ-1)
• a torque mglsinθ about the point O,
• the bob starts swinging to the left
from position 1
• At position 2, all of its potential energy will
be converted into kinetic energy
• Due to Inertia it wont stop!
• mglsinθ now acts till pendulum stops at 3
• So on and so forth
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DOFs
• The minimum number of independent coordinates required
to determine completely the positions of all parts of a
system at any instant of time defines the number of degrees
of freedom of the system
• Pendulum is 1 DOF System!
• Why?
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External Force
• Types of Mechanical Vibrations
i.
Free vibration
• If a system, after an initial disturbance, is left to
vibrate on its own
• Any Example?
ii. Forced vibration
• If a system is subjected to an external force (often, a
repeating type of force)
• Any Example?
• resonance
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Resistance
i. Un-Damped
• no energy is lost or dissipated in friction or other
resistance during oscillation
• Any Example?
ii. Damped
• no energy is lost or dissipated in friction or other
resistance during oscillation
• Any Example?
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Component Behavior
i. Linear
• the basic components of a vibratory system behave
linearly
ii. Non-Linear
• the basic components of a vibratory system behave
non-linearly
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Predictability
i. Deterministic
• the value or magnitude of the excitation (force or
motion) is known
ii. Random
• the value or magnitude of the excitation (force or
motion) is not known
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