In the Name of Allah, the Gracious, the Merciful.

In the Name of Allah, the Gracious,

the Merciful

Engineering Dynamics

By

Muhammad Ishfaq Khan

Lecture 1

UTSI-NSBE Thesis Defense Worskhop 3

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. UTSI-NSBE Thesis Defense Worskhop 6

History

• first musical instruments

• 4000 B.C. with 1st Musical Instument

• Pythagoras’s Monotone (582 507 B.C.)

• 1st Scientific Investigation!


• 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 wind-

induced vibration

• Aero planes Vanished due to fatigue caused by wind on the wing and chassis vibrations


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


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 forthUTSI-NSBE Thesis Defense Worskhop 12

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?


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


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?


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


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


In the Name of Allah, the Gracious, the Merciful.

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