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MBB4023 Vibration

Semester May 2011

Lecture 1

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What is vibration?

Vibration = Any motion that repeats itself afteran interval of time is called vibration oroscillation.

For example:1.The swinging of a pendulum.2.The motion of a plucked string.

The study of vibration deals with the study of oscillatory motions of bodies and the

forces associated with them.

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Basic component of vibration

Vibratory System consists of:1) spring or elasticity - a means for storing potential energy

2) mass or inertia - a means for storing kinetic energy

3) Damper - a means by which the energy is gradually lost

Involves transfer of potential energy to kinetic

energy and kinetic energy to potential energy alternately.

Note: If the systems is damped, some energy is dissipated in

each cycle of vibration which eventually all of it will disappear.

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Basic concept of vibration

At point 1: KE = 0PE = mgl(1 cos )

At point 2: PE = 0KE = Max

At point 3: KE = 0PE = PE @ 1

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1. Free Vibration:

A system is left to vibrate on its own after an initial disturbance and

no external force acts on the system. E.g. simple pendulum

2. Forced Vibration:

A system that is subjected to a repeating external force. E.g.

oscillation arises from diesel engines

- Resonanceoccurs when the frequency of the

external force coincides with one of the natural

frequencies of the system

Classification of Vibration

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1. Undamped Vibration:

When noenergy is lost or dissipated in friction or other resistance

during oscillations

2. Damped Vibration:

When anyenergy is lost or dissipated in friction or other resistance

during oscillations

3. Linear Vibration:

When allbasic components of a vibratory system, i.e. the spring, the

massand the damper behave linearly

Classification of Vibration

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Classification of Vibration

Nonlinear Vibration:

If anyof the components behave nonlinearly. Most vibration

systems behave nonlinearly with increasing amplitude.

Deterministic Vibration:

If the value or magnitude of the excitation (force or motion)

acting on a vibratory system is known at any given time

Nondeterministic or random Vibration:

When the value of the excitation at a given time cannot be

predicted. The response is also random and can be described

only in terms of statistical quantities.

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Step 1: Mathematical Modeling

Transform the physical system to model using gradual

refinement method.

Represent all important vibration components of the

system for the purpose of deriving the mathematical

equations governing the behavior of the system.

Vibration Analysis Procedure

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Step 2: Derivation of Governing Equations

Using the principles of dynamics and free body diagram, derive the

equations that describe the vibration of the system.

Equations are in the form of Ordinary Differential Equation (ODE) for the

discrete system and Partial Differential Equation (PDE) for a

continuous systems.

Methods:

1. Newtons 2nd Law of Motion

2. DAlemberts Principle

3. Energy method.

Vibration Analysis Procedure

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Step 3: Solution of the Governing Equations

Purpose - To determine the response of the vibrating system interms of vibration amplitude, frequency, phase angle, speed,

acceleration etc.

Methods used:Standard methods of solving differential equations.Laplace tranform.

Matrices.Numerical method.

Vibration Analysis Procedure

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Step 4: Interpretation of the Results

The solution provides the data required for vibration

interpretation. If analysis is done during the design stage, thevibration analysis result may cause some design change.

Vibration Analysis Procedure

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Definition and Terminology

Cycle - The movement of a vibrating body from its equilibriumposition to its extreme position in one direction back to its

equilibrium position and then to its extreme position in other

direction and back to its equilibrium position.

Amplitude - The maximum displacement of a vibrating body from

its equilibrium position. The unit is mm.

Period, - Time taken to complete one cycle of motion. Unit is

sec.

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Definition and Terminology

Frequency, f - The number of cycle per unit time, Hz. Itis also a reciprocal of period.

Phase Angle, - Angle between two oscillation, rad.

Natural frequency - Frequency with which the physical

system oscillate after initial disturbance without external

force applied, n and the unit is rad/sec.

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Definition and Terminology

Degree of Freedom

Minimum number of independent coordinates required to

determine the positions of all parts of a system at any instant oftime.

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Examples of single degree-of-freedom systems:

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