Lecture 1 - Introduction to Vibrations

ME 347 Mechanical Vibrations Introduction to Vibrations

Dr. Conchúr Ó Brádaigh

January 2012

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College of Engineering & Informatics

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Basic Concepts of Vibration

ME 347 Mechanical Vibrations - Introduction to Vibrations

Any motion that repeats itself after an interval of time is called vibration or oscillation. Examples are the swinging of a pendulum or the vibration of a plucked string. The theory of vibration deals with the study of oscillatory motions of bodies and the forces associated with them.


Basic Concepts of Vibration


A vibratory system, in general, includes a means for storing potential energy (spring or elasticity); a means for storing kinetic energy (mass or inertia); and a means by which energy is gradually lost (damper) The vibration of a system involves the transfer of its potential energy to kinetic energy, and of kinetic energy to potential energy. If the system is damped, some energy is dissipated in each cycle of vibration and must be replaced by an external source if steady-state vibration is to be achieved


Example System – Simple Pendulum


1-2: Potential energy converted to kinetic energy (max. at 2) 2-3: Kinetic energy converted to potential energy (max. at 3 & 1) Damping from air resistance means that pendulum will eventually stop.


Number of Degrees of Freedom


The minimum number of independent co-ordinates required to determine completely the positions of all parts of a system at any instant in time defines the number of degrees of freedom of the system.

Single degree of freedom systems



2-Degree of Freedom Systems


(c) θ & X or x, y & X (note x2 +y2 = l2)

(a) x1 & x2 (b) θ1 & θ2

School Institute Name to go here

3-Degree of Freedom Systems


(a) x1, x2 & x3

(c) θ1, θ2 & θ3

(b) θ1, θ2 & θ3 or x1, x2, x3 & y1, y2, y3 (note xi

2 +yi2 = li2, i = 1,2,3)

Some systems, especially those involving continuous elastic members, have an infinite number of degrees of freedom (DOF), e.g. the cantilever beam shown. Systems with a limited number of DOF are called discrete or lumped systems, those with an infinite number of DOF are called continuous or distributed systems.

Infinite Degree of Freedom Systems



Vibration Analysis Procedure


1. Mathematical Model of System

2. Derivation of Governing equations

3. Solution of the Governing Equations

4. Interpretation of the Results


Mathematical Model of a Motorcycle



Simplest model considers one equivalent mass, spring and damping constants

More refined model separates out the mass of each wheel from that of the vehicle and rider, and gives stiffness and damping properties to each strut, and stiffnesses to each wheel

Mathematical Model of a Motorcycle (cont.)



The model can be refined further by providing for elasticity & damping of the rider/seat.

This model could then be simplified by combining the strut and wheel properties to make a simpler 2 DOF system.

Spring Elements



A spring is said to be linear if the elongation or reduction in length x is related to the applied force, F as:

where k is the spring constant, rate or stiffness The work done in deforming a spring is stored as strain or potential energy in the spring:

Example: Spring Constant of a Rod



Example 1.3 (Rao). Find the equivalent spring constant of a uniform rod of length l, cross-sectional area A, and Young’s Modulus E, subjected to an axial tensile or compressive force, F as shown:

Example: Spring Constant of a Rod



Example: Spring Constant of a Cantilever Beam



Example 1.4 (Rao). Find the equivalent spring constant of a cantilever beam subjected to a concentrated load F at its end, as shown below

k = ?

Example: Spring Constant of a Cantilever Beam



Solution:

Combinations of Springs


Case 1. Springs in Parallel

2 springs in parallel:

n springs in parallel: College of Engineering & Informatics

Combinations of Springs


Case 2. Springs in Series For 2 springs in series: For n springs in series:


Example: Torsional Spring Constant of a Propellor Shaft



Example 1.6 (Rao). Determine the torsional spring constant of the steel propeller shaft shown below

Example: Equivalent Spring Constant of Hoisting Drum



Example 1.7 (Rao). Determine the equivalent spring constant of the system shown below (do yourselves)

Example: Equivalent Spring Constant of a Crane (do yourselves)



Example: Equivalent Spring Constant of a Rigid Bar Connected by Springs



Combinations of Masses


In many practical applications, several masses appear in combination. For a simple analysis, we can replace these masses by a single equivalent mass. Case 1: Translational Masses Connected by a Rigid Bar


Combinations of Masses


Case 2: Translational and Rotational Masses Coupled Together


Equivalent Mass of a System



Example 1.11 (Rao). Find the equivalent mass of the system shown below, where the rigid link 1 is attached to the pulley and rotates with it.

Equivalent Mass of a System



Example 1.12 (Rao). A cam-follower mechanism is used to convert the rotary motion of a shaft into the oscillating notion of a valve. The system comprises:

Pushrod of mass mp Rocker arm of mass mr and mass moment of inertia Jr about its CG Valve of mass mv and a valve spring of negligible mass

Find the equivalent mass of the system at (i) point A and (ii) point C

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Lecture 1 - Introduction to Vibrations