Mechanical Vibrations Week 1

8/3/2019 Mechanical Vibrations Week 1

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EM412MECHANICALVIBRATIONS

Text : Mechanical Vibrations 4th (Edition) by S.S. Rao

INSTRUCTOR:

MAJ. DR. KUNWAR FARAZ AHMED

Week 1Introduction and Basic Concepts

Chapter 1: Fundamentals of Vibration- Basic Concepts

- Classification

- Vibration Elements

- Harmonic Motion


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To help the students understand thebasic concept of Mechanical

Vibrations.

Objective


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Use Newton's Second Law and free body diagram approach to model

vibratory systems

Solve differential equations and eigenvalue problems for

determining the dynamic response (with correct units) of vibratory

systems Understand the physical and mathematical significance of:natural

frequencies andmode shapes,free andforced response resonance,

damping ,superposition , lumped parameter versuscontinuous

systems , linear versusnon-linear systems

Use appropriate analytical, numerical and computational tools

Understand experimental and data analysis techniques

Design mechanical systems with prescribed vibratory performance.

Synopsis

This Course is designed to teach students how to :


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Sessional 1 20 %

Sessional 2 20 %

Quiz 6 %

Assignment 4 %

Final Exam 50 %

Evaluation


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Class attendance is highly recommended as material may be

presented which departs from the text.

Homework problems will be assigned regularly and should be

done within one week from the assigned date.

I will assign two MATLAB BASED projects as special

assignments.

Consulting/studying in teams is encouraged. However, each

team member must work on all parts of the homework

All exams will be closed book (Formula sheet will only be

provided for the final exam).

Conduct

How this Class will be Conducted:


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Vibration Definition

Introduction

Vibration : Any motion that repeats itself after an interval

of time is called vibration or oscillation. The swinging of a

pendulum and the motion of a plucked string are typical

examples of vibration. The study of vibration deals with the

study of oscillatory motions of bodies and the forcesassociated with them

Encyclopaedia Britannica : Periodic back-and-forth motion

of the particles of an elastic body or medium, commonlyresulting when almost any physical system is displaced

from its equilibrium condition and allowed to respond to

the forces that tend to restore equilibrium.


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Why study vibration?

Introduction

Vibrations can lead to excessivedeflections and failure on

the machines and structures

To reduce vibration throughproper design of machinesand their mountings

To utilize profitably in several consumer and industrialapplications (quartz oscillator for computers)

To improve the efficiency of certain machining, casting,forging & welding processes

To stimulate earthquakes for geological research andconduct studies in design of nuclear reactors.


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Simple vibration systems?

Introduction


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Basic Concepts of Vibration

Introduction

Vibration = any motion that repeats itself after an interval

of time

Vibratory System consists of:

o spring or elasticity (means of storing potential energy)

o mass or inertia (means of storing kinetic energy)o Damper (means by which energy is lost)

Involves transfer ofpotential energy tokinetic energy and

vice versa


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Basic Concepts of Vibration

Introduction

Degree of Freedom (d.o.f) =minimum number of

independent coordinates required to determine completely

the positions of all parts of a system at any instant of time

Examples ofsingle degree-of-freedom systems:

Motion of this simple pendulum

can be stated in term of either or

Cartesian coordinatesx andy

Converts potential energy to

kinetic energy and vice versa


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Examples of single degree-of-freedom systems:

Introduction


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Examples of two degree-of-freedom systems:

Introduction


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Examples of three degree-of-freedom systems:

Introduction


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Example of Infinite-number-of-degrees-of-freedom system:

Introduction

Infinite number of degrees of freedom system are

termedcontinuous ordistributedsystems

Finite number of degrees of freedom are termeddiscrete or lumpedparameter systems

More accurate results obtained by increasing

number of degrees of freedom


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Classification of Vibration

Introduction

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

Forced Vibration: A system that is subjected to a repeating

external force. E.g.oscillation arises from diesel engines

- Resonance occurs when the frequency of the external

force coincides with one of the natural frequencies of thesystem


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Introduction

UndampedVibration: When no energy is lost or dissipated

in friction or other resistance during oscillations

DampedVibration: When any energy is lost or dissipated infriction or other resistance during oscillations

Linear Vibration: When all basic components of a

vibratory system, i.e. the spring, the mass and the damper

behave linearly


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Introduction

Nonlinear Vibration: Ifany of the components behave

nonlinearly

Deterministic Vibration: If the value or magnitude of theexcitation (force or motion) acting on a vibratory system is

known at any given time

Nondeterministic orrandom Vibration: When the value of

the excitation at a given time cannot be predicted


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Introduction

Examples of deterministic and random excitation:


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Vibration Analysis Procedure

Introduction

Step 1: Mathematical Modeling

Step 2: Derivation of Governing Equations

Equations of motion in form of ODEs for discrete and

PDEs for continuous systems. Use Newtons secondlaw, DAlemberts Principle or conservation of energy

Step 3: Solution of the Governing Equations

Laplace Transforms, Matrices or Numerical Methods

Step 4: Interpretation of the Results


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Example of the modeling of a forging hammer

Introduction


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Spring and Elastic Elements

Introduction

Spring force is proportional to the amount ofdeformation and is given byF = k x

The work done (U) in deforming the spring is stored

as strain or potential energy in the spring is given by

Actually springs are non-linear and follow the firstequation only up to the yield point of the material.

We approximate the string to be linear

2

2

1xkU


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Introduction

Linearization process:

for small values the higher order terms are neglected

sinceF = F(x*)

F = kx

k is the linearized spring constant atx*

Elements like beams also behave like springs

...)(!2

1)()()(

2

2

2**

**

xdx

Fdx

dx

dFxFxxFFF

xx

)()(*

* xdxdFxFFF

x

3

33

,3 l

EIWk

EI

Wl

st

st

m

x (t)E, A, I

l


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Introduction

Case I: Springs in Parallel

Case II: Springs in Series

neqsteqstnstst kkkkwherekWkkkW ..,,... 2121

neq

steqkkkk

wherekW 1..111,,21

st

k1 k2

st

1

2

k2

k1


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Introduction

Free body diagram and equations of motion


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Discrete masses:

- Point mass: Has translation only, therefore kinetic energy is

- Rigid body: Has both translation and rotation, thereforekinetic energy is

Inertia (Mass)Elements

Introduction


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Equivalent Mass: For systems with 1DOF, equivalent mass is

something that conceptually is very similar to the equivalent spring

idea

Equations Of Motion: determined following four steps:

- Step 1: Identify the displacement variable of interest- Step2: Write down the defining kinematic constraints

- Step3: Get equivalent mass/moment of inertia

Kinetic energy of actual system and that of the simplified 1-DOFsystem (expressed in terms of the time derivative of displacementvariable of interest) should be the same

- Step4: Get equivalent force/torque

Equate virtual power between actual system and the simplified1-DOF system in terms of the displacement variable of interest


Introduction


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Case I : Translational Masses Connected by a Rigid Bar: Start with asystem with four masses (see (a)). Find the equivalent mass as in (b).

Equivalent mass can be assumed to be located anywhere (here it is at the

same location asm1)


Introduction

1

1

3

31

1

2

2 , xl

l

xxl

l

x

1xxeq

Equating K.Es22

33

2

22

2

112

1

2

1

2

1

2

1eqeqxmxmxmxm

3

2

1

3

2

2

1

2

1 ml

lm

l

lmmeq


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Case II : Translational and Rotational Masses Coupled Together: For the1DOF system below, find its equivalent mass

- Equivalent Translational Mass: If the generalized coordinate that

captures this degree of freedom is the displacementx

- Equivalent Rotational Mass : If the generalized coordinate that

captures this degree of freedom is the angle


Introduction

2

0

2

2

1

2

1 JxmT

2

2

1eqeqeq xmT

R

xandxxeq

,

2

0

22

2

1

2

1

2

1

R

xJxmxmeq

2

0

R

Jmmeq

Rxandeq

,

20

22

2

1

2

1

2

1 JRmJeq

2

0 mRJJeq


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In real life, systems dont vibrate forever, or if they do,there should be something pumping energy into thesystem

Energy initially associated with an oscillatory motion isgradually converted to heat and/or sound. Thismechanism is known as damping

Most common damping mechanism:

- Viscous Damping- Coulomb friction

- Material or Solid or Hysteretic Damping

Damping Elements

Introduction


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Viscous Damping: Damping force is proportional to the

velocity of the vibrating body in a fluid medium such as air,

water, gas, and oil.

Resistance offered by the fluid to the moving body causes

energy to be dissipated. Amount of energy dissipated

depends on:

- Fluid viscosity

- Vibration frequency

- Relative velocity of the vibrating body wrt that of the

fluid ( Typically damping force is proportional to

relative velocity)

- Shape (geometry) characteristics

Damping Elements: Viscous Damping

Introduction


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Symbols used:

fluid viscosity shear stress dev. in the fluid layer at a distance y of the fixed plate vplate relative horizontal velocity; no velocity in the vertical direction

uvelocity of intermediate fluid layers; assumed to change linearly

Damping Elements: Viscous Damping between Parallel Plates

Introduction


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Introduction

Damping Elements: Viscous Damping between Parallel Plates

Shear Stress ( ) developed in the fluid layer at a distancey

from the fixed plate is:

where du/dy= v/h is the velocity gradient

Shear or Resisting Force (F) developed at the bottom surface

of the moving plate is:

where A is the surface area of the moving plate.

is the damping constant


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Coulomb or Dry Friction Damping: Damping force is

constant in magnitude but opposite in direction to that of

the motion of the vibrating body between dry surfaces

Several other friction models are in use beside Coulomb

friction (see, for instance, LuGre model)

We will stick to the Coulomb model

- Damping force is constant in magnitude and opposite to

relative velocity between bodies in contact

- Proportional to the normal contact force between bodies

- Caused by rubbing surfaces that are dry or without

sufficient lubrication

Damping Elements: Coulomb Damping

Introduction


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Equations of Motion for

FBD:

Introduction

Instantaneous

direction of motionm

W

N

F= N

Fs= k x

Direction is

opposite to that of

motion. Always.



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Finding the solution is not difficult, but tricky

Well see in Chapter 2 that it should assume an expression ofthe form:

35

The solution looks like that for half of period, then A and B change, since the

direction of the force changes

Revisited in chapter 2.

Introduction



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Material or Solid or Hysteretic Damping: Energy is absorbed ordissipated by material during deformation due to friction between

internal planes Materials are deformed, energy is absorbed and

dissipated by the material

Friction between internal planes, which slip and slide as thedeformations take place

Stress-strain diagram shows hysteresis loop,

Damping Elements: Hysteretic Damping

Introduction

Area of this loop denotes energy lost

per cycle due to damping

Rubber-like materials do this without

permanent deformation


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Example 1.1: Mathematical Model of a Motorcycle

Introduction

Figure 1.18(a) shows a motorcycle with a rider.- Develop a sequence of three mathematical models of

the system for investigating vibration in the vertical

direction.

- Consider the elasticity of the tires, elasticity anddamping of the struts (in the vertical direction),

masses of the wheels, and elasticity, damping, and

mass of the rider.


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Example 1.1 Solution

Introduction

We start with the simplest model and refine itgradually.

- When the equivalent values of the mass, stiffness,

and damping of the system are used, we obtain a

single degree of freedom model (1 DOF) of the

motorcycle with a rider as indicated in Fig.

1.18(b). In this model, the equivalent stiffness (keq)

includes the stiffness of the tires, struts, and rider.

The equivalent damping constant (ceq) includes

the damping of the struts and the rider. The

equivalent mass includes the mass of the wheels,

vehicle body and the rider.


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Introduction

This model can be refined byrepresenting the masses of wheels,

elasticity of tires, and elasticity and

damping of the struts separately, as

shown in Fig. 1.18(c). In this model,the mass of the vehicle body (mv) and

the mass of the rider (mr) are shown

as a single mass, mv+mr .When the

elasticity (as spring constant kr) and

damping (as damping constant Cr) of

the rider are considered, the refined

model shown in Fig. 1.18(d) can be

obtained.


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Introduction

Note that the models shown inFigs. 1.18(b) to (d) are not

unique. For example, by

combining the spring

constants of both tires, themasses of both wheels, and

the spring and damping

constants of both struts as

single quantities, the model

shown in Fig. 1.18(e) can be

obtained instead of Fig.

1.18(c).


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Example 1.5 Equivalent k of a Crane

Introduction

Theboom AB of crane is a

uniform steel bar of length 10

m and x-section area of 2,500

mm2.

A weight W is suspended whilethe crane is stationary. Steel

cable CDEBF has x-sectional

area of 100 mm2.

Neglecting effect of cableCDEB, find equivalent spring

constant of system in the

vertical direction.


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Introduction

A vertical displacement x of pt B will cause the spring k2(boom) to deform byx2 =x cos 45 and the springk1 (cable)

to deform by an amountx1 =x cos (90 ). Length of cableFB, l1 is as shown.


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Introduction

The angle satisfies the relation:

The total potential energy (U):

Potential Energy of the equivalent spring is:

By setting hence:


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Periodic Motion: motion that repeats itself after an interval

of time . is called the period of the function

Harmonic Motion: a particular form of periodic motion

represented by a sine or cosine function

Very Important Observation: Periodic functions can beresolved into a series of sine and cosine functions of shorter

and shorter periods

f

t

Harmonic Motion: General Concepts

Introduction


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Introduction

Harmonic Motion: Simple harmonic motion with circular motion

of a point mass


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Harmonic Motion: Vibration frequency and period

Introduction


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Harmonic Motion: Energy in vibration: KE and PE

Introduction


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The motion with no friction of the system below (mass-springsystem) leads to a harmonic oscillation.

- Formally discussed in Chapter 2

Plot below shows time evolution of function Nomenclature:

Harmonic Motion: Sinusoidal Wave

Introduction

I d i


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Harmonic Motion

Introduction

If displacementx(t) represented by a harmonic function, sameholds true for the velocity and acceleration:

Quick remarks

- Velocity and acceleration are also harmonic with the same frequency of

oscillation, but lead the displacement by /2 and radians, respectively- For high frequency oscillation ( large), the kinetic energy, since it

depends on , stands to be very large (unless the mass and/or A is

very small)

- Thats why its not likely in engineering apps to see large A associatedwith large

I d i


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Harmonic Motion

Introduction

Complex number representation of harmonic motion:

where i = (1) and a and b denote the real andimaginary x and y components of X, respectively.

I d i


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Harmonic Motion

Introduction

Also, Eqn. (1.36) can be expressed as

Thus,

I t d ti


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Harmonic Motion

Introduction

Operations onHarmonic Functions:- Rotating Vector,

Displacement

Velocity

Acceleration

whereRe denotes the real part

I t d ti


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Harmonic Motion

Introduction

Displacement, velocity, and accelerations as rotatingvectors

Vectorial addition of

harmonic functions

I t d ti


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Example 1.11: Addition of Harmonic Motions

Introduction

Find the sum of the two harmonic motions

Solution:Method 1: By usingtrigonometric relations: Since

the circular frequency is the same for both

And we express the sum as

I t d ti


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Introduction

That is,

That is,

By equating the corresponding coefficients of

costandsinton both sides, we obtain

(E.4)

I t d ti


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Introduction

and

Method 2: By using vectors: For an arbitrary value

oft, the harmonic motions and can bedenotedgraphically as shown in Fig. 1.43. By

adding them vectorially, the resultant vectorcan be found to be

I t d ti


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Introduction

Method 3: By using complex number

representation: the two harmonic motions can be

denoted in terms of complex numbers:

The sum of and can be expressed as

where A and can be determined using Eqs. (1.47)and (1.48) asA = 14.1477 and = 74.5963

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Mechanical Vibrations Week 1

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