Mechanical Vibrations Week 1
Transcript of 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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Classification of Vibration
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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Classification of Vibration
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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Classification of Vibration
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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Spring and Elastic Elements
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
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m
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l
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Spring and Elastic Elements
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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Spring and Elastic Elements
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
Inertia (Mass)Elements
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)
Inertia (Mass)Elements
Introduction
1
1
3
31
1
2
2 , xl
l
xxl
l
x
1xxeq
Equating K.Es22
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2
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2
112
1
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2
1
2
1eqeqxmxmxmxm
3
2
1
3
2
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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
Inertia (Mass)Elements
Introduction
2
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2
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Rxandeq
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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.
Damping Elements: Coulomb Damping
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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
Damping Elements: Coulomb Damping
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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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Example 1.1 Solution
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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Example 1.1 Solution
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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Example 1.5 Solution
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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Example 1.5 Solution
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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Example 1.11 Solution
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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Example 1.11 Solution
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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Example 1.11 Solution
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