MECHANICAL VIBRATION

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MECHANICAL VIBRATION MME4425/MME9510 Prof.
Paul Kurowski
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TEXT BOOKS
REQUIRED
RECOMMENDED
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MME4425b web site http//www.eng.uwo.ca/MME4425b/2
012/ Design Center web site http//www.eng.uwo.
ca/designcentre/
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Software used SolidWorks Design and assembly
of mechanisms and structures SolidWorks
Simulation (add-in to SolidWorks) Structural
analysis Motion Analysis (add-in to
SolidWorks) Kinematic and dynamic analysis of
mechanisms Excel
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SolidWorks 2012 installation and activation
instructions Go to www.solidworks.com/SEK Use
SEK-ID XSEK12 Select release 2012-2013 When
prompted enter serial number for activation
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WHAT IS THE DIFFERENCE BETWEEN DYNAMIC ANALYSIS
AND VIBRATION ANALYSIS?
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DIFFERENCE BETWEEN A MECHANISM AND A STRUCTURE
Structure is firmly supported, mechanism is
not. Structure can only move by deforming under
load. It may be one time deformation when the
load is applied or a structure can vibrate about
its neutral position (point of equilibrium).
Generally a structure is designed to stand
still. Mechanism moves without deforming it
components. Mechanism components move as rigid
bodies. Generally, a mechanism is designed to
move.
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DIFFERENCE BETWEEN A MECHANISM AND A STRUCTURE
STRUCTURES
MECHANISMS
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RIGID BODY MOTION
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RIGID BODY MOTION
How many rigid body motions?
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DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM
Discrete system Mass, stiffness and damping are
separated
Distributed system Mass, stiffness and damping
are NOT separated
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DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM
1DOF.SLDASM 2DOF.SLDASM
Discrete system Mass, stiffness and damping are
separated
Distributed system Mass, stiffness and damping
are NOT separated
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swing arm 01.SLDASM
swing arm 02.SLDASM
Discrete system Mass, stiffness and damping are
separated
Distributed system Mass, stiffness and damping
are NOT separated
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Discrete system Vibration of discrete systems can
be analyzed by Motion Analysis tools such as
Solid Works Motion or by Structural Analysis such
as SolidWorks Simulation based on the Finite
Element Analysis
Distributed system Vibration of distributed
systems must be analyzed by structural analysis
tools such as SolidWorks Simulation based on the
Finite Element Analysis.
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SINGLE DEGREE OF FREEDOM SYSTEM LINEAR VIBRATIONS
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SINGLE DEGREE OF FREEDOM SYSTEM, LINEAR VIBRATIONS
Homogenous equation
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FINDING GENERAL SOLUTION OF HOMOGENEOUS
EQUATION By guessing solution
How to solve this?
We guess solution based on experience that the
solution will be in the form
A magnitude of amplitude ? initial value of
sine function ?n angular frequency
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FINDING GENERAL SOLUTION OF HOMOGENEOUS
EQUATION By guessing solution
?n natural angular frequency found from system
properties
Where A and ? are found from initial conditions
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FINDING GENERAL SOLUTION OF HOMOGENEOUS
EQUATION Using complex numbers method
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FINDING GENERAL SOLUTION OF HOMOGENEOUS
EQUATION Using complex numbers method
We have found two solutions to equation
and
is linear, then the sum of two solutions is also
a solution
Since
Using Eulers relations
The equation can be re-written as
Where A and ? are found from initial conditions
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FINDING GENERAL SOLUTION OF HOMOGENEOUS
EQUATION Using Laplace transformation
Taking Laplace transform of both sides
Using (5), (6)
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Laplace transformation
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Laplace transformation
Inman p 619
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QUANTITIES CHARACTERIZING VIBRATION
Average value of amplitude is
But average value of
is zero.
Therefore, average value of amplitude is not an
informative way to characterize vibration. for
this reason we use mean-square value (variance)
of displacement
Square root of mean square value is root mean
square (RMS). RMS values of are commonly used to
characterize vibration quantities such as
displacement, velocity and acceleration
amplitudes.
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QUANTITIES CHARACTERIZING VIBRATION
Displacement
Velocity
Acceleration
These quantities differ by the order of magnitude
or more, hence it is convenient to use
logarithmic scales.
The decibel is used to quantify how far the
measured signal x1 is above the reference signal
x0
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QUANTITIES CHARACTERIZING VIBRATION
Lines of constant displacement
For a device experiencing vibration in the
frequency range 2-8Hz The maximum acceleration
is 10000mm/s2 The maximum velocity is
400mm/s Therefore the maximum displacement is
30mm
Lines of constant acceleration
Nomogram for specifying acceptable limits of
sinusoidal vibration (Inman p 18)
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LINEAR SDOF
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LINEAR SDOF
10kg mass
Linear spring 400000N/m
Base
SDOF.SLDASM
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LINEAR SDOF
Results of modal analysis
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Trigonometric relationship between the phase,
natural frequency, and initial conditions. Note
that the initial conditions determine the proper
quadrant for the phase.
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PENDULUM SDOF
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PENDULUM SDOF
Galileo Galilei lived from 1564 to 1642. Galileo
entered the University of Pisa in 1581 to study
medicine. According to legend, he observed a lamp
swinging back and forth in the Pisa cathedral. He
noticed that the period of time required for one
oscillation was the same, regardless of the
distance of travel. This distance is called
amplitude. Later, Galileo performed experiments
to verify his observation. He also suggested
that this principle could be applied to the
regulation of clocks.
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PENDULUM SDOF
pendulum 02.SLDPRT
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PENDULUM SDOF
Equations of motion method
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The energy method is suitable for reasonably
simple systems. The energy method may be
inappropriate for complex systems, however. The
reason is that the distribution of the vibration
amplitude is required before the kinetic
energy equation can be derived. Prior knowledge
of the mode shapes is thus required.
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PENDULUM SDOF
Energy method
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TORSIONAL SDOF
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TORSIONAL SDOF
polar moment of inertia of cross-section
disk 01.SLDPRT
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TORSIONAL SDOF
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ROLER SDOF
roler.SLDASM
Inman p 32
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ROLER SDOF
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ROLER SDOF
Inman p 32
k k1 k2 2000N/m m 75.4kg r 0.1m J
0.3770kgm2
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ROLER SDOF
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MASS AT THE END OF BEAM
rotation.SLDASM
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MASS AT THE END OF BEAM
mass 2.7kg
cantilever.SLDPRT
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RING
Ring.SLDASM
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HOMEWORK 1

1. Derive equation of motion of SDOF using energy
   method
2. Find amplitude A and tanF for given x0, v0
3. Find natural frequency of cantilever, l400mm,
   F5mm, E2e11Pa, m2.7kg. Confirm with SW
   Simulation
4. Work with exercises in chapter 19 blue book

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TORSONAL SDOF TRIFILAR
1060 alloy
Model file trifilar.sldasm Configuration trifilar
Model type solid Material as
shown Supports as shown Objectives Find the
natural frequency of trilifar
Fixed support
Custom material E 10MPa ? 1kg/m3 very soft,
very low density
1060 alloy
Restraint in radial direction to force torsional
mode
trifilar.SLDASM
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TORSIONAL SDOF BIFILAR
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TORSIONAL SDOF TRIFILAR
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TORSIONAL SDOF TRIFILAR
Using energy method
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TORSIONAL SDOF TRIFILAR
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TORSONAL SDOF TRIFILAR
Trifilar can be used to find moments of inertia
of objects placed on rotating platform
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spur gear.SLDPRT