1 MECHANICAL VIBRATION MME4425/MME9510 Prof. Paul Kurowski
Jan 12, 2016
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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 sitehttp://www.eng.uwo.ca/MME4425b/2012/
Design Center web sitehttp://www.eng.uwo.ca/designcentre/
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Software used:
SolidWorksDesign 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
Mass, stiffness and damping are separated
Distributed system
Mass, stiffness and damping are NOT separated
DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM
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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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Homogenous equation
SINGLE DEGREE OF FREEDOM SYSTEM, LINEAR VIBRATIONS
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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
Ф – initial value of sine function
ωn – angular frequency
18Where A and Ф are found from initial conditions
FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
By guessing solution
ωn – natural angular frequency
found from system properties
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FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
Using complex numbers method
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and
Since is linear, then the sum of two solutions is also a solution
We have found two solutions to equation
Using Euler’s relations:
The equation can be re-written as:
Where A and Ф are found from initial conditions
FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
Using complex numbers method
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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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2
[ ( )] ( ) (0)
[ ( )] ( ) (0) (0)
L x t sX s x
L x t s X s sx x
Laplace transformation
23Inman p 619
Laplace transformation
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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.
QUANTITIES CHARACTERIZING VIBRATION
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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
QUANTITIES CHARACTERIZING VIBRATION
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QUANTITIES CHARACTERIZING VIBRATION
For a device experiencing vibration in the frequency range 2-8Hz:
The maximum acceleration is 10000mm/s^2
The maximum velocity is 400mm/s
Therefore the maximum displacement is 30mm
Lines of constant displacementL
ine
s o
f co
nst
an
t a
cce
lera
tion
Nomogram for specifying acceptable limits of sinusoidal vibration (Inman p 18)
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LINEAR SDOF
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Linear spring400000N/m
10kg mass
Base
SDOF.SLDASM
LINEAR SDOF
29Results of modal analysis
LINEAR SDOF
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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.
34pendulum 02.SLDPRT
PENDULUM SDOF
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2 sin 0
0
1
2
2
ml mgl
l g
g
l
gf
l
lT
g
Equations of motion method
PENDULUM SDOF
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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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Energy method
2
2
2
.
( ) 0
1( )
2( cos )
1( ( ) ( cos )) 02
sin 0
sin 0
0
T U const
dT U
dt
T m l
U mg l l
dm l mg l l
dt
ml mgl
l g
l g
g
l
PENDULUM SDOF
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TORSIONAL SDOF
41disk 01.SLDPRT
TORSIONAL SDOF
4
2
2(1 )
torsion
T JG
L
J r
EG
JGk
L
0torsionI k
polar moment of inertia of cross-section
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TORSIONAL SDOF
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2J mr
43Inman p 32
ROLER SDOF
roler.SLDASM
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ROLER SDOF
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2/
k
m J r
k = k1 + k2 = 2000N/m
m = 75.4kg
r = 0.1m
J = 0.3770kgm2
2
20004.2 /
75.4 0.38 / 0.10.66
rad s
f Hz
ROLER SDOF
Inman p 32
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ROLER SDOF
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rotation.SLDASM
MASS AT THE END OF BEAM
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MASS AT THE END OF BEAM
cantilever.SLDPRT
mass 2.7kg
3
3EI
kl
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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 tanΦ for given x0, v0
3. Find natural frequency of cantilever, l=400mm, Φ=5mm, E=2e11Pa, m=2.7kg. Confirm with SW Simulation
4. Work with exercises in chapter 19 – blue book
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Model file trifilar.sldasm
Configuration trifilar
Model type solid
Material as shown
Supports as shown
Objectives
Find the natural frequency of trilifar
TORSONAL SDOF TRIFILAR
1060 alloy
Custom material
E = 10MPaρ = 1kg/m3
very soft, very low density
1060 alloy
Fixed support
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
0.845 M kg mass of platform0.1 R m radius of platform
0.004225 J kgm^2 mass moment of inertia of platform
0.1 R m radius of attachment of wires0.5 L m length of wires
9.81 g m/s 2̂ gravitatonal acelleration
6.26 rad/s natural frequency1.00 Hz natural frequency
56Trifilar can be used to find moments of inertia of objects placed on rotating platform
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2P
MASS P
M gRJ J
L
TORSONAL SDOF TRIFILAR
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2
2
2
P
P
PP
M gR
J L
M gRJ
L
spur gear.SLDPRT