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/2012/
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
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How many rigid body motions?
DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM
Discrete system
Distributed system
Mass, stiffness and damping
are separated
Mass, stiffness and damping
are NOT separated
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DISCRETE SYSTEM VS. DISTRIBUTED SYSTEM
1DOF.SLDASM
2DOF.SLDASM
Discrete system
Distributed system
Mass, stiffness and damping
are separated
Mass, stiffness and damping
are NOT separated
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swing arm 01.SLDASM
swing arm 02.SLDASM
Discrete system
Distributed system
Mass, stiffness and damping
are separated
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
Ф – initial value of sine function
ωn – angular frequency
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FINDING GENERAL SOLUTION OF HOMOGENEOUS EQUATION
By guessing solution
ωn – natural angular frequency
found from system properties
Where A and Ф 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
Since
is linear, then the sum of two solutions is also a solution
Using Euler’s relations:
The equation can be re-written as:
Where A and Ф 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
L[ x(t )]  sX ( s)  x(0)
L[ x(t )]  s X ( s)  sx(0)  x(0)
2
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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 acceleration
Lines of constant displacement
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
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Nomogram for specifying acceptable limits of sinusoidal vibration (Inman p 18)
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
ml 2  mgl sin   0
l  g  0

g
l
1
f 
2
T  2
g
l
l
g
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36
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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
T  U  const .
d
(T  U )  0
dt
1
T  m(l ) 2
2
U  mg (l  l cos  )
d 1
( m(l ) 2  mg (l  l cos  ))  0
dt 2
ml 2  mgl sin   0
l  g sin   0
l  g  0

g
l
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TORSIONAL SDOF
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TORSIONAL SDOF
I  ktorsion  0
T
JG


L
 4
J r
2
polar moment of inertia
of cross-section
E
G
2(1   )
ktorsion
disk 01.SLDPRT
JG

L
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TORSIONAL SDOF
1 2
J  mr
2
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ROLER SDOF
roler.SLDASM
Inman p 32
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ROLER SDOF
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ROLER SDOF
k

m  J / r2
Inman p 32
k = k1 + k2 = 2000N/m
m = 75.4kg
r = 0.1m
J = 0.3770kgm2
2000

 4.2rad / s
2
75.4  0.38 / 0.1
f  0.66 Hz
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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
3EI
k 3
l
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 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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TORSONAL SDOF TRIFILAR
1060 alloy
Model file
trifilar.sldasm
Configuration
trifilar
Model type
solid
Material
as shown
Supports
as shown
Fixed support
Objectives
Find the natural frequency of trilifar
Custom material
E = 10MPa
ρ = 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
0.845 M
0.1 R
0.004225 J
kg
m
kgm^2
mass of platform
radius of platform
mass moment of inertia of platform
0.1 R
0.5 L
9.81 g
m
m
m/s^2
radius of attachment of wires
length of wires
gravitatonal acelleration
6.26
1.00
rad/s
Hz
natural frequency
natural frequency
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TORSONAL SDOF TRIFILAR
J MASS
M P gR 2

 JP
2
L
Trifilar can be used to find moments of inertia of objects placed on rotating platform
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
M P gR 2
JPL
M P gR 2
JP 
2L
spur gear.SLDPRT
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