Chapter 2: Response to harmonic excita7on
Introduces the important concept of
resonance
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Mechanical
Engineering ME 5514
1
Harmonic excita7on of undamped systems
Consider the usual spri
Chapter 5 Design
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Acceptable vibration levels (ISO)
Vibration isolation
Vibration absorbers
Effects of damping in absorbers
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1
Examples:
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Mechanical Engineering at Virginia Tech
2
5.1 Acceptable lev
Some"Review:"Window"4.2"
Orthonormal"Vectors"
similar"to"the"unit"vectors"of"staBcs"and"dynamics
"
x1 and x 2 are both normal if x1T x1 = 1 and x T2 x 2 = 1
and are orthogonal if x1T x2 = 0
This is abbreviated by
0 if i j
x x j = ij =
1 if i = j
T
i
A se
Section 4.6 Modal Analysis of the Forced
Response
Extending
chapters 2 and 3
to more then one
degree of
freedom
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Forced Response: the response of an mdof system to a
forcing term
k1
c1
x1
m1
F1
x2
k2
c2
m
Chapter 6 Distributed Parameter Systems
Extending the first 5 chapters
(particularly Chapter 4) to
systems with distributed mass
and stiffness properties:
Strings, rods and beams
DARPAS Project: MOIRE
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1
Di
Chapter 4: Mul.ple Degree of Freedom
Systems
The Millennium bridge required
many degrees of freedom to model
and design with.
Extending the first 3 chapters to more then one
degree of freedom
The rst step in analyzing mul.ple degrees of
Consider a block subject to an impulse at 1me t = 0 with an
ini1al posi1on and an ini1al velocity
F (t )
x(t)
m
t
t ' =0
m!x! = F = F (t ), x (0) = x0 , x! (0 ) = v0
m!x!(t ' ) dt ' =
t
t ' =0
F (t ' ) dt ' = F
F
x! (t ) = + v0
mx!
2.4 Base excita2on
Important class of vibra2on analysis
Preven2ng excita2ons from passing from a vibra2ng base
through its mount into a structure
Vibra2on isola2on
Vibra2ons in your car
Satellite opera2on
Disk d
Sec$on 4.5 Systems with Viscous Damping
Extending the rst 4 sec0ons to included
the eects of viscous damping (dashpots)
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Viscous Damping in MDOF Systems
Two basic
2.8 Numerical Simula3on and Design
Four things we can do computationally to help
solve, understand and design vibration problems
subject to harmonic excitation
Symbolic manipulation
Plotting of the time response
Solution and plotting of the time
ME 5514 Vibra0on of Mechanical Systems
Instructors:
Prof. Tarazaga e-mail: [email protected]
Dr. Ipar Ferhat e-mail: [email protected]
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Classes themselves have certain dynamics
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Mechanical Engineering ME 5
Sec+on 2.2 Harmonic Excita+on of Damped
Systems
40
30
X (dB)
20
10
0
-10
-20
0
0.5
1
r
1.5
2
Extending resonance and response
calcula+on to damped systems
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ME5514 Mechanical Engineering
1
H
6.4 Torsional Vibrations
x,t)
x
x,t)+d
d
xdx
d
dx, from calculus
x
(x,t)
GJ
, from solid mechanics
x
G=shear modulus
J =polar moment of area cross section
Summing moments on the element dx Combining these expressions yields;
2 (x,t)
dx J
dx
2
x
t
Whe