Mech Dynamics 14.5 L03 Damping
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Transcript of Mech Dynamics 14.5 L03 Damping
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14.5 Release
Lecture 3
Damping
ANSYS MechanicalLinear and Nonlinear Dynamics
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Damping in Dynamic Analyses
Topics covered:
A. Damping definition
B. Types damping
C. General Equation of Motion
D. Viscous damping in single-DOF vibration
E. Damping Matrices
i. Full Transient and Damped Modal Analyses
ii. Full Harmonic Analysis
iii. Mode-Superposition Analysis
F. Numerical Damping
G. Workshop 2
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A. Damping Definition
Damping is an energy-dissipation mechanism that causes vibrations todiminish over time and eventually stop.
e.g. vibrational energy that is converted to heat or sound
In damping, the energy of the vibrating system is dissipated by various
mechanisms, and often more than one mechanism may be present at thesame time.
The amount of damping may depend on the material, the velocity ofmotion, and/or the frequency of vibration.
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B. Types of Damping
Damping can be classified as: Viscous damping (e.g. dashpot, shock absorber)
Material / Solid / Hysteretic damping (e.g. internal friction)
Coulomb or dry-friction damping (e.g. sliding friction)
Numerical damping (artificial damping)
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The non-linear governing equation for the Transient Dynamic Analysis is:
[M]: is structural mass matrix u : is nodal acceleration vector
[C]: is structural damping matrix u : is nodal velocity vector
[K]: is structural stiffness matrix {u}: is nodal displacement vector
{F}: is the load vector (t): is time
appliedstiffnessdampinginertia
)(
FFFF
tFuuKuCuM
C. General Equation of Motion
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Equation of motion for a SDOF system with damping,under free vibration
where: =
is the un-damped natural frequency
= is the critical damping(threshold between oscillatory and non-oscillatory behavior).
= is the damping ratio
(ratio of the damping in a system to the critical damping)
=
is the frequency of damped oscillation.
0 ukucum
02
2
uuu nn
D. Viscous damping in single-DOF vibration
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The value of [C]can be input directly as element damping when a springelement is used (Details section of Spring connection).
. Viscous damping in 1 DOF vibration
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The complete expression for the structural damping matrix, [C], is
dampingGyroscopic
1
dampingElement
1
dampingStructural
1
dampingMass
1
ge
mbma
N
l
l
N
k
k
N
j
j
m
j
N
i
i
m
i
GC
KKMMC
D. Damping Matrices
1. Transient (FULL) Analysis and Damped Modal Analysis
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Alpha damping and Beta damping are used to define Rayleigh dampingconstants and . The damping matrix [C] is calculated by using these
constants to multiply the mass matrix [M] and stiffness matrix [K]:
KMC
22
i
i
i
. Structural Damping Matrix [C]
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In many practical structural problems, alpha damping (or mass damping) may be
ignored ( = 0). In such cases, you can evaluate from known values of iand i, as
with a given value of damping, the damping ratio
is directly proportional to
frequency, i.e., lower frequencies will be damped less and higher frequencies will bedamped more (rigid body damping is ignored).
. Structural Damping Matrix [C]
2
2 or
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with a given value of damping, the damping ratio is inverselyproportional to frequency, i.e., lower frequencies will be damped more and
higher frequencies will be damped less
. Structural Damping Matrix [C]
2
2 or
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22
i
i
2
2
1
1
4
4
ff
ff
. Structural Damping Matrix [C]
To specify both and for a given damping ratio , it is commonly assumedthat the sum of the and terms is nearly constant over a range of
frequencies. Therefore, given and a frequency range 1 to 2, twosimultaneous equations can be solved for and :
f1 f2
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The value of and can be input using the following:
[1] Mater ial-dependent damping value
(Mass-Matrix Damping Multiplier, and k-Matrix Damping
Multiplier)
. Structural Damping Matrix [C]
mbma N
jj
m
j
N
ii
m
i
KMC11
22
i
i
i
Equivalent damping
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[2] Direct ly as glo bal dampin g value(Details section of Analysis Settings)
. Structural Damping Matrix [C]
KMC
Damped Modal Full Transient
22
i
i
i
Equivalent damping
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Equation of motion for a SDF system with
damping
tfkuucum sin
21
nd
222 21 nnkfu
2
1
1
2tan
n
n
/
uk/f
u
Damping Matrices
2. Full Harmonic Analysis
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Material damping is inherently present in a material (energy isdissipated by internal friction), so it is typically considered in a dynamic
analysis.
Energy dissipated by internal friction in a real system does not depend
on the cyclic frequency.
The simplest device to represent it is to assume the damping force is
proportional to velocity and inversely proportional to frequency
g = constant structural damping ratio
. Material / Solid / Hysteretic dampingin Harmonic Analysis
KgC
2
gEquivalent damping
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Equation of motion for a SDF system with
material damping
tfkuugk
um
sin2
nd
222 21 gkf
u
n
21
1
2tan
n
g
Unlike viscous damping,
frequency of damped oscillation
ddoes not change
. Material / Solid / Hysteretic dampingin Harmonic Analysis
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The complete expression for the structural damping matrix, [C], is
gis constant damping.
dampingicViscoelast
1
dampingGyroscopic
1
dampingElement
1
dampingStructural
1
dampingMass
1
1
22
vge
mb
ma
N
l
m
N
l
l
N
k
k
N
j
jj
m
j
N
i
i
m
i
CGC
KgKg
MMC
... Structural Damping Matrix [C]
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The value of g, and can be input using the following:
[1] Mater ial-dependent damping value
(Mass-Matrix Damping Multiplier, and k-Matrix Damping
Multiplier)
. Structural Damping Matrix [C]
mbma N
j jj
m
j
N
i i
m
i
KgMC11
2 g
i
ii
22
Equivalent damping
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[2] Direct ly as glo bal dampin g value
(Details section of Analysis Settings)
. Structural Damping Matrix [C]
KgMC
2 g
i
i
i
22
Equivalent damping
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Damping Matrices
3. Mode-Superposition Analysis
The Damping Controls for Harmonic Response, Transient Structural, ResponseSpectrum, and Random Vibration analyses now support all constant dampingratios, stiffness matrix
22
i
i
md
ii
Stiff.
Coef.
Mass
Coef.Constant
ratio
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Not true damping.
Artificially controls numerical noise produced by the higher frequencies of astructure.
Stabilizes the numerical integration scheme by damping out the unwanted highfrequency modes.
The default value of 10% will damp-out spurious high frequencies and is asensible value to try initially.
Use the lowest possible value that damps out nonphysical response withoutsignificantly affecting the final solution.
Available only in transient structural.
High-frequency
response
Primary
Frequency
undamped
F. Numerical Damping
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In summary, Workbench allows the following four inputs for damping:
Alpha and Beta damping Global or material-dependent.
Defines the mass matrix multiplier and stiffness matrix multiplier for damping.
Element damping (viscous)
Defines the damping coefficients (c) directly.
Damping ratio g(solid) Global or material-dependent.
Available only in Full harmonic analysis, and mode-sup analyses (harmonic,
transient, spectrum)
Defines the ratio of actual damping to critical damping.
Does not affect the frequency of damped oscillation d.
Numerical damping (artificial)
Defines the amplitude decay factor obtained through a modification of the time-
integration scheme.
Available only in transient analysis.
NOTE: The effects are cumulative if set in conjunction.
H. Damping --- Summary
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Introduction to ANSYS
Mechanical
14.5 Release
Workshop 3
Damping in Mechanical WB