Mech Dynamics 14.5 L03 Damping

7/24/2019 Mech Dynamics 14.5 L03 Damping

1/24

2012 ANSYS, Inc. March 28, 2013 1 Release 14.5

14.5 Release

Lecture 3

Damping

ANSYS MechanicalLinear and Nonlinear Dynamics


2/24


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


3/24


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.


4/24


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)


5/24


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


6/24


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


7/24 2012 ANSYS, Inc. March 28, 2013 7 Release 14.5

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



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



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]



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).


2

2 or


11/24


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


2

2 or


12/24


22

i

i

2

2

1

1

4

4

ff

ff


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


13/24


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)


mbma N

jj

m

j

N

ii

m

i

KMC11

22

i

i

i

Equivalent damping


14/24


[2] Direct ly as glo bal dampin g value(Details section of Analysis Settings)


KMC

Damped Modal Full Transient

22

i

i

i

Equivalent damping


15/24


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


16/24


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


17/24


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


18/24


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]


19/24


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)


mbma N

j jj

m

j

N

i i

m

i

KgMC11

2 g

i

ii

22

Equivalent damping


20/24


[2] Direct ly as glo bal dampin g value

(Details section of Analysis Settings)


KgMC

2 g

i

i

i

22

Equivalent damping


21/24


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


22/24


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


23/24


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


24/24

2012 ANSYS Inc March 28 2013 24 Release 14 5

Introduction to ANSYS

Mechanical

14.5 Release

Workshop 3

Damping in Mechanical WB

Documents

Mech Dynamics 14.5 L03 Damping