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Noise and Vibration Control

2. Damping

Indian Institute of Technology Roorkee

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What is damping?

Damping is a phenomenon by which mechanicalenergy is dissipated (usually converted as thermalenergy) in dynamic systems.

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2.1 Types of damping

Three primary mechanisms of damping

Internal damping of material

Structural damping at joints and interface

Fluid damping through fluid -structure interactions

Two types of external dampers can be added to a

mechanical system to improve its energy dissipation

characteristics:

Active dampers require external source of power

Passive dampers Does not


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MATERIAL (Internal) Damping

Internal damping originates from energy dissipation

associated with: --microstructure defects (grain boundaries &

impurities),

-thermo elastic effects (caused by local temperature

gradients)

-eddy-current effects (ferromagnetic materials),

-dislocation motion in metals, etc.

Types of Internal damping:

-Viscoelastic damping-Hysteretic damping


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Area of hysteresis loop is energy dissipation per cycle of

motion- termed as per-unit-volume damping capacity (d).

Figure 1A typical hysteresis loop for mechanical damping [4]


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Viscoelastic model

Kelvin vigot modelStress strain relationship is given by

E young's modulus

E* - visco elsatic parameter

Damping capacity per unit volume


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Kelvin vigot model

Subjected to harmonic sinusoidal excitation

By substituting the equations we get

dv depends on excitation frequency


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Other two models of viscoelastic

models

Maxwells model

Standard linear solid model


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Hysteretic Damping

Damping forces does not significantly depend on the

frequency of oscillation of strain. (frequency of harmonicmotion )

Damping capacity per unit vol is generally represented as dh

For this case n =2


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Hysteretic Damping

By considering harmonic motion at frequency

The above equation 2.1 becomes

Simple model for the viscoelastic material is given by


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Experimental results indicate

that for most structural

materials such as steel or

Aluminium the energy loss is:- independent of the frequency

-proportional to amplitude

squared.


C

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Well functioning shock absorber


N i d Vib ti C t l

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Poor functioning shock absorber(Primarily structural damping)


N i d Vib ti C t l

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2.2 Structural damping

Structural damping is a result of the mechanical-energy

dissipation caused by rubbing friction resulting from relativemotion between components and by impacting or intermittent

contact at the joints in a mechanical system or structure

Energy dissipation caused by rubbing is usually representedby a Coulomb-friction model. Energy dissipation caused by

impacting, however, should be determined from the

coefficient of restitution of the two members that are in

contact.



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A typical hysteresis loop for structural damping

Figure 5 Some representative hysteresis loops [4] (a) Structuraldamping, (b) Coulomb friction, (c) Simplified structural damping model



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Structural damping

A simplified model for structural damping caused by local

deformation can be given by

The corresponding hysterisis loop is given by



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2.3 Fluid Damping

Consider a mechanical component moving in a fluid medium.

The direction of relative motion is shown parallel to the y-axisin Figure , Local displacement of the element relative to the

surrounding fluid is denoted by q(x,z,t). The resulting drag

force per unit area of projection on the x-z plane is denoted

by fd .This resistance is the cause of mechanical-energy

dissipation in fluid damping. It is usually expressed as



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Fluid damping -A body moving in a fluid medium.

Figure 6 A body moving in a fluid medium [4]

Direction of relative motion parallel to y-axis

q(x,z,t) local displacement of the element relative to surrounding fluid



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Mechanics of fluid damping.

Figure 7 Mechanics of fluid damping [4]

Resulting drag force per unit area of projection on the x-z plane fd


q& 2 sgn ( )fd = 0.5 cd q&

Where - fluid density

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Damping Classification



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Representation of Damping in vibration Analysis

M is mass (inertia) matrix

K is Stiffness matrixf(t) is forcing function vector

d damping force vector (nonlinear function of x and )

N dof mechanical systemIts motion- represented by vector x of n generalized coordinates xiEqns of motion expressed in vector-matrix form:

x&

Where C =cm M + ck K cm inertial damping matrix

ck stiffness damping matrix

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Damping Models



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Equivalent damping ratios



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2.4 Loss factor

Damping capacity of a device is energy dissipated in a

complete cycle

Specific damping capacity D is given by ratio of

Loss factor



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Loss factor is approximately given as



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Damping parameters



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2.5 Measurement of damping

Damping can be represented by various parameters (such

as specified damping capacity loss factor Q-factor and

damping ratio) a and models ( such as viscous, hysteretisis ,structural and fluid)

A major difficulty arises because usually it is not possible

to isolate various t types of damping( e.g. material. structural,

and fluid) from an overall measurement. Further more,

damping measurements must be conducted under actual

operating conditions for them to be realistic'



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There are two general ways by which damping

measurements can be made:

1. Time-Response Method and

2. frequency-response methods.

The basic difference between the two types of

measurements is that the first type uses a time-response

record of the system to estimate damping, where as the

second type uses a frequency-response record.



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Logarithmic method

when a single degree-of-freedom oscillatory system withviscous damping is excited by an impulse input or an initial

condition excitation its response takes the form of a time

delay



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Logarithmic method

Figure 8 Impulse response of a simple oscillator [4]



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Logarithmic method

damped natural frequency is given by

If the response at t = ti is denoted by y, and the

response at

Then, the logarithmic decrement (per unit cycle) is given b y



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Logarithmic method

Damping ratio can be expressed as

Pre radian logarithmic decrement

Finally the equation obtained is



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Step Response Method

This is also a time-response method If a unit-step

excitations applied to the single-degree-of freedom

oscillatons by system of equations response is given by

Figure 9 Step response of a simple oscillator [4]



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Response is given by

First peak ( peak time )

Peak time (peak value) Mp

Percentage overshoot PO



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Damping ratio is computed using appropriate relation from

the following equations



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Hysteresis Loop method

Depending on inertial and elastic conditions the hystersis

loop will change but the work done in the conservative forces

will be zero consequently work done will be equal to energy

dissipated by damping only

without normalizing with respect to mass, the energydissipation per hysteresis loop of viscous damping is



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Hysteresis Loop method


The energy dissipation per hysteresis loop of hysteretic

damping is

Initial max potential energy is

The loss factor of hysteretic damping is given by

the equivalent damping ratio for hysteretic damping is

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An experimental hysteresis loop of a damping material.

Figure 10 An experimental hysteresis loop of a damping material [4]



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Magnification Factor method

consider the single-degree-of-freedom oscillatory system

with viscous damping , the magnification factor is

Plot of this value and the peak value of magnitude occurs in

the denominator

Resulting solution of resonant frequency is



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Magnification Factor method


Figure 11 The Magnification Factor method of damping

measurement applied to a single dof system [4]

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Use of bode plots

Multi degree freedom system model damping can be

estimated by bode plots

Figure 12 Magnification Factor method applied to a multi dof system [4]



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Bandwidth method

The bandwidth method of damping measurement is based

on frequency response

The peak magnitude is given by equation for low damping.

Bandwidth (half-power) is defined as the width of the

frequency-response magnitude curve when the magnitude is1/sqrt2 times the peak value.



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Bandwidth method

w Frequency expressed as

w2of quadratic equation is



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Bandwidth method


Figure 13 Bandwidth method of damping measurement applied to

a single dof system [4]

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Bandwidth method

The damping ratio can be estimated by using band width in

the relation



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Bandwidth method

For small (in comparison to 1)

For the i th mode the damping ratio is given by:



B d idth th d f d i t i lti

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Bandwidth method of damping measurement in multi-degree freedom system

Figure 14 Bandwidth method of damping measurement in multi-degree freedom system



Effect of vibration amplitude on damping in structures

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Effect of vibration amplitude on damping in structures.

Figure 15 Effect of vibration amplitude on damping in structures [4]



Overview of measurement method for

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Overview of measurement method fordamping



Overview of measurement method for

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Overview of measurement method fordamping


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Slip / interface damping mechanism



Interface damping

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Interface damping


Simplified model is coulomb dry friction model

f= frictional force that opposes the motionr = normal reaction force between the sliding surfaces

v= relative velocity between the sliding surfaces

=coefficient of friction'

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Thank you


2 damping[1]

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