1 Free Vibration Damping for Class

7/27/2019 1 Free Vibration Damping for Class

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SDOF Definitions

lumped mass

stiffness proportionalto displacement

damping proportional to

velocity linear time invariant

2nd order differentialequations

Assumptions

m

k c

x(t)


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Equation of Motion - Natural Frequency

Newtons Second Law:

(written about static equilibrium position)

Circularfrequencygiven as:

=+= )t(fkxxmmaf && (2.2.2)

mk

n = (2.2.3)


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Equation of Motion written in standard form

has the general solution

where A and B are two necessary constantsdetermined from the initial conditions of

displacement and velocity

(2.2.4)

(2.2.5)tcosBtsinAx nn +=

0xx2n =+&&


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This results in

and the frequency can be shown in relation tomass and stiffness or the static displacement ofthe system as

(2.2.6)

(2.2.8)

tcos)0(xtsin)0(x

x nnn

+

=&

=

= g

2

1m

k2

11f


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Energy Method

In a conservative system, the total energy is

constant. Kinetic energy is stored in the mass interms of velocity and potential energy is stored asstrain energy in the spring

From conservation of energy, an equilibrium onstate 1 and state 2 exists - and at the extremesthe maximums result in

(2.3.1)ttanconsUT =+ ( ) 0UTdt

d=+

2211 UTUT +=+ MAXMAX UT =

(2.3.2)

(2.3.3) (2.3.5)


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Rayleigh Method: Effective Mass

The kinetic energy can be written in terms of

velocity for a specific point in the system as

where the effective mass is at some point

If the stiffness is known at that point then thenatural frequency can be calculated as

(2.4.1)

(2.4.2)

2eff2

1 xmT &=

effn m

k=


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Virtual Work - Equilibrium of Bodies

If a system is in equilibrium under the action of a

set of forces is given a virtual displacement, thevirtual work done by the foces will be zero

(1) a virtual displacement is a small variationof the coordinate (must be compatiblewith constraints of the system)

(2) virtual work is work done by all activeforces in avirtual displacement (nogeometry change which implies forces on

system remain unchanged)


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Viscously Damped Free Vibration

The equation of motion for the SDOF with viscous

damping included is

both a homogeneous and particular solution exist

(2.6.2) =++= )t(fkxxcxmmaf &&&


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Homogenous Solution

Homogenous equation is

and assuming an exponential solution form gives

which results in

which yields the characteristic equation

(2.6.3)

(2.6.4)

0kxxcxm =++ &&&

(2.6.4)

(2. 6.5)

0ekcsms st2 =++

0mks

mcs2 =++

m

k

m2

c

m2

cs

2

2,1

=


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Homogenous Solution

This results in a general solution given as

Three distinct solutions result from this general

solution.

(2.6.7)tsts

21 BeAex +=

(1) underdamped

(2) overdamped(3) critically damped

11=


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Homogenous Solution

Some general definitions are:

Critical Damping

% Critical Damping

Roots

(2.6.9)

(2.6.10)

(2.6.11)

km2m2

m

km2c nc ===

cc

c=

1s 2nn2,1 =


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The general solution can be rewritten as

and with adjustingsome terms

the poles can be

observed as seenin the figure uponchan in dam in

(2.6.12)

(2.6.13)

Homogenous Solution

)t(fm

1xx2x 2nn =++ &&&

2

n

2,11i

s=


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Oscillatory Solution (3 different forms)

and the general homogeneous solution becomes

(2.6.14)

(2.6.16)

(2.6.17

(2. 6.18

Homogenous Solution - Underdamped 1


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The damped exponential response

( ) ( )

+

+= t1cos)0(xt1sin

1

)0(x)0(xex n

2n

2

2n

ntn &

Homogenous Solution - Underdamped 1


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Overdamped solution is

with coefficients A and B as

(2.6.19)

(2.6.20)

(2.6.21)

Homogenous Solution - Overdamped 1>

t1t1n

2

n

2

BeAex+ +=

12

)0(x1)0(xA

2

n

n2

++=

&

12

)0(x1)0(xB

2n

n2

=

&


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The critically damped solution is

and the solution becomes

(2.6.22)

(2.6.23)

Homogenous Solution - Critically Damped 1=

( ) tneBtAx +=

( )t

n

t

n

n

et)0(x)0(x

e)0(xx

+

+=

&


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Logarithmic Decrement

From the damped vibration solution, the amount of

damping in the system can be expressed as

and the log decrement becomes

and for small damping

(2.7.2)

(2.7.3)

dn)t(

tdn

d1n

1n

eln

e

eln ===

+

21

2

=

(2.7.4) 2


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Logarithmic Decrement

Plot of Damping Estimates

Log Decrement


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Coulomb Damping

Sliding or dry friction produces a force which is

opposite to the velocity and its direction changeswith each half cycle

or

the decay in amplitude

per cycle is therefore

( ) 0)XX(FXXk21

11d

2

1

2

1 =+

( )d11

FXXk2

1=

d21 F4XXk = (2.8.1)


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Review - SDOF Definitions

lumped mass

stiffness proportionalto displacement

damping proportional to

velocity linear time invariant

2nd order differentialequations

Assumptions

m

k c

x(t)


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Review - SDOF Equations

Equation of Motion

)t(fxkdt

dxc

dt

xdm

2

2

=++ )t(fxkxcxm =++ &&&

Characteristic Equation

Roots or poles of the characteristic equation

0kscsm 2 =++

m

k

m2

c

m2

cs

2

2,1 +

=

or


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Review - SDOF Definitions

Poles expressed as

Damping Factor

Natural Frequency

% Critical Damping

Critical Damping

Damped NaturalFre uenc

POLE

CONJUGATE

j

n

d( ) d

2

n

2

nn2,1 js ==

n=

mk

n=

ccc=

nc

m2c =2

nd 1 =

1 Free Vibration Damping for Class

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