Dynamic Analysis Using FEA

8/7/2019 Dynamic Analysis Using FEA

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Outline:

Dynamic analysis Element mass matrix Problem 1: Free vibration using MATLAB Modal analysis

Modal analysis of undamped systemModal analysis of damped system

Problem 2: Forced vibration using MATLAB


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Dynamic analysis

Inertia force to be included

Equations of motion:x

xyx fyxt

u+

+

=

2

2

y

xyyf

xytv +

+

=

2

2


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Euler-Bernoulli beam:

Element mass matrix: =

l

Te dxNANM0

][

=

22

22

422313221561354

313422

135422156

420

][

llllll

llll

ll

AlM

e

=

0000

0201

0000

0102

6][

AlM

e

Timoshenko beam:

Element mass matrix


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Frame element:

=

2

2

22

2

2

2222

11

2

2

22

2

2

2222

11

42203130

22156013540

00200

31304220

13540221560

00002

][

mllmmllm

lmmlmm

mm

mllmmllm

lmmlmm

mm

M

e


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=

201010

020101

102010

010201101020

010102

12][

AM

e

Linear triangular element:

Bi-linear element:

=

40201020

04020102

20402010

02040201

10204020

01020402

20102040

02010204

36][

A

Me


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Equations of motion for dynamic analysis:

)}({}]{[}]{[ tFdKdM =+

For free vibration :

0}]{[}]{[ =+ dKdM

The eigen value problem:

0}]}{[]([ 2 = dMK

where is the angular natural frequency in rad/s and is the mode shape.d


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Problem 1: Free vibration using MATLAB

Find the natural frequency of a frame of L-shaped which is made of two beams oflength of 1 m each. Both beams have cross section of 0.01 m by 0.01m. The elasticModulus is 100 GPa. The beam has mass density of 1000 kg/m3.


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Problem 1: Free vibration using MATLABFind the natural frequency of a

frame of L-shaped which is madeof two beams of length of 1 meach. Both beams have crosssection of 0.01 m by 0.01m. Theelastic Modulus is 100 GPa. Thebeam has mass density of 1000

kg/m3.

Mode

no.

Natural

frequency

1 34

2 92

3 455

4 667

5 1458


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Modal analysis What is modal analysis?

A technique used to determine a structures vibration characteristics:

Natural frequencies

Mode shapes

Mode participation factors (how much a given mode participates in

a given direction)

Most fundamental of all the dynamic analysis types.

Allows the design to avoid resonant vibrations or to vibrate at a

specified frequency (speakers, for example).

Gives engineers an idea of how the design will respond to different

types of dynamic loads.Helps in calculating solution controls (time steps, etc.) for other

dynamic analyses.Recommendation:Because a structures vibration characteristics

determine how it responds to any type of dynamic load, always perform

a modal analysis first beforetrying any other dynamic analysis.


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Modal analysis of undamped system

The governing differential equation of motion for a n degree of

freedom linear second order system:

[ ]{ } [ ]{ } { }FdKdM =+The form of response or solution can be assumed as: ( ){ } { } tietd =

{ } - mode shape (eigen vector) and natural frequencyThe general solution to be linear combination of each mode:

( ){ } { } { } { }

ti

nn

titin

ececectd

+++=

.......

21

2211

For free vibration with {F}=0, [ ] [ ] { } 02 =+ tieKM

}]{[}]{[2

iiiKM =


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Multiply on both sides,{ }Ti { } [ ]{ } { } [ ]{ }

i

T

ii

T

ii KM =2

According to the general property of mass and stiffness matrix in the form

{ } [ ]{ } ,0>XMX T { } [ ]{ } ,0XKX T { } 0Xfor

As the eigen vectors are orthogonal with respect to mass and stiffness

matrices

jifor { } [ ]{ } [ ]{ } 0==i

T

ji

T

jKM

The orthogonality of eigen vectors provides

{ } [ ]{ } 1=iT

iM [ ]{ }

2}{ii

T

jK =

After normalizing the eigen vector, { } [ ]{ }=d

[ ] [ ]n ,......,, 21=


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[ ][ ]{ } [ ][ ]{ } { }FKM =+

Premultiplying on both sides[ ]T

[ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] { }FKM TTT =+

{ } { } { } { }i

T

ifFdiag ==+ ][ 2

In other words, the system equations are decoupled, iiii f=+ 2


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Modal analysis of damped system

The governing differential equation of motion for a n degree of

freedom linear second order system:

[ ]{ } [ ]{ } { }FdKdCdM =++ }]{[

Proportional or Rayleighs damping:

[ ] [ ] [ ]KMC +=

{ } [ ]{ } ,0=jT

i C ji

]][[][ CTand becomes a diagonal matrix

Governing differential equation of motion: iiiiiii f=++ 2

2

Application of Laplace transform yields,

( )( ) ( ) ( ) ( )

222

020

iii

iiiiii

ss

sfss

++

+++=


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Problem 2: Forced vibration using MATLAB

Beam: Cantilever beam

Force applied: A harmonic force of magnitude 1 N at the tip of thebeam and excitation frequency varies from 0 to 3000 rad/s


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Mode

no.

Natural

frequency

1 65

2 402

3 1135


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Questions

?

?

?

?

?


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Thank You

Dynamic Analysis Using FEA

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