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Bernhard Bettig

Mechanical Design Research Lab

Mechanical Engineering - Engineering Mechanics Dept.

Michigan Technological University

Web site: http://www.me.mtu.edu/~mdrl

Rotordynamics Unit 7:

Time Integration and NaturalFrequency Calculations

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Time IntegrationTime Integration takes as input:

- the FE Model:

- the initial position, velocity and

acceleration for each node:

- a time step increment t

- parameters for the integrationscheme

RKUUCUM =++

UUU 000 ,,

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Time Integration

The output of the time integration is:

- the position, velocity andacceleration for each node, at each

time step: UUU ttt ,,

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Time Integration

One Time Integration scheme is

known as the Central Difference

method.

It assumes that:

And:

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Time Integration

If we substitute those expressions

into:

then:

However, this scheme is

conditionally stable, requiring:

ncr

Ttt = Tn is the smallest period of

the structure

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Time Integration MHYFECS uses another scheme, the

Newmark method

It assumes that:

and are parameters that can be determined to

obtain integration accuracy and stability.

= and = were proposed to provide anunconditionally stable scheme.

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Time Integration

Performing substitutions similar to the

Central Difference Method leads to the

following procedure*:

* Taken from K.J. Bathe, Finite Element Procedures, Prentice Hall

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Time Integration

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Natural Frequency Calculation

Given the homogeneous dynamic

equations:

assume a solution:

Substituting the solution into the eq.:

0KuuCuM =++

ptptpt eppee2,, === uuUu

( ) 0UKCM =++pp2

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Natural Frequency CalculationThis equation can be converted into a

standard eigenvalue form:

For each mode, there is a solution

with real and imaginary parts:

=

0

0

U

U

I0

0I

CKMK

I p

p

1011

jbap = 1=j

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Natural Frequency Calculation

The natural frequency is given by theimaginary part:

n= b

The damping ratio is given by:

This must be positive for the mode tobe stable.

22 ba

a

+

=

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Mode Shape CalculationOnce thep = a+bj is known for a mode, we

can calculate the mode shape.

The mode shape also has real and

imaginary parts:

Substitutingp into the equation on slide 9

and separating real and imaginary parts:

imagreal UUU +=

( ) ( )[ ][ ]( )( ) ( )[ ][ ][ ][ ] 0UUBA

0UUCMKCM0UUKCM

=++

=+++++=+++++

imagreal

imagreal

imagreal

jj

jbabjabajjbajabba

)()(

)2(2

22

22

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Mode Shape Calculation

real part: AUreal BUimag = 0

imaginary part: AUimag +BUreal = 0

( )( ) bab

aba

CMB

KCMA

+=++=

2

22where:

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Mode Shape Calculation

Solving,

( ) 0UBBAA

BUAU

=+

=

imag

imagreal

1

1

However scaling is required. Since

the problem is homogeneous we

have an infinite number of solutions,each scaled differently.

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Mode Shape Calculation

We pick one nodal displacement and

set it to 1.

=

=

=+=

01

1,

1

2221

1211

1

2221

12111

0U

C

CC

UU

C

CCBBAAC

imag

imag

imag

C

C

12111 CUC =imag

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Mode Shape CalculationThe procedure therefore is:

1)

2)

3)

4) Normalizing so that the maximumdisplacement is equal to one.

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111 CCU =imag

=1

1imag

imag

UU

imagreal BUAU1=