L5 determination of natural frequency & mode shape

Noise and VibrationsNoise and Vibrations (BDC4013)(BDC4013)

DR MUHD HAFEEZ ZAINULABIDINUniversiti Tun Hussein Onn Malaysia

Chapter 5 – Computational/Numerical MethodsChapter 5 – Computational/Numerical Methods Determination of Natural Frequencies and Mode ShapesDetermination of Natural Frequencies and Mode Shapes

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Necessity to Use Computational MethodNecessity to Use Computational Method

In two degrees of freedom system, solving the natural frequencies can be conducted by simply calculating the root of the second order polynomial.

4 2 0A B Cω ω+ + =

By assuming

Then the natural frequencies can be found

( ) 0222 =++ CBA nn ωω

3

Classical MethodsClassical Methods

Standard Matrix Iteration Method Dunkerly’s Method Rayleigh’s Method Holzer’s Method

4

Standard Matrix IterationStandard Matrix Iteration

[ ] { } [ ] { } 0M x K x+ =&&Considering a general equation of motion

Assuming harmonic motion

( ) sin( )i ix t X tω=

Equation to solve [ ] { } [ ] { }2 0M X K Xω− + =

5

Standard Matrix IterationStandard Matrix Iteration(two possible solution)(two possible solution)

[ ] { } [ ] { }2 0M X K Xω− + =

multiply [ ] 1K

−multiply [ ] 1

M−

[ ] [ ] { } [ ] [ ] { }1 12 0K M X K K Xω − −− + =

[ ] [ ] { } [ ] { }12 0K M X I Xω −− + =

[ ] [ ] { } { }1

2

1K M X X

ω− =

[ ] [ ] { } [ ] [ ] { }1 12 0M M X M K Xω − −− + =

[ ] { } [ ] [ ] { }12 0I X M K Xω −− + =

[ ] [ ] { } { }1 2M K X Xω− =

Converge to lowest nat freq Converge to highest nat freq

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Standard Matrix IterationStandard Matrix Iteration(Solution procedures to obtain the lowest nat freq)(Solution procedures to obtain the lowest nat freq)

(1) Identify matrix [K] and [M]

(2) Calculate [K]-1

(3) Define the initial trial vector {X} and convergence criteria

(4) Multiply [K]-1 [M] {X} = 1/ω2{Xnew}

(5) Normalized the result {Xnew}/{X1new)

(6) Check the convergence , use for a new trial {X}

(7) When it is converged

normalized 1

2 Xn

=ω

7

Standard Matrix IterationStandard Matrix Iteration(Solution procedures to obtain the highest nat freq)(Solution procedures to obtain the highest nat freq)

(1) Identify matrix [K] and [M]

(2) Calculate [M]-1

(3) Define the initial trial vector {X} and convergence criteria

(4) Multiply [M]-1 [K] {X} = ω2{Xnew}

(5) Normalized the result {Xnew}/{X1new)

(6) Check the convergence , use for a new trial {X}

(7) When it is converged

normalized 2 Xn =ω

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Example Problem 9-1(a)Example Problem 9-1(a)

k2

x1

x2

m1

m2

k1

k3

Calculate the fundamental (lowest) natural frequency and the corresponding mode shapes.

k1=10N/m k2=20N/m k3=15N/m

m1 = 1.2 kg m2 = 2.7 kg

9

Example Problem 9-1(b)Example Problem 9-1(b)

k2

x1

x2

m1

m2

k1

k3

Calculate the highest natural frequency and the corresponding mode shapes.

k1=10N/m k2=20N/m k3=15N/m

m1 = 1.2 kg m2 = 2.7 kg

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Dunkerly’ FormulaDunkerly’ Formula

Dunkerly’ formula is searching for the fundamental (lowest) natural frequency.

It is based on [K]-1 multiplication

[ ] [ ] { } [ ] { }12 0K M X I Xω −− + =

[ ] [ ] { } [ ] { } { }1

2

10K M X I X

ω− − =

[ ] [ ] [ ] { } { }1

2

10K M I X

ω− − =

[ ] [ ] [ ]2

10a M I

ω− =

[ ] [ ]1K a

− =

11

Dunkerly’ FormulaDunkerly’ Formula(calculation procedures)(calculation procedures)

2 2 2 211 22

1 1 1 1

n nnω ω ω ω≈ + + +L : fundamental (lowest) natural frequencynω

nnnn

n

k

mω ≈

(1) Identify k11, k22, knn, m1, m2, mn

(2) Calculate natural frequency of the individual component

(3) Predict the fundamental natural frequency of the system

n nn

: natural frequency of a SDOF system

consisting m and spring of stiffness knnω

12

Example Problem 9-2Example Problem 9-2

k2

x1

x2

m1

m2

k1

k3

Predict the fundamental natural frequency using Dunkerly method

k1=10N/m k2=20N/m k3=15N/m

m1 = 1.2 kg m2 = 2.7 kg

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14

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Example Problem 9-3Example Problem 9-3

0.5m 0.5m 0.5m 0.5m

5 kg 4 kg7 kgE=207 GPa

I=12 10-6 m4

Predict the fundamental natural frequency of the system using Dunkerly method

Solution to Problem 9.3Solution to Problem 9.3

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From the known formula for the deflection of a simply supported beam, the flexibility influence coefficients can be found.

( )2

0 4348

,Deflection 22 LxxL

EI

Pxv ≤≤−−=

EI

La

EI

Laa

3

22

3

3311 48

1 and

256

3 ===

EI

Lmmm 3321

21 256

3

48

1

256

31

++≈

ω

rad/s 56.11111 =ω

( ) ( ) ( )( )( )69

3

21 101210207

2

256

43

48

71

256

531−××

++≈

ω

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Rayleigh MethodRayleigh Method

This method predicts the fundamental (lowest) natural frequency

This method based on energy method

21

2T mx= &

21

2V kx=

{ } [ ] { }1

2

TT x M x= & &

{ } [ ] { }1

2

TV x K x=

18

Rayleigh QuotientRayleigh Quotient

{ } [ ] { }1

2

TT x M x= & & { } [ ] { }1

2

TV x K x=

{ } { }{ } { }

sin( )

cos( )

x X t

x X t

ω

ω ω

=

= −&

{ } [ ] { } 2max

1

2

TT X M X ω= { } [ ] { }max

1

2

TV X K X=

max maxT V=

{ } [ ] { }{ } [ ] { }

2

T

T

X K X

X M Xω =

19

Rayleigh MethodRayleigh Method(Calculation procedures)(Calculation procedures)

Identify [K] and [M] Select any trial vector mode {X} Predict the fundamental natural

frequency based on the Rayleigh Quotient

{ } [ ] { }{ } [ ] { }

2

T

T

X K X

X M Xω =

20

Example problem 9-4Example problem 9-4

k2

x1

x2

m1

m2

k1

k3

Predict the fundamental natural frequency using Rayleigh method

k1=10N/m k2=20N/m k3=15N/m

m1 = 1.2 kg m2 = 2.7 kg

21

22

Holzer MethodHolzer Method

1 1 1 1 2

2 2 1 2 1 2 2 3

3 3 3 3 2

( )

( ) ( )

( )

t

t t

t

I k

I k k

I k

θ θ θθ θ θ θ θθ θ θ

= − −

= − − − −

= − −

&&

&&

&&

21 1 1 1 2

22 2 1 2 1 2 2 3

23 3 3 3 2

( )

( ) ( )

( )

t

t t

t

I k

I k k

I k

ωωω

Θ = Θ − Θ

Θ = Θ − Θ + Θ − Θ

Θ = Θ − Θ+

2

1

0n

i ii

Iω=

Θ =∑)cos( φωθ +Θ= tii

Assume

23

Holzer MethodHolzer Method(calculation)(calculation)

( )

21 1

2 11

22 3 2 2 1 2 1 2 2

21 2 2

3 2 2 12 2

2 21 1 2 2

3 22 2

2

3 2 1 1 2 22

( )

( )

t

t t t

t

t t

t t

t

I

k

k k k I

k I

k k

I I

k k

I Ik

ω

ωω

ω ω

ω

ΘΘ = Θ −

Θ = Θ + Θ − Θ − Θ

ΘΘ = Θ + Θ − Θ −

Θ ΘΘ = Θ − −

Θ = Θ − Θ + Θ

21 1 1 1 2

22 2 1 2 1 2 2 3

( )

( ) ( )

t

t t

I k

I k k

ωω

Θ = Θ − Θ

Θ = Θ − Θ + Θ − Θ

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Holzer MethodHolzer Method(calculation procedures)(calculation procedures)

Set initial ω=0 and set the sweep increment of ω with a value Δω Station 1:

X1=1 (or Θ1=1), calculate M1=ω2m1X1 (or ω2I1Θ1)

Station 2:

Calculate X2 (or Θ 2), calculate M2=M1+ ω2m2X2 (or ω2I2Θ2)

Station 3:

Calculate X3 (or Θ 3), calculate M3=M2+ ω2m3X3 (or ω2I3Θ3)

Station n:

Calculate Xn (or Θ n), calculate Mn=Mn-1+ ω2mnXn (or ω2InΘn)

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I1=2 kg m2

I2=4 kg m2

Kt=4 MNm/rad

Calculate the natural frequencies and mode shapes

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27

28

Holzer MethodHolzer Method(summary calculation)(summary calculation)

Torsion Translation

( )

1

21 1

2 11

2

3 2 1 1 2 22

2 1

111

1

2,3,

t

t

i

i i k kkti

I

k

I Ik

Ik

i n

ω

ω

ω −

−=−

Θ =

ΘΘ = Θ −

Θ = Θ − Θ + Θ

Θ = Θ − Θ ÷

=

∑L

( )

1

21 1

2 11

2

3 2 1 1 2 22

2 1

111

1

2,3,

i

i i k kki

X

m XX X

k

X X m X m Xk

X X m Xk

i n

ω

ω

ω −

−=−

=

= −

= − +

= − ÷

=

∑L

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I1=2 kg m2

I2=4 kg m2

I3=2 kg m2

kt1=3 MNm/rad

Kt2=2 MNm/rad

Calculate the natural frequencies and mode shapes

Health & Medicine

L5 determination of natural frequency & mode shape