[PPT] - A semi- -analytical mathematical formulation as a analytical PowerPoint Presentation

SLIDE 1

A semi A semi-

analytical mathematical formulation as a

analytical mathematical formulation as a tool for the dynamic analysis of tool for the dynamic analysis of catenary catenary shaped shaped slender structures slender structures

by by Ioannis Ioannis K.

K. Chatjigeorgiou

Chatjigeorgiou chatzi@naval.ntua.gr chatzi@naval.ntua.gr

National Technical University of Athens School of Naval Architecture and Marine Engineering Division of Marine Structures

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2 2

Dynamic equilibrium in normal direction Dynamic equilibrium in normal direction

nd

F w s T s EI t v M + − ∂ ∂ + ∂ ∂ − = ∂ ∂ φ φ φ cos

3 3

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3 3

Separation of variables Separation of variables

) , ( ) ( ) , (

1

t s T s T t s T + =

nd

F w s T s EI t v M + − ∂ ∂ + ∂ ∂ − = ∂ ∂ φ φ φ cos

3 3

) , ( ) ( ) , (

1

t s s t s φ φ φ + =

t q t q dC s q EI s q w s q T s T s q T t q M

Dn

∂ ∂ ∂ ∂ − ∂ ∂ − ∂ ∂ + ∂ ∂ + + ∂ ∂ = ∂ ∂ ρ φ φ 2 1 sin d d

4 4 2 2 1 1 2 2 2 2

Uncoupled Uncoupled-

undamped

undamped system system – – free vibrations free vibrations

4 4 2 2 2 2

) ( sin ) ( s q EI s q s w s q s T t q M ∂ ∂ − ∂ ∂ + ∂ ∂ = ∂ ∂ φ

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4 4

Natural frequencies and mode shapes Natural frequencies and mode shapes -

Free

Free vibrations vibrations Nondimensional Nondimensional form form

( ) ( ) x

y x x y x x y K y ∂ ∂ + ∂ ∂ + ∂ ∂ − = ∂ ∂ β α τ

2 2 4 4 2 2

Harmonic motions Harmonic motions

( ) ( )

) ( d ) ( d d ) ( d d ) ( d

2 2 2 4 4

= + + + − x y x x y x x x y x x x y K ω β α

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5 5

Simplified model Simplified model – – untensioned untensioned vertical structure vertical structure

Triantafyllou Triantafyllou, MS and , MS and Triantafyllou Triantafyllou, GS (1991), The paradox of a , GS (1991), The paradox of a hanging string: an explanation using singular perturbations, Jou hanging string: an explanation using singular perturbations, Journal of rnal of Sound and Vibration, 148, 343 Sound and Vibration, 148, 343-

351

351

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6 6

Simplified model Simplified model – – vertical structure under tension: vertical structure under tension: effective tension at the top and bending stiffness of the effective tension at the top and bending stiffness of the same order of magnitude same order of magnitude

Chatjigeorgiou Chatjigeorgiou, IK (2006), Solution of the boundary layer problems for , IK (2006), Solution of the boundary layer problems for calculating the natural modes of riser type slender structures, calculating the natural modes of riser type slender structures, Proc Proc OMAE 2006 Conf, Hamburg, Germany, OMAE2006 OMAE 2006 Conf, Hamburg, Germany, OMAE2006-

92390

92390

( ) ( ) {

} { }

( ) ( ) ( ) ( ) [ ]

{ }

( ) ( ) { ( ) ( ) [ ]

}

) 1 ( ) 1 /( 1 / 1 / ) 1 ( 1 1 / 1 2 2

1 2 1 1 2 1 2 1 2 1 1 2 1 2 / ) 1 ( 1 2 / 2 1 2

x T a x z z T z z a e x T c e x T x T x T a y

x T x T c

− + + Ω + + − + Ω − + − + − + + − + + + − + Ω + + Ω =

− + − −

χ κ χ κ χ κ χ κ κ κ κ κ ε ε κ κ

ε ε

Y J Y J Y J Y J Y J

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7 7

Semi Semi-

analytical formulation of the solution: Can be used in

analytical formulation of the solution: Can be used in both time and frequency domain both time and frequency domain – – Requires the calculation Requires the calculation

f mode shapes.
f mode shapes. Application of the WKB* method

Application of the WKB* method

( ) ( )

) ( d ) ( d d ) ( d d ) ( d

2 2 2 4 4

= + + + − x y x x y x x x y x x x y K ω β α Introduction of a perturbation coordinate z= Introduction of a perturbation coordinate z=ε εx, x, ε ε=K<<1 =K<<1

( ) ( )

) ( d ) ( d d ) ( d d ) ( d

2 2 2 2 4 4 5

= + + + − z y z z y z z z y z z z y ω εβ α ε ε

Two possible solutions Two possible solutions

( ) ε

z u

z y e = ) (

( ) ε

z u

z y

i

e ) ( = Logan JD, Logan JD, Applied Mathematics Applied Mathematics, Wiley , Wiley Interscience Interscience, second edition , second edition (1997) (1997)

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8 8

Perturbation expansion and equation of the perturbation Perturbation expansion and equation of the perturbation terms terms

( )

( )(

)

( )

6 3 4

2 2 4 2 2 3 3 4

= + + + ′ + + ′ + ′ ′ + ′ ′ + ′ ′ ′ − ω υ β υ υ ε α ευ υ υ ε υ υ ε υ υ ε υ ε z z

u′ = υ

( ) ( ) ( )

( )

2 1

ε ευ υ υ O z z z + + =

( ) ( ) ( ) ( ) ( )

( )

2

2 2 1 1 2 4

= + + + + + + ′ + − ε ω υ εβ υ β υ υ εα υ α υ εα ευ O z z z z z

( ) ( )

2 2

= + + ω υ β υ α z z

( ) ( ) ( )

2

1 1 4

= + + ′ + − υ β υ υ α υ α υ z z z ( ) ( ) ( ) ( )

z z z z α α ω β β υ 2 4

2 2

− ± − =

( ) ( ) ( )

z z z β υ α υ α υ υ + ′ − =

4 1

2

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9 9

Final solution for the mode shapes Final solution for the mode shapes

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − + − + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − + − =

∫ ∫ ∫ ∫

x x x x

C C C C x y

2 2 4 2 2 3 2 2 2 2 2 1

2 4 exp 2 4 exp 2 4 exp 2 4 exp ) ( ξ ξ α ξ α ω ξ β ξ β ξ ξ α ξ α ω ξ β ξ β ξ ξ α ξ α ω ξ β ξ β ξ ξ α ξ α ω ξ β ξ β d i d i d d

Natural frequencies and unknown constants are Natural frequencies and unknown constants are determined by the specified boundary conditions determined by the specified boundary conditions

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Mode shapes for normal motions and curvature Mode shapes for normal motions and curvature

E. Passano and C.M. Larsen,

Efficient analysis of a catenary riser, Proc OMAE 2006 Conf, OMAE paper 92308, (2006), Hamburg, Germany.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x (s/L) Static term

vertical coordinate tension angle curvature/10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.5 −1 −0.5 0.5 1 1.5 2

x yo(x)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1500 −1000 −500 500 1000

x d2yo(x)/dx2

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Solution in the time domain Solution in the time domain – – Fundamental assumption: Fundamental assumption: linear variation of tangential motions* linear variation of tangential motions*

*V.J. Papazoglou, S.A. Mavrakos and M.S. Triantafyllou, Nonlinear cable response and model testing in water, Journal of Sound and Vibration, 140(1990), 103-115.

( ) ( ) ( ) ( )

τ τ τ ν φ τ ν β α τ ∂ ∂ ∂ ∂ − ∂ ∂ ⋅ + ⋅ + ∂ ∂ + ∂ ∂ + ∂ ∂ − = ∂ ∂ y y b x y r x r x y x x y x x y K y

a a 2 2 2 2 4 4 2 2

d d

Using Using orthogonality

rthogonality of mode shapes
f mode shapes

( ) ( )

( )

∫ ∑ ∑ ∑

= = =

− Μ Ω + Ν + Λ Ω =

1 1 1 ) ( 1

d cos cos x C b r r

k N j j j N j j j k N j j kj a kj k a k

ϕ ϕ η ϕ η η τ τ η & & & &

Solution in the time domain suing Solution in the time domain suing Runge Runge-

Kutta

Kutta

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Steady state response for bending vibrations for constant Steady state response for bending vibrations for constant amplitude 20cm and variable frequency 1.0, 0.5 amplitude 20cm and variable frequency 1.0, 0.5 rad/s rad/s

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 −1.5 −1 −0.5 0.5 1 1.5 2 x 10

−3

x t y(x,t)

0.2 0.4 0.6 0.8 1 0.5 1 1.5 −4 −3 −2 −1 1 2 3 4 x 10

−3

x t y(x,t)

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13 13

Steady state response for bending moments for constant Steady state response for bending moments for constant amplitude 20cm and variable frequency 1.0, 0.5 amplitude 20cm and variable frequency 1.0, 0.5 rad/s rad/s

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6

x t y’’(x,t)

0.2 0.4 0.6 0.8 1 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

x t y’’(x,t)

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14 14

Parametric excitation of the structure Parametric excitation of the structure r ra0

a0=20cm,

=20cm, ω ω=0.5rad/s =0.5rad/s

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Parametric excitation of the equivalent Parametric excitation of the equivalent undamped undamped system system – – Stable (left) and unstable response (right) Stable (left) and unstable response (right)

5 10 15 20 25 30 35 −4 −3 −2 −1 1 2 3 4 5 x 10

−5

time first generalized variable

0.5 1 1.5 2 2.5 3 3.5 4 −50 −40 −30 −20 −10 10 20 30 40 50

time generalized variables

p pa0

a0=2mm,

=2mm, ω ω=0.5rad/s =0.5rad/s p pa0

a0=20cm,

=20cm, ω ω=0.5rad/s =0.5rad/s

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16 16

Parametric excitation of the structure, Parametric excitation of the structure, undamped undamped system system-

r

ra0

a0=2mm,

=2mm, ω ω=0.5rad/s =0.5rad/s

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Properties of instabilities Properties of instabilities – – Application of Application of Floquet Floquet’ ’s s theory theory Undamped Undamped equivalent of governing system equivalent of governing system

( ) ( )

( )

∫ ∑ ∑ ∑

= = =

− Μ Ω + Ν + Λ Ω =

1 1 1 ) ( 1

cos cos x C b r r

k N j j j N j j j k N j j kj a kj k a k

d ϕ ϕ η ϕ η η τ τ η & & & &

( ) ( ) ( ) ( )

∑

=

+ =

N j j kj k k

f g

1

τ η τ τ τ η & &

Periodic variation of parametric and forcing coefficients Periodic variation of parametric and forcing coefficients

k N k

u u

+

= &

( ) ( )

∑

= +

+ =

N j j kj k N k

u f g u

1

τ τ &

N k K , 2 , 1 =

( )

[ ]

( )

[ ]

τ τ G F + = u u &

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18 18

Matrix form of periodic coefficients Matrix form of periodic coefficients

Fundamental matrix solution Fundamental matrix solution each of the columns is a solution each of the columns is a solution

( )

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = 1 1 1

2 1 2 22 21 1 12 11 NN N N N N

f f f f f f f f f F τ

( )

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =

2 2 2 1 1 1 N N N

g g g g g g g g g G τ

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =

MM M M M M

u u u u u u u u u U

2 1 2 22 12 1 12 11

] [

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19 19

Differential equation in Matrix form Differential equation in Matrix form

Unknown is the matrix [U]. Unknown is the matrix [U]. [U( [U(τ τ+T)] is also a fundamental matrix +T)] is also a fundamental matrix solution solution

( ) [ ][ ] ( ) [ ]

τ τ G U F U + = ] [ &

[U( [U(τ τ+T)]=[A][U( +T)]=[A][U(τ τ)] )] Assumption [U( Assumption [U(τ τ)]=[P][V( )]=[P][V(τ τ)] )]

( )

[ ] [ ] [ ][ ]

( )

[ ] [ ]

( )

[ ]

τ τ τ V B V P A P T V = = +

−1

[P] is chosen so [B] is a Jordan canonical form [P] is chosen so [B] is a Jordan canonical form

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =

M

B λ λ λ ] [

2 1

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20 20

Stability or instability depends on the value of Stability or instability depends on the value of λ λ

Why? Why?

( ) ( )

τ λ τ

j j j

T v v = + ( ) ( )

τ λ τ

j n j j

nT v v = +

1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 3.5 4

rder of eigenvalue

abs of eigenvalue

6.22 15.56 31.12 43.57

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21 21

Advantages and disadvantages of Advantages and disadvantages of Floquet Floquet’ ’s s theory theory

Does not require the calculation of the transformation

Does not require the calculation of the transformation matrix [P] matrix [P]

Requires the numerical calculation of matrix [A]

Requires the numerical calculation of matrix [A]

Requires al lot of effort for the derivation of the

Requires al lot of effort for the derivation of the transition conditions transition conditions

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22 22

Alternative method. Application of the method of multiple Alternative method. Application of the method of multiple scales scales The transition curves are obtained by* The transition curves are obtained by*

A.H. A.H. Nayfeh Nayfeh and D.T. and D.T. Mook Mook, , Nonlinear Oscillations Nonlinear Oscillations, Wiley , Wiley Interscience Interscience, , John Wiley and Sons, Inc (1979) John Wiley and Sons, Inc (1979)

⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ − Φ − − Φ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + Φ + − + Φ − ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + Φ − Φ ± + = Ω

∑ ∑ ∑

≠ = ≠ = = N m j j j m jn j N n j j j n jm j N j j m n jn j m n jm j m n nm nm m n 1 2 2 1 2 2 1 2 2 2 2 2 2 / 1

) 2 ( ) 2 ( 1 1 4 1 2 1 ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ε ε ω ω

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23 23

Transition curves between stability and instability for Transition curves between stability and instability for p pa0

a0/L=0.00005, 0.0001, 0.001.

/L=0.00005, 0.0001, 0.001.

2 4 6 8 10 12 14 16 18 0.05 0.1 0.15 2 4 6 8 10 12 14 16 18 0.05 0.1 0.15

2 4 6 8 10 12 14 16 18 20 0.05 0.1 0.15

A semi A semi-

analytical mathematical formulation as a tool for the dynamic analysis of tool for the dynamic analysis of catenary catenary shaped shaped slender structures slender structures

by by Ioannis Ioannis K.

Chatjigeorgiou chatzi@naval.ntua.gr chatzi@naval.ntua.gr

National Technical University of Athens School of Naval Architecture and Marine Engineering Division of Marine Structures

Dynamic equilibrium in normal direction Dynamic equilibrium in normal direction

F w s T s EI t v M + − ∂ ∂ + ∂ ∂ − = ∂ ∂ φ φ φ cos

Separation of variables Separation of variables

) , ( ) ( ) , (

t s T s T t s T + =

F w s T s EI t v M + − ∂ ∂ + ∂ ∂ − = ∂ ∂ φ φ φ cos

) , ( ) ( ) , (

t s s t s φ φ φ + =

Uncoupled Uncoupled-

undamped system system – – free vibrations free vibrations

Natural frequencies and mode shapes Natural frequencies and mode shapes -

Free vibrations vibrations Nondimensional Nondimensional form form

( ) ( ) x

y x x y x x y K y ∂ ∂ + ∂ ∂ + ∂ ∂ − = ∂ ∂ β α τ

Harmonic motions Harmonic motions

( ) ( )

) ( d ) ( d d ) ( d d ) ( d

= + + + − x y x x y x x x y x x x y K ω β α

Simplified model Simplified model – – untensioned untensioned vertical structure vertical structure

351

Simplified model Simplified model – – vertical structure under tension: vertical structure under tension: effective tension at the top and bending stiffness of the effective tension at the top and bending stiffness of the same order of magnitude same order of magnitude

92390

( ) ( ) {

} { }

( ) ( ) ( ) ( ) [ ]

{ }

( ) ( ) { ( ) ( ) [ ]

}

Semi Semi-

analytical formulation of the solution: Can be used in both time and frequency domain both time and frequency domain – – Requires the calculation Requires the calculation

Application of the WKB* method

( ) ( )

( ) ( )

) ( d ) ( d d ) ( d d ) ( d

= + + + − z y z z y z z z y z z z y ω εβ α ε ε

Two possible solutions Two possible solutions

z y e = ) (

( ) ε

z y

e ) ( = *Logan JD, *Logan JD, Applied Mathematics Applied Mathematics, Wiley , Wiley Interscience Interscience, second edition , second edition (1997) (1997)

Perturbation expansion and equation of the perturbation Perturbation expansion and equation of the perturbation terms terms

( )

( )(

)

( )

6 3 4

= + + + ′ + + ′ + ′ ′ + ′ ′ + ′ ′ ′ − ω υ β υ υ ε α ευ υ υ ε υ υ ε υ υ ε υ ε z z

u′ = υ

( ) ( ) ( )

( )

ε ευ υ υ O z z z + + =

( ) ( ) ( ) ( ) ( )

( )

2

= + + + + + + ′ + − ε ω υ εβ υ β υ υ εα υ α υ εα ευ O z z z z z

( ) ( )

= + + ω υ β υ α z z

( ) ( ) ( )

2

= + + ′ + − υ β υ υ α υ α υ z z z ( ) ( ) ( ) ( )

( ) ( ) ( )

z z z β υ α υ α υ υ + ′ − =

2

Final solution for the mode shapes Final solution for the mode shapes

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

∫ ∫ ∫ ∫

Natural frequencies and unknown constants are Natural frequencies and unknown constants are determined by the specified boundary conditions determined by the specified boundary conditions

Mode shapes for normal motions and curvature Mode shapes for normal motions and curvature

Efficient analysis of a catenary riser, Proc OMAE 2006 Conf, OMAE paper 92308, (2006), Hamburg, Germany.

Solution in the time domain Solution in the time domain – – Fundamental assumption: Fundamental assumption: linear variation of tangential motions* linear variation of tangential motions*

*V.J. Papazoglou, S.A. Mavrakos and M.S. Triantafyllou, Nonlinear cable response and model testing in water, Journal of Sound and Vibration, 140(1990), 103-115.

( ) ( ) ( ) ( )

Using Using orthogonality

( ) ( )

( )

e ) ( = Logan JD, Logan JD, Applied Mathematics Applied Mathematics, Wiley , Wiley Interscience Interscience, second edition , second edition (1997) (1997)

A.H. A.H. Nayfeh Nayfeh and D.T. and D.T. Mook Mook, , Nonlinear Oscillations Nonlinear Oscillations, Wiley , Wiley Interscience Interscience, , John Wiley and Sons, Inc (1979) John Wiley and Sons, Inc (1979)