transverse vibration of slender sandwich beams with viscoelastic inner layer via a galerkin-type...

September 10-14, 2012 | Vienna, Austria

Transverse vibration of slender sandwich beams with viscoelastic inner layer via a

Galerkin-type state-space approach

Evangelos Ntotsios

[email protected]

Alessandro Palmeri

[email protected]

School of Civil and Building Engineering

Loughborough University, U.K.


Motivations Composite beam systems, made of two parallel Bernulli-Euler beams

continuosly connected by a Winkler type viscoelastic layer, enable to:

Approximate the dynamic behaviour of structural sandwich panels

Hohe J. et al. 2004. Palmeri A., 2010. Arikoglu A. & Ozkol I. 2010.

Study the performance of Continuous Dynamic Vibration Absorber (CDVA)

Aida T. et al. 1992.

Capture the dynamics of floating-slab railways tracks

Hussein M.F.M. & Hunt H.E.M. 2006.


Outline

Statement of the problem Euler-Bernouli beam theory Rate-dependent rheological model

Galerkin-type approximation

Kinetic and potential energy

Lagrangian equations of motion Undamped vibrations Dumped vibrations

State-space formulation

Numerical examples


Model under investigation

( , )f z t

1( , )u z t

2 ( , )u z t

( , )v z t0d

z

y

1 1 1 1, , ,E A I

2 2 2 2, , ,E A I

inn inn, ( )m k

L

0K

1K 1 1 1C K

Standar linear solid model

2, , ,( ) ( ) 0 ( 1, 2; 1, , )r j r j r jz a z r j n

20, 0, 0,( ) ( ) 0 ( 1, , )j j jz a z j n

T1 1 1, 1,

1

( ) ( ) ( ) ( )n

j jj

z t z q t

q

T2 2 2, 2,

1

( ) ( ) ( ) ( )n

j jj

z t z q t

q

T0 0 0, 0,

1

( ) ( ) ( ) ( )n

j jj

z t z q t

q

Axial modes for the outer beams:

Tranverse modes for the whole composite beam:


Standard Linear Solid (SLS) model

0( ) ( ) ( )F k

0 0( ) ( ) ( ) ( ) ( )dt

F t t t t s s s

0K

1K 1 1 1C K

F F0

1/0 0 1( ) e ( )tt K K t U 1

0 0 11

j( )

1 j

k K K

Relaxationfunction

Storage modulus

Loss modulus


Standard Linear Solid (SLS) model

10 1 1 1

1

( )( ) ( ) ( ) ; ( ) ( )

tF t K t K t t t

By introducing an additional state variable, l1, taken as the ideal strain in the Maxwell’s element, the reaction force experienced by the SLS can be represented1 through a first-order linear differential equation:

0K

1K 1 1 1C K

F F0

1 Palmeri A. et al. 2003.


Total kinetic energy

0 1 2( ) ( ) ( ) ( )T t T t T t T t

2

0

1( ) ( , ) d ( 1,2)

2

L

r r rT t u z t z r

T ( , ) T (0,0)0 0 0

1 1( ) ( ) ( ) 1, 2 and ( ) ( ) ( )

2 2r r

r r rT t t t r T t t t q Μ q q Μ q

The approximate kinematics of the double-beam system enables to express the total kinetic energy T, in the form:

20 0 0

1( ) ( , ) d

2

LT t v z t z


Total potential energy

0 1 2( ) ( ) ( ) ( )V t V t V t V t

2

0

2 20 0 0 00 0

1( ) ( , ) d ( 1,2)

21 1

( ) ( , ) d " ( , ) d2 2

L

r r r r

L L

V t E A u z t z r

V t EI v z t z K u z t z

T ( , )1( ) ( ) ( ) 1, 2

2r r

r r rV t t t r q K q

Analogously, the total potential energy V, in the form:

0 1 2 0( , ) ( , ) ( , ) ( , )u z t u z t u z t d v z t

T (0,0) T (0,0) T (1,1) T (2,2)0 0 0 0 0 1 1 2 2

T (0,1) T (0,2) T (1,2)0 1 0 2 1 2

1( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2

2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ]

V t t t t t t t t t

t t t t t t

q K q q K q q K q q K q

q K q q K q q K q


Lagrange’s equations of motion

Once all the sources of kinetic and potential energies are expressed as functions of generalized displacements and velocities, the Lagrange’s equations ruling the undamped vibration can be written as:

where the array of the generalized forces , is associated with the array of the generalized displacements :

d( ) ( ) ( ) ( 0,1,2)

d ( ) ( ) rr r

t t t it t t

Qq q

L L

( ) ( ) ( )t T t V t L

0 0000

( ) ( , ) ( , )d ( , ) ( )d( )

LL

t f z t v z t z f z t z zt

Qq

0Q

0q


Mass and stiffness matrices

The Lagrange’s equations of motion can be reduced to a more compact matrix form:

in which:( ) ( ) ( )t t t Mu Ku F

(0,0)

(1,1)

(2,2)

;

M 0 0

M 0 M 0

0 0 M

(0,0) (0,0) (0,1) (0,2)

(0,1) (1,1) (1,1) (1,2)

(0,2) (1,2) (2,2) (2,2)

K K K K

K K K K K

K K K K

0 0

1 1

2 2

( ) ( )

( ) ( ) ; ( ) ( )

( ) ( )

t t

t t t t

t t

q Q

u q F Q

q Q

3 1n 3 1n


Modal analysis

Since M and K constitute a pair of real-valued symmetric matrices, they can be simultaneously diagonalized through the classical eigenproblem:

The approximate j-th modal shapes is given by:

G being the transformation matrix so defined:

2 T,;j j j j k j k Mx K x x Mx

0, 1, 2,( ) ( ) ( ) ( ) ( )j j j j jz v z v z v z z

v x

T0

T1

T2

( )

( ) ( )

( )

z

z z

z

0 0

0 0

0 0

3 3n


Undamped vibration

The undamped vibration of the double-beam system under consideration are ruled in the modal space by:

where:

2 T( ) ( ) ( )t t t θ Ω θ X F

T 11( ) ( ) ( ) ( )mt t t t X u

1diag , , m Ω

1 mX x x


Damped vibration

Including the energy dicipation, pure viscous damping in the outer beams and rate-dependent viscoelastic damping provided by the inner layer is initially introduced in a mixed time-frequency domain:

in which is the viscous damping matrix and is the influence matrix of the inner layer:

inn 0 inn( ) ( ) ( ) ( ) ( )t t k K t t Mu Cu K L u F

(0,0) (0,1) (0,2)

(0,1) (1,1) (1,2)inn

0(0,2) (1,2) (2,2)

1

K

K K K

L K K K

K K K

CinnL


Damped vibration

By using the modal transformation of variables :

in which is the modal influence matrix and

is the modal viscous damping.

Then, in analogy with the case of the one-dimensional SLS model, the effects of the complex-valued rate-dependent stiffness is accounted as:

2 Tinn inn 0( ) ( ) ( ) ( ) ( ) ( )t t t k K t t Ξ Ω B X F

Tinn innB X L X

inn 0 0 0 1 1

11

1

( ) ( ) ( ) ( ) ( )

( )( ) ( )

k K t t K t K t

tt t

( ) ( )t tu X

TΞ X C X


State-space equations of motion

15

Finally, the equations of motion are posed in a convenient state-space matrix form:

in which:

( ) ( ) ( )t t t y Dy GF

1

2 T1

11

( )

( ) ( ) ; ;

( )

m m m m m m n

inn

m m m m m n

t

t t K

t

O I O O

y D Ω Ξ B G X

O I I O


Numerical applications: Modal properties

FEM AnalyticalMode 1 3.15 3.14

Mode 2 12.57 12.57

Mode 3 28.28 28.27

Mode 4 50.26 50.27

Mode 5 78.52 78.54

Mode 6 111.69 111.80

Mode 7 112.14 111.80

Mode 8 113.03 113.10

0 5 10-0.05

0

0.05 3.14 Hz

0 5 10-0.05

0

0.05 12.57 Hz

0 5 10-0.05

0

0.05 28.27 Hz

0 5 10-0.05

0

0.05 50.27 Hz

0 5 10-0.05

0

0.05 78.54 Hz

0 5 10-0.05

0

0.05 111.80 Hz

0 5 10-0.05

0

0.05 111.80 Hz

0 5 10-0.05

0

0.05 113.10 Hz

1 2

3 4

5 6

7 8

Modal frequencies (Hz)

Modal shapes

[m]z [m]z


Numerical applications: Damped vibration

0 20 40 60 80 100

10-10

10-5

Frequency (Hz)|H

0()|

undamped system

damped systemAbsolute values for the Frequency Respose Functions at the L/6 position for the undamped and damped systems

0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Time (sec)

v(L/

6,t)

[m

]

0 5 10 15 20 25-2

0

2

Time (sec)

f 0(t)

[KN

]

Time-histories of damped transverse vibration at L/6 subjected to sweep function excitation f0(t):


Conclusions and future work

A general method for studying free and forced transverse vibrations of a double-beam system has been presented coping with the general case of arbitrary boundary conditions and the rate-dependent behaviour of the viscoelastic inner layer described through the SLS model.

Extension of the proposed method to more sophisticated linear viscoelastic models, e.g. the generalized Maxwell’s model and the Laguerre’s polynomial approximation can be straightforward addressed, while experimental data from dynamical tests on composite beams included in the project “TREViS: Tailoring Nano-Reinforced Elastomers to Vibrating Structures” supported by the EPSRC, will explore the accuracy and versatility of the proposed scheme.

Suitable set of shape functions and associated generalized coordinates have been defined and Lagrange’s equations of motion have been arranged in a compact state-space form where a set of additional internal variables, associated with rate-dependent rheology of the inner layer, has been appended to the classical state variables.

First Grant EP/I033924/1; TREViS: Tailoring Nano-Reinforced Elastomers to Vibrating Structures


Thank You!

transverse vibration of slender sandwich beams with viscoelastic inner layer via a galerkin-type...

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