8/3/2019 Investigasi Tentang Perilaku Dinamik Alami Dari Struktur Kurva Akibat Perubahan Bentuk Kurva_Transportasi Mariti

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The governing equations of dynamic behaviour of a curved beam is derived as follows:

The solutions of eigenvalue are formulated as follows:

The simplified perturbation equation is finally derived as follows:

Considering the boundary conditions, the eigenvalue problem can be solved based on alat beam solution, where the transverse displacement is expressed as follows:

where w h is the homogeneous solution of a flat beam and w p is the particular solutionontaining the curvature term

Numerical Models

B. HU

School of Engineering Sciences, University of Southampton, UK

An Investigation on the natural dynamic behaviours of curved structures affected bychanges in curvatures

Fig, 1: FEM model of circular beam and s-shaped strip

Various curved strips can be modelled based on a flat one if the curvature is a linear function of k = f ( s )

Curved StructuresBackground

Proposition

Perturbation Approach

SupervisorProf. J T Xing

Prof. R A Shenoi

Natural Vibration Numerical results agree with theoretical

ones The curvature affects the natural

frequency significantly The mode shape transition occurs when

the curvature increases to a certainlarge degree

The lower natural frequencies areaffected by the curvature much morethan the higher ones

Fig. 2: Effect of the curvature on the natural frequency of a circular curved beam

Fig. 3: Mode shape transi tions for the circular curved beam

Applications

Fig. 4: Effects of the curvature on the natural frequency of the composite laminat ed curved beams (a) circular beam (b) s-shaped strip

Fig. 5: Lateral vibra tion mode shapes of (a) curved circular beam, (b) s-shaped strip

Composite laminated curved beams S-glass/proxy [0/90]s, [0/90/90/0], [45/-45/-45/45]

Mode shapes performing transition characteristics with the curvature increasing Curvature produces different effects on symmetric and anti-symmetric modes The ratio between the axial and bending stiffness decides the dynamic behaviour of

the curved structure

The curvature adjusts the spatial status, stiffness, stability and capacity of loading of curved structuresComplex curved structures are analyzed only by numerical approachesMore characteristics of dynamic behaviours need to be revealed and explained

The perturbation approach is further developed to analyze curved beamsNovel applications for various curved beams and shells are suggestedCurved beams with different curvatures are modelled by the numerical method

Conclusions

Fluid Structure InteractionsResearch Group,

School of EngineeringSciences

( ) ( )susun

nn

==

0

,

( ) ( )swswn

nn

==

0

,

( ) ( )ssn

nn

==

0

,

= 1

02

4 sd wk k w

sw

ph www +=

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.005

0.01

0.015

0.02

0.025

Curvature parameter b( 1 /2)

N o n - d

i m e n s i o n a l f r e q u

e n c y

FEM symmetric modesAsymptotic curvesAnalytical result

3rd symmetric mode

2nd symmetric mode

1st symmetric mode

anti-symmetric modes

Asymptoticcurves

Subtended angle (degree)

1.71

7.64

29.1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

2

4

6x 10

-4 First symmetric mode

Curvatureparameterb( 1 /2)

N o n - d

i m e n s

i o n a

l f r e q u e n c y

[0/90/90/0][0/90/0/90][45/-45/-45/45]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5x 10

-3 First anti-symmetric mode

Curvatureparameterb( 1 /2)

N o n - d

i m e n s

i o n a

l f r e q u e n c y

[0/90/90/0][0/90][45/-45/-45/45]

k =small parameter

usw

k wsk

us

k k wsu

s

2

22 =

++

+

wsw

k wsk

us

k wsu

k =

++

2

22

2

2

Curved beams with discontinued varying curvatures Based on the perturbation solution, a curved beam with various curvatures can be solved

analytically

Dynamic behaviour of two dimensional thin shell structures The energy method is adopted to obtain the analytical solution of the lateral vibration

Fig. 6: Effects of the curvature on the natural frequency of a circular curved

beam

The lateral vibration The numerical study shows the natural

frequencies of the lateral vibration changesfollowing the different curvatures

It also shows the difference in mode shapesbetween the circular curved strip and the s-shaped strip