Transverse Vibration of double cracked beam using Assumed mode method

Seyyed Jamal Addin Mostafavi yazdi M.S Student: Faculty of Aerospace Engineering

K.N.Toosi University Tehran, Iran

Mostafavi@sina.kntu.ac.ir

Saeid Irani Assistant Professor: Faculty of Aerospace Engineering

K.N.Toosi University Tehran, Iran

Irani@kntu.ac.ir

Abstract— this paper presents transverse vibration of double cracked beam using assumed mode method. First we described the assumed mode method for transverse vibration of continuous systems. Assumed mode method has been used for continuous models, Also Assumed mode method may be used to derive the equations of motion of other linearly elastic systems. For the Bernoulli-Euler beam the strain energy and kinetic energy have been derived. General coordinates of model are assumed as function of time and coordinate. The admissible function that considered must be satisfied the boundary conditions. Then the mass and stiffness matrix have been obtained from strain and kinetic energy. Only steps those are actually required in using the Assumed-Modes Method to arrive at the equations of motion of an N-DOF model of a continuous system.

Transverse vibration; Assumed mode;double crack;shape function

I. INTRODUCTION Fracture is a problem that society has faced for as long as there have been man-made structures. The problem may actually be worse today than in previous centuries, because more can go wrong in our complex technological society. Major airline crashes, for instance would not be possible without modern aerospace technology. Cracks found in structural elements have various causes. They may be fatigue cracks that Take place under service conditions as a result of the limited fatigue strength. They may also be due to mechanical defects, as in the case of turbine blades of jet turbine engines. In these engines the cracks are caused by sand and small stones sucked from the surface of the runway. The Influence on the Eigen frequencies and Eigen modes of changes in the geometry of a Structure are in many respects important to know. Evaluate the risk that a structure runs into resonance due to a developing crack or to compute the size of defects from known shifts in the resonance frequencies. The differential equation and associated boundary conditions for a nominally uniform Bernoulli-Euler beam containing one or more pairs of symmetric cracks are derived by S. CHRISTIDES and A. D. S.BARR [1]. An approximate Galerkin solution to the one-dimensional cracked beam theory

developed by Christides and Barr for the free bending motion of beams with pairs of symmetric open cracks is suggested. M. H. H. SHEN AND C. PIERRE [2] found that the Christides and Barr original solution was not fully converged and that cracks render the convergence of the Galerkin’s procedure very slow by affecting the continuity characteristics of the solution of the boundary value Problem. The forced vibrations of the beam and the effects of the crack locations and sizes on the vibration behavior of the structure are studied by W. M. Ostachowiczto and M. Krawczuk [3] and they analyzed Effects of crack on natural frequency of cracked cantilever beam [4]. M. Krawczuk [5] used finite element Method to model cracked rotating beam. A continuous cracked beam vibration theory is developed by T. G. CHONDROS& A. D. DIMAROGONAS [6] for the lateral vibration of cracked Euler Bernoulli beams with single-edge or double-edged open cracks. D.Y. Zheng, N.J. Kessissoglou [7] used 128-point (1D) Gauss quadrature to find overall flexibility matrix of cracked element and compute the natural frequencies and mode shapes of a cracked beam are obtained using the finite Element method.

II. ASSUMED MODE METHOD The Principle of Virtual Displacements can be extended to produce a generalized discrete-parameter model, or simply generalized-parameter model, of a continuous system in a manner that approximates the deformation of the system. The procedure is referred to as the Assumed-Modes Method. It is employed here to create an SDOF generalized-parameter model and is extended to MDOF systems. A few definitions are needed at the outset. • A continuous system is a system whose deformation is described by one or more functions of one, two, or three spatial variables and time. For example, the deformation of the cantilever beam in Fig. 1 is specified in terms of the deflection curve ( , )v x t of the neutral axis.

978-1-4244-3628-6/09/$25.00 ©2009 IEEE 156

Figure 1. Geometric boundary conditions and vinual displacements

• A geometric boundary condition is a specified kinematical constraint placed on displacement and/or slope on portions of the boundary of a body.

• A virtual displacement of a continuous system is an infinitesimal, imaginary change in the displacement function(s) consistent with all geometric boundary conditions.

Figure 1 illustrates the foregoing definitions. The deformation of the propped cantilever beam in this figure is given by the deflection curve ( , )v x t . The geometric boundary conditions are the cantilevered (i.e., fixed) end at 0x = and the prop that prevents vertical motion of the beam at x L= . These geometric boundary conditions are given by the following three equations:

(0, ) 0 (0, ) 0 ( , ) 0v t v t v L t′= = = (1) The dashed curve shows a possible virtual displacement,

( , )v x tδ of the beam. The only condition on ( , )v x tδ is that it satisfy the same geometric boundary conditions as ( , )v x t ; that is:

(0, ) (0, ) ( , ) 0v t v t v L tδ δ δ′= = = (2) Note that ( , ) [ ( , )]v x t v x tδ δ= is not a function of time, as is

( , )v x t . However, the notation ( , )v x tδ means an arbitrary small change of configuration relative to the configuration of the beam at time t. • An admissible function is a function that satisfies the geometric boundary conditions of the system under consideration and possesses derivatives of order at least equal to that appearing in the strain energy expression for the system. • An assumed mode (also called a shape function) ( )xψ is an admissible function that is selected by the user for the purpose of approximating the deformation of a continuous system. To create a generalized-parameter SDOF model of a continuous system, a single assumed mode is used.For example. The deflection curve of a beam may be approximated by

( , ) ( ) ( )vv x t x q tψ= (3)

Any admissible function may be employed as the shape function ( )xψ , but a shape that can be expected to be similar to the shape of the deforming structure should be chosen.

The time-dependent function ( )v

q t in Eq.3 is called the generalized displacement coordinate for this SDOF model; the subscript v identifies it as a generalized coordinate related to the physical displacement ( , )v x t . It will be determined as the solution to an ordinary differential equation, just as in the case of lumped-parameter models.

A. Selection of Shape Functions

It is by choosing the functions ( )i

xψ that the analyst defines

the N-DoF assumed-modes model. The shape functions ( )i

xψ : 1. Must form a linearly independent set. 2. Must possess derivatives up to the order appearing in the strain energy V. 3. Must satisfy all prescribed boundary conditions, that is, all displacement-type boundary conditions. Functions that satisfy these three conditions are called admissible functions. As an example, consider the propped cantilever beam in Fig. 2. The assumed-modes approximation of the transverse displacement of this beam would be written

1

( , ) ( ) ( )N

i ii

v x t x q tψ=

= ∑ (4)

Where each shape function ( )i

xψ must satisfy the boundary conditions

(0) (0) ( ) 0i i i Lψ ψ ψ′= = = (5)

Since

(0, ) (0, ) ( , ) 0t v t L tv v′= = = (6) For all t. Since the strain energy expression for a Bernoulli-Euler beam contains v"(x, t), the second derivative of the transverse displacement , each ( )

ixψ must be a continuous

function of x, and its first derivative with respect to x must be continuous; that is, the beam can have no abrupt changes of displacement or slope. It is not necessary that the iψ also satisfy the natural boundary conditions, that is, force-type boundary conditions, such as the vanishing of moment M (L, t) at the right end of the propped cantilever beam in Fig.3 However, in cases where it is possible to obtain functions ( )

ixψ that satisfy both prescribed

and natural boundary conditions, such functions may be used in Eq. 4.

Figure 2. Boundary condition example

157

B. Assumed-Modes Method: Bending of Bernoulli-Euler Beams

For the Bernoulli-Euler beam the strain energy is given by

2

0

1( )

2

L

v EI v dx′′= ∫ (7)

Where ( , )v x t refers to transverse displacement, as in Fig. 2, and where the flexural rigidity EI may be a function of x. The kinetic energy is given by

2

0

1( )

2

L

T A v dxρ= ∫ i (8)

Let the assumed-modes expression for ( , )v x t be

1

( , ) ( ) ( )N

i ii

v x t x v tψ=

= ∑ (9)

With the generalized coordinates labeled ( )iv t .When this expression is substituted into Eqs.7 and 8, the following expressions for stiffness and mass coefficients are obtained:

0

L

ij i jk EI dxψ ψ′′ ′′= ∫ (10)

0

L

ij i jm A dxρ ψ ψ= ∫ (11)

The generalized forces due to a distributed transverse force p(x, t), shown in Fig. 3, are given by

0

( ) ( , ) ( )L

i y ip t p x t x dxψ= ∫ (12)

C. Procedure for the Assumed-Modes Method Although we have gone through Lagrange's Equations to

Arrive at Eq 13;

( )Mu ku p t+ = (13)

Seen that the only steps that are actually required in using the Assumed-Modes Method to arrive at the equations of motion of an N-DOF model of a continuous system are:

1. Select a set of N admissible functions, ( )i

xψ .

2. Compute the coefficients ijk of the stiffness matrix by using Eq. 10.

3. Compute the coefficients ijm of the mass matrix by using Eq. 11.

4. Determine expressions for the generalized forces Pi (t) corresponding to the applied force p(x, t) by using Eq. 12.

5. Form the equations of motion by using Eq. 13.[8]

III. ASSUMED MODE METHOD FOR DOUBLE EDGE CRACKED BEAM

A. Modeling The beam modeled Euler Bernoulli and simply supported

with double edge cracks. Cracks are always open. Beam divided by cracks into sub beams, as in Fig 3, each beam has own length. The global coordinate considered for each sub beam.

B. Shape function According to procedure for assumed mode method, for

transverse deflection of beam, first, shape function must be selected. The shape function that is selected is based on global coordinates (Fig.4) and satisfies the condition of selection of shape function.

For each sub beam, the transverse deflection is,

11

( ) ( )N

i ii

v x v tψ=

= ∑ (14)

21

( ) ( )N

i ii

v x v tφ=

= ∑ (15)

Where ( )xψ and ( )xφ are approximated by series of normalized characteristic beam functions given by [9]

1 1 1 111 1

1 21 1 1

( ) 3 2(sin( ) sinh( ))N N

n nn n

n n

x xxx

L L L

λ λψ γ− −

−= =

= + +∑ ∑ (16)

1 2 1 222 1

1 22 2 2

( ) 3 2(sin( ) sinh( ))N N

n nn n

n n

x xxx

L L L

λ λφ γ− −

−= =

= + +∑ ∑ (17)

Figure 3. Double dge cracked beam modeling

158

nλ Calculated from this equation:

cos( ) sinh( ) sin( ) cosh( ) 0λ λ λ λ− = (18)

11

1

sin( )sinh( )

nn

n

λγλ

−−

−

= (19)

TABLE I. CALCULATED RESULTS OF EQ18,19

n nλ nγ

1 3.9266 -0.02787

2 7.06858 0.001204

3 10.2102 -0.000052

4 13.3518 0.2248617E-5

5 16.4995 -0.971679E-7

After determination of shape function, the Mass matrix and stiffness matrix can be calculated from Eq.10, 11.

C. Numerical example&results Consider a simply supported beam with length ( 0.575L = )

and divided by 2 sub beam with equal length by cracks. Cracks are open and located at the middle of the beam.

For each sub-beam 5(n=5) degree of freedom considered and the shape functions for left part of beam:

11 1

1

( ) 3 xxL

ψ =

1 1 1 1 12 1 1

1 1 1

( ) 3 2(sin( ) sinh( ))x x xxL L L

λ λψ γ= + +

1 1 1 1 1 2 1 2 13 1 1 2

1 1 1 1 1

( ) 3 2(sin( ) sinh( )) 2(sin( ) sinh( ))x x x x xxL L L L L

λ λ λ λψ γ γ= + + + +

1 1 1 1 1 2 1 2 14 1 1 2

1 1 1 1 1

3 1 3 13

1 1

( ) 3 2(sin( ) sinh( )) 2(sin( ) sinh( ))

2(sin( ) sinh( ))

x x x x xxL L L L L

x xL L

λ λ λ λψ γ γ

λ λγ

= + + + + +

+

1 1 1 1 1 2 1 2 15 1 1 2

1 1 1 1 1

3 1 3 1 4 1 4 13 4

1 1 1 1

( ) 3 2(sin( ) sinh( )) 2(sin( ) sinh( ))

2(sin( ) sinh( )) 2(sin( ) sinh( ))

x x x x xxL L L L L

x x x xL L L L

λ λ λ λψ γ γ

λ λ λ λγ γ

= + + + + +

+ + +

By substituting above shape functions into Eq.10 and Eq.11, stiffness and mass matrix calculated.

Similar way is used to compute mass and stiffness matrix of right sub-beam. Because the crack is located at the middle of cracked beam, right sub-beam is similar to left hand side beam and the mass and stiffness matrix are equal to left hand side of beam. One important point that must be considered, the shape function of right sub-beam is written for local coordinate.

The eigenvectors for left sub-beam are:

By multiplying eigenvectors and mode shapes, mode shapes of cracked beam calculated. Mode shapes of cracked beam are plotted:

Figure 4. First mode shape

Figure 5. second mode shape

159

Figure 6. Third mode shape

IV. CONCLUSION Approximation methods used to replace continuous

systems with discrete systems. Assumed mode method is One of this approximation methods. Formulation of these methods is similar to Rayleigh Ritz method. Benefit of this method, using shape functions that satisfied boundary conditions and these shape functions used for Galerkin and Rayleigh-Ritz method. By replacing these shape functions in assumed mode formulation, stiffness and mass matrix calculated. Due to cracks, stiffness of structure is decreased and vibration properties changed. By comparison between intact beam and cracked beam, vibration properties of cracked beam decreased.

REFERENCES [1] S. Christides and A. D. S. Barr., “One-dimensional theory of cracked

Bernoulli-Euler beams”. 1984 International Journal of the Mechanical Sciences 26,639-648, 1984

[2] M.H.H. Shen and C. Pierre , “Natural Modes of Bernoulli-Euler Beams with Symmetric Cracks”. Journal of Sound and vibration 138, 115-134, 1990

[3] W. M. Ostachowicz And M. Krawczuk, “Vibration Analysis of cracked beam”. Journal of Computers&Structure, Vol.36, No.2, 245-250, 1990

[4] W. M. Ostachowicz And M. Krawczuk, “Natural Analaysis Of The Effect Of Cracks On The Natural Frequences Of a Cantillever Beam”. Journal of Sound and vibration 151, 191-201, 1991

[5] M. Krawczuk, “Natural Vibration of cracked Rotating Beam”. ACTA MECHANICA 99, 35-48, 1993.

[6] T.G. CHONDROS., A.D.DIMAROGONAS., A Continuous cracked beam Vibration. International Journal of sound and Vibration .V.215,17-34

[7] D.Y. Zheng, N.J. Kessissoglou, “Free vibration analysis of a cracked beam by finite”. Journal of Sound and vibration 273, 457-475, 2004

[8] Roy R. Craig, Jr., and Andrew J. Kurdila, Fundamentals of Structural Dynamics. 2nd ed., John Wiley & Sons, Inc, 2006.

[9] .H.p.Lee and T.Y.Ng, Singapore, Dynamic response of a cracked beam subject to a moving load, Acta Mechanica, V.106,1994,pp.221-230.

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