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49
Free Vibration Analysis of a Rectangular Plate with
Multiple Holes Using Independent Coordinate Coupling
Method
Rajesh Chinthanippula
1, Anusha Battiprolu
2, Vamsi Duvuri
3 Mechanical Engineering1,2,3, WISTM (of A.U, Visakhapatnam, India,12,3Email: [email protected], [email protected], [email protected]
Abstract- Rectangular plates with different types of holes like rectangular hole, circular hole or elliptical hole are regularly used in both modern and classical mechanical, civil, and aerospace engineering. A rectangular plate with circular hole has particular purpose of uses. Presence of hole results in change of the geometrical shape of the plate which results in the change in vibration characteristics of the plate. The existence of a hole in a plate results in a significant change in mode shapes and the natural frequencies of the structure. Therefore, in this study, free vibration analysis is carried for a rectangular plate with circular and square holes. Ritz method was used widely in literature. For rectangular plates with circular holes, use of Ritz method poses evaluation of complex integrals because of variation in coordinate systems of circular and rectangular domains. To overcome this problem, a method called ICCM can be used. This method facilitates the use independent coordinates for both the circular and rectangular domains with a transformation matrix one domain to the other.
Index Terms-Independent coordinates coupling method (ICCM), Ritz method, vibration characteristics, Cartesian coordinate system, ANSYS.
1. INTRODUCTION
Plates with holes are commonly encountered in engineering applications. Natural vibration analysis of plates with holes plays a key role in mechanical, civil as well as ocean engineering and naval architecture. Rectangular plates with point supports can model several structures of realistic interest, such as slabs supported on columns, solar panels or printed circuit boards supported at a few points. Holes are introduced to provide access, reduce weight, and alter the dynamic response of structures. Furthermore, structure borne noise generated by machinery such as the diesel engines, generators, gearboxes and auxiliary machinery are also radiated by these plate structures and should be concealed in the various operating conditions. Holes of various shapes are often created in structural components due to many practical requirements such as venting, lightening the structure, changing the structural resonant frequency, and inspection/access ports. Therefore, in this study, free vibration analysis is carried for a rectangular plate with circular and square holes. There are various methods to carry out free vibration analysis of such structures either by the Rayleigh-Ritz method or the finite element method. Ritz method was used widely in literature. The Rayleigh-Ritz method is a successful method when the rectangular plate has a rectangular hole. For rectangular plates with circular holes, use of Ritz method poses evaluation of complex integrals because of variation in coordinate systems of circular and rectangular domains and the allowable functions for the rectangular hole domain do not permit closed-form integrals. The finite element method is a resourceful tool for structural vibration analysis and therefore, can be applied to any of the cases mentioned
above. But it does not permit qualitative analysis and requires massive computational time.
To overcome this problem, a method called Independent Coordinate Coupling Method (ICCM) can be used. This method facilitates the use independent coordinates for both the circular and rectangular domains with a transformation matrix one domain to the other. Therefore, in this work ICCM is used for the analysis of a rectangular plate with square and circular holes. The ICCM is advantageous because it does not need to use a complex integration process to determine the total energy of the plate with a hole. The procedure for the free vibration analysis is demonstrated through a numerical example and the numerical results are validation with the results obtained using a finite element package ANSYS.
2. RAYLEIGH-RITZ METHOD
2.1 Applying the Rayleigh-Ritz approach to rectangular plate
Let us consider a rectangular plate with side lengths a in the X-direction and b in the Y direction. From the vibration theory of thick plates, the strain energy VR and kinetic energy TR of an elastic isotropic
rectangular plate in the Cartesian coordinate can be written as follows:
∫ ∫ Eq (1)
∫ ∫ *( ) (
)
((
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50 Fig.1 Rectangle plate with local axis
Where ωr= ωr(x,y,t) represents the deflection of the
plate h is the thickness, where
( )is the plate flexural rigidity, ν is the Poisson’s ratio, E be the Young’s modulus, and ρ be the mass density of material. By taking the following non-dimensional variables and the assumed mode method, the deflection of the plate can be expressed as
( ) ( ) ( ) Eq (3) Where ( ) [ ]is a 1×m matrix consisting of the allowable functions and ( ) [ ] is am×1 vector representing generalized coordinates, in which m is the number of allowable functions used for the approximation of the deflection. Inserting Eq.(3) into Eqs. (1) and (2) results in Eq. (4).
̇ , ̇ Eq(4a,b) where
, Eq (5a,b) in which
∫ ∫ Eq (6a)
∫ ∫ *(
)
(
)
( )
+ “Eq (6b)”
α = a /b represents the aspect ratio of the plate and Represent the non-dimensionalized mass and stiffness matrices respectively. The equation of motion can be derived by replacing Eq. (4) into the Lagrange’s equation and the eigenvalue problem can be expressed as
[ ] Eq (7)
If we use the non-dimensionalized mass and stiffness matrices introduced in Eq. (5), the eigen value problem given by Eq. (7) can be also non-dimensionalized.
[ ] Eq (8)
Where ω is the non-dimensionalized natural frequency, which has the relationship with the natural frequency as follows
√ Eq (9)
In this section, Let us consider the simply-supported case in the X direction. In this case, the eigen function of the uniform beam can be used as an allowable function
√ Eq (10)
In the case of the clamped condition in the X direction, the eigen function of a clamped-clamped uniform beam can be used as
( ) Eq (11)
where =4.730, 7.853, 10.996, 14.137,… and =
In the case of a free-edge condition in the X direction, we can use the eigen function of a free-free uniform beam
X 1 =1 , √ ( )
( ) Eq (12)
For the allowable functions in the y direction, the same method can be applied. The
Combination of different allowable functions results in various boundary conditions. The frequency parameter is obtained by solving the generalized eigen value problem defined by Eq (9).
2.2 Applying the Rayleigh-Ritz approach to circular plate
To obtain the natural frequencies of rectangular plate with circular central hole, It is similar to the rectangular plate, the mass and stiffness matrices are determined. From the vibration theory of circular plates, the strain energy and kinetic energy of an isotropic uniform circular plate withthickness h and radius R can be expressed as follows:
∫ ∫ Eq (13)
∫ ∫ [(
)
( )
( ) ((
) (
( ))) (
(
))] Eq (14)
Fig.2 Circular plate with local axis
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respectively The first three modes represent the rigid-body modes and other modes represent the elastic vibration modes. The characteristic values obtained from Eq. (19b) are tabulated in the work of Kwak and Han(2007). In this case,Λchas the following form
[ )] (20).
3. INDEPENDENT COORDINATE COUPLING METHOD FOR A RECTANGULAR PLATE WITH A RECTANGULAR HOLE
Let us consider again the rectangular plate with a rectangular hole, as shown in Fig. 3. As can be seen from Fig. 3, the local coordinates fixed to the hole domain is introduced. Considering the non-dimensionalized coordinates, , we and potential energies in the hole domain as
̇ ,
We can use the same expression used for the free-edge rectangular plate. Since the local coordinate system is used for the hole domain, we do not have to carry out integration for the hole domain. However, the displacement matching condition between the global and local coordinates should be satisfied inside the hole domain. The displacement matching condition inside the hole domain can be written as
( ) ( ) Eq (27) The relationship between the non-dimensionalized
global and local coordinates can be written as
ξ= , = Eq (28) Taking into account Eqs. (3), (10), (21) and (22), and inserting them into Eq. (27), we can derive
∑ ( ) ( )
∑ ( ) ( ) ( )
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If we express Eq. (31) in the matrix form, we can have Eq (32) Eq. (36), for the eigen value problem, we only needed the transformation matrix . MrKr can be easily
computed according to the edge boundary conditions and Mrh, Krh can be computed from the results of the
free-edge condition of rectangular plate. On the other hand, the calculation of Mrh,Krh based on the global
coordinates is not easy because of integral limits. Compared to the approach based on the global coordinates, the numerical integration for the transformation matrix, Trrh, is easy because the integral
limits are 0 and 1. The process represented by Eqs. (32) and (36) is referred to as the ICCM in the study by Kwak and Han(2007). The ICCM enables us to solve the free vibration problem of the rectangular plate with a rectangular hole more easily than the previous approaches based on the global coordinates do. The benefit of the ICCM becomes clearer when we deal with a circular hole, as will be demonstrated in the next section.
4. INDEPENDENT COORDINATE COUPLING METHOD FOR A RECTANGULAR PLATE WITH MULTIPLE HOLES:
Let us consider a rectangular plate with multiple rectangular holes and a circular hole as shown in Fig. 5. We can easily extend the formulation developed in the previous section to the case of a rectangular plate with multiple rectangular holes and a circular hole. By using
Fig.5 Rectangular plate with multi rectangular holes and a circular hole.
By substituting the result of Eq. in the Eq.7 the non-dimensionalized eigen value problem for the rectangular plate with multiple rectangular holes and a circular hole can be expressed as
[ ]
Where √ is the non-dimensionalized natural frequency
5. CONCLUSIONS
The numerical simulation of plates with holes for free vibration analysis is highly complicated and is found analysis of uniform rectangular and circular plates. In the classical Rayleigh-Ritz method, the hole domains are subtracted in the integrals to calculate potential and kinetic energies associated with rectangular plates with holes. However, the method became much more complicated due to the problems that arise during the numerical computation of integrals. This led to opt for an alternative method for simulation called ICCM method.
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53 of rectangular plates with rectangular and circular
holes.
The results so obtained by using ICCM method are compared with the results obtained by using commercial finite element software ANSYS. It is observed that the results obtained are found to be in good agreement with those obtained using ICCM. Hence, it is concluded that ICCM method can be used effectively for the study and analysis of free vibrations associated with rectangular plates having multiple holes having different shapes.
(A.1) REFERENCES
[1] Aksu, G. & Ali, R. (1976). Determination of Dynamic Characteristics of Rectangular Plates with Holes Using a Finite Difference Formulation. Journal of Sound and Vibration, Vol. 44, (147-158), ISSN0022 460X
[2] Ali, R. &Atwal, S. J. (1980).Prediction of Natural Frequencies of Vibration of Rectangular Plates with Rectangular Holes. Computers and Structures, Vol. 12, No. 9, (819-823), ISSN0045-7949
[3] Avalos, D. R. & Laura, P. A. A. (2003).Transverse Vibrations of Simply Supported Rectangular Plates with Two Rectangular Holes. Journal of Sound and Vibration, Vol.267, (967-977), ISSN0022-460X [4]Cheng, L.; Li, Y. Y. & Yam, L. H. (2003).Vibration Analysis of Annular-Like Plates .Journal of Sound and Vibration, Vol. 262, (1153-1170), ISSN0022-460XEastep, F.E. &Hemmig, F.G. (1978).Estimation of Fundamental Frequency of Non-Circular Plates with Free, Non-Circular Holes. Journal of Sound and Vibration, Vol. 56, No. 2,(155-165), ISSN0022-460X
[5] Hegarty, R.F. &Ariman, T. (1975).Elasto-Dynamic Analysis of Rectangular Plates with Circular Holes.Int. J. Solids Structures, Vol. 11, (895-906), ISSN0020-7683
