Study of free vibrating structures coupled with fluid

 

S. Pathania and P.K. Sharma

Department of Mathematics, National Institute of Technology, Hamirpur (H.P.), 177005, (India)

*Corresponding Author: shwetanithmr@gmail.com, psharma@nitham.ac.in

 

ABSTRACT:

Vibrations of circular plates in contact with inviscid liquid are studied. The wet dynamic displacement of the plate is approximated by combining the orthogonal modal functions of a dry circular plate with a clamped boundary condition. Various boundary conditions for the plate have been observed and the wet dynamic modal functions of the plate are derived by using a compatibility requirement along the contacting surface between the plate and the liquid. The kinetic energy and potential energy of the system is analytically evaluated for plate and incompressible liquid. Rayleigh–Ritz method is used to find the natural frequencies and mode shapes for symmetric and asymmetric modes.

 

KEY WORDS: Free vibration, Circular plate, Velocity potential, Rayleigh-Ritz method, Eigen value

 

 

INTRODUCTION:

The study of vibration of plates are often encountered in engineering applications and their use in machine design, nuclear reactor technology, naval and aerospace structures are quite common.  Lamb (1920) calculated the change in natural frequencies of a thin circular plate clamped along its boundary and placed in the aperture of an infinitely rigid wall in contact with water. Amabili and Pasqualini (1995) studied the influence of Poisson’s ratio upon the free vibrations of free-edge circular plates in vacuum and in contact with liquid. Amabili and Kwack (1999) considered the effect of free-surface wave on the vibration of circular plates resting on free fluid surface. Jeong (2003) examines the hydroelastic vibration of two identical circular plates coupled with a bounded fluid. A pressure-based finite element technique has been developed to analyze the dynamics of a partially filled rigid container with bottom-mounted submerged components by Mitra and Sinhamahapatra (2007). Maleki and Ziyaeifar (2008) investigated the potential of baffles (horizontal ring and vertical blade baffles) in increasing the hydrodynamic damping of sloshing in circular-cylindrical storage tanks.

 

In the present study Kirchhoff theory of plates is used to model the circular thermoelastic  plate. The plate is clamped along the edge of the plate by a rigid cylindrical vessel and one side of the plate is partially in contact with liquid contained in cylinder of length . The fluid is assumed to be inviscid and incompressible. The natural frequencies of the liquid-coupled system are obtained by using the Rayleigh–Ritz method.

 

 

Numerical calculations

The frequency equation derived in the preceding sections involves an infinite and a finite series of algebraic terms. In the theoretical calculation, the expansion term is set at 50, the expanding term for the admissible functions is set at 5 and expanding term for the liquid motion is set at 5 to obtain the converged solution. The plate is assumed to be made of aluminium having radius of 240mm and a thickness of 3mm. The liquid chosen for the purpose of numerical calculations is water. The physical properties of the plate and liquid are as follows: Young’s modulus=69.0GPa, Poisson’s ratio=0.334, density=2.713 ×103 Kg/m3, Thermal conductivity=224 W/m0C, Thermal expansion coefficient=26.28×10-6in/in0C, Specific heat=0.8953 KJ/Kg0C, Velocity of sound=1500m/sec, density of liquid =1000 Kg/m3,Thermal expansion coefficient of liquid =2.54645×10-4in/in0C, specific heat at constant volume=0.99907 J/Kg0C. In Figure 1, the natural frequencies of the wet plate are converged to the infinite liquid case when the liquid thickness ratio is greater than approximately 0.5. It seems that the wet natural frequencies of a structure in contact with water are sensitive to the liquid thickness when the liquid thickness is particularly small. The liquid thickness does not significantly affect the mode shapes even though the natural frequencies are increased with the liquid thickness. The natural frequency of symmetric mode become dispersion less i.e. remains constant with variation in liquid thickness ratio greater than 0.1.  


 

 Figure1: Variation of natural frequency with liquid  thickness ratio for symmetric mode (50% water level).                    

 

 

CONCLUSION:

The liquid thickness effect on the natural frequencies is very pronounced for a thin liquid thickness region. As the thickness of the liquid increases, the normalized natural frequencies increase drastically and converge to saturated values.

 

REFERENCE:

1.        A. Maleki, M. Ziyaeifar, Sloshing damping in cylindrical liquid storage tanks with baffles. Journal of Sound and

2.        Vibration 311(2008) 372–385.

3.        H. Lamb, On the vibrations of an elastic plate in contact with water.  Proceedings of the Royal Society of London A98 (1920) 205 – 216.

4.        K. H. Jeong, Free vibration of two identical circular plates coupled with bounded fluid. Journal of Sound and Vibration 260 (2003) 653–670.

5.        M. Amabili, A. Pasqualini, Natural frequencies and modes of free edge circular plates vibrating in vacuum or in contact with liquid. Journal of Sound and Vibration 188 (1995)685–699.

6.        M. Amabili, M.K. Kwak, Vibration of circular plates on a free fluid surface: effect of surface waves. Journal of Sound and Vibration 226 (1999)407–424.

7.        S .Chandrasekhar, Hydrodynamic and Hydromagnetic Stability. Dover, NewYork  (1961).

8.        S. Mitra, K. P. Sinhamahapatra, Slosh dynamics of liquid-filled containers with submerged components using pressure-based finite element method. Journal of Sound and Vibration 304(2007)361–381.

9.        Yuxin Sun and MasumiSaka, Thermoelastic damping in micro-scale circular plate resonators. Journal of Sound and Vibration 329 (2010) 328–337.

 

 

Received on 05.01.2013                                    Accepted on 14.02.2013        

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Research J. Science and Tech 5(1): Jan.-Mar.2013 page 189-193