Modal analysis of a multi-blade system undergoing rotational motion

A modeling method for the modal analysis of a multi-blade system undergoing rotational motion is presented in this paper. Blades are assumed as cantilever beams and the coupling stiffness which originates from the shroud flexibility is considered for the modeling. To obtain general conclusions from the numerical results, the equations of motion are transformed into a dimensionless form. Dimensionless parameters related to the angular speed, the hub radius, and the coupling stiffness are identified and the effects of the parameters on the modal characteristics of the system are investigated. It is shown that the coupling stiffness especially plays an important role to change the modal characteristics of the system. The range of critical angular speed is also obtained through the numerical analysis. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Ha Seong Lim graduated from Department of Mechanical Engineering at Hanyang University in 2006 and received his Master’s degree in 2008. He is currently a technical engineer in STX Offshore & Shipbuilding Company, Seoul, Korea. Hong Hee Yoo graduated from the Department of Mechanical Design and Production Engineering at Seoul National University in 1980 and received his Master’s degree from the same department in 1982. He received his Ph.D. degree in 1989 from the Department of Mechanical Engineering and Applied Mechanics at the University of Michigan at Ann Arbor, U.S.A. He is currently a professor in the School of Mechanical Engineering in Hanyang University, Seoul, Korea.     

A finite thin circular beam element for out-of-plane vibration analysis of curved beams

A finite thin circular beam element for the out-of-plane vibration analysis of curved beams is presented in this paper. Its stiffness matrix and mass matrix are derived, respectively, from the strain energy and the kinetic energy by using the natural shape functions derived from an integration of the differential equations in static equilibrium. The matrices are formulated with respect to the local polar coordinate system or to the global Cartesian coordinate system in consideration of the effects of shear deformation and rotary inertias. Some numerical examples are analyzed to confirm the validity of the element. It is shown that this kind of finite element can describe quite efficiently and accurately the out-of-plane motion of thin curved beams. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Chang-Boo Kim received his B.S. degree in Mechanical Engineering from Seoul University, Korea in 1973. He then received his D.E.A., Dr.-Ing. and Dr.-es-Science degrees from Nantes University, France in 1979, 1981 and 1984, respectively. Dr. Kim is currently a Professor at the School of Mechanical Engineering at Inha University in Incheon, Korea. His research interests are in the area of vibrations, structural dynamics, and MEMS.     

Bending and vibration of circular cylindrical beams with arbitrary radial nonhomogeneity

This paper presents a new approach for analyzing transverse bending and vibration of circular cylindrical beams with radial nonhomogeneity. The radial nonhomogeneity may be continuous or piecewise-constant, corresponding a functionally graded circular cylinder or a multi-layered circular cylinder, respectively. Different from the Euler-Bernoulli and Timoshenko theories of beams, our analysis considers shear deformation, but does not need to introduce a shear correction factor. Using the shear-stress-free condition at the surface of the cylinder, coupled governing equations for deflection and rotation angle are derived, and then converted to a single governing equation. The influences of gradient index on the deflection and stress distribution for cantilever and simply-supported beams are studied. Natural frequencies of free vibration of a cylindrical beam with circular cross-section are calculated for different power-law gradients. In particular, a circular cylindrical shell may be taken as a special case of a bi-layered cylinder where the material properties of the inmost cylinder vanish. For this case, the natural frequencies for simply-supported and clamped-clamped cylindrical shells are evaluated and compared with those using three-dimensional theory. Our results coincide well with the previous.     

Three-dimensional vibration analysis of thick, tapered rods and beams with circular cross-section

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, tapered rods and beams with circular cross-section. Unlike conventional rod and beam theories, which are mathematically one-dimensional (1-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the rods and beams are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four- digit exactitude is demonstrated for the first five frequencies of the rods and beams. Novel numerical results are tabulated for nine different tapered rods and beams with linear, quadratic, and cubic variations of radial thickness in the axial direction using the 3-D theory. Comparisons are also made with results for linearly tapered beams from 1-D classical Euler–Bernoulli beam theory.     

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.