Free Vibration

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Liaoliang Ke - One of the best experts on this subject based on the ideXlab platform.

  • axisymmetric nonlinear Free Vibration of size dependent functionally graded annular microplates
    2013
    Co-Authors: Liaoliang Ke, Sritawat Kitipornchai, Jie Yang, M A Bradford, Yuesheng Wang
    Abstract:

    In this paper, a non-classical microplate model is developed for the axisymmetric nonlinear Free Vibration analysis of annular microplates made of functionally graded materials (FGMs) based on the modified couple stress theory, Mindlin plate theory and von Karman geometric nonlinearity. This non-classical model is capable of incorporating the microplate model with the length scale parameter, geometric nonlinearity, transverse shear deformation and rotary inertia. By using Hamilton's principle, the higher-order governing equations and boundary conditions for the problem are derived. The differential quadrature (DQ) method is employed to discretise the governing equations, which are then solved by a modified iterative method to obtain the nonlinear frequencies of FGM microplates with different boundary conditions. Numerical results are then presented in both tabular and graphical form to investigate the effects of the length scale parameter, gradient index, inner-to-outer radius ratio and radius-to-thickness ratio on the nonlinear Free Vibration characteristics of FGM microplates. It is found that unlike homogeneous microplates, the FGM microplates display different Vibration behavior at positive and negative amplitudes due to the presence of bending-extension coupling.

  • nonlinear Free Vibration of size dependent functionally graded microbeams
    2012
    Co-Authors: Liaoliang Ke, Jie Yang, Yuesheng Wang, Sritawat Kitipornchai
    Abstract:

    Nonlinear Free Vibration of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and von Karman geometric nonlinearity. The non-classical beam model is developed within the framework of Timoshenko beam theory which contains a material length scale parameter related to the material microstructures. The material properties of FGMs are assumed to be graded in the thickness direction according to the power law function and are determined by Mori-Tanaka homogenization technique. The higher-order nonlinear governing equations and boundary conditions are derived by using the Hamilton principle. A numerical method that makes use of the differential quadrature method together with an iterative algorithm is employed to determine the nonlinear Vibration frequencies of the FGM microbeams with different boundary conditions. The influences of the length scale parameter, material property gradient index, slenderness ratio, and end supports on the nonlinear Free Vibration characteristics of the FGM microbeams are discussed in detail. It is found that both the linear and nonlinear frequencies increase significantly when the thickness of the FGM microbeam is comparable to the material length scale parameter.

  • nonlinear Free Vibration of single walled carbon nanotubes using nonlocal timoshenko beam theory
    2010
    Co-Authors: Jie Yang, Liaoliang Ke, Sritawat Kitipornchai
    Abstract:

    Nonlinear Free Vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based on von Karman geometric nonlinearity and Eringen's nonlocal elasticity theory. The SWCNTs are modeled as nanobeams where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear Vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the nonlinear Free Vibration characteristics of SWCNTs.

  • nonlinear Free Vibration of functionally graded carbon nanotube reinforced composite beams
    2010
    Co-Authors: Liaoliang Ke, Jie Yang, Sritawat Kitipornchai
    Abstract:

    This paper investigates the nonlinear Free Vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Karman geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear Vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, Vibration amplitude, slenderness ratio and end supports on the nonlinear Free Vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, Vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear Free Vibration characteristics of FG-CNTRC beams.

  • nonlinear Free Vibration of embedded double walled carbon nanotubes based on nonlocal timoshenko beam theory
    2009
    Co-Authors: Liaoliang Ke, Jie Yang, Yang Xiang, Sritawat Kitipornchai
    Abstract:

    Nonlinear Free Vibration of embedded double-walled carbon nanotubes (DWNTs) is studied in this paper based on Eringen's nonlocal elasticity theory and von Karman geometric nonlinearity. The effects of the transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The surrounding elastic medium is described as the Winkler model characterized by the spring. The governing equations and boundary conditions are derived by using the Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations, which are then solved by a direct iterative method to obtain the nonlinear Vibration frequencies of nonlocal DWNTs with different boundary conditions. A detailed parametric study is conducted to investigate the influences of nonlocal parameter, length of the tubes, spring constant and end supports on the nonlinear Free Vibration characteristics of DWNTs.

Sritawat Kitipornchai - One of the best experts on this subject based on the ideXlab platform.

  • axisymmetric nonlinear Free Vibration of size dependent functionally graded annular microplates
    2013
    Co-Authors: Liaoliang Ke, Sritawat Kitipornchai, Jie Yang, M A Bradford, Yuesheng Wang
    Abstract:

    In this paper, a non-classical microplate model is developed for the axisymmetric nonlinear Free Vibration analysis of annular microplates made of functionally graded materials (FGMs) based on the modified couple stress theory, Mindlin plate theory and von Karman geometric nonlinearity. This non-classical model is capable of incorporating the microplate model with the length scale parameter, geometric nonlinearity, transverse shear deformation and rotary inertia. By using Hamilton's principle, the higher-order governing equations and boundary conditions for the problem are derived. The differential quadrature (DQ) method is employed to discretise the governing equations, which are then solved by a modified iterative method to obtain the nonlinear frequencies of FGM microplates with different boundary conditions. Numerical results are then presented in both tabular and graphical form to investigate the effects of the length scale parameter, gradient index, inner-to-outer radius ratio and radius-to-thickness ratio on the nonlinear Free Vibration characteristics of FGM microplates. It is found that unlike homogeneous microplates, the FGM microplates display different Vibration behavior at positive and negative amplitudes due to the presence of bending-extension coupling.

  • Free Vibration of size dependent mindlin microplates based on the modified couple stress theory
    2012
    Co-Authors: Yuesheng Wang, Jie Yang, Sritawat Kitipornchai
    Abstract:

    This paper develops a Mindlin microplate model based on the modified couple stress theory for the Free Vibration analysis of microplates. This non-classical plate model contains an internal material length scale parameter related to the material microstructures and is capable of interpreting the size effect that the classical Mindlin plate model is unable to describe. The higher-order governing equations of motion and boundary conditions are derived using the Hamilton principle. The p-version Ritz method is employed to determine the natural frequencies of the microplate with different boundary conditions. A detailed parametric study is conducted to study the influences of the length scale parameter, side-to-thickness ratio and aspect ratio on the Free Vibration characteristics of the microplate. It is found that the size effect is significant when the thickness of microplate is close to the material length scale parameter.

  • nonlinear Free Vibration of size dependent functionally graded microbeams
    2012
    Co-Authors: Liaoliang Ke, Jie Yang, Yuesheng Wang, Sritawat Kitipornchai
    Abstract:

    Nonlinear Free Vibration of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and von Karman geometric nonlinearity. The non-classical beam model is developed within the framework of Timoshenko beam theory which contains a material length scale parameter related to the material microstructures. The material properties of FGMs are assumed to be graded in the thickness direction according to the power law function and are determined by Mori-Tanaka homogenization technique. The higher-order nonlinear governing equations and boundary conditions are derived by using the Hamilton principle. A numerical method that makes use of the differential quadrature method together with an iterative algorithm is employed to determine the nonlinear Vibration frequencies of the FGM microbeams with different boundary conditions. The influences of the length scale parameter, material property gradient index, slenderness ratio, and end supports on the nonlinear Free Vibration characteristics of the FGM microbeams are discussed in detail. It is found that both the linear and nonlinear frequencies increase significantly when the thickness of the FGM microbeam is comparable to the material length scale parameter.

  • nonlinear Free Vibration of single walled carbon nanotubes using nonlocal timoshenko beam theory
    2010
    Co-Authors: Jie Yang, Liaoliang Ke, Sritawat Kitipornchai
    Abstract:

    Nonlinear Free Vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based on von Karman geometric nonlinearity and Eringen's nonlocal elasticity theory. The SWCNTs are modeled as nanobeams where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear Vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the nonlinear Free Vibration characteristics of SWCNTs.

  • nonlinear Free Vibration of functionally graded carbon nanotube reinforced composite beams
    2010
    Co-Authors: Liaoliang Ke, Jie Yang, Sritawat Kitipornchai
    Abstract:

    This paper investigates the nonlinear Free Vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Karman geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear Vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, Vibration amplitude, slenderness ratio and end supports on the nonlinear Free Vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, Vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear Free Vibration characteristics of FG-CNTRC beams.

Jie Yang - One of the best experts on this subject based on the ideXlab platform.

  • axisymmetric nonlinear Free Vibration of size dependent functionally graded annular microplates
    2013
    Co-Authors: Liaoliang Ke, Sritawat Kitipornchai, Jie Yang, M A Bradford, Yuesheng Wang
    Abstract:

    In this paper, a non-classical microplate model is developed for the axisymmetric nonlinear Free Vibration analysis of annular microplates made of functionally graded materials (FGMs) based on the modified couple stress theory, Mindlin plate theory and von Karman geometric nonlinearity. This non-classical model is capable of incorporating the microplate model with the length scale parameter, geometric nonlinearity, transverse shear deformation and rotary inertia. By using Hamilton's principle, the higher-order governing equations and boundary conditions for the problem are derived. The differential quadrature (DQ) method is employed to discretise the governing equations, which are then solved by a modified iterative method to obtain the nonlinear frequencies of FGM microplates with different boundary conditions. Numerical results are then presented in both tabular and graphical form to investigate the effects of the length scale parameter, gradient index, inner-to-outer radius ratio and radius-to-thickness ratio on the nonlinear Free Vibration characteristics of FGM microplates. It is found that unlike homogeneous microplates, the FGM microplates display different Vibration behavior at positive and negative amplitudes due to the presence of bending-extension coupling.

  • Free Vibration of size dependent mindlin microplates based on the modified couple stress theory
    2012
    Co-Authors: Yuesheng Wang, Jie Yang, Sritawat Kitipornchai
    Abstract:

    This paper develops a Mindlin microplate model based on the modified couple stress theory for the Free Vibration analysis of microplates. This non-classical plate model contains an internal material length scale parameter related to the material microstructures and is capable of interpreting the size effect that the classical Mindlin plate model is unable to describe. The higher-order governing equations of motion and boundary conditions are derived using the Hamilton principle. The p-version Ritz method is employed to determine the natural frequencies of the microplate with different boundary conditions. A detailed parametric study is conducted to study the influences of the length scale parameter, side-to-thickness ratio and aspect ratio on the Free Vibration characteristics of the microplate. It is found that the size effect is significant when the thickness of microplate is close to the material length scale parameter.

  • nonlinear Free Vibration of size dependent functionally graded microbeams
    2012
    Co-Authors: Liaoliang Ke, Jie Yang, Yuesheng Wang, Sritawat Kitipornchai
    Abstract:

    Nonlinear Free Vibration of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and von Karman geometric nonlinearity. The non-classical beam model is developed within the framework of Timoshenko beam theory which contains a material length scale parameter related to the material microstructures. The material properties of FGMs are assumed to be graded in the thickness direction according to the power law function and are determined by Mori-Tanaka homogenization technique. The higher-order nonlinear governing equations and boundary conditions are derived by using the Hamilton principle. A numerical method that makes use of the differential quadrature method together with an iterative algorithm is employed to determine the nonlinear Vibration frequencies of the FGM microbeams with different boundary conditions. The influences of the length scale parameter, material property gradient index, slenderness ratio, and end supports on the nonlinear Free Vibration characteristics of the FGM microbeams are discussed in detail. It is found that both the linear and nonlinear frequencies increase significantly when the thickness of the FGM microbeam is comparable to the material length scale parameter.

  • nonlinear Free Vibration of single walled carbon nanotubes using nonlocal timoshenko beam theory
    2010
    Co-Authors: Jie Yang, Liaoliang Ke, Sritawat Kitipornchai
    Abstract:

    Nonlinear Free Vibration of single-walled carbon nanotubes (SWCNTs) is studied in this paper based on von Karman geometric nonlinearity and Eringen's nonlocal elasticity theory. The SWCNTs are modeled as nanobeams where the effects of transverse shear deformation and rotary inertia are considered within the framework of Timoshenko beam theory. The governing equations and boundary conditions are derived by using the Hamilton's principle. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear Vibration frequencies of SWCNTs with different boundary conditions. Zigzag (5, 0), (8, 0), (9, 0) and (11, 0) SWCNTs are considered in numerical calculations and the elastic modulus is obtained through molecular mechanics (MM) simulation. A detailed parametric study is conducted to study the influences of nonlocal parameter, length and radius of the SWCNTs and end supports on the nonlinear Free Vibration characteristics of SWCNTs.

  • nonlinear Free Vibration of functionally graded carbon nanotube reinforced composite beams
    2010
    Co-Authors: Liaoliang Ke, Jie Yang, Sritawat Kitipornchai
    Abstract:

    This paper investigates the nonlinear Free Vibration of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on Timoshenko beam theory and von Karman geometric nonlinearity. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction and estimated though the rule of mixture. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear Vibration frequencies of FG-CNTRC beams with different end supports. A detailed parametric study is conducted to study the influences of nanotube volume fraction, Vibration amplitude, slenderness ratio and end supports on the nonlinear Free Vibration characteristics of FG-CNTRC beams. The results for uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) beams are also provided for comparison. Numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fraction, Vibration amplitude, slenderness ratio, end supports and CNT distribution on the nonlinear Free Vibration characteristics of FG-CNTRC beams.

J.r. Banerjee - One of the best experts on this subject based on the ideXlab platform.

  • Free Vibration analysis for plates with arbitrary boundary conditions using a novel spectral dynamic stiffness method
    2016
    Co-Authors: Xiang Liu, J.r. Banerjee
    Abstract:

    Exact method for modal analysis of plates with arbitrary boundary conditions.Enhancement of the Wittrick-Williams algorithm by resolving the J0 count elegantly.Securing exact solutions for Free Vibration of plates for benchmark purposes.The method has two orders of magnitude higher computational efficiency than the FEM.Discussion and conclusions on a wide range of existing analytical and exact methods. An exact method for Free Vibration analysis of plates with arbitrary boundary conditions is presented. This is achieved by integrating the spectral method into the classical dynamic stiffness method. The formulation satisfies the governing differential equation exactly and any arbitrary boundary conditions are satisfied in a series sense. The Wittrick-Williams algorithm is enhanced with several elegant techniques to obtain solutions. The exactness and computational efficiency of the method are demonstrated by comparing results obtained from other methods. Finally, mathematical and physical insights are gained and significant conclusions are drawn for various analytical methods for Free Vibration analysis of plates.

  • dynamic stiffness formulation and Free Vibration analysis of functionally graded beams
    2013
    Co-Authors: J.r. Banerjee, C W Cheung
    Abstract:

    Abstract The dynamic stiffness method is developed to investigate the Free Vibration behaviour of functionally graded beams. Material properties are assumed to vary continuously in the beam thickness direction according to a power law distribution. Hamilton’s principle is used to derive the governing differential equations of motion and natural boundary conditions in Free Vibration. For harmonic oscillation the differential equations are solved in closed analytical form. The dynamic stiffness matrix is derived by relating the amplitudes of forces to those of the displacements at the beam ends. The Wittrick–Williams algorithm is used as the solution technique when applying the dynamic stiffness matrix to compute the natural frequencies and mode shapes. A parametric study is carried out to demonstrate the effects of the length to thickness ratio and the variation of the power law index parameter. Numerical results are discussed and compared with the published ones wherever possible. Some conclusions are drawn.

  • Free Vibration of a rotating tapered rayleigh beam a dynamic stiffness method of solution
    2013
    Co-Authors: J.r. Banerjee, Dominic R Jackson
    Abstract:

    The dynamic stiffness method for Free Vibration analysis of a rotating tapered Rayleigh beam is developed to investigate its Free Vibration characteristics. The type of taper considered covers a majority of practical cross-sections. The effects of centrifugal stiffening, an outboard force, an arbitrary hub radius and importantly, the rotatory inertia (Rayleigh beam) are included in the analysis. Natural frequencies and mode shapes of some examples are illustrated by using the developed dynamic stiffness matrix and applying the Wittrick-Williams algorithm. The theory is validated by using comparative results in the literature. The effects of slenderness ratio, rotational speed and taper ratio on results are discussed. This is followed by some concluding remarks.

  • Free Vibration of beams carrying spring-mass systems - A dynamic stiffness approach
    2012
    Co-Authors: J.r. Banerjee
    Abstract:

    Free Vibration analysis of beams carrying spring-mass systems is carried out by using the dynamic stiffness method. The eigenvalue problem for the Free Vibration study is formulated by assembling the dynamic stiffness matrices of beam and spring-mass elements. The Wittrick-Williams algorithm is then applied to yield the required natural frequencies and mode shapes of the combined system. Numerical examples are given for a cantilever beam carrying a spring-mass system at the tip. A parametric study is then carried out by varying the mass and stiffness properties of the spring-mass system and the subsequent effects on the natural frequencies and mode shapes are illustrated. The proposed theory can be applied for other boundary conditions of the beam and can be extended to complex structures carrying spring-mass systems. The results are discussed and validated against published literature.

  • Free Vibration of a three layered sandwich beam using the dynamic stiffness method and experiment
    2007
    Co-Authors: J.r. Banerjee, C W Cheung, R Morishima, M Perera, James Njuguna
    Abstract:

    Abstract In this paper, an accurate dynamic stiffness model for a three-layered sandwich beam of unequal thicknesses is developed and subsequently used to investigate its Free Vibration characteristics. Each layer of the beam is idealised by the Timoshenko beam theory and the combined system is reduced to a tenth-order system using symbolic computation. An exact dynamic stiffness matrix is then developed by relating amplitudes of harmonically varying loads to those of the responses. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick–Williams algorithm to carry out the Free Vibration analysis of a few illustrative examples. The accuracy of the theory is confirmed both by published literature and by experiment. The paper closes with some concluding remarks.

K.m. Liew - One of the best experts on this subject based on the ideXlab platform.

  • Free Vibration analysis of moderately thick functionally graded plates by local Kriging meshless method
    2011
    Co-Authors: Ping Zhu, K.m. Liew
    Abstract:

    This paper mainly Presents Free Vibration analyses of metal and ceramic functionally graded plates with the local Kriging meshless method. The Kriging technique is employed to construct shape functions which possess Kronecker delta function property and thus make it easy to implement essential boundary conditions. The eigenvalue equations of Free Vibration problems are based on the first-order shear deformation theory and the local Petrov–Galerkin formulation. The cubic spline function is used as the weight function which vanishes on internal boundaries of local quadrature domains and hence simplifies the implementation. Convergence studies are conducted to examine the stability of the present method. Three types of functionally graded plates – square, skew and quadrilateral plates – are considered as numerical examples to demonstrate the versatility of the present method for Free Vibration analyses.

  • Free Vibration analysis of fluid conveying single walled carbon nanotubes
    2007
    Co-Authors: C D Reddy, S Rajendran, K.m. Liew
    Abstract:

    The effect of fluid flow on the Free Vibration and instability of fluid-conveying single-walled carbon nanotubes is studied. The possibility of developing a technique to measure the mass flow rate of fluid is examined. Atomistic simulations and the continuum beam model are used. Simulations are performed to quantify the inertial, stiffness, Coriolis, and centrifugal forces generated by flow during the Free Vibration. A numerical expression is developed to measure the mass flow rate of the fluid velocities up to 40% of the critical flow velocity. This observation is useful to quantify the mass flow measurement of fluid conveying single-walled carbon nanotubes.

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