Vibration Analysis of Axially Loaded Beams Including Rotary Inertia: An Exact Dynamic Finite Element

2004 ◽  
Author(s):  
Seyed M. Hashemi
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Dynamic Stiffness ◽  
Vibrational Analysis ◽  
Single Frequency ◽  
Rotary Inertia ◽  
Shape Functions ◽  
Bending Vibrations ◽  
Dynamic Finite Element ◽  
Axially Loaded

An ‘exact’ basis function Dynamic Finite Element (DFE) for the free vibrational analysis of axially loaded beams and assemblages composed of beams is presented. The shear deformation is neglected but the Rotary Inertia (RI) effects are taken into consideration. The dynamic trigonometric shape functions for bending vibrations of an axially loaded uniform beam element are first derived in an exact sense. Then, exploiting the Principle of Virtual Work together with the nodal approximations of variables based on these dynamic shape functions, leads to a single frequency dependent Dynamic Stiffness Matrix (DSM) that represents both mass and stiffness properties. A Wittrick-Williams algorithm, based on a Sturm sequence root counting technique, is then used as the solution method. The application of the theory is demonstrated by an illustrative example of cantilever beam where the influence of Rotary Inertia (RI) effect and different axial loads on the natural frequencies of the system is demonstrated by numerical results.

On the Natural Frequency Calculation of Rotating Cantilever Uniform Beams With Coriolis Effects: A Dynamic Finite Element (DFE)

1999 ◽  
Author(s):  
S. M. Hashemi ◽  
M. J. Richard
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Vibrational Analysis ◽  
Single Frequency ◽  
Shape Functions ◽  
Coriolis Forces ◽  
Dynamic Finite Element ◽  
Stiffness Properties ◽  
Frequency Calculation ◽  
Natural Frequency Calculation

Abstract A Dynamic Finite Element (DFE) for vibrational analysis of rotating assemblages composed of beams is presented in which the complexity of the acceleration, due to the presence of gyroscopic, or Coriolis forces, is taken into consideration. The dynamic trigonometric shape functions of uncoupled bending and axial vibrations of an axially loaded uniform beam element are derived in an exact sense. Then, exploiting the Principle of Virtual Work together with the nodal approximations of variables, based on these dynamic shape functions, leads to a single frequency dependent stiffness matrix which is Hermitian and represents both mass and stiffness properties. A Wittrick-Williams algorithm, based on a Sturm sequence root counting technique, is then used as the solution method. The application of the theory is demonstrated by two illustrative examples of vertical and radial beams where the influence of Coriolis forces on natural frequencies of the clamped-free rotating beams is demonstrated by numerical results.

Natural Frequencies of Rotating Uniform Beams with Coriolis Effects

2001 ◽  
Vol 123 (4) ◽  
pp. 444-455 ◽  
Author(s):  
S. M. Hashemi ◽  
M. J. Richard
Keyword(s):  
Solution Method ◽  
Natural Frequencies ◽  
Virtual Work ◽  
Vibrational Analysis ◽  
Single Frequency ◽  
Shape Functions ◽  
Coriolis Forces ◽  
Dynamic Finite Element ◽  
Stiffness Properties ◽  
Dynamic Shape

A Dynamic Finite Element (DFE) for vibrational analysis of rotating assemblages composed of beams is presented in which the complexity of the acceleration, due to the presence of gyroscopic, or Coriolis forces, is taken into consideration. The dynamic trigonometric shape functions of uncoupled bending and axial vibrations of an axially loaded uniform beam element are derived in an exact sense. Then, exploiting the Principle of Virtual Work together with the nodal approximations of variables, based on these dynamic shape functions, leads to a single frequency dependent stiffness matrix which is Hermitian and represents both mass and stiffness properties. A Wittrick-Williams algorithm, based on a Sturm sequence root counting technique, is then used as the solution method. The application of the theory is demonstrated by two illustrative examples of vertical and radial beams where the influence of Coriolis forces on natural frequencies of the clamped-free rotating beams is demonstrated by numerical results.

Vibration Analysis of Composite Wings Undergoing Material and Geometrical Couplings: A Dynamic Finite Element Formulation

Aerospace ◽  
2004 ◽  
Author(s):  
Seyed M. Hashemi ◽  
Stephen Borneman
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Virtual Work ◽  
Laminated Composite ◽  
Element Formulation ◽  
Mechanical Parameters ◽  
Principle Of Virtual Work ◽  
Shape Functions ◽  
Dynamic Finite Element ◽  
Beam Theories

The differential equations governing Geometric-Materially coupled torsion-flexural vibrations of laminated composite wings are first reviewed. Based on the finite element methodology (FEM), Euler-Bernoulli and St-Venant beam theories, the Dynamic Trigonometric Shape Functions (DTSFs) for the beam’s uncoupled displacements are thereafter derived. Exploiting the Principle of Virtual Work (PVW) and interpolating the variables based on the DTSF, the Dynamic Finite Element (DFE) formulations for uniform beams’ coupled vibrations are first developed. The variable geometrical and mechanical parameters are then incorporated in the formulation. The applicability of the DFE method is then demonstrated by two illustrative examples where a Wittrick-Williams root counting technique is used to find the systems’ natural frequencies. The proposed DFE approach can also be advantageously extended to incorporate more complexities, further coupling as well as other geometrical and mechanical parameters in the formulation.

scholarly journals A Bernoulli-Euler Stiffness Matrix Approach for Vibrational Analysis of Spinning Linearly Tapered Beams

1997 ◽  
Author(s):  
S. M. Hashemi ◽  
M. J. Richard ◽  
G. Dhatt
Keyword(s):  
Finite Element ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Vibrational Analysis ◽  
Single Frequency ◽  
Frequency Dependent ◽  
Beam Elements ◽  
Stiffness Properties ◽  
Out Of Plane ◽  
Element Matrix

This paper presents a Dynamic Finite Element (DFE) formulation, based on the Dynamic Stiffness Matrix (DSM) approach, for vibrational analysis of spinning beams. The constituent members are considered to be linearly tapered as well as centrifugally stiffened. A non-dimensional formulation is considered, and the frequency dependent trigonometric shape functions are used to find a single frequency dependent element matrix (called DSM) which has both mass and stiffness properties. An adapted bisection method based on a Sturm sequence root counting technique, is used to find the first four out-of-plane flexural natural frequencies of a cantilevered linearly tapered (in height) beam for different non-dimensional rotating speeds. The results have been compared to those found by finite elements method using Hermite beam elements. Much better convergency rates are found by this method when comparing to conventional finite element methods.

scholarly journals Triply Coupled Bending-Bending-Torsion Vibrational Analysis of Rotating Beams Using Fem and Dynamic Finite Element Formulation

2021 ◽  
Author(s):  
Mohammad Shavezipur
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Convergence Rates ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Element Formulation ◽  
Shape Functions ◽  
Weighted Residual Method ◽  
Torsion Beam ◽  
Dynamic Finite Element

This research presents the numerical analysis of the triply coupled flap-wise, cord-wise and torsional vibrations of flexible rotating blades. Euler-Bernoulli bending and St. Venant torsion beam theories are considered to derive the governing differential equations of motion. Based on Finite Element Methodology (FEM), the cubic "Hermite" shape functions are implemented where the solution of the equations results in a linear engine problem. Then, the Dynamic (frequency dependent) Trigonometric Shape Functions (DISF's) for beam's uncoupled displacements are derived. The application of the Dynamic Finite Element (DFE) approach to the solution of the governing equations is then presented. The DFE formulation, based on the weighted residual method and the DTSF's results in a nonlinear engine problem representing eigenvalues and engine modes of the system. The applicability of the DFE method is then demonstrated by illustrative examples, where a Wittrick-Williams root counting technique is used to find the system's natural frequencies. The DFE approach, an intermediate method between FEM and "Exact" formulation, is characterized by higher convergence rates, and can be advantageously used when multiple natural frequencies and/or higher modes of beam-like structures are to be evaluated.

scholarly journals Triply Coupled Bending-Bending-Torsion Vibrational Analysis of Rotating Beams Using Fem and Dynamic Finite Element Formulation

2021 ◽  
Author(s):  
Mohammad Shavezipur
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Convergence Rates ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Element Formulation ◽  
Shape Functions ◽  
Weighted Residual Method ◽  
Torsion Beam ◽  
Dynamic Finite Element

This research presents the numerical analysis of the triply coupled flap-wise, cord-wise and torsional vibrations of flexible rotating blades. Euler-Bernoulli bending and St. Venant torsion beam theories are considered to derive the governing differential equations of motion. Based on Finite Element Methodology (FEM), the cubic "Hermite" shape functions are implemented where the solution of the equations results in a linear engine problem. Then, the Dynamic (frequency dependent) Trigonometric Shape Functions (DISF's) for beam's uncoupled displacements are derived. The application of the Dynamic Finite Element (DFE) approach to the solution of the governing equations is then presented. The DFE formulation, based on the weighted residual method and the DTSF's results in a nonlinear engine problem representing eigenvalues and engine modes of the system. The applicability of the DFE method is then demonstrated by illustrative examples, where a Wittrick-Williams root counting technique is used to find the system's natural frequencies. The DFE approach, an intermediate method between FEM and "Exact" formulation, is characterized by higher convergence rates, and can be advantageously used when multiple natural frequencies and/or higher modes of beam-like structures are to be evaluated.

scholarly journals New frequency-dependent trigonometric interpolation functions for the dynamic finite element analysis of thin rectangular plates

2021 ◽  
Author(s):  
Supun Jayasinghe ◽  
Seyed M. Hashemi
Keyword(s):  
Finite Element ◽  
Trigonometric Interpolation ◽  
Rectangular Plates ◽  
Structural Components ◽  
Shape Functions ◽  
Element Analysis ◽  
Frequency Dependent ◽  
Analysis And Design ◽  
Dynamic Finite Element ◽  
Semianalytical Method

The Dynamic Finite Element (DFE) formulation is a superconvergent, semianalytical method used to perform vibration analysis of structural components during the early stages of design. It was presented as an alternative to analytical and numerical methods that exhibit various drawbacks, which limit their applicability during the preliminary design stages. The DFE method, originally developed by the second author, has been exploited heavily to study the modal behaviour of beams in the past. Results from these studies have shown that the DFE method is capable of arriving at highly accurate results with a coarse mesh, thus, making it an ideal choice for preliminary stage modal analysis and design of structural components. However, the DFE method has not yet been extended to study the vibration behaviour of plates. Thus, the aim of this study is to develop a set of frequency-dependent, trigonometric shape functions for a 4-noded, 4-DOF per node element as a basis for developing a DFE method for thin rectangular plates. To this end, the authors exploit a distinct quasi-exact solution to the plate governing equation and this solution is then used to derive the new, trigonometric basis and shape functions, based on which the DFE method would be developed.

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