Transverse Vibrations of a Rotating Twisted Timoshenko Beam Under Axial Loading

1993 ◽  
Vol 115 (3) ◽  
pp. 285-294 ◽  
Author(s):  
W.-R. Chen ◽  
L. M. Keer
Keyword(s):  
Timoshenko Beam ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Axial Loading ◽  
Beam Theory ◽  
The Finite Element Method ◽  
Bending Vibrations ◽  
Parametric Studies ◽  
General Eigenvalue Problem ◽  
Twisted Beam

Transverse bending vibrations of a rotating twisted beam subjected to an axial load and spinning about its axial axis are established by using the Timoshenko beam theory and applying Hamilton’s Principle. The equations of motion of the twisted beam are derived in the twist nonorthogonal coordinate system. The finite element method is employed to discretize the equations of motion into time-dependent ordinary differential equations that have gyroscopic terms. A symmetric general eigenvalue problem is formulated and used to study the influence of the twist angle, rotational speed, and axial force on the natural frequencies of Timoshenko beams. The present model is useful for the parametric studies to understand better the various dynamic aspects of the beam structure affecting its vibration behavior.

scholarly journals A study on dynamic response of functionally graded sandwich beams under different dynamic loadings

2018 ◽  
Vol 192 ◽  
pp. 02011
Author(s):  
Wachirawit Songsuwan ◽  
Monsak Pimsarn ◽  
Nuttawit Wattanasakulpong
Keyword(s):  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Functionally Graded ◽  
Transverse Shear Deformation ◽  
Sandwich Beams ◽  
Governing Equations ◽  
Dynamic Loadings ◽  
Parametric Studies

In this research, free and forced vibration of functionally graded sandwich beams is considered using Timoshenko beam theory which takes into account the significant effects of transverse shear deformation and rotary inertia. The governing equations of motion are formulated from Lagrange's equations and they are solved by using The Ritz and Newmark methods. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, etc. on natural frequencies and dynamic deflections of the beams. According to the numerical results, all parametric studies considered in this research have significant impact on free and forced behaviour of the beams; for example, the frequency is low and the dynamic deflection is large for the beams which are hinged at both ends.

scholarly journals FREE VIBRATION OF TWO-DIRECTIONAL FGM BEAMS USING A HIGHER-ORDER TIMOSHENKO BEAM ELEMENT

2018 ◽  
Vol 56 (3) ◽  
pp. 380 ◽  
Author(s):  
Tran Thi Thom ◽  
Nguyen Dinh Kien
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Beam Element ◽  
Higher Order ◽  
Functionally Graded ◽  
The Finite Element Method ◽  
Power Law Distribution ◽  
Graded Material

Free vibration of two-directional functionally graded material (2-D FGM) beams is studied by the finite element method (FEM). The material properties are assumed to be graded in both the thickness and longitudinal directions by a power-law distribution. Equations of motion based on Timoshenko beam theory are derived from Hamilton's principle. A higher-order beam element using hierarchical functions to interpolate the displacements and rotation is formulated and employed in the analysis. In order to improve the efficiency of the element, the shear strain is constrained to constant. Validation of the derived element is confirmed by comparing the natural frequencies obtained in the present paper with the data available in the literature. Numerical investigations show that the proposed beam element is efficient, and it is capable to give accurate frequencies by a small number of elements. The effects of the material composition and aspect ratio on the vibration characteristics of the beams are examined in detail and highlighted.

scholarly journals Investigation on the Dynamics of an On-Board Rotor-Bearing System

2012 ◽  
Author(s):  
Mzaki Dakel ◽  
Sébastien Baguet ◽  
Régis Dufour
Keyword(s):  
Dynamic Behavior ◽  
Fourier Transforms ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Beam Theory ◽  
Rigid Base ◽  
The Finite Element Method ◽  
Mass Unbalance ◽  
Motion Display

In ship and aircraft turbine rotors, the rotating mass unbalance and the different movements of the rotor base are among the main causes of vibrations in bending. The goal of this paper is to investigate the dynamic behavior of an on-board rotor under rigid base excitations. The modeling takes into consideration six types of base deterministic motions (rotations and translations) when the kinetic and strain energies in addition to the virtual work of the rotating flexible rotor components are computed. The finite element method is used in the rotor modeling by employing the Timoshenko beam theory. The proposed on-board rotor model takes into account the rotary inertia, the gyroscopic inertia, the shear deformation of shaft as well as the geometric asymmetry of shaft and/or rigid disk. The Lagrange’s equations are applied to establish the differential equations of the rotor in bending with respect to the rigid base which represents a noninertial reference frame. The linear equations of motion display periodic parametric coefficients due to the asymmetry of the rotor and time-varying parametric coefficients due to the base rotational motions. In the proposed applications, the rotor mounted on rigid/elastic bearings is excited by a rotating mass unbalance associated with sinusoidal vibrations of the rigid base. The dynamic behavior of the rotor is analyzed by means of orbits of the rotor as well as fast Fourier transforms (FFTs).

Active Damping of Vibrations of a Cantilever Beam by Axial Force Control

2007 ◽  
Author(s):  
Thomas Pumho¨ssel ◽  
Horst Ecker
Keyword(s):  
Cantilever Beam ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Open Loop ◽  
Active Damping ◽  
Nonlinear Feedback ◽  
Euler Beam ◽  
Bending Vibrations ◽  
Loop Control ◽  
Damping Of Vibrations

In several fields, e.g. aerospace applications, robotics or the bladings of turbomachinery, the active damping of vibrations of slender beams which are subject to free bending vibrations becomes more and more important. In this contribution a slender cantilever beam loaded with a controlled force at its tip, which always points to the clamping point of the beam, is treated. The equations of motion are obtained using the Bernoulli-Euler beam theory and d’Alemberts principle. To introduce artificial damping to the lateral vibrations of the beam, the force at the tip of the beam has to be controlled in a proper way. Two different methods are compared. One concept is the closed-loop control of the force. In this case a nonlinear feedback control law is used, based on axial velocity feedback of the tip of the beam and a state-dependent amplification. By contrast, the concept of open-loop parametric control works without any feedback of the actual vibrations of the mechanical structure. This approach applies the force as harmonic function of time with constant amplitude and frequency. Numerical results are carried out to compare and to demonstrate the effectiveness of both methods.

scholarly journals Geometrically Nonlinear Analysis for Elastic Beam Using Point Interpolation Meshless Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Cheng He ◽  
Xinhai Wu ◽  
Tao Wang ◽  
Huan He
Keyword(s):  
Meshless Method ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Beam Theory ◽  
The Finite Element Method ◽  
Time Step ◽  
Geometrically Nonlinear Analysis ◽  
Discrete Equations ◽  
Beam Equations ◽  
Point Interpolation

The intrinsic beam theory, as one of the exact beam formulas, is quite suitable to describe large deformation of the flexible curved beam and has been widely used in many engineering applications. Owing to the advantages of the intrinsic beam theory, the resulted equations are expressed in first-order partial differential form with second-order nonlinear terms. In order to solve the intrinsic beam equations in a relative simple way, in this paper, the point interpolation meshless method was employed to obtain the discretization equations of motion. Different from those equations by using the finite element method, only the differential of the shape functions are needed to form the final discrete equations. Thus, the present method does not need integration process for all elements during each time step. The proposed method has been demonstrated by a numerical example, and results show that this method is highly efficient in treating this type of problem with good accuracy.

Dynamic Stability of a Cantilever Shaft-Disk System

1992 ◽  
Vol 114 (3) ◽  
pp. 326-329 ◽  
Author(s):  
Lien-Wen Chen ◽  
Der-Ming Ku
Keyword(s):  
Dynamic Stability ◽  
Dynamic Instability ◽  
Equations Of Motion ◽  
Beam Theory ◽  
The Finite Element Method ◽  
Disk System ◽  
Stability Behavior ◽  
Gyroscopic Moment ◽  
The Mathematical Model ◽  
Cantilever Shaft

The dynamic stability behavior of a cantilever shaft-disk system subjected to axial periodic forces varying with time is studied by the finite element method. The equations of motion for such a system are formulated using deformation shape functions developed from Timoshenko beam theory. The effects of translational and rotatory inertia, gyroscopic moment, bending and shear deformation are included in the mathematical model. Numerical results show that the effect of the gyroscopic term is to shift the boundaries of the regions of dynamic instability outwardly and, therefore, the sizes of these regions are enlarged as the rotational speed increases.

In-Plane Free Vibration of Curved Beams Using Finite Element Method

2007 ◽  
Author(s):  
F. Yang ◽  
R. Sedaghati ◽  
E. Esmailzadeh
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Central Line ◽  
Circular Ring ◽  
Curved Beam ◽  
The Finite Element Method ◽  
Curved Beams ◽  
Element Method

Curved beam-type structures have many applications in engineering area. Due to the initial curvature of the central line, it is complicated to develop and solve the equations of motion by taking into account the extensibility of the curve axis and the influences of the shear deformation and the rotary inertia. In this study the finite element method is utilized to study the curved beam with arbitrary geometry. The curved beam is modeled using the Timoshenko beam theory and the circular ring model. The governing equation of motion is derived using the Extended-Hamilton principle and numerically solved by the finite element method. A parametric sensitive study for the natural frequencies has been performed and compared with those reported in the literature in order to demonstrate the accuracy of the analysis.

Parametric Studies on Bending of Twisted Timoshenko Beams Under Complex Loadings

2012 ◽  
Vol 28 (1) ◽  
pp. N1-N6 ◽  
Author(s):  
W.-R. Chen
Keyword(s):  
Twist Angle ◽  
Width Ratio ◽  
Algebraic Equations ◽  
Equilibrium Equations ◽  
Static Loads ◽  
Parametric Studies ◽  
Element Approach ◽  
Minimum Potential ◽  
Parametric Analyses ◽  
Twisted Beam

ABSTRACTStatic bending of a twisted Timoshenko beam subjected to combined transverse and axial loadings is studied. The equilibrium equations are established in the twist coordinates by applying the principle of minimum potential energy. The governing equations are then reduced into solvable algebraic equations using a finite element approach. The effects of the twist angle, thickness-to-width ratio, length-to-thickness ratio, loading and boundary conditions on the static bending characteristics of the twisted beams are investigated. The present parametric analyses will provide engineers a good insight into the influence of various structural aspects of the twisted beam on its response to different static loads.

Buckling of Standing Tapered Timoshenko Columns with Varying Flexural Rigidity Under Combined Loadings

2016 ◽  
Vol 16 (06) ◽  
pp. 1550017 ◽  
Author(s):  
D.-L. Sun ◽  
X.-F. Li ◽  
C. Y. Wang
Keyword(s):  
Cross Section ◽  
Timoshenko Beam ◽  
Material Properties ◽  
Flexural Rigidity ◽  
Axial Loading ◽  
Beam Theory ◽  
Shear Rigidity ◽  
Initial Value ◽  
Buckling Loads ◽  
The Stability

The stability of a nonuniform column subjected to a tip force and axially distributed loading is investigated based on the Timoshenko beam theory. An emphasis is placed on buckling of a standing column with varying cross-section and variable material properties under self-weight and tip force. Four kinds of columns with different taper ratios are analyzed. A new initial value method is suggested to determine critical tip force and axial loading at buckling. The effectiveness of the method is confirmed by comparing our results with those for Euler–Bernoulli columns for the case of sufficiently large shear rigidity. The effects of shear rigidity, taper ratio, and gravity loading on the buckling loads of a heavy standing or hanging column are examined.

Vibration of infinite Timoshenko beam on Pasternak foundation under vehicular load

2017 ◽  
Vol 20 (5) ◽  
pp. 694-703
Author(s):  
Weili Luo ◽  
Yong Xia
Keyword(s):  
Timoshenko Beam ◽  
Current System ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Surface Irregularity ◽  
Radius Of Gyration ◽  
Winkler Foundation ◽  
Shear Rigidity ◽  
Maximum Displacement ◽  
Line Load

The vibration of beams on foundations under a vehicular load has attracted wide attention for decades. The problem has numerous applications in several fields such as highway structures. However, most of analytical or semi-analytical studies simplify the vehicular load as a concentrated point or distributed line load with the constant or harmonically varying amplitude, and neglect the presence of the vehicle and the road irregularity. This article carries out an analytical study of vibration on an infinite Pasternak-supported Timoshenko beam under vehicular load which is generated by the passage of a quarter car on a road with harmonic surface irregularity. The governing equations of motion are derived based on Hamilton’s principle and Timoshenko beam theory and then are solved in the frequency–wavenumber domain with a moving coordinate system. The analytical solutions are expressed in a general form of Cauchy’s residue theorem. The results are validated by the case of an Euler–Bernoulli beam on a Winkler foundation, which is a special case of the current system and has an explicit form of solution. Finally, a numerical example is employed to investigate the influence of properties of the beam (the radius of gyration and the shear rigidity) and the foundation (the shear viscosity, rocking, and normal stiffness) on the deflected shape, maximum displacement, critical frequency, and critical velocity of the system.

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