Some Solutions of the Timoshenko Beam Equations

1955 ◽  
Vol 22 (4) ◽  
pp. 579-586
Author(s):  
B. A. Boley ◽  
C. C. Chao
Keyword(s):  
Timoshenko Beam ◽  
Laplace Transformation ◽  
Numerical Results ◽  
Beam Model ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Infinite Beam ◽  
Beam Equations

Abstract Solutions are obtained by the method of Laplace transformation for four types of loadings applied to a semi-infinite beam. Numerical results are presented for two of these, both for suddenly applied and gradually varying loads. The effects of shear deformations and rotatory inertia are taken into account according to Timoshenko’s beam model. A comparison with the corresponding results of the Bernoulli-Euler theory are briefly presented.

Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads

2018 ◽  
Vol 25 (3) ◽  
pp. 593-611
Author(s):  
Xiayang Zhang ◽  
Haoquan Liang ◽  
Meijuan Zhao
Keyword(s):  
Timoshenko Beam ◽  
Laplace Transformation ◽  
Impact Load ◽  
Moving Load ◽  
Numerical Results ◽  
Dynamic Responses ◽  
Superposition Method ◽  
Load Case ◽  
Inverse Laplace Transformation ◽  
Damping Effect

This paper, taking the clamped boundary condition as an example, develops Su and Ma's fundamental solutions of the dynamic responses of a Timoshenko beam subjected to impact load. Based on that, a further extension regarding the general moving load case is also established. Kelvin–Voigt damping, whether proportionally or nonproportionally damped, is incorporated into the model, making it more comprehensive than the model of Su and Ma. Numerical inverse Laplace transformation is introduced to obtain the time-domain solution, where Durbin's formula and the corresponding convergence criteria are utilized in numerical experiments. Further, the real modal superposition method is applied at an analytical level to validate the numerical results by applying a proportionally damped condition. Total comparisons are made between the methods by sufficient case studies. The dynamic responses with and without damping effect are computed with wider slenderness to verify the correctness and effectiveness of the numerical results. Furthermore, parametric studies regarding the damping coefficients are performed to explore the nonproportional damping effect. The results show that the structural damping has significant influences on the dynamic behaviors and is especially stronger at small slender ratios. As the damping decreases the inherent frequencies and excites the low-frequency modal components more actively, a resonant phenomenon appears in high slenderness case when the beam experiences a low-speed moving load. Additionally, the computations in the moving load case indicate that the algorithm convergence is preferable when the number of grids exceeds 1000.

Development of an Active Curved Beam Model Using a Moving Frame Method

2016 ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Timoshenko Beam ◽  
Virtual Work ◽  
Three Dimensional ◽  
Moving Frame ◽  
Beam Model ◽  
Principle Of Virtual Work ◽  
Beam Equations ◽  
Moving Frame Method ◽  
Frame Method

In order to develop an active large-deformation beam model for slender, flexible or soft robots, the d’Alembert principle of virtual work is derived for three-dimensional elastic solids from Hamilton’s principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross-sections of a deforming beam. In the derivation of the beam model, Élie Cartan’s moving frame method is utilized. The resulting large-deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate-form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. Finally, to illustrate the deformation due to internal actuation, an active Timoshenko-beam model is derived by linearizing the nonlinear planar equations. For an active, simply-supported Timoshenko-beam, the analytical solution is presented.

Development of an Active Curved Beam Model—Part II: Kinetics and Internal Activation

2017 ◽  
Vol 84 (6) ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Timoshenko Beam ◽  
Virtual Work ◽  
Constitutive Relations ◽  
Differential Forms ◽  
Three Dimensional ◽  
Beam Model ◽  
Cauchy Stress ◽  
Principle Of Virtual Work ◽  
Beam Equations

Utilizing the kinematics, presented in the Part I, an active large deformation beam model for slender, flexible, or soft robots is developed from the d'Alembert's principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross sections of a deforming beam. In the derivation of the beam model, Élie Cartan's moving frame method is utilized. The resulting large deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko beam model is derived by linearizing the nonlinear planar equations. For an active, simply supported Timoshenko beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active C1 beam elements in a nonlinear finite element (FE) code.

A new isogeometric Timoshenko beam model incorporating microstructures and surface energy effects

2020 ◽  
Vol 25 (10) ◽  
pp. 2005-2022 ◽  
Author(s):  
Shuohui Yin ◽  
Yang Deng ◽  
Gongye Zhang ◽  
Tiantang Yu ◽  
Shuitao Gu
Keyword(s):  
Surface Energy ◽  
Elasticity Theory ◽  
Timoshenko Beam ◽  
Couple Stress Theory ◽  
Surface Elasticity ◽  
Numerical Results ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Order Continuity ◽  
Microstructure Effect

A new isogeometric Timoshenko beam model is developed using a modified couple stress theory (MCST) and a surface elasticity theory. The MCST is wildly used to capture microstructure effects that includes only one material length scale parameter, while the Gurtin–Murdoch surface elasticity theory containing three surface elasticity constants is employed to approximate the nature of surface energy effects. A new effective computational approach is presented for the current nonclassical Timoshenko beam model based on isogeometric analysis with high-order continuity basis functions of non-uniform rational B-splines, which effectively fulfills the higher continuity requirements in MCST. To validate the new approach and quantitatively illustrate both the microstructure and surface energy effects, the numerical results obtained from the developed approach for static deflection and natural frequencies of beams are compared with the analytical results available in the literature. Numerical results reveal that both the microstructure effect and surface energy effect should be considered in very thin beams, which also explains the size-dependent behavior.

An Approximate Theory of Lateral Impact on Beams

1955 ◽  
Vol 22 (1) ◽  
pp. 69-76
Author(s):  
B. A. Boley
Keyword(s):  
Sudden Change ◽  
Approximate Theory ◽  
Lateral Impact ◽  
Governing Equations ◽  
Transverse Impact ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Lateral Contraction ◽  
Velocity Of Propagation ◽  
Variation Technique

Abstract The approximate theory derived in this paper describes, by means of a “traveling-wave” approach, the behavior of beams under transverse impact. Lateral impact is considered in detail, namely, one in which a section of the beam undergoes a sudden change in velocity or shear force. The theory considers the effects of shear deformations and of rotatory inertia according to Timoshenko’s model, and that of lateral contraction as suggested by Love. The governing equations and the boundary conditions are developed with the aid of an energy-variation technique. Numerical examples are given in which the behavior of the boundary layer near the point of impact is examined. For one of these the exact solution is available and is in agreement with the present approximate results. Some general considerations concerning the velocity of propagation also are discussed.

Response of a Submerged Cylindrical Shell to an Axially Propagating Step Wave

1965 ◽  
Vol 32 (4) ◽  
pp. 788-792 ◽  
Author(s):  
M. J. Forrestal ◽  
G. Herrmann
Keyword(s):  
Cylindrical Shell ◽  
Steady State ◽  
Wave Front ◽  
Transverse Shear ◽  
Shell Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Axial Coordinate

An infinitely long, circular, cylindrical shell is submerged in an acoustic medium and subjected to a plane, axially propagating step wave. The fluid-shell interaction is approximated by neglecting fluid motions in the axial direction, thereby assuming that cylindrical waves radiate away from the shell independently of the axial coordinate. Rotatory inertia and transverse shear deformations are included in the shell equations of motion, and a steady-state solution is obtained by combining the independent variables, time and the axial coordinate, through a transformation that measures the shell response from the advancing wave front. Results from the steady-state solution for the case of steel shells submerged in water are presented using both the Timoshenko-type shell theory and the bending shell theory. It is shown that previous solutions, which assumed plane waves radiated away from the vibrating shell, overestimated the dumping effect of the fluid, and that the inclusion of transverse shear deformations and rotatory inertia have an effect on the response ahead of the wave front.

Whirling of Composite-Material Driveshafts Including Bending-Twisting Coupling and Transverse Shear Deformation

1993 ◽  
Author(s):  
Charles W. Bert ◽  
Chun-Do Kim
Keyword(s):  
Composite Material ◽  
Shear Deformation ◽  
Timoshenko Beam ◽  
Transverse Shear ◽  
Critical Speed ◽  
Numerical Results ◽  
Transverse Shear Deformation ◽  
First Order ◽  
Shear Deformable

Abstract A simplified theory for predicting the first-order critical speed of a shear deformable, composite-material driveshaft is presented. The shaft is modeled as a Bresse-Timoshenko beam generalized to include bending-twisting coupling. Numerical results are compared with those for both thin and thick walled shell theories and generalized Bernoulli-Euler theory.

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