Dynamics of Beams Using a Geometrically Exact Elastic Rod Approach

2006 ◽  
Author(s):  
Fredy Coral Alamo ◽  
Hans Ingo Weber
Keyword(s):  
Virtual Work ◽  
Fundamental Problem ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Dynamic Responses ◽  
Elastic Rod ◽  
Rod Theory ◽  
Cosserat Rod ◽  
Slender Beams ◽  
Nonlinear Equations Of Motion

The dynamics of a long slender beam, intrinsically straight, is addressed systematically for 3-D problems using the Cosserat rod theory. The model developed allows for bending, extension/compression and torsion, thus enabling the study of the dynamics of various types of elastic deformations. In this work a linear constitutive relation is used, also, the Bernoulli hypothesis is considered and the shear deformations are neglected. The fundamental problem when using any finite element (FE) formulation is the choice of the displacement functions. When using Cosserat rod theory this problem is handled using approximate solutions of the nonlinear equations of motion (in quasi-static sense). These nonlinear displacement functions are functions of generic nodal displacements and rotations. Based on the Lagrangian approach formed by the kinetic and strain energy expressions, the principle of virtual work is used to derive the nonlinear ordinary differential equations of motion that are solved numerically. As an application, a curved rod, formed by many straight elements is investigated numerically. When using the Cosserat rod approach, that take into account all the geometric nonlinearities in the rod, the higher accuracy of the dynamic responses is achieved by dividing the system into a few elements which is much less than the traditional FE methods, this is the main advantage when using this approach. Overall, the Cosserat model provides an accurate way of modelling long slender beams and simulation times are greatly reduced through this approach.

Dynamics of the Elastica With End Mass and Follower Loading

1990 ◽  
Vol 57 (1) ◽  
pp. 203-208 ◽  
Author(s):  
J. M. Snyder ◽  
J. F. Wilson
Keyword(s):  
Internal Pressure ◽  
Large Deformations ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Cantilever Beams ◽  
Static Responses ◽  
Tube Type ◽  
Payload Mass ◽  
Nonlinear Equations Of Motion ◽  
Special Case

Orthotropic, polymeric tubes subjected to internal pressure may undergo large deformations while maintaining linear moment-curvature behavior. Such tubes are modeled herein as inertialess, elastic cantilever beams (the elastica) with a payload mass at the tip and with internal pressure as the eccentric tip follower loading that drives the configurations through large deformations. From the nonlinear equations of motion, dynamic beam trajectories are calculated over a range of system parameters for the special case of a point mass at the tip and a terminated ramp pressure loading. The dynamic responses, which are unique because the loading history and the range of motion are fully defined, are presented in nondimensional form and are compared to static responses presented in a companion study. These results are applicable to the dynamic design of high flexure, tube-type, robotic manipulator arms.

The Nonlinear Vibration of a Vehicle and Passenger System

2001 ◽  
Author(s):  
Ke Yu ◽  
Albert C. J. Luo ◽  
Yuancheng He
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Equations ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Pavement Surface ◽  
Torsional Spring ◽  
Nonlinear Equations Of Motion

Abstract The vibration of passengers in a vehicle traveling on a rough pavement surface is investigated. The nonlinear equations of motion for a vehicle and passenger system with impacts are derived, and the corresponding equilibrium and stability are investigated. The dynamic responses for the vehicle and passenger system with and without impacts are simulated numerically. This investigation shows that the strong torsional spring is required in order to reduce the vibration amplitudes of passengers and to avoid impacts between the vehicle and passenger.

Structural Simplification of Jack-Up Rig and Its Dynamic Responses in Regular Waves

1988 ◽  
Vol 32 (02) ◽  
pp. 134-153
Author(s):  
Jong-Shyong Wu ◽  
Cuann-Yeu Chang
Keyword(s):  
Virtual Work ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Jacobi Method ◽  
Mode Shapes ◽  
Regular Waves ◽  
Static Condensation ◽  
Rigid Joints ◽  
Jack Up

This paper is composed of two main parts: simplification of leg structure of the jack-up rig, and dynamic analysis of the entire rig due to excitation of regular waves. First, the legs of spatial beamlike lattice with rigid joints are replaced by the equivalent beams through application of the theory of static condensation and the principle of virtual work. Then the equations of motion of the entire rig are derived based on the simplified mathematical model, and the natural frequencies and mode shapes are sought by the Jacobi method. Finally, the dynamic behavior of the hinged rig and fixed rig operating in four kinds of water depths (and hence effective leg lengths) and wave heights is studied by means of the mode superposition technique. The phase angles between responses of the legs and the influence on responses of support conditions at the seabed, wave attack angle, and damping ratio are the key points of the investigation.

Nonlinear Response of a Sandwich Plate Subjected to Blast Load

2007 ◽  
Author(s):  
Ali Bas¸ ◽  
Zafer Kazancı ◽  
Zahit Mecitog˘lu
Keyword(s):  
Sandwich Plate ◽  
Virtual Work ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Blast Load ◽  
Thin Plates ◽  
Inertia Effects ◽  
The Time Domain ◽  
The Galerkin Method

Present work includes in-plane stiffness and inertia effects on the motion of a sandwich plate under blast load. The geometric nonlinearity effects are taken into account with the von Ka´rma´n large deflection theory of thin plates. All edges clamped boundary conditions are considered in the analyses. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate solutions are assumed for the space domain and substituted into the equations of motion. Then the Galerkin Method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The results of theoretical analyses are obtained.

Modeling and Adaptive Control for Tracking Wing Trajectories

2013 ◽  
Author(s):  
Ian P. Murphy ◽  
Shirin Dadashi ◽  
Jessica Gregory ◽  
Yu Lei ◽  
Javid Bayandor ◽  
...  
Keyword(s):  
Dynamic Models ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Micro Air Vehicles ◽  
Adaptive Controller ◽  
Flapping Wing ◽  
Lyapunov Analysis ◽  
Unknown Parameters ◽  
Open Problems ◽  
Nonlinear Equations Of Motion

Studies of Micro Air Vehicles (MAVs) have gained increased attention over the past decade, while a significant range of open problems in this emerging field remain unaddressed. This paper highlights the investigations entailing flapping wing vehicles, designed based on inspiration from observations of avian flight. The nonlinear equations of motion of a ground fixed flapping wing robot are derived that incorporates a quasi-steady model of aerodynamics. The equations of motion are developed using Lagrange’s equations and the aerodynamic contributions are formulated using virtual work principles. The aerodynamics are constructed with a quasi-steady state formulation where the functions representing lift and drag coefficients as a function of angle of attack are treated as unknowns. An adaptive controller is introduced that seeks to learn the aerodynamic effects. A Lyapunov analysis of the controller guarantees boundedness of all error terms and asymptotic stability in both the joint position and derivative error. The controllers are simulated using two dynamic models based on flapping wing prototypes designed at Virginia Tech. The numerical experiments validate the Lyapunov analysis and verify that unknown parameters are learned if persistently excited.

scholarly journals Modeling and Fabrication of Soft Actuators Based on Fiber-Reinforced Elastomeric Enclosures

Actuators ◽  
2021 ◽  
Vol 10 (6) ◽  
pp. 127
Author(s):  
Zhi Chen ◽  
Aicheng Zou ◽  
Zhantian Qin ◽  
Xingguo Han ◽  
Tianming Li ◽  
...  
Keyword(s):  
Virtual Work ◽  
Fiber Reinforced ◽  
Complex Environments ◽  
Manufacturing Method ◽  
Soft Actuator ◽  
Rod Theory ◽  
Cosserat Rod ◽  
Proposed Model ◽  
Soft Actuators ◽  
The Relationship

Unlike rigid actuators, soft actuators can easily adapt to complex environments. Understanding the relationship between the deformation of soft actuators and external factors such as pressure would enable rapid designs based on specific requirements, such as flexible, compliant endoscopes. An effective model is demonstrated that predicts the deformation of a soft actuator based on the virtual work principle and the geometrically exact Cosserat rod theory. The deformation process is analyzed for extension, bending, and twisting modules. A new manufacturing method is then introduced. Through any combination of modules, the soft actuator can have a greater workspace and more dexterity. The proposed model was verified for various fiber-reinforced elastomeric enclosures. There is good agreement between the model analysis and the experimental data, which indicates the effectiveness of the model.

A Versatile 3D Vehicle-Track-Bridge Element for Dynamic Analysis of the Railway Bridges under Moving Train Loads

2019 ◽  
Vol 19 (04) ◽  
pp. 1950050 ◽  
Author(s):  
Xiang Xiao ◽  
Wei-Xin Ren
Keyword(s):  
Virtual Work ◽  
Interaction Analysis ◽  
Equations Of Motion ◽  
Time Integration ◽  
Contact Forces ◽  
Dynamic Responses ◽  
Railway Bridges ◽  
Beam Elements ◽  
Track Bridge ◽  
Moving Train

There has been a growing interest to carry out the vehicle–track–bridge (VTB) dynamic interaction analysis using 2D or 3D finite elements based on simplified wheel–rail relationships. The simplified or elastic wheel–rail contact relationships, however, cannot consider the lateral contact forces and geometric shapes of the wheel and rails, and even the occasional jump of wheels from the rails. This does not guarantee a reliable analysis for the safety running of trains over bridges. To consider the wheel–rail constraint and contact forces, this paper proposes a versatile 3D VTB element, consisting of a vehicle, eight rail beam elements, four bridge beam elements, and continuous springs as well as the dampers between the rail and bridge girder. With the 3D VTB element matrices formulated, a procedure for assembling the interaction matrices of the 3D VTB element is presented based on the virtual work principle. The global equations of motion of the VTB interaction system are established accordingly, which can be solved by time integration methods to obtain the dynamic responses of the vehicle, track and bridge, as well as the stability and safety indices of the moving train. Finally, an illustrative example is used to verify the proposed the versatile 3D VTB element for the dynamic interactive analysis of railway bridges under moving train loads.

Planar Vibration of an Elastica Arch: Theory and Experiment

1990 ◽  
Vol 112 (3) ◽  
pp. 374-379 ◽  
Author(s):  
N. C. Perkins
Keyword(s):  
Natural Frequencies ◽  
Numerical Solutions ◽  
Large Range ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Geometrically Nonlinear ◽  
Linear Vibration ◽  
Elastic Rod ◽  
Rod Theory ◽  
Nonlinear Rod Theory

This investigation examines the planar, linear vibration of a deep arch that is described by a simply supported elastica. The arch is formed from an elastic rod that buckles nonlinearly under the action of a large, steady end-load. A theoretical model is proposed that governs the planar response of the rod about a generally curved, pre-stressed equilibrium. The model utilizes a geometrically nonlinear rod theory to describe the planar bending and extension of the rod centerline. The equations of motion are linearized about an elastica equilibrium and numerical solutions for free vibration are determined using a variational formulation of the associated eigenvalue problem. Natural frequencies and mode shapes are computed over a large range of centrally and eccentrically applied end-loads. Results from an experimental modal test provide support for the model.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

Analytical Modelling of the Dynamics of the Four-Bar Mechanism: A Comparison of Some Methods

2018 ◽  
Author(s):  
J. P. Meijaard ◽  
V. van der Wijk
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Force Balance ◽  
Dynamic Force ◽  
Algebraic Equations ◽  
Principle Of Virtual Work ◽  
Principal Vector ◽  
Reaction Forces ◽  
Principal Points

Some thoughts about different ways of formulating the equations of motion of a four-bar mechanism are communicated. Four analytic methods to derive the equations of motion are compared. In the first method, Lagrange’s equations in the traditional form are used, and in a second method, the principle of virtual work is used, which leads to equivalent equations. In the third method, the loop is opened, principal points and a principal vector linkage are introduced, and the equations are formulated in terms of these principal vectors, which leads, with the introduced reaction forces, to a system of differential-algebraic equations. In the fourth method, equivalent masses are introduced, which leads to a simpler system of principal points and principal vectors. By considering the links as pseudorigid bodies that can have a uniform planar dilatation, a compact form of the equations of motion is obtained. The conditions for dynamic force balance become almost trivial. Also the equations for the resulting reaction moment are considered for all four methods.

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